ANALYTICAL RANDOM VIBRATION ANALYSIS OF BOUNDARY-EXCITED THIN RECTANGULAR PLATES · 2016-05-23 · ANALYTICAL RANDOM VIBRATION ANALYSIS OF BOUNDARY-EXCITED THIN RECTANGULAR PLATES

ANALYTICAL RANDOM VIBRATION ANALYSIS OF

BOUNDARY-EXCITED THIN RECTANGULAR PLATES

ASHKAN HAJI HOSSEINLOO*,‡ and FOOK FAH YAP†

School of Mechanical and Aerospace EngineeringNanyang Technological University

50 Nanyang Avenue, Singapore 639798*[email protected]†[email protected]

NADER VAHDATI

Department of Mechanical Engineering

The Petroleum Institute

PO Box 2533 Abu Dhabi, UAE

[email protected]

Received 4 April 2012

Accepted 6 June 2012

Published 5 March 2013

Fatigue life, stability and performance of majority of the structures and systems depend sig-

ni¯cantly ondynamic loadings applied on them. Inmany engineering cases, the dynamic loading is

random vibration and the structure is a plate-like system. Examples could be printed circuitboards or jet impingement cooling systems subjected to random vibrations in harsh military

environments. In this study, the response of thin rectangular plates to random boundary exci-

tation is analytically formulated and analyzed. In the presentedmethod, closed-formmode shapes

are used and some of the assumptions in previous studies are eliminated; hence it is simpler andreduces the computational load. In addition, the e®ects of di®erent boundary conditions, modal

damping and excitation frequency range on dynamic random response of the system are studied.

The results show that increasingboth themodal damping ratio and the excitation frequency range

will decrease the root mean square acceleration and the maximum de°ection of the plate.

Keywords: Random vibration; boundary excitation; rectangular; thin plate.

1. Introduction

Fatigue life, stability and performance of majority of the structures and systems

depend signi¯cantly on dynamic loadings applied on them. A common type of

loading is random excitation which often occurs in earthquakes or other harsh

conditions like those in military environments. According to US Air-Force database,

‡Corresponding author.

International Journal of Structural Stability and DynamicsVol. 13, No. 3 (2013) 1250062 (16 pages)

#.c World Scienti¯c Publishing Company

DOI: 10.1142/S0219455412500629

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http://dx.doi.org/10.1142/S0219455412500629

20% of all the failures observed in electronic equipment are due to vibration problems

which are essentially random in nature.1,2 In order to prevent catastrophic and

fatigue failures or overdesign costs, an analytical random vibration analysis of these

structures seems to be very helpful for their optimum design. Fortunately, many

parts and components of the systems subjected to random vibration are of a rec-

tangular shape. Examples are the walls and °oors of buildings subjected to lateral

and vertical components of an earthquake motion, respectively, or printed circuit

boards subjected to random transportation vibration. Moreover, many of the sys-

tems like the aforementioned examples are base and boundary-excited systems.

Therefore, it seems that an analytical random vibration analysis of boundary-excited

plates can signi¯cantly help the ¯eld.

Random vibration of thin plates has been studied for decades. However, few

studies have focused on boundary or base excitation. Moreover, the analyses done so

far on thin plates are either analytical with the simplest boundary conditions such as

simply supported edges or with substantially simplifying assumptions, or otherwise

the analyses are carried out by numerical or experimental approaches. Huang and

Kececioglu studied the response of a simpli¯ed printed circuit board to random base

excitation.3 They modeled the system as a thin plate with simply supported edges.

They neglected the input velocity terms when calculating cross correlation and cross

spectral density functions. Leibowitz studied the random vibro-acoustic response of a

rectangular clamped or simply supported plate to turbulent excitation.4 In his study,

he used Warburton's method for determining the natural frequencies of the clamped

plate. However, he used the mode shapes corresponding to a simply supported plate.

A year later, Sadasivan et al.5 used a similar but hybrid method where they used the

product of beam functions with clamped ends instead of the simplifying simply

supported mode shapes for the calculation of joint acceptance integrals. Elishakof6

studied random vibration of orthotropic plates with either clamped or simply sup-

ported edges where he used an approximate method for determining the probabilistic

response of the plate. He used a series of approximate modes of vibration which

satis¯ed the boundary conditions but not the ¯eld equations. Furthermore, Miao and

Ouyang7 used a normalized group of generalized Fourier functions which satis¯ed the

clamped boundary conditions to evaluate the response of the system under random

excitation. They concluded that in order to involve higher order modes, many more

terms must be included to give satisfactory results which in turn increase the com-

putational e®ort and time. Later Poltrak8 studied random vibration of a thin plate

with a very speci¯c boundary condition where the plate had mutually perpendicular

clamped and free edges.

