TunedVibrationAbsorber

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Tuned Vibration Absorbers:

Analysis, Visualization, Experimentation, and Design

Dr. R.E. Kielb, Dr. H.P. Gavin, C.J. Dillenbeck

Pratt School of Engineering

Duke University

Durham NC 27708

[email protected] , [email protected]

November 9, 2005

Abstract

A tuned vibration absorber is a relatively small spring-mass oscillator that sup-

presses the response of a relatively large, primary spring-mass oscillator at a partic-

ular frequency. The mass of the tuned vibration absorber is typically a few percentof the mass of the primary mass, but the motion of the tuned vibration absorber

is allowed to be much greater than the expected motion of the primary mass. The

natural freuqency of the tuned vibration absorber is tuned to be the same as the

frequency of excitation. Tuned vibration absorbers are particularly effective when

the excitation frequency is close to the natural frequency of the primary system.

In this web-based experiment, you will use the basic concepts involved in the

analysis and design of a tuned vibration absorber to:

predict the behavior of an experimental tuned vibration absorber,

modify your mathematical model baesd on observed behavior of the tuned

vibration absorber,

use your updated mathematical model for the tuned vibration absorber to

design a better tuned vibration absorber, and

test your re-designed system to verify the re-designed absorber.

All computations for analysis and design can be easily accomplished in Matlab

and all measurements will be accomplished using Dukes Web-based Educational

framework for Analysis, Visualization and Experimentation at

http://weave.duke.edu/weave/ .

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1 Purpose

The purpose of this lesson is to provide experiences in:

1. calculating and measuring natural frequencies and frequency response functions of1 and 2 degree-of-freedom systems.

2. experimentally determining damping of 1 degree-of-freedom systems

3. designing and testing a tuned, lightly-damped vibration absorber

2 Physical Model

The physical model (shown in Figures 1 and 2) utilizes two beams and two lumpedmasses, M and m, to simulate a 2 degree-of-freedom (2dof) oscillator. The masses of thebeams are small as compared to the lumped masses and can be ignored. The length ofthe secondary beam, l, can be adjusted from 0 cm to 29 cm. In the retracted position,l = 0 cm, this model simulates a single degree-of-freedom oscillator with the stiffness ofthe primary beam and a mass of (M+ m). The upper support is assumed to have infinitestiffness. Additional properties are listed in Table 1.

FRONT VIEW

M

m

L

l

DEFORMED VIEW

X(t)

x(t)x(t)

X(t)..

..

SIDE VIEW

X

x

F(t)

Figure 1: Physical lay-out of the tuned vibration absorber experiment.

The stiffness of the primary beam,

K =1

4Ebp

tpL

3,

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Figure 2: Photograph of the physical lay-out.

Table 1: Physical properties of the tuned vibration absorber experiment.Property Primary Absorber Units

Mass 2.0 0.5 kgLength 91.44 variable cmThickness, t 0.635 0.159 cmWidth, b 5.08 2.54 cmModulus, E 73.1 73.1 GPaMass density 2768 2768 kg/m3

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relates the static force F to the static displacement X, F = KX. The stiffness of thesecondary beam ,

k =1

4Ebsts

l3

,

relates forces applied to the absorber mass to the deflection of the absorber beam, (xX).In the experiment, dynamic forcing is applied to the primary mass (M) of the experi-

mental model using a voice-coil actuator. A voice-coil actuator is made from a permanentmagnet and a cylindrical coil of magnet wire with N turns of diameter D. The constantmagnetic flux, B from the permanent magnet passes in the radial direction through thecylindrical coil of magnet wire. When a controlled electrical current, i(t), is applied tothe coil, a force F(t) is induced in the coil, according to

F(t) = NDBi(t) .

The motion of the primary mass, M, and the secondary mass, m, are measured usingmicro-electro-mechanical-sensing (MEMS) accelerometers. The accelerometers transducetheir acceleration to a voltage signal, which is proportional to the accelerations, X(t) andx(t).

3 Types of Excitation

There are three common types of excitation used for controlled vibration testing:

1. sinusoidal with a single frequency or slowly varying frequency;

2. impulsive; or

3. random with many many frequencies acting at the same time.

This WEAVE experiment utilizes excitation of the first type.

4 Theory of Vibrations

4.1 Single Degree of Freedom

The equations of motion of a 1 dof oscillator (see Figure 3), says that the forces due toinertia, mx(t), plus the forces due to viscous energy dissipation, cx(t), plus the forces dueto the structural stiffness, kx(t), must be in equilibrium with the external force, F(t).

mx(t) + cx(t) + kx(t) = F(t)

Dividing both sides of this equation by m, all terms in the equation have units of accel-eration,

x(t) +c

mx(t) +

k

mx(t) =

1

mF(t) .

