Engineering Formular Series - Free Vibration With A Viscous Damper.

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.

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:

1. If < 0, the system is termed underdamped. The roots of the characteristic equation are complex conjugates, corresponding to oscillatory motion with an exponential decay in amplitude. 2. If = 0, the system is termed critically-damped. The roots of the characteristic equation are repeated, corresponding to simple decaying motion with at most one overshoot of the system's resting position. 3. If > 0, the system is termed overdamped. The roots of the characteristic equation are purely real and distinct, corresponding to simple exponentially decaying motion.

To simplify the solutions coming up, we define the critical damping cc, the damping ratio z, and the damped vibration frequency wd as, where the natural frequency of the system wn is given by, Note that wd will equal wn 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; long-winded!), where is the period of the damped vibration.

Critically-Damped 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 cc 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, Tn. 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.

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 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,

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