Numerical Analysis of Dry Friction－Induced Vibration of Moving Slider－Elastic Annular Beam System

0 Introduction

A Friction system usually exists in the Engineering such as braking system，where the friction－induced instable vibration has a negative influence on the braking system，especially the disc－pad system［1］.Many researches have been carried out for this problem with analytical，computational and experimental techniques.Various models including the single－mode approximation，the beam model，the plate model［2］and the finite element model were established to investigate the brake system of rotating brake disc acted by fixed pads.The early achievements on disc brake system are attributed to the dry frictional stick－slip self－excited vibration［3－4］ and unstable structural vibration［5－6］.Stick－slip refers to a fluctuation of friction force or sliding velocity with time or sliding distance changing［7］.

先天性肌性斜颈的患儿，通常在出生后颈部迅速长出类似肿瘤的硬块，数日后会变成蚕豆一样大小。硬块在宝宝出生后2～3周时最大，有的直径可达2cm，所以很容易被妈妈们发现。此外，如果发现宝宝的头总喜欢面向一侧，搬正后很快转回原位，因应该引起父母的重视。

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In view of the stick－slip mechanism，the model of rigid body－rigid transmission belt has been widely used.In 2001，a dynamic system is presented by Galvanetto to get the mechanism of discontinuous bifurcations，where stick－slip vibration can be affected by the non－smooth bifurcations［8］.Hereby，Stribeck－type coefficient is concluded to analyze the system stability and determine the critical speed of a dynamic model with two－degree－of－freedom［9］.

斯洛文尼亚国境内分布着大量石灰岩，千百万年来，积水溶解了岩石中的碳酸钙，形成了由隧道和洞穴构成的“地下迷宫”。最早来此进行地质勘探的德国科学家将这种岩溶地貌称为“喀斯特”，这一名称至今仍在地质学界沿用。最常被人们光顾的洞穴当数斯科契扬溶洞群。这是一个由石笋、钟乳石和石柱组成的童话世界，其地下峡谷的恢宏壮观使之居于世界最著名的喀斯特地貌之列。

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During braking process，the time－dependent nature of the contacting interface in the pad－disc system is important and must be considered even if lower relative speed［5］.From the point of the rotating disc，its vibration is activated by the circumferentially moving action of the rigid pads.Influence of the moving force on brake stability attracts more and more attentions and the moving method is adopted in finite element method and experimental approaches.The deflection of a beam with moving force and the resonance velocity of the moving load are analyzed by Wayou，Tchoukuegno and Woafo［13］.An approximation solution of moving oscillator was investigated by Pesterev and Bergman to describe the variation of displacement and shear force in a one－dimensional distributed system with an arbitrarily varying speed［14］.Chen et al.analyzed a viscoelastic beam moving axially using the method of multiple scales.They found that the instability frequency intervals are influenced by the axial speed and the beam coefficient［15］.Using the finite element method，unstable frequencies of a braking system inspired by moving loads can be obtained based on the system eigenvalue analysis［16］.A linear complex－valued eigenvalue formulation for a disc with moving load is established by Cao et al.to calculate the stationary components of the disc brake［17］.Based on the finite difference method and complex eigenvalue analysis，stability and uncertain parameters analyses［18］are carried out for disc－pad systems with moving－interactions.Accordingly，Ouyang investigated the instability of a brake disc by presenting the relationship between eigenvalue and disc′s rotating speed［15］.

However，up to now，the flexible coupling between the disc and pads and the moving interactions have not been considered simultaneously in one dynamical braking model.This paper deals with a rigid－flexible coupled braking system，and focuses attention mainly on motion of the pads moving on the elastic disc.The disc is simplified as an annular beam and the friction between the beam and pads are determined using a Stribecktype friction model.Then the 1st order Galerkin reduction is applied for the beam based on result of moving load simulation.Influence of acceleration of the disc rotation on the frictional instability of the pad is investigated.

When v decreases to 1，Stribeck－type friction effect induced by self－excited vibration （or stickslip）appears and limit cycles with rather large vibration amplitude for all parts of the system are resulted（see the presented horizontal vibration of the pad in Fig.9）.

1 Dynamic Modals

1.1 Pads and disc

During braking，the pads move circumferentially along the outer edge of the elastic disc.So the disc can be simplified as an annular beam which is supported on uniform and continuous flexible boundary（to simulate the distribution of flexible constraint effect from the inner part of disk）［19－20］.The pad－disc coupling system is shown in Fig.1 （the boundary supporting is not included）.The pads move at relative speed Vand vibrate independently in horizontal directions X1/X2and vertical directions Y1/Y2，and the rotational angles areΦ1/Φ2［21］.For the disk，only its transverse displacement Wdis considered，because it influences the pads′motion directly during braking contact.The contact stiffness involves the tangential ones Kt1and Kt2and the vertical one Ken.Parameters and their values of the system are listed in Table 1.

