motion report final

8/11/2019 Motion Report Final

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Motion Lab Report

Ty Dorman

Ben Wetzel

Brian Zieverink

ME 3870

Lab: Thursday 5:30PM


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Table of Contents

Abstract ......................................................................................................................................................... 3

Introduction ................................................................................................................................................... 3

Experimental Procedure ................................................................................................................................ 5

Results and Discussion ................................................................................................................................. 7

Single Degree of Freedom System ............................................................................................................ 7

Two Degree of Freedom System ............................................................................................................ 13

Investigation of Results ........................................................................................................................... 15

Conclusion .................................................................................................................................................. 16

References ................................................................................................................................................... 18

Appendix ..................................................................................................................................................... 19

Relevant Equations ................................................................................................................................. 19

Sample Calculations ................................................................................................................................ 20

MATLAB Code ......................................................................................................................................... 20

Description of Responsibilities ................................................................................................................ 22

Datasheets .............................................................................................................................................. 23


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Abstract

The presence of vibrations in a mechanical system is often unavoidable. Unfortunately,

vibrations also have a good chance of damaging the mechanical system and limiting its

effectiveness. As such, lots of effort has been put into studying vibration dampening to minimize

or prevent the effects of vibration in a system. In this study, a given spring-mass-damper system

was analyzed to determine its defining characteristics with the intent of designing a vibration

damper that would eliminate vibrations at a determined resonance frequency. To achieve this

objective, the resonance frequency was determined experimentally. A spring and mass were then

selected based on how close their natural frequency was to the resonance frequency of the

system. Upon testing the vibration absorber at the resonance frequency, a reduction in vibration

amplitude of approximately 90 percent was observed. While this result does not meet the

ultimate objective of complete absorption, the results were significant enough to call the

vibration absorber effective at the systems resonance frequency.

Introduction

In any vibrating mass system, it is important to understand the relationship between the mass,

spring, and damper in the system and be able to minimize the oscillations to prevent damage to

the components in and around the system. Understanding how to determine the natural

frequency of the system and reduce the amplitude of oscillation at the natural frequency is

critical in the design of systems consisting of masses, springs, and dampers.

The ultimate goal of this experiment is to reduce the oscillation amplitude of the primary single

degree of freedom system. This will be achieved through calculation of the appropriate mass and

spring combination in order to design a vibration absorber. The physical characteristics of the


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single degree of freedom system will be experimentally determined and then combined with the

vibration absorber to create a two degree of freedom system.

Additionally, after completion of this experiment, the user should be comfortable with writing

LabView programs for data acquisition and have the ability to know when and how to apply least

squares curve fitting.

The single degree of freedom system studied in this experiment consists of a mass and a set of

parallel springs, which are represented inFigure 1by a spring and damper combination.

Figure 1: Single Degree of Freedom Vibrational System (Department of Mechanical Engineering 1)

This system can be represented using the equation below.

| | In the single degree of freedom system, the damping can be estimated by allowing the system to

respond to a single impulse force, collecting position data over a period of time, and then using

the Logarithmic Decrement method between two amplitude peaks.

Once an additional mass and spring are added for vibration damping purposes, the system

becomes a two degree of freedom system as shown inFigure 2.


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Figure 2: Two Degree of Freedom System (Department of Mechanical Engineering 3)

This system can be represented using the following equations

In this experiment, it is assumed that the system will only operate at the resonant frequency of

the primary system. If the system operates at any frequency other than the resonant frequency,

the vibration absorber will not function optimally and could cause the system to oscillate at a

frequency capable of destroying the system.

Experimental Procedure

The beginning of this experiment consisted of gathering data to define the characteristics of the

spring-mass system under analysis. The primary mass (m1) and the unbalanced mass (mu) were

measured in kilograms using a calibrated scale. The eccentricity (e) was measured in meters,

using a linear scale.

The primary spring was placed in a drill chuck that was connected to a LabView program which

measured both force and displacement. The program plotted the force against displacement as


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the spring was compressed by the drill chuck and the slope of that plot was determined to be the

primary spring stiffness (k1).

Another LabView program was used to graph the free response of the spring-mass system. The

system was subjected to an impulse force and the amplitude of oscillation was plotted over time.

