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In previous article we discussed about the undamped free vibrating system. In this article we are going to talk about the free vibrations with damping for a SDOF(Single Degree of Freedom) system.
For such a system, the excitation force F(t) is kept zero, while the viscous damping is available.
The differential equation of motion for such a system can be written as follows:
m.z'' + C.z' + K.z = 0 [ F(t) = 0]
Here, 'C' is the damping constant or force per unit velocity. The solution of the above equation gives us a value of 'z' with which we can interpret the following results:
In this case the motion of the system is not exponential but is an exponential subsidence. Because of the relatively large damping, so much energy is dissipated by the damping force that there is not sufficient kinetic energy left to carry the mass and pass the equilibrium position. Physically this means a relatively large damping and the system is said to be over damped.
In this case the system is called critically damped system. It is similar to the previous case, but the difference is that the amplitude may become negative as shown in the figure.
In previous article we discussed about the undamped free vibrating system. In this article we are going to talk about the free vibrations with damping for a SDOF(Single Degree of Freedom) system.
For such a system, the excitation force F(t) is kept zero, while the viscous damping is available.
The differential equation of motion for such a system can be written as follows:
m.z'' + C.z' + K.z = 0 [ F(t) = 0]
Here, 'C' is the damping constant or force per unit velocity. The solution of the above equation gives us a value of 'z' with which we can interpret the following results:
- Case 1: (C/2m) > K/m
- Case 2: (C/2m)^2 = K/m
In this case the system is called critically damped system. It is similar to the previous case, but the difference is that the amplitude may become negative as shown in the figure.
The damping constant obtained from this case is known as critical damping constant and the ratio of the actual damping to the critical damping constant is known as the damping ratio.
- Case 3: (C/2m)^2 < K/m
In this case the solution for displacement 'z' can be written in a sinusoidal form, and the frequency of therefore damped system is known as damped frequency.
From above discussions we can say the a freely vibrating SDOF with viscous damping is over damped if damping ratio is greater than one, critically damped if equal to one and underdamped when less than one.
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