S232953 Mark Cruickshank Prac 2 MDOF ENG432.pdf.pdf

Mark Cruickshank S232953 Charles Darwin University ENG432 Dynamics of Engineering Prac2: Three Degree of Freedom Systems

School of Engineering and Information Technology ASSESSMENT COVER SHEET

Student Name Mark Cruickshank Student ID S232953 Assessment Title Lab2 Three degrees of freedom systems Unit Number and Title ENG432 Dynamics of Engineering Systems Lecturer/Tutor John Motagner Date Submitted 10/5/2015 Date Received

Office use only KEEP A COPY Please be sure to make a copy of your work. If you have submitted assessment work electronically make sure you have a backup copy. PLAGIARISM Plagiarism is the presentation of the work of another without acknowledgement. Students may use a limited amount of information and ideas expressed by others but this use must be identified by appropriate referencing. CONSEQUENCES OF PLAGIARISM Plagiarism is misconduct as defined under the Student Conduct By-Laws. The penalties associated with plagiarism are designed to impose sanctions on offenders that reflect the seriousness of the Universitys commitment to academic integrity. * By submitting this assignment and cover sheet electronically, in whatever form you are deemed to have made the declaration set out above.

I declare that all material in this assessment is my own work except where there is a clear acknowledgement and reference to the work of others. I have read the Universitys Academic and Scientific Misconduct Policy and understand its implications.* http://www.cdu.edu.au/governance/documents/3.3academicandscientificmisconduct.doc. Signed: Mark Cruickshank .Date: 10/5/2015


ENG432 Dynamics of Engineering Systems

Prac2: 3 Degrees of Freedom Systems

Mark Cruickshank S232953


!

Table!of!Contents!

1.!Objective.............................................................................................................................. 4!2.!Assumptions ........................................................................................................................ 5!3.!Known!input!values.............................................................................................................. 5!Table!3.1!!Known!input!values ............................................................................................................................................ 6!4.!Determining!the!spring!stiffness........................................................................................... 6!4.1!Spring!stiffness!!Method!1 ............................................................................................................................................. 7!

4.1.1$Table$Spring$A ....................................................................................................................................................................7!4.1.2$Table$Spring$B ....................................................................................................................................................................7!4.1.3$Table$Spring$C.....................................................................................................................................................................7!4.1.4$Table$Spring$D ....................................................................................................................................................................7!4.1.5$Graph$Spring$A$$Weight$versus$displacement ....................................................................................................8!4.1.6!Graph!Spring!B!!Weight!versus!displacement ................................................................................................... 9!4.1.7$Graph$Spring$C$$Weight$versus$displacement ....................................................................................................9!4.1.8$Graph$Spring$D$$Weight$versus$displacement................................................................................................. 10!4.2!Spring!stiffness!!Method!2 .......................................................................................................................................... 10!4.2.1$Results$table$$Spring$A............................................................................................................................................... 10!4.2.2$Results$table$$Spring$B............................................................................................................................................... 10!4.2.3$Results$table$$Spring$C ............................................................................................................................................... 11!4.2.4$Results$table$$Spring$D............................................................................................................................................... 11!4.3!Method!1!and!Method!2!comparisons ..................................................................................................................... 11!4.3.1$Method$1$and$Method$2$comparison$table ......................................................................................................... 11!4.3.2!Variation!between!Method!1!and!Method!2 ...................................................................................................... 11!

5.!Initial!displacements!of!the!sliders ..................................................................................... 12!5.1!Free!body!diagram ........................................................................................................................................................... 12!5.2!Equations!of!motion!for!each!Mode .......................................................................................................................... 13!5.2.1$Equation$of$motion$system$1..................................................................................................................................... 13!........................................................................................................................................................................................................... 13!5.2.4!Stiffness!&!Mass!Matrices .......................................................................................................................................... 13!5.3.1$Mass$Matrix ...................................................................................................................................................................... 13!5.3.3$Stiffness$Matrix................................................................................................................................................................ 14!5.3.5$Combining$All$Matrices ............................................................................................................................................... 14!5.4!Determining!the!initial!displacements .................................................................................................................... 14!

