TABLE OF CONTENTS

1 Introduction.................................................................................................................................................................3

2 Theory............................................................................................................................................................................ 3

3 Description of apparatus........................................................................................................................................6

4 Procedure of experiments.....................................................................................................................................6

5 Results, calculation and discussion...................................................................................................................7

5.1 Lightly damped translational oscillations:...........................................................................................7

5.2 Effect of damping..........................................................................................................................................10

5.3 Coupled translational and rotational oscillations results...........................................................11

6 Reference....................................................................................................................................................................12

7 Appendix.................................................................................................................................................................... 12

Abstract:

The main reason of this lab is to investigate how damping can affect to the frequency and using log decrement to find damping ratio, how neglecting mass of spring can affect to the error and behaviour of a system of coupled harmonic oscillator.

From the experiments, the theory is confirmed that in 1 DOF system there is only 1 natural frequency and in 2 DOF system, its motion can be expressed as a 2nd order differential equation which should have 2 natural frequencies. These 2 natural frequencies are translational frequency and rotational frequency, which show the behaviour of coupling.

The lab also shows the effect by varying the mass of 1 DOF system on the natural frequency and how a change in inertia would affect the rotational oscillations, and then follow-on effect of the beating phenomenon.

Data analysis using Matlab-Fast Fourier Transform is used in this lab as to illustrate how it can be used effectively to find natural frequencies in the system. However in few cases, it is hard to distinguish visually two frequencies on the frequency spectrum so the pulse graph method from the displacement vs time graph can be used to find period between each pulse then the beating frequency.

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1 INTRODUCTION

The main reason of this lab session is to provide some practical experience of harmonic oscillations and more specifically of the dynamic responses of a mass spring system that exhibits both translational and rotational modes of vibration.

The single Wilberforce pendulum, using in this experiment, is a pendulum that couples between longitudinal and torsional oscillations resulting in complete transfer of energy between translational and rotational harmonic motion. Two types of harmonic motions can be initiated by displacing the brass mass, or the pendulum bob that is equipped with a threaded crossbar extending on both sides with nuts mounted on it so its inertia can be adjusted.

In the first part of the experiment, we only concern about translational motion, which is one degree of freedom, to find its key parameters and study the effect of damping. So in this part, we need to determine the vertical positions of the masses at its equilibrium points.

In the 2nd part of the experiment, we concern both translational and rotational motion, which is two degree of freedom, to investigate beating behaviour and coupling between motions. So for this part, we need to determine both vertical positions and angular positions.

2 THEORY

2.1 ONE DEGREE OF FREEDOM

2.2 TW O

DEGREE OF FREEDOM Two degree of freedom system requires two independent coordinates to describe its motion. The coordinates can be explained by examining the following example:

Two coordinates are:

X1 is the displacement of mass M1 from its equilibrium position. X2 is the displacement of mass M2 from its equilibrium position.

In our case where the pendulum couples between translational and rotational oscillation, our two coordinates are the vertical and angular displacement.

2

It requires only one independent coordinate to describe the motion of the mass at many points of time. That coordinate is the displacement of mass from its equilibrium point. A simple oscillator is an example of one degree of freedom system:

2.3 EFFECT OF DAMPING Damping is produced when energy stored in the oscillation is dissipated, that slowing/decreasing the motion/amplitude gradually to its steady state. In our experiment, the damping effect was made by friction of air.

Critical damping cc is defined as the value of damping constant c for which the radical becomes 0:

( cc

2m )2

− km

=0

cc=2m√ km

=2√km=2m ωn

Then we can define Damping Ratio :

ζ= ccc

= c2mωn

The damping ratio determines the damping effect on a system:

Critically damped ( = 1) - The amplitude of oscillation will reach zero in a very short time. Underdamped (0 < < 1) – The amplitude of oscillation will decay slowly. Overdamped ( > 1) – There will be no oscillation.

2.4 LOGARITHMIC DECREMENT We have following formulae:

δ=1nln( x i

xi+1)

ζ= δ

√(2π )2+δ 2

2.5 COUPLED TRANSLATIONAL AND ROTATIONAL OSCILLATIONS: As above we have expression for translational oscillation’s natural frequency:

f n=12π √ k

m

In rotational oscillation, we have another expression for natural frequency:

f n=12π √ k

I

Where I = moment of inertia of mass.

