Aerodynamic Effects in a Dropped Ping-Pong Ball Experiment

This paper addresses aerodynamic modeling issues related to a simple experiment in which a pingpong ball is dropped from rest onto a table surface. From the times between the ball-table impacts, the initial drop height and the coefficient of restitution can be determined using a model that neglects aerodynamic drag. The experiment prompts questions about modeling the dynamics of a simple impact problem, including the importance of accounting for aerodynamic effects. Two nonlinear aerodynamic models are discussed in the context of experimental results

A Simple Dynamics Experiment Based on Acoustic Emission

This paper describes a simple experiment well suited for an undergraduate course in mechanical measurements and/or dynamics, in which physical information is extracted from an acoustic emission signature. In the experiment, a ping–pong ball is dropped onto a hard table surface and the audio signal resulting from the ball–table impacts is recorded. The times between successive bounces, or “flight times”, are used to determine the height of the initial drop and the coefficient of restitution of the impact. The experiment prompts questions about modeling the dynamics of a simple impact problem, including the use of the coefficient of restitution and the importance of accounting for aerodynamic effects

A Mass-Spring-Damper Model of a Bouncing Ball

The mechanical properties of a vertically dropped ball, represented by an equivalent mass-spring-damper model, are shown to be related to impact parameters. In particular, the paper develops relationships connecting the mass, stiffness and damping of a linear ball model to the coefficient of restitution and the contact time of the ball with the surface during one bounce. The paper also shows that the ball model parameters are functions of quantities readily determined in an experiment: (i) the height from which the ball is dropped from rest, (ii) the number of bounces, and (iii) the time elapsing between dropping the ball and the ball coming to rest. For a ball with significant bounce, approximate expressions are derived for the model parameters as well as for the natural frequency and damping ratio. Results from numerical and experimental studies of a bouncing ping-pong ball are presented

Design of PID Controllers Satisfying Gain Margin and Sensitivity Constraints on a Set of Plants

This paper presents a method for the design of PID-type controllers, including those augmented by a filter on the D element, satisfying a required gain margin and an upper bound on the (complementary) sensitivity for a finite set of plants. Important properties of the method are: (i) it can be applied to plants of any order including non-minimum phase plants, plants with delay, plants characterized by quasi-polynomials, unstable plants and plants described by measured data, (ii) the sensors associated with the PI terms and the D term can be different (i.e., they can have different transfer function models), (iii) the algorithm relies on explicit equations that can be solved efficiently, (iv) the algorithm can be used in near real-time to determine a controller for on-line modification of a plant accounting for its uncertainty and closed-loop specifications, (v) a single plot can be generated that graphically highlights tradeoffs among the gain margin, (complementary) sensitivity bound, low-frequency sensitivity and high-frequency sensor noise amplification, and (vi) the optimal controller for a practical definition of optimality can readily be identified

Optimal Design of Low Order Controllers Satisfying Sensitivity and Robustness Constraint

The set of all stabilizing controllers of a given low order structure that guarantee specifications on the gain margin, phase margin and a bound on the sensitivity corresponds to a region in n-dimensional space defined by the coefficients of the controllers. For several practical criteria defined in the paper it is shown that the optimal controller lies on the surface of that region. Moreover, it is shown how to reduce that region to avoid actuator saturation during operation

Robust PI Controller Design Satisfying Sensitivity and Uncertainty Specifications

This paper presents a control design method for determining proportional-integral-type controllers satisfying specifications on gain margin, phase margin, and an upper bound on the (complementary) sensitivity for a finite set of plants. The approach can be applied to plants that are stable or unstable, plants given by a model or measured data, and plants of any order, including plants with delays. The algorithm is efficient and fast, and as such can be used in near real-time to determine controller parameters (for online modification of the plant model including its uncertainty and/or the specifications). The method gives an optimal controller for a practical definition of optimality. Furthermore, it enables the graphical portrayal of design tradeoffs in a single plot, highlighting the effects of the gain margin, complementary sensitivity bound, low frequency sensitivity and high frequency sensor noise amplification

