## ANLYSIS OF THE DYNAMIC BEHAVIOR AND PERFORMANCE OF A VIBRATION ISOLATION SYSTEM WITH GEOMETRIC NONLINEAR FRICTION DAMPING 1)

Liu Xingtian2), Chen Shuhai, Wang Jiadeng, Shen Junfeng

Laboratory of Space Mechanical and Thermal Integrative Technology, Shanghai Institute of Satellite Engineering, Shanghai 201109, China

 基金资助: 国家自然科学基金资助项目.  51505294国家自然科学基金资助项目.  51875363

Received: 2018-09-10   Accepted: 2019-02-26   Online: 2019-03-18

2)刘兴天,高级工程师,主要研究方向：卫星结构设计,结构动力学与振动控制.E-mail:xtliu509@126.com Abstract

In vibration isolation field, nonlinear vibration isolation system catch more attention than linear system because of the better vibration isolation performance. In this paper, a novel nonlinear vibration isolation system with geometric nonlinear friction damping is proposed by add two friction damper that perpendicular to the movement direction of the isolated object. The absolute and relative displacement transmissibility of such kind of vibration isolation system are studied in this paper. Different from the friction damper which usually assuming that the friction force is constant, the friction force studied in this paper is proportional to the displacement of the isolated mass by configuring two linear friction dampers perpendicular to the moving direction of the mass. The mathematical model of the friction damping and the forced vibration of the system are established. The dynamic equation is solved by using Harmonic Balance Method (HBM) subsequently by making some simplification. The result solved by HBM is verified numerically. The performance of the nonlinear vibration isolation system is compared with that of a linear one by the performance index defined by absolute and relative transmissibility. The geometric nonlinear friction can offer small or large friction damping depends on the relative displacement, therefore, the nonlinear friction force can improve the transmissibility for both absolute and relative displacement at resonance and the higher frequencies region if the damping values are chosen carefully which surpass a traditional Kevin vibration isolator model. Meanwhile, the nonlinear vibration isolation system can enlarge the application region for different excitation amplitude and avoid the system failure though the responses of the isolated mass is amplified at low frequency. The vibration isolation system with the configuration of the friction damper proposed is very suitable for both resonance and higher frequencies vibration control. The conclusions given are of importance when design and choosing the friction damping parameters.

Keywords： geometric nonlinear ; isolator ; transmissibility ; friction damping ; passive vibration isolation

Liu Xingtian, Chen Shuhai, Wang Jiadeng, Shen Junfeng. ANLYSIS OF THE DYNAMIC BEHAVIOR AND PERFORMANCE OF A VIBRATION ISOLATION SYSTEM WITH GEOMETRIC NONLINEAR FRICTION DAMPING 1). Chinese Journal of Theoretical and Applied Mechanics[J], 2019, 51(2): 371-379 DOI:10.6052/0459-1879-18-302

## 1 摩擦阻尼和隔振系统建模

### 图 1 Fig.1   Schematic of the vibration isolation system with geometric nonlinear friction damping

$$f_v = 2F_{{\rm f}1} \frac{\left| z \right|\left| \dot {z} \right|^{-1}}{\sqrt {L^2 + z^2} }\dot {z}$$

$$f_v \approx \frac{2F_{f1} }{L}\left| z \right| {\rm sgn} \dot {z}$$

$$m\ddot {z} + c\dot {z} + \left( {F_{{\rm f}2} + \frac{2F_{{\rm f}1} }{L}\left| z \right|} \right) {\rm sgn} \dot {z} + kz = m\omega ^2y_0 \sin \left( {\omega t} \right)$$

$${u}" + 2\zeta {u}' + \left( {\lambda _1 + \lambda _2 \left| u \right|} \right){\rm sgn} {u}' + u = \tilde {y}_0 \Omega ^2\sin (\Omega t)$$

