Semi-analytical solution for a system with clearance nonlinearity and periodic excitation

Abstract

Many dynamical systems such as gears, tire-pavement, automotive brakes, and cam-follower have clearance nonlinearity and excitation, which are periodic in nature. It is essential to accurately predict the steady-state response of these systems using contact-mechanics-based model for understanding their nonlinear dynamic behavior. Among the methods available to theoretically solve the system’s nonlinear governing equation(s), a semi-analytical technique such as the harmonic balance method (HBM) is preferred over numerical approaches for various reasons, including accuracy. An HBM formulation that can predict the fundamental, sub-, and super-harmonic solutions is presented here. As multiple variants of HBM exist in the literature, this work focuses on comparatively evaluating the most appropriate variant for the system under consideration. Since the system has multiple discontinuities in terms of contact stiffness and damping forces, these have to be smoothed precisely to be utilized in the HBM. Hence, a novel smoothing function was proposed and evaluated against other existing smoothing functions in literature based on various criteria. Next, the most applicable HBM variant was selected with reference to steady-state solutions from numerical methods. The predictions from the selected HBM variant were validated against the results furnished in the literature for a similar system. Finally, the nonlinear frequency response of the system with multiple discontinuities was estimated using the selected HBM and found to be in good agreement with numerical results.

Appendices

Appendix

A: Values of parameters

Table 3 Value of parameters used in the study (Reference: [1])

B: Hyperbolic smoothing function

In reference to fig. B.1, assuming two continuous lines 1 and 2 (Eq. 18) that are taken from Eq. (4).

$$\begin{aligned} Line\ 1:&\ F_\textrm{s} - k_\textrm{c} (q-b) = 0 \end{aligned}$$

(18a)

$$\begin{aligned} Line\ 2:&\ F_\textrm{s} = 0 \end{aligned}$$

(18b)

Fig. 20

Hyperbolic function; Key: lines from the nonlinear force $F_\textrm{s}$; $\cdot \cdot \cdot$ curves with $u = - 2\epsilon ^2 k_\textrm{c}^2 b^2$; - - curves with $u = - \epsilon ^2 k_\textrm{c}^2 b^2$

The family of hyperbolic curves for which lines 1 and 2 are asymptotic, given as Eq. (19),

$$\begin{aligned} F_\textrm{s} [F_\textrm{s} - k_\textrm{c} (q-b)] + u = 0 \end{aligned}$$

(19)

Introducing a non-dimensional arbitrary constant $\epsilon$ (tuning parameter) to define a point closer to the point of nonlinearity (b,0). Assuming Eq. (19) passes through a point $(b-\epsilon b,k_\textrm{c} \epsilon b)$, marked as *. On substituting the point $(b-\epsilon b,k_\textrm{c} \epsilon b)$ in Eq. (19), the value of u is given as,

$$\begin{aligned} u = - 2\epsilon ^2 k_\textrm{c}^2 b^2 \end{aligned}$$

(20)

A reduction in u by a factor of 2 as given by Eq. (21) improves the accuracy of smoothed function as shown in Fig. B.1.

$$\begin{aligned} u = - \epsilon ^2 k_\textrm{c}^2 b^2 \end{aligned}$$

(21)

Thus, the family of hyperbolas is given by,

$$\begin{aligned} F_\textrm{s}[F_\textrm{s}- k_\textrm{c}(q-b)] - \epsilon ^2 k_\textrm{c}^2 b^2 = 0 \end{aligned}$$

(22)

On solving Eq. (22) for $F_\textrm{s}$, Eq. (23) can be obtained.

$$\begin{aligned} F_{s1,s2} = \frac{k_\textrm{c}}{2} (q-b \pm \sqrt{(q-b)^2 + 4\epsilon ^2 b^2} ) \end{aligned}$$

(23)

Using curve with the positive sign as smoothed function (curve 1, as marked in Fig. B.1) and is given by,

$$\begin{aligned} Curve\ 1: \varGamma _s = \frac{k_\textrm{c}}{2} (q-b + \sqrt{(q-b)^2 + 4\epsilon ^2 b^2} ) \nonumber \\ \end{aligned}$$

(24)