Analytical Modeling and Simulation

 

 

 

 

 

 

Research has been initiated in three areas of analysis. First, a rotor model has been developed which will serve as a testbed for dither control strategies. Many different types of models have been used in the past to model brake rotors and computer disks; see for example the review articles of Kinkaid, et al. [1] and Mottershead [2]. There are two general categories of disc models; those that use a modal description and those that use a finite-element model. While the latter is more general, the former was adopted in the initial phase of this investigation for simplicity's sake. In particular, the disc rotor was modeled by a thin, stationary, clamped-free annular plate.  The model can be summarized by the following equations:

                                       (1)

where

 

, E is the elastic modulus, h is the thickness, n is the Poisson's ratio, r is the mass density, and F(r,q,t) is the force per unit area applied by the brake pad, or by other external means. The term is the biharmonic operator, given in polar coordinates as

 

 

It is assumed that the plate is clamped at its inner radius, r = b:

 

 at r = b                                              (2a, 2b)

 

At the plate's outer radius, r = a, boundary conditions of zero moment and zero shear are applied:

 

                                    (3)

 

 

or

 

                        (4)

 

 

The exact solution to the annular-plate equations were developed by Vogel and Skinner [3].  Their solution was thoroughly verified and implemented into a Matlab computer program. Samples of the modeshapes for a clamped-free annular plate with b/a = 0.5 are shown in Figures 1a-1c (i designates the number of nodal circles, including the clamped location along the inner radius, and j designates the number of nodal diameters.)

 

Figure 1a: i =1, j = 2 mode of clamped-free annular plate with b/a = 0.5.

Figure 1b: i =2, j = 0 mode of clamped-free annular plate with b/a = 0.5.

 

Figure 1c: i =2, j = 3 mode of clamped-free annular plate with b/a = 0.5.

 

Although the model is not a highly-accurate representation of an actual brake rotor, it satisfies the main requirements for this initial phase of the investigation. Most importantly, it supports a fairly rich modal content. In other words, it is extendable to include many modes if necessary, and can include modes near the squeal frequency and near the dither frequency. Since the dither frequencies can be quite high, the model uses the exact modes of a clamped free annular plate, having an arbitrary number of nodal circles and nodal diameters. In contrast, mode-based models usually utilize the exact dependence on q (in terms of azimuthal harmonics cos(nq) and sin(nq)) but only use a cubic approximation for the radial dependence of each mode. Another requirement satisfied by this model is that it displays the correct qualitative trends in the natural frequencies as a function of thickness, inner and outer radii, number of nodal diameters and number of nodal circles.  Although disk rotation is not explicitly included, previous researchers have shown that this effect is small in typical brake rotor applications.

 

Once the eigenfunctions have been found, the displacement of any point of the plate can be expanded as:

 

                   (5)

 

where Rij(r) contains the radial dependence of the eigenfunction, and where = 0 for i = 1, 2, … The eigenfunctions are assumed to be normalized so that:

 

 

and

 

 

(similar relations hold for sin.)  Orthogonality of the eigenfunctions implies that

 


 for j¹q and/or  i¹p

 

 for all i,j,p,q


 for j¹q and/or  i¹p

 

 for all i,j,p,q

 

 

The forcing F(r,q,t) on the right hand side of equation (1) provides a means by which the annular plate model can be coupled to a contacting dynamic system. For example, one can consider a rotating spring-mass-damper system as shown in Figure 2, similar to the one studied by Iwan and Stahl [4].

 

 

Figure 2: Annular plate with rotating spring-mass-damper system

 

In this case, F(r,q,t) can be expressed as:

 

                      (6)

 

where d( ) denotes a Dirac delta function. At this point, the modal expansion (5) can be substituted into (6) with the result being substituted into (1). Invoking the orthogonality of the eigenfunctions would result in a set of coupled, linear ordinary differential equations in time. Unfortunately, these equations would have periodic coefficients, making the determination of stability more difficult to ascertain.

 

Alternatively, Iwan and Stahl [4] introduced a rotating reference frame, where the location of the spring-mass-damper system was fixed at f = 0: 

 

                                                                      (7)

 

Using the angle f, expansion (5) can be re-written in the form:

 

 

                   (8)

 

It must be realized that derivatives with respect to t must include the effect of the rotating frame; in light of (7) we have:

 

                                                      (9)

 

and

 

                                              (10)

 

Substitution of (9) into (1) and (6) yields:

 

           (11)

 

 

when the expansion (8) is substituted into (10) and orthogonality of the eigenfunctions is invoked, a set of linear ordinary differential equations in time are obtained. Unlike before, though, the equations have constant coefficients, so stability can be determined by a simple eigenvalue calculation. The form of the differential equations is

 

 

        p =1,2,…; q =0,1,2,                   (12)

 

                      p =1,2,…; q =0,1,2,…                           (13)

 

 

where, again, the generalized coordinate Bp0 may be discarded for p = 1, 2, … It is noted equations (12) and (13) constitute Ntot = (Nr)(2Nq+1) linear, second-order differential equations with constant coefficients.  A vector of generalized coordinates, y, may be defined as

 

 

Then, a state vector x can be formed from y and its time derivative:

 

 

Using x, equations (12) and (13) can be easily cast in the form:

 

                                                                         (14)

 

where [A(W)] is a 2Ntotx2Ntot matrix function of the rotation speed, W.  The eigenvalues of (14) can be computed as a function of speed and checked for stability.

 

 

It is known that friction can provide a destabilizing effect in rotational systems, even when the slip velocity is constant. To see this, consider the inclusion of a friction follower force, which is constant in magnitude but which remains tangent to the local contact surface at all times. The friction force adds the following term to the right-hand side of equation (11):

 

                                                     (15)

 

where Fq is a constant magnitude friction force parallel to the surface of the plate. At this point the expansion (8) into (15), multiplying by Rpq(r) cos(qf) and integrating over the area of the plate yields the following term to be added to the right-hand-side of (12):

 

                                                              (16a)

 

where Bpq = 0 when q = 0.  Similarly, multiplying by Rpq(r) sin(qf) and integrating over the area of the plate yields the following term to be added to the right-hand-side of (12):

 

                                                              (16b)

 

 

Future investigation:

- Inclusion of a dither excitation to understand its effect on the stability,

- Inclusion of a ‘non constant sliding’ friction force in the model,

- Optimization of the dither parameters for best control.

 

 

 

Note:

"Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF)."