6.!Determining!the!natural!frequencies!of!each!mode............................................................ 15!7.!Plot!of!each!mode!vs!time ................................................................................................. 15!7.1!Response!Mode!1 .............................................................................................................................................................. 15!7.2!Response!Mode .................................................................................................................................................................. 16!7.3!Response!Mode!3 .............................................................................................................................................................. 17!7.4!Response!all!Modes!!Subplot ..................................................................................................................................... 17!8.!Conclusion ......................................................................................................................... 18!


1.!Objective!

The objective of this laboratory is to demonstrate three modes of vibrations for a three-

degrees of freedom mass-spring system. We expect a three-degree of freedom system

to behave differently according to the natural frequencies of each mass/node.

To summarize we would expect three conditions for a three-degree of freedom system:

Condition 1: At this frequency all masses move in the same direction, this would be the

smallest frequency. This is illustrated in the below figure.

Condition 2: At this frequency we would expect the end mass to move in the opposite

directions as opposed to the middle and furthers end mass. The two outer masses

should move with the same amplitude but in opposite directions if the stiffness of each

spring is the same.

Condition 3: At this frequency (the fastest of the other 2) the two outer masses move in

the same direction with the same amplitude if the stiffness of each spring is the same.

We have used a vacuum to assist in minimizing the friction between the track and the

slider(s), thus for this laboratory we have assumed there is no friction (by using the

vacuum). This laboratory will involve determining the following:


The weights of each slider(s)

The spring-stiffness of each of the four springs

The natural frequencies for the three individual sliders

The initial displacement distances(s) for each of the sliders.

We should be able to determine the initial displacement by calculating the mode ratios

then setting initial displacements based on these calculations. Finally we can conclude

into how accurate the experimental results are in comparison to the theoretically

determined.

Due to being an external student of this unit, I am unable to complete the practical

testing associated with this laboratory, however I can conclude on how the physical

testing would behave with respect to my results.

2.!Assumptions! As previously mentioned, we have assumed that there is no friction between the track

and the slider; therefore this system will have no damping. Friction has been eliminated

due to the vacuum suction that is applied to the track.

We also assume the following:

All equipment measuring scales are properly calibrated, thus our measurement

tolerances are very minimal.

That all springs act in a linear manner.

3.!Known!input!values! All known input values for this experiment are displayed in the below table:

The masses of the springs and sliders have been determined by measuring these on a

set of digital scales, we would assume an error of no more than +/- 0.001kg.


The lengths of the un-stretched springs have been determined visually, by using a metric

ruler; again we would assume an error of no more than +/- 0.002M or 2mm.

Table!3.1!!Known!input!values!No. Description

Value Unit

1 Mass carrier 0.0519 kg 2 Un-stretched length spring A 0.3180 M 3 Un-stretched length spring B 0.2920 M 4 Un-stretched length spring C 0.1880 M 5 Un-stretched length spring D 0.0760 M 6 Mass slider A 0.36615 Kg 7 Mass slider B 0.27480 Kg 8 Mass slider C 0.36850 Kg

4.!Determining!the!spring!stiffness! Determining the spring stiffness for each spring was calculated using two methods.

The first method (Method 1) involved attaching a weight to the spring and measuring the

displacement, the results were reordered.

The second method (Method 2) was determined by determining the period over 10

oscillations. The following equations express how this was calculated (Palm 2009):

T = 2

=2T

=k

mobject=2T

%

& '

(

) *

k = 2T%

& '

(

) * 2mobject

As we have assumed that the springs act in a linear way, we would expect the

relationship between force and displacement to act in a linear way, provided the spring is

within the elastic zone (Hooks Law).


These calculations can be seen in the attached Microsoft Excel spreadsheet.