3

2.6 MOMENT OF INERTIA OF MASS: The pendulum bob, used in the 3rd experiment, is equipped with a threaded crossbar extending on both sides with nuts mounted on it so its inertia can be adjusted. The pendulum bob rotated about z-axis. Formulae used to find inertia in this case (axis of rotation: z-axis)

Circular hoop on top:

I z=mr2

Solid cylinder (Pendulum bob):

I z=m r2

2

Solid cylinder (Nuts):

I y/ z=mr2

4+ m L2

12

Solid rod (Crossbar):

I y/ z=m L2

12

2.7 PARALLEL AXIS THEOREM : I z=∑ I cm+m d2

Where Icm = Inertia around z-axis of the object through its centre of mass; d = perpendicular distance between object’s centre of mass and axis of rotation

2.8 BEATING FREQUENCY When two waveforms with different frequencies interact on each other, there will be alternating constructive and destructive interference. So the resulting waveform will establish its own new frequency. This can be called as beating behaviour:

We can use following 2 DOF system as an example:

After solving for X1(t) and X2(t), we have:

X1=2 A cos [ ( ω1+ω2 )2

t ] cos[ (ω1−ω2 )2

t ]X2=2 A sin [ (ω1+ω2 )

2t ]sin [ (ω1−ω2 )

2t ]

Where

4

(ω1+ω2 )2

=Vibration frequency

(ω1−ω2 )2

=beat frequency

3 DESCRIPTION OF APPARATUS

The Wilberforce pendulum apparatus used in the experiments is supplied by PASCO (model ME-8091). The mass is first attached to the end of the spring, suspended from a clamp system. Then we can displace the mass and make it oscillate. The vertical position, velocity and acceleration of the Wilberforce pendulum are measured with a Motion sensor placed below the pendulum. A laser and a laser switch measure the angular speed in the torsional mode as the spokes of the wheel break the laser beam. A force sensor attached to the end of the spring measures the spring force as the pendulum oscillates. All the data from sensors is then captured by the Data studio software. To vary the mass, additional nuts were used with or without the initial mass to have different sets of data. For damping experiment, additional paper plate was added to the bottom of the mass in the place of photogate wheel to increase the surface area and make the damping effect due to air resistance more effectively. For coupling oscillations, crossbar pendulum bob was used with known distance between the nuts and pendulum’s centre of mass to vary the frequency of the oscillation.

4 PROCEDURE OF EXPERIMENTS

4.1 LIGHTLY DAMPED TRANSLATION OSCILLATIONS The first experiment is about 1-DOF system which is measuring the vertical position of the mass. After hanging the brass bob, displacing it vertically downward to set the system into translational oscillation. Force sensor and Motion sensor will then record and send the data to Data studio. To find stiffness constant, the spring was pulled for a known length (10cm) and would increase by 10cm after each 10s and the force sensor measured the force. The mass was varied to have different sets of data and graphs to have more accuracy in determining spring stiffness. The natural frequency for each mass was found by using Matlab-Fast Fourier Transform (FFT).

4.2 EFFECT OF DAMPING Attaching a paper plate to the end of the mass and it still was a 1 DOF system. Procedure in part (a) was repeated but only with the brass bob then collecting the data from Data Studio. Frequency of the damped system and its damping ratio via log decrement were then found by using Matlab-FFT.

4.3 COUPLED TRANSLATIONAL AND ROTATIONAL OSCILLATIONS: The pendulum bob with crossbar was used in this part. The distance between the nuts and the centre of mass was first recorded and then we can estimate inertia from that. The bob was displaced

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and the system was set in motion, along with force sensor and motion sensor, laser sensor would help to record the angular velocity and then send data to Data Studio. By varying the distance between the nuts and centre of mass, beating behaviour could be observed. From Matlab-FFT, we can see how the dominant frequencies in the spectra of the recorded signals vary with inertia. From that, we can estimate the beating frequency and the associated pair of natural frequencies that result in this.

5 RESULTS, CALCULATION AND DISCUSSION

5.1 LIGHTLY DAMPED TRANSLATIONAL OSCILLATIONS:

5.1.1 Determine natural frequency from Data Studio From Matlab-FFT, we can determine the natural frequency of oscillation with brass bob attached only (234g) that is 0.8398 Hz. Refer Appendix.

