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Research Papers

# The Bounds on the Coefficients of Restitution for the Frictional Impact of Rigid Pendulum Against a Fixed Surface

[+] Author and Article Information
V. A. Lubarda

Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411vlubarda@ucsd.edu

For small angle $φ=arctan(a/b)$ there is no rebound if friction is sufficiently large $(μ≥tan φ)$; see Ref. 8, p. 180.

The coefficient $κ$ also depends on the material properties and the incidence angular velocity $ω−$, affecting the nature of the deformation, but such dependence cannot be determined within a rigid body mechanics (10).

The two definitions yield different expressions in many (but not all) frictional impact problems in which there is a change in the slip direction during the impact process. For example, Newton’s and Poisson’s definitions of the coefficient of normal restitution, as well as the energetic definition, to be discussed in Sec. 4, for a spinning disk striking a rough horizontal surface are all equivalent $(κ=κ̂=η)$.

From experimental or numerical finite element method evaluations, it may be anticipated that $η$ depends on the material properties, the radius of pendulum’s local curvature in the contact region, and the incidence angular velocity affecting the extent of inelastic deformation in the region of contact.

The notations used in Ref. 8 are $pc=τ0$, $pf=τ1$, $e∗=η$, $r1=a$, and $r3=b$.

If $η$ is assumed to be given, then the kinematic and kinetic coefficients of normal restitution depend on $η$ and the geometric parameter $μb/a$, accounting for friction and the pendulum impact configuration, represented by the ratio $b/a$. Clearly, the higher the value of $η$ (less dissipation by the deformation), the higher the coefficients $κ$ and $κ̂$, and thus the higher the rebound $(ω+)$.

The tangential stiffnesses of the pendulum and the surface are assumed to be infinite, so that the elastic energy is entirely due to deformation in the normal direction.

Thus, $η$ is here the arithmetic mean of $κ$ and $κ̂$, i.e., $η=(κ+κ̂)/2$, which is in agreement with the exact geometric mean relationship $η=(κκ̂)1/2$, to first order terms in $μb/a$.

J. Appl. Mech 77(1), 011006 (Sep 24, 2009) (7 pages) doi:10.1115/1.3172198 History: Received October 02, 2008; Revised April 12, 2009; Published September 24, 2009

## Abstract

Upper bounds on Newton’s, Poisson’s, and energetic coefficients of normal restitution for the frictional impact of rigid pendulum against a fixed surface are derived, demonstrating that the upper bound on Newton’s coefficient is smaller than 1, while the upper bound on Poisson’s coefficient is greater than 1. The upper bound on the energetic coefficient of restitution, which is a geometric mean of Newton’s and Poisson’s coefficients of normal restitution, is equal to 1. Lower bound on all three coefficients is equal to zero. The bounds on the tangential impact coefficient, defined by the ratio of the frictional and normal impulses, are also derived. Its lower bound is negative, while its upper bound is equal to the kinetic coefficient of friction. Simplified bounds in the case of a nearly vertical impact are also derived.

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## Figures

Figure 1

A rigid pendulum, suspended from a frictionless pin O, strikes a fixed horizontal surface with the incidence angular velocity ω−. The componential reactions at the contact point with the coordinates (a,−b) are N and F.

Figure 3

Bilinear variation of the angular velocity ω=ω(τ), according to Eq. 9. The slopes in the compression and restitution phases of the impact are (a+μb)/J0 and (a−μb)/J0, respectively.

Figure 2

Schematic time variation of the normal and friction components of the reactive force during the impact of duration t1. The normal impulse up to an arbitrary time t is τ=∫0tNdt. The friction component of the reactive force is related to the normal component by the Amontons–Coulomb law of sliding friction F=−μN sgn(t−t0), where t0 is the time at which sliding changes its direction, and μ is the coefficient of kinetic friction.

Figure 6

A nearly vertical impact (b⪡a) of a rigid pendulum against a rough horizontal surface. The componential reactions at the contact point are N and F.

Figure 5

(a) The force ratio F/N versus the normal impulse τ. (b) Routh’s impact diagram showing the variation of the tangential impulse f versus the normal impulse τ.

Figure 4

(a) Two impact configurations of the rigid pendulum against a fixed surface, corresponding to two different values of the angle φ. (b) The variation of the Newton coefficient of normal restitution κ̂ with φ (in radians), in the case of an elastic impact (η=1), with different coefficients of friction μ. The far left curve is for μ=0.1, and the subsequent curves toward the right are for μ=1/3,0.5,2/3,1. As φ→π/2, the coefficient κ̂=−ω+/ω−→1 for all μ (passive friction for vertical impact).

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