ERKEL Daniel Vibrating Beam Laboratory Report (Updated With Name of Labgroup on 07.04.2013)

7/29/2019 ERKEL Daniel Vibrating Beam Laboratory Report (Updated With Name of Labgroup on 07.04.2013)

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DEPARTMENT OF MECHANICAL ENGINEERING

CODE AND TITLE OF COURSEWORKCourse code:

MECH 3004

Title:

VIBRATING BEAM LAB

STUDENT NAME: ERKEL, DANIEL

DEGREE AND YEAR: EBF, 3rd YEAR

LAB GROUP: AM4

DATE OF LAB. SESSION: 08/03/2013

DATE COURSEWORK DUE FOR SUBMISSION: 26/04/2013

ACTUAL DATE OF SUBMISSION: 01/04/2012

LECTURERS NAME: DR YURIY SEMENOV

PERSONAL TUTORS NAME: DR KEVIN DRAKE

RECEIVED DATE AND INITIALS:

I confirm that this is all my own work (if submitted electronically, submission will be taken asconfirmation that this is your own work, and will also act as student signature)

Signed: Daniel Erkel


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Daniel Erkel Vibrating Beam

Contents

1 Introduction 21.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Experimental Procedure 32.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Results from the Experiment 53.1 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Analysis and Discussion of Results 74.1 Comparison with Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.1 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.1.2 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1.3 Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Changing the Material of the Beam and a Few Words on Earthquake Engineering . . 11

5 Conclusion 12

6 Appendix 156.1 Complete Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2 Excel Calculation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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Applied Mechanics - Report on Coursework

Vibrating Beam

Daniel Erkel, 3rd year, EBF

Abstract

The present report discusses the results of a laboratory experiment conducted on the subjectof vibration of beams. In the course of the experiment a suspended beam (with one end fixedand one end free) was excited using an actuator controlled from a computer. The vibrationwas recorded on the computer and analysed in this report. The first three normal modes of thebeam were found, with the corresponding natural frequencies and mode shapes. The experimentalresults were then compared with the results of a theoretical analysis, based on the Bernoulli-Eulerbeam-model. Good correlation was found with the greatest error still being small, only 5%. Thereport also discusses the position of the nodes found investigating each mode shape during theexperiment, and again finds good correlation with theory. Finally the effect of changing thematerial of the beam and the effect of adding mass is investigated and the possible use of thismethod for vibration control is discussed, with concluding remarks on the use of the method incivil engineering.

1 Introduction

1.1 Objectives

The laboratory experiment presented in the followings aimed at demonstrating the behaviour of

a vertically suspended beam excited by an actuator hence providing means for the comparison ofresults obtained from a theoretical analysis with experimental values. Changing the frequency ofexcitation, the beam vibrated in different normal modes (or a certain combination of amplitudes ofall modes, if the frequency did not match the natural frequency associated with a particular mode).Registering the displacement during the vibration using a laser distance measure connected to thecomputer made it possible to observe the frequency and amplitude of the vibration and plot thesevariables on the computer screen. The plots obtained this way were then used to determine thenatural frequencies of the beam and find normal modes. The experiment also showed how the mass,inertia and the material of the vibrating body affect its vibration. These objectives were formalisedin the three parts of the experiment:

Part I Determining the beams natural frequencies

Part II Assessing the mode shape associated with each natural frequency

Part III Altering the natural frequencies of the beam

1.2 Background

On a macroscopic scale, real life objects can be modeled as elastic continua [1]. In the case of simple(simple meant on a macroscopic scale) vertical structures, such as skyscrapers, the simple beam is amodel that is capable of well predicting the dynamic behaviour of the body under certain conditions.The model is useful for example for predicting the physical response given by the building to theexcitation produced by an earthquake. Equally useful is this model for giving a simple prediction

of how the structure of an airplane wing will behave, vibrating in turbulent air. For this reason,studying the vibration of beams is vital for engineers to understand the vibration of many complexstructures in real life.

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1.3 Theory

The formal and thorough presentation of the complete theoretical background of this experiment isomitted here, as this would, among others, include the derivation of equations used and this is beyondthe scope of the report. Also, the introduction of the entire underlying theoretical background is notrequired for understanding the observations made here. All vital equations and fragments of theoryare presented along with the calculations and in the discussion part of the report. The theory ofbeam bending can be found in (for example) Ref. [2], while the equations used here are derived inRef. [3] or in in Timoshenkos famous Vibration Problems in Engineering (Ref. [4]).

