Recall, for a spring-mass-damper system (damped) under free vibration, the general equation of motion is

     m d2y/dt2 + c dy/dt + ky = 0

Similarly, for horizontal motion,



Development of Motion Equation

However, it is common for a damped system to have an external harmonic force acting on the system. For example, unbalanced rotating motors will cause sever forces on the mounting brackets. This system is called forced vibrations.

A forced vibration is one in which the system is excited by an external, time-varying force P, called a forcing function. In this case


In general, forcing functions are periodic, and since any periodic function can be expressed as a Fourier series, it is convenient to look at a forcing function of the type

     P(t) = Posinωt     or      P(t) = Pocosωt

Considering the sin term, the differential equation is


The solution of this differential equations consists of two parts, the complementary or homogenous solution,

     x(t) = xc(t) + xP(t)


The complementary solution is the general solution of the homogeneous equation as presented in the damped free vibrations section. Recall, for an underdamped system, the solution is


Since the forcing function is sin, the particular solution will be of the form

     xP(t) = Dsin(ωt - φ)

Substituting the particular solution into the differential equation and take the appropriate derivatives gives

     D[(k-mω2)cosφ + cω sinφ] sinωt
          + D[cω cosφ - (k-mω2)sinφ] cosωt = Posinωt

The sin and cos terms can be separated to get the two equations

     D[(k-mω2)cosφ + cω sinφ] = Posinωt

     D[cω cosφ - (k-mω2)sinφ] = 0

Now the constants, D and φ, can be obtained as



The D and φ terms can also be written using ωn and ζ terms, giving


To better understand the solution, it is convenient to consider the ration of the static deflection δP,


to the dynamic deflection, D. This ratio can be calculated as


A plot for several damping ratios, ζ, is shown below.

Notice that when the external force frequency is close to the system natural frequency, systems with low damping have large displacement amplitudes. This is commonly known as resonance.

The total solution for the underdamped case is

where the constants A and B must be determined from the initial conditions.


Effect of Various Damping Factors
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