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Vibration isolation using a hybrid lever-type isolation system with an X-shape supporting structure
International Journal of Mechanical Sciences 98 (2015) 169–177
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Vibration isolation using a hybrid lever-type isolation system with an X-shape supporting structure Chunchuan Liu a, Xingjian Jing a,n, Fengming Li b a b
Department of Mechanical Engineering, Hong Kong Polytechnic University, Hong Kong, China School of Mechanical Engineering, Beijing University of Technology, Beijing, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 18 December 2014 Received in revised form 27 March 2015 Accepted 16 April 2015 Available online 25 April 2015
This study presents some novel results about analysis and design of low-frequency or broadbandfrequency vibration isolation using a hybrid lever-type isolation system with an X-shape supporting structure in passive or semi-active control manners. It is shown that the system has inherent nonlinear stiffness and damping properties due to structure geometrical nonlinearity. Theoretical analysis reveals that the hybrid isolation system can achieve very good ultra-low-frequency isolation through a significantly-improved anti-resonance frequency band (by designing structure parameters). Noticeably, the system can realize a uniformly-low broadband vibration transmissibility, which has never been reported before. Cases studies show that the system can work very well with good isolation performance subject to multi-tone and random excitations. The results provide a new innovative approach to passive or semi-active vibration control (e.g., via a simple linear stiffness control) for many engineering problems with better ultra-low/broadband-frequency vibration suppression. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Vibration isolation Geometrical nonlinearity Anti-resonant frequency Nonlinear damping
1. Introduction Vibration control is often a critical and difficult issue in many engineering practices. A typical example can be seen in microvibration control of on-orbit spacecraft. Micro-vibration can be produced by various mechanical parts in on-orbit spacecraft such as cryocoolers, mobile mirrors, and reaction/momentum wheel assemblies, mainly appearing in the frequency range from less than 1 Hz up to 1 kHz [1,2]. Because of the tiny environmental damping in aerospace, micro-vibration may exist for a long time, which can thus degrade working environment of sensitive instruments in onboard spacecraft. Usually, passive isolation techniques are commonly employed to suppress vibration [3–6]. The vibration energy can be dissipated through passive damping, and high frequency disturbances can be suppressed by the designed low dynamic stiffness of isolator systems. Passive vibration isolation is known for its high reliability, easy implementation and low development cost, while may not be effective for low frequency vibration. Recently, some nonlinear vibration isolation methods have been developed in the literature [7–15,22–26], which demonstrate excellent vibration isolation performance both in high and low frequency range, i.e., high static but low dynamic stiffness. This can be
n
Corresponding author. E-mail address: [email protected] (X.J. Jing).
http://dx.doi.org/10.1016/j.ijmecsci.2015.04.012 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
achieved through nonlinear stiffness design of a isolation system leading to different nonlinear stiffness properties such as quasi-zero stiffness [7–11] and negative stiffness [14,15]. However, in some cases complicated nonlinear phenomena could also happen if system parameters are not well designed, incurring potentially worse stability problems. Recently, a passive scissor-like structured isolation system was proposed and studied in [33,16], where an excellent high-static-low-dynamic stiffness property can be achieved by flexibly designing structural parameters. To address low frequency vibration isolation, active control methods are often used [17–21]. Different isolation systems with different active control strategies can be designed for various requirements of low frequency vibration isolation. Although excellent low frequency isolation performance could be obtained through active isolation techniques, high energy cost, high development expense, actuation saturation, stability issues, and/or complexity in implementation etc. could occur. Alternatively, the semi-active control method can be employed to realize excellent vibration transmissibility at resonant frequency and above through some simple nonlinear damping systems [22–26]. But ultra-low frequency vibration control may still be a problem that cannot be addressed with pure damping control methods. System inertia can be changed in a lever-type isolation system, which can reduce resonant frequency and produce an anti-resonant frequency [27–29]. Since the anti-resonant frequency can be designed into low frequency region (smaller than the resonant
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x M
z1
z2
l11
l22
m1
l21
l12
kl y
m2
Fz 2
l l
θ Base
Fig. 1. The hybrid SLS lever-type vibration isolation system.
frequency), the ultra-low frequency isolation effect can therefore be achieved with the lever-type isolation system [27]. The inertia force with the system can be increased by changing mass ratio and lever length ratio, which can be employed to reduce the supporting spring force. Therefore, the dynamic stiffness of the system can be reduced but a relative high static stiffness maintained [29]. However, only designing system inertia as mentioned, the difference between the resonant and anti-resonant frequencies is relatively very small, and thus the ultra-low-frequency isolation performance is rather limited. It is noticed that Jo and Yabuno [30,31] proposed a new type of vibration absorbers utilizing quadratic nonlinear coupling to reduce the vibration amplitude. The main idea of the vibration reduction is based on mode coupling such that the vibration amplitude only around the natural frequency of the master vibration system can be reduced [30]. In this paper, a novel X-shape structure strengthened hybrid lever-type vibration isolation system is investigated, which combines the features of the scissor-like structure and lever-type structure together, and is aimed at exploring inherent structure nonlinearity for superior low-/broadband-frequency vibration isolation/control performance in passive and/or semi-active manners. The mathematical modeling of the system is conducted with the harmonic balance method (HBM) for analysis of the nonlinear dynamics to derive the vibration transmissibility. It is systematically shown that this hybrid isolation system can demonstrate much better ultra-low frequency vibration isolation with designable antiresonance frequency and lower transmissibility in a relative larger frequency range by designing structure parameters. Noticeably, the system can also achieve obvious vibration suppression covering the full frequency range with proper structure parameter selection and a simple linear stiffness feedback control. These advantageous performances should be of more significance in engineering practice. To demonstrate the effectiveness of the proposed hybrid structure system, vibration isolation subject to multi-tone and random excitations at the base is also demonstrated as case studies. The rest of this paper is organized as follows. The design of Xshape structural hybrid lever-type isolation system is introduced in Section 2. The mathematic modeling and analysis are carried out in Section 3, where the inherent nonlinearity is discussed in detail. Then, vibration isolation performance with parameter sensitivity analysis is studied in Section 4. Section 5 shows
dynamic responses of the novel hybrid lever-type isolation system under multi-zone and random excitations. Finally, a conclusion is drawn.
2. The X-shape structured hybrid lever-type isolation system The hybrid lever-type vibration isolation system supported by an X-shape structure is shown in Fig. 1. The supporting structure is an n-layer scissor-like or X-shape structure with horizontal spring [16]. Each layer has two rods combined with one joint, and length of each rod is 2l. Angle of the rod with respect to horizontal line is θ, and the angle displacement of the joints is represented by ϕ. The restoring force in the horizontal spring is linear with a stiffness kl . The n-layer X-shape structure can provide high static and low dynamic stiffness [16], which is very important to obtain good and stable isolation both in the low and high frequency range. The absolutely motions of the isolation body M and base are represented as x and y. z1(t) and z2(t) are the absolutely displacements of the attached mass m1 and m2 in the levers type I and type II [27,28]. The mass m1 is located on the type I lever, where the lever ratio α1 ¼l11/l12, while, type II lever is attached to m2, which has the lever ratio α2 ¼ l21/l22. The simple linear stiffness feedback control is applied to the mass m2, i.e., F z2 ¼ kz2 z2 .
