How to Match Theoretical and Experimenta

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HOW TO MATCH THEORETICAL AND EXPERIMENTAL BOUNDARYCONDITIONS OF A CANTILEVER BEAM

Thiago Ritto 1*, Romulo Aguiar1, Rubens Sampaio1, and Edson Cataldo2

1Mechanical Engineering DepartmentPUC-Rio

Rua Marques de Sao Vicente, 225, Gavea, CEP: 22453-900, RJ, Brazile-mail: [email protected], [email protected], [email protected]

2 Applied Mathematics Department, Graduate Program in Telecommunications Engineering,Graduate Program in Mechanical Engineering

UFFRua Mario Santos Braga, S/N, Centro, Niteroi, CEP: 24020-140, RJ, Brazil,

e-mail: [email protected]

Keywords: boundary condition, Timoshenko beam, experimental modal analysis, finite ele-

ments

ABSTRACT

The purpose of this paper is to describe how to match experimental and theoretical frequencies

of a cantilever beam through a prescribed error tolerance. In some cases there is a clear dif-

ference between a clamped boundary condition in a real system or in an experiment (called in

this article as e-clamped condition) and the clamped condition as regularly modeled in theory

(the t-clamped condition). In such occasions the e-clamped (experimental) might not follow the

t-clamped (theoretical) condition. Actually, it hardly will. Depending on the e-clamped bound-

ary condition the error between the natural frequencies of t-clamped and e-clamped might be

greater than 10%. This work presents a theoretical model that relaxes the boundary condition

of zero rotation. One parameter, a torsional stiffness at the fixed end of the beam, is used inorder to fit the natural frequencies of the numerical simulations with the experimental data.

The idea is to identify the torsional stiffness that minimizes the error between the numerical and

experimental frequencies. For the numerical simulations, the Finite Element Method (with Tim-

oshenko beam element) is used, and for the experiments, multiple rubber layers are mounted

to the clamped side of the beam so that different e-clamped conditions can be analyzed and

reproduced if necessary. The numerical and the experimental results show excellent agreement.

1. INTRODUCTION

In structural dynamics modeling, the clamped end is used as a boundary condition of a mount,

joint, etc. The theoretical clamped boundary condition implies that no movement in any direc-tion nor rotation is allowed, mentioned in this work as t-clamped condition [1, 2]. However,

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in the field or in a laboratory experiment, this clamped condition, mentioned throughout this

work as e-clamped, might not respect the same restriction of no movement or rotation, as in the

t-clamped condition. Sometimes the mount has some flexibility and due to such characteristic

the entire numerical simulation might not have a good agrement with experimental data. Never-

theless, this small change in the flexibility of a clamped boundary condition may substantiallychange the dynamic response of the entire system, such as natural frequencies and vibration

modes.

The present work proposes an alternative way to model the e-clamped boundary condition

of a cantilever beam, because a simple boundary condition might not be that simple [3]. One

parameter (torsional stiffness) is introduced in the clamped side in order to fit the numerical

simulations with the experiments. This work shows that such minor change in the boundary

condition is enough to represent the experimental data. In the context of Finite Element Model

Updating there are some studies of this kind as, for instance, [4, 5].

A similar idea was used by [6] in the impact of the temperature field on vibration of beams.

Due to the relatively high possible temperature variations in beams it was developed a newmodel for the clamped and the simply supported beam. To obtain better analytical results in

comparison with experimental data, a spring was added and the comparison between the ana-

lytical model and experimental data showed good agreement.

To identify the optimal value of the torsional stiffness that minimizes the error between nu-

merical and experimental results an optimization procedure is performed. The results achieved

herein might be used to change the dynamical characteristics of a structure, avoiding unwanted

frequencies. One can add some flexibility into a mount, for instance, and change the charac-

teristic of the boundary condition in order to change the system natural frequencies. With such

action it is possible to change the dynamical characteristics avoiding undesirable frequencies

with a procedure that can be less expensive than change the system itself.

2. EXPERIMENTAL APPARATUS AND RESULTS

2.1 Test rig apparatus and data acquisition methodology

The experimental apparatus attempts to obtain the frequency response of the cantilever beam

when the e-clamped condition is changed. The experimental apparatus is shown in Fig. 1.

(a) (b)

Figure 1. Test rig: a) photo; b) scheme.

The experiment is composed by a simple cantilever beam, supported by a cast iron bench.

The beam is made of aluminum, with a 511 mm length and a retangular section of 30.7 by3.04 mm. The mount is composed by steel couplings, which tries to guarantee the boundary

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condition of zero displacement and zero rotation by pressing the aluminum beam against both

couplings, and the mount is fixed at the cast iron rig, using 7/16 inch bolts for all couplings.

