Research ArticleCompliance Matrix of a Single-Bent Leaf Flexure fora Modal Analysis

Nghia-Huu Nguyen1 Moo-Yeon Lee2 Ji-Soo Kim1 and Dong-Yeon Lee1

1School of Mechanical Engineering Yeungnam University Gyeongsan 712-749 Republic of Korea2Department of Mechanical Engineering Dong-A University Busan 604-714 Republic of Korea

Correspondence should be addressed to Dong-Yeon Lee dyleeynuackr

Received 22 March 2015 Accepted 4 May 2015

Academic Editor Chao Tao

Copyright copy 2015 Nghia-Huu Nguyen et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We present a compliance matrix for a single-bent leaf flexure (SBLF) that shows the relationships between the deformations andthe six-axis loads applied to the SBLF Higher-order beam theory that considers the variable shear and warping effect is consideredin bending The partially restrained warping at the junction between elements is also considered in torsion The total strain energyis calculated and the complete compliance matrix is derived by using Castiglianorsquos second theorem Sensitivity analyses over thecompliance elements are performed and verified via finite element analysis (FEA) The results show that the derived complianceelements are in good agreementwith FEAwith errors of less than 76We suggest that theoretical compliance elements consideringvariable shear andwarping in bending and partially restrainedwarping in torsion give highly accurate design equations representingthe compliant mechanism of the SBLF The present work could be used in a modal analysis of a single-bent leaf flexure

1 Introduction

Flexure guides are used in precision engineering especiallyin nanoscanner devices because of their smooth elastic andno-friction characteristics Flexure guides provide short ormoderate ranges of travel due to their elastic deformationcharacteristics [1] Newflexure guidewhich has largeworkingrange and nanoresolution is hot topic in the recent yearsSeveral types of flexure guides have been described in theliterature including hinge flexures [2ndash7] and leaf-springflexures [8ndash13] A single-bent leaf flexure investigated in thisstudy is an L-shaped leaf-spring mechanism The SLBF hasremarkable advantages for flexure such as a moderate rangeof deflection and easily obtained uniform spring materialMany studies have focused on flexure guides Formulas arederived for the stiffness of a parallel leaf-spring flexure in[8] A displacement reduction mechanism based on torsionleaf-spring hinges was developed in [9] An orthoplanarspring design that operates by raising or lowering its platformrelative to the base with no rotation is presented in [10] Aplanar positioningmechanismwith three degrees of freedom

using a flexure guide is presented in [11] New flexures weredesigned and developed with a large range and compactdimensions as described in [1 12 14]

The compliance matrix is often used to express the load-deformation relationship for a linearly elastic system It hasbeen applied in many previous studies in the precision field[3 4 15ndash17] to derive the equations of motion for generalflexure mechanisms The compliance matrix for a prismaticbeam is presented in [3] but the shear deformation and thewarping effect in bending and torsion were not consideredThe constant shear deformation is considered in bending thewarping is not included in torsion [4 15] The warping wasconsidered in the torsion theory however it was applied tothe beams with a fully fixed (restrained) end [18 19] Thepartially restrained warping in torsion was investigated in[20] and was applied to truck chassis frames The generalhigher-order beam theory (HBT) was presented in [21ndash23]and Levinson [21] proposed an equation for the deflectionof a beam that includes the shear and warping in bendinganalysis Relationships between Levinson beam theory andclassical Euler-Bernoulli theory are presented and applied

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 672831 10 pageshttpdxdoiorg1011552015672831

2 Shock and Vibration

y

x

Element ①

②

l

l

b

t

z

FxTx

My

Fy

Fz

Mz

(a)

Moving body

Single-bent leaf flexure

x

y120579z

(b)

Figure 1 (a) Schematic diagram of a SBLF (b) Example of the planar scanner using SBLF

to examples by Reddy et al [22] However these modelsare usually applied to the cantilever beam or the simplysupported beam models not to the SBLF The compliancematrix of a bent beam microsuspension was presented in[24] The beammodel is similar to the SBLF model howeveronly the in-plane deformations were derived and the sheardeformations were not considered in the bending analysis

In this study we analyzed and derived a compliancematrix that expresses the relationship between the deforma-tions and applied loads of the SBLF by using Castiglianorsquossecond theorem The HBT of Levinson is applied in thebending analysis The partially restrained warping at thejunction between elements is introduced and the full andfree warping is also investigated in the torsional analysis Thetheoretical results are verified by finite element analysis (FEA)results The good agreement between the results of the twomethods validates the accuracy of the theoretical equations

2 Generalized Modeling of the SBLF

Figure 1(a) shows a model of the SBLF that consists of twoleaf flexure elements One end is fixed and other end has sixcomponents of loading three forces (119865

119909119865119910 and119865

119911) and three

moments (119879119909 119872119910 and 119872

119911) Dimensions 119897 119887 and 119905 are the

length width and thickness respectively Under the 6-axisloads axial bending shear and torsional deformations willoccur in both elements of the SBLF Figure 1(b) shows oneexample of the SBLF in a planar nanoscanner which includesfour SBLFs symmetrically connected with the four corners ofthe square moving body to ensure smooth and nonparasiticerror motion of the system In this study the deformationsdue to the six components of loading need to be analyzed anddetermined in the compliance matrix form

21 Derivation of the Total Strain Energy Castiglianorsquos secondtheorem was used to find the translational and rotationaldeformations of the SBLF Thus the deformations due to

loads are defined by the partial derivative of the total strainenergy (SE) with respect to the loads as follows

120575119894=120597119880

120597119865119894

120579119894=

120597119880

120597119872119894

(1)

where119880 is the total strain energy that is stored in the flexure120575119894is the deflection due to the force119865

119894 and 120579

119894is the rotation due

to the moment119872119894 When the six components of loading are

applied the total SE stored in two elements of the SBLF canbe classified as the extensional shear bending and torsionaldeformation energies

The extensional SE (119880119886) in element 1 due to the axial force

119865119909are

1198801198861119865119909

= int

119897

0

119865119909

2

2119860119864119889119909 (2)

In element 2 the extensional SE (119880119886) due to the axial force 119865

119910

is given by

1198801198862119865119910

= int

119897

0

119865119910

2

2119860119864119889119910 (3)

where 119860 = 119887119905 is the cross-sectional area and 119864 is the elasticmodulus (Youngrsquos modulus)

In the bending and shear analyses the shear deformationsdue to the transverse loads are usually neglected in Euler-Bernoulli beam theory (EBT) Accordingly the EBT canpredict the deflection accurately only for a thin beam (ratio119897119887 gt 20) Timoshenko beam theory considers the addi-tional constant shear deformation Thus TBT shows moreprecise analysis results than EBT However for thick beams(119897119887 lt 20) the higher-order beam theory that considersthe variable shear deformation and the warping in bendingshould be applied In the HBT of Levinson the variable sheardeformation and warping in bending are considered and the

Shock and Vibration 3

deflection is described in [21] Therefore the shear SE (119880119904) in

element 1 due to the shear forces 119865119910 119865119911is given by

1198801199041119865119910

= int

119897

0

119865119910

2

1198972

4119864119868119911

(1 + ]) (1199052

1198972)(

119909

119897) 119889119909

1198801199041119865119911

= int

119897

0

119865119911

2

1198972

4119864119868119910

(1 + ]) (1198872

1198972)(

119909

119897) 119889119909

(4)

where ] is Poissonrsquos ratio In element 2 the shear SE (119880119904) due

to the shear forces 119865119909 119865119911is given by

1198801199042119865119909

= int

119897

0

119865119909

2

1198972

4119864119868119911

(1 + ]) (1199052

1198972)(

119910

119897) 119889119910

1198801199042119865119911

= int

119897

0

119865119911

2

1198972

4119864119868119910

(1 + ]) (1198872

1198972)(

119910

119897) 119889119910

(5)

The bending SE (119880119887) in element 1 due to 119865

119911 119865119910119872119910 and119872

119911

is given by

1198801198871= int

119897

0

(119872119910minus 119865119911119909)2

2119864119868119910

119889119909 +

(119872119911+ 119865119910119909)2

2119864119868119911

119889119909 (6)

In element 2 the bending SE (119880119887) due to 119865

119910 119865119909 119865119911 119879119909 and

119872119911is given by

1198801198872= int

119897

0

(119872119911+ 119865119910119897 + 119865119909119910)2

2119864119868119911

119889119910 +(119879119909minus 119865119911119910)2

2119864119868119909

119889119910 (7)

where 119868119909= 119868119910= 1199051198873

12 and 119868119911= 1198871199053

12 is the inertiamomentabout the 119909 119910 and 119911 axes respectively

A partially restrained warping torsion analysis at the jointbetween elements 1 and 2 is presented in [25] The warpingrestraint factor (119870) is adopted to represent the degree ofwarping restraint In the present study the torsional SE (119880

119905)

was defined in element 1 due to moment 119879119909as follows [25]

1198801199051=1

2119864119862119908int

119897

0

(12057910158401015840

119909)2

119889119909 +1

2119866119869119909int

119897

0

(1205791015840

119909)2

119889119909 (8)

where

120579119909=

119879119909

120572119866119869119909

(120572119909

+ (1 minus 119870) (minus sinh120572119909 minus tanh120572119897 + tanh120572119897 cosh120572119909)) (9)

In element 2 the torsional SE (119880119905) due to moments 119865

119911119897 and

119872119910is as follows

1198801199052=1

2119864119862119908int

0

minus119897

(12057910158401015840

119910)2

119889119910 +1

2119866119869119910int

0

minus119897

(1205791015840

119910)2

119889119910 (10)

where

120579119910=

119872119910119888

120572119866119869119910

[120572119910 minus sinh120572119910

minus (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897sinh120572119897

)) cosh120572119910

+ (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897sinh120572119897

))]

(11)

where119872119910119888= 119872119910minus 119865119911119897

The torsion constant 119869119909of a bar with a rectangular

cross section can be determined by Equation (161) in [26]Therefore

119869119909=1198871199053

3[1 minus 063

119905

119887]

119869119909= 119869119910

(12)

Warping constant 119862119908

with respect to the shear center isdefined by Equation (743) in [27] as the warping momentof inertia

119862119908= int120596

2

119889119860 =(119887119905)3

144 (13)

where 120596 is the warping function and 120572 = radic119866119869119910119864119862119908is the

torsion-bending constant (119898)119870 the warping restraint factor is introduced to express

the degree of restraint from the warping at the joint of ele-ments 1 and 2 Equations (9) and (11) show that when 119870 =

1 free warping occurs at end 119909 = 0 or (119910 = minus119897) and fullyrestrained warping occurs when119870 = 0

The total SE (119880) of a SBLF with six-axis loads at free endwas summarized as follows

119880 = 1198801+ 1198802 (14)

where1198801is the total SE of element 1119880

1= 1198801198861+1198801199041+1198801198871+1198801199051

and1198802is the total SE of element 2119880

2= 1198801198862+1198801199042+1198801198872+1198801199052

From (1) the translational displacements due to forces119865119909

119865119910 and 119865

119911according to the 119909 119910 and 119911 axes respectively are

given by

120575119909=120597119880

120597119865119909

=119865119909119897

119860119864+1198651199091198971199052

(1 + ])2119864119868119911

+1

119864119868119911

(1198721199111198972

2+1198651199091198973

3+

1198651199101198973

2)

120575119910=120597119880

120597119865119910

=

119865119910119897

119860119864+

1198651199101198971199052

(1 + ])2119864119868119911

+1

119864119868119911

(31198721199111198972

2+

41198651199101198973

3+1198651199091198973

2)

4 Shock and Vibration

120575119911=120597119880

120597119865119911

=1198651199111198971198872

(1 + ])2119864119868119910

minus1198721199111198972

2119864119868119910

+1198651199111198973

3119864119868119910

minus1198791199091198972

2119864119868119909

+1198651199111198973

3119864119868119909

+

(1198651199111198972

minus119872119910119897) (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870

119905+ 120572119897119870

2

119905)

120572119866119869119910

(15)

The rotational displacements due to moments 119879119909 119872119910 and

119872119911about the 119909 119910 and 119911 axes respectively are given by

120579119909=120597119880

120597119879119909

=2119879119909119897 minus 1198651199111198972

2119864119868119909

+

119879119909(1198702 sinh120572119897 minus sinh120572119897 + 120572119897 cosh120572119897)

120572119866119869119909cosh120572119897

120579119910=

120597119880

120597119872119910

=

119872119910119897

119864119868119910

minus1198651199111198972

2119864119868119910

+

(119872119910minus 119865119911119897) (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870

119905+ 120572119897119870

2

119905)

120572119866119869119910

120579119911=

120597119880

120597119872119911

=1

119864119868119911

(2119872119911119897 +

31198651199101198972

2+1198651199091198972

2)

(16)

where

119870119905= (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897

sinh120572119897)) (17)

22 Compliance Matrix of the SBLF The relationshipbetween the translational and rotational displacements andthe applied forces and moments at the free end of the SBLF isformulated as follows

120575 = [119862] 119891

119891 = [119870] 120575

(18)

where 120575 = [120575119909120575119910120575119911120579119909120579119910120579119911]119879 119891 = [119865

119909119865119910119865119911119879119909119872119910119872119911]119879

[119862] is the compliance matrix or flexibility matrix [119870] is thestiffness matrix 119865

119899and 120575

119899are the force and translational

displacement with respect to the 119899-axis and 119872119899and 120579

119899

are the moment and rotational displacement about the 119899-axis respectively The compliance matrix [119862] includes 11independent elements as follows

120575119909

120575119910

120575119911

120579119909

120579119910

120579119911

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119862120575119909119865119909

119862120575119909119865119910

0 0 0 119862120575119909119872119911

119862120575119910119865119909

119862120575119910119865119910

0 0 0 119862120575119910119872119911

0 0 119862120575119911119865119911

119862120575119911119879119909

119862120575119911119872119910

0

0 0 119862120579119909119865119911

119862120579119909119879119909

0 0

0 0 119862120579119910119865119911

0 119862120579119910119872119910

0

119862120579119911119865119909

119862120579119911119865119910

0 0 0 119862120579119911119872119911

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119865119909

119865119910

119865119911

119879119909

119872119910

119872119911

119862120575119909119865119909

= (120597119880

120597119865119909

)1

119865119909

=119897

119860119864+1198971199052

(1 + ])4119864119868119911

+1198973

3119864119868119911

119862120575119909119865119910

= (120597119880

120597119865119909

)1

119865119910

=1198973

2119864119868119911

119862120575119909119872119911

= (120597119880

120597119865119909

)1

119872119911

=1198972

2119864119868119911

119862120575119910119865119909

= (120597119880

120597119865119910

)1

119865119909

=1198973

2119864119868119911

119862120575119910119865119910

= (120597119880

120597119865119910

)1

119865119910

=119897

119860119864+1198971199052

(1 + ])4119864119868119911

+41198973

3119864119868119911

119862120575119910119872119911

= (120597119880

120597119865119910

)1

119872119911

=31198973

2119864119868119911

119862120575119911119865119911

= (120597119880

120597119865119911

)1

119865119911

=1198971198872

(1 + ])2119864119868119910

+1198973

3119864119868119910

+1198973

3119864119868119909

+

1198972

(2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120575119911119879119909

= (120597119880

120597119865119911

)1

119879119909

= minus1198972

2119864119868119909

119862120575119911119872119910

= (120597119880

120597119865119911

)1

119872119910

= minus1198972

2119864119868119910

minus

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119909119865119911

= (120597119880

120597119879119909

)1

119865119911

= minus1198972

2119864119868119909

119862120579119909119879119909

= (120597119880

120597119879119909

)1

119879119909

=119897

119864119868119909

Shock and Vibration 5

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

12

14

FEATheoryError

4 6 8 10 12 14 16 18 20

Fy sensitivity with length l

l (mm)

120575y

(mm

)

(a) 120575119910-119897

FEATheoryError

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

120575y

(mm

)

t (mm)02 04 06 08 10

Fy sensitivity with thickness t

(b) 120575119910-119905

FEATheoryError

00

02

04

06

08

10

Erro

r (

)

2

4

6

8

10

12

14

120575y

(mm

)

b (mm)2 3 4 5 6 7 8

Fy sensitivity with width b

(c) 120575119910-119887

Figure 2 Variation of deflection 120575119910according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119910

+

(1198702 sinh120572119897 minus sinh120572119897 + 120572119897 cosh120572119897)

120572119866119869119909cosh120572119897

119862120579119910119865119911

= (120597119880

120597119872119910

)1

119865119911

= minus1198972

2119864119868119910

minus

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119910119872119910

= (120597119880

120597119872119910

)1

119872119910

=119897

119864119868119910

+

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119911119865119909

= (120597119880

120597119872119911

)1

119865119909

=1198972

2119864119868119911

119862120579119911119865119910

= (120597119880

120597119872119911

)1

119865119910

=31198973

2119864119868119911

119862120579119911119872119911

= (120597119880

120597119872119911

)1

119872119911

=2119897

119864119868119911

(19)

