Optimization of Vibration Absorbers: A Graphical Method^ for Use on Idealized Systems With Restricted Damping

Abstract

can be used on complex real-world problems. It can also provide additional insight into the physical behavior of a transient system. References 1 Aggarwal, T. C, and Hasz, J. R., "Designing Optimum Dampers Against Self-Excited Chatter," ASME Paper No. 68-WA/ Prod-25. 2 Bartel, D. L., Haug, E. J., and Rim, K., "The Optimum Design of Spatial Frames Using the Method of Constrained Steepest Descent With State Equations," to be published in Journal of Engineering for Industry, TRANS. ASME, Series B, Vol. 93, 1971. 3 Falcon, K. C, Stone, B. J., Simcock, W. D., anaSAndrew, C, "Optimization of Vibration Absorbers: A Graphical Method^ for Use on Idealized Systems With Restricted Damping," Journal of Mechanical Engineering Science, Vol. 9, No. 5, 1967, pp. 374-381. 4 The authors have presented two applications of an optimization program involving low order linear mechanical systems. The results are of some interest although the paper is hard to follow due to misprints (Figs. 1 and 4 seem to be interchanged) and unusual notation (s is called a state vector while z, a vector of displacements and velocities, though typically called a state vector in dynamics, is not called a state vector in the paper). The reference to "time domain optimization" in the title may be misleading. Another way to describe what the authors have done is by the phrase "parameter optimization" since the computer program apparently varied a damping parameter and a spring constant to optimize some aspects of the transient response of a particular 2 degree-of-freedom linear system model subject to inequality constraints. In related work, a group at I.I.T. investigated several possible methods of optimizing vibratory systems. Included were not only parameter optimization but also a true time domain optimization in which dynamic programming techniques were used to determine the optimal time history of forces which would achieve a minimum of a performance criterion subject to constraints, independent of the manner in which the force would actually be realized The authors have demonstrated their ability to achieve parameter optimization using a gradient technique, but it is not entirely clear that the method should be used on "complex realworld problems." The authors' examples are hardly complex nor do they necessarily represent the real world. Would anyone realistically construct a car bumper by trying to match it to a linear spring and dashpot combination? The potential of the technique might be better illustrated by using it to optimize nonlinear devices which, though suboptimal in the true time domain sense, yield responses closer to optimal than could be achieved with linear devices. Even in the vibration absorber problem, the proposed criterion of time-optimal energy dissipation is not so easy to justify. This criterion evidently yields a different optimal system for every different initial condition and indeed for every choice of percent energy remaining, t. This phenomenon might be explained by realizing that the response of the system in question can at least roughly be considered the sum of the responses of two normal modes. Each modal velocity is described 2 Professor and Graduate Student, respectively, Department of Mechanical Engineering, University of California, Davis, Calif. 3 Numbers in brackets designate Additional References at end of discussion. by an exponentially decaying sinusoid whose initial value is a function of all initial conditions. The rate of exponential decay for each mode can be different functions of the design parameters. For e ->-1 the tradeoff between the fast decay of a mode with high initial conditions and the slow decay of the mode with lower initial conditions is very critical. In the extreme one could hypothesize the mode with large initial value decaying very rapidly and the low initial condition mode never decaying in order for values of 6 close to one to be achieved in optimum time. The exact nature of the trade off is a function of initial conditions and e. When e ->-0, the responses of both modes are forced to be small as soon as possible. When t-*T for this case, the difference due to initial conditions in the values of the modal responses must be small because the responses are small. Hence, initial conditions have little effect, and the main concern of the optimization process becomes making both responses small as soon as possible. This is the most reasonable criteria for optimization. Surely for most vibration absorber design techniques one desires a useful criterion which will produce a single isolation design which is optimal for a broad class of inputs or initial conditions. The authors' results for e ->• 0 suggest the sort of result in optimal linear regulator design in which an infinite time integral square criteria yields an optimal design independent of initial conditions. Finally, the paper illustrates the difficulties in interpretation which often arise in parameter optimization. Computed optimal parameters may be nearly useless unless supplemented by an understanding of the influence of small changes in the optimization criterion on the system parameters. In equation In another instance, the authors have taken a specific result and made a rather broad generalization from it which may not be justified. The statement that "an optimum steady state absorber will also be nearly optimum for transient conditions when e is small" surely must be qualified. Though the statement is true for the authors' specific case, many other constraints and criteria might be used, and it would be amazing if the statement were universally true. Only when the systems remain entirely in the domain of linear optimum systems can one expect simple relations between optimal systems designed on the basis of transient and forced response [3]