Besides analytical methods, a great proportion of the studies on random vibration

of plates are carried out by the help of either numerical and modeling software or

experimental tests. Maymon9 developed an analytical method to evaluate the ran-

dom response of deterministic and indeterministic structures to random loadings.

However, for his method to function e±ciently, he required closed-form solutions for

A. H. Hosseinloo, F. F. Yap & N. Vahdati

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natural frequencies, mode shapes and stress functions of the structure. Since these

functions were not available in closed-form even for a simple clamped thin plate, at

least at that time, he used the numerical Taylor expansion method. Finite element,

¯nite strip and experimental methods are some other methods that have been used to

evaluate and analyze random response of rectangular plates or plate-like structures

such as printed circuit boards or printed wiring boards.2,10�14

Problems involving rectangular plates fall into three distinct categories15: (a)

plates with all edges simply supported; (b) plates with a pair of opposite edges simply

supported; (c) plates which do not fall into any of the above categories. According to

the literature there is hitherto no analytical random vibration study done that deals

with boundary-excited thin plates with boundary conditions falling into the third

category. All random plate vibration studies are made with the assumption that the

edges are simply supported or are done numerically or empirically, or otherwise

complex series expansion methods for mode shape approximation which are not

exact and yet computationally heavy, are used. In 2009, Xing and Liu16 found exact

closed-form solutions for mode shapes of some of the boundary conditions in the third

category for the very ¯rst time. As mentioned, there is a gap in the analytical random

vibration analysis of plate-like structures with boundary conditions falling in the

third category. Furthermore, exact mode shapes of some of the boundary conditions

of this category are recently (at the time when this paper was written) developed.

Therefore, the authors were motivated to do an analytical study on random vibra-

tion of boundary-excited thin plates.

Application-wise, the main motivation of the current study (in addition to the

applications mentioned earlier) is the random vibration analysis of jet impingement

cooling systems. In this cooling system, there are nozzle and impingement plates. The

edges of the thin rectangular plates are assumed to be clamped and the plates sub-

jected to random boundary excitation. The vibration of either of the nozzle or

impingement plates can deteriorate the overall thermal performance of the sys-

tem.17,18 Therefore, it is essential to study the random vibration response of these

plates for further protection and thermal performance maintenance using di®erent

vibration isolation and protection techniques.

2. Theoretical Analysis

2.1. Mathematical formulation

Figure 1 shows a thin rectangular plate with general boundary conditions. a and b

are the length and width of the plate, and wbðtÞ and wðx; y; tÞ are displacements of

the boundary and de°ection (displacement relative to the boundary) of an arbitrary

point on the plate, respectively.

The governing dynamic equation of the plate is given by

�@ 2wa

@t2þ c

@w

@tþDr4w ¼ 0; ð1Þ

Analytical Random Vibration Analysis of Boundary-Excited Thin Rectangular Plates

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where wa is absolute displacement of the plate and is given by

waðx; y; tÞ ¼ wbðtÞ þ wðx; y; tÞ: ð2ÞIn Eq. (1), � and c are areal density and viscous damping coe±cient per unit area,

respectively and D is the °exural rigidity of the plate. By substituting Eq. (2) into

Eq. (1) and rearranging the equation, one obtains

�@ 2w

@t2þ c

@w

@tþDr4w ¼ pðtÞ; ð3Þ

where,

pðtÞ ¼ ��w::bðtÞ: ð4ÞIn this study, three di®erent boundary conditions are studied, namely CCCC, SSCC

and SCCC where letters C and S represent clamped and simply supported boundary

conditions, respectively. The eigenvalue problem of these boundary conditions can be

shown to be self-adjoint,19 and thus the eigenfunctions of the eigenvalue problem will

be orthogonal. In other words, if the eigenfunctions of the mnth and klth modes are

designated as mn and kl, the following equations as orthogonality of the eigen-

functions hold true, ZR

� mnðx; yÞ klðx; yÞdxdy ¼ mmn�mn;kl ð5Þ

and ZR

c mnðx; yÞ klðx; yÞdxdy ¼ cmn�mn;kl; ð6Þ

where mmn and cmn are called modal mass and modal damping, respectively.