The frequency response is obtained [7] using an impulse modal hammer (Endevco Model

30927). The system response is captured by two mini-accelerometers (Endevco, Model 25B),

one positioned at the free end of the beam and the other at the middle. All signals (hammer andaccelerometers) are preconditioned before analyzed by the signal processor (HP 35650).

Figure 2. Mount photo.

The specification of all sensors and the modal hammer are shown in Tab. 1.

Table 1. Hammer and sensors specs.

Accelerometer 1 - 25B SN BL47

Sensitivity 4.7902 mV/gWeight 0.2 10−3 KgMeasure Range ±1000 gResonance frequency 50 kH z Accelerometer 2 - 25B SN BL557

Sensitivity 4.7707 mV/gWeight 0.2 10−3 KgMeasure Range ±1000 g

Resonance frequency 50 kH z Impulse Modal Hammer - 30927 SN 1660

Sensitivity 99.7 mV/lbMax impulse 1000 lbf Range 50 lb

The methodology applied here is to obtain the frequency response of the beam as the bound-

ary condition at the clamped end is varied. This boundary condition is varied by applying

multiple rubber layers at the mount, as shown on the sketch of Fig. 3. The number of layers

vary from 0 (no rubber layer) to 10 rubber layers.

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(a) (b)

Figure 3. Rubber layer: (a) Scheme; (b) Photo (1 layer).

For each boundary condition the frequency response of the cantilever beam is obtained using

the modal hammer. The modal test is ran two times, each time is an average of six distinct

impulses applied by the hammer in different points of the beam.

Finally the experimental results will be used to validate the mathematical modeling of the

cantilever beam, with the objective to predict the natural frequencies for a given number of

layers.

2.2 Experimental results

As mentioned before, for each boundary condition the modal tests are ran twice, with each test

an average of six impulses applied in different points of the beam. The natural frequencies are

obtained from the magnitude signal of both accelerometers and confirmed by the phase chart.Figure 4 shows the frequency response of the cantilever beam with no rubber layers.

Figure 4. Frequency response, 0 rubber layers, data from accelerometer # 1, two modal tests.

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Figure 4 shows the response of the accelerometer #1 in both experimental data acquisitions.

All experimental data was acquired and analyzed from accelerometers #1 and #2, but once the

response of accelerometer #1 gives a better resolution than accelerometer #2, the results shown

in this article are from accelerometer #1. Figure 4 shows 6 major peaks, corresponding to the

first six natural frequencies, the first natural frequency at 9 Hz and the sixth natural frequencyaround 790 Hz. It is also noticeable the presence of several minor sharp peaks in multiples of

60 Hz. Those peaks correspond to the local electric supply and its harmonics, and those peaks

should not be considered for the modal analysis.

Figure 5 shows the frequency response in four distinct boundary conditions: 1, 3, 6 and 10

rubber layers.

(a) (b)

(c) (d)

Figure 5: Frequency response: (a) 1 rubber layer; (b) 3 rubber layers; (c) 6 rubber layers; (d) 10

rubber layers.

From the charts of all boundary conditions it is possible to obtain the natural frequencies for

each case and compare each frequency of each particular vibration mode with the variation of

the boundary condition. This is shown in Tab. 2.

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Table 2. Natural frequencies - changing boundary conditions.

natural fre-

quency (Hz) 1st 2nd 3rd 4th 5th 6th

rubber layers0 9.25 58.80 163.75 321.50 534.50 793.20

1 9.00 57.00 160.90 315.60 524.75 778.75

2 9.00 57.00 160.75 315.25 525.40 780.00

3 9.00 56.75 160.00 314.00 523.00 777.25

4 9.00 56.75 160.15 314.20 523.00 777.00

6 9.00 56.65 159.90 313.60 522.75 775.75

8 9.00 56.50 159.75 313.20 521.50 773.50

10 9.00 56.25 158.75 311.50 519.25 770.75

Observing the system frequency response and the data shown in Tab. 2, it can be noticed that

there is no variation of the first frequency with the presence of the rubber layers, but it is worth

it to remark that the accuracy of the experiment is 0.25Hz, what makes it difficult to distinguish

9.0Hz from 9.1Hz, for example.

The higher the number of rubber layers more flexible the boundary conditions becomes and

lower the natural frequencies, compared to the condition without rubber layers. A comparison

between the opposite sides of the experiment, i.e., the frequency response of the beam without

rubber layers on the clamped end and the system with 10 rubber layers is shown in Fig. 6.

(a) (b)

Figure 6: Frequency response, boundary condition comparison. Thin line: 0 rubber layers;thick line: 10 rubber layers. a) 0-200 Hz, b) 300-800Hz

3. MODEL OF THE CLAMPED BOUNDARY CONDITION

3.1 The FE model and the clamped boundary condition

Consider the beam element shown in Fig. 7.

