基于ANSYS的汽車發(fā)動(dòng)機(jī)連桿有限元分析
基于ANSYS的汽車發(fā)動(dòng)機(jī)連桿有限元分析,基于,ansys,汽車發(fā)動(dòng)機(jī),連桿,有限元分析
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譯文題目： Analysis of disc brake of
gyroscope
陀螺儀盤形制動(dòng)系統(tǒng)分析
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Analysis of disc brake of Gyroscope
Abstract
In this paper, the dynamic instability of a car brake system with a rotating disc in contact with two stationary pads is studied. For actual geometric approximation, the disc is modeled as a hat-disc shape structure by the finite element method. From a coordinate transformation between the reference and moving coordinate systems, the contact kinematics between the disc and pads is described. The corresponding gyroscopic matrix of the disc is constructed by introducing the uniform planar-mesh method. The dynamic instability of a gyroscopic non-conservative brake system is numerically predicted with respect to system parameters. The results show that the squeal propensity for rotation speed depends on the vibration modes participating in squeal modes. Moreover, it is highlighted that the negative slope of friction coefficient takes an important role in generating squeal in the in-plane torsion mode of the disc.
Keywords： Gyroscopic; Disc brake; Brake squeal; Mode-coupling
1. Introduction
Disc brake squeal has been investigated by many researchers for several decades. Much valuable information on squeal mechanisms has been accumulated throughout the research. Kinkaid et al. [1] presented the overview on the various disc brake squeal studies. Ouyang et al. [2] published the review article focused on the numerical analysis of automotive disc brake squeal. They have shown that one major approach on brake squeal study is the linear stability analysis. From the linearized equations of motion, the real parts of eigenvalues have been calculated for determining the equilibrium stability. In the literature, there are two major directions on the linear squeal analysis: the complex eigenvalue analysis of the static steady- sliding equilibrium [3–8] and the stability analysis of rotating brake system [9–12,14]. The stability analysis at the static steady-sliding equilibrium of the stationary disc and pads provides the squeal mechanism as mode-merging character in the friction–frequency domain. Parti- cularly, Huang et al. [6] used the eigenvalue perturbation method to develop the necessary condition for mode-merging without the direct eigensolutions. Kang et al. [7] derived the closed-form solution for mode-merging between disc doublet mode pair. Due to the stationary disc assumption, the finite element (FE) method has been easily implemented as referred to the review article [2]. Alternately, Cao et al. [13] studied the moving load effect from a FE disc brake model with moving pads, where the disc was stationary, and therefore, the gyroscopic effects were neglected. Giannini et al. [15,16] validated the mode-merging behavior as squeal onset by using the experimental squeal frequencies. On the other hand, the stability of a rotating disc brake has been investigated in the analytical manner. The rotating disc brake system has been modeled as a ring [10] and an annular plate [12] in point contact with two pads, and an annular plate subject to distributed frictional traction [9]. With inclusion of gyroscopic effect, the real parts of eigenvalues have been solved with respect to system parameters. Due to the complexity of the rotating disc modeling, however, a rotating FE disc brake model has not been developed yet.
Fig. 1. Hat-disc brakesystem.
Fig. 2. Coordinate systemoftherotatingdisc,reference(y) andlocal(c) coordinates.
Recently, Kang et al. [14] developed a theoretical disc brake model in the comprehensive manner. The disc brake model consists of a rotating annular plate in contact with two stationary annular sector plates. The comprehensive analysis explained the stability character influenced by mode-coupling and gyroscopic effect, and provided the physical background on the approxima- tions and mechanisms used in the previous squeal literature. However, it still contains limitations on examining brake squeal mechanisms since the annular plate approximation does not represent all of modal behaviors existing on the physical disc brake, for example, the in-plane mode and hat mode of the disc. In this paper, the methodology of constructing a rotating FE disc brake model is developed. Consequently, it enables us to examine the squeal mechanisms in the physical FE brake model subject to rotation effects. The global contact model [10] describing the contact kinematics under the undeformed config- uration is utilized to develop contact modeling between the rotating disc and two stationary pads. From the assumed mode method, the equations of motion of the friction-engaged brake system are derived. The numerical results demonstrate several squeal modes and explain the corresponding squeal mechanisms.