Since the plate is homogenous, � and c are constant over the plate's area and thus,

cmn ¼ c

�mmn: ð7Þ

In Eqs. (5) and (6), R is the integration domain (that is the plate's surface) and �mn;kl

is the Kronecker delta function and is de¯ned as,

�mn;kl ¼1; mn ¼ kl

0; mn 6¼ kl

�: ð8Þ

Fig. 1. Thin rectangular plate with coordinates and general boundary conditions.


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Now the solution to Eq. (3) is assumed to have the form

wðx; y; tÞ ¼X1m¼1

X1n¼1

mnðx; yÞqmnðtÞ; ð9Þ

where qmnðtÞ is the time varying function.

By substituting Eq. (9) into Eq. (3) and then multiplying both sides of the

resultant equation by klðx; yÞ and integrating over the plate's surface, one obtains

€qmnðtÞ þ �mn _qmnðtÞ þ !2mnqmnðtÞ ¼

1

mmn

ZR

mnðx; yÞpðtÞdxdy; ð10Þ

where !mn is the undamped natural frequency of themnth mode of the plate and �mn

is the frequency bandwidth of the mnth mode and is de¯ned as,

�mn ¼ cmn

mmn

¼ 2�mn!mn; ð11Þ

where �mn is the modal damping ratio of the mnth mode.

The frequency response function of the plate at point ðx; yÞ, Hwðx; y; !Þ to the

input wbðtÞ can be calculated by applying a unit harmonic input as

wbðtÞ ¼ ei!t: ð12ÞThe substitution of Eq. (12) into Eq. (4) and in view of Eq. (10), one obtains

€qmnðtÞ þ �mn _qmnðtÞ þ !2mnqmnðtÞ ¼

�!2Imn

mmn

ei!t; ð13Þ

where

Imn ¼ZR

mnðx; yÞdxdy: ð14Þ

The solution of Eq. (13) may be written as

qmnðtÞ ¼ Hmnð!ÞImnei!t; ð15Þ

where

Hmnð!Þ ¼�!2

mmnð�!2 þ �mni!þ !2mnÞ

: ð16Þ

By substituting Eq. (15) into Eq. (9), one can ¯nd the frequency response function of

the plate's de°ection Hwðx; y; !Þ,

Hwðx; y; !Þ ¼X1m¼1

X1n¼1

mnðx; yÞHmnð!ÞImn: ð17Þ

Consequently, according to Eq. (2) the frequency response function of the plate's

absolute displacement Hwaðx; y; !Þ isHwa

ðx; y; !Þ ¼ 1þHwðx; y; !Þ: ð18Þ


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In view of the frequency response functions and the power spectral density of the

input, one can ¯nd the variances of the plate's de°ection and acceleration at di®erent

points. The input is considered ergodic and stationary with one-sided displacement

power spectral density (PSD) of Swbð!Þ and zero mean value. Hence, the mean square

de°ection of the plate at point ðx; yÞ can be calculated as

E½w2ðx; y; tÞ� ¼Z 1

0

jHwðx; y; !Þj2Swbð!Þd!: ð19Þ

Similarly, the mean square of the plate absolute acceleration at this point is given by

E½ €w 2aðx; y; tÞ� ¼

Z 1

0

!4jHwaðx; y; !Þj2Swb

ð!Þd!: ð20Þ

In Eqs. (19) and (20), the input excitation is shown as the power spectral density of

the input displacement while PSD is usually measured in terms of acceleration rather

than displacement. However, it is very easy to relate these two functions,

Swbð!Þ ¼ S €wb

ð!Þ!4

: ð21Þ

2.2. Eigenvalue equations and eigenfunctions

In the equations derived in Sec. 2.1, mnðx; yÞ is the undamped normal mode shape

or eigenfunction of the mnth mode. The mode shapes depend on the system and its

boundary condition. In this study, three boundary conditions are considered, namely

CCCC, SSCC and SCCC as illustrated in Fig. 2. The exact closed-form mode shapes

for these boundary conditions were ¯rst derived in 200916 and are summarized in

Table 1. The eigenvalue equations and normal mode shapes of these three boundary

conditions are listed in Table 1. Note that the normal mode shape of the CCCC plate

presented by Xing and Liu in Table 2 of their paper16 is slightly wrong but it is re-

derived and corrected herein.

