Damping

In physics, damping is an effect that reduces the amplitude of oscillations in an oscillatory system (except for mass-dominated systems where √2), particularly the harmonic oscillator. This effect is linearly related to the velocity of the oscillations. This restriction leads to a linear differential equation of motion, and a simple analytic solution.

In mechanics, damping may be realized using a dashpot. This device uses the viscous drag of a fluid, such as oil, to provide a resistance that is related linearly to velocity. The damping force F_c is expressed as follows:

where c is the viscous damping coefficient, given in units of newton seconds per meter (N s/m) or simply kilograms per second. In engineering applications it is often desirable to linearize non-linear drag forces. This may by finding an equivalent work coefficient in the case of harmonic forcing. In non-harmonic cases, restrictions on the speed may lead to accurate linearization.

Generally, damped harmonic oscillators satisfy the second-order differential equation:

where ω₀ is the undamped angular frequency of the oscillator and ζ is a constant called the damping ratio.

The value of the damping ratio ζ determines the behavior of the system. A damped harmonic oscillator can be:

• Overdamped (ζ > 1): The system returns (exponentially decays) to equilibrium without oscillating. Larger values of the damping ratio ζ return to equilibrium more slowly.
• Critically damped (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating. This is often desired for the damping of systems such as doors.
• Underdamped (0 < ζ < 1): The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.
• Undamped (ζ = 0): The system oscillates at its natural resonant frequency (ω_o).