3. Mathematical modeling and analysis Due to function of the levers, the motion of the two attached mass m1 and m2 can be expressed as: z1 ¼ α1 y ðα1 1Þx;
z2 ¼ α2 x ðα2 1Þy:
ð1Þ
So the kinetic energy of the hybrid isolation system is written as, 1 1 T ¼ M x_ 2 þ ½m1 ðα1 1Þ2 þ m2 α22 x_ 2 ½m1 α1 ðα1 1Þ þ m2 α2 ðα2 1Þx_ y_ 2 2 1 þ ½m1 α21 þm2 ðα2 1Þ2 y_ 2 ; 2
ð2Þ
C. Liu et al. / International Journal of Mechanical Sciences 98 (2015) 169–177
The potential energy of the isolation system is given by: 1 V ¼ kl z2 : 2
ð3Þ
where z is the horizontal displacement. The dynamic equation can be determined using the Hamilton principle, and the virtual work δw is given by: _~ x~ c z_ δz c n φδφ δw ¼ c1 xδ þ F z2 δz2 ; 2 3 j_
ð4Þ
where c1 and c2 are the damping coefficients in the vertical and horizontal direction, c3 is the rotational fiction coefficient of each joint, nj is the joints number of the X-shape structure, and x~ ¼ x y is the relative displacement. Fz2 is the active linear stiffness feedback control force based on the absolutely displacement z2, which is denoted as: F z2 ¼ kz2 z2 ;
ð5Þ
From Eqs. (1)–(5) and using the Hamilton principle, the dynamic equation of the hybrid lever-type vibration isolation system can be obtained as: " 2 dz dx~ dz þ c1 þc2 M x€~ þ ½m1 ðα1 1Þ2 þ m2 α22 x€~ þ kl z dx~ dx dx~ 2 # dφ x_~ þ kz2 α22 x~ þ c 3 nj dx~ ¼ M y€ þ ½m1 ðα1 1Þ m2 α2 y€ kz2 α2 y;
ð6Þ
where the relationship between displacements ϕ, z and x~ can be given by (see also [16]): ~ l sin θ þ x=ð2nÞ θ; φ ¼ arctan l cos θ z=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ~ z ¼ 2l cos θ 2 l ½l sin θ þ x=ð2nÞ : ð7Þ dz dx~ Using 2the Taylor series expansion for the terms kl zdx~ dx , dφ and dx~ , Eq. (6) can be changed as:
dz 2 dx~
½1 þ μ2 α22 þμ1 ðα1 1Þ2 M x€~ þ ðkz2 α22 þβ1 Þx~ þ β2 x~ 2 þ β3 x~ 3 þ c3 nj ζ 1 Þx~ x_~ þ ðc2 ζ 5 þ c3 nj ζ 2 Þx~ 2 x_~ ¼ M½μ1 ðα1 1Þ ðμ2 α2 þ 1Þy€ kz2 α2 y;
ð8Þ
where μ1 ¼ m1 =M and μ2 ¼ m2 =M are the mass ratio of attached mass m1 and m2 with respect to isolated mass M, the coefficients β1, β2, β3, β4, ζ0, ζ1, ζ2, ζ3, ζ4, and ζ5 are the Taylor series expansion terms, which are listed in the Appendix. Define the dimensionless variables as: t^ ¼ ω1 t;
η1 ¼ βk1 ; l
η2 ¼ βk2 ; l
η3 ¼ βk3 ; l
η4 ¼ βk4 ; l
γ 1 ¼ kkz2 ; l
ðc1 þ c2 ζ 3 þ c3 nj ζ 0 Þ ðc2 ζ 4 þ c3 nj ζ 1 Þ ðc2 ζ 5 þ c3 nj ζ 2 Þ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ξ1 ¼ ; ξ2 ¼ ; ξ3 ¼ ; 2 Mkl 2 Mkl 2 Mkl pffiffiffiffiffiffiffiffiffiffiffi ^ Eq. (8) can be where ω1 ¼ kl =M , and denoting ðÞ0 ¼ dðÞ=dt, rewritten as: ~ þ ðγ 1 α22 þ η1 Þx~ þ η2 x~ 2 þ η3 x~ 3 þ η4 x~ 4 þ 2ξ1 x~ 0 ½1 þ μ1 ðα1 1Þ2 þ μ2 α22 x″ þ 2ξ2 x~ x~ 0 þ 2ξ3 x~ 2 x~ 0 ¼ ½μ1 ðα1 1Þ ðμ2 α2 þ 1Þy″ γ 1 α2 y:
ð9Þ
Taking the base excitation as the harmonic excitation ^ where Ω ¼ ω0 =ω1 and ω0 is the external exciting y ¼ A0 cos Ωt, frequency. Through the Harmonic Balance Method (HBM), the relative motion can be given generally as [32]: x~ ¼ B0 þ B1 cos ðΩt^ þ ϕÞ;
where B0 is the bias term. Substituting Eq. (10) into the dimensionless dynamic Eq. (9), it gives: ! η B2 3 3η B4 η2 B20 þ η3 B30 þ η4 B40 þ η1 B0 þ 2 1 þ η3 B0 B21 þ 3η4 B20 B21 þ 4 1 ¼ 0; 2 2 8 ð11Þ n
ðη1 þ γ 1 α22 ÞB1 ½1 þ μ1 ðα1 1Þ2 þ μ2 α22 Ω2 B1 þ 2η2 B0 B1 þ 3η3 B20 B1 3 þ 4η4 B30 B1 þ ðη3 B31 þ 4η4 B0 B31 Þ 4 ¼ f½1 þ μ2 α2 μ1 ðα1 1ÞΩ2 γ 1 α2 gA0 cos ϕ;
ð10Þ
ð12Þ
ð2ξ1 ΩB1 þ 2ξ2 ΩB0 B1 þ2ξ3 ΩB1 B20 þ ξ3 ΩB31 Þ ¼ f½1 þ μ2 α2 μ1 ðα1 1ÞΩ2 γ 1 α2 gA0 sin ϕ; The displacement transmissibility can be achieved as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 uB þ A2 þ 2B1 A0 cos ϕ 0 T ¼t 1 ; A20
ð13Þ
ð14Þ
where B1 and ϕ are the amplitude and phase angle of the relative ~ and they can be determined by Eqs. (11)–(13). motion x, From the linear part of dynamic equation, the anti-resonant frequency Ωa and resonant frequency Ωr can be determined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η1 þ γ 1 α22 η1 þ γ 1 α2 ðα2 1Þ ; Ωr ¼ ; Ωa ¼ ½μ1 α1 ðα1 1Þ þ μ2 α2 ðα2 1Þ ½1 þ μ1 ðα1 1Þ2 þ μ2 α22 ð15Þ The anti-resonant frequency Ωa can be designed to be smaller than the resonant frequency Ωr by properly choosing structure parameters such as μ1, μ2, α1, α2, γ1 and η1. This will produce excellent isolation performance in low frequency range. In order to validate the accuracy of the first order harmonic response with the HBM, the second order harmonic response is also used to analyze the nonlinear dynamic Eq. (9), which can be assumed as follows: x~ ¼ B0 þB1 cos ðΩt^ þϕ1 Þ þB2 cos ð2Ωt^ þ ϕ2 Þ;
þ β4 x~ 4 þ ðc1 þ c2 ζ 3 þc3 nj ζ 0 Þx_~ þðc2 ζ 4
171
ð16Þ
Substituting Eq. (16) into the dynamic Eq. (9), five equations on the variables B0, B1, B2, ϕ1, and ϕ2 can be determined, which can be solved by the iterative method using the Maple software. The absolute displacement transmissibility of the proposed isolation system is computed with the first order and second order harmonics as shown in Eq. (2). The system parameters are taken as follows: kl ¼ 100 N/m, M¼1kg, c1 ¼0.5, c2 ¼0.1, c3 ¼0.1, μ1 ¼μ2 ¼ 0.1, α1 ¼3.0, α2 ¼1.2, n¼2, θ¼451 and γ1 ¼3.0. nj ¼ 3n, and l¼ 0.2 m, and the base excitation amplitude A0 ¼ 0.006 m. In Fig. 2, the displacement transmissibility of the fundamental amplitude is calculated with the HBM using the first and second order harmonic responses. As shown in Fig. 2, the results with only the first order harmonic response are the same as those using the first two order harmonic responses. It can be further checked that the higher order harmonics have very little effect on the dynamic response of the nonlinear system (9). Therefore, only the first order harmonic response is used in the following discussions.