2 Experimental Procedure

2.1 Apparatus

Detailed diagrams showing the equipment used in the experiment can be found in the Appendix partof the present report. The dimensions of the beam are presented on the following diagram:

Figure 1: Diagram showing the beam used in the experiment (dimensions are given in mm)

The beam was fixed with clamps at two points, at the top, where the beam was suspended from,and at a point somewhat lower. The clamps used at the latter were connected to the actuator and astrain gauge. The actuator, controlled by an amplified signal coming from a computer, was a strongelectric motor, exciting the beam, inducing the vibration. Between the actuator and the clamps onthe beam, there was a strain gauge to measure the force exerted on the beam. A laser distancemeasuring device registered the amplitude and frequency of the vibration at two different pointsand fed back the data to the computer. This data was then plotted on the computer monitor. Thebeam was fixed in a cabinet with a glass door for added safety.

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2.2 Procedure

The signal controlling the vibration was sent to the actuator from a computer next to the cabinetwith the beam (a software named LabVIEW was used on the computer). The digital signal convertedto analogue (voltage) travelled through an amplifier and moved the electric motor in the actuator,which in turn induced the forced vibration of the beam. The vibration, as said before, was registeredusing laser distance measuring devices fixed at two points. The experiment consisted of three parts:

Part I Determining the beams natural frequencies

a) The dimensions of the beam recorded (this was given in a table provided as a handoutfor the experiment [5]), a frequency-sweep was carried out. The different frequencies andcorresponding amplitudes were registered on the computer on a frequency response plot,from which approximate values for the natural frequencies can be found.

b) To locate the natural frequencies more precisely (with an accuracy of1Hz), a number ofdifferent frequencies were chosen to stimulate the beam and the amplitudes were recordedfrom the computer, writing both down in an Excel table.

Part II Assessing the mode shape associated with each natural frequencya) The beam was simulated at each of the natural frequencies so that the associated mode

shape could be observed.

b) The position or positions of the nodes (on the mode shape) were found visually andobserved by touch.

Part III Altering the natural frequencies of the beam

a) An additional weight was attached with a clamp to the point where lower node of thesecond mode shape was found on the beam to observe how this affects the natural fre-quency.

b) A frequency-sweep similar to that in the first part of the experiment was performed bythe demonstrator the results of which were again registered on the computer. The resultswere then compared to the those obtained without the added weight.

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3 Results from the Experiment

3.1 Part I

a) The result of the first frequency-sweep is presented on the following two graphs:

Figure 2: Frequency response graph for the first frequency-sweep with no extra mass attached to thebeam

From the graph it can be observed, that the natural frequencies of the first three modes areapproximately 2Hz, 18Hz and 47Hz.

b) The beam was then stimulated at a number of different frequencies (chosen to be close to thenatural frequencies observed from the graph previously) and the corresponding amplitudes areshown in the following table:

Table 1: Frequency response

f (Hz) Amplitude (A/Aex)

First natural frequency2.7 32.8 6

2.9 203 8

Second natural frequency17 218 919 1.5

Third natural frequency46 1.247 1.4

47.5 1.648 1.3

49 1.4

The amplitudes are given as a ratio, normalized by the amplitude multiplied by the gain,

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a common form of representation (see Ref. [6]). It can be seen that the values previouslyobserved from the graph are close to the exact (determined with an accuracy of1Hz) values.

3.2 Part II

The mode shapes were observed and the nodes located by pushing a loosely held pen against thebeam and moving until a point was found where there was no vibration. The exact location of thesenodes are given in the following table (from [5]):

Table 2: Table showing the location of the nodes for the different modes

Mode Node positions (cm)

1st 2nd 1 N/A2 1483 92 162

3.3 Part III

In the third part of the experiment a mass was attached at the point of the (second) node of thesecond mode shape and a frequency-sweep was conducted again. The plot resulting from this ispresented below:

Figure 3: Frequency response graph for the first frequency-sweep with extra mass attached to thebeam

It can be seen from this plot that the natural frequencies of the system have decreased compared tothe beam with no mass.