3 FEA Verification

In this study FEA was conducted by using Pro-Mechanicacommercial software (Wildfire 5 PTC Corp MA USA) toverify the results obtained by theory and to validate thereliability of the calculated results The default values of theSBLF were chosen as follows length 119897 = 10mm width 119887 =

4mm and thickness 119905 = 05mm The forces and moments

6 Shock and Vibration

Table 1 Comparison between theory and the FEA at the default values of the SBLF

Displacements FEA Theory Error ()Shear and Warp No shear and No warp S and W No S and NoW

120575119909(mm) 116210 116466 116183 022 minus002

120575119910(mm) 044183 046480 046451 494 488

120575119911(mm) 021102 020825 022556 131 645

120579119909(rad) 000239 000233 000251 219 504

120579119910(rad) 000210 000209 000227 047 743

120579119911(rad) 000673 000697 000697 343 343

Table 2 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under 119865119909 119865119910 and 119865

119911

119897 (mm) FEA Theory Error () t (mm) FEA Theory Error () b (mm) FEA Theory Error ()(a) Deflection 120575

119909(mm)

5 013819 014550 502 025 877255 929028 557 2 228306 232366 17565 030485 031934 454 0325 397715 422907 596 26 174683 178743 2278 057071 059507 409 04 217394 226869 418 32 141120 145229 28395 095908 099619 372 0475 130778 135502 349 38 118188 122298 33611 149324 154623 343 055 084862 087301 279 44 101548 105621 386125 219646 226869 318 0625 058257 059507 210 5 088936 092946 43214 309203 318708 298 07 041772 042366 140 56 079058 082988 474155 420323 432493 281 0775 031008 031227 070 62 071120 074957 51217 555339 570573 267 085 023676 023676 000 68 064607 068343 547185 716579 735301 255 0925 018508 018377 minus071 74 059170 062802 57820 906374 929028 244 1 014756 014550 minus142 8 054565 058091 607

(b) Deflection 120575119910(mm)

5 005369 005809 758 025 355785 371567 425 2 090181 092903 29365 011959 012759 627 0325 162113 169129 415 26 069073 071464 3358 022485 023785 547 04 086939 090720 417 32 055848 058064 38295 037879 039827 489 0475 051931 054178 415 38 046800 048896 42911 059062 061825 447 055 033455 034901 414 44 040226 042229 474125 086949 090720 416 0625 022798 023785 415 5 035238 037161 51714 122458 127453 392 07 016225 016931 417 56 031329 033180 558155 166503 172963 374 0775 011953 012477 420 62 028185 029969 59517 219995 228192 359 085 009057 009458 424 68 025604 027324 629185 283849 294080 348 0925 007025 007339 428 74 023449 025109 66120 358976 371567 339 1 005557 005809 433 8 021624 023226 690

(c) Deflection 120575119911(mm)

5 002203 002113 409 025 160255 158055 137 2 052645 051994 12465 005270 005127 272 0325 074212 073006 163 26 037023 036566 1248 010341 010146 189 04 040469 039777 171 32 028218 027863 12695 017912 017672 134 0475 024617 024151 190 38 022569 022270 13311 028479 028210 095 055 016164 015831 207 44 018640 018372 144125 042541 042263 065 0625 011239 010988 224 5 015750 015497 16114 060595 060338 043 07 008169 007972 242 56 013538 013290 184155 083137 082938 024 0775 006152 005992 261 62 011792 011542 21217 110665 110569 009 085 004770 004636 281 68 010379 010126 244185 143675 143737 minus004 0925 003788 003675 301 74 009214 008956 28120 182663 182945 minus015 1 003071 002973 322 8 008239 007975 321

Shock and Vibration 7

00

05

10

15

20

25

30

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

Fz sensitivity with length l

l (mm)4 6 8 10 12 14 16 18 20

minus2

120575z

(mm

)

(a) 120575119911-119897

FEATheoryError

00

05

10

15

20

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with thickness t

t (mm)02 04 06 08 10

120575z

(mm

)

(b) 120575119911-119905

FEATheoryError

00

01

02

03

04

05

06

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with width b

b (mm)2 3 4 5 6 7 8

120575z

(mm

)

(c) 120575119911-119887

Figure 3 Variation of deflection 120575119911according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119911

applied to the SBLF are 119865119909= 10N 119865

119910= 119865119911= 1N and 119879

119909=

119872119910= 119872119911= 1NmmThewarping restraint factor was chosen

to be 119870 = 05 The material selected in this simulation isaluminum Al 6061 In the parametric analysis the sensitivityanalysis was performed to determine the agreement betweenFEA and theory with the variation of length 119897 = 5 to 20mmwidth 119887 = 2 to 8mm and thickness 119905 = 025 to 1mmrespectively If the errors between the two methods are lowerthan 10 the results can usually be accepted and used fromthe viewpoint of the engineering design

Table 1 shows the comparison between the theory andFEA results at the default values of the SBLF Equations (15)ndash(16) were applied to find the translational and rotationaldisplacements by the theoretical method that considers thevariable shear deformation in the bending and the warpingin both the bending and torsion and the partially restrained

warping at the joint of elements 1 and 2 The derivedtheoretical equations showed a good agreement with the FEAresults all of the errors of these simulationswere less than 5whereas the results of without shear and warp effect showedhigher errors up to 74 as presented in Table 1 The resultsof in-plane deformations 120575

119909 120575119910 and 120579

119911in this study were

consistent with results of the equations in [24] Althoughthese results were showing the good agreement at the defaultvalues the sensitive parameters analysis was also conductedto ensure the reliability of the derived equations

In the research of the reference [28] we analyzed thedisplacement of a SBLF under transverse loading by theEuler-Bernoulli Timoshenko and third-order beam theoryand the results showed that the third-order beam theoryof Levinson which considers the shear and warp effect inbending gave the good results In this study the above beam

8 Shock and Vibration

Table 3 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under loads 119879119909119872119910 and119872

119911

119897 (mm) FEA Theory Error () 119905 (mm) FEA Theory Error () 119887 (mm) FEA Theory Error ()(a) Rotation 120579

119909(rad)

5 000114 000108 559 025 001795 001762 186 2 000568 000561 13465 000152 000145 404 0325 000833 000815 215 26 000404 000397 1668 000189 000183 307 04 000455 000445 228 32 000312 000306 19195 000226 000221 240 0475 000277 000270 250 38 000254 000248 21311 000263 000258 191 055 000183 000178 271 44 000213 000208 236125 000301 000296 153 0625 000127 000123 289 5 000183 000179 26114 000338 000334 123 07 000093 000090 303 56 000161 000156 288155 000375 000371 099 0775 000070 000068 316 62 000143 000138 31817 000412 000409 078 085 000054 000052 325 68 000128 000124 349185 000450 000447 060 0925 000043 000042 333 74 000116 000112 38320 000487 000485 045 1 000035 000034 339 8 000106 000102 419

(b) Rotation 120579119910(rad)

5 000089 000084 477 025 001605 001583 134 2 000540 000533 13365 000126 000122 308 0325 000744 000732 160 26 000376 000371 1358 000163 000159 211 04 000407 000399 177 32 000285 000281 13995 000200 000197 149 0475 000248 000243 201 38 000228 000224 14611 000237 000235 106 055 000163 000159 223 44 000188 000185 159125 000274 000272 074 0625 000114 000111 246 5 000158 000155 17714 000312 000310 049 07 000083 000081 269 56 000136 000133 200155 000349 000348 029 0775 000063 000061 292 62 000118 000116 22917 000386 000385 013 085 000049 000047 316 68 000104 000101 262185 000423 000423 minus001 0925 000039 000037 341 74 000092 000090 29920 000460 000461 minus012 1 000032 000030 365 8 000082 000080 339

(c) Rotation 120579119911(rad)

5 000328 000348 571 025 005405 005573 302 2 001365 001393 20065 000432 000453 470 0325 002462 002537 294 26 001047 001072 2318 000535 000557 403 04 001320 001361 295 32 000848 000871 26695 000638 000662 358 0475 000789 000813 294 38 000711 000733 30111 000741 000766 325 055 000508 000523 293 44 000612 000633 337125 000845 000871 302 0625 000346 000357 293 5 000537 000557 37214 000948 000975 284 07 000246 000254 293 56 000477 000498 406155 001051 001080 271 0775 000182 000187 294 62 000430 000449 43917 001153 001184 261 085 000138 000142 296 68 000391 000410 470185 001256 001289 253 0925 000107 000110 298 74 000358 000377 50120 001359 001393 247 1 000084 000087 301 8 000330 000348 531

theory was also used and the translational displacements 120575119909

120575119910 and 120575

119911under 119865

119909 119865119910 and 119865

119911according to the variations of

length 119897 thickness 119905 and width 119887 respectively are presentedin Table 2 Figure 2 shows typical simulation results underforce 119865

119910 The errors in these investigations are lower (76)

The sensitive analysis of 119865119911is also presented in Figure 3 All

the errors are lower (5) These results reveal the accuracyof the derived theoretical equations for the translationaldisplacements of SBLF

The comparison results among the fully restrained par-tially restrained and free warping in torsion analysis of SBLFwere investigated in [25] and the partially restrained warpingwith factor 119870 = 05 was applied in this torsion simulationTable 3 shows the variation of rotational displacements 120579

119909 120579119910

and 120579119911due to moments 119879

119909 119872119910 and 119872

119911with the variation

of parameters 119897 119905 and 119887 of the SBLF respectively It canbe observed that the results of theory and FEA are in goodagreement with the errors lower than 6 Figure 4 presentsthe typical variation of rotation 120579

119909according to 119897 119905 and 119887

under 119879119909 The maximum error is lower than 6

In summary the relationship between the deformationand the applied loads for the SBLF is presented in compliancematrix form The variable shear deformation and warpingeffect were considered and partial restraint was introducedin the analysis All results were verified by FEA with strongagreement between the two methods The errors were lowerthan 76 These results demonstrate the high accuracy andreliability of the proposed theoretical equations

Shock and Vibration 9

0000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

4 6 8 10 12 14 16 18 20

Tx sensitivity with length l

l (mm)

(a) 120579119909-119897

0000

0005

0010

0015

0020

0025

0030

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

t (mm)02 04 06 08 10

Tx sensitivity with thickness t

(b) 120579119909-119905

2 3 4 5 6 7 80000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

Tx sensitivity with width b

b (mm)

(c) 120579119909-119887

Figure 4 Variation of rotation 120579119909according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under load 119879

119909

4 Conclusion

In this study a compliance matrix that expresses the rela-tionship between the deformations and applied loads ofthe SBLF was analyzed and derived by using Castiglianorsquossecond theorem In bending analysis higher-order beamtheory was applied wherein variable shear deformation andthe warping were considered in the calculated formulas forthe shear forces At the joint of two elements of the SBLFthe partially restrained warping in torsion was analysed withconsideration of the warping restraint factor The theoreticalresults were verified by FEA at both the default and thesensitive values The results indicate that there is strongagreement between the two methods with errors below

76 This suggests the accuracy of the proposed theoreticalequations and that they can be used in the precision machinedesign

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Nghia-Huu Nguyen and Moo-Yeon Lee equally contributedto this work

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

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Control Scienceand Engineering

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

y

x

Element ①

②

l

l

b

t

z

FxTx

My

Fy

Fz

Mz

(a)

Moving body

Single-bent leaf flexure

x

y120579z

(b)

Figure 1 (a) Schematic diagram of a SBLF (b) Example of the planar scanner using SBLF

to examples by Reddy et al [22] However these modelsare usually applied to the cantilever beam or the simplysupported beam models not to the SBLF The compliancematrix of a bent beam microsuspension was presented in[24] The beammodel is similar to the SBLF model howeveronly the in-plane deformations were derived and the sheardeformations were not considered in the bending analysis

In this study we analyzed and derived a compliancematrix that expresses the relationship between the deforma-tions and applied loads of the SBLF by using Castiglianorsquossecond theorem The HBT of Levinson is applied in thebending analysis The partially restrained warping at thejunction between elements is introduced and the full andfree warping is also investigated in the torsional analysis Thetheoretical results are verified by finite element analysis (FEA)results The good agreement between the results of the twomethods validates the accuracy of the theoretical equations

2 Generalized Modeling of the SBLF

Figure 1(a) shows a model of the SBLF that consists of twoleaf flexure elements One end is fixed and other end has sixcomponents of loading three forces (119865

119909119865119910 and119865

119911) and three

moments (119879119909 119872119910 and 119872

119911) Dimensions 119897 119887 and 119905 are the

length width and thickness respectively Under the 6-axisloads axial bending shear and torsional deformations willoccur in both elements of the SBLF Figure 1(b) shows oneexample of the SBLF in a planar nanoscanner which includesfour SBLFs symmetrically connected with the four corners ofthe square moving body to ensure smooth and nonparasiticerror motion of the system In this study the deformationsdue to the six components of loading need to be analyzed anddetermined in the compliance matrix form

21 Derivation of the Total Strain Energy Castiglianorsquos secondtheorem was used to find the translational and rotationaldeformations of the SBLF Thus the deformations due to

loads are defined by the partial derivative of the total strainenergy (SE) with respect to the loads as follows

120575119894=120597119880

120597119865119894

120579119894=

120597119880

120597119872119894

(1)

where119880 is the total strain energy that is stored in the flexure120575119894is the deflection due to the force119865

119894 and 120579

119894is the rotation due

to the moment119872119894 When the six components of loading are

applied the total SE stored in two elements of the SBLF canbe classified as the extensional shear bending and torsionaldeformation energies

The extensional SE (119880119886) in element 1 due to the axial force

119865119909are

1198801198861119865119909

= int

119897

0

119865119909

2

2119860119864119889119909 (2)

In element 2 the extensional SE (119880119886) due to the axial force 119865

119910

is given by

1198801198862119865119910

= int

119897

0

119865119910

2

2119860119864119889119910 (3)

where 119860 = 119887119905 is the cross-sectional area and 119864 is the elasticmodulus (Youngrsquos modulus)

In the bending and shear analyses the shear deformationsdue to the transverse loads are usually neglected in Euler-Bernoulli beam theory (EBT) Accordingly the EBT canpredict the deflection accurately only for a thin beam (ratio119897119887 gt 20) Timoshenko beam theory considers the addi-tional constant shear deformation Thus TBT shows moreprecise analysis results than EBT However for thick beams(119897119887 lt 20) the higher-order beam theory that considersthe variable shear deformation and the warping in bendingshould be applied In the HBT of Levinson the variable sheardeformation and warping in bending are considered and the

Shock and Vibration 3

deflection is described in [21] Therefore the shear SE (119880119904) in

element 1 due to the shear forces 119865119910 119865119911is given by

1198801199041119865119910

= int

119897

0

119865119910

2

1198972

4119864119868119911

(1 + ]) (1199052

1198972)(

119909

119897) 119889119909

1198801199041119865119911

= int

119897

0

119865119911

2

1198972

4119864119868119910

(1 + ]) (1198872

1198972)(

119909

119897) 119889119909

(4)

where ] is Poissonrsquos ratio In element 2 the shear SE (119880119904) due

to the shear forces 119865119909 119865119911is given by

1198801199042119865119909

= int

119897

0

119865119909

2

1198972

4119864119868119911

(1 + ]) (1199052

1198972)(

119910

119897) 119889119910

1198801199042119865119911

= int

119897

0

119865119911

2

1198972

4119864119868119910

(1 + ]) (1198872

1198972)(

119910

119897) 119889119910

(5)

The bending SE (119880119887) in element 1 due to 119865

119911 119865119910119872119910 and119872

119911

is given by

1198801198871= int

119897

0

(119872119910minus 119865119911119909)2

2119864119868119910

119889119909 +

(119872119911+ 119865119910119909)2

2119864119868119911

119889119909 (6)

In element 2 the bending SE (119880119887) due to 119865

119910 119865119909 119865119911 119879119909 and

119872119911is given by

1198801198872= int

119897

0

(119872119911+ 119865119910119897 + 119865119909119910)2

2119864119868119911

119889119910 +(119879119909minus 119865119911119910)2

2119864119868119909

119889119910 (7)

where 119868119909= 119868119910= 1199051198873

12 and 119868119911= 1198871199053

12 is the inertiamomentabout the 119909 119910 and 119911 axes respectively

A partially restrained warping torsion analysis at the jointbetween elements 1 and 2 is presented in [25] The warpingrestraint factor (119870) is adopted to represent the degree ofwarping restraint In the present study the torsional SE (119880

119905)

was defined in element 1 due to moment 119879119909as follows [25]

