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

, where

is the damping coefficient, such that

represents critical damping. The natural underdamped frequency is given by

, where

is the stiffness and

is the mass. The damped frequency of the system, defined for

, is

. The frequency ratio is

, where

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

, where

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

or

. 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

to

.

When the loading frequency

is set to zero, only

is used as the force

. This allows a constant force loading,

. For example, by setting

and

, 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.

[1] M. Paz and W. Leigh,

*Structural Dynamics: Theory and Computation*, 5th ed., Boston: Kluwer Academic Publishers, 2004.

[2] W. T. Thomson,

*Theory of Vibration with Applications*, Englewood Cliffs, NJ: Prentice-Hall, 1972.

[3] R. W. Clough and J. Penzien,

*Dynamics of Structures*, New York: McGraw-Hill, 1975.

[4] R. K. Vierck,

*Vibration Analysis*, Scranton, PA: International Textbook Company, 1967.

[5] A. A. Shabana,

*Theory of Vibration*, Vol. 1, New York: Springer-Verlag, 1991.

[6] B. Morrill,

*Mechanical Vibrations*, New York: Ronald Press, 1957.