5.1.2 Determine stiffness of the spring (k) and compare with estimate obtained using the force sensor

0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

f(x) = 7.1379776205543 x + 0.0064547585926

Weight-extension

Extension (m)

Wei

ght

(N)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-2

0

2

4

f(x) = − 6.67257142857143 x + 2.70406666666667

Force gauge reading vs extension

Extension (m)

Forc

e (N

)

So we have:

Spring stiffness calculated ¿ load /extension kmass=7.138N /m

Spring stiffness calculated ¿ Force sensor k sensor=6.6726N /m

We can say that they are quite close to each other.

5.1.3 Natural frequency of oscillation for each mass used in part ii Natural frequency (for brass bob and 2 nuts) = 0.752 Hz

Natural frequency (for brass bob and 3 nuts) = 0.6738 Hz and 1.348 Hz

Natural frequency (for 2 nuts and 3 nuts) = 0.8984 Hz

5.1.4 Compare the experimentally derived frequencies with the ones estimated theoretically

Mass (kg) Natural Frequency (Hz) Percentage Difference (%)6

Mass / extension

Force Sensor

Natural Frequencyfrom Matlab

Mass / extension

Force Sensor

0.2 0.9508 0.9193 0.8984 -5.83 -2.330.234 0.8790 0.8499 0.8398 -4.67 -1.200.2952 0.7826 0.7567 0.752 -4.07 -0.62Calculation of Natural Frequency in each case and their difference using Kmass , Ksensor

Set of data from brass bob and 3 nuts is decided not to be used as there are 2 natural frequencies, which show that it was 2-DOF system. However, in this experiment we only concern about 1-DOF system.

To find Natural frequency (Hz) and its percentage difference, following formula is used:

f n=12π √ k i

m j

for j=1,2,3

5.1.5 Correct for the mass of spring and estimate the error introduced by neglecting it

Mass of the spring = 48.5g = 0.0485kg

However, we need to consider the effective mass of the spring because not all the whole length of the spring would contribute to the oscillation.

So mass of the spring:

M spring=0.04853

We have new set of data:

Mass (g) Mass (kg) Weight (N) Length (cm) Extension (m)No mass + Spring 16.17 0.01617 0.158595 11.5 02 Nuts + 3 Nuts+ Spring 216.17 0.21617 2.120595 38.5 0.27Brass Bob + Spring 250.17 0.25017 2.454135 44 0.325Brass Bob + 2 Nuts + Spring 311.37 0.31137 3.054507 51.7 0.402Brass Bob + 3 Nuts + Spring 388.97 0.38897 3.815763 62.8 0.513

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0.25 0.3 0.35 0.4 0.45 0.5 0.55012345

f(x) = 7.06474492832026 x + 0.194308789559102

Weight (with spring) - extension

Extension (m)

Wei

ght

(N)

We have new spring stiffness constant:

Spring stiffness calculated ¿ load with spring∧extension

k mass+ spring=7.0647N /m

Mass (kg)

Natural Frequency (Hz) Percentage Difference (%)

Mass + Spring/ extension

Force Sensor + spring

Natural Frequencyfrom Matlab

Mass+spring / extension

Force Sensor +spring

0.216167 0.9099 0.8842 0.8984 1.28 1.580.250167 0.8458 0.8220 0.8398 0.71 2.120.311367 0.7581 0.7368 0.752 0.81 2.03

Calculation of Natural Frequency in each case and their difference using Kmass+spring , Ksensor

Using k3 and mass of the spring added to the mass, natural frequencies for each case were recalculated. Then new percentage differences were found.

f n=12π √ k i

m j+mspring

3

for j=1,2,3

Discussion For Mass vs Extension:

Load(kg)

Load+spring(kg)

fn (Hz)Mass/extension

Percentage difference (%)

fn (Hz)Mass+Spring/extension

Percentage difference(%)

0.2 0.216167 0.9508 -5.83 0.9099 1.280.234 0.250167 0.8790 -4.67 0.8458 0.710.2952 0.311367 0.7826 -4.07 0.7581 0.81

In any cases, using Kmass or Kmass+spring, the percentage difference still doesn’t exceed 6% so we can say the experiment agrees with theory as experimentally derived frequencies are all close to estimated theoretically frequencies.8

For using Kmass+spring, the percentage difference is noticeably smaller than the values from using Kmass only. We can say that as considering the mass of spring is negligible can contribute a big error in our calculation.

