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About the Tool

The tool VB-10 plots the time response of one-degree-of-freedom systems to four types of periodic forces. The tool also determines the natural frequency and damping status of the system.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.


Example: A one-degree-of-freedom system subject to periodic force

The input in the tool is

After clicking "Run", the results will be shown as follows in another page.

The natural frequencies and damping status are also shown as follows.

Circular natural frequency, wn = 2.23606798 (rad/s)

Natural frequency in Hertz, fn = 0.35588127 (Hz)

Natural Period, Tn = 1/fn = 2.80992589 (sec)

-0.33333333 + j (-2.21108319) ,-0.33333333 + j (2.21108319)

This is an underdamped System (0 < damping ratio < 1)

Damping ratio, C/(2*M*wn) = 0.1490712

Damped frequency, wd = 2.21108319 (rad/s)

Subject Review

A one-degree-of-freedom system

Fig. 1 A 1-DOF system

Response of One-Degree-of-Freedom Systems to Periodic Force (Fourier Series)

The equation of motion of a 1-DOF system, as shown in Fig. 1, is in the form of


T: Period

F(t) is a periodic force , which can be expanded in Fourier series as


The superposition, the sready-state response is given by

where xj and yj are the solutions of the following equations

If the steady-state response of the system exists (i.e. c≠0 or ωn≠0), the steady-state response can be approximated by a truncated series of 2n+1 terms


Solution of four types of periodic forces are summarized in following table.

Solution of four types of periodic forces


1. Den Hartog, J. P., 1985, Mechanical Vibrations, Dover Publications, New York.

2. Timoshenko, S., 1990, Vibration Problems in Engineering, John Wiley & Sons; 5th edition.

3. Yang, B., 2005, Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes, Elsevier Science, Boston.