From this plot, the logarithmic decrement was determined and subsequently used to calculate the

damping coefficient.

A rotating unbalanced mass was placed on the primary mass as shown in Figure 3below.

Figure 3

A frequency sweep was performed by using a motor to rotate the unbalanced mass at varying

frequencies between 4 and 10 Hertz. An oscilloscope was used to measure the amplitude and

frequency of the accelerometer for each input motor frequency. The frequency at which the

amplitude was the greatest was determined to be the resonance frequency.

Design of a vibration absorber began once the resonance frequency was determined. The natural

frequency of the vibration absorber, which most effectively reduced the amplitude of vibration,

was calculated and then analysis of the available masses and springs began. The masses were

measured using a scale and the spring constants were measured using the same procedure used to

determine the primary spring stiffness. The combination of springs and masses which had a


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Figure 5: Linear Least Squares Fit for Force vs. Displacement Data

The purpose of collecting this data was to determine the stiffness of the spring that was to be

used in the experiment. The following values were experimentally determined from the Single

Degree of Freedom System:

; ; The primary mass was measured on a calibrated scale and k was determined from the slope of

the Least Squares Fit line. Table 1presents the frequency data acquired from the single degree

of freedom system.


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Table 1: Single Degree of Freedom Frequency Accelerometer Data

Motor Frequency

(Hz)

Accelerometer Frequency

(Hz)

Percent Difference

(%)

Amplitude of Accelerometer

(V)

10.000 10.000 0.00 0.6775

9.000 9.259 2.80 0.6425

8.000 7.692 4.00 0.6025

7.000 6.944 0.80 0.5525

6.000 6.098 1.60 0.6325

5.000 5.000 0.00 0.9600

4.625 4.630 0.10 2.4750

4.500 4.630 2.80 2.7688

4.375 4.630 5.50 2.8375

4.250 4.425 3.95 1.3125

4.125 4.310 4.30 0.6188

4.000 4.202 4.80 0.5850

From this table it can be seen that the frequencies for the motor speed readout almost match the

frequencies from the accelerometer readout. The percent difference between the frequencies is at

most 5.50 % which an average percent difference of 2.55%. Thus it can be concluded that the

frequencies are practically the same.Figure 6 shows the amplitude vs. frequency of the single

degree of freedom system. Represented is the experimental data and the simulated vibration data.

The MATLAB code used to create this figure can be found in the Appendix.


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Figure 6: Amplitude vs. frequency of 1DOF system for the experimental and simulated vibrations

The amplitude for the data was originally in units of volts. From the accelerometer specification

sheet [3], it was found that a sensitivity could be used to convert from volts to units of

acceleration due to gravity, or gs. Then, units of gs was converted to . Next theacceleration could be divided by the frequency squared to give the amplitude in meters. The data

fromTable 1,specifically the frequency of the accelerometer and the amplitude, was utilized to

create the frequency vs. amplitude plot ofFigure 6.


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Figure 7: Experimental Data of Accelerometer Frequency vs. Accelerometer Amplitude

FromTable 1 andFigure 7 it is concluded that the experimental natural frequency is at 4.630 Hz.

Given this observation the experimental natural frequencycan be calculated. Theexperimental natural frequency was found to be 29.09 rad/s. See Sample Calculation 2 for a

worked out solution. The free response data of the single degree of freedom system is displayed

inFigure 8.The damping ratio was calculated using logarithmic decrement. FromFigure 8,two

different peaks were chosen. The peaks were spaced three time periods apart. Using Equation 7,

the damping ratio was calculated to be 0.004046. See Sample Calculation 4 for a worked out

solution.


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Figure 8: Experimental Data from Free Response of System

The natural frequency found from the experiment would be more accurate than the one

determined theoretically. The experimental natural frequency reflects the system as it naturally

is, whereas, the theoretical natural frequency was found based off the measured spring stiffness

and the measured mass of the system. The actual mass of the system we tested was not

measured, but a similar system was weighed and used for the mass of our system. This could

result in errors of the actual mass of the system we tested. In addition, the spring stiffness was

found for a spring that was similar to the ones used in testing, but the actual springs used in

testing were not used to find their respective stiffness. There could have been slight differences

in spring stiffness, which would cause errors in the calculation of the natural frequency. Also, the

theoretical natural frequency doesnt account for fluctuations and other forces acting on the

system. Thus, it is seen that the experimental natural frequency more accurately reflects the

system than the theoretical one does.