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4 Analysis and Discussion of Results

4.1 Comparison with Theory

4.1.1 Part I

The natural frequency of the nth mode can be obtained using the following equation (from Ref. [3]):

n = (aL)2

EI

mL4

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(1)

EI is the flexural rigidity of the beam (with Ebeing the Youngs modulus and I the second momentof inertia of the beam) [2], m is the mass per unit length (also equal to A), L the length and finallyaL is the solution of the frequency equation (see Ref. [4]), the values of which are given in the nexttable with the results of the theoretical analysis. The density and Youngs modulus of the materialof the beam in the first set of calculations, aluminum, can be found in engineering handbooks anddatabases, such as The Engineering ToolBox1. The moment of inertia of the beam can be calculatedusing the following equation (from Ref. [2]):

I=bh3

12(2)

From this, the variables used in equation (1) are collected in the following table:

Table 3: Variables used in equation (1)

b (m) 0.0762h (m) 0.012L (m) 1.9I (m4) 1.10E-08E (GPa) 69 (kg/m3) 2700m (kg/m) 2.47

The aL and a calculated from this is given for the first three modes in the following table:

Table 4: Solutions to the frequency equation (see Ref. [4])

Mo de 1 Mo de 2 Mo de 3

aL 1.875 4.694 7.855a 0.987 2.471 4.134

The theoretical natural frequencies were calculated for the first three modes using the above values.These are compared with results obtained in the experiment:

1The Engineering ToolBox - www.engineeringtoolbox.com, downloaded: March 30, 2013

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Table 5: Comparison of natural frequencies obtained in the experiment and through theoreticalanalysis

f (Hz)

Mode Theoretical Experimental Error in percentage

1 2.71 2.9 6.84%2 17.01 18 5.81%3 47.64 47.5 0.29%

It can be seen that the error in all three three cases is below 10% and thus perceivable but notsignificant. In the case of the first and second modes, the difference is around 6-7%, whereas in thecase of the third mode it is less than 0.3%. There are several reasons for these differences. Firstlythe beam is modeled using the Bernoulli-Euler beam theory, which approximates the infinitesimallysmall differential element of the beam as a parallelogram and does not include rotational inertia. Thismodel however is known to break down at high frequencies, when the wiggling of the beam is toolarge, or in other words, the deformations at a microscopic scale are greater and shear deformation

effects are more dominant [7]. Here what is seen however is that the difference is greater for smallerfrequencies, therefore using the more accurate Timoshenko beam-theory could increase the accuracyof the results, but not significantly. Another possible source of error is related to the materialproperties chosen for the calculations, which may not be completely accurate, as the density orthe Youngs modulus of the type of aluminum used for the beam could be (in fact it is likely tobe) slightly different. The error from this however may be insignificant. The first one of the morelikely sources of error is the non-uniformity of the material. While it was assumed that the beamis uniform, this is very unlikely, in fact, it is physically impossible. Small impurities in the materialcan alter the behaviour of the beam. The second likely cause for the difference, and possibly themost important one, is that resulting from the damping being ignored. The beam here is assumed tovibrate freely with no damping. In reality however this is not the case. Firstly there is a structural

or hysteretic damping, that is in fact inversely proportional to the frequency, therefore it could wellexplain the decreasing difference [3]. Secondly, the actuator also has a certain internal damping asthe electric motor within the machine definitely has a damping coefficient. There can be viscousdamping, if there is a hydraulic component, or Coulomb damping if there is friction between themetallic parts within the actuator. Also, if the clamps did not hold completely tight, this could havecaused further damping. Finally errors in measurement and errors in the calculations have to betaken into consideration as well. The list of measurement errors ranges from errors that may haveoccurred measuring the length of the beam to errors in certain components of the equipment, suchas the laser distance measuring device or the strain gauge, Small errors may have also resulted fromtruncation of decimals in the calculations.

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4.1.2 Part II

The mode shapes were plotted using the following equation (see Ref. [4] also found in Ref. [5]):

(x) = A1

cos ax cosh ax

cos aL + cosh aL

sinaL + sinhaL(sinax sinhax)

(3)

Where A1 is an arbitrary constant that is set equal to 1 here. Using this equation, the followinggraph was plotted:

Figure 4: Mode shapes plotted using equation (3)

The equation (and consequently the plot) can be used to find the nodes analytically. These arecompared with experimental results in the following table:

Table 6: Comparison of the results for the location of the nodes from the experiment and thetheoretical analysis

Mode Node positions (cm) Error No de positions (cm) Error

1st 2nd

Theoretical Experimental Theoretical Experimental

1 N/A N/A2 148 149 0.68%3 92 95.5 3.80% 162 164.9 1.79%

Error in locating the nodes can have similar reasons as the error in the natural frequencies foundpreviously. In this case it is likely that the most significant contributing factor was the measurementerror, if the node was located manually (not using the laser distance measuring device).

