Tuned Vibration Absorbers: Analysis, Visualization, Experimentation, and Design Dr. R.E. Kielb, Dr. H.P. H.P. Gavin, C.J. Dillenbeck 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 suppresses the response of a relatively large, primary spring-mass oscillator at a particular frequency. The mass of the tuned vibration absorber is typically a few percent of the mass of the primary mass, but the motion of the tuned vibration absorber is allowed allowed to be much much greater greater than the expected motion motion of the primary 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: •
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predict the behavior of an experimental tuned vibration absorber, modify modify your your mathemat mathematical ical model baesd baesd on observ observed ed behavior behavior of the tuned tuned vibration absorber, use your your updated updated mathemat mathematical ical model for the tuned tuned vibratio vibration n absorber absorber to design a better tuned vibration absorber, and test your re-designed system to verify the re-designed absorber.
All computation computationss for analysis analysis and design can be easily easily accompli accomplished shed in Matlab Matlab and all measurements will be accomplished using Duke’s Web-based Educational framework for Analysis, Visualization and Experimentation at http://weave.duke.edu/weave/ .
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Purpose
The purpose of this lesson is to provide experiences in: 1. calculating and measuring natural frequencies and frequency response functions of 1 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
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Physical Model
The physical model (shown in Figures 1 and 2) utilizes two beams and two lumped masses, M and m, to simulate a 2 degree-of-freedom (2dof) oscillator. The masses of the beams are small as compared to the lumped masses and can be ignored. The length of the 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 of the primary beam and a mass of (M + m). The upper support is assumed to have infinite stiffness. Additional properties are listed in Table 1. £ ¤
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X(t)
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F(t) l
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x
FRONT VIEW
x(t)
SIDE VIEW
x(t)
DEFORMED VIEW
Figure 1: Physical lay-out of the tuned vibration absorber experiment. The stiffness of the primary beam, 1 tp K = Eb p 4 L
2
3
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Figure 2: Photograph of the physical lay-out.
Table 1: Physical properties of the tuned vibration absorber experiment. Property
Mass Length Thickness, t Width, b Modulus, E Mass density
Primary
Absorber
Units
2.0 91.44 0.635 5.08 73.1 2768
0.5 variable 0.159 2.54 73.1 2768
kg cm cm cm GPa kg/m3
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relates the static force F to the static displacement X , F = KX . The stiffness of the secondary beam , 1 ts 3 k = Eb s , 4 l relates forces applied to the absorber mass to the deflection of the absorber beam, ( x − X ). In the experiment, dynamic forcing is applied to the primary mass ( M ) of the experimental model using a voice-coil actuator . A voice-coil actuator is made from a permanent magnet and a cylindrical coil of magnet wire with N turns of diameter D. The constant magnetic flux, B from the permanent magnet passes in the radial direction through the cylindrical coil of magnet wire. When a controlled electrical current, i(t), is applied to the coil, a force F (t) is induced in the coil, according to
F (t) = NπDBi(t) . The motion of the primary mass, M , and the secondary mass, m, are measured using micro-electro-mechanical-sensing (MEMS) accelerometers. The accelerometers transduce ¨ (t) and their acceleration to a voltage signal, which is proportional to the accelerations, X x¨(t).
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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 4.1
Theory of Vibrations Single Degree of Freedom
The equations of motion of a 1 dof oscillator (see Figure 3), says that the forces due to inertia, m¨ ˙ plus the forces due x(t), plus the forces due to viscous energy dissipation, cx(t), to the structural stiffness, kx(t), must be in equilibrium with the external force, F (t). ˙ + kx(t) = F (t) m¨ x(t) + cx(t) Dividing both sides of this equation by m, all terms in the equation have units of acceleration, 1 c k ˙ + x(t) = F (t) . x¨(t) + x(t) m m m 4
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m
k
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x
Figure 3: A single degree of freedom spring-mass-damper oscillator, with dynamic forcing, F (t) and dynamic reponse x(t).
√
2 n
Substituting ω = k/m, and ζ = c/(2 mk), ˙ + ωn2 x(t) = x¨(t) + 2ζωn x(t)
1 F (t) . m
¯ If the external forcing is harmonic (sinusoidal), then F (t) = F cos(ωt) where ω is the frequency of the external forcing. The forcing frequency is not necessarily equal to the natural frequency, ωn . The natural frequency depends on the structural mass and stiffness while the forcing frequency is independent of the structural properties. When linear elastic structures are sinusoidally excited, they tend to respond sinusoidally at the same frequency as the excitation frequency, ω. See Figure 4. ¯(ω)cos(ωt − φ(ω)) x(t) = x where φ(ω) is the phases shift between the excitation, F (t) and the response, x(t). The
x
2π/ω
x(t)
F
t F(t)
φ/ω
2π/ω
Figure 4: Sinusoidal forcing, F (t), and sinusoidal response, x(t), tend to have the same ¯ and a phase difference φ. The amplitude frequency, ω, but different amplitudes (¯x and F ) ¯ and the phase difference φ both depend on the frequency of forcing, ω. ratio x ¯/F amplitude of the response is proportional to the amplitude of the forcing, and also depends on the frequency of the forcing. x¯(ω) =
(1 − Ω1/k) + (2ζ Ω) F¯ , 2 2
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2
where Ω = ω/ωn is the frequency ratio. Note that if the forcing frequency is zero, then the resulting displacement would be static, xst = F¯ /k. The phase shift if given by the formula, 2ζ Ω φ(ω) = tan−1 . 1 − Ω2
o i t a r
e d u t i n g a m
) s e e r g e d ( e s a h p
10 9 8 7 6 5 4 3 2 1 0 0
0.5
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135 90 45 0 frequency ratio
Figure 5: Frequency response functions of a single degree of freedom spring-mass-damper system. 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, Ω ≈ 1, and the damping is small, ζ 1, the response can become extremely large. The maximum dynamic amplification, x¯(ω)/xst or Q, is Q=
1 √ 2ζ 1 −
ζ 2
For low levels of damping, Q ≈ 21ζ . Therefore, by knowing the value of the peak amplification, Q, the damping ratio may be estimated, ζ ≈ 2Q. A second way to estimate damping in lightly damped structures is via the half-power-bandwidth method. If ω 1 and ¯ (ω)/xst = Q/sqrt2, then the damping ω2 are frequencies below and above ωn at which x ratio is approximately (ω2 − ω1 )/(ω1 + ω2 ). If the forcing frequency is known be a particular value, then tuned vibration absorbers can effectively reduct the dynamic response of the system to be essentially zero. The 6
design of tuned vibration absorbers requires an understanding of two-degree-of-freedom systems.