For using force sensor:

Load(kg)

Load+ spring(kg)

fn (Hz)ForceSensor

Percentage difference(%)

fn (Hz)Force Sensor + Spring

Percentage difference(%)

0.2 0.216167 0.9193 -2.33 0.8842 1.580.234 0.250167 0.8499 -1.20 0.8220 2.120.2952 0.311367 0.7567 -0.62 0.7368 2.03To calculate natural frequency from data from force sensor, only spring stiffness Ksensor was used. In both case, there is no percentage difference that exceed 2.5% so again this confirms theory.

However as considering mass of spring to be not negligible, in overall the percentage difference turned out to be bigger. This can be due to error in from using formula of effective mass, as force sensor would have a better accuracy.

5.2 EFFECT OF DAMPING

5.2.1 Frequency of damped system (using 234g Brass bob) Refer to appendix for graph

f ndamped=0.8105Hz

From above we have fn for lightly damped system:

f nslightly damped=0.8398Hz

5.2.2 Damping ratio by applying the log decrement method Refer to appendix

Damping ratio calculated with the mean of , = 0.012666

Damping ratio calculated with the maximum of n, = 0.0085716

Discussion

By applying log decrement method on the data we got from the 1st experiment for the brass bob only, we can find damping ratio for the slightly damped system:

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Damping ratio calculated with the mean of , = 0.0024463

Damping ratio calculated with the maximum of n, = 0.0022418

From above, it is believed that the plate contributed to the damping effect that leads to the higher in damping ratio (0.012666 > 0.0024463). This also confirms theory: as higher the damping ratio, frequency would get lower.

Percentagedifference of damping ratio=0.012666−0.00244630.012666

×100

Percentagedifference of dampingratio=80.7%

Percentagedifference of frequency=0.8398−0.81050.8398

×100=3.48%

However apart from the big difference percentage in damping ratio (80.7%), the difference between two frequencies as shown is so small (3.48%) so we can conclude that:

It is reasonable to consider damping in the slightly damped system is negligible as it would not result in a big error in calculation but this cannot be applied on the damped system as the percentage difference of damping ratio between them are too big.

5.3 COUPLED TRANSLATIONAL AND ROTATIONAL OSCILLATIONS RESULTS 5.3.1 Result and calculation of inertia

Dimensions

Components Mass (kg)

Radius (m) Length (m)Distance from centre (m)Inner Centre Outer

Cylinder Bob 0.234 0.015 0.037 0 0 0Cylinder Mount 0.011 0.0045 0.019 0 0 0

Cylindrical Masses0.0055 0.0045 0.01 0.02675 0.03500 0.045875

Photogate Wheel 0.0075 0.05 - 0 0 0Crossbar 0.006 - 0.01 0 0 0

Mass Moment of Inertia (by axis Z)

Components Mass Moment of Inertia (kg/m2)Cylinder Bob 2.6325×10-5

Cylinder Mount 1.11375×10-7

Cylindrical Mass 7.3677×10-8

10

Photogate Wheel 1.875×10-5Crossbar 0.5×10-7

Mass Moment of Inertia with Parallel Axis Theorem for Cylindrical Mass

Cylindrical Mass Inner Centre OuterMass Moment of Inertia (kg/m2) 4.0093×10-6 6.8112×10-5 1.1649×10-5

Total Mass Moment of Inertia

Cylindrical Mass Inner Centre OuterMass Moment of Inertia (kg/m2) 5.8205×10-5 6.3809×10-5 7.3483×10-5

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Estimation of Natural Translational, Rotational and Beating Frequencies

Position of Cylindrical Mass

Translational Frequencies, F1 (Hz)

Rotational Frequencies, F2 (Hz)

Beating Frequencies (Hz)

Inner 0.8008 0.7812 0.0196Centre 0.8008 0.7496045 0.0511955Outer 0.8008 0.694225982 0.106574018

Discussion:

Referring to appendix for frequency spectrum of nuts at centre and outer position, only one frequency can be observed. So to acquire beating frequency, period between each peak of “beating” are marked (as in appendix). Mean of period are later found and beating frequency is:

1average period

Then 2nd frequency can be found by: f 1−beating frequency

Behaviour of 2 DOF system can be easier to observe when the nuts at nearest to the centre of pendulum and as it goes toward the free end, it would be harder to observe.

From above, we can see as the cylinder masses (nuts) going toward the free end, the mass moment of inertia of the system increases. That would result in a lower rotational frequency (conservation of angular momentum). However there is no change in translational frequency which support the theory that vertical translational oscillation is independent on inertia of system.

For the nuts at inner position (closest to the centre of pendulum), we have low inertia resulting in higher rotational frequency and no change in translational frequency. The associated frequency spectrum illustrates 2 distinct peaks, with 1 being more dominant than the other. So the beating frequency in this case can be easily observed.