4.1!Spring!stiffness!!Method!1! Method 1 his was determined by using small increments of added weight and recording

the displacement. Small increments of mass(s) were used to ensure accurate results.

The results for each given spring are displayed in the below tables:

4.1.1!Table!Spring!A!Spring A

No. Mass [kg] Weight [N] Distance [m] Displacement [m] 1 0 0 0.318 0 2 0.05032 0.4936392 0.531 0.213 3 0.05542 0.5436702 0.566 0.248 4 0.06052 0.5937012 0.596 0.278 5 0.06562 0.6437322 0.628 0.31

4.1.2!Table!Spring!B!Spring B

No. Mass [kg] Weight [N] Distance [m] Displacement [m] 1 0 0 0.292 0 2 0.05032 0.4936392 0.541 0.249 3 0.05542 0.5436702 0.573 0.281 4 0.06052 0.5937012 0.603 0.311 5 0.06562 0.6437322 0.632 0.34

4.1.3!Table!Spring!C!Spring C

No. Mass [kg] Weight [N] Distance [m] Displacement [m] 1 0 0 0.188 0 2 0.05032 0.4936392 0.35 0.162 3 0.05542 0.5436702 0.368 0.18 4 0.06052 0.5937012 0.388 0.2 5 0.06562 0.6437322 0.405 0.217 6 0.07072 0.6937632 0.425 0.237 7 0.07582 0.7437942 0.443 0.255

4.1.4!Table!Spring!D!Spring D

No. Mass [kg] Weight [N] Distance [m] Displacement [m] 1 0 0 0.076 0 2 0.05032 0.4936392 0.15 0.074 3 0.05542 0.5436702 0.158 0.082


4 0.06052 0.5937012 0.165 0.089 5 0.06562 0.6437322 0.173 0.097 6 0.07072 0.6937632 0.181 0.105 7 0.07582 0.7437942 0.189 0.113

From each of the above tables, weight was plotted against displacement in Microsoft

Excel, a trend line was set through the origin to obtain the spring stiffness, these can be

seen in the below graphs.

4.1.5!Graph!Spring!A!!Weight!versus!displacement!


!!!

4.1.6%Graph%Spring%B%%Weight%versus%displacement

!

4.1.7!Graph!Spring!C!!Weight!versus!displacement!


4.1.8!Graph!Spring!D!!Weight!versus!displacement!

4.2!Spring!stiffness!!Method!2! The results for method 2 are displayed in the below tables:

4.2.1!Results!table!!Spring!A!Spring A

No Time [sec] Period [T] Wn [rad/sec] Mass [kg] K [N/m] 1 12.15 1.215 5.171345932 0.0502 1.342 2 12.28 1.228 5.116600413 0.0502 1.314 3 12.37 1.237 5.079373733 0.0502 1.295 Average 1.317

4.2.2!Results!table!!Spring!B!Spring B

No Time [sec] Period [T] Wn [rad/sec] Mass [kg] K [N/m] 1 11.75 1.175 5.347391751 0.0502 1.435 2 11.47 1.147 5.477929649 0.0502 1.506 3 11.69 1.169 5.374837731 0.0502 1.450 Average 1.464


4.2.3!Results!table!!Spring!C!Spring C

No Time [sec] Period [T] Wn [rad/sec] Mass [kg] K [N/m] 1 9 0.9 6.981317008 0.0502 2.447 2 8.96 0.896 7.012483602 0.0502 2.469 3 8.88 0.888 7.07565913 0.0502 2.513 Average 2.476

4.2.4!Results!table!!Spring!D!Spring D

No Time [sec] Period [T] Wn [rad/sec] Mass [kg] K [N/m] 1 8 0.8 7.853981634 0.1002 6.181 2 7.78 0.778 8.07607366 0.1002 6.535 3 7.87 0.787 7.983717036 0.1002 6.387

Average 6.368

4.3!Method!1!and!Method!2!comparisons! As seen in the below table, the results for both methods are displayed, as we can see all

of the springs for both methods showed a linear relationship between force and

displacement.