1198801199051=1

2119864119862119908int

119897

0

(12057910158401015840

119909)2

119889119909 +1

2119866119869119909int

119897

0

(1205791015840

119909)2

119889119909 (8)

where

120579119909=

119879119909

120572119866119869119909

(120572119909

+ (1 minus 119870) (minus sinh120572119909 minus tanh120572119897 + tanh120572119897 cosh120572119909)) (9)

In element 2 the torsional SE (119880119905) due to moments 119865

119911119897 and

119872119910is as follows

1198801199052=1

2119864119862119908int

0

minus119897

(12057910158401015840

119910)2

119889119910 +1

2119866119869119910int

0

minus119897

(1205791015840

119910)2

119889119910 (10)

where

120579119910=

119872119910119888

120572119866119869119910

[120572119910 minus sinh120572119910

minus (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897sinh120572119897

)) cosh120572119910

+ (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897sinh120572119897

))]

(11)

where119872119910119888= 119872119910minus 119865119911119897

The torsion constant 119869119909of a bar with a rectangular

cross section can be determined by Equation (161) in [26]Therefore

119869119909=1198871199053

3[1 minus 063

119905

119887]

119869119909= 119869119910

(12)

Warping constant 119862119908

with respect to the shear center isdefined by Equation (743) in [27] as the warping momentof inertia

119862119908= int120596

2

119889119860 =(119887119905)3

144 (13)

where 120596 is the warping function and 120572 = radic119866119869119910119864119862119908is the

torsion-bending constant (119898)119870 the warping restraint factor is introduced to express

the degree of restraint from the warping at the joint of ele-ments 1 and 2 Equations (9) and (11) show that when 119870 =

1 free warping occurs at end 119909 = 0 or (119910 = minus119897) and fullyrestrained warping occurs when119870 = 0

The total SE (119880) of a SBLF with six-axis loads at free endwas summarized as follows

119880 = 1198801+ 1198802 (14)

where1198801is the total SE of element 1119880

1= 1198801198861+1198801199041+1198801198871+1198801199051

and1198802is the total SE of element 2119880

2= 1198801198862+1198801199042+1198801198872+1198801199052

From (1) the translational displacements due to forces119865119909

119865119910 and 119865

119911according to the 119909 119910 and 119911 axes respectively are

given by

120575119909=120597119880

120597119865119909

=119865119909119897

119860119864+1198651199091198971199052

(1 + ])2119864119868119911

+1

119864119868119911

(1198721199111198972

2+1198651199091198973

3+

1198651199101198973

2)

120575119910=120597119880

120597119865119910

=

119865119910119897

119860119864+

1198651199101198971199052

(1 + ])2119864119868119911

+1

119864119868119911

(31198721199111198972

2+

41198651199101198973

3+1198651199091198973

2)

4 Shock and Vibration

120575119911=120597119880

120597119865119911

=1198651199111198971198872

(1 + ])2119864119868119910

minus1198721199111198972

2119864119868119910

+1198651199111198973

3119864119868119910

minus1198791199091198972

2119864119868119909

+1198651199111198973

3119864119868119909

+

(1198651199111198972

minus119872119910119897) (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870

119905+ 120572119897119870

2

119905)

120572119866119869119910

(15)

The rotational displacements due to moments 119879119909 119872119910 and

119872119911about the 119909 119910 and 119911 axes respectively are given by

120579119909=120597119880

120597119879119909

=2119879119909119897 minus 1198651199111198972

2119864119868119909

+

119879119909(1198702 sinh120572119897 minus sinh120572119897 + 120572119897 cosh120572119897)

120572119866119869119909cosh120572119897

120579119910=

120597119880

120597119872119910

=

119872119910119897

119864119868119910

minus1198651199111198972

2119864119868119910

+

(119872119910minus 119865119911119897) (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870

119905+ 120572119897119870

2

119905)

120572119866119869119910

120579119911=

120597119880

120597119872119911

=1

119864119868119911

(2119872119911119897 +

31198651199101198972

2+1198651199091198972

2)

(16)

where

119870119905= (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897

sinh120572119897)) (17)

22 Compliance Matrix of the SBLF The relationshipbetween the translational and rotational displacements andthe applied forces and moments at the free end of the SBLF isformulated as follows

120575 = [119862] 119891

119891 = [119870] 120575

(18)

where 120575 = [120575119909120575119910120575119911120579119909120579119910120579119911]119879 119891 = [119865

119909119865119910119865119911119879119909119872119910119872119911]119879

[119862] is the compliance matrix or flexibility matrix [119870] is thestiffness matrix 119865

119899and 120575

119899are the force and translational

displacement with respect to the 119899-axis and 119872119899and 120579

119899

are the moment and rotational displacement about the 119899-axis respectively The compliance matrix [119862] includes 11independent elements as follows

120575119909

120575119910

120575119911

120579119909

120579119910

120579119911

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119862120575119909119865119909

119862120575119909119865119910

0 0 0 119862120575119909119872119911

119862120575119910119865119909

119862120575119910119865119910

0 0 0 119862120575119910119872119911

0 0 119862120575119911119865119911

119862120575119911119879119909

119862120575119911119872119910

0

0 0 119862120579119909119865119911

119862120579119909119879119909

0 0

0 0 119862120579119910119865119911

0 119862120579119910119872119910

0

119862120579119911119865119909

119862120579119911119865119910

0 0 0 119862120579119911119872119911

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119865119909

119865119910

119865119911

119879119909

119872119910

119872119911

119862120575119909119865119909

= (120597119880

120597119865119909

)1

119865119909

=119897

119860119864+1198971199052

(1 + ])4119864119868119911

+1198973

3119864119868119911

119862120575119909119865119910

= (120597119880

120597119865119909

)1

119865119910

=1198973

2119864119868119911

119862120575119909119872119911

= (120597119880

120597119865119909

)1

119872119911

=1198972

2119864119868119911

119862120575119910119865119909

= (120597119880

120597119865119910

)1

119865119909

=1198973

2119864119868119911

119862120575119910119865119910

= (120597119880

120597119865119910

)1

119865119910

=119897

119860119864+1198971199052

(1 + ])4119864119868119911

+41198973

3119864119868119911

119862120575119910119872119911

= (120597119880

120597119865119910

)1

119872119911

=31198973

2119864119868119911

119862120575119911119865119911

= (120597119880

120597119865119911

)1

119865119911

=1198971198872

(1 + ])2119864119868119910

+1198973

3119864119868119910

+1198973

3119864119868119909

+

1198972

(2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120575119911119879119909

= (120597119880

120597119865119911

)1

119879119909

= minus1198972

2119864119868119909

119862120575119911119872119910

= (120597119880

120597119865119911

)1

119872119910

= minus1198972

2119864119868119910

minus

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119909119865119911

= (120597119880

120597119879119909

)1

119865119911

= minus1198972

2119864119868119909

119862120579119909119879119909

= (120597119880

120597119879119909

)1

119879119909

=119897

119864119868119909

Shock and Vibration 5

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

12

14

FEATheoryError

4 6 8 10 12 14 16 18 20

Fy sensitivity with length l

l (mm)

120575y

(mm

)

(a) 120575119910-119897

FEATheoryError

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

120575y

(mm

)

t (mm)02 04 06 08 10

Fy sensitivity with thickness t

(b) 120575119910-119905

FEATheoryError

00

02

04

06

08

10

Erro

r (

)

2

4

6

8

10

12

14

120575y

(mm

)

b (mm)2 3 4 5 6 7 8

Fy sensitivity with width b

(c) 120575119910-119887

Figure 2 Variation of deflection 120575119910according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119910

+

(1198702 sinh120572119897 minus sinh120572119897 + 120572119897 cosh120572119897)

120572119866119869119909cosh120572119897

119862120579119910119865119911

= (120597119880

120597119872119910

)1

119865119911

= minus1198972

2119864119868119910

minus

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119910119872119910

= (120597119880

120597119872119910

)1

119872119910

=119897

119864119868119910

+

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119911119865119909

= (120597119880

120597119872119911

)1

119865119909

=1198972

2119864119868119911

119862120579119911119865119910

= (120597119880

120597119872119911

)1

119865119910

=31198973

2119864119868119911

119862120579119911119872119911

= (120597119880

120597119872119911

)1

119872119911

=2119897

119864119868119911

(19)

3 FEA Verification

In this study FEA was conducted by using Pro-Mechanicacommercial software (Wildfire 5 PTC Corp MA USA) toverify the results obtained by theory and to validate thereliability of the calculated results The default values of theSBLF were chosen as follows length 119897 = 10mm width 119887 =

4mm and thickness 119905 = 05mm The forces and moments

6 Shock and Vibration

Table 1 Comparison between theory and the FEA at the default values of the SBLF

Displacements FEA Theory Error ()Shear and Warp No shear and No warp S and W No S and NoW

120575119909(mm) 116210 116466 116183 022 minus002

120575119910(mm) 044183 046480 046451 494 488

120575119911(mm) 021102 020825 022556 131 645

120579119909(rad) 000239 000233 000251 219 504

120579119910(rad) 000210 000209 000227 047 743

120579119911(rad) 000673 000697 000697 343 343

Table 2 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under 119865119909 119865119910 and 119865

119911

119897 (mm) FEA Theory Error () t (mm) FEA Theory Error () b (mm) FEA Theory Error ()(a) Deflection 120575

119909(mm)

5 013819 014550 502 025 877255 929028 557 2 228306 232366 17565 030485 031934 454 0325 397715 422907 596 26 174683 178743 2278 057071 059507 409 04 217394 226869 418 32 141120 145229 28395 095908 099619 372 0475 130778 135502 349 38 118188 122298 33611 149324 154623 343 055 084862 087301 279 44 101548 105621 386125 219646 226869 318 0625 058257 059507 210 5 088936 092946 43214 309203 318708 298 07 041772 042366 140 56 079058 082988 474155 420323 432493 281 0775 031008 031227 070 62 071120 074957 51217 555339 570573 267 085 023676 023676 000 68 064607 068343 547185 716579 735301 255 0925 018508 018377 minus071 74 059170 062802 57820 906374 929028 244 1 014756 014550 minus142 8 054565 058091 607

(b) Deflection 120575119910(mm)

5 005369 005809 758 025 355785 371567 425 2 090181 092903 29365 011959 012759 627 0325 162113 169129 415 26 069073 071464 3358 022485 023785 547 04 086939 090720 417 32 055848 058064 38295 037879 039827 489 0475 051931 054178 415 38 046800 048896 42911 059062 061825 447 055 033455 034901 414 44 040226 042229 474125 086949 090720 416 0625 022798 023785 415 5 035238 037161 51714 122458 127453 392 07 016225 016931 417 56 031329 033180 558155 166503 172963 374 0775 011953 012477 420 62 028185 029969 59517 219995 228192 359 085 009057 009458 424 68 025604 027324 629185 283849 294080 348 0925 007025 007339 428 74 023449 025109 66120 358976 371567 339 1 005557 005809 433 8 021624 023226 690

(c) Deflection 120575119911(mm)

5 002203 002113 409 025 160255 158055 137 2 052645 051994 12465 005270 005127 272 0325 074212 073006 163 26 037023 036566 1248 010341 010146 189 04 040469 039777 171 32 028218 027863 12695 017912 017672 134 0475 024617 024151 190 38 022569 022270 13311 028479 028210 095 055 016164 015831 207 44 018640 018372 144125 042541 042263 065 0625 011239 010988 224 5 015750 015497 16114 060595 060338 043 07 008169 007972 242 56 013538 013290 184155 083137 082938 024 0775 006152 005992 261 62 011792 011542 21217 110665 110569 009 085 004770 004636 281 68 010379 010126 244185 143675 143737 minus004 0925 003788 003675 301 74 009214 008956 28120 182663 182945 minus015 1 003071 002973 322 8 008239 007975 321

Shock and Vibration 7

00

05

10

15

20

25

30

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

Fz sensitivity with length l

l (mm)4 6 8 10 12 14 16 18 20

minus2

120575z

(mm

)

(a) 120575119911-119897

FEATheoryError

00

05

10

15

20

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with thickness t

t (mm)02 04 06 08 10

120575z

(mm

)

(b) 120575119911-119905

FEATheoryError

00

01

02

03

04

05

06

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with width b

b (mm)2 3 4 5 6 7 8

120575z

(mm

)

(c) 120575119911-119887

Figure 3 Variation of deflection 120575119911according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119911

applied to the SBLF are 119865119909= 10N 119865

119910= 119865119911= 1N and 119879

119909=

119872119910= 119872119911= 1NmmThewarping restraint factor was chosen

to be 119870 = 05 The material selected in this simulation isaluminum Al 6061 In the parametric analysis the sensitivityanalysis was performed to determine the agreement betweenFEA and theory with the variation of length 119897 = 5 to 20mmwidth 119887 = 2 to 8mm and thickness 119905 = 025 to 1mmrespectively If the errors between the two methods are lowerthan 10 the results can usually be accepted and used fromthe viewpoint of the engineering design

Table 1 shows the comparison between the theory andFEA results at the default values of the SBLF Equations (15)ndash(16) were applied to find the translational and rotationaldisplacements by the theoretical method that considers thevariable shear deformation in the bending and the warpingin both the bending and torsion and the partially restrained

warping at the joint of elements 1 and 2 The derivedtheoretical equations showed a good agreement with the FEAresults all of the errors of these simulationswere less than 5whereas the results of without shear and warp effect showedhigher errors up to 74 as presented in Table 1 The resultsof in-plane deformations 120575

119909 120575119910 and 120579

119911in this study were

consistent with results of the equations in [24] Althoughthese results were showing the good agreement at the defaultvalues the sensitive parameters analysis was also conductedto ensure the reliability of the derived equations

In the research of the reference [28] we analyzed thedisplacement of a SBLF under transverse loading by theEuler-Bernoulli Timoshenko and third-order beam theoryand the results showed that the third-order beam theoryof Levinson which considers the shear and warp effect inbending gave the good results In this study the above beam

8 Shock and Vibration

Table 3 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under loads 119879119909119872119910 and119872

119911

119897 (mm) FEA Theory Error () 119905 (mm) FEA Theory Error () 119887 (mm) FEA Theory Error ()(a) Rotation 120579

119909(rad)

5 000114 000108 559 025 001795 001762 186 2 000568 000561 13465 000152 000145 404 0325 000833 000815 215 26 000404 000397 1668 000189 000183 307 04 000455 000445 228 32 000312 000306 19195 000226 000221 240 0475 000277 000270 250 38 000254 000248 21311 000263 000258 191 055 000183 000178 271 44 000213 000208 236125 000301 000296 153 0625 000127 000123 289 5 000183 000179 26114 000338 000334 123 07 000093 000090 303 56 000161 000156 288155 000375 000371 099 0775 000070 000068 316 62 000143 000138 31817 000412 000409 078 085 000054 000052 325 68 000128 000124 349185 000450 000447 060 0925 000043 000042 333 74 000116 000112 38320 000487 000485 045 1 000035 000034 339 8 000106 000102 419

(b) Rotation 120579119910(rad)

5 000089 000084 477 025 001605 001583 134 2 000540 000533 13365 000126 000122 308 0325 000744 000732 160 26 000376 000371 1358 000163 000159 211 04 000407 000399 177 32 000285 000281 13995 000200 000197 149 0475 000248 000243 201 38 000228 000224 14611 000237 000235 106 055 000163 000159 223 44 000188 000185 159125 000274 000272 074 0625 000114 000111 246 5 000158 000155 17714 000312 000310 049 07 000083 000081 269 56 000136 000133 200155 000349 000348 029 0775 000063 000061 292 62 000118 000116 22917 000386 000385 013 085 000049 000047 316 68 000104 000101 262185 000423 000423 minus001 0925 000039 000037 341 74 000092 000090 29920 000460 000461 minus012 1 000032 000030 365 8 000082 000080 339

(c) Rotation 120579119911(rad)

5 000328 000348 571 025 005405 005573 302 2 001365 001393 20065 000432 000453 470 0325 002462 002537 294 26 001047 001072 2318 000535 000557 403 04 001320 001361 295 32 000848 000871 26695 000638 000662 358 0475 000789 000813 294 38 000711 000733 30111 000741 000766 325 055 000508 000523 293 44 000612 000633 337125 000845 000871 302 0625 000346 000357 293 5 000537 000557 37214 000948 000975 284 07 000246 000254 293 56 000477 000498 406155 001051 001080 271 0775 000182 000187 294 62 000430 000449 43917 001153 001184 261 085 000138 000142 296 68 000391 000410 470185 001256 001289 253 0925 000107 000110 298 74 000358 000377 50120 001359 001393 247 1 000084 000087 301 8 000330 000348 531

theory was also used and the translational displacements 120575119909

120575119910 and 120575

119911under 119865

119909 119865119910 and 119865

119911according to the variations of

length 119897 thickness 119905 and width 119887 respectively are presentedin Table 2 Figure 2 shows typical simulation results underforce 119865

