Assessing the Reliability of some Classical Mechanical Vibration Designs via Simulation Software

This work is part of a series on problems which aid students in achieving a better understanding of underlying engineering principles and a better appreciation for the limitations of linear physical modeling in dynamics. Another issue worthy of attention is how robust some designs are based on linear modeling. The problems treated here (3 in all) do not have analytical solutions and have only become tractable due to the widespread availability and early exposure in introductory mathematics classes to simulation software such as MAPLE, MATLAB etc. MAPLE is employed here. The first problem is meant to enhance students understanding of stability. It concerns a spring-mass system vibrating in a slot in a horizontal disk rotating with a prescribed motion. It is shown that for certain spin-up speeds, instabilities can develop if the system parameters are not chosen properly. Effects of spring non-linearity on these instabilities are explored. An area that students should be aware of is the reliability of designs based on linear models. A passive vibration absorber is revisited and it is shown that the classical choice of system parameters may not work if spring non-linearities are included. Choices that do work are given. Finally a problem involving "vibration cancellation" is studied. The response of a linear single degree of freedom spring-mass system to a pulse can be made identically zero for all times greater than a certain one by the application of a second pulse with a suitable phase difference. Some effects of spring non-linearities on the linear model predictions are given. Assessment was achieved by noting students better and fuller understanding of the basics. Introduction Other articles on the use of simulation in engineering education exist. See for example, the work of Fraser et al. 1 on simulation in fluid mechanics. Questions from the Fluid Mechanics Concepts Inventory 2 (FMCI) identified some student conceptual difficulties. A simulation involving these concepts was developed and its efficacy was addressed using a second administration of the FMCI. A recent work of Kieffer et al. 3 explored the use of simulation in helping students achieve a better understanding of materials science concepts. They used a survey and student performance to assess impact. This latter point is also the main assessment of the current work. It is the authors’ experience that exposure to simulation, such as the ones at hand, leads to a better and fuller understanding of the basics. This paper is one of an ongoing series (see references 4, 5, 6, 7, ) on the role of mathematical software in furthering the depth of understanding of the dynamics of mechanical systems. A major theme of the current work is the effect of non-linearities. In particular, one of the issues addressed is how robust are design parameters obtained from linear models. The problems treated here (3 in all) do not have analytical solutions and have only become tractable due to the widespread availability and early exposure in introductory mathematics P ge 22247.2 classes to simulation software such as MAPLE , MATLAB 10 etc. MAPLE is employed here. The first problem is meant to enhance students understanding of stability. It concerns a springmass system vibrating in a smooth slot in a horizontal disk rotating with a prescribed motion. It is shown that for certain spin-up speeds, instabilities can develop for certain values of the system parameters. Effects of spring non-linearity on these instabilities are explored. An area that students should be aware of is the reliability of designs based on linear models. A passive vibration absorber is revisited and the classical choice of system parameters is investigated for a case where spring non-linearities are included. Finally a problem involving "vibration cancellation" is studied. The response of a linear single degree of freedom spring-mass system to a pulse can be made identically zero for all times greater than a certain one by the application of a second pulse with a suitable phase difference. Some effects of spring non-linearities on the linear model predictions are given. Physical Examples Spin-up Stability An interesting and informative example is that of a particle, restrained by a spring, vibrating in a smooth slot in a rotating platform. Intuitively, there is competition between the stabilizing spring force and the destabilizing centrifugal force and a basic question is how that scenario plays out. Shown in Figure 1 is a mass moving in a slot and connected to a spring, the whole system rotating in a horizontal plane at a constant rate. Applying Newton’s law expressed in polar coordinates leads to: 2 [( ) (2 ) ] s r r F u Nu m R R u R R u θ θ θ θ θ − + = − + +