source.nb
source.cdf
source code in HTML
source code in PDF

## Vibration analysis of free response of second order system

July 24 2013 compiled on — Wednesday July 06, 2016 at 08:38 AM
This demonstration allows you to view different aspects of the response of a single degree of freedom system due to either harmonic force or an impulse. The differential equation of the system is where f(t) can be either or and an impluse

is the mass of the system, is the damping coefficient, is the stiffness, is a constant that represents the magnitude of the force and is the radial frequency of the applied force. The response is plotted as a function of time for under-damped, critical damping and over-damped cases.

You can select to view the transient response, the steady state response or the total response. The dynamic magnification factor and phase of the response relative to the force is displayed. Different test cases are avaliable to view many different loading conditions.

You can vary the system parameters (mass, damping, and stiffness) and simulate the response. The analytical solution is displayed at the top of the plot for the cases of underdamping, critically damping, and overdamp. In addition, a standard physical model of mass-spring-damper is run at the same time as the response plot is updated.

You can set the initial conditions for initial position and speed using the sliders. To observe the impulse response, setting the checkbox causes the initial conditions to become and , which by definition will make the response the same as the impulse response.

The underdamped response of a second-order system is given by

The critically damped system has the response

And the overdamped system has the response

In the above, and are found from the initial conditions, is the natural frequency in rad/sec, is the damped natural frequency in rad/sec, and is the damping coefficient.

For the underdamped case, the damped period of oscillation is given by and the time constant is given by . Both are in seconds.