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Advanced Vibration ( 306همج)

VIBRATIONS OF CANTILEVER BEAM

(CONTINUOUS SYSTEM)

Ahmed Barakat

Mohamad Gamal Abd ElAziz

Mohamad Add ElAziz Ahmed

Essam Ewis Shabaan

Weam Elsahar

Date Performed: December 26, 2011

Supervisor: Dr. Wael Akl

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Table of Contents

1- Introduction: ................................................................................................................ 3

2- Objective of the Experiment: ....................................................................................... 3

3- Experiment setup: ........................................................................................................ 4

3-1 Used equipment: ................................................................................................... 4

3-2 Accelerometer fixation: ....................................................................................... 5

3-3 Beam specimens: .................................................................................................. 6

3-4 Data acquisition and display: ............................................................................... 7

4- Experiment Procedure: ................................................................................................ 9

5- Results: ...................................................................................................................... 10

6- Observation:............................................................................................................... 12

7- Verification: ............................................................................................................... 14

8- Conclusion: ................................................................................................................ 15

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1- Introduction:

When a dynamic system is subjected to a steady-state harmonic excitation, it is forced to

vibrate at the same frequency as that of the excitation. The harmonic excitation can be given

in many ways like with constant frequency and variable frequency or a swept-sine frequency,

in which the frequency changes from the initial to final values of frequencies with a given

time-rate (i.e., ramp). If the frequency of excitation coincides with one of the natural

frequencies of the system, a condition of resonance is encountered and dangerously large

oscillations may result, which results in failure of major structures, i.e., bridges, buildings, or

airplane wings etc. Hence, the natural frequency of the system is the frequency at which the

resonance occurs. At the point of resonance the displacement of the system is a maximum.

Thus calculation of natural frequencies is of major importance in the study of vibrations.

Because of friction & other resistances vibrating systems are subjected to damping to some

degree due to dissipation of energy. Damping has very little effect on natural frequency of the

system, and hence the calculations for natural frequencies are generally made on the basis of

no damping. Damping is of great importance in limiting the amplitude of oscillation at

resonance. The relative displacement configuration of the vibrating system for a particular

natural frequency is known as the eigen function in the continuous system. For every natural

frequency there would be a corresponding eigen function. The mode shape corresponding to

lowest natural frequency (i.e. the fundamental natural frequency) is called as the fundamental

(or the first) mode. The displacements at some points may be zero. These points are known as

nodes. Generally for higher modes the number of nodes increases. The mode shape changes

for different boundary conditions of the beam.

After determination of the first 3 natural frequencies (fundamental mode, 2nd

mode and

3rd

mode) and also the mode shapes we can now define the dynamic properties of the

experimented cantilever beam. This data will lead us to verify the "free undamped" vibration

equation of the cantilever beam, although the experiment is done on a forced vibration system

but we used this system in defining the common data with the "free undamped" system i.e

natural frequencies and mode shapes.

2- Objective of the Experiment:

The aim of the experiment is to analyze the forced vibrations of the continuous cantilever

beam, the phenomena of resonances, the phase of the vibration signal and to obtain the

fundamental natural frequency and damping ratio of the system, and compare the results with

theoretically calculated values.

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3- Experiment setup:

Fig. 1 (system setup)

The continuous system will be simulated using metal beam specimens and a force exciting

shaker, the sample required to be tested is bolted with the moving head of the shaker

simulating a cantilever beam fixation and an accelerometer is then waxed on the tested beam.

A data acquisition system (NI-PXI) is used to provide the shaker’s amplifier with the

excitation signal & acquire the signal from the accelerometer. Also a NI-PXI system with an

embedded controller will be used to process the acquired data and display the results on the

LABVIEW interface.

3-1 Used equipment:

1- Piezoelectric Accelerometer Miniature DeltaTron® Accelerometers Type 4519-002 (see

the appendix for the data sheet)

Specifications:

Voltage Sensitivity (@ 160Hz) 10 mV/g ±10%

Measuring Range ± 500g

Mounted Resonance Frequency kHz 45

Amplitude Response ±10% (typical)

Frequency range 0.5 to 20000 Hz

Shaker

Shaker

actuation signal

Amplifier

Accelerometer

NI-PXI system

Measured signal

from

accelerometer

To display

LABVIEW interface

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2- Signal force 100W Power Amplifier Model PA100E (see the appendix for the data sheet)

Specifications:

Output voltage 10V

Output current 10A

Signal to noise ratio: > -70dB

Power response –1dB 10kHz

Frequency response –3dB 20kHz

Distortion 20Hz to 10kHz <0.75% thd

3- Inertial Shaker Gearing & Watson V20, 1008 N (see the appendix for the data sheet)

4- NI measurements system that consists of :

a- NIPXI-019-1-1 Analyzer chassis

b- NI pxi-8186 Embedded Controller

c- NI PXI-4472B 8 Ch, 24-Bit, Vibration-Optimized Dynamic Signal Acquisition

Module

d- NI PXI-6733 High-Speed Analog Output -- 1 MS/s, 16-Bit, 8 Channels

3-2 Accelerometer fixation:

The Miniature DeltaTron® Accelerometer Type 4519-002 was sufficient to capture the

required signal within the expected amplitude & frequency range also as it weighs only 1.5 g

the accelerometer will not act as a lumped mass added on the beam. The accelerometer was

fixed on a certain position on each beam sample to avoid matching the nodes of the first 3

modes.

