Differential Equations and Linear Algebra, 4th Edition

5.6 Forced Oscillations and Resonance

In Section 5.4 we derived the differential equation

$$mx″+cx\text{′}+kx=F(t)$$ (1)

that governs the one-dimensional motion of a mass m that is attached to a spring (with constant k) and a dashpot (with constant c) and is also acted on by an external force F(t). Machines with rotating components commonly involve mass-spring systems (or their equivalents) in which the external force is simple harmonic:

$$F(t)=F_0\cos \omega t\text{or}F(t)=F_0\sin \omega t,$$ (2)

where the constant $F_0$ is the amplitude of the periodic force and $\omega$ is its circular frequency.

For an example of how a rotating machine component can provide a simple harmonic force, consider ...