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In previous article we discussed about the Free vibrations with the viscous damping, in which we discussed three different cases when the system is overdamped, critically damped and underdamped.
In today's article we will discuss about the forced vibrations of single degree freedom system. There are cases when vibrations caused by the rotating parts of machines cause steady state periodic exciting forces in the system. This exciting force can be written as follows:
F(t) = F*.sin(w.t)
The equation of motion for a SDOF having a damping of 'C' subjected to such an exciting force can be written as follows:
m.z'' + C.z' + K.z = F(t)
The equation is a linear, 2nd order differential, nonhomogeneous equation. The solution of this equation consists of two parts, namely: (i) complementary function, and (ii) particular integral. Complementary function is obtained by considering no forcing function.
Therefore its equation can be written as follows:
m.z'' + C.z' + K.z = 0
the solution for the above equation has already been discussed in the previous article. Now the particular integral can be found by assuming an exciting force F(t), under action here.
m.z'' + C.z' + K.z = F(t)
This can be shown for the particular integral, that the system will vibrate harmonically with the same frequency as that of the exciting force.
The complete solution is obtained by adding the complementary function and the particular integral. The complementary function is an exponentially decaying function and will die out soon, and the motion will be described by only the particular integral. There the system as a whole will vibrate harmonically with the same frequency as the forcing.
Let, F` be the amplitude of the exciting force;
K is the spring constant, and
Az is the amplitude of the displacement, then it can be shown that
Az = Magnification factor * (F`/K)
F`/K describes the static state amplitude, while magnification factor will depend upon the frequency ratio and the damping ratio of the system.
In previous article we discussed about the Free vibrations with the viscous damping, in which we discussed three different cases when the system is overdamped, critically damped and underdamped.
In today's article we will discuss about the forced vibrations of single degree freedom system. There are cases when vibrations caused by the rotating parts of machines cause steady state periodic exciting forces in the system. This exciting force can be written as follows:
F(t) = F*.sin(w.t)
The equation of motion for a SDOF having a damping of 'C' subjected to such an exciting force can be written as follows:
m.z'' + C.z' + K.z = F(t)
The equation is a linear, 2nd order differential, nonhomogeneous equation. The solution of this equation consists of two parts, namely: (i) complementary function, and (ii) particular integral. Complementary function is obtained by considering no forcing function.
Therefore its equation can be written as follows:
m.z'' + C.z' + K.z = 0
the solution for the above equation has already been discussed in the previous article. Now the particular integral can be found by assuming an exciting force F(t), under action here.
m.z'' + C.z' + K.z = F(t)
This can be shown for the particular integral, that the system will vibrate harmonically with the same frequency as that of the exciting force.
The complete solution is obtained by adding the complementary function and the particular integral. The complementary function is an exponentially decaying function and will die out soon, and the motion will be described by only the particular integral. There the system as a whole will vibrate harmonically with the same frequency as the forcing.
Let, F` be the amplitude of the exciting force;
K is the spring constant, and
Az is the amplitude of the displacement, then it can be shown that
Az = Magnification factor * (F`/K)
F`/K describes the static state amplitude, while magnification factor will depend upon the frequency ratio and the damping ratio of the system.
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