Fig. 2 (fixation of the accelerometer on the beam specimen using wax)

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3-3 Beam specimens:

The following different beam specimens were used as beam cantilevers:

Specimens Width(W)

[cm]

Thickness(T)

[mm]

Length(L)

[mm] Material

Weight

[grams]

Steel ruler 2.65 0.5 320 Steel 32.8

Aluminum

beam 1 1.75 0.77 320 Aluminum 9.41

Aluminum

beam 2 1.85 1.8 320 Aluminum 23.41

Aluminum

beam 3 4.3 1.7 320 Aluminum 58.2

Table 1 (Materials & dimensions of the specimens)

Fig. 3 (beam specimens with different materials and dimensions)

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3-4 Data acquisition and display:

Fig. 4 (Data acquisition system)

The NI-PXI data acquisition system receives voltage signal from the accelerometer and using

its embedded controller with the aid of software the data can be analyzed in frequency

domain (i.e., using FFT) and the data can displayed on a screen. Also this system provides the

amplifier with excitation signal through one of the analog outputs.

Fig. 5 (The signal amplifier used for the shaker)

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Fig. 6 (software interface page for selecting the excitation output parameters)

Fig. 7 (software interface page for selecting the data display)

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4- Experiment Procedure:

1- Choose a beam of a particular material (steel or aluminum), dimensions (L, W, T)

2- bolt one end of the beams on the shaker as a cantilever beam support (see Fig. 2)

3- Place an accelerometer (using wax) at the predetermined place on the cantilever

beam, to measure the forced vibration response (acceleration)

4- Make a proper connection of accelerometer with data acquisition card in the NI-PXI

system to capture the vibration data.

5- Make a proper connection of the shaker with the amplifier and from the amplifier to

the analog output of the NI-PXI data acquisition card

6- In the LABVIEW interface adjust the output settings to generate a uniform white

noise from the shaker.

7- Start the experiment by giving the forced signal to the exciter and allow the beam to

force vibrate.

8- Display the results in frequency domain (the designed LABVIEW code performs a

FFT over the captured signal & displays the results in frequency domain)

9- Export the results to an excel sheet

10- Repeat the whole experiment for different material, dimensions

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5- Results:

The results were obtained for different beam specimens, the following plots shows the

frequency vs. magnitude behavior for each of the experimented specimen.

Fig. 8 (frequency vs. magnitude Steel ruler Beam1)

Fig. 9(frequency vs. magnitude Aluminum beam 2)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 100 200 300 400 500 600

Mag

nit

ud

e

Frequency [Hz]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 100 200 300 400 500 600

Mag

nit

ud

e

Frequency [Hz]

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Fig.10(frequency vs. magnitude Aluminum beam 3)

Fig.11 (frequency vs. magnitude Aluminum beam 4)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600

Mag

nit

ud

e

Frequency [Hz]

0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 400 500 600

Mag

nit

ud

e

Frequency [Hz]

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From the previous plots we can obtain the first three natural frequencies of each system to be

compared to the analytically calculate results

Specimens Fn1 [Hz] Fn2 [Hz] Fn3 [Hz]

Steel ruler 4 25 69

Aluminum beam 1 5 28 70

Aluminum beam 2 11 69 180

Aluminum beam 3 13 81 182

Table 2 (the first three natural frequencies of each beam specimen)

6- Observation:

Using the same obtained natural frequencies we excited each beam specimen to verify the

obtained values the following figures shows the different mode shapes for different

specimens.

Fig. 12(beam excited with the first natural frequency)

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Fig. 13(beam excited with the second natural frequency)

Fig. 14(beam excited with the third natural frequency)

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7- Verification:

From a previously performed calculations using Hamilton’s principle we can deduce the

frequency equation for a fixed-free constrained beam as follows:

1)()cos( LCoshL

Using MATLAB function fsolve to find βL in this nonlinear equation:

The following MATALB code was used to generate the solutions:

function result = beam(y)

result=((cos(y).*cosh(y))+1);

end

x=1:10; fsolve(@beam,x);

Fig. 15(MATLAB code execution)

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Solutions are:

(βL)1=1.8751 , (βL)2=4.6941 , (βL)3=7.8548 , (βL)4=10.9955 ……………….

The next step is to find the relation representing the natural frequencies of the beam function

in its physical properties.

m

EIf

2

2

I = moment of inertia of the beam’s section

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3bTI

E = modulus of elasticity of the beam’s material

(Esteel=200 Gpa, EAluminum=69 Gpa(29))

m = mass per unit length of the beam

Using all of the previous relations & substituting each beam specimen properties we can

calculate the first three natural frequencies for each.

Specimens Fn1 [Hz] Fn2 [Hz] Fn3 [Hz]

Steel ruler 4 25 69.9

Aluminum beam 1 4.4 27.7 77

Aluminum beam 2 10.3 64.6 181

Aluminum beam 3 14 88 160

3- Conclusion:

Comparing the calculated natural frequencies with the measured ones showed a great

resemblance for the steel specimen while in the other aluminum specimens using EAluminum as

69 Gpa showed a noticeable deviation from the measured values we suggest that the

used aluminum material properties wasn’t a correct assumption.

b

T