The authors designed a novel type of dynamic vibration absorber, called periodic vibration absorber (PVA), for mechanical systemssubjected to periodic excitation. Since the periodic rather than single harmonic excitation is the most occurring case in mechanicalsystems, the design of PVA is hence of engineering significance. The PVA designed in this paper is composed of a dual-beaminterconnected with a discrete spring in between. By appropriately choosing the design parameters, the PVA can be of resonancefrequencies in integer multiples of the base frequency such that the PVA can absorb significant amount of higher harmonicsin addition to the base harmonic. The designed PVA was first experimentally verified for its resonance frequencies. The PVAimplemented onto a mechanical system was then tested for its absorption ability. From both tests, satisfying agreement betweenexperiments and numerical calculations has been obtained. The sensitivities of the design variables, such as the discrete spring’sstiffness and location, were discussed as well.The parameters’ sensitivities provided us with the PVA’s adjustable room for excitationfrequency’smismatch. Numerical results showed that within 3%of frequencymismatch, the PVA still performed better than a singleDVA via adjusting the spring’s constant and location. All the results proved that the novel type of PVA could be a very effectivedevice for vibration reduction of mechanical systems subjected to periodic excitation.
1. Introduction
Dynamic vibration absorber (DVA), also known as tunedmass damper (TMD) [1], has been proven to be a usefuldevice for mechanical vibration attenuation. A conventionalDVA, composed of a spring-mass-damper (SMD), is mostlymounted to a primary structure, as shown in Figure 1(a)[2], to absorb the vibration of one single (tonal) frequency.The fundamental design of an SMD can be seen from manyvibration textbooks and is not addressed here.Themore elab-orate works on SMD fall into the category of damper design.Brock [3] derived an optimum Lanchester damper. Ozer andRoyston [4] extended Den Hartog’s method to multi-DOFstructure and derived the optimal dampers and mountinglocations. They further employed Sherman-Morrison matrixinversion formula to calculate the optimal parameters for adamped multi-DOF absorber system [5]. Ren [6] introducedthe so-called ground-hook DVA as shown in Figure 1(b).Of the same mass ratio and under harmonic excitation, theground-hook DVA appeared to have better absorption than
the traditional one. Wong and Cheung, similarly, concludedthat the ground-hookDVA’s vibration suppression is superiorto the traditional one particularly when the excitation comesfrom the ground motion [7].
In order to enhance the DVA’s absorption ability, numer-ous papers have been aimed at different research aspects, suchas control rules derivation, structure’s properties variation,special material and different DVA’s combination. Details ofthese researches may be referenced to Sun et al. [8], wherethey surveyed the gradual development of the passive, adap-tive, and active tuned vibration absorber. Chen and Xu [9]discussed a DVA comprised of not only mass-spring-viscousdamper but feedback control force to suppress broadbandvibration and their results showed that the response had beenreduced by 90%. Burdisso and Heilmann [10] developed ahybridDVA for vibration control as shown in Figure 1(c).TheDVA comprised two reaction masses and in between therewas a passive/semiactive/active damper. This hybrid DVAhas been proven to have better suppression effect than anordinary SMD, particularly for broadband vibration.
Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 571421, 11 pageshttp://dx.doi.org/10.1155/2014/571421
2 Shock and Vibration
k
M
K
x1
x2
Traditional DVA
m
C
(a)
C
Ground-hook DVA
k
M
K
x1
x2m
(b)
M
Kx1
x2 x3
k1
m1m2
Active force
k2
(c)
C
C1 C2
M
K
x1
x2 x3
k1
m1m2
k2
(d)
Figure 1: Four often-seen DVA’s, (a) conventional, (b) ground-hook, (c) dual-mass, and (d) multimode.
