An object will not vibrate if there is no restoring force causing it to want to return to its equilibrium position. If this force is proportional to the distance from equilibrium it is a linear restoring force and obeys Hooke’s law. Hooke’s law says that if we double the displacement from equilibrium, the force acting to return the object to the equilibrium position also doubles. If the displacement is one third as big the force is one third as big and so on. Most springs obey Hooke’s law; the more you stretch the spring, the larger the force.
What if the force is not proportional to the displacement? Such a force is called a non-linear force which does not obey Hooke’s law. An example is the modern compound bow used in archery. A system of pulleys causes the force to be the smallest when the displacement is greatest. This makes it easier for the archer to hold the bow at maximum displacement while he or she aims at the target. Non-linear forces can be quite complicated but fortunately most forces involved in sound and musical instruments are close enough to linear that we can ignore non-linear effects. The few times a non-linear force acts will be explicitly mentioned; in all other cases you can assume the forces are linear.
The simplest of all vibrations occurs when there is a Hooke’s law force and no friction acts. This type of motion is called simple harmonic motion and will be the model we will use for vibrations in musical instruments. A free hanging mass on a spring and a pendulum swinging with low amplitude approximately obey simple harmonic motion.
If friction acts the motion will gradually stop. This is called damped harmonic motion. To maintain a constant vibration when there is friction, a period force must be applied. Harmonic motion that has damping and an applied period force is called damped, driven harmonic motion and will be discussed further in the next chapter.