$$u = u_0 \sin \left( {\Omega \tau + \theta } \right)$$

$$-u_0 \Omega ^2\sin \left( {\Omega \tau + \theta } \right) + 2\zeta \Omega u_0 \cos \left( {\Omega \tau + \theta } \right) + \left( {\lambda _1 + \lambda _2 u {\rm sgn} \left( u \right)} \right) {\rm sgn} \left( {u}' \right) + u_0 \sin \left( {\Omega \tau + \theta } \right) = \tilde {y}_0 \Omega ^2\sin (\Omega t)$$

$$Sgn[sin ( \Omega \tau + \theta] \approx \ A_{01} + A_{11} cos ( \Omega \tau + \theta) + A_{21} sin ( \Omega \tau + \theta) \approx \frac{4}{π}sin ( \Omega \tau + \theta)$$

$-u_0 \Omega ^2 + u_0 = \tilde {y}_0 \Omega ^2\cos \theta$

$2\zeta \Omega u_0 + \frac{4}{ π }\lambda _1 + \frac{4}{ π ^2}\lambda _2 u_0 =-\tilde {y}_0 \Omega ^2\sin \theta$

$$( \Omega ^4-2\Omega ^2 + 1 + 4 \zeta ^2 \Omega ^2 + \frac{16}{π^2} \zeta \lambda _2 \Omega + \frac{16 \lambda _2^2 }{ π^4})u_0^2 + \frac{16 \lambda _1^2 }{ π^2} + ( \frac{32}{ π^3} \lambda _1 \lambda _2 + \frac{16}{ π }\zeta \lambda _1 \Omega)u_0-\tilde {y}_0^2 \Omega ^4 = 0$$

$u_0 = \Bigg [-\frac{32}{{π}^3}\lambda _1 \lambda _2 + \qquad \sqrt {\left( {1-\Omega ^2} \right)^2\Bigg( 4\tilde {y}_0^2 \Omega ^4-\frac{64}{{π}^2}\lambda _1^2 \Bigg ) + \frac{64\lambda _2^2 }{{π}^4}\tilde {y}_0^2 \Omega ^4} \Bigg ] \Bigg/ \qquad \Bigg [2\Bigg ( \Omega ^4-2\Omega ^2 + 1 + \frac{16\lambda _2^2 }{{π}^4} \Bigg ) \Bigg ]$

$$T_{\rm a} = \sqrt {u_0^2 + 2y_0 u_0 \cos \theta + y_0^2 } / y_0$$

## 2 仿真与结果

### 图2 Fig.2   Results comparison when$\lambda _1 = 0$,$\lambda _2 = 0$ for HBM and numerical method

### 图3 Fig.3   Results comparison when$\lambda _1 = 0.3$,$\lambda _2 = 0.5$ for HBM and numerical method

### 图4 Fig.4   Effect of$\lambda _1$ on the absolute transmissibility of the vibration isolation system when$\lambda _2 = 0$

### 图5 Fig.5   Effect of$\lambda _1$ on the relative transmissibility of the vibration isolation system when$\lambda _2 = 0$

### 图6 Fig.6   Effect of$\lambda _{2}$ on the absolute transmissibility of the vibration isolation system when$\lambda _{1} = 0$

### 图7 Fig.7   Effect of$\lambda _{2}$ on the relative transmissibility of the vibration isolation system when$\lambda _{1} = 0$

### 图8 Fig.8   Comparison of the absolute transmissibility for the two kinds of vibration isolation system with the same resonance factor

### 图9 Fig.9   Comparison of the relative transmissibility for the two kinds of vibration isolation system with the same resonance factor

### 图10 Fig.10   Effect of the combination of$\lambda _1$ and$\lambda _2$ on the absolute transmissibility of the nonlinear vibration isolation system

### 图11 Fig.11   Effect of the combination of$\lambda _1$ and$\lambda _2$ on the relative transmissibility of the nonlinear vibration isolation system

### 图12 Fig.12   Effect of the excitation amplitude on the absolute transmissibility of the constant friction vibration isolation system