As to multifrequency vibration suppression, Sun et al.[11] studied the differences in vibration absorption betweenthe conventional DVA, the state-switched absorber (SSA),and dual-DVA (Figure 1(d)). Under the optimization process,their results showed that dual-DVA had very close perfor-mance as the SSA and both were, as expected, superior to theconventional DVA in multifrequency vibration suppression.The conclusion for dual-DVA design is as simple as tuningeach DVA to a frequency wanted to be absorbed. Hill andSnyder [12] designed a dual mass absorber to suppress thevibration atmultiple frequencies, which consisted of two rods(smooth and threaded) supporting two equal masses (bells)on both sides.This device was able to tune the first six naturalfrequencies, mixed in bending and torsion modes. Thenatural frequencies were yet restricted to be tuned in pairs,that is, first and second together, and so on. In 2006, Wangand Cheng [13] used the impedance technique to designa multifrequency absorber by varying single beam’s severalcross-sectional areas such that the beam’s natural frequen-cies coincided with the designated frequencies. Although a
geometrically nonuniform beam could theoretically attainany desired multifrequency, it however required very tediouscalculations and complicated shaping in manufacturing.
In real world, most of the mechanical systems are sub-jected to periodic rather than simple harmonic excitation.Mathematically, periodic excitation is composed of a seriesof infinite harmonics in integer multiples of base frequency.Nevertheless, only are the first few components significantin mechanical vibration. In general cases, the first three har-monics contain about 90% of the overall excitation. To sup-press periodic excitation, onemay employ themultifrequencyDVA technique [11] and tune three DVA’s frequencies to thefirst three harmonics. The consequence, yet, will be signifi-cantmass loading to themain system.Themass loading effectmay be reduced by lowering the DVA’s mass ratio but the costwould be the absorber highly sensitive to excitation frequencyvariation.The present investigation is hence motivated by thenecessity of developing a simple, passive PVA of relativelylow mass ratio for periodic excitation. The derived dual-beam PVA (Figure 2) in this paper will prove itself to meet
Shock and Vibration 3
X
Y
M
F
PVA
k
xs
K/2 K/2
Figure 2: The schematic diagram of primary system mounted witha dual-beam PVA.
k
Decomposed
Figure 3: Decomposition of the dual-beam PVA.
the goal and to provide significant applicability for vibrationengineers.
2. Frequency Equation of PVA
Figure 2 schematically shows the designedPVAmounted on aprimary system to resist periodic excitation.𝑀 and𝐾 denotethe primary system’smass and stiffness, respectively.ThePVAis composed of two cantilever beams (dual-beam) with anintermediate spring of constant 𝑘. The intermediate springis connected at the position of 𝑥
𝑠. For simplicity, though not
necessary, the two beams’ length is assumed to be the same.𝜌𝑖, 𝐴𝑖, 𝐸𝑖, and 𝐼
𝑖, 𝑖 = 1, 2, stand for the 𝑖th beam’s density,
cross-sectional area, Young’s modulus, and area moment ofinertia, respectively.
Utilizing the structure combination technique and thereceptance method [14, 15], this PVA can be divided into twoparts, as shown in Figure 3, and the corresponding frequencyequation is
𝛼 (𝜔) + 𝛽 (𝜔) = 0, (1)
where 𝛼(𝜔) is the receptance of the first cantilever beam andcan be expressed [15] as
𝛼 (𝜔) =
𝑀1
∑
𝑛=1
Φ2
𝑛(𝑥𝑠)
𝜌1𝐴1(𝜔2
1𝑛− 𝜔2) ∫
𝐿
0
Φ2𝑛(𝑥) 𝑑𝑥
. (2)
Similarly, 𝛽(𝜔) stands for the receptance of the second can-tilever beam plus spring,
𝛽 (𝜔) =
𝑀2
∑
𝑛=1
Φ2
𝑛(𝑥𝑠)
𝜌2𝐴2(𝜔2
2𝑛− 𝜔2) ∫
𝐿
0
Φ2𝑛(𝑥) 𝑑𝑥
+1
𝑘, (3)
whereΦ𝑛(𝑥) is a cantilever beam’s 𝑛th mode shape. 𝜔
𝑖𝑛is the
𝑖th beam’s 𝑛th natural frequency. 𝑀1and 𝑀
2, respectively,
are the mode numbers of two beams. Note that the cantileverbeam is assumed to be of Bernoulli-Euler’s model and 𝜔 (or𝜔𝑝, 𝑝 = 1, 2, . . .) denotes the PVA’s natural frequency to be
solved.The design criterion is to make the PVA’s first few natural
frequencies be in integer multiples of the base frequencyof periodic excitation; that is, 𝜔
𝑝= 𝑝𝜔
𝑓, 𝑝 = 1, 2, . . . .