119910 The errors in these investigations are lower (76)

The sensitive analysis of 119865119911is also presented in Figure 3 All

the errors are lower (5) These results reveal the accuracyof the derived theoretical equations for the translationaldisplacements of SBLF

The comparison results among the fully restrained par-tially restrained and free warping in torsion analysis of SBLFwere investigated in [25] and the partially restrained warpingwith factor 119870 = 05 was applied in this torsion simulationTable 3 shows the variation of rotational displacements 120579

119909 120579119910

and 120579119911due to moments 119879

119909 119872119910 and 119872

119911with the variation

of parameters 119897 119905 and 119887 of the SBLF respectively It canbe observed that the results of theory and FEA are in goodagreement with the errors lower than 6 Figure 4 presentsthe typical variation of rotation 120579

119909according to 119897 119905 and 119887

under 119879119909 The maximum error is lower than 6

In summary the relationship between the deformationand the applied loads for the SBLF is presented in compliancematrix form The variable shear deformation and warpingeffect were considered and partial restraint was introducedin the analysis All results were verified by FEA with strongagreement between the two methods The errors were lowerthan 76 These results demonstrate the high accuracy andreliability of the proposed theoretical equations

Shock and Vibration 9

0000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

4 6 8 10 12 14 16 18 20

Tx sensitivity with length l

l (mm)

(a) 120579119909-119897

0000

0005

0010

0015

0020

0025

0030

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

t (mm)02 04 06 08 10

Tx sensitivity with thickness t

(b) 120579119909-119905

2 3 4 5 6 7 80000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

Tx sensitivity with width b

b (mm)

(c) 120579119909-119887

Figure 4 Variation of rotation 120579119909according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under load 119879

119909

4 Conclusion

In this study a compliance matrix that expresses the rela-tionship between the deformations and applied loads ofthe SBLF was analyzed and derived by using Castiglianorsquossecond theorem In bending analysis higher-order beamtheory was applied wherein variable shear deformation andthe warping were considered in the calculated formulas forthe shear forces At the joint of two elements of the SBLFthe partially restrained warping in torsion was analysed withconsideration of the warping restraint factor The theoreticalresults were verified by FEA at both the default and thesensitive values The results indicate that there is strongagreement between the two methods with errors below

76 This suggests the accuracy of the proposed theoreticalequations and that they can be used in the precision machinedesign

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Nghia-Huu Nguyen and Moo-Yeon Lee equally contributedto this work

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

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Shock and Vibration 3

deflection is described in [21] Therefore the shear SE (119880119904) in

element 1 due to the shear forces 119865119910 119865119911is given by

1198801199041119865119910

= int

119897

0

119865119910

2

1198972

4119864119868119911

(1 + ]) (1199052

1198972)(

119909

119897) 119889119909

1198801199041119865119911

= int

119897

0

119865119911

2

1198972

4119864119868119910

(1 + ]) (1198872

1198972)(

119909

119897) 119889119909

(4)

where ] is Poissonrsquos ratio In element 2 the shear SE (119880119904) due

to the shear forces 119865119909 119865119911is given by

1198801199042119865119909

= int

119897

0

119865119909

2

1198972

4119864119868119911

(1 + ]) (1199052

1198972)(

119910

119897) 119889119910

1198801199042119865119911

= int

119897

0

119865119911

2

1198972

4119864119868119910

(1 + ]) (1198872

1198972)(

119910

119897) 119889119910

(5)

The bending SE (119880119887) in element 1 due to 119865

119911 119865119910119872119910 and119872

119911

is given by

1198801198871= int

119897

0

(119872119910minus 119865119911119909)2

2119864119868119910

119889119909 +

(119872119911+ 119865119910119909)2

2119864119868119911

119889119909 (6)

In element 2 the bending SE (119880119887) due to 119865

119910 119865119909 119865119911 119879119909 and

119872119911is given by

1198801198872= int

119897

0

(119872119911+ 119865119910119897 + 119865119909119910)2

2119864119868119911

119889119910 +(119879119909minus 119865119911119910)2

2119864119868119909

119889119910 (7)

where 119868119909= 119868119910= 1199051198873

12 and 119868119911= 1198871199053

12 is the inertiamomentabout the 119909 119910 and 119911 axes respectively

A partially restrained warping torsion analysis at the jointbetween elements 1 and 2 is presented in [25] The warpingrestraint factor (119870) is adopted to represent the degree ofwarping restraint In the present study the torsional SE (119880

119905)

was defined in element 1 due to moment 119879119909as follows [25]

1198801199051=1

2119864119862119908int

119897

0

(12057910158401015840

119909)2

119889119909 +1

2119866119869119909int

119897

0

(1205791015840

119909)2

119889119909 (8)

where

120579119909=

119879119909

120572119866119869119909

(120572119909

+ (1 minus 119870) (minus sinh120572119909 minus tanh120572119897 + tanh120572119897 cosh120572119909)) (9)

In element 2 the torsional SE (119880119905) due to moments 119865

119911119897 and

119872119910is as follows

1198801199052=1

2119864119862119908int

0

minus119897

(12057910158401015840

119910)2

119889119910 +1

2119866119869119910int

0

minus119897

(1205791015840

119910)2

119889119910 (10)

where

120579119910=

119872119910119888

120572119866119869119910

[120572119910 minus sinh120572119910

minus (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897sinh120572119897

)) cosh120572119910

+ (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897sinh120572119897

))]

(11)

where119872119910119888= 119872119910minus 119865119911119897

The torsion constant 119869119909of a bar with a rectangular

cross section can be determined by Equation (161) in [26]Therefore

119869119909=1198871199053

3[1 minus 063

119905

119887]

119869119909= 119869119910

(12)

Warping constant 119862119908

with respect to the shear center isdefined by Equation (743) in [27] as the warping momentof inertia

119862119908= int120596

2

119889119860 =(119887119905)3

144 (13)

where 120596 is the warping function and 120572 = radic119866119869119910119864119862119908is the

torsion-bending constant (119898)119870 the warping restraint factor is introduced to express

the degree of restraint from the warping at the joint of ele-ments 1 and 2 Equations (9) and (11) show that when 119870 =

1 free warping occurs at end 119909 = 0 or (119910 = minus119897) and fullyrestrained warping occurs when119870 = 0

The total SE (119880) of a SBLF with six-axis loads at free endwas summarized as follows

119880 = 1198801+ 1198802 (14)

where1198801is the total SE of element 1119880

1= 1198801198861+1198801199041+1198801198871+1198801199051

and1198802is the total SE of element 2119880

2= 1198801198862+1198801199042+1198801198872+1198801199052

From (1) the translational displacements due to forces119865119909

119865119910 and 119865

119911according to the 119909 119910 and 119911 axes respectively are

given by

120575119909=120597119880

120597119865119909

=119865119909119897

119860119864+1198651199091198971199052

(1 + ])2119864119868119911

+1

119864119868119911

(1198721199111198972

2+1198651199091198973

3+

1198651199101198973

2)

120575119910=120597119880

120597119865119910

=

119865119910119897

119860119864+

1198651199101198971199052

(1 + ])2119864119868119911

+1

119864119868119911

(31198721199111198972

2+

41198651199101198973

3+1198651199091198973

2)

4 Shock and Vibration

120575119911=120597119880

120597119865119911

=1198651199111198971198872

(1 + ])2119864119868119910

minus1198721199111198972

2119864119868119910

+1198651199111198973

3119864119868119910

minus1198791199091198972

2119864119868119909

+1198651199111198973

3119864119868119909

+

(1198651199111198972

minus119872119910119897) (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870

119905+ 120572119897119870

2

119905)

120572119866119869119910

(15)

The rotational displacements due to moments 119879119909 119872119910 and

119872119911about the 119909 119910 and 119911 axes respectively are given by

120579119909=120597119880

120597119879119909

=2119879119909119897 minus 1198651199111198972

2119864119868119909

+

119879119909(1198702 sinh120572119897 minus sinh120572119897 + 120572119897 cosh120572119897)

120572119866119869119909cosh120572119897

120579119910=

120597119880

120597119872119910

=

119872119910119897

119864119868119910

minus1198651199111198972

2119864119868119910

+

(119872119910minus 119865119911119897) (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870

119905+ 120572119897119870

2

119905)

120572119866119869119910

120579119911=

120597119880

120597119872119911

=1

119864119868119911

(2119872119911119897 +

31198651199101198972

2+1198651199091198972

2)

(16)

where

119870119905= (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897

sinh120572119897)) (17)

22 Compliance Matrix of the SBLF The relationshipbetween the translational and rotational displacements andthe applied forces and moments at the free end of the SBLF isformulated as follows

120575 = [119862] 119891

119891 = [119870] 120575

(18)

where 120575 = [120575119909120575119910120575119911120579119909120579119910120579119911]119879 119891 = [119865

119909119865119910119865119911119879119909119872119910119872119911]119879

[119862] is the compliance matrix or flexibility matrix [119870] is thestiffness matrix 119865

119899and 120575

119899are the force and translational

displacement with respect to the 119899-axis and 119872119899and 120579

119899

are the moment and rotational displacement about the 119899-axis respectively The compliance matrix [119862] includes 11independent elements as follows

120575119909

120575119910

120575119911

120579119909

120579119910

120579119911

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119862120575119909119865119909

119862120575119909119865119910

0 0 0 119862120575119909119872119911

119862120575119910119865119909

119862120575119910119865119910

0 0 0 119862120575119910119872119911

0 0 119862120575119911119865119911

119862120575119911119879119909

119862120575119911119872119910

0

0 0 119862120579119909119865119911

119862120579119909119879119909

0 0

0 0 119862120579119910119865119911

0 119862120579119910119872119910

0

119862120579119911119865119909

119862120579119911119865119910

0 0 0 119862120579119911119872119911

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119865119909

119865119910

119865119911

119879119909

119872119910

119872119911

119862120575119909119865119909

= (120597119880

120597119865119909

)1

119865119909

=119897

119860119864+1198971199052

(1 + ])4119864119868119911

+1198973

3119864119868119911

119862120575119909119865119910

= (120597119880

120597119865119909

)1

119865119910

=1198973

2119864119868119911

119862120575119909119872119911

= (120597119880

120597119865119909

)1

119872119911

=1198972

2119864119868119911

119862120575119910119865119909

= (120597119880

120597119865119910

)1

119865119909

=1198973

2119864119868119911

119862120575119910119865119910

= (120597119880

120597119865119910

)1

119865119910

=119897

119860119864+1198971199052

(1 + ])4119864119868119911

+41198973

3119864119868119911

119862120575119910119872119911

= (120597119880

120597119865119910

)1

119872119911

=31198973

2119864119868119911

119862120575119911119865119911

= (120597119880

120597119865119911

)1

119865119911

=1198971198872

(1 + ])2119864119868119910

+1198973

3119864119868119910

+1198973

3119864119868119909

+

1198972

(2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120575119911119879119909

= (120597119880

120597119865119911

)1

119879119909

= minus1198972

2119864119868119909

119862120575119911119872119910

= (120597119880

120597119865119911

)1

119872119910

= minus1198972

2119864119868119910

minus

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119909119865119911

= (120597119880

120597119879119909

)1

119865119911

= minus1198972

2119864119868119909

119862120579119909119879119909

= (120597119880

120597119879119909

)1

119879119909

=119897

119864119868119909

Shock and Vibration 5

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

12

14

FEATheoryError

4 6 8 10 12 14 16 18 20

Fy sensitivity with length l

l (mm)

120575y

(mm

)

(a) 120575119910-119897

FEATheoryError

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

120575y

(mm

)

t (mm)02 04 06 08 10

Fy sensitivity with thickness t

(b) 120575119910-119905

FEATheoryError

00

02

04

06

08

10

Erro

r (

)

2

4

6

8

10

12

14

120575y

(mm

)

b (mm)2 3 4 5 6 7 8

Fy sensitivity with width b

(c) 120575119910-119887

Figure 2 Variation of deflection 120575119910according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119910

+

(1198702 sinh120572119897 minus sinh120572119897 + 120572119897 cosh120572119897)

120572119866119869119909cosh120572119897

119862120579119910119865119911

= (120597119880

120597119872119910

)1

119865119911

= minus1198972

2119864119868119910

minus

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119910119872119910

= (120597119880

120597119872119910

)1

119872119910

=119897

119864119868119910

+

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119911119865119909

= (120597119880

120597119872119911

)1

119865119909

=1198972

2119864119868119911

119862120579119911119865119910

= (120597119880

120597119872119911

)1

119865119910

=31198973

2119864119868119911

119862120579119911119872119911

= (120597119880

120597119872119911

)1

119872119911

=2119897

119864119868119911

(19)

3 FEA Verification

In this study FEA was conducted by using Pro-Mechanicacommercial software (Wildfire 5 PTC Corp MA USA) toverify the results obtained by theory and to validate thereliability of the calculated results The default values of theSBLF were chosen as follows length 119897 = 10mm width 119887 =

4mm and thickness 119905 = 05mm The forces and moments

6 Shock and Vibration

Table 1 Comparison between theory and the FEA at the default values of the SBLF

Displacements FEA Theory Error ()Shear and Warp No shear and No warp S and W No S and NoW

120575119909(mm) 116210 116466 116183 022 minus002

120575119910(mm) 044183 046480 046451 494 488

120575119911(mm) 021102 020825 022556 131 645

120579119909(rad) 000239 000233 000251 219 504

120579119910(rad) 000210 000209 000227 047 743

120579119911(rad) 000673 000697 000697 343 343

Table 2 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under 119865119909 119865119910 and 119865

119911

119897 (mm) FEA Theory Error () t (mm) FEA Theory Error () b (mm) FEA Theory Error ()(a) Deflection 120575

119909(mm)

5 013819 014550 502 025 877255 929028 557 2 228306 232366 17565 030485 031934 454 0325 397715 422907 596 26 174683 178743 2278 057071 059507 409 04 217394 226869 418 32 141120 145229 28395 095908 099619 372 0475 130778 135502 349 38 118188 122298 33611 149324 154623 343 055 084862 087301 279 44 101548 105621 386125 219646 226869 318 0625 058257 059507 210 5 088936 092946 43214 309203 318708 298 07 041772 042366 140 56 079058 082988 474155 420323 432493 281 0775 031008 031227 070 62 071120 074957 51217 555339 570573 267 085 023676 023676 000 68 064607 068343 547185 716579 735301 255 0925 018508 018377 minus071 74 059170 062802 57820 906374 929028 244 1 014756 014550 minus142 8 054565 058091 607

(b) Deflection 120575119910(mm)

5 005369 005809 758 025 355785 371567 425 2 090181 092903 29365 011959 012759 627 0325 162113 169129 415 26 069073 071464 3358 022485 023785 547 04 086939 090720 417 32 055848 058064 38295 037879 039827 489 0475 051931 054178 415 38 046800 048896 42911 059062 061825 447 055 033455 034901 414 44 040226 042229 474125 086949 090720 416 0625 022798 023785 415 5 035238 037161 51714 122458 127453 392 07 016225 016931 417 56 031329 033180 558155 166503 172963 374 0775 011953 012477 420 62 028185 029969 59517 219995 228192 359 085 009057 009458 424 68 025604 027324 629185 283849 294080 348 0925 007025 007339 428 74 023449 025109 66120 358976 371567 339 1 005557 005809 433 8 021624 023226 690

(c) Deflection 120575119911(mm)

5 002203 002113 409 025 160255 158055 137 2 052645 051994 12465 005270 005127 272 0325 074212 073006 163 26 037023 036566 1248 010341 010146 189 04 040469 039777 171 32 028218 027863 12695 017912 017672 134 0475 024617 024151 190 38 022569 022270 13311 028479 028210 095 055 016164 015831 207 44 018640 018372 144125 042541 042263 065 0625 011239 010988 224 5 015750 015497 16114 060595 060338 043 07 008169 007972 242 56 013538 013290 184155 083137 082938 024 0775 006152 005992 261 62 011792 011542 21217 110665 110569 009 085 004770 004636 281 68 010379 010126 244185 143675 143737 minus004 0925 003788 003675 301 74 009214 008956 28120 182663 182945 minus015 1 003071 002973 322 8 008239 007975 321

Shock and Vibration 7

00

05

10

15

20

25

30

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

Fz sensitivity with length l

l (mm)4 6 8 10 12 14 16 18 20

minus2

120575z

(mm

)

(a) 120575119911-119897

FEATheoryError

00

05

10

15

20

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with thickness t

t (mm)02 04 06 08 10

120575z

(mm

)

(b) 120575119911-119905

FEATheoryError

00

01

02

03

04

05

06

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with width b

b (mm)2 3 4 5 6 7 8

120575z

(mm

)