4. Results and discussions The vibration isolation performance of the proposed hybrid lever-type isolation system is studied in this section with respect to different structure parameters. For convenience, some parameters are chosen as constant in the analysis as: kl ¼100 N/m,
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Fig. 2. Transmissibility computed by first and second order harmonics.
Fig. 3. Isolation effect in low frequency domain with different SLS layer numbers n.
Fig. 4. Isolation effect in low frequency domain with different assembly angles θ.
Fig. 5. Isolation effect in low frequency domain with different lever ratios α1.
M¼ 1kg, c1 ¼0.5, c2 ¼0.1, c3 ¼ 0.1, nj ¼3n, and l ¼0.1 m. Moreover, the base excitation amplitude A0 ¼ 0.005 m. 4.1. Isolation performance at low frequency In Figs. 3–9, the isolation performance of the proposed hybrid structure system at low frequency (i.e., 0o Ωo1) is investigated, and the sensitivity analysis of some structure parameters is shown. The designable structure parameters mainly are: mass ratio μ1 and μ2, lever ratio α1 and α2, layer number n and assembly angle θ, active stiffness feedback ratio γ1. To consider the influence of a specific structure parameter, other parameters will take constant values in the following analysis without further specification as follows: μ1 ¼μ2 ¼ 0.1, α1 ¼4.0, α2 ¼1.1, n ¼3, θ¼ 301 and γ1 ¼4.0. In Figs. 3 and 4, the isolation effects at low frequency with different X-shape structural layer number and assembly angle are analyzed. As shown in Fig. 3, the anti-resonant and resonant frequencies are about 0.65 and 1.56 for layer number n¼2. The isolation effect at low frequency is very good in that the displacement transmissibility is about 19.5 dB at the beginning of the excitation frequency and the minimal transmissibility is about 33.04 dB at the anti-resonant frequency. For the layer number n¼1, the displacement transmissibility is about 16.5 dB at the beginning of the frequency and the minimal transmissibility is about 22.4 dB at the anti-resonant frequency about 0.79, which shows that the isolation effect for the layer number n¼2 is much better than that of n¼1 in the low frequency domain. For the layer number
Fig. 6. Isolation effect in low frequency domain with different lever ratios α2.
n¼3 and n¼4, the displacement transmissibility at the beginning of the frequency is about 20.2 dB and 20.5 dB, and the minimal transmissibility values are about 37.2 dB and -39.2 dB, respectively. It indicates that the isolation effect of the proposed isolation system at low frequency will become better as the layer number increasing and the anti-resonant frequency and minimal transmissibility are becoming smaller as well.
C. Liu et al. / International Journal of Mechanical Sciences 98 (2015) 169–177
Fig. 7. Isolation effect in low frequency domain with different mass ratios μ1.
Fig. 8. Isolation effect in low frequency domain with different mass ratios μ2.
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angle with respect to the horizontal line. This suggests that the isolation performance at low frequency can be much improved by reducing the dynamic stiffness of the X-shape structure. In Figs. 5 and 6, the displacement transmissibility and isolation effect with different lever ratios α1 and α2 are discussed. As shown in Fig. 5, the displacement transmissibility at the beginning of excitation frequency is the same for different lever ratios α1. However, the antiresonant frequency and corresponding displacement transmissibility are both decreased with the lever ratio α1 increasing. For the lever ratio α1 ¼ 2.0, the displacement transmissibility is almost horizontal before the excitation frequency about 0.88. In Fig. 6, the displacement transmissibility at ultra-low frequency and anti-resonant frequency for the lever ratio α2 ¼1.05 are about 25.1 dB and 39.9 dB, which are smaller than those for the lever ratio α2 ¼1.1. The anti-resonant frequency and corresponding transmissibility become much bigger for larger lever ratio α2. Displacement transmissibility at low frequency is sensitive with the change of the lever ratio α2 as shown in Fig. 6. This indicates that the lever ratio α2 plays much more important role in designing excellent ultra-low frequency isolation. In Figs. 7 and 8, the displacement transmissibility with different attached mass ratios μ1 and μ2 is given. As shown in Fig. 7, the displacement transmissibility at ultra-low excitation frequency is not changed with increase of the mass ratio μ1, but the anti-resonant frequency and corresponding transmissibility become smaller with increasing the mass ratio μ1. Fig. 8 shows that the mass ratio μ2 has almost no influence on the isolation performance in the low frequency range. It can be concluded that the isolation effect can be improved by increasing the mass ratio μ1 but not the mass ratio μ2. From the analysis in Figs. 3–8, excellent ultra-low frequency isolation performance can be obtained by designing passive structure parameters. In Fig. 9, the influence of the active linear stiffness control (i.e., ratio γ1) on isolation effect in the low frequency range is discussed. The displacement transmissibility at ultra-low excitation frequency and anti-resonant frequency is all becoming smaller, and the effective low-frequency isolation range also becoming larger with the increase of the stiffness ratio γ1, while the anti-resonant frequency is becoming bigger simultaneously. These clearly demonstrate that the isolation performance in the low frequency range is sensitively affected by the linear stiffness feedback control. 4.2. Isolation performance over the full frequency range
Fig. 9. Isolation effect in low frequency domain with different active stiffness ratios γ1.
In Fig. 4, the displacement transmissibility at the beginning frequency and anti-resonant frequency for the assembly angle 451 are about 19.07 dB and 34.3 dB, which is bigger than those for the assembly angle 301. Those for the assembly angle 601 are about 16.5 dB and 28.2 dB, which is much bigger than those for the assembly angle 301 and 451. Fig. 4 also shows that the antiresonant frequency becomes larger with increasing the assembly
In Figs. 10–17, the special isolation performance of the proposed hybrid structure system in the whole frequency range is discussed. Fig. 5 already indicates that excellent isolation performance both in low frequency and high frequency range could be obtained by choosing properly the lever ratio α1. To explore this further, in the calculation of the results in Figs. 10–17, the structure parameters are taken as, μ1 ¼ μ2 ¼0.1, α1 ¼1.7, α2 ¼1.1, n ¼3, θ¼301 and γ1 ¼1.0 if no further specification. In Fig. 10, the displacement transmissibility with different active stiffness ratios γ1 is shown. For the active stiffness ratio γ1 ¼1.0, the displacement transmissibility at the ultra-low excitation frequency is about 18.6 dB, and is very similar to that at high frequency about 19.02 dB. The displacement transmissibility almost keeps a flat line over the whole frequency range except those around the resonance frequency which have the peak value equal to 0.035 dB. Comparing with the results in Figs. 2–8, the anti-resonance frequency range disappears. Fig. 10 also shows that the resonant frequency becomes larger but the peak transmissibility is almost not changed with the increase of the active stiffness feedback control. In Figs. 11 and 12, the isolation performance in the whole frequency region is discussed with different lever ratio α1 and ratio α2. As shown in Fig. 11, the resonant frequency and corresponding
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Fig. 10. Isolation performance in whole frequency domain with different stiffness ratios.
Fig. 11. Isolation performance in whole frequency domain with different lever ratios α1.
Fig. 13. Isolation performance in whole frequency domain with different mass ratios μ1.
Fig. 14. Isolation performance in whole frequency domain with different mass ratios μ2.
Fig. 12. Isolation performance in whole frequency domain with different lever ratios α2.