In Table 1, k1 and k2 are de¯ned as

k1 ¼cos�1a� cosh�2a�2

�1sin�1a� sinh�2a

; ð22Þ

Fig. 2. Three di®erent boundary conditions of the plate.


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and

k2 ¼cos �1b� cosh �2b�2�1

sin �1b� sinh �2b: ð23Þ

�1; �2; �1 and �2 are mode dependent unknowns. These unknowns along with the

undamped natural frequency of the plate can be determined by solving a set of ¯ve

nonlinear equations. Two out of the ¯ve equations are the eigenvalue equations in

Table 1 and the other three equations are as follows,

� 21 þ � 2

2 ¼ 2k2

�1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � � 2

1

p�2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ � 2

1

p

8><>: ; ð24Þ

where

k2 ¼ !mn

ffiffiffiffiffiffiffiffiffi�=D

p: ð25Þ

For details of the derivation of the above ¯ve equations, one can refer toRefs. 16 and 20.

In order to solve the nonlinear equations for the ¯ve unknowns, one can use the

Newton's method. According to the eigenvalue equations the intervals of �1a and �1b

can be determined as,

�1a 2 ðm�;m�þ 0:5�Þ; m ¼ 1; 2; . . .

�1b 2 ðn�;n�þ 0:5�Þ; n ¼ 1; 2; . . .: ð26Þ

Thus, the midpoint of these intervals can be considered as the initial values for �1

and �1 in the Newton's method, that is,

�1 ¼ðm�þ �=4Þ

a; m ¼ 1; 2; . . .

�1 ¼ðn�þ �4Þ

b; n ¼ 1; 2; . . .

8>><>>:

: ð27Þ

Table 1. Eigenvalue equations and normal mode shapes of CCCC, SSCC and SCCC plates.

Boundary

condition

Eigenvalue equation Normal mode shape ðx; yÞ

CCCC 1� cos�1a cosh�2a

sin�1a sinh�2a¼ �2

1 � �22

2�1�2

;

1� cos�1b cosh�2b

sin�1b sinh�2b¼ � 2

1 � � 22

2�1�2

� cos�1xþ �2

�1

k1 sin�1xþ cosh�2x� k1 sinh�2x

� �

� cos�1yþ�2�1

k2 sin�1yþ cosh�2y� k2 sinh�2y

� �

SSCC �2 tan�1a ¼ �1 tan�2a;

�2 tan�1b ¼ �1 tan�2bsin�1x� sin�1a

sinh�2asinh�2x

� �

sin�1y�sin�1b

sinh �2bsinh �2y

� �

SCCC �2 tan�1a ¼ �1 tan�2a;

1� cos�1b cosh�2b

sin�1b sinh�2b¼ � 2

1 � � 22

2�1�2

sin�1x� sin�1a

sinh�2asinh�2x

� �

� cos�1yþ�2�1

k2 sin�1yþ cosh�2y� k2 sinh�2y

� �


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By making use of Eq. (24) and the initial values for �1 and �1, one can ¯nd for the

initial values of the other three unknowns,

�2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� 21 þ 2�2

1

q

�2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

1 þ 2� 21

q

k ¼ ffiffiðp�2

1 þ �22Þ=2

8>>>><>>>>:

: ð28Þ

Having solved the ¯ve nonlinear equations, exact natural frequencies and mode

shapes of the three aforementioned boundary conditions are available to use for

random vibration analysis of the plate using the equations derived in Sec. 2.1. The

only unknown parameter left for the random vibration analysis of the system is the

damping of the system which should be either extracted experimentally by modal

analysis or estimated according to the type of the joints and the plate material.

3. Analysis Veri¯cation

In order to verify the analysis and the methodology given in Sec. 2, the response of a

plate evaluated by the proposed analytical method is compared to that of the plate

obtained by the ¯nite element package ANSYS. To this end, a clamped, aluminum,

rectangular thin plate of dimensions 400mm � 200mm � 2mm is modeled in

ANSYS and the results are compared with the analytical approach. Equal modal

damping for all modes is assumed. The input is considered as a white noise base

excitation with a uniform acceleration PSD level of 0.5 g2/Hz over the frequency

range of [20, 2000] Hz. The damping ratio of the plate is varied from 0.5% to 10%

with increments of 0.5%. The root mean square (rms) acceleration and maximum

plate de°ection (3� value) at the center of the plate are plotted in Figs. 3 and 4,

respectively. The de°ection is normalized by the plate's thickness and the

Fig. 3. Normalized rms acceleration of the center point of clamped plate as a function of its modal damping

ratio.