Substituting (2) and (3) into (1), and it is obtained that
𝑀1
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
𝜌1𝐴1(𝜔2
1𝑛− (𝑝𝜔
𝑓)2
) ∫𝐿
0
Φ2𝑛(𝑥) 𝑑𝑥
+
𝑀2
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
𝜌2𝐴2(𝜔2
2𝑛− (𝑝𝜔
𝑓)2
) ∫𝐿
0
Φ2𝑛(𝑥) 𝑑𝑥
+1
𝑘= 0,
𝑝 = 1, 2, . . . ,
(4)
where 𝑥∗𝑠= 𝑥𝑠/𝐿 is the normalized spring location and 𝑝𝜔
𝑓
is the PVA’s 𝑝th natural frequency.For simplicity, a dimensionless parameter, 𝛼
𝑛= 𝜔1𝑛/
𝜔11
= 𝜔2𝑛/𝜔21
denoting the known ratios of a cantileverbeam’s 𝑛th natural frequency to its first natural frequency, isintroduced. Equation (4) is then rewritten as
𝑀1
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
𝜌1𝐴1(𝜔2
11𝛼2𝑛− (𝑝𝜔
𝑓)2
) ∫𝐿
0
Φ2𝑛(𝑥) 𝑑𝑥
+
𝑀2
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
𝜌2𝐴2(𝜔2
21𝛼2𝑛− (𝑝𝜔
𝑓)2
) ∫𝐿
0
Φ2𝑛(𝑥) 𝑑𝑥
+1
𝑘= 0.
(5)
We further define two design variables, 𝜛1= 𝜔11/𝜔𝑓and
𝜛2= 𝜔21/𝜔𝑓, denoting the ratios of the first and the second
cantilever beam’s fundamental natural frequency to the basefrequency of excitation. Equation (5) becomes
𝑀1
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
(𝜛2
1𝛼2𝑛− 𝑝2) ∫
𝐿
0
Φ2𝑛(𝑥) 𝑑𝑥
+
𝑀2
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
(𝜛2
2𝛼2𝑛− 𝑝2) ∫
𝐿
0
Φ2𝑛(𝑥) 𝑑𝑥
𝜌1𝐴1
𝜌2𝐴2
+
𝜌1𝐴1𝜔2
𝑓
𝑘= 0.
(6)
4 Shock and Vibration
0.48
0.47
0.46
0.45
0.70 0.80 0.90 1.00
𝜇 = 2.0
𝜇 = 3.0
𝜇 = 4.0
x∗s
𝜛1
(a)
1.95
1.90
1.85
1.80
1.75
1.70
𝜛2
𝜇 = 2.0
𝜇 = 3.0
𝜇 = 4.0
0.70 0.80 0.90 1.00
x∗s
(b)
1.20
1.00
0.80
0.60
0.40
0.20
k∗
𝜇 = 2.0
𝜇 = 3.0
𝜇 = 4.0
0.70 0.80 0.90 1.00
x∗s
(c)
Figure 4: Variations of design parameters to different 𝑥∗𝑠.
Computer
ONO SOKKI DS-2000Multichannel data station
LV-1710, ONO SOKKILaser vibrometer
MeasureLaser
Hammer
Figure 5: The schematic diagram of PVA experimental test.