A Mass-Spring-Damper Model of a Bouncing Ball (Conference proceeding)

The mechanical properties of a vertically dropped ball, represented by an equivalent mass-spring-damper model, are related to the coefficient of restitution and the time of contact of the ball during one bounce with the impacting surface. In addition, it is shown that the coefficient of restitution and contact time of a single bounce are related to the total number of bounces and the total time elapsing between dropping the ball and the ball coming to rest. For a ball with significant bounce, approximate expressions for model parameters, i.e., stiffness and damping or equivalently natural frequency and damping ratio, are developed. Experimentally based results for a bouncing pingpong ball are presented

Machine Design Experiments Using Gears to Foster Discovery Learning

Machine Design Experiments Using Gears to Foster Discovery Learning For the typical undergraduate engineering student the topic of gears is introduced and discussed in several courses. Early exposure may be in a physics course or in a first dynamics course,where gear pairs are presented as an idealized means to change speed ratios and torque ratios.They are used for mechanical advantage or to achieve desired speed, and the focus is usually on kinematics. Since gears have inertia they store kinetic energy and are part of the dynamic equations of motion of mechanisms and machines. For mechanical engineering students, gears are a core component studied in courses such as \u27kinematics and dynamics of mechanisms\u27 and \u27machine design\u27, where the nomenclature and design equations are developed for various types of gears. There may be exposure to real gears in a mechanical engineering laboratory; more often, students may see gears passed around in class and as part of demonstrations.In this paper we describe new experiments that were designed to provide mechanical engineering students with discovery learning experiences with gears and mechanical systems using gears.The suite of practical experiments presents students with a range of challenges that require them to analyze, measure, design, and fabricate gears. Activities in the experiments include: (1) Identifying gear types (spur, helical, bevel, etc.) and appropriate applications (automotive transmissions and differentials, drills, gear head motors). (2) Disassembling and re-assembling a kitchen mixer (with design and manufacturing questions related to its gears). (3) Disassembling and re-assembling an automotive HVAC baffle sub-assembly (with measurement of train ratios, and design and manufacturing questions related to its gears). (4) Designing the gear mechanism for driving the minute and hour hands of a gear clock given a known yet arbitrary drive speed. Fabricating the gears of the clock via rapid prototyping (3D printing), assembling the clock, and then testing the timing accuracy.In addition to reporting the details of the experiments, we share experiences of students and teaching assistants in their use and effectiveness. We provide insights into how well students became familiar with types and nomenclature of gears and understood the applicability of different gears to actual real-world problems. The intent of the experiments is to effectively enhance mechanical engineering students\u27 awareness of gears and expand their knowledge and confidence in the use of gears in machine and mechanism design

Design of Predictive Controllers by Dynamic Programming and Neural Networks

This paper proposes a method for the design of predictive controllers for nonlinear systems. The method consists of two phases, a solution phase and a learning phase. In the solution phase, dynamic programming is applied to obtain a closed-loop control law. In the learning phase, neural networks are used to simulate the control law. This phase overcomes the curse of dimensionality problem that has often hindered the implementation of control laws generated by dynamic programming. Experimental results demonstrate the effectiveness of the metho

Robust PI Controller Design Satisfying Gain and Phase Margin Constraints

This paper presents a control design algorithm for determining PI-type controllers satisfying specifications on gain margin, phase margin, and an upper bound on the (complementary) sensitivity for a finite set of plants. Important properties of the algorithm are: (i) it can be applied to plants of any order including plants with delay, unstable plants, and plants given by measured data, (ii) it is efficient and fast, and as such can be used in near real-time to determine controller parameters (for on-line modification of the plant model including its uncertainty and/or the specifications), (iii) it can be used to identify the optimal controller for a practical definition of optimality, and (iv) it enables graphical portrayal of design tradeoffs in a single plot (highlighting tradeoffs among the gain margin, complementary sensitivity bound, low frequency sensitivity and high frequency sensor noise amplification)