Fig. 15. Isolation performance in whole frequency domain with different layer numbers.
transmissibility value change little as the change of the lever ratio α1. The displacement transmissibility at ultra-low excitation frequency keeps the same for different lever ratios α1, but the transmissibility at high frequency increases for bigger lever ratio α1. In Fig. 12, the transmissibility at ultra-low frequency and the peak transmissibility for lever ratio α2 ¼1.2 are about 14.5 dB and
3.01 dB, and those for lever ratio α2 ¼ 1.3 are about 12.1 dB and 6.6 dB, which are much larger than those for the lever ratio α2 ¼1.1. In Figs. 13 and 14, the effects of the attached mass ratio μ1 and μ2 on the isolation performance in the whole frequency region are shown. In Fig. 13, the peak transmissibility values for the mass ratio μ1 ¼0.05 and μ1 ¼0.15 are about 1.92 dB and 1.16 dB, which are
C. Liu et al. / International Journal of Mechanical Sciences 98 (2015) 169–177
Fig. 16. Isolation performance in whole frequency domain with different assembly angles θ.
175
Fig. 18. Response in time domain under multi-tone excitation for different isolation systems 1.
5. Vibration isolation subject to multi-tone and random excitations
Fig. 17. Isolation performance in whole frequency domain with different damping coefficients. Table 1 Parameters used in the hybrid isolation system. Parameters
μ1
μ2
α1
α2
n
θ
γ1
Low frequency Full frequency
0.1 0.1
0.1 0.1
4.0 1.7
1.1 1.1
3 3
301 301
4.0 1.0
much bigger than those for mass ratio μ1 ¼0.1. The transmissibility at high frequency becomes larger as increasing the mass ratio μ1. As shown in Fig. 14, the displacement transmissibility is almost not changed for different mass ratios μ2, except that the resonant frequency is reduced with increase of the mass ratio μ2. In Figs. 15 and 16, the effects of the layer number and assembly angle on the isolation performance in the whole frequency domain are given. Fig. 15 shows that the displacement transmissibility at ultra-low frequency and resonant frequency decreases with increase of layer number n, while increases at high frequency. In Fig. 16, the transmissibility at ultra-low frequency and resonant frequency decreases with decrease of the assembly angle θ, while increases at high frequency simultaneously. These suggest different structure parameters should be chosen for different vibration isolation requirements. In Fig. 17, the displacement transmissibility is almost not changed with different damping coefficients. However, the effective isolation frequency range is much improved as decreasing the damping coefficient but with increasing peak value.
The vibration isolation performance of the proposed hybrid isolation system subject to multi-tone and random excitations is studied, compared with corresponding hybrid isolation system without the X-shape structure. The X-shape structure can provide specially-designed nonlinearity in stiffness and damping. In order to study the influence of these nonlinear stiffness and damping properties on the performance of the proposed isolation system, traditional springs and viscous dampers are used to replace the X-shape structure for comparison with the same stiffness and damping coefficients as those in the X-shape structure. As discussed before, the parameters of the proposed hybrid isolation system for vibration isolation in low or full frequency ranges can be chosen as in Table 1, which are referred to as low-frequency parameter set and full-frequency parameter set, respectively. In Figs. 18 and 19, dynamic responses in time domain for different isolation systems under multi-tone excitations are shown, where initial phase angles are assumed to zeros. The base multi-tone excitation in Fig. 18 is taken as: amplitude z1 ¼ 0.005, z2 ¼0.004, z3 ¼ 0.003, and frequency ω01 ¼3 Hz, ω02 ¼ 5 Hz, and ω03 ¼6 Hz. Fig. 18 indicates that the responses for the proposed hybrid isolation system with low-frequency parameter set are much smaller than those for that without X-shape supporting structure, demonstrating excellent low frequency vibration isolation performance. In Fig. 20, dynamic response in time domain under multi-zone excitation (z1 ¼ 0.005, z2 ¼ 0.004, z3 ¼0.003, ω01 ¼4 Hz, ω02 ¼14 Hz, and ω03 ¼24 Hz) is shown. It shows that the responses of the hybrid isolation system without X-shape structure are much larger (thus of worse performance) than those for the proposed hybrid isolation system with full-frequency parameter set. In Figs. 20 and 21, the dynamic responses with different vibration isolation systems subject to base white noise acceleration excitations are discussed. The power spectral density function and mean value of base white noise acceleration excitations are taken as s0 ¼0.001 and zero, respectively. But the frequency ranges for the low frequency and broadband white noise excitations are taken as (0–5 Hz) and (4–100 Hz). As shown in Figs. 20 and 21, the dynamic responses for the designed hybrid isolation system under the low frequency or broadband white noise excitations are much smaller than those of the corresponding hybrid isolation system without X-shape supporting structure. It can be concluded that excellent vibration isolation performance can be obtained with the proposed hybrid structure system subject to both multi-tone and random excitations. Note that the
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6. Conclusions
Fig. 19. Response in time domain under multi-tone excitation for different isolation systems 2.
A novel hybrid lever-type isolation system strengthened by an X-shape supporting structure is proposed, which combines the advantageous features of the scissor-like structure and lever-type structures together. System inherent nonlinear stiffness and damping characteristics are explored to significantly improve vibration isolation/control performance in passive and/or semi-active manners. It is shown that the proposed hybrid isolation system can achieve very good ultra-low-frequency vibration isolation with tunable antiresonance frequency and low-frequency transmissibility by designing structure parameters. Moreover, a uniformly-low isolation transmissibility over full frequency range can also be realized with proper values of structure parameters. To the best of our knowledge, this full-frequency vibration suppression performance has not been reported before. The isolation performances mentioned above are verified by case studies with comparisons subject to multi-tone and random excitations. The results demonstrate again that the inherent nonlinear stiffness and damping characteristics introduced by the X-shape structure play an important and beneficial role in vibration isolation and control. Further study will focus on prototyping and experimental testing of the proposed hybrid isolation system.
Acknowledgment The authors would like to thank the handling editors and anonymous reviewers for your time and effort in reviewing this paper. The authors also gratefully acknowledge the support from a NSFC Project (No 61374041) of China, a GRF Project of HK RGC (Ref no.15206514), a CAST-funded Project (H-ZG2D), and internal Competitive Research Grants of Hong Kong Polytechnic University.
Appendix A
Fig. 20. Response under low frequency white noise excitation for different isolation systems.
The coefficients β1, β2, β3, β4, ζ0, ζ1, ζ2, ζ3, ζ4, ζ5 in Eq. (8) are listed as follows: β1 ¼
kl tan 2 θ ; n2
ða:1Þ
β2 ¼
3kl tan θ ; 4n3 l cos 3 θ
ða:2Þ
β3 ¼ β4 ¼ ζ0 ¼ ζ1 ¼
ζ2 ¼ Fig. 21. Response under broad-band frequency white noise excitation for different isolation systems.
hybrid isolation system without the X-shape supporting structure is a pure linear system. The comparisons above demonstrate also that the nonlinear stiffness and damping characteristics introduced by the X-shape supporting structure are very crucial for the performance improvement and very beneficial in vibration control.