Table 1 Parameters

Symbol Physical meaning Parameter value X/Φ/sgn Horizontal coordinate/Angle coordinate/Sign function X1/Y1/Φ1 Horizontal/vertical/angle displacements of upper pad X2/Y2/Φ2 Horizontal/vertical/angle displacements of lower pad Wd/V/a Transverse displacement/speed/deceleration of disc P1/P2 Contact pressure of the upper/lower pad F1/F2 Contact friction force of the upper/lower pad Ωx/Ωy/Ωφ Natural frequencies of the one pad μ1/μ2 Contact friction coefficients between upper/lower pad and disc Kx/Ky Horizontal/vertical stiffness of pads Ks/（kN·m·rad－1） Angle stiffness of pads 3.5×104 Cx/Cy/（N·s·m－1） Horizontal/vertical damping coefficients of pads 100 Cs/（N·s·m－1·rad－1） Angle damping coefficient of pads 200 c/（N·s·m－1） Damping coefficient of disc 200 Kt1/Kt2/（kN·m－1） Tangential contact stiffness between pads and disc 3.0×104 Ken/（kN·m－1） Vertical contact stiffness between pads and disc 3.0×104 M/kg/Is/（kg·m2） Mass/rotational inertia of pads 0.1/0.25 ρ/（kg·m－3） Density of disc 7.8×103 E/GPa/A/m2 Young′s modulus of disc/Area of the disc 196/8×10－4 q/m/I/m4 Thickness of pads/Rotational inertia of beam （disc） 0.01/1.7×10－8 fs/fc Maximum/minimum static friction coefficient 0.5/0.25 vs/（m·s－2） Stribeck velocity 0.5 L/m/h/m/b/m Length/Thickness of the beam/Length of pad 1.00/0.016/0.10 Λ1/Λ2 Displacements of the pads moving to disc

Fig.1 Pad－disc coupling system

2.1 两组患儿疗效对比 经过治疗，观察组患儿的总有效率(87.50%)明显高于对照组(67.50%)，其差异具有统计学意义(χ2=4.591，P=0.032)。见表2。

1.2 Non－dimensional Form

To rewrite the equations conveniently，introduce the dimensionless variables and parameters

1.3 Friction

Considering a forcemoves along a straight beam.To analyze its response under the moving load，the beam is discretized into pelements.Let dt be the unit time and dxthe unit space，the nodal force is subsequently allocated in every unit time.

The Stribeck effect refers to a phenomenon that the friction coefficient decreases.The negative slope in relation between the friction and relative velocity is known as the main reason of friction instability.The Stribeck－type friction coefficientμ（vr）can be expressed as

where fcand fsare the minimum and the maximum static friction coefficients respectively，vsis the Stribeck velocity，and vrthe relative velocity.Whenδ＝1，Eq.（6）is also known as Tustin index model.The friction coefficient varies with relative velocity as shown in Fig.2.

Fig.2 Static friction and Coulomb friction with Stribeck effect

2 Moving Interaction and Galerkin Reduction

2.1 Moving load simulation

Many models have been proposed to describe the friction properties，such as the coulomb friction（before occurrence of relative displacement between the contact parts），static friction（no relative velocity from static to relative motion）and－Stribeck effect，etc.

研究人员逐步揭示环偶极子产生机理，指出环偶极子是由电极子及磁极子之间相互作用破坏近场电环境平衡而产生的，非对称电流能增强环偶极子强度，是产生高Q值的原因[25，28]。可依据单元结构建立基于耦合LC电路的物理模型进行定量分析。为了定量分析环偶极矩强度，根据体积电流密度分布，利用多极散射理论得到多极子的散射强度，如电偶极矩、磁偶极矩和环偶极矩。该方法已成功应用于微波、太赫兹和光学波段环偶极矩的计算，计算公式为：

For the moving load acted system，two conditions will be dealt with during calculation：

（1）Cdx ＝vdt（Cis a positive integer）.In this case，the load moves from one discrete point to the Cth one after each time step dt.