Utilizing the orthogonal property of beam’s mode shapesand one normalized equation, ∫𝐿
0
Φ2
𝑛(𝑥)𝑑𝑥 = 𝐿, (6) is sim-
plified to be
𝑀1
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
(𝜛2
1𝛼2𝑛− 𝑝2)
+
𝑀2
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
(𝜛2
2𝛼2𝑛− 𝑝2)
𝜌1𝐴1
𝜌2𝐴2
+
𝐿𝜌1𝐴1𝜔2
𝑓
𝑘= 0.
(7)
Subsequently, further reduction yields
𝑀1
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
(𝜛2
1𝛼2𝑛− 𝑝2)
+
𝑀2
∑
𝑛=1
Φ2
𝑛(𝑥∗
𝑠)
(𝜛2
2𝛼2𝑛− 𝑝2) 𝜇
+1
𝑘∗= 0, (8)
where 𝜇 = (𝜌2𝐴2)/(𝜌1𝐴1) denotes another design variable,
the mass ratio of the two beams. The normalized springconstant 𝑘∗ = 𝑘(𝜛
2
1𝐿3
)/(𝛽4
1𝐸1𝐼1) is the last design variable,
where 𝛽1is the first mode wave number of single cantilever
beams; that is, Φ1= (sinh𝛽
1𝑥 − sin𝛽
1𝑥) − 𝜆
1(cosh𝛽
1𝑥 −
cos𝛽1𝑥). Equation (8) presents the PVA’s frequency equation
in a dimensionless form.Mathematically, we have derived thedesign problem as an explicit function of five design variables(𝜛1, 𝜛2, 𝑘∗, 𝑥∗
𝑠, and 𝜇); that is, 𝑓(𝜛
1, 𝜛2, 𝑘∗
, 𝑥∗
𝑠, 𝜇) = 0. It
implies that themaximumnumber of PVA’s frequencies couldbe tuned up to five; that is, PVA’s first five natural frequenciesmight be determined via appropriate selection of the fivedesign variables. It is yet unnecessary and nonrealistic to tuneso many frequencies since the first three harmonics usuallycontain more than 90% of the excitation. To tune higherharmonics more closely it will sacrifice the accuracy of lowerones, which are yet the most important. In the followingexamples, we will tune the PVA only up to the first threenatural frequencies; that is, 𝑝 = 1 ∼ 3.
3. Simulations and Experimental Verification
From (8), it is seen that there is room to set two out offive design variables as known values. Prior to doing that,it is helpful to realize the intercorrelation between all designvariables. We first set the mass ratio (𝜇), once at a time, at aspecific value and look into the correlations between 𝑥
∗
𝑠and
the other three parameters (𝜛1, 𝜛2, and 𝑘
∗). The curves aredrawn in Figure 4. It is seen that all parameters vary with 𝑥
∗
𝑠
in a nonlinear trend. From the shown curves, it is obvious that𝜛1and 𝑘
∗ are more sensitive to 𝑥∗
𝑠. 𝜛2curves are rather flat
relative to𝑥∗𝑠variations.𝜛
2, yet, showsmuch larger sensitivity
(curves farther apart) to 𝜇’s change than 𝜛1and 𝑘∗ do. These
correlations shown in Figure 4 provide us with a reference todetermine the design variables, although not in an optimalsense. For example, one may first select appropriate 𝜇 and𝜛2(most sensitive); then, from its corresponding 𝑥
∗
𝑠one
Shock and Vibration 5
0.00 70.00 140.00 210.00 280.00 350.001
10
1E + 2
1E + 3
1E + 4
1E + 5
FRF
(Hz)
70Hz
138Hz
209Hz
(a)
0.00 35.00 70.00 105.00 140.00 175.00
1E + 6
(Hz)
1
10
1E + 2
1E + 3
1E + 4
1E + 5
FRF
35Hz
69Hz
109Hz
(b)
Figure 6: Experimental FRF of (a) specimen A and (b) specimen B.
Table 1: The geometrical and material properties of PVA.