2. Derivation of equations of motion
The disc part of a brake system is modeled as a hat-disc shape structure as shown in Fig. 1. The hat-disc is subject to the clamped boundary condition at the inner rotating shaft and the free boundary condition at the outer radius. Owing to the complexity of the geometry, the finite element method is utilized for modal analysis. The disc rotation with constant speed（Ω）generates friction stresses over the contact with two stationary pads loaded by pre-normal load （N0）. The friction material of the pad is modeled as the uniform contact stiffness (kc), where contact stresses are defined on the global contact model. Centrifugal force is neglected due to the slow rotation in the brake squeal problem. In order to describe the contact kinematics, the displacement vectors of the disc and top pad are expressed in the reference coordinates (Fig. 2), respectively, such that
where the superscripts, p1 and p2 denote the top and bottom pads, respectively, and the disc displacement is also defined in the local coordinates(Fig. 2):
As shown in Fig. 3, the contact point P of friction material of the top pad is assumed to be in contact with P of the disc and laterally fixed with R of the top pad, which results in
Fig. 3. Contact kinematicsatacontactpoint P (or P0) in the global contact model:(a)contact displacements; (b) contact forces. P0 of friction material of the top pad is in contact with P of the disc.
The velocity vectors of the disc and top pad are obtained from the following time-derivatives. First, the position vectors of the disc are expressed in the local coordinates as
For describing the direction vector of friction force, the contact velocity vector of the disc is derived by taking the time-derivative in Eq. (6) in the reference coordinates:
where the coordinate transformation is given by the differentia- tion in the local coordinates such that
Since the brake pad is stationary, the contact velocity vector at P’ of the top pad is simply the partial time-derivative of Eq. (4):
From Coulomb’s law of friction, contact friction force is expressed as
where the normal load is the sum of pre-stress (p0=N0/Ac) and the normal load variation:
and the relative velocity at top contact is given by
In order to capture the negative slope effect, the continuous friction curve [14] is used such that
Where μs,μk and α are the control paramerers determining the magnitude and the slope of the friction coefficient, and the friction coefficient is assumed to be uniform and calculated at the centroid of the contact area (rctr).
The transverse vibrations of the disc and pad components are expressed in the modal expansion form of N= ( Nd + 2Np ) truncated modes using the assumed mode method:
are the nth transverse mode shape functions obtained from the eigenfunctions of the top pad, disc and bottom pad components, respectively. The radial and tangential vibrations, can be written in the modal expansion form associated with the corresponding mode shape functions as well. The modal coordinates are rearranged in the vector form for the following discretization:
From the discretization of Lagrange equation by modal coordinates, the friction-coupled equations of motion are given by
Where U is the total strain energy of the uncoupled component disc and two pads, and
Here Vd and Vp are the volumes of the disc and pad, respectively. In the similar manner of obtaining the virtual work and contact strain energy at the top contact, on the bottom contact can be derived as well. The direction vector of friction force at the top contact is linearized by Taylor expansion at the steady sliding equilibrium such that
where h:o:t denotes the higher order terms. Here is associated with frictional follower force as explained in [11] and neglected in the neglected in the subsequent analysis due to the insignificance of the frictional follower force as referred to [5], [10,11] and [14].
Using the finite element method, the transverse mode shape functions are discretized in the matrix form
where the lengths of their columns correspond to the numbers of nodes in the component FE model. The radial and tangential mode functions are also denoted as and .
From the mass-normalization and the linearization at the steady-sliding equilibrium of Eq. (23), the homogeneous part of the linearized equations of motion takes the (N×N) matrix form such that
where the system matrices are described in Eq. (37) and Eqs. (A.1)–(A.7) of Appendix A. Substituting into Eq. (36) and solving Re(λ) and Im(λ) of the characteristic equation result in the determination of the modal stability and frequency. Here the physical meaning of each system matrix of Eq. (36) is provided in the following. is the gyroscopic matrix to be described in Eq. (37), [C] is the structural modal damping matrix, and is the negative slope matrix. The negative friction-slope effect can be referred to [17]. is the radial dissipative matrix stemming from in Eq. (32). Also, [ω2] is the natural frequency matrix of the disc and pad components, is the contact stiffness matrix. Of the non-symmetric stiffness matrix, is the non-symmetric non-conservative work matrix produced by friction-couple. is derived from the in-plane frictional follower force associated with in Eq. (32), but neglected in the subsequent analysis due to its insignificance [14] as well. The local contact model [10] incorporated with the frictional follower forces can be referred to Kang et al. [14], where the frictional follower force effects were shown to be marginal due to the dominant role of [B] in the numerical and analytical manners.