（2）dx ＝Cvdt（Cis a positive integer）.In this case，after each time step dt，the load moves at a place between the discrete points e and e＋1.Dividing the space step dx into C equal subspaces，the load moves at the next sub－point after every time step dt.When the loadF槇 moves at the ith one，（e＋i/Cdx）wherei＝1，2，…，C ，the load will be resolved to the points e and e＋1as ＝respectively，see Fig.3.

地铁的动力照明负荷主要包括照明荧光灯、空调、自动扶梯、风机、水泵、消防设备等，这些设备大多属于非线性设备，必然会对电网造成谐波影响。动力照明负荷主要产生5、7次谐波。[2]

Fig.3 Moving load simulation

Calculation shows that the natural frequencies areΩx＝3.1Hz，Ωy＝2.7Hz andΩφ＝1.3Hz for horizontal，vertical and angle direction motions of the pads，and 1.25Hz（the first order）and 2.50Hz（the second order）for transverse motion of the disk，respectively.One finds that in spectrums under higher speed sustained moving interaction，say v ＝2，motions of the pad and the disc in all directions are mixed with disc′s frequency in steady state，because the vibration frequencies are their natural frequencies and disc′s frequency in corresponding directions，respectively.However，as the moving speed v decreases to 1，the friction－induced limit cycle vibration occurs for the pad，and common frequency 3.05Hz appears in all directions of the pad′s motion.It should be noted that even though the frictional instability has happened for the pad，the disc still vibrates in the first－order mode.So the equation of motion of the disc is discretized to the first－order mode by Galerkin model reduction in the following section.

where 1，i and LAare the leftmost，the ith node and rightmost node in beam′s horizontal direction，respectively.j and j＋1present the jth and（j＋1）th time moment，respectively.

Fig.4 Continuous flexible boundary discretized as p springs

The disc partial differential Eq.（5）is solved by using the finite difference method and the ordinary differential Eq.（4）is solved by Runge－Kutta method，respectively.And the dynamical behavior of the braking system with moving actions between the parts can be simulated with the same time step dt＝0.000 1and space step dx ＝0.02.The transverse response of disc solved by finite difference method is transferred to contact pressure and friction force items of pads，and then Eq.（4）can be solved.Takeσ珋＝vt，then the transverse response of the beam is induced by successive change of the contact position between the pad and disc.Let Kx＝5.0×104 kN/m，K＝3.0×

y 104 kN/m，v＝2and 1，and simulation shows that both the x direction response of the pad and the transverse response of the annular disc are periodic vibration as shown in Figs.5，6.The spectra of motions are also presented in the figures.It should be noted that except the difference of the equilibriums in velocity axis，the friction－induced dynamics of two pads are same with each other.So we present only the upper pad′s motion in this paper.

Fig.5 Response of the system under moving interactions as relative－equilibrium occuring for the pad（v ＝2）

Fig.6 Response of the system under moving interactions as friction－induced limit cycle occuring for the pad（v ＝1）

Corresponding to the moving load simulation mentioned above，the uniform and continuous flexible boundary of the beam should be discretized as p elastic springs with stiffness k，as shown in Fig.4.k depends on the material properties and p＝50in the following analysis.Then the boundary and continual conditions can be expressed as

2.2 Approximate reduction equations

Take the trial function（Eq.（8））to approximate the first－order transverse vibration of the disc.

Substitute Eq.（8）into Eq.（5）and multiply the weighted function on both sides，then integrate it from 0to Lin x，so the ordinary differential equation of transverse vibration of the disc is resulted as

Approximation of first－order mode functionto the desired solution of transverse deflection of the disk at low speed.The time－varying function items included珋σ（t）in Eq.（9）reflect or interpret the fact that the space position of contacting load between the pads and disc varies with time t，after the vibration description for the beam is changed from a partial differential equation to an ordinary differential one.Because the stiffness value varies periodically with time，the natural frequency of the pad′s horizontal direction cannot keep in a fixed value but fluctuates periodically near 2.71Hz during integration，as shown in Fig.7.Accordingly，the system equilibrium positions will also fluctuate slightly（see the example of the angle equilibrium of pad in the vicinity of－2.678 6×10－6）.

Fig.7 Fluctuations

Fig.8 Bifurcation of horizontal direction of pad

Eqs.（4，5）are solved using the Runge－Kutta method，and the dynamical behavior of the braking system with moving interaction between parts can be investigated numerically.The complex eigenvalue analysis is carried out referred to Ref.［22］and the eigenvalues are 0.22＋19.58iat v＝1and－0.27＋19.58iat v＝2（assuming sign functions are positive）.The bifurcation diagram in Fig.8reveals horizontal motion of the pad under braking process with the decrease of speed v.The points are sampled when the velocity of horizontal motion is zero.The instability of motion of the pad happens and the stick－slip limit cycle is resulted at v＝1.25，which is also known as the critical speed.