(c) 120575119911-119887

Figure 3 Variation of deflection 120575119911according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119911

applied to the SBLF are 119865119909= 10N 119865

119910= 119865119911= 1N and 119879

119909=

119872119910= 119872119911= 1NmmThewarping restraint factor was chosen

to be 119870 = 05 The material selected in this simulation isaluminum Al 6061 In the parametric analysis the sensitivityanalysis was performed to determine the agreement betweenFEA and theory with the variation of length 119897 = 5 to 20mmwidth 119887 = 2 to 8mm and thickness 119905 = 025 to 1mmrespectively If the errors between the two methods are lowerthan 10 the results can usually be accepted and used fromthe viewpoint of the engineering design

Table 1 shows the comparison between the theory andFEA results at the default values of the SBLF Equations (15)ndash(16) were applied to find the translational and rotationaldisplacements by the theoretical method that considers thevariable shear deformation in the bending and the warpingin both the bending and torsion and the partially restrained

warping at the joint of elements 1 and 2 The derivedtheoretical equations showed a good agreement with the FEAresults all of the errors of these simulationswere less than 5whereas the results of without shear and warp effect showedhigher errors up to 74 as presented in Table 1 The resultsof in-plane deformations 120575

119909 120575119910 and 120579

119911in this study were

consistent with results of the equations in [24] Althoughthese results were showing the good agreement at the defaultvalues the sensitive parameters analysis was also conductedto ensure the reliability of the derived equations

In the research of the reference [28] we analyzed thedisplacement of a SBLF under transverse loading by theEuler-Bernoulli Timoshenko and third-order beam theoryand the results showed that the third-order beam theoryof Levinson which considers the shear and warp effect inbending gave the good results In this study the above beam

8 Shock and Vibration

Table 3 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under loads 119879119909119872119910 and119872

119911

119897 (mm) FEA Theory Error () 119905 (mm) FEA Theory Error () 119887 (mm) FEA Theory Error ()(a) Rotation 120579

119909(rad)

5 000114 000108 559 025 001795 001762 186 2 000568 000561 13465 000152 000145 404 0325 000833 000815 215 26 000404 000397 1668 000189 000183 307 04 000455 000445 228 32 000312 000306 19195 000226 000221 240 0475 000277 000270 250 38 000254 000248 21311 000263 000258 191 055 000183 000178 271 44 000213 000208 236125 000301 000296 153 0625 000127 000123 289 5 000183 000179 26114 000338 000334 123 07 000093 000090 303 56 000161 000156 288155 000375 000371 099 0775 000070 000068 316 62 000143 000138 31817 000412 000409 078 085 000054 000052 325 68 000128 000124 349185 000450 000447 060 0925 000043 000042 333 74 000116 000112 38320 000487 000485 045 1 000035 000034 339 8 000106 000102 419

(b) Rotation 120579119910(rad)

5 000089 000084 477 025 001605 001583 134 2 000540 000533 13365 000126 000122 308 0325 000744 000732 160 26 000376 000371 1358 000163 000159 211 04 000407 000399 177 32 000285 000281 13995 000200 000197 149 0475 000248 000243 201 38 000228 000224 14611 000237 000235 106 055 000163 000159 223 44 000188 000185 159125 000274 000272 074 0625 000114 000111 246 5 000158 000155 17714 000312 000310 049 07 000083 000081 269 56 000136 000133 200155 000349 000348 029 0775 000063 000061 292 62 000118 000116 22917 000386 000385 013 085 000049 000047 316 68 000104 000101 262185 000423 000423 minus001 0925 000039 000037 341 74 000092 000090 29920 000460 000461 minus012 1 000032 000030 365 8 000082 000080 339

(c) Rotation 120579119911(rad)

5 000328 000348 571 025 005405 005573 302 2 001365 001393 20065 000432 000453 470 0325 002462 002537 294 26 001047 001072 2318 000535 000557 403 04 001320 001361 295 32 000848 000871 26695 000638 000662 358 0475 000789 000813 294 38 000711 000733 30111 000741 000766 325 055 000508 000523 293 44 000612 000633 337125 000845 000871 302 0625 000346 000357 293 5 000537 000557 37214 000948 000975 284 07 000246 000254 293 56 000477 000498 406155 001051 001080 271 0775 000182 000187 294 62 000430 000449 43917 001153 001184 261 085 000138 000142 296 68 000391 000410 470185 001256 001289 253 0925 000107 000110 298 74 000358 000377 50120 001359 001393 247 1 000084 000087 301 8 000330 000348 531

theory was also used and the translational displacements 120575119909

120575119910 and 120575

119911under 119865

119909 119865119910 and 119865

119911according to the variations of

length 119897 thickness 119905 and width 119887 respectively are presentedin Table 2 Figure 2 shows typical simulation results underforce 119865

119910 The errors in these investigations are lower (76)

The sensitive analysis of 119865119911is also presented in Figure 3 All

the errors are lower (5) These results reveal the accuracyof the derived theoretical equations for the translationaldisplacements of SBLF

The comparison results among the fully restrained par-tially restrained and free warping in torsion analysis of SBLFwere investigated in [25] and the partially restrained warpingwith factor 119870 = 05 was applied in this torsion simulationTable 3 shows the variation of rotational displacements 120579

119909 120579119910

and 120579119911due to moments 119879

119909 119872119910 and 119872

119911with the variation

of parameters 119897 119905 and 119887 of the SBLF respectively It canbe observed that the results of theory and FEA are in goodagreement with the errors lower than 6 Figure 4 presentsthe typical variation of rotation 120579

119909according to 119897 119905 and 119887

under 119879119909 The maximum error is lower than 6

In summary the relationship between the deformationand the applied loads for the SBLF is presented in compliancematrix form The variable shear deformation and warpingeffect were considered and partial restraint was introducedin the analysis All results were verified by FEA with strongagreement between the two methods The errors were lowerthan 76 These results demonstrate the high accuracy andreliability of the proposed theoretical equations

Shock and Vibration 9

0000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

4 6 8 10 12 14 16 18 20

Tx sensitivity with length l

l (mm)

(a) 120579119909-119897

0000

0005

0010

0015

0020

0025

0030

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

t (mm)02 04 06 08 10

Tx sensitivity with thickness t

(b) 120579119909-119905

2 3 4 5 6 7 80000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

Tx sensitivity with width b

b (mm)

(c) 120579119909-119887

Figure 4 Variation of rotation 120579119909according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under load 119879

119909

4 Conclusion

In this study a compliance matrix that expresses the rela-tionship between the deformations and applied loads ofthe SBLF was analyzed and derived by using Castiglianorsquossecond theorem In bending analysis higher-order beamtheory was applied wherein variable shear deformation andthe warping were considered in the calculated formulas forthe shear forces At the joint of two elements of the SBLFthe partially restrained warping in torsion was analysed withconsideration of the warping restraint factor The theoreticalresults were verified by FEA at both the default and thesensitive values The results indicate that there is strongagreement between the two methods with errors below

76 This suggests the accuracy of the proposed theoreticalequations and that they can be used in the precision machinedesign

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Nghia-Huu Nguyen and Moo-Yeon Lee equally contributedto this work

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Shock and Vibration

120575119911=120597119880

120597119865119911

=1198651199111198971198872

(1 + ])2119864119868119910

minus1198721199111198972

2119864119868119910

+1198651199111198973

3119864119868119910

minus1198791199091198972

2119864119868119909

+1198651199111198973

3119864119868119909

+

(1198651199111198972

minus119872119910119897) (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870

119905+ 120572119897119870

2

119905)

120572119866119869119910

(15)

The rotational displacements due to moments 119879119909 119872119910 and

119872119911about the 119909 119910 and 119911 axes respectively are given by

120579119909=120597119880

120597119879119909

=2119879119909119897 minus 1198651199111198972

2119864119868119909

+

119879119909(1198702 sinh120572119897 minus sinh120572119897 + 120572119897 cosh120572119897)

120572119866119869119909cosh120572119897

120579119910=

120597119880

120597119872119910

=

119872119910119897

119864119868119910

minus1198651199111198972

2119864119868119910

+

(119872119910minus 119865119911119897) (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870

119905+ 120572119897119870

2

119905)

120572119866119869119910

120579119911=

120597119880

120597119872119911

=1

119864119868119911

(2119872119911119897 +

31198651199101198972

2+1198651199091198972

2)

(16)

where

119870119905= (119870 tanh120572119897 minus (1 minus 119870)(1 minus cosh120572119897

sinh120572119897)) (17)

22 Compliance Matrix of the SBLF The relationshipbetween the translational and rotational displacements andthe applied forces and moments at the free end of the SBLF isformulated as follows

120575 = [119862] 119891

119891 = [119870] 120575

(18)

where 120575 = [120575119909120575119910120575119911120579119909120579119910120579119911]119879 119891 = [119865

119909119865119910119865119911119879119909119872119910119872119911]119879

[119862] is the compliance matrix or flexibility matrix [119870] is thestiffness matrix 119865

119899and 120575

119899are the force and translational

displacement with respect to the 119899-axis and 119872119899and 120579

119899

are the moment and rotational displacement about the 119899-axis respectively The compliance matrix [119862] includes 11independent elements as follows

120575119909

120575119910

120575119911

120579119909

120579119910

120579119911

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119862120575119909119865119909

119862120575119909119865119910

0 0 0 119862120575119909119872119911

119862120575119910119865119909

119862120575119910119865119910

0 0 0 119862120575119910119872119911

0 0 119862120575119911119865119911

119862120575119911119879119909

119862120575119911119872119910

0

0 0 119862120579119909119865119911

119862120579119909119879119909

0 0

0 0 119862120579119910119865119911

0 119862120579119910119872119910

0

119862120579119911119865119909

119862120579119911119865119910

0 0 0 119862120579119911119872119911

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119865119909

119865119910

119865119911

119879119909

119872119910

119872119911

119862120575119909119865119909

= (120597119880

120597119865119909

)1

119865119909

=119897

119860119864+1198971199052

(1 + ])4119864119868119911

+1198973

3119864119868119911

119862120575119909119865119910

= (120597119880

120597119865119909

)1

119865119910

=1198973

2119864119868119911

119862120575119909119872119911

= (120597119880

120597119865119909

)1

119872119911

=1198972

2119864119868119911

119862120575119910119865119909

= (120597119880

120597119865119910

)1

119865119909

=1198973

2119864119868119911

119862120575119910119865119910

= (120597119880

120597119865119910

)1

119865119910

=119897

119860119864+1198971199052

(1 + ])4119864119868119911

+41198973

3119864119868119911

119862120575119910119872119911

= (120597119880

120597119865119910

)1

119872119911

=31198973

2119864119868119911

119862120575119911119865119911

= (120597119880

120597119865119911

)1

119865119911

=1198971198872

(1 + ])2119864119868119910

+1198973

3119864119868119910

+1198973

3119864119868119909

+

1198972

(2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120575119911119879119909

= (120597119880

120597119865119911

)1

119879119909

= minus1198972

2119864119868119909

119862120575119911119872119910

= (120597119880

120597119865119911

)1

119872119910

= minus1198972

2119864119868119910

minus

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119909119865119911

= (120597119880

120597119879119909

)1

119865119911

= minus1198972

2119864119868119909

119862120579119909119879119909

= (120597119880

120597119879119909

)1

119879119909

=119897

119864119868119909

Shock and Vibration 5

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

12

14

FEATheoryError

4 6 8 10 12 14 16 18 20

Fy sensitivity with length l

l (mm)

120575y

(mm

)

(a) 120575119910-119897

FEATheoryError

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

120575y

(mm

)

t (mm)02 04 06 08 10

Fy sensitivity with thickness t

(b) 120575119910-119905

FEATheoryError

00

02

04

06

08

10

Erro

r (

)

2

4

6

8

10

12

14

120575y

(mm

)

b (mm)2 3 4 5 6 7 8

Fy sensitivity with width b

(c) 120575119910-119887

Figure 2 Variation of deflection 120575119910according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119910

+

(1198702 sinh120572119897 minus sinh120572119897 + 120572119897 cosh120572119897)

120572119866119869119909cosh120572119897

119862120579119910119865119911

= (120597119880

120597119872119910

)1

119865119911

= minus1198972

2119864119868119910

minus

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119910119872119910

= (120597119880

120597119872119910

)1

119872119910

=119897

119864119868119910

+

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119911119865119909

= (120597119880

120597119872119911

)1

119865119909

=1198972

2119864119868119911

119862120579119911119865119910

= (120597119880

120597119872119911

)1

119865119910

=31198973

2119864119868119911

119862120579119911119872119911

= (120597119880

120597119872119911

)1

119872119911

=2119897

119864119868119911

(19)

3 FEA Verification

In this study FEA was conducted by using Pro-Mechanicacommercial software (Wildfire 5 PTC Corp MA USA) toverify the results obtained by theory and to validate thereliability of the calculated results The default values of theSBLF were chosen as follows length 119897 = 10mm width 119887 =

4mm and thickness 119905 = 05mm The forces and moments

6 Shock and Vibration

Table 1 Comparison between theory and the FEA at the default values of the SBLF

Displacements FEA Theory Error ()Shear and Warp No shear and No warp S and W No S and NoW

120575119909(mm) 116210 116466 116183 022 minus002

120575119910(mm) 044183 046480 046451 494 488

120575119911(mm) 021102 020825 022556 131 645

120579119909(rad) 000239 000233 000251 219 504

120579119910(rad) 000210 000209 000227 047 743

120579119911(rad) 000673 000697 000697 343 343

Table 2 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under 119865119909 119865119910 and 119865

119911

119897 (mm) FEA Theory Error () t (mm) FEA Theory Error () b (mm) FEA Theory Error ()(a) Deflection 120575

119909(mm)

5 013819 014550 502 025 877255 929028 557 2 228306 232366 17565 030485 031934 454 0325 397715 422907 596 26 174683 178743 2278 057071 059507 409 04 217394 226869 418 32 141120 145229 28395 095908 099619 372 0475 130778 135502 349 38 118188 122298 33611 149324 154623 343 055 084862 087301 279 44 101548 105621 386125 219646 226869 318 0625 058257 059507 210 5 088936 092946 43214 309203 318708 298 07 041772 042366 140 56 079058 082988 474155 420323 432493 281 0775 031008 031227 070 62 071120 074957 51217 555339 570573 267 085 023676 023676 000 68 064607 068343 547185 716579 735301 255 0925 018508 018377 minus071 74 059170 062802 57820 906374 929028 244 1 014756 014550 minus142 8 054565 058091 607

(b) Deflection 120575119910(mm)

5 005369 005809 758 025 355785 371567 425 2 090181 092903 29365 011959 012759 627 0325 162113 169129 415 26 069073 071464 3358 022485 023785 547 04 086939 090720 417 32 055848 058064 38295 037879 039827 489 0475 051931 054178 415 38 046800 048896 42911 059062 061825 447 055 033455 034901 414 44 040226 042229 474125 086949 090720 416 0625 022798 023785 415 5 035238 037161 51714 122458 127453 392 07 016225 016931 417 56 031329 033180 558155 166503 172963 374 0775 011953 012477 420 62 028185 029969 59517 219995 228192 359 085 009057 009458 424 68 025604 027324 629185 283849 294080 348 0925 007025 007339 428 74 023449 025109 66120 358976 371567 339 1 005557 005809 433 8 021624 023226 690

(c) Deflection 120575119911(mm)

5 002203 002113 409 025 160255 158055 137 2 052645 051994 12465 005270 005127 272 0325 074212 073006 163 26 037023 036566 1248 010341 010146 189 04 040469 039777 171 32 028218 027863 12695 017912 017672 134 0475 024617 024151 190 38 022569 022270 13311 028479 028210 095 055 016164 015831 207 44 018640 018372 144125 042541 042263 065 0625 011239 010988 224 5 015750 015497 16114 060595 060338 043 07 008169 007972 242 56 013538 013290 184155 083137 082938 024 0775 006152 005992 261 62 011792 011542 21217 110665 110569 009 085 004770 004636 281 68 010379 010126 244185 143675 143737 minus004 0925 003788 003675 301 74 009214 008956 28120 182663 182945 minus015 1 003071 002973 322 8 008239 007975 321

Shock and Vibration 7

00

05

10

15

20

25

30

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

Fz sensitivity with length l

l (mm)4 6 8 10 12 14 16 18 20

minus2

120575z

(mm

)

(a) 120575119911-119897

FEATheoryError

00

05

10

15

20

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with thickness t

t (mm)02 04 06 08 10

120575z

(mm

)

(b) 120575119911-119905

FEATheoryError

00

01

02

03

04

05

06

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with width b

b (mm)2 3 4 5 6 7 8

120575z

(mm

)