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acceleration is normalized by the rms acceleration of the excitation. According to

both ¯gures, analytical results agree well with those obtained from ANSYS, thereby

verifying the proposed method.

4. Numerical Results and Discussion

The same aluminum plate considered for the veri¯cation in Sec.3 is used here too for

presentation of numerical results. The plate has the same dimensions of 400mm �200mm � 2mm, where the long edge is parallel to the x axis, Young's modulus of

70GPa and Poisson ratio of 0.33. The modal damping ratio is considered constant

and equal to 3% unless otherwise speci¯ed. Three di®erent boundary conditions,

namely CCCC, SCCC and SSCC are considered.

Frequency response functions (FRF) of the plate's midpoint relative acceleration

for the three boundary conditions are plotted in Fig. 5 over the frequency range of

[20, 2000] Hz. It is very interesting that many modes (natural frequencies), obtained

from solving the ¯ve nonlinear equations, are missing in the FRF. For instance, for

the clamped plate there are seventeen modes in the frequency range of [20, 2000] Hz

where out of which only six are clearly shown in Fig. 5. This is due to the nature of

the boundary excitation. Referring to Eqs. (14) and (17), it can be stated that those

modes that have a nodal line passing through the point where FRF is being calcu-

lated or those that have zero mode shape integral will not show up in the FRF. The

latter condition includes the modes that the signed volume under their mode shapes

is zero. For the clamped plate, there are 11 modes with this property out of the ¯rst

17 modes making them disappear from the FRF.

Next, the plate is subjected to random boundary excitation with uniform PSD

level of 0.5 g2/Hz over the frequency range of [20, 2000] Hz. The selected frequency

range is the range commonly found in many military applications. Furthermore, PSD

Fig. 4. Normalized maximum de°ection of the center point of clamped plate as a function of its modal

damping ratio.


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level is set to 0.5 g2/Hz so as to achieve almost the upper bound limit of excitation

acceleration in many military applications, that is approximately 30 g. Figures 6 to 9

show the normalized rms acceleration and normalized maximum de°ection of dif-

ferent points of the clamped plate in surface and contour forms. These ¯gures are not

plotted for the other two boundary conditions since they have almost the same shape

and trend and they di®er only in the values. However, di®erent boundary conditions

will be compared later. According to Figs. 6 and 8, the ¯rst few modes contribute to

the acceleration response of the plate while Figs. 7 and 9 show that the dominant

mode for the plate de°ection is only the ¯rst mode. Referring to Eq. (21), this can be

Fig. 5. Frequency response function of the plate's midpoint j €W ðsÞ€W bðsÞj

� �.

Fig. 6. Normalized rms acceleration of the clamped plate over its surface.


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justi¯ed since the PSD of the de°ection of a point on the plate decreases with power

four of the frequency as the frequency increases. Furthermore, referring to Figs. 6 to

9, the critical point where the plate's maximum de°ection and maximum accelera-

tion occur is the center of the plate. Hence, for the other ¯gures the response of the

plate only at its center is considered.

Di®erent materials have di®erent structural damping. Moreover, damping of a

structure with a selected material itself can vary due to the change in joints,

Fig. 8. Normalized rms acceleration contour of clamped plate.

Fig. 7. Normalized maximum de°ection of the clamped plate over its surface.


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temperature, and humidity or even due to aging. In order to observe the e®ect of

modal damping on the plate's response, it is varied from 0.5% to 10% with incre-

ments of 0.5% and the response of the plate is evaluated in terms of normalized rms

acceleration and normalized maximum de°ection. Since it was found that the critical

point for the plate's response is its center for both de°ection and acceleration, the

response of the plate is computed only at its center. Furthermore, to investigate the

e®ect of boundary conditions on the response of the plate, the plate's response is

plotted for the three boundary conditions.

Figures 10 and 11 show the normalized rms acceleration and normalized max-

imum de°ection of the center point of the plate, respectively. As it can be seen from

the ¯gures, both acceleration and de°ection of the plate decrease with respect to

increasing modal damping ratio. Moreover, referring to these ¯gures, the CCCC

plate has the largest acceleration and the SSCC plate has the smallest whereas the

SSCC plate has the largest de°ection and the CCCC plate has the smallest.

