UC Berkeley

In this dissertation, the dynamics of three classic mechanical systems are examined using a combination of numerical and analytical methods. The three systems are a rolling sphere, a pair of rolling cylinders, and a stack of blocks. The kinematics and dynamics of each of these systems are governed by a set of constraints. For the sphere and cylinders the complexities of their dynamics are governed by a set of non-integrable (non-holonomic) constraints, while the complexity of the stack of blocks can be attributed to stick-slip phenomena, impacts, and a time-varying set of integrable constraints. For each of these classic systems, we establish new results.

Consider a rigid body rolling with one point in contact with a fixed surface. Now suppose that the instantaneous point of contact traces out a closed path. As a demonstration of a phenomenon known as holonomy, the body will typically not return to its original orientation. The simplest demonstration of this phenomenon in rigid body dynamics occurs in the motion of a rolling sphere and finds application to path planning and reorientation of spherical robots. Motivated by recent works of Bryant and Johnson, we establish expressions for the change in orientation (i.e., holonomy) of a rolling sphere after its center of mass completes a rectangular path. The holonomy in this case can be quantified using an angle of rotation and an axis of rotation. We use numerical methods to show that all possible changes in orientation are possible using a single rectangular path. Based on the Euler angle parameterization of a rotation, we develop a more intuitive method to achieve a desired orientation using three rectangular paths. With regards to applications, the paths we discuss can be employed to achieve any desired reorientation of a spherical robot.

The next mechanical system we examine was inspired by a common, yet hazardous, method of transporting cylindrical tanks used to carry compressed gas. The method involves rolling both tanks at opposite angles of inclination to the vertical. By propelling one of the tanks while maintaining point contact between the tanks, both tanks can be moved such that their centers of mass move in a straight line. The purpose of our work is to explore this locomotion mechanism. First, the problem of supporting an inclined cylinder in point contact with a rough surface is examined. The analysis shows that dependent on the geometry of the cylinder and the coefficient of static friction, a wide range of angles of inclination are feasible. The presence of non-integrable constraints on the motion of the rolling cylinder is explored using the concept of a holonomy. The problem of transporting two cylinders using the aforementioned mechanism is then analyzed with the help of Frobenius' integrability criterion for constraints and numerical simulations. Our result demonstrate the surprising mechanical advantage of transporting a pair of cylinders, the range of possible angles of inclination, and the forces needed to sustain the motion.

The third mechanical system of interest is a collection of two-dimensional blocks stacked vertically. The surfaces of the blocks are rough. Of particular interest is the case where the bottom block in the stack is driven by simple harmonic motion. In the ensuing motion, a typical block in the stack can be at rest, sliding, rotating, or sliding and rotating with respect to the block underneath it. A single block in motion on a rough plane is well-known from studies in the 1980s to exhibit complex dynamics. The complexity of the dynamics of a stack of blocks dramatically increases as the number of blocks increases. In addition, the challenges to numerically investigate the dynamics are considerable. In this dissertation, we adapt a nonsmooth generalized-alpha method for systems with frictional contact to compute the dynamics of the stack. From the simulations we observe that high-frequency excitations of the bottom block tend to stabilize the stack. Our simulations also reveal the existence of an abundance of distinct solutions stemming from a unique initial configuration and excitation of the bottom block. Many, but not all, of these motions result in the toppling of the stack of blocks: a result that illustrates the surprisingly complex dynamics of a simple mechanical system and has application to robotic manipulation of stacks of objects.