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Free vibration (no external force) of a single degree-of-freedom system with viscous damping can be illustrated as,

Damping that produces a damping force proportional to the mass's velocity is commonly referred to as "viscous damping", and is denoted graphically by a dashpot.

For an unforced damped SDOF system, the general equation of motion becomes,

with the initial conditions,

This equation of motion is a second order, homogeneous, ordinary differential equation (ODE). If all parameters (mass, spring stiffness, and viscous damping) are constants, the ODE becomes a linear ODE with constant coefficients and can be solved by the Characteristic Equation method. The characteristic equation for this problem is,

which determines the 2 independent roots for the damped vibration problem. The roots to the characteristic equation fall into one of the following 3 cases:

To simplify the solutions coming up, we define the critical damping c_c, the damping ratio z, and the damped vibration frequency w_d as,

where the natural frequency of the system w_n is given by,

Note that w_d will equal w_n when the damping of the system is zero (i.e. undamped). The time solutions for the free SDOF system is presented below for each of the three case scenarios.

To obtain the time solution of any free SDOF system (damped or not), use the SDOF Calculator.

When < 0 (equivalent to < 1 or < ), the characteristic equation has a pair of complex conjugate roots. The displacement solution for this kind of system is,

An alternate but equivalent solution is given by,

The displacement plot of an underdamped system would appear as,

Note that the displacement amplitude decays exponentially (i.e. the natural logarithm of the amplitude ratio for any two displacements separated in time by a constant ratio is a constant; long-winded!),

where is the period of the damped vibration.

When = 0 (equivalent to = 1 or = ), the characteristic equation has repeated real roots. The displacement solution for this kind of system is,

The critical damping factor c_c can be interpreted as the minimum damping that results in non-periodic motion (i.e. simple decay).

The displacement plot of a critically-damped system with positive initial displacement and velocity would appear as,

The displacement decays to a negligible level after one natural period, T_n. Note that if the initial velocity v₀ is negative while the initial displacement x₀ is positive, there will exist one overshoot of the resting position in the displacement plot.

When > 0 (equivalent to > 1 or > ), the characteristic equation has two distinct real roots. The displacement solution for this kind of system is,

The displacement plot of an overdamped system would appear as,

The motion of an overdamped system is non-periodic, regardless of the initial conditions. The larger the damping, the longer the time to decay from an initial disturbance.

If the system is heavily damped, , the displacement solution takes the approximate form,