Fig. 4. Transverse modal vector at z=zk interpolated by the uniform planar=mesh method: (a) modal vector on the irregular mesh; (b) modal vector interpolated on the uniform planar mesh in the polar coordinates
In the finiteele ment approach, several technical difficulties are encountered in calculating Eqs. (27)–(31) numerically and summarized as:
The mesh of the contact area between the disc and pad should be identical in order to connect the finite contact force elements on the same contact positions of the mating parts.
Td requires the numerical θ-derivatives of modal vectors.
Particularly, the gyroscopic matrix is given by
Where
In order to resolve the above, the hat-disc and each pad should be uniformly meshed in the cylindrical coordinates by ANSYS (or any other pre-processing FE software). In general, this task is tricky and not recommended for the practical purpose. Alternately, the uniform discretization will be achieved by interpolating the modal vectors of irregular meshes onto those of uniform meshes. The only pre-requisite for this task is to discretize the disc geometry in the axial direction (as Fig. 1) generating the planar mesh on each layer perpendicular to the axis, where the planar meshes are not yet uniform. Then, the modal vectors assigned to the planar mesh of each layer are interpolated onto those of the uniform mesh in polar coordinates by MATLAB, which will be referred to the uniform planar-mesh method. Fig. 4 illustrates how the modal vector on the irregular planar mesh is interpolated onto that of the uniform planar mesh. For the mode shape shown on the irregular mesh (Fig. 5a), the interpolated modal vector on the uniform planar meshes assigned to the top surfaces of the rotor and hat parts is demonstrated as in Figs. 5b and c.
From the uniform planar mesh in the cylindrical coordinates, the numerical θ-derivatives of the nth mode vector can be calculated at (ri, θj, zk), for example,
are the numbers of the nodes of the hat-disc, respectively, in the cylindrical coordinates (r, θ, z ). Fig.6 demonstrates the several θ-derivative modal vectors of the hat-disc at a given zk.
Fig. 5. The uniform planar-mesh method: (a) mode shape on their regular mesh; (b) modal vector of A on the uniform planar mesh; (c) modal vector of B on the uniform planar mesh. A: top rotor surface; B: top hat surface.
In order to assign the finite contact force element to each finite element of the disc and pad contacts at the same location, the planar mesh taken in the disc contact surface is defined on the pad contact surface as well. Moreover, the modal vectors on the pad contact are interpolated onto those of the defined planar mesh. Connecting the finite contact force element between the disc and pad is referred to Fig. 7 and [18]. As a result, the numerical volume and area integrations are available in such a way that
Where Mc, Mp denote the number of the nodes, respectively, of the contact area and pad in the cylindrical coordinates, and ?, gd, gp are the quantities associated with modal vectors interpolated on the uniform planar mesh.
Fig.6. Derivatives of a transverse modal vector (nth mode, n=19) at zk=4mm.
(a) ， (b) ，(c) ，(d) 。
As previously mentioned, one of the major differences between the current model and the previous gyroscopic annular plate models is the vibration modes obtained from the different disc geometry. In Kang et al. [14], the only vibration modes of the disc are the transverse modes of the annular plate. In automotive applications, however, the transverse mode approximation cannot capture the general modal behavior, for example, of the in-plane mode, hat-mode, and so forth. Fig.8 illustrates several vibration modes of the FE hat-disc model that the annular plate approx- imation is not likely to capture.
Fig. 7. Scheme of the contact mesh: (a) irregular mesh; (b) the contact forces assigned to （rj，θj） on the identical uniform mesh of the contact surface
3. Numerical results
The numerical simulation is conducted for the system para- meters given in Tables 1 and 2. The following numerical stability analysis will be divided into two sections: constant friction coefficient and velocity-dependent friction coefficient. Fig. 9 illustrates the friction–velocity curves used in the subsequent numerical analysis. The velocity-dependent friction coefficient is linearized in the form of negative slope matrix [Ns], whereas the negative slope effect disappears under the assumption of the constant friction coefficient. The natural frequencies obtained through modal analysis at kc=0 can be found from those at K=0[%] in Fig.10. Here, 3431 and 5720 Hz 3431 and 5720 Hz correspond to the first bending and torsion mode, respectively, of the pad, and 1365 Hz corresponds to the in-plane torsion mode of the disc.