We can see that Method 1 (determined by measuring the deflection per given weight)

appears to have a larger magnitude then that of method 2. The last column is the

absolute difference between both methods. Whilst conducting Method 2, we found that

determining the exact place where the oscillation peaks/finishes and the timing between

these periods considerably hard to maintain, thus we would assume this difference in

results is a related to this.

4.3.1!Method!1!and!Method!2!comparison!table!

Spring Stiffness Comparison: Method 1 and Method 2

Spring type

Result - Method 1 [N/m]

Result - Method 2 [N/m]

Abbs difference [N/m]

A 1.9225 1.317 0.6052 B 1.9225 1.464 0.4585 C 2.9609 2.476 0.4847 D 6.6253 6.368 0.2577

4.3.2!Variation!between!Method!1!and!Method!2!


Firstly, both these experiments were conducted based on human judgment, thus we can

illustrate that due to human judgment there would be a discrepancy with results. Also

due to the age of the springs, there are some springs that are deformed IE have been

loaded with weight or possibly stretched past their elastic limit, that may have also

contributed between any discrepancies in results. These deformations are displayed in

the below photo:

!4.3.2a!Photo!of!the!springs!used!

The above photo is springs D to A, with spring D being the springs to the top of the

photo.

5.!Initial!displacements!of!the!sliders!

5.1!Free!body!diagram! A free body diagram of the system can be seen in the below figure:


5.2!Equations!of!motion!for!each!Mode! From the above free body diagram, we can represent the system by using Newtons

second law of motion, as per the following equations:

5.2.1!Equation!of!motion!system!1!

m1 x 1 = k2 x2 x1( ) k1x1 m1 x 1 = (k1 + k2)x1 + k2x2 m1 x 1 + (k1 + k2)x1 k2x2 = 0

!

m2 x 2 = k2 x2 x1( ) + k3(x3 x2)m2 x 2 + k2 x2 x1( ) k3(x3 x2) = 0 m2 x 2 k1x1 + (k2 + k2)x2 k3x3 = 0

!

m3 x 3 = k4 x3 k3(x3 x2)m3 x 3 + k4 x3 + k3(x3 x2) = 0 m3 x 3 + k3x2 + (k3 + k4 )x2 = 0

5.2.4!Stiffness!&!Mass!Matrices! From the above three equation the following matrices can be found to form the below

equation:

M[ ] ! x + k[ ] ! x = 0

5.3.1!Mass!Matrix!


m1 0 00 m2 00 0 m3

"

#

$ $ $

%

&

' ' '

!

5.3.3!Stiffness!Matrix!

k1 + k2 k2 0k2 k2 + k3 k30 k3 k3 + k4

#

$

% % %

&

'

( ( (

5.3.5!Combining!All!Matrices!

m1 0 00 m2 00 0 m3

"

#

$ $ $

%

&

' ' '

x 1 x 2 x 3

"

#

$ $ $

%

&

' ' ' =

k1 + k2 k2 0k2 k2 + k3 k30 k3 k3 + k4

#

$

% % %

&

'

( ( (

x`x2x3

"

#

$ $ $

%

&

' ' '

5.4!Determining!the!initial!displacements!

We have used the attached Matlab file to determine the mode ratios for each of the three

modes. The initial displacements are based on these ratios. For each mass per given

mode the initial displacement can be seen in the below table in meters.

No. Mass no. Displacement [m] Mode Ratios from

Matlab

MODE 1 1 Mass A 0.05 1

2 Mass B 0.045185 0.9037

3 Mass C 0.017925 0.3585 MODE 2

1 Mass A -0.042025 -0.8405

2 Mass B 0.04351 0.8702

3 Mass C 0.034695 0.6939

MODE 3

1 Mass A 0.02077 0.2077

2 Mass B -0.08737 -0.8737

3 Mass C 0.10671 1.0671


The negative values indicate that the mass is moved in the opposite direction. The last

column of the above table is the mode ratio. This has been multiplied by the same

number to form the initial displacement per each mode.