ð5 4 cos 2 θÞkl 2
8n4 l cos 6 θ
;
5 sin θð5 2 cos 2θÞkl 3
64n5 l cos 8 θ 1 2
4n2 l cos 2 θ tan θ 3
4n3 l cos 3 θ
ða:3Þ
;
ða:4Þ
;
ða:5Þ
;
ða:6Þ
ð1 þ 3 sin 2 θÞ 4
8n4 l cos 6 θ
;
ða:7Þ
ζ3 ¼
tan 2 θ ; n2
ða:8Þ
ζ4 ¼
tan θ ; n3 l cos 3 θ
ða:9Þ
ζ5 ¼
ð1 þ 3 sin 2 θÞ 2
2n4 l cos 6 θ
:
ða:10Þ
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Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Vibration isolation using a hybrid lever-type isolation system with an X-shape supporting structure Chunchuan Liu a, Xingjian Jing a,n, Fengming Li b a b
Department of Mechanical Engineering, Hong Kong Polytechnic University, Hong Kong, China School of Mechanical Engineering, Beijing University of Technology, Beijing, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 18 December 2014 Received in revised form 27 March 2015 Accepted 16 April 2015 Available online 25 April 2015
This study presents some novel results about analysis and design of low-frequency or broadbandfrequency vibration isolation using a hybrid lever-type isolation system with an X-shape supporting structure in passive or semi-active control manners. It is shown that the system has inherent nonlinear stiffness and damping properties due to structure geometrical nonlinearity. Theoretical analysis reveals that the hybrid isolation system can achieve very good ultra-low-frequency isolation through a significantly-improved anti-resonance frequency band (by designing structure parameters). Noticeably, the system can realize a uniformly-low broadband vibration transmissibility, which has never been reported before. Cases studies show that the system can work very well with good isolation performance subject to multi-tone and random excitations. The results provide a new innovative approach to passive or semi-active vibration control (e.g., via a simple linear stiffness control) for many engineering problems with better ultra-low/broadband-frequency vibration suppression. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Vibration isolation Geometrical nonlinearity Anti-resonant frequency Nonlinear damping
1. Introduction Vibration control is often a critical and difficult issue in many engineering practices. A typical example can be seen in microvibration control of on-orbit spacecraft. Micro-vibration can be produced by various mechanical parts in on-orbit spacecraft such as cryocoolers, mobile mirrors, and reaction/momentum wheel assemblies, mainly appearing in the frequency range from less than 1 Hz up to 1 kHz [1,2]. Because of the tiny environmental damping in aerospace, micro-vibration may exist for a long time, which can thus degrade working environment of sensitive instruments in onboard spacecraft. Usually, passive isolation techniques are commonly employed to suppress vibration [3–6]. The vibration energy can be dissipated through passive damping, and high frequency disturbances can be suppressed by the designed low dynamic stiffness of isolator systems. Passive vibration isolation is known for its high reliability, easy implementation and low development cost, while may not be effective for low frequency vibration. Recently, some nonlinear vibration isolation methods have been developed in the literature [7–15,22–26], which demonstrate excellent vibration isolation performance both in high and low frequency range, i.e., high static but low dynamic stiffness. This can be
n
Corresponding author. E-mail address: [email protected] (X.J. Jing).
http://dx.doi.org/10.1016/j.ijmecsci.2015.04.012 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
achieved through nonlinear stiffness design of a isolation system leading to different nonlinear stiffness properties such as quasi-zero stiffness [7–11] and negative stiffness [14,15]. However, in some cases complicated nonlinear phenomena could also happen if system parameters are not well designed, incurring potentially worse stability problems. Recently, a passive scissor-like structured isolation system was proposed and studied in [33,16], where an excellent high-static-low-dynamic stiffness property can be achieved by flexibly designing structural parameters. To address low frequency vibration isolation, active control methods are often used [17–21]. Different isolation systems with different active control strategies can be designed for various requirements of low frequency vibration isolation. Although excellent low frequency isolation performance could be obtained through active isolation techniques, high energy cost, high development expense, actuation saturation, stability issues, and/or complexity in implementation etc. could occur. Alternatively, the semi-active control method can be employed to realize excellent vibration transmissibility at resonant frequency and above through some simple nonlinear damping systems [22–26]. But ultra-low frequency vibration control may still be a problem that cannot be addressed with pure damping control methods. System inertia can be changed in a lever-type isolation system, which can reduce resonant frequency and produce an anti-resonant frequency [27–29]. Since the anti-resonant frequency can be designed into low frequency region (smaller than the resonant
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x M
z1
z2
l11
l22
m1
l21
l12
kl y
m2
Fz 2
l l
θ Base
Fig. 1. The hybrid SLS lever-type vibration isolation system.
frequency), the ultra-low frequency isolation effect can therefore be achieved with the lever-type isolation system [27]. The inertia force with the system can be increased by changing mass ratio and lever length ratio, which can be employed to reduce the supporting spring force. Therefore, the dynamic stiffness of the system can be reduced but a relative high static stiffness maintained [29]. However, only designing system inertia as mentioned, the difference between the resonant and anti-resonant frequencies is relatively very small, and thus the ultra-low-frequency isolation performance is rather limited. It is noticed that Jo and Yabuno [30,31] proposed a new type of vibration absorbers utilizing quadratic nonlinear coupling to reduce the vibration amplitude. The main idea of the vibration reduction is based on mode coupling such that the vibration amplitude only around the natural frequency of the master vibration system can be reduced [30]. In this paper, a novel X-shape structure strengthened hybrid lever-type vibration isolation system is investigated, which combines the features of the scissor-like structure and lever-type structure together, and is aimed at exploring inherent structure nonlinearity for superior low-/broadband-frequency vibration isolation/control performance in passive and/or semi-active manners. The mathematical modeling of the system is conducted with the harmonic balance method (HBM) for analysis of the nonlinear dynamics to derive the vibration transmissibility. It is systematically shown that this hybrid isolation system can demonstrate much better ultra-low frequency vibration isolation with designable antiresonance frequency and lower transmissibility in a relative larger frequency range by designing structure parameters. Noticeably, the system can also achieve obvious vibration suppression covering the full frequency range with proper structure parameter selection and a simple linear stiffness feedback control. These advantageous performances should be of more significance in engineering practice. To demonstrate the effectiveness of the proposed hybrid structure system, vibration isolation subject to multi-tone and random excitations at the base is also demonstrated as case studies. The rest of this paper is organized as follows. The design of Xshape structural hybrid lever-type isolation system is introduced in Section 2. The mathematic modeling and analysis are carried out in Section 3, where the inherent nonlinearity is discussed in detail. Then, vibration isolation performance with parameter sensitivity analysis is studied in Section 4. Section 5 shows
dynamic responses of the novel hybrid lever-type isolation system under multi-zone and random excitations. Finally, a conclusion is drawn.
2. The X-shape structured hybrid lever-type isolation system The hybrid lever-type vibration isolation system supported by an X-shape structure is shown in Fig. 1. The supporting structure is an n-layer scissor-like or X-shape structure with horizontal spring [16]. Each layer has two rods combined with one joint, and length of each rod is 2l. Angle of the rod with respect to horizontal line is θ, and the angle displacement of the joints is represented by ϕ. The restoring force in the horizontal spring is linear with a stiffness kl . The n-layer X-shape structure can provide high static and low dynamic stiffness [16], which is very important to obtain good and stable isolation both in the low and high frequency range. The absolutely motions of the isolation body M and base are represented as x and y. z1(t) and z2(t) are the absolutely displacements of the attached mass m1 and m2 in the levers type I and type II [27,28]. The mass m1 is located on the type I lever, where the lever ratio α1 ¼l11/l12, while, type II lever is attached to m2, which has the lever ratio α2 ¼ l21/l22. The simple linear stiffness feedback control is applied to the mass m2, i.e., F z2 ¼ kz2 z2 .