For the nuts at centre and outer position (further from the centre of pendulum), we have higher inertia resulting in low rotational frequency. So the associated frequency spectrum can not illustrate 2 distinct peaks where f1 is so close to f2. So a “pulse count” is used referring to the displacement vs time of each case.

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6 CONCLUSION

From the experiments, the theory is confirmed that the natural translation frequency of the system is dependent only on stiffness (k) and the mass (m) and rotational translation frequency is dependent on the inertia (I) and mass (m).

Also, the frequency of oscillation decreases as the damping effect increases. From the damping coefficient, one can decide if damping should be neglected.

As the moment of inertia of the system increases, the rotation frequency decreases. Also, the phenomenon of beating is able to observed when their two frequencies are very close to each other, causing the beating period to be high enough to be noticeable.

7 REFERENCE

http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

Common Moments of inertia of

http://content.yudu.com/Library/A1zyct/PascoPhysicsampEngin/resources/151.html

PASCO’s introduction about Wilberforce pendulum

http://hyperphysics.phy-astr.gsu.edu/hbase/sound/beat.html

Introduction of beating

Dr.Balabani ‘s lecture notes

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8 APPENDIX

8.1 LIGHTLY DAMPED TRANSLATIONAL OSCILLATIONS

Different masses were used and each of their mass was measured as below:

Brass Bob: 234g 2 Nuts: 61.2g 3 Nuts: 138.8g

14

Natural frequency (for the brass bob only, 234g) is 0.8398 Hz

Mass (g) Mass (kg) Weight (N) Length (cm)

Extension (m)

No mass 0 0 0 11.5 02 Nuts + 3 Nuts 200 0.2 1.962 38.5 0.27Brass Bob 234 0.234 2.29554 44 0.325Brass Bob + 2 Nuts 295.2 0.2952 2.895912 51.7 0.402Brass Bob + 3 Nuts 372.8 0.3728 3.657168 62.8 0.513

Extension after each 10s (m) Average force at each 10s section (N)

0.1 2

0.2 1.415

0.3 0.713

0.4 0.011

0.5 -0.614

Natural frequency of oscillation for each mass used in part ii

For brass bob and 2 nuts (295.2g):

15

For brass bob and 3 nuts (372.8g):

16

Natural frequency (for brass bob and 2 nuts) = 0.752 Hz

17

For 2 nuts and 3 nuts (200g):

18

Natural frequency (for brass bob and 3 nuts) = 0.6738 Hz and 1.348 Hz

19

%---------------------------------%------------Ex1A-----------------%--------------------------------- clearclcMatrix1A = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/1a Position.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix1A(:,1));x = transpose(Matrix1A(:,2));fs = 10;L=length(t);x=x-mean(x); % Plot the graphfigure(1),plot(t,x,'r');

20

Natural frequency (for 2 nuts and 3 nuts) =

0.8984 Hz

figure(1),title('Displacment vs time (ex1A)');figure(1),ylabel('Displacement(m)');figure(1),xlabel('Time(s)'); % Calculate FFTNFFT=2^nextpow2(L);X1 = fft(x,NFFT)/L;X1= 2*abs(X1(1:NFFT/2+1)); %generate frequency axis;f=fs/2*linspace(0,1,NFFT/2+1); %show this graphicallyfigure(2),plot(f,X1,'b');figure(2),title('Frequency spectrum');figure(2),xlabel('Frequency[Hz]');figure(2),ylabel('x(f)');

%---------------------------------%------------Ex1C(brass+2)-----------------%--------------------------------- clearclcMatrix1C = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/1c brass+2 Position.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix1C(:,1));x = transpose(Matrix1C(:,2));fs = 10; L=length(t);x=x-mean(x); % Plot the graphfigure(1),plot(t,x,'r');figure(1),title('Displacment vs time (ex1C-Brass+2 Nuts)');figure(1),ylabel('Displacement(m)');figure(1),xlabel('Time(s)'); % Calculate FFTNFFT=2^nextpow2(L);X1 = fft(x,NFFT)/L;X1= 2*abs(X1(1:NFFT/2+1)); %generate frequency axis;f=fs/2*linspace(0,1,NFFT/2+1); %show this graphicallyfigure(2),plot(f,X1,'b');figure(2),title('Frequency spectrum');figure(2),xlabel('Frequency[Hz]');figure(2),ylabel('x(f)');