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Two Degree of Freedom System

A vibration absorber was designed to suppress erratic vibrations of the primary system at its

natural frequency. This was accomplished by finding a separate spring and mass that produce a

natural frequency close to the natural frequency of the single degree of freedom system. The

force vs. displacement data for the spring tested for use in the vibration absorber is shown in

Figure 9.

Figure 9: Force vs. displacement data for vibration absorber spring

The slope of the best fit line inFigure 9 represents the spring stiffness. This spring was coupled

with a mass to create a natural frequency close to that of the primary system. The following

values were determined experimentally to create the vibration absorber:

; ; The mass used with the spring was determined using Equation 2 in the Appendix. This mass was

determined to be 0.732 kg, but the closest mass that could be used in the lab was 0.720 kg. A

y = -0.6191x + 32.179

0

5

10

15

20

25

0 5 10 15 20 25 30 35

Force(N)

Displacement (mm)


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sample calculation for can be found in the Appendix as Sample Calculation 5. The vibrationabsorber greatly reduced the amplitude of vibration around the natural frequency of the one

degree of freedom system.Figure 10 shows that around the natural frequency of the system, the

vibration amplitude decreases when the vibration absorber is added.

Figure 10: Amplitude response at different frequencies for the 1DOF and the 2DOF systems

A vibration absorber acts to reduce the amplitude of a system at its natural frequency. This can

be advantageous if the system is expected to operate at its natural frequency. The system would

fluctuate violently when operating at its natural frequency, but the vibration absorber greatly

reduces the amplitude of these vibrations. A disadvantage to having a vibration absorber would

be that there are more moving parts in the system. More moving parts means there is a greater

chance of failure in the system. Also there would be added mass to the system. This could be

problematic if there is a weight requirement for the system. One other disadvantage would be


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that at certain frequencies the system would still vibrate erratically. This could become an issue

if the system were operating at these frequencies.

Investigation of Results

Figure 11 displays the experimental and simulated data responses of the two degree of freedom

system. The amplitude of the system was determined by converting from volts to meters. Sample

calculation 3 shows how this is done. Also the MATLAB code used to generateFigure 11 can be

found in the Appendix.

Figure 11: Amplitude vs. frequency of 2DOF system for experimental and simulated vibrations


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The model does not seem to adequately describe the system. The experimental data looks

similar, but the simulated signal has a smaller amplitude. A two degree of freedom system has

more moving parts and other vibrations going on, and this could cause a response that would not

be expected. So the theoretical response differs from the experimental one for the reason that the

system is not acting under perfect conditions in the lab. As the frequency of operation deviates

from the natural frequency, the motion starts to become more chaotic and the accelerometer

would not know how to process such data. Thus, the accelerometer signal would look less and

less sinusoidal as the frequency moves away from the natural frequency.

Conclusion

The objective of this experiment was to define a given system in terms of mass, spring stiffness,

and damping coefficient. Upon defining the system, the resonance frequency was to be

determined and ultimately, a vibration absorber was to be designed to most effectively minimize

the amplitude of vibration at the determined resonance frequency. Measurements determined that

the system could be defined as having a mass of 2.432 kilograms, a spring stiffness of 971.7

Newtons per meter, and a dampening coefficient of 0.393 Newton-seconds per meter. Also the

resonance frequency was determined to be 4.6296 Hertz. The vibration absorber was designed to

minimize the vibration amplitude and the resonance frequency and those results were pretty well

achieved. Without the absorber, the amplitude was 2.8375 Volts, but with it, the amplitude was

0.39375 Volts. This is a significant reduction in amplitude.

There were a few issues during the duration of the experiment which likely resulted in a

reduction of accuracy in the final results. Measuring the spring stiffness of the spring for the


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vibration absorber proved to be challenging as the spring was rather thin and long. This resulted

in deflection of the spring when the drill chuck was applying pressure to it rather than just

compression. Consequently, the calculated spring stiffness likely contains some error. Also,

when measuring the vibration amplitude of the system with the vibration absorber, the mass and

spring of the vibration absorber were not physically secured to anything. This resulted in the

mass and spring separating from the primary system temporarily when the vibrations were more

violent and likely altering the amplitude of the vibration dampening system at certain input

frequencies. This was not an issue at the resonance frequency though, so our conclusion is not

altered.