3. Mathematical modeling and analysis Due to function of the levers, the motion of the two attached mass m1 and m2 can be expressed as: z1 ¼ α1 y ðα1 1Þx;
z2 ¼ α2 x ðα2 1Þy:
ð1Þ
So the kinetic energy of the hybrid isolation system is written as, 1 1 T ¼ M x_ 2 þ ½m1 ðα1 1Þ2 þ m2 α22 x_ 2 ½m1 α1 ðα1 1Þ þ m2 α2 ðα2 1Þx_ y_ 2 2 1 þ ½m1 α21 þm2 ðα2 1Þ2 y_ 2 ; 2
ð2Þ
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The potential energy of the isolation system is given by: 1 V ¼ kl z2 : 2
ð3Þ
where z is the horizontal displacement. The dynamic equation can be determined using the Hamilton principle, and the virtual work δw is given by: _~ x~ c z_ δz c n φδφ δw ¼ c1 xδ þ F z2 δz2 ; 2 3 j_
ð4Þ
where c1 and c2 are the damping coefficients in the vertical and horizontal direction, c3 is the rotational fiction coefficient of each joint, nj is the joints number of the X-shape structure, and x~ ¼ x y is the relative displacement. Fz2 is the active linear stiffness feedback control force based on the absolutely displacement z2, which is denoted as: F z2 ¼ kz2 z2 ;
ð5Þ
From Eqs. (1)–(5) and using the Hamilton principle, the dynamic equation of the hybrid lever-type vibration isolation system can be obtained as: " 2 dz dx~ dz þ c1 þc2 M x€~ þ ½m1 ðα1 1Þ2 þ m2 α22 x€~ þ kl z dx~ dx dx~ 2 # dφ x_~ þ kz2 α22 x~ þ c 3 nj dx~ ¼ M y€ þ ½m1 ðα1 1Þ m2 α2 y€ kz2 α2 y;
ð6Þ
where the relationship between displacements ϕ, z and x~ can be given by (see also [16]): ~ l sin θ þ x=ð2nÞ θ; φ ¼ arctan l cos θ z=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ~ z ¼ 2l cos θ 2 l ½l sin θ þ x=ð2nÞ : ð7Þ dz dx~ Using 2the Taylor series expansion for the terms kl zdx~ dx , dφ and dx~ , Eq. (6) can be changed as:
dz 2 dx~
½1 þ μ2 α22 þμ1 ðα1 1Þ2 M x€~ þ ðkz2 α22 þβ1 Þx~ þ β2 x~ 2 þ β3 x~ 3 þ c3 nj ζ 1 Þx~ x_~ þ ðc2 ζ 5 þ c3 nj ζ 2 Þx~ 2 x_~ ¼ M½μ1 ðα1 1Þ ðμ2 α2 þ 1Þy€ kz2 α2 y;
ð8Þ
where μ1 ¼ m1 =M and μ2 ¼ m2 =M are the mass ratio of attached mass m1 and m2 with respect to isolated mass M, the coefficients β1, β2, β3, β4, ζ0, ζ1, ζ2, ζ3, ζ4, and ζ5 are the Taylor series expansion terms, which are listed in the Appendix. Define the dimensionless variables as: t^ ¼ ω1 t;
η1 ¼ βk1 ; l
η2 ¼ βk2 ; l
η3 ¼ βk3 ; l
η4 ¼ βk4 ; l
γ 1 ¼ kkz2 ; l
ðc1 þ c2 ζ 3 þ c3 nj ζ 0 Þ ðc2 ζ 4 þ c3 nj ζ 1 Þ ðc2 ζ 5 þ c3 nj ζ 2 Þ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ξ1 ¼ ; ξ2 ¼ ; ξ3 ¼ ; 2 Mkl 2 Mkl 2 Mkl pffiffiffiffiffiffiffiffiffiffiffi ^ Eq. (8) can be where ω1 ¼ kl =M , and denoting ðÞ0 ¼ dðÞ=dt, rewritten as: ~ þ ðγ 1 α22 þ η1 Þx~ þ η2 x~ 2 þ η3 x~ 3 þ η4 x~ 4 þ 2ξ1 x~ 0 ½1 þ μ1 ðα1 1Þ2 þ μ2 α22 x″ þ 2ξ2 x~ x~ 0 þ 2ξ3 x~ 2 x~ 0 ¼ ½μ1 ðα1 1Þ ðμ2 α2 þ 1Þy″ γ 1 α2 y:
ð9Þ
Taking the base excitation as the harmonic excitation ^ where Ω ¼ ω0 =ω1 and ω0 is the external exciting y ¼ A0 cos Ωt, frequency. Through the Harmonic Balance Method (HBM), the relative motion can be given generally as [32]: x~ ¼ B0 þ B1 cos ðΩt^ þ ϕÞ;
where B0 is the bias term. Substituting Eq. (10) into the dimensionless dynamic Eq. (9), it gives: ! η B2 3 3η B4 η2 B20 þ η3 B30 þ η4 B40 þ η1 B0 þ 2 1 þ η3 B0 B21 þ 3η4 B20 B21 þ 4 1 ¼ 0; 2 2 8 ð11Þ n
ðη1 þ γ 1 α22 ÞB1 ½1 þ μ1 ðα1 1Þ2 þ μ2 α22 Ω2 B1 þ 2η2 B0 B1 þ 3η3 B20 B1 3 þ 4η4 B30 B1 þ ðη3 B31 þ 4η4 B0 B31 Þ 4 ¼ f½1 þ μ2 α2 μ1 ðα1 1ÞΩ2 γ 1 α2 gA0 cos ϕ;
ð10Þ
ð12Þ
ð2ξ1 ΩB1 þ 2ξ2 ΩB0 B1 þ2ξ3 ΩB1 B20 þ ξ3 ΩB31 Þ ¼ f½1 þ μ2 α2 μ1 ðα1 1ÞΩ2 γ 1 α2 gA0 sin ϕ; The displacement transmissibility can be achieved as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 uB þ A2 þ 2B1 A0 cos ϕ 0 T ¼t 1 ; A20
ð13Þ
ð14Þ
where B1 and ϕ are the amplitude and phase angle of the relative ~ and they can be determined by Eqs. (11)–(13). motion x, From the linear part of dynamic equation, the anti-resonant frequency Ωa and resonant frequency Ωr can be determined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η1 þ γ 1 α22 η1 þ γ 1 α2 ðα2 1Þ ; Ωr ¼ ; Ωa ¼ ½μ1 α1 ðα1 1Þ þ μ2 α2 ðα2 1Þ ½1 þ μ1 ðα1 1Þ2 þ μ2 α22 ð15Þ The anti-resonant frequency Ωa can be designed to be smaller than the resonant frequency Ωr by properly choosing structure parameters such as μ1, μ2, α1, α2, γ1 and η1. This will produce excellent isolation performance in low frequency range. In order to validate the accuracy of the first order harmonic response with the HBM, the second order harmonic response is also used to analyze the nonlinear dynamic Eq. (9), which can be assumed as follows: x~ ¼ B0 þB1 cos ðΩt^ þϕ1 Þ þB2 cos ð2Ωt^ þ ϕ2 Þ;
þ β4 x~ 4 þ ðc1 þ c2 ζ 3 þc3 nj ζ 0 Þx_~ þðc2 ζ 4
171
ð16Þ
Substituting Eq. (16) into the dynamic Eq. (9), five equations on the variables B0, B1, B2, ϕ1, and ϕ2 can be determined, which can be solved by the iterative method using the Maple software. The absolute displacement transmissibility of the proposed isolation system is computed with the first order and second order harmonics as shown in Eq. (2). The system parameters are taken as follows: kl ¼ 100 N/m, M¼1kg, c1 ¼0.5, c2 ¼0.1, c3 ¼0.1, μ1 ¼μ2 ¼ 0.1, α1 ¼3.0, α2 ¼1.2, n¼2, θ¼451 and γ1 ¼3.0. nj ¼ 3n, and l¼ 0.2 m, and the base excitation amplitude A0 ¼ 0.006 m. In Fig. 2, the displacement transmissibility of the fundamental amplitude is calculated with the HBM using the first and second order harmonic responses. As shown in Fig. 2, the results with only the first order harmonic response are the same as those using the first two order harmonic responses. It can be further checked that the higher order harmonics have very little effect on the dynamic response of the nonlinear system (9). Therefore, only the first order harmonic response is used in the following discussions.