Therefore, it can be concluded that more clamped edges results in more rigidity of the

plate and, hence in larger acceleration and smaller de°ection.

As shown in Fig. 11, although the CCCC plate has smaller de°ection than the

SCCC plate, this di®erence is very small and thus negligible. However, this di®erence

in acceleration is not negligible as seen in Fig. 10. The reason is that for the

de°ection, the ¯rst mode is the dominant mode as discussed earlier, while for the

acceleration a few modes are contributing. Furthermore, the CCCC and the SCCC

plates have very close FRF around the ¯rst mode while they deviate at the higher

modes (referring to Fig. 5), hence resulting in almost equal de°ections but di®erent

accelerations.

Fig. 9. Normalized maximum de°ection contour of clamped plate.


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The upper bound of operating frequency range of some military applications can

go as high as 5,000Hz such as in missiles.2 Consequently, if the application or

operating environment of a device is changed, its excitation frequency will be also

changed. Therefore, it seems useful to investigate the e®ect of the frequency range of

the excitation on the plate's response. However, in order to make it comparable, the

rms acceleration of the excitation (input power) is kept constant. To this end, the

starting frequency range of the excitation is ¯xed at 20Hz and the upper bound is

increased from 520Hz to 5020Hz with increments of 250Hz. The PSD level is uni-

form for any frequency range and it decreases as the frequency range increases so as

to keep the rms acceleration level of the excitation (input power) constant. The

modal damping ratio of the plate is considered constant at 3%.

Fig. 11. Normalized maximum de°ection of center point of plate as a function of its modal damping ratio

for the three boundary conditions.

Fig. 10. Normalized rms acceleration of center point of plate as a function of its modal damping ratio forthe three boundary conditions.


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Figures 12 and 13 show the normalized rms acceleration and normalized max-

imum de°ection of the center point of the plate as a function of the excitation

frequency range length. According to the ¯gures, as the frequency range broadens the

acceleration and the de°ection of the plate decrease. This is due to the fact that when

the frequency range broadens and the input power is kept constant, spectral power is

decreased for the low frequency modes and is added to the high frequency modes. In

general, higher modes contribute less than the lower ones to a system's response

especially to its de°ection. Therefore, broadening the excitation frequency range

decreases the acceleration and de°ection of the plate.

Fig. 13. Normalized maximum de°ection of centere point of plate as a function of the excitation frequency

range length for the three boundary conditions.

Fig. 12. Normalized rms acceleration of centere point of plate as a function of the excitation frequency

range length for the three boundary conditions.


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5. Summary and Conclusion

In this study, an analytical random vibration analysis of boundary-excited thin

rectangular plates with di®erent boundary conditions was tackled. First the problem

was formulated and the mathematical equations were derived. Three boundary

conditions, namely CCCC, SCCC and SSCC whose closed-form mode shapes were

not found until 2009, were considered. The closed-form mode shapes of the CCCC

plate were corrected and then along with the other two boundary conditions were

used for the evaluation of the random response of the plates.

The study proves that many modes of the plate will not contribute to its

response due to the mathematics and nature of the boundary excitation. It

also shows that the center point of the plate is the critical point in terms of

maximum de°ection and maximum acceleration. Moreover, the study shows that

the very low frequency modes particularly the ¯rst mode is the dominant mode

for the de°ection of the plate while for the acceleration the ¯rst few modes will

contribute.

The e®ect of modal damping ratio, excitation frequency range, and boundary

condition were also studied. The results show that increasing both the modal

damping ratio and the excitation frequency range will decrease the rms acceleration

and the maximum de°ection of the plate. Furthermore, the results show that adding

the clamped edge will increase the rigidity of the plate as compared to the simply

supported edge and, hence increases the rms acceleration and decreases the max-

imum de°ection of the plate.

The novel method presented herein eliminates some of the assumptions of the

previous studies in the literature, yet simpli¯es the mathematics and, hence reduces

the computational e®ort. Moreover, the results presented herein can serve as

benchmark for future references.

Acknowledgment

The authors would like to gratefully acknowledge the ¯nancial support of the

Defence Science Organization (DSO).

References

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Documents

ANALYTICAL RANDOM VIBRATION ANALYSIS OF BOUNDARY-EXCITED THIN RECTANGULAR PLATES · 2016-05-23 · ANALYTICAL RANDOM VIBRATION ANALYSIS OF BOUNDARY-EXCITED THIN RECTANGULAR PLATES