(a)
Specimen AFirst beam (SUS304) Second beam (Al)
Density (kg/m3) 𝜌1= 7800 𝜌
2= 2710
Modulus (×109 N/m2) 𝐸1= 200 𝐸
2= 69
Thickness (mm) ℎ1= 0.88 ℎ
2= 3.68
Width (mm) 𝑏1= 20.0 𝑏
2= 27.57
Length (mm) 𝐿1= 175.0 𝐿
2= 175.0
Spring constant: 𝑘 = 0.791KN/m; connecting location: 𝑥𝑠= 171.5mm.
𝑓1= 70Hz.
(b)
Specimen BFirst beam (SUS304) Second beam (Al)
Density (kg/m3) 𝜌1= 7800 𝜌
2= 2710
Modulus (×109 N/m2) 𝐸1= 200 𝐸
2= 69
Thickness (mm) ℎ1= 1.19 ℎ
2= 4.97
Width (mm) 𝑏1= 25.0 𝑏
2= 34.47
Length (mm) 𝐿1= 250.0 𝐿
2= 250.0
Spring constant: 𝑘 = 0.8301KN/m; Connecting location: 𝑥𝑠= 245.0mm.
𝑓1= 35Hz.
can continue for suitable 𝜛1and 𝑘
∗. The above-mentionedprocess is just one of many possibilities. Two examples solvedby the above process are illustrated and the calculations aregiven in Table 1.
A simple experiment is then set up to verify if the obtainedPVA (Table 1) has integer multiples of the base frequencyas we desired. Figure 5 shows the setup of experimentalapparatus. The displacement and corresponding FRFs of the
Table 2: Simulated response amplitudes due to periodic excitationof 𝑓 = 120Hz.
Wave Without absorber With DVA With PVA
Square wave 0.33 0.012 0.004(−28.8 dB) (−38.3 dB)
Saw-tooth 0.18 0.026 0.007(−16.8 dB) (−28.2 dB)
Table 3: Simulated and experimental results with a saw-tooth waveof 𝑓 = 55Hz.
Without absorberexperiment
With PVAsimulation
With PVAexperiment
0.014 0.001 0.0015(−22.9 dB) (−19.4 dB)
PVA are picked up and transferred by an ONO SOKKILaser Vibrometer (LV-1710). Figure 6 shows the FRFs of twospecimens and it is seen that the first three natural frequenciesare very close to integer multiples of the base frequency, one70Hz and one 35Hz. The PVAs’ first resonance frequenciescoincided precisely with the base frequency but the secondand the third showed some errors (3.80% at most). The firstthree modes of specimen A are sketched in Figure 7 and it isseen that the first beam deforms more significantly than thesecond one for its larger flexibility (Table 1).
Now, implement the PVA onto the main system andanalytically calculate the vibration reduction of the mainsystemdue to periodic excitation.Thedata of themain systemare chosen as follows: 𝑀 = 0.5Kg, 𝐾 = 6.125 kN/m, and𝑓𝑛= 17.6Hz. Unit square wave loading is first discussed. Note
6 Shock and Vibration
Seco
nd b
eam
Firs
t bea
m
Base Base
Base
[Number 1] [Number 2]
[Number 3]
Seco
nd b
eam
Firs
t bea
m
Seco
nd b
eam
Firs
t bea
m
Figure 7: The calculated first three modes of PVA.
that in the following simulations, the excitation frequencyis set at 120Hz throughout the paper unless compared toexperiments, in which the excitation is set at 55Hz. Figure 8shows three simulated response amplitudes of the primarysystem for (i) with no absorber, (ii) with a single DVAtuned at 𝜔DVA = 𝜔
𝑓, and (iii) with the designed PVA.