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Figure 7. Beam element.

In the Finite Element Method, the clamped boundary condition is usually modeled as u1a|x=0 =u1v|x=0 = u1r|x=0 = 0, see Fig. 8(a). The model proposed in this paper relax the condition

u1r|x=0 = 0. Instead of imposing no rotational displacement, a torsional stiffness is incorpo-

rated to the system, Fig. 8(b).

x

(a)

x(b)

Figure 8: a) Clamped boundary condition usually applied in Finite Element Method, u1a|x=0 =u1v|x=0 = u1r|x=0 = 0; b) u1r|x=0 = 0, torsional stiffness kt.

The Finite Element Method was used to discretize a cantilever beam, clamped in one end

and free in the other. The Timoshenko beam element used has three degrees of freedom (axial,

vertical and rotational) at each node (Fig. 7). The variational form of strain energy is given by:

δH = L0

[δu

a(EAu

a) + δu

v(EI u

v) + δu

r(GIu

r)]dx . (1)

Where is the derivative with respect to x (d/dx). The degrees of freedom are: the axial dis-

placement (ua); the vertical displacement (uv); and the rotation (ur). E is the elastic modulus,

G is the shearing modulus, I is the inertia momentum, L is the beam length and A is the cross

section area. The virtual work done by inertial forces (δT ) and damping forces (δD) are the

following:

{δT,δD} = L

0{δua(ρAua) + δuv(ρAuv) + δur(ρI ur),−δuacua− δuvcuv−δurcur}dx . (2)

The damping forces are taking into account as a Rayleigh damping proportional to the mass,

where c is the damping constant. The element matrices are the following ones:

[M (e)] = ρA h

420

140 0 0 170 0 00 156 22h 0 54 −13h0 22h 4h2 0 13h −3h2

70 0 0 140 0 00 54 13h 0 156 −22h0 −13h −3h2 0 −22h 4h2

; [C ] = c

ρA[M ]

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[K (e)] = ρ E 1+m

A(1 + m)/l 0 0 −A(1 + m)/l0 12I/l3 6I/l2 00 6I/l2 4I (1 + m/4)/l 0−A(1 + m)/l 0 0 A(1 + m)/l

0 −12I/l3

−6I/l2

00 6I/l2 2I (1 − m/2)/l 0

0 0−12I/l3 6I/l2

−6I/l2 2I (1 − m/2)/l0 012I/l3 −6I/l2

−6I/l2 4I (1 + m/4)/l

.

Where m = 12

h2

EI

GAks , ks is the shear correction factor and l is the element size.

The essential boundary conditions at x = 0 are given by ua|x=0 = 0 and uv|x=0 = 0. The

system is discretized using the Finite Element Method, and after assembling the matrices, one

can write:

[M ]u(t) + [C ]u(t) + [K ]u(t) = f (t) . (3)

Where [M ], [C ] and [K ] are the mass, damping and stiffness matrices, which are real and

positive-definite. The external force is represented by vector f (t) = (f 1a, f 1v, f 1r, f 2a, f 2v, f 2r, . . . , f nr)and the displacements (u1a, u1v, u1r, u2a, u2v, u2r, . . . , unr) are the components of the vector

u(t).

The data used for the numerical simulation were: c = 10 N/(m.s); E = 2e11 Pa; ρ = 7850kg/m3; ν = 0.3; G = 0.5E/(1 + ν ); ks = 5/6; L = 511 mm, b = 30.7mm; h = 3.04mm.

Figure 9 shows the beam modeled.

Figure 9. Beam modeled.

3.2 Natural frequencies

To calculate the natural frequencies without damping the following eigenvalue problem must

be solved:

(ω2i [M ] + [K ])Φi = 0 , (4)

where ωi is the i-th natural frequency, [M ] and [K ] are the mass and stiffness matrices andΦi is the vibration mode.

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To determine the number of elements needed, the error of the sixth natural frequency is

calculated (because the higher frequency obtained through experiment was the sixth). It was

necessary 32 elements to have an error lower than 0.1%.

The i-th natural frequency with damping is calculated in the following way:

ωdi = ωi

1 − ξ 2i , (5)

where ξ i is the damping rate of the i-th mode, which can be calculated using the normal

modes (Φ) and the damping matrix ([C ]):

ξ i = ΦT

i [C ]Φi

2ωi

. (6)

4. IDENTIFICATION PROCEDURE

Since the torsional stiffness is not known, its value must be identified [8], this is an inverse

problem [9]. The cost function chosen to be minimized was the following:

minkt

6i=1

(ωdi(kt) − ωexpi)2

ω2expi

, (7)

where ωexpi is the i-th natural frequency obtained through experiment. The function fmin-

search of MATLAB was used to solve the optimization problem. Table 3 shows the torsional

stiffness obtained for different number of layers.