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References

Department of Mechanical Engineering.ME3870 Discrete Vibrating Systems Lab Manual.Classroom

distribution.


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Appendix

Relevant Equations

Single Degree of Freedom System: (Department of Mechanical Engineering 1)

() (1)

(2)

(3)

(4)

Two Degree of Freedom System: (Department of Mechanical Engineering 3)

()() (5) ()() (6)

Logarithmic Decrement: (Department of Mechanical Engineering 25)

(7)


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Axy=0;n=length(F); %number of data pointsfori=1:n %calculation of Ax

Ax = Ax + x(i);endfori=1:n %calculation of Ay

Ay = Ay + F(i);endfori=1:n %calculation of Axx

sum = (x(i))^2;Axx = Axx + sum;

endfori=1:n %calculation of Axy

sum = x(i) * F(i);Axy = Axy + sum;

endm = ((n*Axy) - (Ax*Ay)) / ((n*Axx) - (Ax^2)); %calc. of slopeb = ((Axx*Ay) - (Ax*Axy)) / ((n*Axx) - (Ax^2)); %calc. of interceptfori=1:n %store values of Force using the least squares fir method

F_ls(i) = m * x(i) + b;

endplot(x,F,'b-')hold onplot(x,F_ls,'r--')xlabel('Displacement (mm)')ylabel('Force (N)')legend('Static Calibration Data','Least Squares Fit')

Experimental and Simulated Signals for Amplitude vs. Frequency of the 1DOF Sysytem:

%Frequency and Amplitude data from the experimentfreq=[10.000 9.000 8.000 7.000 6.000 5.000 4.625 4.500 4.375 4.250 4.1254.000];amp=[0.00401 0.00443 0.00602 0.00678 0.01007 0.02272 0.06832 0.07643 0.078330.03966 0.01971 0.01961];

%Transfer function for simulated dataw= [0:.1:(400*(2*pi/60))];f=w./(2*pi);j= sqrt(-1);s=j.*w;y=abs((.0029.*s.^2)./((2.432.*s.^2)+(0.3933.*s)+(2*971.7)));

%Plot both experimental response and simulated responseplot(freq,amp,'b-')hold onplot(f,y,'r--')xlabel('Motor Frequency (Hz)')ylabel('Amplitude (m)')legend('Experimental Data','Simulated Data')hold off


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Experimental and Simulated Signals for Amplitude vs. Frequency of the 2DOF Sysytem:

%Frequency and Amplitude data from the experimentfreq=[10.20400 9.43400 7.81250 7.14290 6.09760 5.00000 4.06500 4.504504.62960 4.80770];amp=[0.00375 0.00371 0.00550 0.00742 0.01170 0.03225 0.00797 0.00973 0.01087

0.01517];

%Transfer function for simulated dataw= [0:.1:(400*(2*pi/60))];f=w./(2*pi);j= sqrt(-1);s=j.*w;m1=2.374;m2=0.720;b1=0.3933;b2=5.002;k1=2*971.7;k2=619.1;mu=.058;e=.050;F1=mu.*e.*(s.^2); %F1=mu*e*(d2A/dt)

%Using equations 7 and 8, we can find y1 and y2

y1=abs((F1.*((m2.*s.^2)+(b2.*s)+(k2)))./((((m1.*s.^2)+((b1+b2).*s)+(k1)+(k2)).*((m2.*s.^2)+(b2.*s)+(k2)))-(((b2.*s)+k2).^2)));

%Plot both experimental response and simulated responseplot(freq,amp,'b-')hold on

plot(f,y1,'r--')xlabel('Accelerometer Frequency (Hz)')ylabel('Amplitude (m)')legend('Experimental Data','Simulated Data')hold off

Description of Responsibilities

Ty Dorman: Experimental Procedure; Conclusion; Abstract

Ben Wetzel: Compile final report; Introduction; Relevant Equations; References

Brian Zieverink: Results and Discussion; MATLAB code; Charts and Tables


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Datasheets

See attached Accelerometer data sheet

motion report final

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