Mechanical and Electromagnetic Vibrations and Waves

Chapter 3

Forced Oscillations

Oscillations are said to be forced if the oscillator is permanently subject to external forces as well as internal restoring forces and friction forces. In this chapter we assume that the oscillator is linear; this is the case for mechanical systems subject to ordinary forces and electric circuits formed by ideal resistors, inductors and capacitors and connect.ed to generators. Then, the superposition principle is valid: if several forces act simultaneously on a linear system, the resulting motion is the superposition of the motions produced by each force acting separately.

In this chapter we study the forced oscillations of a system with one or several degrees of freedom especially in the case of a simple harmonic excitation force. The response of the system is particularly important if the excitation frequency is close to the natural frequency or one of the normal frequencies of the system; this phenomenon is called resonance. This study is important because several systems may be modeled as harmonic oscillators, and because the Fourier theory allows us to consider any excitation force as a superposition of harmonic forces.

3.1. Transient regime and steady regime

The equation of an oscillator with one degree of freedom, which is subject to an external force F(t), is μ ü = −Ku − b + F(t) where μ is its inertial parameter, −Ku is the restoring force ...