Table 3. Torsional stiffness results

N0 of Layers 0 1 2 3 4 6 8 10kt (kN/m) 2.8 1.2 1.2 1.1 1.1 1.1 1.0 0.9

5. COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTS

The first numerical simulation performed considered the t-clamped condition, u1a|x=0 = u1v|x=0= u1r|x=0 = 0. The error between this result and the experiment with no rubber layers is shown

in Tab. 4, second column.

Even without the presence of rubber layers, the error between the simulation (t-clamped) and

the experiment is higher than 2.6%. Nevertheless, if the boundary condition is modeled with

a torsional spring, kt = 2.8e3N/m, the error drops considerably, as shown on Tab. 4, third

column.

Table 4. No rubber layer. Percent error for kt = 2.8e3N/m

0 layer clamped condition kt = 2.8e3N/mnatural freq error(%) error(%)

1st 2.65 0.65

2nd 2.40 0.46

3rd 1.73 0.14

4th 1.50 0.31

5th 0.89 0.866th 1.51 0.19

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Relaxing the condition ur = 0 and using a single extra parameter, kt, it is possible to obtain

excellent approximations. Either in an experiment or in any engineering application, it is not

simple to precisely assure the clamped boundary condition, even in this controlled test, in which

is noted that the clamped condition is not ideal. However, by simply adding a torsional stiffness,

the results become satisfactory. Mounting a rubber layer on the clamped side it is expected thatmount flexibility, therefore a smaller kt will be needed to match simulation with experimental

results.

The error between the numerical and experimental results is synthesized on Tab. 5. Depend-

ing on the number of rubber layers, a different kt needs to be fitted, so the best match between

the simulation and the experiment is achieved.

Table 5. Error between numerical and experimental results

Error(%)

Layers 0 1 2 3 4 6 8 10

kt (kN/m) 2.8 1.2 1.2 1.1 1.1 1.0 1.0 0.9

1st 0.65 0.80 0.80 0.39 0.39 0.26 0.06 0.49

2nd 0.46 0.07 0.07 0.16 0.16 0.23 0.33 0.13

3rd 0.14 0.49 0.39 0.22 0.32 0.26 0.30 0.69

4th 0.31 0.36 0.25 0.11 0.17 0.06 0.06 0.38

5th 0.86 0.74 0.87 0.63 0.63 0.66 0.52 0.80

6th 0.19 0.08 0.08 0.07 0.10 0.20 0.40 0.16

In order to attempt to establish a relation between the torsional stiffness kt and the number of

rubber layers at the mount, Fig. 13 shows how kt decreases with the number of layers.

Figure 10. Layer versus kt.

Except for the first point, which is the torsional stiffness for the boundary condition of no

rubber layers, the relationship between the torsional stiffness and the number of rubber layers

seems linear. Figure 11 shows the linear relationship between the number of layers and kt.

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Figure 13. FRF, from 0 to 2500 Hz.

To investigate a little more the proposed modeling, three different rubber materials were

tested: hard rubber, soft rubber, foam rubber. All the three cases were performed with one

layer. For the foam rubber the layer was mounted only in one surface of the beam. The results

are in Tab. 6.

Table 6: Percent error between the numerical and the experimental results for different materials

Error(%)Natural Freq. hard rubber soft rubber foam rubber

kt (kN/m) 1.3 1.1 1.1

1st 1.30 0.52 0.64

2nd 0.15 0.36 0.29

3rd 0.48 0.44 0.66

4th 0.47 0.34 0.27

5th 0.88 0.56 0.31

6th 0.65 0.21 0.29

The only error above 1% was in the first natural frequency of the hard rubber. But oneknows that the precision of the experiment is 0.25Hz, so if ω1 equals 9.05, for instance (instead

of 9.00Hz), the error would be 0.08%. These results are again very motivating. The proposed

model works well even for different materials.

6. CONCLUDING REMARKS

An alternative to model the clamped boundary condition of a cantilever beam was proposed

in this work. Only one parameter (torsional stiffness) was needed to well fit the numerical

results with the experimental ones. It is very difficult to find a perfect clamped condition in

experiments or in an engineering application, while relaxing the condition ur|x=0 = 0 (no

rotational displacement) and using a torsional spring (kt) instead, seems to be a good idea tomodel the clamped boundary condition of a cantilever beam. The numerical results shown

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very good agreement with the experimental results (errors lower than 1%). The relationship

between the number of rubber layers and kt is almost linear for the rubber used in the experience

herein. In this way, it is possible to use these results to change the dynamical characteristics of

a structure, avoiding unwanted frequencies.

ACKNOWLEDGMENTS

The authors acknowledge the financial support of CNPQ, CAPES, and FAPERJ.

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How to Match Theoretical and Experimenta