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c

km

F

x

Figure 3: A single degree of freedom spring-mass-damper oscillator, with dynamic forcing,F(t) and dynamic reponse x(t).

Substituting 2n = k/m, and = c/(2

mk),

x(t) + 2nx(t) + 2nx(t) =

1

m

F(t) .

If the external forcing is harmonic (sinusoidal), then F(t) = Fcos(t) where is thefrequency of the external forcing. The forcing frequency is not necessarily equal to thenatural frequency, n. The natural frequency depends on the structural mass and stiffnesswhile the forcing frequency is independent of the structural properties.

When linear elastic structures are sinusoidally excited, they tend to respond sinu-soidally at the same frequency as the excitation frequency, . See Figure 4.

x(t) = x()cos(t ())

where () is the phases shift between the excitation, F(t) and the response, x(t). The

x

2/

F(t)

2/

/

x(t)

Ft

Figure 4: Sinusoidal forcing, F(t), and sinusoidal response, x(t), tend to have the samefrequency, , but different amplitudes (x and F) and a phase difference . The amplituderatio x/F and the phase difference both depend on the frequency of forcing, .

amplitude of the response is proportional to the amplitude of the forcing, and also dependson the frequency of the forcing.

x() =1/k

(1 2)2 + (2)2F ,

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where = /n is the frequency ratio. Note that if the forcing frequency is zero, thenthe resulting displacement would be static, xst = F /k. The phase shift if given by theformula,

() = tan1

2

1 2

.

01

23456789

10

0 0.5 1 1.5 2 2.5

magnituderatio

0

45

90

135

0 0.5 1 1.5 2 2.5

phase(degrees)

frequency ratio

Figure 5: Frequency response functions of a single degree of freedom spring-mass-dampersystem. Damping ratios are 0.05, 0.10, 0.15, 0.20, and 0.25. Top: magnitude amplification.Bottom: phase shift.

Note that when the forcing frequency is close to the resonant frequency, 1, andthe damping is small, 1, the response can become extremely large. The maximumdynamic amplification, x()/xst or Q, is

Q =1

21 2

For low levels of damping, Q 12

. Therefore, by knowing the value of the peak am-plification, Q, the damping ratio may be estimated, 2Q. A second way to estimatedamping in lightly damped structures is via the half-power-bandwidth method. If 1 and2 are frequencies below and above n at which x()/xst = Q/sqrt2, then the dampingratio is approximately (2 1)/(1 + 2).

If the forcing frequency is known be a particular value, then tuned vibration absorberscan effectively reduct the dynamic response of the system to be essentially zero. The

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design of tuned vibration absorbers requires an understanding of two-degree-of-freedomsystems.

4.2 Two Degree of Freedom: Primary and Absorber Systems

For the primary and absorber system shown in Figure 1, two dynamic equations of motiondescribe the motion of the primary mass M and the absorber mass m,

MX(t) + CX(t) cx(t) + KX(t) kx(t) = F(t),

andmx(t) + c(x(t) X(t)) + k(x(t)X(t)) = 0 .

These two equations may be written in matrix form as follows.

M 00 m

X(t)x(t)

+

C+ c cc c

X(t)x(t)

+

K+ k kk k

X(t)x(t)

=

10

F(t) .

If F(t) is sinusoidal,F(t) = F cos(t) ,

then the response tends also to be sinusoidal ,

X(t)x(t)

=

Xcos(t ())x cos(t ())

Substituting F(t) = F eit, X(t) = Xeit, and x(t) = xeit, into the equation above, andsolving for the complex amplitudes, X and x, leads to the following expressions

X =m2 + ci + k

(M2 + (C+ c)i + (K+ k))(m2 + ci + k) (ci + k)2 F ,

and

x =ci + k

(M2 + (C+ c)i + (K+ k))(m2 + ci + k) (ci + k)2 F ,

which you should be able to derive.If c is small, then, in order to make the amplitude of motion of the primary mass, X

equal to zero, it is sufficient to set m2 + k equal to zero. In other words, if the naturalfrequency of the absorber,

k/m equals the forcing frequency, , then the motion of the

primary mass will tend to zero.Figure 6 shows the frequency response function for a particular set of numerical values

for the masses, stiffnesses and damping rates. Note that

k/m/2/ 2.5 Hz, and thatthe motion of the primary mass at 2.5 Hz (solid line) is practically zero.