6.!Determining!the!natural!frequencies!of!each!mode! The natural frequencies for each mode has been calculated using Matlab, please refer to

the back of this report for the Matlab program

The natural frequencies for all three masses can be seen in the below table.

Mass item no. Natural frequency

Mass A 2.4555 rad/sec

Mass B 2.4496 rad/sec

Mass C 6.560 rad/sec

7.!Plot!of!each!mode!vs!time! Each of the three modes response in the time domain was plotted against time, these

can be seen in the below diagrams.

!

!

!

!

!

!

!

!

7.1!Response!Mode!1!


7.2!Response!Mode!!


7.3!Response!Mode!3!

!

!

!

7.4!Response!all!Modes!!Subplot!


All three response have been put into the above sub-plot, to illustrate how the behavior

of the system changed as the initial displacement changed increased and/or

decreased.

8.!Conclusion! The spring rates from Method 1 were chosen due to the lower level of error whilst

measuring. As previously stated, being an external student for this unit, the as calculated

initial distances of all three modes has not been physically tested, thus we can only

assume that these results would be within the range as expected. We would expect to

see some small amount of error due to the surface having a minimal amount of friction;

therefore I would assume the system would have a small amount of damping as a result.

The responses of the modes appear to behave as expected, although we have not

physically tested these results we can see at the lowest frequency the masses move in

the same direction, this mode also appear to have small amount of phase shift and even

amplitudes. The amplitudes are not of the same magnitude due to the springs having

different stiffness values; the last spring that attaches to mass 3 has the highest stiffness

value, approximately 2.5 times the stiffness of springs that attach to mass 1 and 2. The

response of mode 2 appeared to have the most accurate period, with minimal phase

shift.

As demonstrated, in this part of the experiment and from a design perspective, we can

see how the natural frequency of the systems components, the stiffness of the springs

and the initial displacements of the masses can have an affect on multiple degrees of

freedom systems behavior. The results would have looked quite different if the springs

were placed into different positions IE changing spring D with spring B.

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Contents

Known inputs

Matrics

Test K matrix for determininant.

Eigenvalue formation

Mode coordintates to physical coordinates

The below are for the three cases:

The below values will determine the displacment offsets %% note these are only ratios!!:

Printing values to the screen

Setting dx/dt matrices

Time vecotor

Create unit vectors for x01, x02 and x03

Solving for the ODE's

Plotting the vectors

Plotting the Responses for all modes into a subplot

close all; clear all; clc

Known inputs

% Stiffness input values [N/m]:k1 = 1.9225; % this is spring Ak2 = 1.9225; % this is spring Bk3 = 2.9609; % this is spring Ck4 = 6.6253; % this is spring D

% Mass input values for the sliders [kg]m1 = 0.36615; % this is mass Am2 = 0.27480; % this is mass Bm3 = 0.36850; % this is mass C

Matrics

K = [k1+k2 -k2 0;-k2 k2+k3 -k3;0 -k3 k3+k4]; % This is the stiffness matrix

M = [m1 0 0; 0 m2 0;0 0 m3]; % This is the mass matrix

Test K matrix for determininant.

% If det = 0, [K] does not have an inverse.

detK = det(K);

% This has a determinant thus we can proceed with Eigenvalue formation:

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Eigenvalue formation

[P,Lambda] = eig(inv(M)*K); % This is the eigen vector

omega = sqrt(Lambda)*[1 1 1]'; %This is the natural frequency for each given slider

P=P./(P(1,:):P(1,:):P(1,:)); % This is the mode matrice

Mode coordintates to physical coordinates

The below are for the three cases:

ro1 = [1 0 0]'; % Assuming mode 1 equals one and remianing modes (2&3) are equal to zeroro2 = [0 1 0]'; % Assuming mode 2 equals one and remianing modes (1&3) are equal to zeroro3 = [0 0 1]'; % Assuming mode 3 equals one and remianing modes (1&2) are equal to zero