To have a fair comparison, DVA’s mass is equal to PVA’s.Figure 8(a) compares the responses of (i) and (iii) and theabsorption effects of PVA are very significant. The reasonwe did not superimpose (ii) in Figure 8(a) is because underthe same scale it is difficult to see the differences between(ii) and (iii). Instead, Figure 8(b) enlarges the responses of(ii) and (iii) and the differences between them reflect thecontribution of higher harmonics. Note that the responseamplitudes shown in the above figures, afterwards as well,are all in a dimensionless form by normalizing them withrespect to the static displacement; that is, 𝑋/𝑋
𝑠, 𝑋𝑠
=
𝐹/𝐾. Simulations for unit saw-tooth wave loading at thesame frequency are illustrated in Figure 9 and similar results
are obtained. From Figures 8(b) and 9(b), one may noticethat the higher harmonics of saw-tooth and square-wavehave different frequencies even though they are of the samebase frequency. This can be explained after Fourier seriesexpansion of the square-wave and saw-tooth functions,
𝑓 (𝑡) =
∞
∑
𝑛=1
𝐵𝑛sin (𝑛𝜔
𝑓𝑡) ,
𝐵𝑛=
{{{{
{{{{
{
2
𝑛𝜋(1 − cos 𝑛𝜋) , for squarewave,
2
𝑛𝜋(1 − cos 𝑛𝜋
2) , for saw-toothwave.
(9)
Since we used odd functions in both cases, the secondharmonic (𝑛 = 2) of the square wave is zero by itself but notin the saw-tooth case and the third harmonics for both casesare of the same magnitude. That explains why Figure 8(b)showed the 3rd harmonic and Figure 9(b) showed the 2nd
Shock and Vibration 7
0.50
0.25
0.00
−0.25
−0.50
Dim
ensio
nles
s res
pons
e
Without PVAWith PVA
0.00 0.01 0.02 0.03 0.04 0.05
Time (s)
(a)
0.02
0.01
0.00
−0.01
−0.02
Dim
ensio
nles
s res
pons
eWith single mass DVAWith PVA
0.00 0.01 0.02 0.03 0.04 0.05
Time (s)
(b)
Figure 8: Simulation of PVA effect due to square wave loading for (a) overall absorption and (b) higher harmonics absorption.
0.20
0.10
0.00
−0.10
−0.20
Dim
ensio
nles
s res
pons
e
Without PVAWith PVA
0.00 0.01 0.02 0.03 0.04 0.05
Time (s)
(a)
0.03
−0.03
0.02
0.00
−0.02
Dim
ensio
nles
s res
pons
e
With single mass DVAWith PVA
0.00 0.01 0.02 0.03 0.04 0.05
Time (s)
(b)
Figure 9: Simulation of PVA effect due to saw-tooth wave loading for (a) overall absorption and (b) higher harmonics absorption.
in the curves of (ii). Table 2 compares the absorption effectsof all the simulated cases. In a square-wave case, a DVA isgood enough to reduce the system response amplitude by28.8 dB but just 16.8 dB in the case of saw-tooth because thesquare-wave originally contains no second harmonic. The
PVA yet reduces the system responses by 38.3 dB and 28.2 dB,respectively, for the square-wave and the saw-tooth, contrastto the DVA, 9.5 dB, and 11.4 dB more.
Experiments follow to verify the above simulations.Figure 10(a) shows the photo and Figure 10(b) shows the
8 Shock and Vibration
(a)
Computer
Agilent 33120A, function/arbitrary
Shaker
Measure
LV-1710, ONO SOKKILaser vibrometer
ONO SOKKI DS-2000Multichannel data station
Laser
PA30E, linear power amplifier
(b)
Figure 10: (a) Photo of experimental setup of main system with PVA and (b) schematic diagram.
0.00 0.05 0.10 0.15 0.20
0.02
0.01
0.00
−0.01
−0.02
Time (s)Without PVAWith PVA
Dim
ensio
nles
s res
pons
e
Figure 11: Experimental data of time response.
schematic diagram of apparatus setup. Figure 11 is the exper-imental time responses of the system with and without PVAunder saw-tooth excitation of frequency 55Hz. It is observedthat themain system’s vibration is drastically reduced by PVA.Figure 12 compares the experimental data of Figure 11 and thesimulation results. Note that at the first glance there seemsa huge discrepancy between experiment and simulation butthorough inspection will reveal that the major differencecomes from the nonzero DC term in the experimental data.