(c) 120575119911-119887

Figure 3 Variation of deflection 120575119911according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119911

applied to the SBLF are 119865119909= 10N 119865

119910= 119865119911= 1N and 119879

119909=

119872119910= 119872119911= 1NmmThewarping restraint factor was chosen

to be 119870 = 05 The material selected in this simulation isaluminum Al 6061 In the parametric analysis the sensitivityanalysis was performed to determine the agreement betweenFEA and theory with the variation of length 119897 = 5 to 20mmwidth 119887 = 2 to 8mm and thickness 119905 = 025 to 1mmrespectively If the errors between the two methods are lowerthan 10 the results can usually be accepted and used fromthe viewpoint of the engineering design

Table 1 shows the comparison between the theory andFEA results at the default values of the SBLF Equations (15)ndash(16) were applied to find the translational and rotationaldisplacements by the theoretical method that considers thevariable shear deformation in the bending and the warpingin both the bending and torsion and the partially restrained

warping at the joint of elements 1 and 2 The derivedtheoretical equations showed a good agreement with the FEAresults all of the errors of these simulationswere less than 5whereas the results of without shear and warp effect showedhigher errors up to 74 as presented in Table 1 The resultsof in-plane deformations 120575

119909 120575119910 and 120579

119911in this study were

consistent with results of the equations in [24] Althoughthese results were showing the good agreement at the defaultvalues the sensitive parameters analysis was also conductedto ensure the reliability of the derived equations

In the research of the reference [28] we analyzed thedisplacement of a SBLF under transverse loading by theEuler-Bernoulli Timoshenko and third-order beam theoryand the results showed that the third-order beam theoryof Levinson which considers the shear and warp effect inbending gave the good results In this study the above beam

8 Shock and Vibration

Table 3 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under loads 119879119909119872119910 and119872

119911

119897 (mm) FEA Theory Error () 119905 (mm) FEA Theory Error () 119887 (mm) FEA Theory Error ()(a) Rotation 120579

119909(rad)

5 000114 000108 559 025 001795 001762 186 2 000568 000561 13465 000152 000145 404 0325 000833 000815 215 26 000404 000397 1668 000189 000183 307 04 000455 000445 228 32 000312 000306 19195 000226 000221 240 0475 000277 000270 250 38 000254 000248 21311 000263 000258 191 055 000183 000178 271 44 000213 000208 236125 000301 000296 153 0625 000127 000123 289 5 000183 000179 26114 000338 000334 123 07 000093 000090 303 56 000161 000156 288155 000375 000371 099 0775 000070 000068 316 62 000143 000138 31817 000412 000409 078 085 000054 000052 325 68 000128 000124 349185 000450 000447 060 0925 000043 000042 333 74 000116 000112 38320 000487 000485 045 1 000035 000034 339 8 000106 000102 419

(b) Rotation 120579119910(rad)

5 000089 000084 477 025 001605 001583 134 2 000540 000533 13365 000126 000122 308 0325 000744 000732 160 26 000376 000371 1358 000163 000159 211 04 000407 000399 177 32 000285 000281 13995 000200 000197 149 0475 000248 000243 201 38 000228 000224 14611 000237 000235 106 055 000163 000159 223 44 000188 000185 159125 000274 000272 074 0625 000114 000111 246 5 000158 000155 17714 000312 000310 049 07 000083 000081 269 56 000136 000133 200155 000349 000348 029 0775 000063 000061 292 62 000118 000116 22917 000386 000385 013 085 000049 000047 316 68 000104 000101 262185 000423 000423 minus001 0925 000039 000037 341 74 000092 000090 29920 000460 000461 minus012 1 000032 000030 365 8 000082 000080 339

(c) Rotation 120579119911(rad)

5 000328 000348 571 025 005405 005573 302 2 001365 001393 20065 000432 000453 470 0325 002462 002537 294 26 001047 001072 2318 000535 000557 403 04 001320 001361 295 32 000848 000871 26695 000638 000662 358 0475 000789 000813 294 38 000711 000733 30111 000741 000766 325 055 000508 000523 293 44 000612 000633 337125 000845 000871 302 0625 000346 000357 293 5 000537 000557 37214 000948 000975 284 07 000246 000254 293 56 000477 000498 406155 001051 001080 271 0775 000182 000187 294 62 000430 000449 43917 001153 001184 261 085 000138 000142 296 68 000391 000410 470185 001256 001289 253 0925 000107 000110 298 74 000358 000377 50120 001359 001393 247 1 000084 000087 301 8 000330 000348 531

theory was also used and the translational displacements 120575119909

120575119910 and 120575

119911under 119865

119909 119865119910 and 119865

119911according to the variations of

length 119897 thickness 119905 and width 119887 respectively are presentedin Table 2 Figure 2 shows typical simulation results underforce 119865

119910 The errors in these investigations are lower (76)

The sensitive analysis of 119865119911is also presented in Figure 3 All

the errors are lower (5) These results reveal the accuracyof the derived theoretical equations for the translationaldisplacements of SBLF

The comparison results among the fully restrained par-tially restrained and free warping in torsion analysis of SBLFwere investigated in [25] and the partially restrained warpingwith factor 119870 = 05 was applied in this torsion simulationTable 3 shows the variation of rotational displacements 120579

119909 120579119910

and 120579119911due to moments 119879

119909 119872119910 and 119872

119911with the variation

of parameters 119897 119905 and 119887 of the SBLF respectively It canbe observed that the results of theory and FEA are in goodagreement with the errors lower than 6 Figure 4 presentsthe typical variation of rotation 120579

119909according to 119897 119905 and 119887

under 119879119909 The maximum error is lower than 6

In summary the relationship between the deformationand the applied loads for the SBLF is presented in compliancematrix form The variable shear deformation and warpingeffect were considered and partial restraint was introducedin the analysis All results were verified by FEA with strongagreement between the two methods The errors were lowerthan 76 These results demonstrate the high accuracy andreliability of the proposed theoretical equations

Shock and Vibration 9

0000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

4 6 8 10 12 14 16 18 20

Tx sensitivity with length l

l (mm)

(a) 120579119909-119897

0000

0005

0010

0015

0020

0025

0030

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

t (mm)02 04 06 08 10

Tx sensitivity with thickness t

(b) 120579119909-119905

2 3 4 5 6 7 80000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

Tx sensitivity with width b

b (mm)

(c) 120579119909-119887

Figure 4 Variation of rotation 120579119909according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under load 119879

119909

4 Conclusion

In this study a compliance matrix that expresses the rela-tionship between the deformations and applied loads ofthe SBLF was analyzed and derived by using Castiglianorsquossecond theorem In bending analysis higher-order beamtheory was applied wherein variable shear deformation andthe warping were considered in the calculated formulas forthe shear forces At the joint of two elements of the SBLFthe partially restrained warping in torsion was analysed withconsideration of the warping restraint factor The theoreticalresults were verified by FEA at both the default and thesensitive values The results indicate that there is strongagreement between the two methods with errors below

76 This suggests the accuracy of the proposed theoreticalequations and that they can be used in the precision machinedesign

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Nghia-Huu Nguyen and Moo-Yeon Lee equally contributedto this work

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

12

14

FEATheoryError

4 6 8 10 12 14 16 18 20

Fy sensitivity with length l

l (mm)

120575y

(mm

)

(a) 120575119910-119897

FEATheoryError

0

1

2

3

4

Erro

r (

)

2

4

6

8

10

120575y

(mm

)

t (mm)02 04 06 08 10

Fy sensitivity with thickness t

(b) 120575119910-119905

FEATheoryError

00

02

04

06

08

10

Erro

r (

)

2

4

6

8

10

12

14

120575y

(mm

)

b (mm)2 3 4 5 6 7 8

Fy sensitivity with width b

(c) 120575119910-119887

Figure 2 Variation of deflection 120575119910according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119910

+

(1198702 sinh120572119897 minus sinh120572119897 + 120572119897 cosh120572119897)

120572119866119869119909cosh120572119897

119862120579119910119865119911

= (120597119880

120597119872119910

)1

119865119911

= minus1198972

2119864119868119910

minus

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119910119872119910

= (120597119880

120597119872119910

)1

119872119910

=119897

119864119868119910

+

119897 (2120572119897 minus 2 sinh120572119897 + (2 cosh120572119897 minus cosh 2120572119897 minus 1)119870119905+ 120572119897119870

2

119905)

120572119866119869119910

119862120579119911119865119909

= (120597119880

120597119872119911

)1

119865119909

=1198972

2119864119868119911

119862120579119911119865119910

= (120597119880

120597119872119911

)1

119865119910

=31198973

2119864119868119911

119862120579119911119872119911

= (120597119880

120597119872119911

)1

119872119911

=2119897

119864119868119911

(19)

3 FEA Verification

In this study FEA was conducted by using Pro-Mechanicacommercial software (Wildfire 5 PTC Corp MA USA) toverify the results obtained by theory and to validate thereliability of the calculated results The default values of theSBLF were chosen as follows length 119897 = 10mm width 119887 =

4mm and thickness 119905 = 05mm The forces and moments

6 Shock and Vibration

Table 1 Comparison between theory and the FEA at the default values of the SBLF

Displacements FEA Theory Error ()Shear and Warp No shear and No warp S and W No S and NoW

120575119909(mm) 116210 116466 116183 022 minus002

120575119910(mm) 044183 046480 046451 494 488

120575119911(mm) 021102 020825 022556 131 645

120579119909(rad) 000239 000233 000251 219 504

120579119910(rad) 000210 000209 000227 047 743

120579119911(rad) 000673 000697 000697 343 343

Table 2 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under 119865119909 119865119910 and 119865

119911

119897 (mm) FEA Theory Error () t (mm) FEA Theory Error () b (mm) FEA Theory Error ()(a) Deflection 120575

119909(mm)

5 013819 014550 502 025 877255 929028 557 2 228306 232366 17565 030485 031934 454 0325 397715 422907 596 26 174683 178743 2278 057071 059507 409 04 217394 226869 418 32 141120 145229 28395 095908 099619 372 0475 130778 135502 349 38 118188 122298 33611 149324 154623 343 055 084862 087301 279 44 101548 105621 386125 219646 226869 318 0625 058257 059507 210 5 088936 092946 43214 309203 318708 298 07 041772 042366 140 56 079058 082988 474155 420323 432493 281 0775 031008 031227 070 62 071120 074957 51217 555339 570573 267 085 023676 023676 000 68 064607 068343 547185 716579 735301 255 0925 018508 018377 minus071 74 059170 062802 57820 906374 929028 244 1 014756 014550 minus142 8 054565 058091 607

(b) Deflection 120575119910(mm)

5 005369 005809 758 025 355785 371567 425 2 090181 092903 29365 011959 012759 627 0325 162113 169129 415 26 069073 071464 3358 022485 023785 547 04 086939 090720 417 32 055848 058064 38295 037879 039827 489 0475 051931 054178 415 38 046800 048896 42911 059062 061825 447 055 033455 034901 414 44 040226 042229 474125 086949 090720 416 0625 022798 023785 415 5 035238 037161 51714 122458 127453 392 07 016225 016931 417 56 031329 033180 558155 166503 172963 374 0775 011953 012477 420 62 028185 029969 59517 219995 228192 359 085 009057 009458 424 68 025604 027324 629185 283849 294080 348 0925 007025 007339 428 74 023449 025109 66120 358976 371567 339 1 005557 005809 433 8 021624 023226 690

(c) Deflection 120575119911(mm)

5 002203 002113 409 025 160255 158055 137 2 052645 051994 12465 005270 005127 272 0325 074212 073006 163 26 037023 036566 1248 010341 010146 189 04 040469 039777 171 32 028218 027863 12695 017912 017672 134 0475 024617 024151 190 38 022569 022270 13311 028479 028210 095 055 016164 015831 207 44 018640 018372 144125 042541 042263 065 0625 011239 010988 224 5 015750 015497 16114 060595 060338 043 07 008169 007972 242 56 013538 013290 184155 083137 082938 024 0775 006152 005992 261 62 011792 011542 21217 110665 110569 009 085 004770 004636 281 68 010379 010126 244185 143675 143737 minus004 0925 003788 003675 301 74 009214 008956 28120 182663 182945 minus015 1 003071 002973 322 8 008239 007975 321

Shock and Vibration 7

00

05

10

15

20

25

30

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

Fz sensitivity with length l

l (mm)4 6 8 10 12 14 16 18 20

minus2

120575z

(mm

)

(a) 120575119911-119897

FEATheoryError

00

05

10

15

20

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with thickness t

t (mm)02 04 06 08 10

120575z

(mm

)

(b) 120575119911-119905

FEATheoryError

00

01

02

03

04

05

06

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with width b

b (mm)2 3 4 5 6 7 8

120575z

(mm

)

(c) 120575119911-119887

Figure 3 Variation of deflection 120575119911according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119911

applied to the SBLF are 119865119909= 10N 119865

119910= 119865119911= 1N and 119879

119909=

119872119910= 119872119911= 1NmmThewarping restraint factor was chosen

to be 119870 = 05 The material selected in this simulation isaluminum Al 6061 In the parametric analysis the sensitivityanalysis was performed to determine the agreement betweenFEA and theory with the variation of length 119897 = 5 to 20mmwidth 119887 = 2 to 8mm and thickness 119905 = 025 to 1mmrespectively If the errors between the two methods are lowerthan 10 the results can usually be accepted and used fromthe viewpoint of the engineering design

Table 1 shows the comparison between the theory andFEA results at the default values of the SBLF Equations (15)ndash(16) were applied to find the translational and rotationaldisplacements by the theoretical method that considers thevariable shear deformation in the bending and the warpingin both the bending and torsion and the partially restrained

warping at the joint of elements 1 and 2 The derivedtheoretical equations showed a good agreement with the FEAresults all of the errors of these simulationswere less than 5whereas the results of without shear and warp effect showedhigher errors up to 74 as presented in Table 1 The resultsof in-plane deformations 120575

119909 120575119910 and 120579

119911in this study were

consistent with results of the equations in [24] Althoughthese results were showing the good agreement at the defaultvalues the sensitive parameters analysis was also conductedto ensure the reliability of the derived equations

In the research of the reference [28] we analyzed thedisplacement of a SBLF under transverse loading by theEuler-Bernoulli Timoshenko and third-order beam theoryand the results showed that the third-order beam theoryof Levinson which considers the shear and warp effect inbending gave the good results In this study the above beam

8 Shock and Vibration

Table 3 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under loads 119879119909119872119910 and119872

119911

119897 (mm) FEA Theory Error () 119905 (mm) FEA Theory Error () 119887 (mm) FEA Theory Error ()(a) Rotation 120579

119909(rad)

5 000114 000108 559 025 001795 001762 186 2 000568 000561 13465 000152 000145 404 0325 000833 000815 215 26 000404 000397 1668 000189 000183 307 04 000455 000445 228 32 000312 000306 19195 000226 000221 240 0475 000277 000270 250 38 000254 000248 21311 000263 000258 191 055 000183 000178 271 44 000213 000208 236125 000301 000296 153 0625 000127 000123 289 5 000183 000179 26114 000338 000334 123 07 000093 000090 303 56 000161 000156 288155 000375 000371 099 0775 000070 000068 316 62 000143 000138 31817 000412 000409 078 085 000054 000052 325 68 000128 000124 349185 000450 000447 060 0925 000043 000042 333 74 000116 000112 38320 000487 000485 045 1 000035 000034 339 8 000106 000102 419

(b) Rotation 120579119910(rad)

5 000089 000084 477 025 001605 001583 134 2 000540 000533 13365 000126 000122 308 0325 000744 000732 160 26 000376 000371 1358 000163 000159 211 04 000407 000399 177 32 000285 000281 13995 000200 000197 149 0475 000248 000243 201 38 000228 000224 14611 000237 000235 106 055 000163 000159 223 44 000188 000185 159125 000274 000272 074 0625 000114 000111 246 5 000158 000155 17714 000312 000310 049 07 000083 000081 269 56 000136 000133 200155 000349 000348 029 0775 000063 000061 292 62 000118 000116 22917 000386 000385 013 085 000049 000047 316 68 000104 000101 262185 000423 000423 minus001 0925 000039 000037 341 74 000092 000090 29920 000460 000461 minus012 1 000032 000030 365 8 000082 000080 339

(c) Rotation 120579119911(rad)

5 000328 000348 571 025 005405 005573 302 2 001365 001393 20065 000432 000453 470 0325 002462 002537 294 26 001047 001072 2318 000535 000557 403 04 001320 001361 295 32 000848 000871 26695 000638 000662 358 0475 000789 000813 294 38 000711 000733 30111 000741 000766 325 055 000508 000523 293 44 000612 000633 337125 000845 000871 302 0625 000346 000357 293 5 000537 000557 37214 000948 000975 284 07 000246 000254 293 56 000477 000498 406155 001051 001080 271 0775 000182 000187 294 62 000430 000449 43917 001153 001184 261 085 000138 000142 296 68 000391 000410 470185 001256 001289 253 0925 000107 000110 298 74 000358 000377 50120 001359 001393 247 1 000084 000087 301 8 000330 000348 531

theory was also used and the translational displacements 120575119909

120575119910 and 120575

119911under 119865

119909 119865119910 and 119865

119911according to the variations of

length 119897 thickness 119905 and width 119887 respectively are presentedin Table 2 Figure 2 shows typical simulation results underforce 119865

