Definition 
Free vibration (no external force) of a single degreeoffreedom 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. 
Time Solution for Damped SDOF Systems  
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.  
Underdamped Systems 
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; longwinded!),
where is the period of the damped vibration. 
CriticallyDamped Systems 
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 nonperiodic motion (i.e. simple decay). The displacement plot of a criticallydamped 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 v_{0} is negative while the initial displacement x_{0} is positive, there will exist one overshoot of the resting position in the displacement plot. 
Overdamped Systems 
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 nonperiodic, 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,