The below values will determine the displacment offsets %% note these are only ratios!!:

Xo1 = P*ro1; % This will give the physical coordinates - MODE 1Xo2 = P*ro2; % This will give the physical coordinates - MODE 2Xo3 = P*ro3; % This will give the physical coordinates - MODE 3

Printing values to the screen

Lambda % Print on screenomega % Print on screenP % Print on screenXo1 % Print on screenXo2 % Print on screenXo3 % Print on screenK % Print on screendetK % Print on screen

Lambda =

5.7560 0 0 0 15.9370 0 0 0 32.5930

omega =

2.3992 3.9921 5.7090

P =

1.0000 -0.8405 0.2077

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0.9037 0.8702 -0.8737 0.3585 0.6939 1.0671

Xo1 =

1.0000 0.9037 0.3585

Xo2 =

-0.8405 0.8702 0.6939

Xo3 =

0.2077 -0.8737 1.0671

K =

3.8450 -1.9225 0 -1.9225 4.8834 -2.9609 0 -2.9609 9.5862

detK =

110.8574

Setting dx/dt matrices

dx1dt = @(t1,x1) [x1(4), x1(5), x1(6), (k2*x1(2)-(k1+k2)*x1(1))/m1, (k2*x1(1)-(k2+k3)*x1(2)+k3*x1(3))/m2, (k3*x1(2)-(k2+k4)*x1(3))/m3]';dx2dt = @(t2,x2) [x2(4), x2(5), x2(6), (k2*x2(2)-(k1+k2)*x2(1))/m1, (k2*x2(1)-(k2+k3)*x2(2)+k3*x2(3))/m2, (k3*x2(2)-(k2+k4)*x2(3))/m3]';dx3dt = @(t3,x3) [x3(4), x3(5), x3(6), (k2*x3(2)-(k1+k2)*x3(1))/m1, (k2*x3(1)-(k2+k3)*x3(2)+k3*x3(3))/m2, (k3*x3(2)-(k2+k4)*x3(3))/m3]';

Time vecotor

tspan = [0 10]; % creating time vector

Create unit vectors for x01, x02 and x03

% I have manually copied and pasted this vector and added three zeros to

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% ensure matrix is dimmensionally correct. There is probablly another easier wat to do this.

x010 = [0.05; 0.045185; 0.017925; 0; 0; 0];x020 = [-0.042025; 0.04351; 0.034695;0;0;0];x030 = [0.02077; -0.08737; 0.10671;0;0;0];

Solving for the ODE's

[t1,x1] = ode45(dx1dt,tspan,x010);[t2,x2] = ode45(dx2dt,tspan,x020);[t3,x3] = ode45(dx3dt,tspan,x030);

Plotting the vectors

figure(1) % new figureplot(t1,x1(:,1:3)); % this is for mode 1title('Time domain response for Mode 1');xlabel('Time [sec.]');ylabel('Amplitude');legend('x1', 'x2', 'x3')grid on



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Plotting the Responses for all modes into a subplot

figure(4) % new figure

subplot(3,1,1); % this is for mode 1plot(t1,x1(:,1:3));title('Time domain response for Mode 1');xlabel('Time [sec.]');ylabel('Amplitude');legend('x1', 'x2', 'x3')grid on

subplot(3,1,2); % This is for mode 2plot(t2,x2(:,1:3));title('Time domain response for Mode 2');xlabel('Time [sec.]');ylabel('Amplitude');legend('x1', 'x2', 'x3')grid on

subplot(3,1,3); % This is for mode 3plot(t3,x3(:,1:3));title('Time domain response for Mode 3');xlabel('Time [sec.]');ylabel('Amplitude');legend('x1', 'x2', 'x3')grid on

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S232953 Mark Cruickshank Prac 2 MDOF ENG432.pdf.pdf

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