EJSS SHM model with resonance showing Amplitude vs frequency graphs

δx[i]δt=vx[i]

δvx[i]δt=−km(x[i]−l)−bvx[i]m+Asin(2πf[i]t)m

where the terms

−km(x[i]−l) represents the restoring force component as a result of the spring extending and compressing.

−bvx[i]m represents the damping force component as a result of drag retarding the mass's motion.

+Asin(2πf[i]t)m represents the driving force component as a result of a external periodic force acting the mass m.

Forced oscillations

Forced oscillations are oscillations that are subjected to a periodic driving force provided by an external agent such as motor or a push by a person etc.

Resonance is an interesting phenomenon that occurs when driving force frequency matches that of the system's natural oscillating frequency resulting in a motion that reaches some maximum amplitude.

Resonance

In physics, resonance is the tendency of a system to oscillate with greater amplitude at some frequencies than at others. Frequencies at which the response amplitude is a relative maximum are known as the system's resonant frequencies, or resonance frequencies. At these frequencies, even small periodic driving forces +Asin(2πf[i]t)m can produce large amplitude oscillations, because the system stores vibrational energy.

Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a spring mass system). However, there are some losses from cycle to cycle, called damping. When damping is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple, distinct, resonant frequencies.