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Vibration analysis of free response of second order system

Nasser M. Abbasi

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 mu ′′ + cu ′ + ku = f(t)  where f(t) can be either F sin (ωt )  or and an impluse F δ(t)

m  is the mass of the system, c  is the damping coefficient, k  is the stiffness, F  is a constant that represents the magnitude of the force and ω  is the radial frequency of the applied force. The response u(t)  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 y(0) = 0  and  ′
y (0) = 1  , which by definition will make the response the same as the impulse response.

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

y(t) = exp(ζωnt)(A cos(ωdt)+  B sin(ωdt))

The critically damped system has the response

y(t) = (A + Bt) exp(− ζω t)

And the overdamped system has the response

                  ∘ ------                   ∘ ------
y(t) = A exp((− ζ +  ζ2 − 1)ωnt )+ B exp((− ζ +  ζ2 − 1)ωnt )

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

For the underdamped case, the damped period of oscillation is given by Td = 2ωπ
      d  and the time constant is given by τ = -1-
    ζωn  . Both are in seconds.