119910 The errors in these investigations are lower (76)

The sensitive analysis of 119865119911is also presented in Figure 3 All

the errors are lower (5) These results reveal the accuracyof the derived theoretical equations for the translationaldisplacements of SBLF

The comparison results among the fully restrained par-tially restrained and free warping in torsion analysis of SBLFwere investigated in [25] and the partially restrained warpingwith factor 119870 = 05 was applied in this torsion simulationTable 3 shows the variation of rotational displacements 120579

119909 120579119910

and 120579119911due to moments 119879

119909 119872119910 and 119872

119911with the variation

of parameters 119897 119905 and 119887 of the SBLF respectively It canbe observed that the results of theory and FEA are in goodagreement with the errors lower than 6 Figure 4 presentsthe typical variation of rotation 120579

119909according to 119897 119905 and 119887

under 119879119909 The maximum error is lower than 6

In summary the relationship between the deformationand the applied loads for the SBLF is presented in compliancematrix form The variable shear deformation and warpingeffect were considered and partial restraint was introducedin the analysis All results were verified by FEA with strongagreement between the two methods The errors were lowerthan 76 These results demonstrate the high accuracy andreliability of the proposed theoretical equations

Shock and Vibration 9

0000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

4 6 8 10 12 14 16 18 20

Tx sensitivity with length l

l (mm)

(a) 120579119909-119897

0000

0005

0010

0015

0020

0025

0030

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

t (mm)02 04 06 08 10

Tx sensitivity with thickness t

(b) 120579119909-119905

2 3 4 5 6 7 80000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

Tx sensitivity with width b

b (mm)

(c) 120579119909-119887

Figure 4 Variation of rotation 120579119909according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under load 119879

119909

4 Conclusion

In this study a compliance matrix that expresses the rela-tionship between the deformations and applied loads ofthe SBLF was analyzed and derived by using Castiglianorsquossecond theorem In bending analysis higher-order beamtheory was applied wherein variable shear deformation andthe warping were considered in the calculated formulas forthe shear forces At the joint of two elements of the SBLFthe partially restrained warping in torsion was analysed withconsideration of the warping restraint factor The theoreticalresults were verified by FEA at both the default and thesensitive values The results indicate that there is strongagreement between the two methods with errors below

76 This suggests the accuracy of the proposed theoreticalequations and that they can be used in the precision machinedesign

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Nghia-Huu Nguyen and Moo-Yeon Lee equally contributedto this work

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

Table 1 Comparison between theory and the FEA at the default values of the SBLF

Displacements FEA Theory Error ()Shear and Warp No shear and No warp S and W No S and NoW

120575119909(mm) 116210 116466 116183 022 minus002

120575119910(mm) 044183 046480 046451 494 488

120575119911(mm) 021102 020825 022556 131 645

120579119909(rad) 000239 000233 000251 219 504

120579119910(rad) 000210 000209 000227 047 743

120579119911(rad) 000673 000697 000697 343 343

Table 2 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under 119865119909 119865119910 and 119865

119911

119897 (mm) FEA Theory Error () t (mm) FEA Theory Error () b (mm) FEA Theory Error ()(a) Deflection 120575

119909(mm)

5 013819 014550 502 025 877255 929028 557 2 228306 232366 17565 030485 031934 454 0325 397715 422907 596 26 174683 178743 2278 057071 059507 409 04 217394 226869 418 32 141120 145229 28395 095908 099619 372 0475 130778 135502 349 38 118188 122298 33611 149324 154623 343 055 084862 087301 279 44 101548 105621 386125 219646 226869 318 0625 058257 059507 210 5 088936 092946 43214 309203 318708 298 07 041772 042366 140 56 079058 082988 474155 420323 432493 281 0775 031008 031227 070 62 071120 074957 51217 555339 570573 267 085 023676 023676 000 68 064607 068343 547185 716579 735301 255 0925 018508 018377 minus071 74 059170 062802 57820 906374 929028 244 1 014756 014550 minus142 8 054565 058091 607

(b) Deflection 120575119910(mm)

5 005369 005809 758 025 355785 371567 425 2 090181 092903 29365 011959 012759 627 0325 162113 169129 415 26 069073 071464 3358 022485 023785 547 04 086939 090720 417 32 055848 058064 38295 037879 039827 489 0475 051931 054178 415 38 046800 048896 42911 059062 061825 447 055 033455 034901 414 44 040226 042229 474125 086949 090720 416 0625 022798 023785 415 5 035238 037161 51714 122458 127453 392 07 016225 016931 417 56 031329 033180 558155 166503 172963 374 0775 011953 012477 420 62 028185 029969 59517 219995 228192 359 085 009057 009458 424 68 025604 027324 629185 283849 294080 348 0925 007025 007339 428 74 023449 025109 66120 358976 371567 339 1 005557 005809 433 8 021624 023226 690

(c) Deflection 120575119911(mm)

5 002203 002113 409 025 160255 158055 137 2 052645 051994 12465 005270 005127 272 0325 074212 073006 163 26 037023 036566 1248 010341 010146 189 04 040469 039777 171 32 028218 027863 12695 017912 017672 134 0475 024617 024151 190 38 022569 022270 13311 028479 028210 095 055 016164 015831 207 44 018640 018372 144125 042541 042263 065 0625 011239 010988 224 5 015750 015497 16114 060595 060338 043 07 008169 007972 242 56 013538 013290 184155 083137 082938 024 0775 006152 005992 261 62 011792 011542 21217 110665 110569 009 085 004770 004636 281 68 010379 010126 244185 143675 143737 minus004 0925 003788 003675 301 74 009214 008956 28120 182663 182945 minus015 1 003071 002973 322 8 008239 007975 321

Shock and Vibration 7

00

05

10

15

20

25

30

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

Fz sensitivity with length l

l (mm)4 6 8 10 12 14 16 18 20

minus2

120575z

(mm

)

(a) 120575119911-119897

FEATheoryError

00

05

10

15

20

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with thickness t

t (mm)02 04 06 08 10

120575z

(mm

)

(b) 120575119911-119905

FEATheoryError

00

01

02

03

04

05

06

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with width b

b (mm)2 3 4 5 6 7 8

120575z

(mm

)

(c) 120575119911-119887

Figure 3 Variation of deflection 120575119911according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119911

applied to the SBLF are 119865119909= 10N 119865

119910= 119865119911= 1N and 119879

119909=

119872119910= 119872119911= 1NmmThewarping restraint factor was chosen

to be 119870 = 05 The material selected in this simulation isaluminum Al 6061 In the parametric analysis the sensitivityanalysis was performed to determine the agreement betweenFEA and theory with the variation of length 119897 = 5 to 20mmwidth 119887 = 2 to 8mm and thickness 119905 = 025 to 1mmrespectively If the errors between the two methods are lowerthan 10 the results can usually be accepted and used fromthe viewpoint of the engineering design

Table 1 shows the comparison between the theory andFEA results at the default values of the SBLF Equations (15)ndash(16) were applied to find the translational and rotationaldisplacements by the theoretical method that considers thevariable shear deformation in the bending and the warpingin both the bending and torsion and the partially restrained

warping at the joint of elements 1 and 2 The derivedtheoretical equations showed a good agreement with the FEAresults all of the errors of these simulationswere less than 5whereas the results of without shear and warp effect showedhigher errors up to 74 as presented in Table 1 The resultsof in-plane deformations 120575

119909 120575119910 and 120579

119911in this study were

consistent with results of the equations in [24] Althoughthese results were showing the good agreement at the defaultvalues the sensitive parameters analysis was also conductedto ensure the reliability of the derived equations

In the research of the reference [28] we analyzed thedisplacement of a SBLF under transverse loading by theEuler-Bernoulli Timoshenko and third-order beam theoryand the results showed that the third-order beam theoryof Levinson which considers the shear and warp effect inbending gave the good results In this study the above beam

8 Shock and Vibration

Table 3 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under loads 119879119909119872119910 and119872

119911

119897 (mm) FEA Theory Error () 119905 (mm) FEA Theory Error () 119887 (mm) FEA Theory Error ()(a) Rotation 120579

119909(rad)

5 000114 000108 559 025 001795 001762 186 2 000568 000561 13465 000152 000145 404 0325 000833 000815 215 26 000404 000397 1668 000189 000183 307 04 000455 000445 228 32 000312 000306 19195 000226 000221 240 0475 000277 000270 250 38 000254 000248 21311 000263 000258 191 055 000183 000178 271 44 000213 000208 236125 000301 000296 153 0625 000127 000123 289 5 000183 000179 26114 000338 000334 123 07 000093 000090 303 56 000161 000156 288155 000375 000371 099 0775 000070 000068 316 62 000143 000138 31817 000412 000409 078 085 000054 000052 325 68 000128 000124 349185 000450 000447 060 0925 000043 000042 333 74 000116 000112 38320 000487 000485 045 1 000035 000034 339 8 000106 000102 419

(b) Rotation 120579119910(rad)

5 000089 000084 477 025 001605 001583 134 2 000540 000533 13365 000126 000122 308 0325 000744 000732 160 26 000376 000371 1358 000163 000159 211 04 000407 000399 177 32 000285 000281 13995 000200 000197 149 0475 000248 000243 201 38 000228 000224 14611 000237 000235 106 055 000163 000159 223 44 000188 000185 159125 000274 000272 074 0625 000114 000111 246 5 000158 000155 17714 000312 000310 049 07 000083 000081 269 56 000136 000133 200155 000349 000348 029 0775 000063 000061 292 62 000118 000116 22917 000386 000385 013 085 000049 000047 316 68 000104 000101 262185 000423 000423 minus001 0925 000039 000037 341 74 000092 000090 29920 000460 000461 minus012 1 000032 000030 365 8 000082 000080 339

(c) Rotation 120579119911(rad)

5 000328 000348 571 025 005405 005573 302 2 001365 001393 20065 000432 000453 470 0325 002462 002537 294 26 001047 001072 2318 000535 000557 403 04 001320 001361 295 32 000848 000871 26695 000638 000662 358 0475 000789 000813 294 38 000711 000733 30111 000741 000766 325 055 000508 000523 293 44 000612 000633 337125 000845 000871 302 0625 000346 000357 293 5 000537 000557 37214 000948 000975 284 07 000246 000254 293 56 000477 000498 406155 001051 001080 271 0775 000182 000187 294 62 000430 000449 43917 001153 001184 261 085 000138 000142 296 68 000391 000410 470185 001256 001289 253 0925 000107 000110 298 74 000358 000377 50120 001359 001393 247 1 000084 000087 301 8 000330 000348 531

theory was also used and the translational displacements 120575119909

120575119910 and 120575

119911under 119865

119909 119865119910 and 119865

119911according to the variations of

length 119897 thickness 119905 and width 119887 respectively are presentedin Table 2 Figure 2 shows typical simulation results underforce 119865

119910 The errors in these investigations are lower (76)

The sensitive analysis of 119865119911is also presented in Figure 3 All

the errors are lower (5) These results reveal the accuracyof the derived theoretical equations for the translationaldisplacements of SBLF

The comparison results among the fully restrained par-tially restrained and free warping in torsion analysis of SBLFwere investigated in [25] and the partially restrained warpingwith factor 119870 = 05 was applied in this torsion simulationTable 3 shows the variation of rotational displacements 120579

119909 120579119910

and 120579119911due to moments 119879

119909 119872119910 and 119872

119911with the variation

of parameters 119897 119905 and 119887 of the SBLF respectively It canbe observed that the results of theory and FEA are in goodagreement with the errors lower than 6 Figure 4 presentsthe typical variation of rotation 120579

119909according to 119897 119905 and 119887

under 119879119909 The maximum error is lower than 6

In summary the relationship between the deformationand the applied loads for the SBLF is presented in compliancematrix form The variable shear deformation and warpingeffect were considered and partial restraint was introducedin the analysis All results were verified by FEA with strongagreement between the two methods The errors were lowerthan 76 These results demonstrate the high accuracy andreliability of the proposed theoretical equations

Shock and Vibration 9

0000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

4 6 8 10 12 14 16 18 20

Tx sensitivity with length l

l (mm)

(a) 120579119909-119897

0000

0005

0010

0015

0020

0025

0030

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

t (mm)02 04 06 08 10

Tx sensitivity with thickness t

(b) 120579119909-119905

2 3 4 5 6 7 80000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

Tx sensitivity with width b

b (mm)

(c) 120579119909-119887

Figure 4 Variation of rotation 120579119909according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under load 119879

119909

4 Conclusion

In this study a compliance matrix that expresses the rela-tionship between the deformations and applied loads ofthe SBLF was analyzed and derived by using Castiglianorsquossecond theorem In bending analysis higher-order beamtheory was applied wherein variable shear deformation andthe warping were considered in the calculated formulas forthe shear forces At the joint of two elements of the SBLFthe partially restrained warping in torsion was analysed withconsideration of the warping restraint factor The theoreticalresults were verified by FEA at both the default and thesensitive values The results indicate that there is strongagreement between the two methods with errors below

76 This suggests the accuracy of the proposed theoreticalequations and that they can be used in the precision machinedesign

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Nghia-Huu Nguyen and Moo-Yeon Lee equally contributedto this work

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

00

05

10

15

20

25

30

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

Fz sensitivity with length l

l (mm)4 6 8 10 12 14 16 18 20

minus2

120575z

(mm

)

(a) 120575119911-119897

FEATheoryError

00

05

10

15

20

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with thickness t

t (mm)02 04 06 08 10

120575z

(mm

)

(b) 120575119911-119905

FEATheoryError

00

01

02

03

04

05

06

Erro

r (

)

0

2

4

6

8

10Fz sensitivity with width b

b (mm)2 3 4 5 6 7 8

120575z

(mm

)

(c) 120575119911-119887

Figure 3 Variation of deflection 120575119911according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under 119865

119911

applied to the SBLF are 119865119909= 10N 119865

119910= 119865119911= 1N and 119879

119909=

119872119910= 119872119911= 1NmmThewarping restraint factor was chosen

to be 119870 = 05 The material selected in this simulation isaluminum Al 6061 In the parametric analysis the sensitivityanalysis was performed to determine the agreement betweenFEA and theory with the variation of length 119897 = 5 to 20mmwidth 119887 = 2 to 8mm and thickness 119905 = 025 to 1mmrespectively If the errors between the two methods are lowerthan 10 the results can usually be accepted and used fromthe viewpoint of the engineering design

Table 1 shows the comparison between the theory andFEA results at the default values of the SBLF Equations (15)ndash(16) were applied to find the translational and rotationaldisplacements by the theoretical method that considers thevariable shear deformation in the bending and the warpingin both the bending and torsion and the partially restrained

warping at the joint of elements 1 and 2 The derivedtheoretical equations showed a good agreement with the FEAresults all of the errors of these simulationswere less than 5whereas the results of without shear and warp effect showedhigher errors up to 74 as presented in Table 1 The resultsof in-plane deformations 120575

119909 120575119910 and 120579

119911in this study were

consistent with results of the equations in [24] Althoughthese results were showing the good agreement at the defaultvalues the sensitive parameters analysis was also conductedto ensure the reliability of the derived equations

In the research of the reference [28] we analyzed thedisplacement of a SBLF under transverse loading by theEuler-Bernoulli Timoshenko and third-order beam theoryand the results showed that the third-order beam theoryof Levinson which considers the shear and warp effect inbending gave the good results In this study the above beam

8 Shock and Vibration

Table 3 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under loads 119879119909119872119910 and119872

119911

119897 (mm) FEA Theory Error () 119905 (mm) FEA Theory Error () 119887 (mm) FEA Theory Error ()(a) Rotation 120579

119909(rad)

5 000114 000108 559 025 001795 001762 186 2 000568 000561 13465 000152 000145 404 0325 000833 000815 215 26 000404 000397 1668 000189 000183 307 04 000455 000445 228 32 000312 000306 19195 000226 000221 240 0475 000277 000270 250 38 000254 000248 21311 000263 000258 191 055 000183 000178 271 44 000213 000208 236125 000301 000296 153 0625 000127 000123 289 5 000183 000179 26114 000338 000334 123 07 000093 000090 303 56 000161 000156 288155 000375 000371 099 0775 000070 000068 316 62 000143 000138 31817 000412 000409 078 085 000054 000052 325 68 000128 000124 349185 000450 000447 060 0925 000043 000042 333 74 000116 000112 38320 000487 000485 045 1 000035 000034 339 8 000106 000102 419

(b) Rotation 120579119910(rad)