However，the use of rigid belt model ignores the interaction between the brake pad and disc in transverse direction.To overcome this shortage，elastic or flexible brake disc models are adopted in recent investigations.A braking system with a flexible thick plate for disc and two continuous beams for pads by using the Mindlin′s theory is established by Beloiu and Ibrahim to account for the influence of flexible belt on the braking behaviors.The braking noise and response in time and frequency domains are investigated analytical－ly and experimentally by considering the influence of non－linearity and randomness of contact forces［10］.Nayfeh，Jilani and Manzione reduced the order of a circular flexible uniform thickness disk and analyzed its dynamical behavior analytically［11］.Recently，an elastic annular disc model is adopted for a pre－loaded mass－damp－spring system with separation and reattachment by Li，Ouyang and Guan to investigate friction－induced vibration using the improved model［12］.They found that separation often occurs in low speed，which is caused by friction during the unstable vibration.Larger in－plane stiffness and pre－load bring earlier separation and instability.In addition，the frequency of disc is increased with the effect of separation.However，the contact effect between the slider and disc and internal resonances were not considered.

3 Numerical Analyses

3.1 Non－internal resonances

To reveal the variation of the coupling system dynamics during velocity decrease process，the time histories，spectra（Fig.5（a）and Fig.6（a）），and phase portraits（Fig.9）of the selected vibration characteristics of the upper pad are presented at the speeds v＝2and v＝1，respectively.When v＝2，vibrations of pads will be damped，and a quasi－periodic motion occurs in this non－stationary case，which may be attributed to sustained oscillation of the contacting force resulted from the disc transverse deflection.The amplitude of upper pad in xdirection keeps vibrating with small amplitude，until changing to be an equilibrium at about 2.5×10－5 .

The values referred in this paper are shown in Table 1.

3.2 Acceleration of disc rotation

Fig.10shows the time histories of braking processes of the pads and disc at deceleration珔a＝－0.005and珔a＝－0.002，respectively.One finds that the higher the braking deceleration is，the faster the disc speed reaches to zero and the shorter the stick－slip vibration lasts.Obviously，the stick－slip vibration with high amplitude is more extensively at lower speed，where small"jitter"phenomena happens many times.As the speed approaches zero，the pads cannot stop vibration immediately.In fact，the vibration lasts for a short time under action of the relative friction force between the pads and disc.

The equations of motion of the pad－disc coupling system are deduced as andσ＝σ（T）represents the position of the loads varying with the time.

Fig.9 Phase portraits of the pad

4 Conclusions

Fig.10 Dynamic responses of the upper pad

The disk－pads coupling braking system with－Stribeck－type frictional interaction with each other is investigated in this paper.The disk is simplified as an annular elastic beam and its vibration under action of moving loads is simulated using the finite difference method.Then Galerkin reduction is used for the disk vibration equation and numerical simulations are carried out for the 7－DOF equations.Conclusions are summarized in the following.

（1）Mainly the first－order mode vibration of the annular beam can be induced by frictional moving loads.By using the Galerkin reduction，the first－order mode vibration equation of the disk is resulted with time－varying stiffness，which reflects the moving contact between the pads and disk.As a result，the system equilibrium positions and natural frequency of the disc will fluctu－ate periodically during time integration.

（2）As the disc speed decreases below the critical one，the relative equilibrium of the pad in the disc loses its stability and stick－slip type vibration will be resulted in all directions′movements.As a counterpart of the pads，the disc vibrates also with large amplitude transversely.

李莉觉得许峰现在也知道，他们之前在一起的时光，现在变得很微妙，他们大度地演绎了一场高尚爱情。现在看来，那种幸福就像悬在空中。

Lim，KB[3]等提出了一种水-热存储系统用来对空气源热泵进行除霜，结果表明，该装置可以提高18.6%的能量利用效率。卜萃文[4]等对重庆某大学学生公寓直热式空气源热泵热水系统的运行时段及回水管网进行了改进。结果表明，改造后的系统与原有系统相比，冬季、过渡季节、夏季分别能达到40.5%、38.6%、28.7%的节能率，节能效果显著。

（3）During non－stationary braking process，braking deceleration shortens the period of stickslip vibration but enlarges the vibration amplitude.In addition，small"jitter"phenomena can happen many times during process.

Acknowledgement

This work is supported by the National Natural Science Foundation of China （Nos.51575378，11272228and 11332008）.

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