The Treatment of Damping Forces in Vibration Theory

Summary

This paper is the first of a series of three which are concerned with the subject of “ hysteretic damping.” This type of damping, in a simple system with one degree of freedom, is like the familiar “ viscous damping ” in that it implies a resisting force which is in phase with velocity; but it is unlike viscous damping in that the magnitude of the force is not proportional to the velocity but to the displacement. When a system has n degrees of freedom, hysteretic damping implies that damping forces exist which are proportional to relative displacement but which are in phase with relative velocity.

From a physical standpoint, hysteretic damping may give a better representation of the facts when the damping arises from the internal friction of solid materials. On the side of theory, it raises considerations which it is the purpose of these three papers to elucidate. It may be said, at the outset, that the notion of hysteretic damping raises no great mathematical difficulty; on the contrary, a main reason for presenting the theory is that it appears to allow of a much simpler discussion (than does viscous damping) of the nature of steady damped oscillation of systems having n degrees of freedom.

In the first paper, the purpose is discussed of mathematical theories of damping in vibration theory. It is concluded that the theory of “ hysteretic damping ” is a useful one since it provides an alternative to the fiction of “ viscous ” damping while retaining the mathematical linearity of equations of motion. The word “ hysteretic” is proposed for use in this sense instead of the previously used adjective, namely “ structural.” “ Complex damping ” is related to hysteretic damping in a way which is explained.

The theory is given for forced oscillations of a system with one degree of freedom. It is shown that free vibration cannot be treated satisfactorily unless the definition of hysteretic damping is widened in some way to cover non-harmonic motion.