4.2
Two Degree of Freedom: Primary and Absorber Systems
For the primary and absorber system shown in Figure 1, two dynamic equations of motion describe the motion of the primary mass M and the absorber mass m, ¨ (t) + C X ˙ (t) − cx(t) ˙ + KX (t) − kx(t) = F (t), M X and
˙ (t)) + k(x(t) − X (t)) = 0 . ˙ − X m¨ x(t) + c(x(t)
These two equations may be written in matrix form as follows.
M 0 X ¨ (t) C + c −c X ˙ (t) K + k 0
m
x¨(t)
+
−c
c
If F (t) is sinusoidal,
˙ x(t)
+
−k
−k
k
X (t) 1 x(t)
=
0
F (t) .
¯ cos(ωt) , F (t) = F
then the response tends also to be sinusoidal ,
X (t) X cos(ωt ¯ − Φ(ω)) x(t)
=
x¯ cos(ωt − φ(ω))
¯ iωt , and x(t) = x Substituting F (t) = F¯ eiωt , X (t) = Xe ¯eiωt , into the equation above, and ¯ and x¯, leads to the following expressions solving for the complex amplitudes, X 2
¯= X
+ ciω + k F¯ , (−M ω2 + (C + c)iω + (K + k))(−mω 2 + ciω + k) − (ciω + k)2
x¯ =
ciω + k F¯ , 2 2 2 (−M ω + (C + c)iω + (K + k))(−mω + ciω + k) − (ciω + k)
and
−mω
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 −mω2 + k equal to zero. In other words, if the natural frequency of the absorber, k/m equals the forcing frequency, ω, 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/π ≈ 2.5 Hz, and that the motion of the primary mass at 2.5 Hz (solid line) is practically zero.
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e d u t i n g a m
) s e e r g e d ( e s a h p
35 30 25 20 15 10 5 0 1
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270 180 90 0 frequency (Hertz)
Figure 6: Frequency response functions of a tuned vibration absorber system, solid line = ¯ ; dashed line = absorber mass, x primary 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
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Tutorial Flow
The following flow-charts will lead you through a set of steps required for conducting your WEAVE experiment. These flow-charts provide information with which you will make calculations and will perform experiments. You will then use your calculations to design a 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 the mass is M + m. (a) Calculate the stiffness of the primary beam, K ; (b) Calculate what you would expect the natural frequency to be, ωp = (rad/sec) given values for the stiffness and masses; and
K/(M + m)
(c) Make a plot of the amplitude and phase of the frequency response function ¨ (t). from F (t) to X 2. Oscillator with a tuned vibration absorber. The secondary beam length, l is 18 cm. 8
(a) Calculate the stiffness of the secondary beam, k; (b) Calculate what you would expect the absorber natural frequency to be, ωa = k/m (rad/sec) given values for the stiffness and masses;
(c) Calculate what you would expect the natural frequencies to be, given values for 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 function ˙ (t), assuming light damping; and from F (t) to X (e) Make a plot of the amplitude and phase of the frequency response function from 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 of the 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 the mass is M + m. (a) Run an experiment with sinusoidal forcing starting at 0.5 Hz and ending at 10 Hz. (b) Determine the frequency at which the response is the maximum. Does this correspond to the previously predicted value? If not, then modify the value for the primary system stiffness, K , in order to match the experimentally-derived frequency. Is the experimental value for K larger or smaller than the one used to predict the natural frequency? What would explain this difference? (c) Using the half-power-bandwidth method, determine the damping ratio of the primary system. From this damping ratio, determine C . (d) Make a plot of the amplitude and phase of the frequency response function ¨ (t) using the experimentally-derived values of C and K . How from F (t) to X does 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 10 Hz. Print out the plot of the experimental results. (b) Determine the frequency at which the response of the primary mass is the minimum. Does this correspond to the previously predicted value? If not, then modify the value for the absorber system stiffness, k, in order to match the experimentally-derived frequency. Is the experimental value for K larger or smaller than the one used to predict the natural frequency? What would explain this difference? 9
(c) Find the damping ratio of the absorber system using the half-power bandwidth method. 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 function ˙ (t), using the experimentally-derived values. How does this from F (t) to X compare to the predicted frequency response function? (e) Make a plot of the amplitude and phase of the frequency response function from ˙ using the experimentally-derived values. How does this compare F (t) to x(t), to 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 of the absorber beam, l, required to suppress the primary mass motion at a frequency that is 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 value found in the previous step. Run an experiment and determine the frequency at which the tuned vibration absorber resulted in minimal motion of the primary mass. Print out the plot of the experimental results. Was this frequency close to the one you designed for? What would explain the difference between the actual absorber frequency and your designed absorber frequency?
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