Figure 12: Comparison of the simulation and experimental data.
If one shifts the DC bias, the maximum amplitudes of thesimulation and the experiment are, respectively, 0.001 and0.0015. Table 3 illustrates the data and the difference is about3.5 dB. This difference seems not to be negligible but incontrast to the overall effect of 19.4 dB, this discrepancyis acceptable. This discrepancy can be attributed to thefollowing causes. First, the PVA did not perfectly match thehigher harmonics (3.8% error) and secondly all the analyseswere based on undamped situation. Though no intendeddamper was added in the primary system or PVA, there
Shock and Vibration 9
0.02
0.01
0
−0.01
−0.02
Varia
tion
of n
atur
al fr
eque
ncy
0 1 2 3 4Change of spring location (%)
−4 −3 −2 −1
1st natural frequency2nd natural frequency3rd natural frequency
(a)
0 1 2 3 4Change of spring location (%)
0.02
0.01
0
−0.01
−0.02
Varia
tion
of n
atur
al fr
eque
ncy
−4 −3 −2 −1
1st natural frequency2nd natural frequency3rd natural frequency
(b)
Figure 13: PVA’s natural frequencies vary with (a) spring’s stiffness 𝑘∗ and (b) spring’s location 𝑥∗
𝑠.
0
0.05
0.1
0 2E − 3 4E − 3 6E − 3 8E − 3
−0.1
Time (s)
Dim
ensio
nles
s res
pons
e
0% mismatch
−0.05
0.01
1% mismatch (adjusts k∗)1% mismatch (no adjustment)
Figure 14: System response to small mismatch frequency with (a) one variable and (b) two-variable adjustment.
10 Shock and Vibration
0.00 1.00 2.00 3.00 4.00
0.01
0.1
Loading frequency ratio (𝜔/𝜔f)
1E − 9
1E − 8
1E − 7
1E − 6
1E − 5
1E − 4
1E − 3
Resp
onse
PVA’s natural frequency
Figure 15: FRF of the combined structure.
exists damping in the real world. A small amount of dampingwould shift the best absorption frequency and reduces theattenuation effects.
Though the simulation has shown PVA’s excellent vibra-tion absorbability, to meet engineering applications, the PVAhas to be adjustable to slight excitation frequency variationwithout reconstructing PVA. Reviewing all of the designvariables, 𝑘
∗ and 𝑥∗
𝑠, the spring’s stiffness and position
appear to be the two easiest ones for adjustment withoutreassembling the structure. Figure 13 illustrates how the PVA’sfirst three resonance frequencies vary with (a) the springstiffness and (b) the spring location. The ordinates are thedimensionless frequency variations; that is, Δ𝜔
𝑖/𝜔𝑖, 𝑖 =
1, 2, 3. As anticipated, both the spring stiffness and locationchange PVA’s resonance frequencies monotonically, that is,changing 𝑘
∗ and 𝑥∗
𝑠, will shift all the first three frequencies
in the same direction but of different sensitivity. For instance,the first frequency (𝑝 = 1) is the most sensitive to 𝑘∗ but thethird one (𝑝 = 3) is the most sensitive to 𝑥
∗
𝑠. Since all the
sensitivities are not in proportion, it is unable to tune 𝑘∗, 𝑥∗𝑠
and simultaneously retain the first three natural frequenciesin exact integer multiples as wanted. The objective is toreduce the response to a maximum amount; therefore, thefirst harmonic should be adjusted to the least error. Figure 14shows some of the solutions to reducing the mismatch effect.Figure 14(a) compares the system responses of perfect match,1% frequency mismatch without any adjustment and withsingle 𝑘
∗ adjustment. It is seen that the system’s vibrationlevel increased by mismatch and was obviously attenuated bysimply tuning the spring stiffness constant (3.78% increase).Figure 14(b) compares the differences of tuning one andtwo variables. With simultaneous adjustment of 𝑘∗ and 𝑥
∗
𝑠,
the response amplitude is further reduced. From the aboveillustration we are confident that the designed PVA offerssatisfactory periodic vibration absorbability and can be tunedto correspond to slightly external frequency variations.