Fig.8. Several vibration modes of the hat-disc.
Table 1 Nominal values of disc parameters.
3.1. The constant friction coefficient assumption
Under the assumption of the constant friction coefficient, the squeal mechanisms associated with the flutter modes are investigated by the eigenvalue sensitivity analysis. The contact stiffness and the friction coefficient are chosen for the parameters of the following sensitivity analysis. Fig.10 demonstrates the frequency loci of the friction-coupled system (kc≠0，μ=0.42) with respect to the contact stiffness variation at Ω=5rad/s. In the stiffness–frequency domain, the unstable frequency loci are identified by marking on the frequency loci corresponding to the eigenvalues having positive real parts. The mode shapes asso- ciated with the unstable frequency loci are used for examining the squeal character. The pad rigid modes and the third disc transverse doublet mode pair are found to participate in the squeal modes as shown in Figs. 11a and 12a. It is relevant to track the frequency and real part loci with respect to friction coefficient for demonstrating the mode-coupling between two closely spaced modes in the following.
Table 2 Nominal values of pad parameters.
Fig.9. Constant friction coefficient (broken line: μ=0.42) and velocity-dependent friction coefficient (solid line: μs=0.5,μk=0.32,α=1.0)
Fig.11 illustrates the binary flutter modes stemming from two rigid modes of the pad. From the mode-merging character shown in the eigenvalue loci without rotation effects, it is found that the mode-coupling of the binary mode is engaged enough to causes the modal instability by splitting the branches of Re(λ). The rotation effects influence on equilibrium stability by altering the eigenvalue loci of the rotation-free disc approximation [14] (the disc model without rotation effects). Particularly, the modification of Re(λ) loci is attributed to the modal damping separation. In theμ-Re(λ) domain, the radial dissipative effect rotates the loci of Re(λ) clockwise around the pivot point (μ= 0) ncreasing tcritical μ, whereas the separation between the two modal radial dissipative (viscous damping) terms destabilizes the steady-sliding equilibrium by strengthening the splitting of Re(λ), which is called ‘‘viscous damping instability [19]’’. This eigenvalue perturbation due to rotation effects has been analytically investigated in [14].
Fig.10 Stability map in the stiffness-frequency domain for constant friction coefficient; the mark ‘?！?denotes Re(λn)>0 for the nth mode of the friction-coupled system (kc≠0，μ=0.42，N=92), ξn=0.002，K[%]=100*kc/knom。
Fig.12 demonstrates that the modes 13,14 of Fig. 10 are associated with the 3rd disc transverse doublet pair and their mode-coupling generates the modal destabilization. The rotation effects on the mode-coupled eigenvalues are also seen in Figs,12b and c. For the disc doublet mode pair, the gyroscopic destabilizing effect is further involved due to disc rotation. The gyroscopic effect has been shown to strengthen the splitting of Re(λ) as well [14].
The increase of rotation speed Ω is shown to change the modal stability map of Fig. 10 by comparison with that of Fig.13. Due to the gyroscopic destabilization on the nonconservative brake system, the additional squeal modes appear at the higher speed. Fig. 14 illustrates the mode shapes of the squeal modes newly appearing at Ω=20rad/s, but being stable at Ω=5rad/s. By using the comprehensive analytical model, Kang et al. [14] has explained that the gyroscopic destabilization stems from the gyroscopic frictional mode-coupling. Since the gyroscopic fric- tional mode-coupling is proportional toΩ, it can be said that the newly appearing squeal modes in Fig. 13 have the sufficient amount of friction-coupling due to the increase of Ω. Therefore, the rotation-dependent Re(λ) of the of the rotating disc brake system may be the subject in predicting squeal occurrence with accuracy
Fig.11. Eigenvalue loci of modes 25,26 at K = 100 [％] in Fig.10 with rotation effects (solid line) and w/o rotation effects (broken line): (a) mode shapes; (b) frequency loci; (c) real part loci.
Fig.12. Eigenvalue loci of modes 13,14 at K = 100 [％
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