4. Results and discussions The vibration isolation performance of the proposed hybrid lever-type isolation system is studied in this section with respect to different structure parameters. For convenience, some parameters are chosen as constant in the analysis as: kl ¼100 N/m,
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Fig. 2. Transmissibility computed by first and second order harmonics.
Fig. 3. Isolation effect in low frequency domain with different SLS layer numbers n.
Fig. 4. Isolation effect in low frequency domain with different assembly angles θ.
Fig. 5. Isolation effect in low frequency domain with different lever ratios α1.
M¼ 1kg, c1 ¼0.5, c2 ¼0.1, c3 ¼ 0.1, nj ¼3n, and l ¼0.1 m. Moreover, the base excitation amplitude A0 ¼ 0.005 m. 4.1. Isolation performance at low frequency In Figs. 3–9, the isolation performance of the proposed hybrid structure system at low frequency (i.e., 0o Ωo1) is investigated, and the sensitivity analysis of some structure parameters is shown. The designable structure parameters mainly are: mass ratio μ1 and μ2, lever ratio α1 and α2, layer number n and assembly angle θ, active stiffness feedback ratio γ1. To consider the influence of a specific structure parameter, other parameters will take constant values in the following analysis without further specification as follows: μ1 ¼μ2 ¼ 0.1, α1 ¼4.0, α2 ¼1.1, n ¼3, θ¼ 301 and γ1 ¼4.0. In Figs. 3 and 4, the isolation effects at low frequency with different X-shape structural layer number and assembly angle are analyzed. As shown in Fig. 3, the anti-resonant and resonant frequencies are about 0.65 and 1.56 for layer number n¼2. The isolation effect at low frequency is very good in that the displacement transmissibility is about 19.5 dB at the beginning of the excitation frequency and the minimal transmissibility is about 33.04 dB at the anti-resonant frequency. For the layer number n¼1, the displacement transmissibility is about 16.5 dB at the beginning of the frequency and the minimal transmissibility is about 22.4 dB at the anti-resonant frequency about 0.79, which shows that the isolation effect for the layer number n¼2 is much better than that of n¼1 in the low frequency domain. For the layer number
Fig. 6. Isolation effect in low frequency domain with different lever ratios α2.
n¼3 and n¼4, the displacement transmissibility at the beginning of the frequency is about 20.2 dB and 20.5 dB, and the minimal transmissibility values are about 37.2 dB and -39.2 dB, respectively. It indicates that the isolation effect of the proposed isolation system at low frequency will become better as the layer number increasing and the anti-resonant frequency and minimal transmissibility are becoming smaller as well.
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Fig. 7. Isolation effect in low frequency domain with different mass ratios μ1.
Fig. 8. Isolation effect in low frequency domain with different mass ratios μ2.
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angle with respect to the horizontal line. This suggests that the isolation performance at low frequency can be much improved by reducing the dynamic stiffness of the X-shape structure. In Figs. 5 and 6, the displacement transmissibility and isolation effect with different lever ratios α1 and α2 are discussed. As shown in Fig. 5, the displacement transmissibility at the beginning of excitation frequency is the same for different lever ratios α1. However, the antiresonant frequency and corresponding displacement transmissibility are both decreased with the lever ratio α1 increasing. For the lever ratio α1 ¼ 2.0, the displacement transmissibility is almost horizontal before the excitation frequency about 0.88. In Fig. 6, the displacement transmissibility at ultra-low frequency and anti-resonant frequency for the lever ratio α2 ¼1.05 are about 25.1 dB and 39.9 dB, which are smaller than those for the lever ratio α2 ¼1.1. The anti-resonant frequency and corresponding transmissibility become much bigger for larger lever ratio α2. Displacement transmissibility at low frequency is sensitive with the change of the lever ratio α2 as shown in Fig. 6. This indicates that the lever ratio α2 plays much more important role in designing excellent ultra-low frequency isolation. In Figs. 7 and 8, the displacement transmissibility with different attached mass ratios μ1 and μ2 is given. As shown in Fig. 7, the displacement transmissibility at ultra-low excitation frequency is not changed with increase of the mass ratio μ1, but the anti-resonant frequency and corresponding transmissibility become smaller with increasing the mass ratio μ1. Fig. 8 shows that the mass ratio μ2 has almost no influence on the isolation performance in the low frequency range. It can be concluded that the isolation effect can be improved by increasing the mass ratio μ1 but not the mass ratio μ2. From the analysis in Figs. 3–8, excellent ultra-low frequency isolation performance can be obtained by designing passive structure parameters. In Fig. 9, the influence of the active linear stiffness control (i.e., ratio γ1) on isolation effect in the low frequency range is discussed. The displacement transmissibility at ultra-low excitation frequency and anti-resonant frequency is all becoming smaller, and the effective low-frequency isolation range also becoming larger with the increase of the stiffness ratio γ1, while the anti-resonant frequency is becoming bigger simultaneously. These clearly demonstrate that the isolation performance in the low frequency range is sensitively affected by the linear stiffness feedback control. 4.2. Isolation performance over the full frequency range
Fig. 9. Isolation effect in low frequency domain with different active stiffness ratios γ1.
In Fig. 4, the displacement transmissibility at the beginning frequency and anti-resonant frequency for the assembly angle 451 are about 19.07 dB and 34.3 dB, which is bigger than those for the assembly angle 301. Those for the assembly angle 601 are about 16.5 dB and 28.2 dB, which is much bigger than those for the assembly angle 301 and 451. Fig. 4 also shows that the antiresonant frequency becomes larger with increasing the assembly
In Figs. 10–17, the special isolation performance of the proposed hybrid structure system in the whole frequency range is discussed. Fig. 5 already indicates that excellent isolation performance both in low frequency and high frequency range could be obtained by choosing properly the lever ratio α1. To explore this further, in the calculation of the results in Figs. 10–17, the structure parameters are taken as, μ1 ¼ μ2 ¼0.1, α1 ¼1.7, α2 ¼1.1, n ¼3, θ¼301 and γ1 ¼1.0 if no further specification. In Fig. 10, the displacement transmissibility with different active stiffness ratios γ1 is shown. For the active stiffness ratio γ1 ¼1.0, the displacement transmissibility at the ultra-low excitation frequency is about 18.6 dB, and is very similar to that at high frequency about 19.02 dB. The displacement transmissibility almost keeps a flat line over the whole frequency range except those around the resonance frequency which have the peak value equal to 0.035 dB. Comparing with the results in Figs. 2–8, the anti-resonance frequency range disappears. Fig. 10 also shows that the resonant frequency becomes larger but the peak transmissibility is almost not changed with the increase of the active stiffness feedback control. In Figs. 11 and 12, the isolation performance in the whole frequency region is discussed with different lever ratio α1 and ratio α2. As shown in Fig. 11, the resonant frequency and corresponding
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Fig. 10. Isolation performance in whole frequency domain with different stiffness ratios.
Fig. 11. Isolation performance in whole frequency domain with different lever ratios α1.
Fig. 13. Isolation performance in whole frequency domain with different mass ratios μ1.
Fig. 14. Isolation performance in whole frequency domain with different mass ratios μ2.
Fig. 12. Isolation performance in whole frequency domain with different lever ratios α2.