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4.1.3 Part III

Comparing the plot resulting from the second frequency-sweep with the mass attached to the beamit is clearly visible that the first and the third modes natural frequencies decreased, while the secondis approximately the same. This seems obvious if one observes the equation of the natural frequency,where the mass is in the denominator of the fraction with the flexural rigidity (and hence the inertia)in the numerator and also considers the location of the added mass (the node of the second mode).Because the mass was placed at a point which is stationary in the second mode, it did not affectthat natural frequency, while it changed the other too.Adding mass to the vibrating structure provides an effective way to control the natural frequency ofit, and makes it possible to in a way tune the structure by affecting its inertia. This can be veryimportant if for example one knows the frequency of the vibration induced by the wind blowing ata building at a given speed (where the vortices shed by the building lead to its vibration) or thefrequency of the vibration induced by an earthquake. The method however does not eliminate theresonance, or significantly lower the amplitude (it can be seen from the two plots that the highestpeak of the three modes is approximately equal) therefore further safety measures are necessary.Nevertheless if one knows the average strength of earthquakes in a given area or the wind blowing,this method can help preventing the collapse of a building. A more advanced method is discussedin the next section.

Figure 5: Comparison of the two frequency-sweeps - the graph on the top is the one without themass, the one on the bottom is the sweep performed with the mass. The green lines are vertical andthe red arrows show the change in natural frequency in the case of the first and the third normalmodes

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4.2 Changing the Material of the Beam and a Few Words on Earthquake Engi-

neering

Before discussing more sophisticated methods to control the vibration of large structures such asbuildings, first the effect of changing the beams material is investigated. If the beam would be madeof steel, that would effect both the inertia of the beam (the mass) and the elasticity, by changing theYoungs modulus of the beam. Therefore the effect on the natural frequency is not straightforward,as the natural frequency is determined by an interplay between the two. The following table showsthe effect of changing the material on the natural frequency of the beam from aluminum to ASTM-A36 steel (the properties of which were obtained from The Engineering ToolBox once again):

Table 7: The effect of changing the material on the natural frequency of the beam

f(Hz)

Mode Steel Aluminum Error in percentage

1 2.72 2.71 0.17%

2 17.04 17.01 0.17%3 47.72 47.64 0.17%

It can be seen that the balance between inertia and elasticity is almost unaffected, with the differencebetween the results for natural frequencies obtained from theoretical analysis being an insignificant0.17%. However using a more advanced beam-theory (the Timoshenko beam-model [4]) or FiniteElement Analysis would probably yield different results. The position of the nodes do not change,as the equation describing the mode shapes is not affected by the flexural rigidity or the mass.As described in the previous section, adding extra mass provides means to control the naturalfrequency of a structure, building. Lowering the natural frequency can help to prevent a disastrousresonance catastrophe. A more sophisticated way of controlling the vibration of a building however

is using an active mass. An example of a similar system is that of the Roseau Tower, an offshorestructure, where the added mass is an underwater tank with porous boundaries, through which watercan move. Since it is underwater, the water in the tank does not add extra load to the structure, itonly affects it when the tower starts swaying. Also, by using water, there is an active control as thewater floating inside the tank also oscillates and can absorb the energy of the vibrating tower [8].Another example for this type of motion control is the tuned mass damper used in many buildings,such as the Shanghai World Finance Center. The active mass here is not fixed rigidly to the structure,but instead suspended from flexible joints, or cables in a frame inside the building. This systemvibrates with a phase difference compared to the buildings natural frequency and thus not onlyhelps tuning it through the extra added mass, but also absorbs the energy of the vibration [9]. Theimage on the next page shows the structure of the tuned mass damper used in the Shanghai World

Finance Center, and the building itself.

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Figure 6: Images showing the iconic Shanghai World Finance Center, often mockingly referred to asthe bottle opener (on the right) and the tuned mass damper built in the building, and its position onone of the floor-plans (on the left) - The images were downloaded from www.sh.chinanews.com.cn,

www.thetoptenz.net on March 29, 2013, the one with the floor plan is from Ref. [9])

5 Conclusion

The experiment successfully demonstrated the vibration of a suspended beam and helped the un-derstanding of the important concept of natural frequency, mode shapes, nodes and the control ofnatural frequency and vibration. Theoretical calculations corresponded well with experimental re-sults and showed that the simple Bernoulli-Euler beam-model can accurately model the behaviour ofsimple structures vibrating at the first few normal modes. A more accurate model may be necessaryat more complex structures, and in the case of large buildings Finite Element Analysis is what canbe sufficiently accurate. Overall the experiment was successful and the present report found good

correlation between theory and experimental results.

