By F. S. Beckman
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We observe at this point, without going into the mathematical details, that the case of the forced pendulum also merits consideration: the motion of a pendulum subjected not only to the force of gravity and the force of friction, but also to another force which is a function of time. There are a great number of examples of oscillating systems that are similar to that of a forced pendulum: all of those in which a system is subjected to the action of a restoring force that makes it make periodic oscillations, but also undergoes the action of another external variable force.
In Chapter 4, however, before entering into the merits of the cited models, we will continue to examine what is meant by the linearity of a model, with the assistance of some technical details. 4 Linearity in models A vast quantity of different system dynamics can be found in both natural and social sciences. However, they all have several common characteristics which enable them to be described in fairly simple and, above all, very general mathematical terms. We are referring to the dynamic processes typical of systems close to a conﬁguration of stable equilibrium.
The linear hypothesis is in the assumption that the restoring force and the size of the angle are directly proportional at any given moment during oscillation. This means, for example, that if we hit the pendulum with double the force, then the maximum amplitude of the oscillation, as long as it does not become ‘too’ large, doubles; in this way, we can assume that the oscillation consists of the sum of two oscillations caused by two equal initial forces, applied separately, that are summed together.