Our calculations revealed, not shown in this paper, thatthe adjustment yielded poor results for the 3rd harmonic asthe mismatch exceeded 3%. It is yet still assured that the first
frequency can always be adjusted to meet the base frequency𝜔𝑓even for larger excitation frequency variation and the PVA
still performs better than a single DVA. This can be verifiedby comparing the response amplitudes. From Table 2, theamplitude with a DVA is 0.026 and the maximum amplitudein Figure 14(b) is around 0.02, less than DVA’s. Figure 15shows the FRF of the combined primary-PVA system. Thedash lines denote the PVA’s resonance frequencies and theybecome antiresonance frequencies of the combined system.
4. Conclusions
In this paper, a periodic vibration absorber (PVA) of dual-beam type is for the first time ever designed, analyticallydiscussed, and experimentally verified. The PVA consists oftwo cantilever beams interconnected with an intermediatediscrete spring. When the spring is appropriately chosen andlocated, the PVA can very effectively attenuate any periodicexcitation. The frequency equation of the designed PVAwas theoretically derived from the receptance method andsubsequently arranged in a general, dimensionless form interms of five design variables. Enforcing the PVA’s resonancefrequencies in integer multiples and solving the frequencyequation, the PVA’s parameters were determined accordingto the excitation base frequency. This paper demonstratedexamples of setting the PVA’s first three resonance frequenciesin integer multiples of the base frequency and the resultsappeared to be accurate by experimental verification.
The responses of the primary system with/without thedesigned PVA were calculated in simulations. As expected,the results showed PVA’s excellent absorption effect to peri-odic excitation. Experiments followed to verify the theoreticalcalculations and satisfactory agreement has been obtained.From the shown examples, PVA could improve the responseamplitude 9.5∼11.4 dB more, compared to a single DVA. Theerror of PVA’s absorption between simulation and experimentwas about 3.5 dB that might be attributed to the designvariables’ variations and the ignored damping existed instructures. The ability of adjusting spring’s stiffness andlocation to compensate the mismatch of excitation frequencywas studied as well. The results showed that PVA could bewell tuned if mismatch was less than 3% and the tuned PVAstill performed better than a single DVA.The derived PVA inthis paper is useful for the audience in vibration engineeringand is believed to provide an efficient and effective tool forsuppressing periodic vibration of structures.
Nomenclature
𝐴𝑖: Cross-sectional area of 𝑖th beam
𝐸𝑖: Young’s modulus of 𝑖th beam
𝐼𝑖: Area moment of inertia of 𝑖th beam
𝑘: Stiffness of an intermediate spring𝑘∗: Normalized spring constant𝐾: Stiffness of the primary system𝐿: Length of beam𝑀: Mass of the primary system𝑥𝑠: Spring location
𝑥∗
𝑠: Normalized spring location, 𝑥∗
𝑠= 𝑥𝑠/𝐿
Shock and Vibration 11
𝛼(𝜔): Receptance of the first cantilever beam𝛼𝑛: Ratios of cantilever beam’s 𝑛th natural
frequency to its first natural frequency𝛽(𝜔): Receptance of the second cantilever beam
plus spring𝛽1: First mode coefficient of a single cantilever
beam𝜔 : PVA’s natural frequency𝜔𝑖𝑛: 𝑛th natural frequency of 𝑖th beam
𝜔𝑓: Fundamental frequency of the periodic
excitation𝜛𝑖: Ratios of the 𝑖th beam’s first natural
frequency to the fundamental frequencyof the excitation
𝜌𝑖: Density of 𝑖th beam
𝜇: Mass ratio of the two beamsΦ𝑛: 𝑛th mode of a cantilever beam.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
The authors are grateful to the National Science Council forits support of this research under the Grant no. NSC 98-2811-E-011-001.
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