The equation of motion of a second-order linear system of mass
with harmonic applied loading is given by the differential equation
. There are 12 different analytical solutions depending on whether damping or loading is present and, if so, whether the system is underdamped, critically damped or overdamped.
The solution for each of the 12 cases was derived analytically and shown in the Demonstration, subject to the user's choice. Following are definitions of the relevant parameters. All units are in SI.
The damping ratio is
is the damping coefficient, such that
represents critical damping. The natural underdamped frequency is given by
is the stiffness and
is the mass. The damped frequency of the system, defined for
. The frequency ratio is
is the forcing frequency. The dynamic magnification factor
is the ratio of the steady state response to the static response. The static response is given by
is the force magnitude. The time constant is
and the damped period of oscillation is
When the system is undamped and the load is harmonic, resonance occurs when
. When the system is underdamped and the load is again harmonic, practical resonance occurs when
and the corresponding maximum magnification factor is
. You can force the loading frequency to be equal to the natural frequency by clicking the button located to the right of the slider used to input the loading frequency. The forcing frequency
is expressed in Hz, but converted to radians per second internally.
This Demonstration also shows plots of the phase of the response
relative to loading
. The phase of the response lags behind loading by an angle
, which is plotted in the complex plane on an Argand diagram. The phase angle ranges from
When the loading frequency
is set to zero, only
is used as the force
. This allows a constant force loading,
. For example, by setting
, a step response is obtained. To make the loading zero, the slider
is set equal to zero.
The Demonstration contains a number of pre-configured test cases to illustrate different loading conditions, such as beating phenomenon, resonance, practical response, impulse response, and step responses for different damping values.
 M. Paz and W. Leigh, Structural Dynamics: Theory and Computation
, 5th ed., Boston: Kluwer Academic Publishers, 2004.
 W. T. Thomson, Theory of Vibration with Applications
, Englewood Cliffs, NJ: Prentice-Hall, 1972.
 R. W. Clough and J. Penzien, Dynamics of Structures
, New York: McGraw-Hill, 1975.
 R. K. Vierck, Vibration Analysis
, Scranton, PA: International Textbook Company, 1967.
 A. A. Shabana, Theory of Vibration
, Vol. 1, New York: Springer-Verlag, 1991.
 B. Morrill, Mechanical Vibrations
, New York: Ronald Press, 1957.