5 000089 000084 477 025 001605 001583 134 2 000540 000533 13365 000126 000122 308 0325 000744 000732 160 26 000376 000371 1358 000163 000159 211 04 000407 000399 177 32 000285 000281 13995 000200 000197 149 0475 000248 000243 201 38 000228 000224 14611 000237 000235 106 055 000163 000159 223 44 000188 000185 159125 000274 000272 074 0625 000114 000111 246 5 000158 000155 17714 000312 000310 049 07 000083 000081 269 56 000136 000133 200155 000349 000348 029 0775 000063 000061 292 62 000118 000116 22917 000386 000385 013 085 000049 000047 316 68 000104 000101 262185 000423 000423 minus001 0925 000039 000037 341 74 000092 000090 29920 000460 000461 minus012 1 000032 000030 365 8 000082 000080 339

(c) Rotation 120579119911(rad)

5 000328 000348 571 025 005405 005573 302 2 001365 001393 20065 000432 000453 470 0325 002462 002537 294 26 001047 001072 2318 000535 000557 403 04 001320 001361 295 32 000848 000871 26695 000638 000662 358 0475 000789 000813 294 38 000711 000733 30111 000741 000766 325 055 000508 000523 293 44 000612 000633 337125 000845 000871 302 0625 000346 000357 293 5 000537 000557 37214 000948 000975 284 07 000246 000254 293 56 000477 000498 406155 001051 001080 271 0775 000182 000187 294 62 000430 000449 43917 001153 001184 261 085 000138 000142 296 68 000391 000410 470185 001256 001289 253 0925 000107 000110 298 74 000358 000377 50120 001359 001393 247 1 000084 000087 301 8 000330 000348 531

theory was also used and the translational displacements 120575119909

120575119910 and 120575

119911under 119865

119909 119865119910 and 119865

119911according to the variations of

length 119897 thickness 119905 and width 119887 respectively are presentedin Table 2 Figure 2 shows typical simulation results underforce 119865

119910 The errors in these investigations are lower (76)

The sensitive analysis of 119865119911is also presented in Figure 3 All

the errors are lower (5) These results reveal the accuracyof the derived theoretical equations for the translationaldisplacements of SBLF

The comparison results among the fully restrained par-tially restrained and free warping in torsion analysis of SBLFwere investigated in [25] and the partially restrained warpingwith factor 119870 = 05 was applied in this torsion simulationTable 3 shows the variation of rotational displacements 120579

119909 120579119910

and 120579119911due to moments 119879

119909 119872119910 and 119872

119911with the variation

of parameters 119897 119905 and 119887 of the SBLF respectively It canbe observed that the results of theory and FEA are in goodagreement with the errors lower than 6 Figure 4 presentsthe typical variation of rotation 120579

119909according to 119897 119905 and 119887

under 119879119909 The maximum error is lower than 6

In summary the relationship between the deformationand the applied loads for the SBLF is presented in compliancematrix form The variable shear deformation and warpingeffect were considered and partial restraint was introducedin the analysis All results were verified by FEA with strongagreement between the two methods The errors were lowerthan 76 These results demonstrate the high accuracy andreliability of the proposed theoretical equations

Shock and Vibration 9

0000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

4 6 8 10 12 14 16 18 20

Tx sensitivity with length l

l (mm)

(a) 120579119909-119897

0000

0005

0010

0015

0020

0025

0030

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

t (mm)02 04 06 08 10

Tx sensitivity with thickness t

(b) 120579119909-119905

2 3 4 5 6 7 80000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

Tx sensitivity with width b

b (mm)

(c) 120579119909-119887

Figure 4 Variation of rotation 120579119909according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under load 119879

119909

4 Conclusion

In this study a compliance matrix that expresses the rela-tionship between the deformations and applied loads ofthe SBLF was analyzed and derived by using Castiglianorsquossecond theorem In bending analysis higher-order beamtheory was applied wherein variable shear deformation andthe warping were considered in the calculated formulas forthe shear forces At the joint of two elements of the SBLFthe partially restrained warping in torsion was analysed withconsideration of the warping restraint factor The theoreticalresults were verified by FEA at both the default and thesensitive values The results indicate that there is strongagreement between the two methods with errors below

76 This suggests the accuracy of the proposed theoreticalequations and that they can be used in the precision machinedesign

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Nghia-Huu Nguyen and Moo-Yeon Lee equally contributedto this work

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

Table 3 Theoretical and FEA results with various lengths 119897 thicknesses 119905 and widths 119887 under loads 119879119909119872119910 and119872

119911

119897 (mm) FEA Theory Error () 119905 (mm) FEA Theory Error () 119887 (mm) FEA Theory Error ()(a) Rotation 120579

119909(rad)

5 000114 000108 559 025 001795 001762 186 2 000568 000561 13465 000152 000145 404 0325 000833 000815 215 26 000404 000397 1668 000189 000183 307 04 000455 000445 228 32 000312 000306 19195 000226 000221 240 0475 000277 000270 250 38 000254 000248 21311 000263 000258 191 055 000183 000178 271 44 000213 000208 236125 000301 000296 153 0625 000127 000123 289 5 000183 000179 26114 000338 000334 123 07 000093 000090 303 56 000161 000156 288155 000375 000371 099 0775 000070 000068 316 62 000143 000138 31817 000412 000409 078 085 000054 000052 325 68 000128 000124 349185 000450 000447 060 0925 000043 000042 333 74 000116 000112 38320 000487 000485 045 1 000035 000034 339 8 000106 000102 419

(b) Rotation 120579119910(rad)

5 000089 000084 477 025 001605 001583 134 2 000540 000533 13365 000126 000122 308 0325 000744 000732 160 26 000376 000371 1358 000163 000159 211 04 000407 000399 177 32 000285 000281 13995 000200 000197 149 0475 000248 000243 201 38 000228 000224 14611 000237 000235 106 055 000163 000159 223 44 000188 000185 159125 000274 000272 074 0625 000114 000111 246 5 000158 000155 17714 000312 000310 049 07 000083 000081 269 56 000136 000133 200155 000349 000348 029 0775 000063 000061 292 62 000118 000116 22917 000386 000385 013 085 000049 000047 316 68 000104 000101 262185 000423 000423 minus001 0925 000039 000037 341 74 000092 000090 29920 000460 000461 minus012 1 000032 000030 365 8 000082 000080 339

(c) Rotation 120579119911(rad)

5 000328 000348 571 025 005405 005573 302 2 001365 001393 20065 000432 000453 470 0325 002462 002537 294 26 001047 001072 2318 000535 000557 403 04 001320 001361 295 32 000848 000871 26695 000638 000662 358 0475 000789 000813 294 38 000711 000733 30111 000741 000766 325 055 000508 000523 293 44 000612 000633 337125 000845 000871 302 0625 000346 000357 293 5 000537 000557 37214 000948 000975 284 07 000246 000254 293 56 000477 000498 406155 001051 001080 271 0775 000182 000187 294 62 000430 000449 43917 001153 001184 261 085 000138 000142 296 68 000391 000410 470185 001256 001289 253 0925 000107 000110 298 74 000358 000377 50120 001359 001393 247 1 000084 000087 301 8 000330 000348 531

theory was also used and the translational displacements 120575119909

120575119910 and 120575

119911under 119865

119909 119865119910 and 119865

119911according to the variations of

length 119897 thickness 119905 and width 119887 respectively are presentedin Table 2 Figure 2 shows typical simulation results underforce 119865

119910 The errors in these investigations are lower (76)

The sensitive analysis of 119865119911is also presented in Figure 3 All

the errors are lower (5) These results reveal the accuracyof the derived theoretical equations for the translationaldisplacements of SBLF

The comparison results among the fully restrained par-tially restrained and free warping in torsion analysis of SBLFwere investigated in [25] and the partially restrained warpingwith factor 119870 = 05 was applied in this torsion simulationTable 3 shows the variation of rotational displacements 120579

119909 120579119910

and 120579119911due to moments 119879

119909 119872119910 and 119872

119911with the variation

of parameters 119897 119905 and 119887 of the SBLF respectively It canbe observed that the results of theory and FEA are in goodagreement with the errors lower than 6 Figure 4 presentsthe typical variation of rotation 120579

119909according to 119897 119905 and 119887

under 119879119909 The maximum error is lower than 6

In summary the relationship between the deformationand the applied loads for the SBLF is presented in compliancematrix form The variable shear deformation and warpingeffect were considered and partial restraint was introducedin the analysis All results were verified by FEA with strongagreement between the two methods The errors were lowerthan 76 These results demonstrate the high accuracy andreliability of the proposed theoretical equations

Shock and Vibration 9

0000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

4 6 8 10 12 14 16 18 20

Tx sensitivity with length l

l (mm)

(a) 120579119909-119897

0000

0005

0010

0015

0020

0025

0030

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

t (mm)02 04 06 08 10

Tx sensitivity with thickness t

(b) 120579119909-119905

2 3 4 5 6 7 80000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

Tx sensitivity with width b

b (mm)

(c) 120579119909-119887

Figure 4 Variation of rotation 120579119909according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under load 119879

119909

4 Conclusion

In this study a compliance matrix that expresses the rela-tionship between the deformations and applied loads ofthe SBLF was analyzed and derived by using Castiglianorsquossecond theorem In bending analysis higher-order beamtheory was applied wherein variable shear deformation andthe warping were considered in the calculated formulas forthe shear forces At the joint of two elements of the SBLFthe partially restrained warping in torsion was analysed withconsideration of the warping restraint factor The theoreticalresults were verified by FEA at both the default and thesensitive values The results indicate that there is strongagreement between the two methods with errors below

76 This suggests the accuracy of the proposed theoreticalequations and that they can be used in the precision machinedesign

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Nghia-Huu Nguyen and Moo-Yeon Lee equally contributedto this work

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

0000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

4 6 8 10 12 14 16 18 20

Tx sensitivity with length l

l (mm)

(a) 120579119909-119897

0000

0005

0010

0015

0020

0025

0030

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

t (mm)02 04 06 08 10

Tx sensitivity with thickness t

(b) 120579119909-119905

2 3 4 5 6 7 80000

0001

0002

0003

0004

0005

0006

Erro

r (

)

0

2

4

6

8

10

FEATheoryError

120579x

(rad

)

Tx sensitivity with width b

b (mm)

(c) 120579119909-119887

Figure 4 Variation of rotation 120579119909according to (a) length 119897 (b) thickness 119905 and (c) width 119887 under load 119879

119909

4 Conclusion

In this study a compliance matrix that expresses the rela-tionship between the deformations and applied loads ofthe SBLF was analyzed and derived by using Castiglianorsquossecond theorem In bending analysis higher-order beamtheory was applied wherein variable shear deformation andthe warping were considered in the calculated formulas forthe shear forces At the joint of two elements of the SBLFthe partially restrained warping in torsion was analysed withconsideration of the warping restraint factor The theoreticalresults were verified by FEA at both the default and thesensitive values The results indicate that there is strongagreement between the two methods with errors below

76 This suggests the accuracy of the proposed theoreticalequations and that they can be used in the precision machinedesign

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Nghia-Huu Nguyen and Moo-Yeon Lee equally contributedto this work

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

Acknowledgment

This researchwas supported in part by Basic Science ResearchProgram through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (NRF-2013R1A1A4A01009657)

References

[1] E-J Park J Shim D-Y Lee and J Lee ldquoA double-bent planarleaf flexure guide for a nano-scannerrdquo Journal of the KoreanPhysical Society vol 57 no 61 pp 1581ndash1588 2010

[2] K Kim D Ahn and D Gweon ldquoOptimal design of a 1-rotational DOF flexure joint for a 3-DOFH-type stagerdquoMecha-tronics vol 22 no 1 pp 24ndash32 2011

[3] T T Y Koseki T Tanikawa N Koyachi and T Arai ldquoKinematicanalysis of a translational 3-dof micro-parallel mechanismusing the matrix methodrdquo Advanced Robotics vol 16 no 3 pp251ndash264 2002

[4] J H Kim S H Kim and Y K Kwak ldquoDevelopment andoptimization of 3-D bridge-type hinge mechanismsrdquo Sensorsand Actuators A Physical vol 116 no 3 pp 530ndash538 2004

[5] J W Ryu D-G Gweon and K S Moon ldquoOptimal design of aflexure hinge basedXY120593wafer stagerdquo Precision Engineering vol21 no 1 pp 18ndash28 1997

[6] N Lobontiu and E Garcia ldquoAnalytical model of displacementamplification and stiffness optimization for a class of flexure-based compliant mechanismsrdquo Computers amp Structures vol 81no 32 pp 2797ndash2810 2003

[7] Y Li and Q Xu ldquoA novel piezoactuated XY stage withparallel decoupled and stacked flexure structure for micro-nanopositioningrdquo IEEE Transactions on Industrial Electronicsvol 58 no 8 pp 3601ndash3615 2011

[8] D M Brouwer J P Meijaard and J B Jonker ldquoLarge deflectionstiffness analysis of parallel prismatic leaf-spring flexuresrdquoPrecision Engineering vol 37 no 3 pp 505ndash521 2013

[9] M Hayashi and M Fukuda ldquoGeneration of nanometer dis-placement using reduction mechanism consisting of torsionalleaf spring hingesrdquo International Journal of Precision Engineer-ing and Manufacturing vol 13 no 5 pp 679ndash684 2012

[10] J J Parise L L Howell and S PMagleby ldquoOrtho-planar linear-motion springsrdquo Mechanism and Machine Theory vol 36 no11-12 pp 1281ndash1299 2001

[11] S Fukada and K Nishimura ldquoNanometric positioning over aone-millimeter stroke using a flexure guide and electromagneticlinear motorrdquo International Journal of Precision Engineering andManufacturing vol 8 pp 49ndash53 2007

[12] Q S Xu ldquoDesign and development of a compact flexure-basedXY precision positioning system with centimeter rangerdquo IEEETransactions on Industrial Electronics vol 61 no 2 pp 893ndash9032014

[13] X-P S Su and H S Yang ldquoDesign of compliant microleveragemechanismsrdquo Sensors and Actuators A Physical vol 87 no 3pp 146ndash156 2001

[14] Y K Yong S S Aphale and S O R Moheimani ldquoDesignidentification and control of a flexure-based XY stage for fastnanoscale positioningrdquo IEEE Transactions on Nanotechnologyvol 8 no 1 pp 46ndash54 2009

[15] J-J Kim Y-M Choi D Ahn B Hwang D-G Gweon andJ Jeong ldquoA millimeter-range flexure-based nano-positioning

stage using a self-guided displacement amplification mecha-nismrdquo Mechanism and Machine Theory vol 50 pp 109ndash1202012

[16] S Xiao Y Li and Q Meng ldquoMobility analysis of a 3-PUU flexure-based manipulator based on screw theory andcompliance matrix methodrdquo International Journal of PrecisionEngineering and Manufacturing vol 14 no 8 pp 1345ndash13532013

[17] N Lobontiu ldquoCompliance-based matrix method for modelingthe quasi-static response of planar serial flexure-hinge mecha-nismsrdquo Precision Engineering vol 38 no 3 pp 639ndash650 2014

[18] M Kujawa ldquoTorsion of restrained thin-walled bars of openconstraint bisymmetric cross-sectionrdquo Tast Quarterly vol 16pp 5ndash15 2011

[19] E J Sapountzakis ldquoBars under torsional loading a generalizedbeam theory approachrdquo ISRN Civil Engineering vol 2013Article ID 916581 39 pages 2013

[20] A H Al-HaKeem Structural analysis of truck chassis framesunder longitudinal loads considering bimoment effects [PhDthesis] Cranfield Institute of Technology Cranfield UK 1991

[21] M Levinson ldquoAnew rectangular beam theoryrdquo Journal of Soundand Vibration vol 74 no 1 pp 81ndash87 1981

[22] J N Reddy C M Wang G T Lim and K H Ng ldquoBendingsolutions of Levinson beams and plates in terms of the classicalTheoriesrdquo International Journal of Solids and Structures vol 38no 26-27 pp 4701ndash4720 2001

[23] C M Wang J N Reddy and K H Lee Shear DeformableBeams and Plates Relationships with Classical Solutions ElsevierScience Oxford UK 2000

[24] N Lobontiu andEGarciaMechanics ofMicroelectromechanicalSystems Kluwer Academic New York NY USA 2005

[25] N H Nguyen B D Lim and D Y Lee ldquoTorsional analysis of asingle-bent leaf flexurerdquo Structural Engineering and Mechanicsvol 54 no 1 pp 189ndash198 2015

[26] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill 2nd edition 1951

[27] W D Pilkey Analysis and Design of Elastic Beams Computa-tional Methods John Wiley amp Sons New York NY USA 2002

[28] N H Nguyen B D Lim and D Y Lee ldquoDisplacement analysisof a single-bent leaf flexure under transverse loadrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16 no4 pp 749ndash754 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of