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0

5

10

15

2025

30

35

1 1.5 2 2.5 3 3.5 4 4.5 5

magnitude

0

90

180

270

1 1.5 2 2.5 3 3.5 4 4.5 5

phas

e(degrees)

frequency (Hertz)

Figure 6: Frequency response functions of a tuned vibration absorber system, solid line =primary mass, X; dashed line = absorber mass, x. K=5000 N; C=1 N/m/s; M=10 kg;k=500 N/m; c=2 N/m/s; m=2 kg

5 Tutorial Flow

The following flow-charts will lead you through a set of steps required for conducting yourWEAVE experiment. These flow-charts provide information with which you will makecalculations and will perform experiments. You will then use your calculations to designa tuned vibration absorber, which you will then be able to test.

5.1 Pre-test analysis

1. Single degree of freedom oscillator, The secondary beam length, l is zero and themass is M + m.

(a) Calculate the stiffness of the primary beam, K;

(b) Calculate what you would expect the natural frequency to be, p =

K/(M + m)

(rad/sec) given values for the stiffness and masses; and

(c) Make a plot of the amplitude and phase of the frequency response functionfrom F(t) to X(t).

2. Oscillator with a tuned vibration absorber. The secondary beam length, l is 18 cm.

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(a) Calculate the stiffness of the secondary beam, k;

(b) Calculate what you would expect the absorber natural frequency to be, a =k/m (rad/sec) given values for the stiffness and masses;

(c) Calculate what you would expect the natural frequencies to be, given valuesfor the stiffness and masses (note that this is an eigen-value problem);

(d) Make a plot of the amplitude and phase of the frequency response functionfrom F(t) to X(t), assuming light damping; and

(e) Make a plot of the amplitude and phase of the frequency response functionfrom F(t) to x(t), assuming light damping.

5.2 Run WEAVE experiments to confirm the pre-test analysis

These experiments will be conducted on-line through the web-site http://weave.duke.edu/weave/Each experiment will take about 30 seconds to 50 seconds to execute. Visual display ofthe experimental data and a digital data downloads for post-test analysis are possible.

1. Single degree of freedom oscillator, The secondary beam length, l is zero and themass is M + m.

(a) Run an experiment with sinusoidal forcing starting at 0.5 Hz and ending at 10Hz.

(b) Determine the frequency at which the response is the maximum. Does thiscorrespond to the previously predicted value? If not, then modify the value forthe primary system stiffness, K, in order to match the experimentally-derivedfrequency. Is the experimental value for K larger or smaller than the one usedto predict the natural frequency? What would explain this difference?

(c) Using the half-power-bandwidth method, determine the damping ratio of theprimary system. From this damping ratio, determine C.

(d) Make a plot of the amplitude and phase of the frequency response functionfrom F(t) to X(t) using the experimentally-derived values of C and K. Howdoes this compare to the predicted frequency response function?

2. Oscillator with a tuned vibration absorber. The secondary beam length, l is 18 cm.

(a) Run an experiment with sinusoidal forcing starting at 0.5 Hz and ending at 10Hz. Print out the plot of the experimental results.

(b) Determine the frequency at which the response of the primary mass is theminimum. Does this correspond to the previously predicted value? If not,then modify the value for the absorber system stiffness, k, in order to matchthe experimentally-derived frequency. Is the experimental value for K largeror smaller than the one used to predict the natural frequency? What wouldexplain this difference?

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(c) Find the damping ratio of the absorber system using the half-power bandwidthmethod. Use this value to approximate the damping constant of the absorber,c.

(d) Make a plot of the amplitude and phase of the frequency response functionfrom F(t) to X(t), using the experimentally-derived values. How does thiscompare to the predicted frequency response function?

(e) Make a plot of the amplitude and phase of the frequency response function fromF(t) to x(t), using the experimentally-derived values. How does this compareto the predicted frequency response function?

5.3 Re-design the tuned vibration absorber.

Given the information developed in the first two steps above, try to predict the length ofthe absorber beam, l, required to suppress the primary mass motion at a frequency thatis 125 percent of the absorber frequency found in the previous analysis. Show your work.

5.4 Re-test the re-designed tuned vibration absorber.

Again, using the web-based experiment, set the length of the absorber beam to the valuefound in the previous step. Run an experiment and determine the frequency at whichthe tuned vibration absorber resulted in minimal motion of the primary mass. Print outthe plot of the experimental results. Was this frequency close to the one you designedfor? What would explain the difference between the actual absorber frequency and your

designed absorber frequency?

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