### 图13 Fig.13   Effect of the excitation amplitude on the relative transmissibility of the constant friction vibration isolation system

### 图14 Fig.14   Effect of the excitation amplitude on the absolute transmissibility of the nonlinear friction vibration isolation system

### 图15 Fig.15   Effect of the excitation amplitude on the relative transmissibility of the nonlinear friction vibration isolation system

### 图16 Fig.16   Effect of the combination of system parameters on the absolute transmissibility of the vibration isolation system

### 图17 Fig.17   Effect of the combination of system parameters on the relative transmissibility of the vibration isolation system

## 3 结 论

(1) 仅就引入几何非线性而言,变阻尼摩擦系数和恒定摩擦系数对绝对传递率和相对传递率的影响类似,即增大变摩擦阻尼系数使系统共振传递率变优而高频衰减变差;

(2) 由于变阻尼摩擦系统的摩擦力和位移相关,因此纯变阻尼摩擦系统无频率"锁定"现象,且该系统能在保持共振峰控制的同时,保持高频衰减;

(3) 与变阻尼摩擦系数相比,在小幅值激励的情况下,恒定摩擦阻尼系统对激励的隔振效果变差,隔振器应用频率范围变小,严重可导致隔振器失效;

(4)通过合理的参数选择,可以使几何非线性系统在共振传递率和高频衰减效果均优于恒定摩擦阻尼系统,同时可以显著避免因激励幅值过小时产生的"锁住"现象,提高系统的可靠性.

The authors have declared that no competing interests exist.

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On the relative and absolute transmissibility of a vibration isolation system subjected to base excitation

//XXI Conference with International Participation, Noise and Vibration, Serbia, Tara, 2008

URL In this paper a one-degree of freedom passive vibration isolator system which is subject to harmonic base excitation is analyzed. The isolator is modeled as a parallel combination of a stiffness and damping element with cubic non-linearity. The method of averaging is used to obtain the steady-state harmonic response. A parametric analysis is conducted in order to investigate the influence of the system parameters on the relative and absolute transmissibility of the system from the viewpoint of possible improvement of the transmissibility of a system with linear viscous damping. .

López I, Busturia J, Nijmeijer H .

Energy dissipation of a friction damper

Journal of Sound and Vibration, 2003,278(3):539-561

URL In this paper the energy dissipated through friction is analysed for a type of friction dampers used to reduce squeal noise from railway wheels. A one degree-of-freedom system is analytically studied. First the existence and stability of a periodic solution are demonstrated and then the energy dissipated per cycle is determined as a function of the system parameters. In this way the influence of the mass, natural frequency and internal damping of the friction damper on the energy dissipation is established. It is shown that increasing the mass and reducing the natural frequency and internal damping of the friction damper maximizes the dissipated energy.

Berger E .

Friction modeling for dynamic system simulation

Applied Mechanics Review, 2002,55(6):535-577

URL The working of friction modeling tools for dynamic system simulation were discussed. It was observed that in engineering applications, success of models in predicting experimental results depended on the frictional models. It was also observed that the success of frictional model relied on the system dynamic models. Various frictional models from the vast field of engineering models were examined and specific focus was laid on lumped-parameter system models, continuum system models and forced system models.

Stein G, Zahoransky R, Mucka P .

On dry friction modelling and simulation in kinematically excited oscillatory systems