Fig. 15. Isolation performance in whole frequency domain with different layer numbers.
transmissibility value change little as the change of the lever ratio α1. The displacement transmissibility at ultra-low excitation frequency keeps the same for different lever ratios α1, but the transmissibility at high frequency increases for bigger lever ratio α1. In Fig. 12, the transmissibility at ultra-low frequency and the peak transmissibility for lever ratio α2 ¼1.2 are about 14.5 dB and
3.01 dB, and those for lever ratio α2 ¼ 1.3 are about 12.1 dB and 6.6 dB, which are much larger than those for the lever ratio α2 ¼1.1. In Figs. 13 and 14, the effects of the attached mass ratio μ1 and μ2 on the isolation performance in the whole frequency region are shown. In Fig. 13, the peak transmissibility values for the mass ratio μ1 ¼0.05 and μ1 ¼0.15 are about 1.92 dB and 1.16 dB, which are
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Fig. 16. Isolation performance in whole frequency domain with different assembly angles θ.
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Fig. 18. Response in time domain under multi-tone excitation for different isolation systems 1.
5. Vibration isolation subject to multi-tone and random excitations
Fig. 17. Isolation performance in whole frequency domain with different damping coefficients. Table 1 Parameters used in the hybrid isolation system. Parameters
μ1
μ2
α1
α2
n
θ
γ1
Low frequency Full frequency
0.1 0.1
0.1 0.1
4.0 1.7
1.1 1.1
3 3
301 301
4.0 1.0
much bigger than those for mass ratio μ1 ¼0.1. The transmissibility at high frequency becomes larger as increasing the mass ratio μ1. As shown in Fig. 14, the displacement transmissibility is almost not changed for different mass ratios μ2, except that the resonant frequency is reduced with increase of the mass ratio μ2. In Figs. 15 and 16, the effects of the layer number and assembly angle on the isolation performance in the whole frequency domain are given. Fig. 15 shows that the displacement transmissibility at ultra-low frequency and resonant frequency decreases with increase of layer number n, while increases at high frequency. In Fig. 16, the transmissibility at ultra-low frequency and resonant frequency decreases with decrease of the assembly angle θ, while increases at high frequency simultaneously. These suggest different structure parameters should be chosen for different vibration isolation requirements. In Fig. 17, the displacement transmissibility is almost not changed with different damping coefficients. However, the effective isolation frequency range is much improved as decreasing the damping coefficient but with increasing peak value.
The vibration isolation performance of the proposed hybrid isolation system subject to multi-tone and random excitations is studied, compared with corresponding hybrid isolation system without the X-shape structure. The X-shape structure can provide specially-designed nonlinearity in stiffness and damping. In order to study the influence of these nonlinear stiffness and damping properties on the performance of the proposed isolation system, traditional springs and viscous dampers are used to replace the X-shape structure for comparison with the same stiffness and damping coefficients as those in the X-shape structure. As discussed before, the parameters of the proposed hybrid isolation system for vibration isolation in low or full frequency ranges can be chosen as in Table 1, which are referred to as low-frequency parameter set and full-frequency parameter set, respectively. In Figs. 18 and 19, dynamic responses in time domain for different isolation systems under multi-tone excitations are shown, where initial phase angles are assumed to zeros. The base multi-tone excitation in Fig. 18 is taken as: amplitude z1 ¼ 0.005, z2 ¼0.004, z3 ¼ 0.003, and frequency ω01 ¼3 Hz, ω02 ¼ 5 Hz, and ω03 ¼6 Hz. Fig. 18 indicates that the responses for the proposed hybrid isolation system with low-frequency parameter set are much smaller than those for that without X-shape supporting structure, demonstrating excellent low frequency vibration isolation performance. In Fig. 20, dynamic response in time domain under multi-zone excitation (z1 ¼ 0.005, z2 ¼ 0.004, z3 ¼0.003, ω01 ¼4 Hz, ω02 ¼14 Hz, and ω03 ¼24 Hz) is shown. It shows that the responses of the hybrid isolation system without X-shape structure are much larger (thus of worse performance) than those for the proposed hybrid isolation system with full-frequency parameter set. In Figs. 20 and 21, the dynamic responses with different vibration isolation systems subject to base white noise acceleration excitations are discussed. The power spectral density function and mean value of base white noise acceleration excitations are taken as s0 ¼0.001 and zero, respectively. But the frequency ranges for the low frequency and broadband white noise excitations are taken as (0–5 Hz) and (4–100 Hz). As shown in Figs. 20 and 21, the dynamic responses for the designed hybrid isolation system under the low frequency or broadband white noise excitations are much smaller than those of the corresponding hybrid isolation system without X-shape supporting structure. It can be concluded that excellent vibration isolation performance can be obtained with the proposed hybrid structure system subject to both multi-tone and random excitations. Note that the
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6. Conclusions
Fig. 19. Response in time domain under multi-tone excitation for different isolation systems 2.
A novel hybrid lever-type isolation system strengthened by an X-shape supporting structure is proposed, which combines the advantageous features of the scissor-like structure and lever-type structures together. System inherent nonlinear stiffness and damping characteristics are explored to significantly improve vibration isolation/control performance in passive and/or semi-active manners. It is shown that the proposed hybrid isolation system can achieve very good ultra-low-frequency vibration isolation with tunable antiresonance frequency and low-frequency transmissibility by designing structure parameters. Moreover, a uniformly-low isolation transmissibility over full frequency range can also be realized with proper values of structure parameters. To the best of our knowledge, this full-frequency vibration suppression performance has not been reported before. The isolation performances mentioned above are verified by case studies with comparisons subject to multi-tone and random excitations. The results demonstrate again that the inherent nonlinear stiffness and damping characteristics introduced by the X-shape structure play an important and beneficial role in vibration isolation and control. Further study will focus on prototyping and experimental testing of the proposed hybrid isolation system.
Acknowledgment The authors would like to thank the handling editors and anonymous reviewers for your time and effort in reviewing this paper. The authors also gratefully acknowledge the support from a NSFC Project (No 61374041) of China, a GRF Project of HK RGC (Ref no.15206514), a CAST-funded Project (H-ZG2D), and internal Competitive Research Grants of Hong Kong Polytechnic University.
Appendix A
Fig. 20. Response under low frequency white noise excitation for different isolation systems.
The coefficients β1, β2, β3, β4, ζ0, ζ1, ζ2, ζ3, ζ4, ζ5 in Eq. (8) are listed as follows: β1 ¼
kl tan 2 θ ; n2
ða:1Þ
β2 ¼
3kl tan θ ; 4n3 l cos 3 θ
ða:2Þ
β3 ¼ β4 ¼ ζ0 ¼ ζ1 ¼
ζ2 ¼ Fig. 21. Response under broad-band frequency white noise excitation for different isolation systems.
hybrid isolation system without the X-shape supporting structure is a pure linear system. The comparisons above demonstrate also that the nonlinear stiffness and damping characteristics introduced by the X-shape supporting structure are very crucial for the performance improvement and very beneficial in vibration control.
ð5 4 cos 2 θÞkl 2
8n4 l cos 6 θ
;
5 sin θð5 2 cos 2θÞkl 3
64n5 l cos 8 θ 1 2
4n2 l cos 2 θ tan θ 3
4n3 l cos 3 θ
ða:3Þ
;
ða:4Þ
;
ða:5Þ
;
ða:6Þ
ð1 þ 3 sin 2 θÞ 4
8n4 l cos 6 θ
;
ða:7Þ
ζ3 ¼
tan 2 θ ; n2
ða:8Þ
ζ4 ¼
tan θ ; n3 l cos 3 θ
ða:9Þ
ζ5 ¼
ð1 þ 3 sin 2 θÞ 2
2n4 l cos 6 θ
:
ða:10Þ
C. Liu et al. / International Journal of Mechanical Sciences 98 (2015) 169–177
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