In most mechanical systems, there is some type of damping effect when vibrations occur. It could be air or fluid resistance, shock absorber, or even molecule interactions. This damping effect can be effectively modeled as a force that is proportional to the objects velocity. This type of device is usually called a damper.

Spring-Mass-Damper System

Development of the Equation of Motion

Similar to the simple mass-spring system (undamped) presented in the previous section, the equations of motion of an mass-spring-damper system (damped) can be developed by summing forces. Gravity effects will cancel out similar to the damped system, and are not included in this derivation.

    F_S + F_d = -ma_y

substituting F_S = ky and F_d = cv = c(dy/dt) gives,

Recall, solving any differential equation requires that a general solution is first assumed and then initial conditions are used to find the constants.

For an undamped system, both sin and cos functions were used in the solution. For the damped system, it is more convenient to use an exponential form as,

     y(t) = De^st

where A is an arbitrary constant, and s is a characteristic parameter. Substituting this into the equation of motion will give

     (ms² + cs + k ) De^st = 0

For nontrivial solutions (D ≠ 0), the two possible roots, s₁ and s₂, are

and the general solution is

where A and B are constants that are determined from initial or boundary conditions. When the square root term is equal to zero, the two roots are equal which gives critical damping, c_c, as

To help simplify the equations another constant is used, the damping ratio or damping factor, ζ. This term denotes the severity of the damping.

Now, the roots can be simplified to

Depending on the term within the radical, the system is said to be overdamped, underdamped or critically damped.

     ζ = 1       Critically Damped
     ζ > 1       Overdamped
     ζ < 1       Underdamped

Critically Damped

For ζ = 1, the roots s₁ and s₂ are equal,

     s₁ = s₂ = -ω_n

The general solution to the differential equation is

The coefficients A and B are

For an overdamped system, general solution is

The constants A and B are determined from the initial conditions,

Underdamped

When ζ < 1, underdamped system, the roots are

Here, ω_d is referred to as the damped frequency,

Combining with the general solution gives

This can be simplified to give

where the constants C and D are