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MEEG 311 Lab ReportLab #2

Kevin Eckenhoff

Abstract-The actual and theoretical values of the phase angle and amplification factor of a spring-damper

system under forced harmonic motion were compared. Although the phase angles matched, theamplification factors were off. They trended together, but their magnitudes had a large amount of error.

Introduction-In this lab, the validity of the theory underlying forced harmonic motion of damped-spring

systems was investigated. y using a computer to measure the motion of the system, the measuredvalues for the amplification factor and the phase angle. This can then be compared to the theoreticalvalues, which would indicate the validity of the theory.

!nder a regular force, a spring e"tends a certain amount. #hen a spring-damper systeme"periences a harmonic forcing load, its reaction is also harmonic. It has an amplification factor, whichrelates the amplitude of motion to the displacement under a non-harmonic load of the same amplitude. Italso has a phase angle which shifts its motion away from the forcing function, although it shares theangular fre$uency. These values can be calculated given parameters, but they must ultimately becompared to the measured values to prove the validity of the theory

Results -%or each trial, & peaks were chosen such that all three lay in the steady state portion of the

motion. These values were then transformed from the encoder units to mm. The magnitudes of these peaks were then averaged to find the amplitude of the forced motion.

Trial '()mm* '+)mm* '&)mm* '()mm* '+)mm* '&)mm* 'ave

Trial ( /0& /0/ &.1(1 &. 2/ &. 1 &. 0Trial + 0(( 0( 0 1 &.0 & . ( &.01/ &.0 1Trial & (1& (1 + (1+0 2.20 2./ + 2.2 1 2./ 2Trial ( /+ ( /2 ( 2/ .( 1 .+ & .(2& .(Trial 1 ((2 ((/( ((/ 1. /+ 1.(+ 1.(&& 1.(Trial 2 /+& /( /(& &.(2( &.(++ &.(( &.(&Trial / (( (( 1 (( / 1. + 1. 2 1. (1 1. (Trial +(+/ +(( +( 0.& 0.++2 0.+(/ 0.+Trial 0 (1 (1 / (1&1 2.0+2 2./2 2./(+ 2. (Trial ( 20/ 200 221 &. &. 12 +.0 &.


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Table (3 The measurements of the amplitude of motion are shown. Three peaks in the steady state portion of motion were chosen for each trial. These were then transformed from encoder units to mm.The average amplitude for each trial was also calculated.

The ne"t step was to calculate the static deflection of the the system. This was done by dividing

%o by the k value derived in the previous lab. This was e$ual to / 45m. Thus the static deflection wase$ual to .+00 mm.The value of the natural fre$uency was then calculated. This was done by using the formula wn6

)k5m*7)(5+*. The mass for group ( is (.+&, while the mass of group + is (./+. Thus the naturalfre$uency for group ( is +2./+, while the fre$uency for group + is ++.10. %or each trial, the amplificationfactor can then be calculated by diving the amplitude by the static deflection. The value for r can then befound as the ratio between the forcing fre$uency and the natural fre$uency.

Trial ')mm* 8 r

Trial ( &. / (+. &+ .+&1Trial + &.0/ 0 (&./ ( .11&Trial & 2.201/ +&.(+ ./ 1Trial .(/&2 + .++ .0Trial 1 1.(+ 0 (/.2(1 (.(/1Trial 2 &.(+12 ( . 1 .+/Trial / 1. ( 1 (/.+ 0 .11&Trial 0.+&+ &(. 0 . &Trial 0 2./12+ +&. 1 (.((+Trial ( +.011 ( .&1 (.&0

Table &3 The values for 8 are shown. These were derived by diving the amplitude by the static

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

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Magnification Factor Versus Frequency Ratio

Group 1Group 2

Frequency Ratio

M a g n

i f i c a

t i o n

F a c

t o r


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deflection. The fre$uency ratio was found by dividing the fre$uency for each trial by the naturalfre$uency of the configuration.

Table (3 9hows the amplification factor plotted against the fre$uency ratio. oth groups of trials showthe same trends in their lines, indicating that the relationship between fre$uency ratio and amplification isaccurately shown.

The time to reach steady state was measured by recording the time of the first peak in the steadystate region. 4e"t, the time difference of the forcing function and the response was analy:ed bycomparing the time that each signal reached :ero on the approach of the positive side. #hen thisdifference was multiplied by the forcing fre$uency, the phase angle was calculated.

Trial

Time to9teady9tate)s* Tmotion)s* Tforce)s*

Time;ifference)s*

eta(6 .+ , and>eta+6.+2. The values of 8 were very different for both methods of solution, indicating a large amountof error. This may be due to the fact that any error in solving for r will cause a very large error inamplification, especially for values around r6(. The measured values of the phase angle were alsocompared with those calculated from e$uation &. These speak to the accuracy of the theory.


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%igure +- amplification %actor vs. %re$uency ?atio. 9hows the relationship between the theoretical andactual values for the amplification factor. Although both graphs trend together, there seems to be a largeamount of error in the magnitude. The ma"imum value for 8 is when r 6.0.

%igure &- 9hows the amplification factor versus fre$uency ratio for group two, both measured andtheoretical. This data suffers from the same amplification error as figure +.

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Magnification Factor Vs. Frequency Ratio

GG

M a g n

i f i c a

t i o n

F a c

t o r

0

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Magnification Factor Vs. Frequency Ratio

GG

M a g n

i f i c a t i o n

F a c

t o r


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%igure 3 The relationship between the measured and theoretical values for 8 are shown. They aree"tremely close, and thus speak to the accuracy of the theory, as well as the measurements.

%igure 13 9hows the relationship between the theoretical and actual phase angles. ;ue to their closenature, it can be concluded that the theory is indeed accurate.

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# ase $ng!e Vs. Frequency Ratio

Group 1Group 1

# a s e

$ n g

! e

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

3

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"

8

# ase $ng!e Vs. Frequency Ratio

Group 2Group 2 T eoretica!

Frequency Ratio

# a s e

$ n g

! e

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