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References

[1] G. Genta, Vibration Dynamics and Control. Mechanical Engineering Series, Springer, 2009.

[2] J. M. Gere, Mechanics of Materials. Brooks/Cole - Thomson Learning, 6th ed., 2004.

[3] S. Rao, Mechanical Vibrations. Addison-Wesley, 1995.

[4] J. W. Weaver, S. Timoshenko, and D. Young, Vibration Problems in Engineering. A Wiley-Interscience publication, John Wiley & Sons, 1990.

[5] Dr Yuriy Semenov - University College London, E466: Vibrating Beam - Laboratory Handout.

[6] D. Seborg, D. Mellichamp, T. Edgar, and I. Francis J. Doyle, Process Dynamics and Control.John Wiley & Sons, 2010.

[7] B. Tongue, Principles Of Vibration 2E C. Oxford University Press, Incorporated, 2002.

[8] S. Chakrabarti, Handbook of Offshore Engineering. No. v. 1 in Handbook of Offshore Engineering,Elsevier, 2005.

[9] W. Shi, J. Shan, and X. Lu, Modal identification of shanghai world financial center both fromfree and ambient vibration response, Engineering Structures, vol. 36, pp. 1426, 2012. cited By(since 1996) 0.

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List of Figures

1 Diagram showing the beam used in the experiment (dimensions are given in mm) . . 32 Frequency response graph for the first frequency-sweep with no extra mass attached

to the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Frequency response graph for the first frequency-sweep with extra mass attached to

the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Mode shapes plotted using equation (3) . . . . . . . . . . . . . . . . . . . . . . . . . 95 Comparison of the two frequency-sweeps - the graph on the top is the one without

the mass, the one on the bottom is the sweep performed with the mass. The greenlines are vertical and the red arrows show the change in natural frequency in the caseof the first and the third normal modes . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 Images showing the iconic Shanghai World Finance Center, often mockingly referredto as the bottle opener (on the right) and the tuned mass damper built in the building,and its position on one of the floor-plans (on the left) - The images were downloadedfrom www.sh.chinanews.com.cn, www.thetoptenz.net on March 29, 2013, the one withthe floor plan is from Ref. [9]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Complete diagram showing all components of the system . . . . . . . . . . . . . . . . 15

List of Tables

1 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Table showing the location of the nodes for the different modes . . . . . . . . . . . . 63 Variables used in equation (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Solutions to the frequency equation (see Ref. [4]) . . . . . . . . . . . . . . . . . . . . 75 Comparison of natural frequencies obtained in the experiment and through theoretical

analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Comparison of the results for the location of the nodes from the experiment and the

theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 The effect of changing the material on the natural frequency of the beam . . . . . . 11

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6 Appendix

6.1 Complete Diagrams

Figure 7: Complete diagram showing all components of the system

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6.2 Excel Calculation Examples

(See next page)

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1

=1*(COS(H$3*$A3)-COSH(H$3*$A3)-((COS(H$2)+COSH(H$2))/(SIN(H$2)+SINH(H$2))*(SIN($A3*H$3)-SINH($A3*H$3))))




















=1*(COS(H$3*$A23)-COSH(H$3*$A23)-((COS(H$2)+COSH(H$2))/(SIN(H$2)+SINH(H$2))*(SIN($A23*H$3)-SINH($A23*H$3))))=1*(COS(H$3*$A24)-COSH(H$3*$A24)-((COS(H$2)+COSH(H$2))/(SIN(H$2)+SINH(H$2))*(SIN($A24*H$3)-SINH($A24*H$3))))








































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b (m) 0.0762

h (m) 0.012

L (m) 1.9

I (m^4) =(B2*B3^3)/(12)

E (GPa) 200

\rho (kg/m^3) 7800m (kg/m) =B7*B2*B3

Mode 1 Mode 2 Mode 3

aL 1.875 4.694 7.855

a =B11/$B$4 =C11/$B$4 =D11/$B$4

Mode Steel Aluminum Error in percentage

1 =(B11)^2*SQRT(($B$6*$B$5*10^9)/( 2.71424321071458 =ABS((B16-C16)/B16)

2 =(C11)^2*SQRT(($B$6*$B$5*10^9)/( 17.0110995684569 =ABS((B17-C17)/B17)

3 =(D11)^2*SQRT(($B$6*$B$5*10^9)/(47.636362865886 =ABS((B18-C18)/B18)

Variables used in equation (1)

omega

Documents

ERKEL Daniel Vibrating Beam Laboratory Report (Updated With Name of Labgroup on 07.04.2013)