Investigation of Fundamental Principles of Rigid Body Impact Mechanics

In impact mechanics, the collision between two or more bodies is a common, yet a very challenging problem. Producing analytical solutions that can predict the post-collision motion of the colliding bodies require consistent modeling of the dynamics of the colliding bodies. This dissertation presents a new method for solving the two and multibody impact problems that can be used to predict the post-collision motion of the colliding bodies. Also, we solve the rigid body collision problem of planar kinematic chains with multiple contacts with external surfaces. In the first part of this dissertation, we study planar collisions of Balls and Cylinders with an emphasis on the coefficient of restitution (COR). We conduct a set of experiments using three types of materials; steel, wood, and rubber. Then, we estimate the kinematic COR for all collision pairs. We discover unusual variations among the Ball-Ball (B-B) and Ball-Cylinder (B-C) CORs. We propose a discretization method to investigate the cause of the variations in the COR. Three types of local contact models are used for the simulation including: rigid body, bimodal linear, and bimodal Hertz models. Based on simulation results, we discover that the bimodal Hertz model produce collision outcomes that has the greatest agreement with the experimental results. In addition, our simulations show that softer materials need to be segmented more than harder ones. Softer materials are materials with smaller stiffness values than harder ones. Moreover, we obtain a relationship between the stiffness ratio and the number of segments of softer material to produce the physically most accurate B-C CORs. We validate this relationship and the proposed method by conducting two additional sets of experiments. In the second part of the dissertation, we study planar collisions of hybrid chains of balls and cylinders with an emphasis on the post-impact velocities. We use three types of materials including steel, wood, and rubber. We perform the collision experiments of balls and cylinders for three, four, and five-body chains and obtain their corresponding pre- and post-impact velocities. We propose a discretization method to accurately calculate the post-impact velocities of the colliding bodies in the chain. We use the bimodal Hertz contact force model and employ the Ball-Ball COR at the contacting segments to analyze the impact dynamics of the colliding objects. A relationship between the stiffness ratio and the number of segments of softer material is used to determine the required number of segments for each ball and cylinder connected in the chain. The simulation runs show that by using this relationship, we obtain the greatest agreement of the post-impact velocities of the colliding bodies with the experimental results. Finally, we consider the rigid body collision problem of particle based multi-branch kinematic chains with external surfaces. One end of the chain strikes an impact surface while other ends are resting on contact surfaces. The chain consists of two types of primitive building units, a mass with a revolute joint, and a connecting rod. A solution to the problem was presented before. Yet, the uniqueness and existence of solutions were not proven for the general case. In this chapter, we use the linear and angular momentum principles with a set of complementary equations to analytically prove that the solution exists and it is unique. The task is to find critical configuration conditions such that a contact mass has zero normal velocity, and normal impulse. We present a mathematical development that expresses the normal velocities and impulses at the contacting ends in terms of the normal impulse at the impact point. The approach not only proves uniqueness and existence, but also yields precise conditions to detect the so called critical configurations of the chain. When the chain has a critical configuration, the normal velocity and impulse at a contacting end become simultaneously equal to zero. We apply the proposed methods to obtain critical configurations of two numerical examples: a three- and a five-mass chains. Finally, we experimentally verify the existence of the critical configuration for a three-mass chain on flat and inclined surfaces with different inclination angles