%---------------------------------%------------Ex1C(Brass+3Nuts)-----------------%---------------------------------

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clearclcMatrix1C = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/1c brass+3 Position.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix1C(:,1));x = transpose(Matrix1C(:,2));fs = 10; L=length(t);x=x-mean(x); % Plot the graphfigure(1),plot(t,x,'r');figure(1),title('Displacment vs time (ex1C-Brass+3Nuts)');figure(1),ylabel('Displacement(m)');figure(1),xlabel('Time(s)'); % Calculate FFTNFFT=2^nextpow2(L);X1 = fft(x,NFFT)/L;X1= 2*abs(X1(1:NFFT/2+1)); %generate frequency axis;f=fs/2*linspace(0,1,NFFT/2+1); %show this graphicallyfigure(2),plot(f,X1,'b');figure(2),title('Frequency spectrum');figure(2),xlabel('Frequency[Hz]');figure(2),ylabel('x(f)');

%---------------------------------%------------Ex1C(2+3)-----------------%--------------------------------- clearclcMatrix1C = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/1c +2+3 Position.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix1C(:,1));x = transpose(Matrix1C(:,2));fs = 10; L=length(t);x=x-mean(x); % Plot the graphfigure(1),plot(t,x,'r');figure(1),title('Displacment vs time (ex1C-2+3 Nuts)');figure(1),ylabel('Displacement(m)');figure(1),xlabel('Time(s)'); % Calculate FFTNFFT=2^nextpow2(L);X1 = fft(x,NFFT)/L;X1= 2*abs(X1(1:NFFT/2+1));

22

%generate frequency axis;f=fs/2*linspace(0,1,NFFT/2+1); %show this graphicallyfigure(2),plot(f,X1,'b');figure(2),title('Frequency spectrum');figure(2),xlabel('Frequency[Hz]');figure(2),ylabel('x(f)');

23

8.2 EFFECT OF DAMPING 8.2.1 Damped system

24

f ndamped=0.8105Hz

Damping ratio by applying the log decrement method

Damping ratio calculated with the mean of , = 0.012666

Damping ratio calculated with the maximum of n, = 0.0085716

8.2.1 Lighltly damped system

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%--------------------------------------------------------------------------%--------- Ex2---------%%--------------------------------------------------------------------------clear clc Matrix2 = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/2 Position.txt','\t',2,0);% creating a matrix from datat2 = transpose(Matrix2(:,1)); x2= transpose(Matrix2(:,2));Fs=10; %The sampling rate is 10hzL2=length(t2);x2=x2-mean(x2); % Plot the graphfigure(1),plot(t2,x2,'r');figure(1),title('Displacment vs time (ex2)')figure(1),ylabel('Displacement[m]');figure(1),xlabel('Time[s]'); % Calculate FFTNFFT2 = 2^nextpow2(L2);X2 = fft( x2 , NFFT2 ) / L2;X2 = 2*abs(X2(1:NFFT2/2+1));

26

%generate frequency axis;f2 = Fs/2*linspace(0,1,NFFT2/2+1); %Displaying the results graphicallyfigure(2),plot(f2,X2,'b');figure(2),title('Frequency Spectrum');figure(2),xlabel('Frequency[Hz]');figure(2),ylabel('X(f)'); %--------------------------------------------------------------------------%Calculating damping ratio%-------------------------------------------------------------------------- [pks,locs] = findpeaks(x2); %locates the peaks and their locationsnum_rows = size(pks,2); %This function finds the total number of maxium locatedlimit = num_rows - 1; %This function sets the limit for the loop for n=1:(limit) %Calculating Logarithim Decrement step1 = 1/n; % 1/n step2 = pks(1)/pks(n+1); % x1/x(1+n) delta(n) = step1*log(step2); % (1/n)log(x1/x(1+n)end %Calculating damping ratio using Logarithim Decrement ratio meanmean_delta = mean(delta); %Calculating the mean of Logarithim Decrementstep4 = (2*3.14)^2; %2pi^2 step5 = mean_delta^2; %Logarithim Decrement^2step6 = step4+step5; %2pi^2+Logarithim Decrement^2step7 = sqrt(step6); % Square root of 2pi^2+Logarithim Decrement^2 Damp_ratio_using_mean_delta = mean_delta/step7; % Logarithim Decrement / (Square root of 2pi^2+Logarithim Decrement^2) %Calculating damping ratio using Logarithim Decrement at last position last_delta_pos = size(delta,2) ; %This function calculates maxium entries of Logarithim Decrementstep4_1 = (2*3.14)^2; %2pi^2 step5_1 = delta(last_delta_pos)^2; %Logarithim Decrement^2 step6_1 = step4_1 + step5_1; %2pi^2+Logarithim Decrement^2step7_1 = sqrt(step6_1); % Square root of 2pi^2+Logarithim Decrement^2 Damp_ratio = delta(last_delta_pos)/step7_1; % Logarithim Decrement / (Square root of 2pi^2+Logarithim Decrement^2) % Show the valuesfigure(3),text(0.1,0.85,'Exercise 2');figure(3),text(0.1,0.80 ,['Damping ratio using mean of Logarithim Decrement ratio ' num2str(Damp_ratio_using_mean_delta)]);figure(3),text(0.1,0.75,['Damping ratio ' num2str(Damp_ratio)]); %--------------------------------------------------------------------------%--------- Ex2(lightlydamped)---------%%--------------------------------------------------------------------------clear clc