Journal of Sound and Vibration, 2008,311(1-2):74-96

URL This paper deals with the analysis and simulation of a general single degree of freedom (sdof) oscillatory system with idealised linear viscous damper and dry friction. For dry friction modelling the phenomenological macro-slip approach is employed, described in mathematical form either by the signum function approach or by the physically correct stick lip approach assuming switching phenomena on a short time scale. Both approaches are illustrated first using a steady-state harmonic acceleration excitation with constant amplitude and then a stationary random acceleration excitation, corresponding to a field-measured excitation in a vehicle. The differences in the two approaches are highlighted, indicating that the physically correct stick lip approach describes the friction phenomenon better than the standard signum approach. The signum approach is prone to false numerical oscillations completely distorting the acceleration response signal in comparison to measured suspension system response. The acceleration transmissibility response is analysed in respect to the dry friction force magnitude, employing stationary random excitation. A sdof oscillatory system without viscous damping, subjected to both stationary random acceleration and harmonic acceleration is analysed, too. It is shown that such a system can be used without serious practical problems; however, no implications on its performance from the analysis under harmonic constant amplitude acceleration excitation can be made.

Zhao D, Zhang W, Ma R et al.

Research on a new damper and its application in vibration control of a building

Industrial Construction, 2006,36(2):1-5

URL It is explored the vibrational energy dissipation theory of a new dry friction damper and its application in vibration control of a building.The research shows that the new dampers have traditional dampers′ merit and overcome the shortcoming that can only offer invariable friction.At the same time,they can be controlled by the signal of the vibration of the building.So the ability of energy dissipation and vibration control effect is augmented.The simulation using finite element program proved that the vibration of the building is reduced with the use of this damper.

Displacement-proportional friction (DPF) in base isolation

Earthquake Engineering and Structural Dynamics, 1987,15(7):799-813

Ferri A .

Friction Damping and Isolation Systems

Journal of Vibration and Acoustics, 1995,117(B):196-206

Ferri A, Whiteman W .

Free response of a system with negative viscous damping and displacement-dependent dry friction damping

Journal of Sound and Vibration, 2007,306(3-5):400-418 A stability analysis is conducted of an autonomous single-degree-of-freedom system damped with negative viscous damping and a displacement-dependent Coulomb friction force. The geometry of the dry friction damping element yields a friction force that grows linearly with the system displacement. The most direct application of this system is in the study of a turbomachinery blade with shroud interfaces designed to achieve this geometry. Recent studies have shown that the damping of systems with this type of displacement-dependent dry friction force resembles linear structural damping and suggests that this arrangement may be an effective means of flutter suppression in these turbine and fan blade applications. For this study, the inclusion of negative viscous damping is used in order to approximate destabilizing aerodynamic forces. An exact analysis is conducted to determine the stability of this autonomous system. Results show that energy losses from the displacement-dependent dry friction damper are large enough to achieve local and even global stability under certain conditions.

Whiteman W, Ferri A .

Displacement-dependent dry friction damping of a beam-like structure

Journal of Sound and Vibration, 1996,198(3):313-329

URL The flexural vibration of a beam-like structure damped with a displacement dependent Coulomb friction force has been examined. Due to the geometry of the dry friction damping element, the friction force grows linearly with the beam's transverse displacement. Harmonic excitation of the system is analyzed using first order harmonic balance and then using an "exact" time domain method. The stick-slip behaviour of the system has also been studied. Even though the only source of damping is dry friction, the system was seen to exhibit damping characteristics similar to linear structural damping. The equivalent natural frequency and equivalent viscous damping were also investigated, and found to depend on the amplitude of response and on the design parameters of the frictional interface. In particular, the overall damping levels as well as the amount of stick-slip motion are seen to depend strongly on the amount of displacement dependence. Another significant result was the appearance of two stable steady-state solutions at certain parameter values. The results suggest that the overall characteristics of mechanical systems may be improved by properly configuring frictional interfaces to allow normal forces to vary with displacement.

Tang B, Brennan M .

A comparison of two nonlinear damping mechanisms in a vibration isolator

Journal of Sound and Vibration, 2013,332(3):510-520

URL 78 The influence of two nonlinear dampers on vibration transmissibility are compared. 78 An isolator with geometric nonlinear damping force is studied. 78 Analytical expressions and numerical simulations are used for the analysis. 78 An isolation system with geometrically nonlinear damping has some advantages compared to a linear damper.

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