# forced oscillation

## Forced oscillation

A response of a mechanical or electrical system in reaction to an external signal.

A simple electrical *RLC* circuit (illus. *a*) consists of a resistor with resistance *R* (measured in ohms), an inductor with inductance *L* (measured in henrys), and a capacitor with capacitance *C* (measured in farads). The dynamics relating the input voltage, *u*(*t*), to the current, *y*(*t*), passing through the resistor are described by Eq. (1). Equation (1) (1) states that the input voltage is equal to the sum of the voltage across the inductor, the voltage across the resistor, and the voltage across the capacitor, where the voltage across the inductor is the product of its inductance (*L*) and the rate of change of the current through the inductor; the voltage across the resistor is the product of its resistance (*R*) and the current passing through it; and the voltage across the capacitor is the integral over time of the current through the capacitor (that is, the charge on the capacitor plates) divided by the capacitance (*C*).

A fundamental property of differential equations states that the response of a differential equation to a periodic input can be decomposed as a sum of two responses. The first one, called the zero-input response or free oscillation, is due to initial energy stored in the circuit and decays eventually to zero. The second one, due to the voltage input *u*(*t*), converges to a periodic signal with the same frequency as *u*(*t*). The latter is referred to as the forced oscillation or the steady-state response. The decaying rate of the free oscillation depends on the time constant of the circuit which is determined by the values of *R*, *L*, and *C* and the structure of the circuit. *See* Time constant

Similarly, an analogous mechanical system, a simple spring-mass-damper system (illus. *b*), consists of a body with mass *M*, which is attached to a wall by a spring with spring constant *k*, and rests on a horizontal surface over which it moves with friction coefficient *r*. The dynamic equation that relates the force applied to the body, *f*(*t*), to the body's displacement, *y*(*t*), is given by Eq. (2).

Analogous to the *RLC* circuit case, application of a sinusoidal force *f*(*t*) results eventually in a forced oscillation of the displacement *y*(*t*) that is also a sinusoidal function. The magnitude and the phase of the displacement *y*(*t*) depends on the complex mechanical impedance that is a function of the mass (*M*), the spring constant (*k*), and the friction coefficient (*r*). The exact evaluation is similar to the *RLC* circuit case. *See* Mechanical impedance, Oscillation, Vibration