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Matrix2 = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/1a Position.txt','\t',2,0);% creating a matrix from datat2 = transpose(Matrix2(:,1)); x2= transpose(Matrix2(:,2));Fs=10; %The sampling rate is 10hzL2=length(t2);x2=x2-mean(x2); % Plot the graphfigure(1),plot(t2,x2,'r');figure(1),title('Displacment vs time (ex2-lightlydamped)')figure(1),ylabel('Displacement[m]');figure(1),xlabel('Time[s]'); % Calculate FFTNFFT2 = 2^nextpow2(L2);X2 = fft( x2 , NFFT2 ) / L2;X2 = 2*abs(X2(1:NFFT2/2+1)); %generate frequency axis;f2 = Fs/2*linspace(0,1,NFFT2/2+1); %Displaying the results graphicallyfigure(2),plot(f2,X2,'b');figure(2),title('Frequency Spectrum');figure(2),xlabel('Frequency[Hz]');figure(2),ylabel('X(f)'); %--------------------------------------------------------------------------%Calculating damping ratio%-------------------------------------------------------------------------- [pks,locs] = findpeaks(x2); %locates the peaks and their locationsnum_rows = size(pks,2); %This function finds the total number of maxium locatedlimit = num_rows - 1; %This function sets the limit for the loop for n=1:(limit) %Calculating Logarithim Decrement step1 = 1/n; % 1/n step2 = pks(1)/pks(n+1); % x1/x(1+n) delta(n) = step1*log(step2); % (1/n)log(x1/x(1+n)end %Calculating damping ratio using Logarithim Decrement ratio meanmean_delta = mean(delta); %Calculating the mean of Logarithim Decrementstep4 = (2*3.14)^2; %2pi^2 step5 = mean_delta^2; %Logarithim Decrement^2step6 = step4+step5; %2pi^2+Logarithim Decrement^2step7 = sqrt(step6); % Square root of 2pi^2+Logarithim Decrement^2 Damp_ratio_using_mean_delta = mean_delta/step7; % Logarithim Decrement / (Square root of 2pi^2+Logarithim Decrement^2) %Calculating damping ratio using Logarithim Decrement at last position

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last_delta_pos = size(delta,2) ; %This function calculates maxium entries of Logarithim Decrementstep4_1 = (2*3.14)^2; %2pi^2 step5_1 = delta(last_delta_pos)^2; %Logarithim Decrement^2 step6_1 = step4_1 + step5_1; %2pi^2+Logarithim Decrement^2step7_1 = sqrt(step6_1); % Square root of 2pi^2+Logarithim Decrement^2 Damp_ratio = delta(last_delta_pos)/step7_1; % Logarithim Decrement / (Square root of 2pi^2+Logarithim Decrement^2) % Show the valuesfigure(3),text(0.1,0.85,'Exercise 2 lightly damped');figure(3),text(0.1,0.80 ,['Damping ratio using mean of Logarithim Decrement ratio ' num2str(Damp_ratio_using_mean_delta)]);figure(3),text(0.1,0.75,['Damping ratio ' num2str(Damp_ratio)]);

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8.3 COUPLED TRANSLATIONAL AND ROTATIONAL OSCILLATIONS

Inner Position:

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Centre position:

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Outer Position

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%---------------------------------%------------Ex3(inner-displacement)-----------------%--------------------------------- clearclcMatrix3b = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/3b Inner Position.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix3b(:,1));x = transpose(Matrix3b(:,2));fs = 10; L=length(t);x=x-mean(x); % Plot the graphfigure(1),plot(t,x,'r');figure(1),title('Displacment vs time (ex3B-inner)');figure(1),ylabel('Displacement(m)');figure(1),xlabel('Time(s)'); % Calculate FFTNFFT=2^nextpow2(L);X1 = fft(x,NFFT)/L;X1= 2*abs(X1(1:NFFT/2+1)); %generate frequency axis;f=fs/2*linspace(0,1,NFFT/2+1); %show this graphicallyfigure(2),plot(f,X1,'b');figure(2),title('Frequency spectrum');

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figure(2),xlabel('Frequency[Hz]');figure(2),ylabel('x(f)'); %---------------------------------%------------Ex3(inner-angularvelocity)-----------------%--------------------------------- clearclcMatrix3b = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/3b inner Period.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix3b(:,1));p = transpose(Matrix3b(:,2));fs = 10; num=size(p,2); for n=1:(num) w(n)=2*3.14/p(n);end L=length(t); % Plot the graphfigure(1),plot(t,w,'g');figure(1),title('Angular velocity vs time (ex3B-inner)');figure(1),ylabel('Angular velocity(rad/s)');figure(1),xlabel('Time(s)'); %---------------------------------%------------Ex3(centre-displacement)-----------------%--------------------------------- clearclcMatrix3b = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/3b Centre Position.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix3b(:,1));x = transpose(Matrix3b(:,2));fs = 10; L=length(t);x=x-mean(x); % Plot the graphfigure(1),plot(t,x,'r');figure(1),title('Displacment vs time (ex3B-centre)');figure(1),ylabel('Displacement(m)');figure(1),xlabel('Time(s)'); % Calculate FFTNFFT=2^nextpow2(L);X1 = fft(x,NFFT)/L;X1= 2*abs(X1(1:NFFT/2+1)); %generate frequency axis;f=fs/2*linspace(0,1,NFFT/2+1); %show this graphically

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figure(2),plot(f,X1,'b');figure(2),title('Frequency spectrum');figure(2),xlabel('Frequency[Hz]');figure(2),ylabel('x(f)'); %---------------------------------%------------Ex3(centre-angularvelocity)-----------------%--------------------------------- clearclcMatrix3b = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/3b centre Period.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix3b(:,1));p = transpose(Matrix3b(:,2));fs = 10; num=size(p,2); for n=1:(num) w(n)=2*3.14/p(n);end L=length(t); % Plot the graphfigure(1),plot(t,w,'g');figure(1),title('Angular velocity vs time (ex3B-centre)');figure(1),ylabel('Angular velocity(rad/s)');figure(1),xlabel('Time(s)');

%---------------------------------%------------Ex3(outer-displacement)-----------------%--------------------------------- clearclcMatrix3b = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/3b outer Position.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix3b(:,1));x = transpose(Matrix3b(:,2));fs = 10; L=length(t);x=x-mean(x); % Plot the graphfigure(1),plot(t,x,'r');figure(1),title('Displacment vs time (ex3B-outer)');figure(1),ylabel('Displacement(m)');figure(1),xlabel('Time(s)'); % Calculate FFTNFFT=2^nextpow2(L);X1 = fft(x,NFFT)/L;X1= 2*abs(X1(1:NFFT/2+1));

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%generate frequency axis;f=fs/2*linspace(0,1,NFFT/2+1); %show this graphicallyfigure(2),plot(f,X1,'b');figure(2),title('Frequency spectrum');figure(2),xlabel('Frequency[Hz]');figure(2),ylabel('x(f)'); %---------------------------------%------------Ex3(outer-angularvelocity)-----------------%--------------------------------- clearclcMatrix3b = dlmread('/Users/Nguyen/Documents/MATLAB/DC1A/Dynamics Lab DC1 Group 7/3b outer Period.txt','\t',2,0);% creating a matrix from datat = transpose(Matrix3b(:,1));p = transpose(Matrix3b(:,2));fs = 10; num=size(p,2); for n=1:(num) w(n)=2*3.14/p(n);end L=length(t); % Plot the graphfigure(1),plot(t,w,'g');figure(1),title('Angular velocity vs time (ex3B-outer)');figure(1),ylabel('Angular velocity(rad/s)');figure(1),xlabel('Time(s)');

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