Single Degree Forced Vibrations
Steady State Response due to Harmonic Oscillation :
Consider a spring-mass-damper system as shown in figure. The equation of motion of this system subjected to a harmonic force F sin ωt can be given by
where, m , k and c are the mass, spring stiffness and damping coefficient of the system, F is the amplitude of the force, w is the excitation frequency or driving frequency.
Force polygon
The steady state response of the system can be determined by solving equation in many different ways. Here a simpler graphical method is used which will give physical understanding to this dynamic problem. From solution of differential equations it is known that the steady state solution (particular integral) will be of the form
As each term of equation represents a forcing term viz., first, second and third terms, represent the inertia force, spring force, and the damping forces. The term in the right hand side of equation is the applied force. One may draw a close polygon as shown in figure considering the equilibrium of the system under the action of these forces. Considering a reference line these forces can be presented as follows.
(i) Spring force = kx = kX sin (ωt – Ø). (This force will make an angle ωt – Ø with the reference line, represented by line OA).
(ii) Damping force = 
(This force will be perpendicular to the spring force, represented by line AB).
(iii) Inertia force = 
(this force is perpendicular to the damping force and is in opposite direction with the spring force and is represented by line BC).
(iv) Applied force = F sin ωt which can be drawn at an angle ωt with respect to the reference line
and is represented by line OC.
From equation (1), the resultant of the spring force, damping force and the inertia force will be the applied force, which is clearly shown in figure.
It may be noted that till now, we don't know about the magnitude of X and Ø which can be easily computed from Figure. Drawing a line CD parallel to AB, from the triangle OCD of Figure,
As the ratio F/k is the static deflection (X0)of the spring, Xk/F = X/X0 is known as the magnification factor or amplitude ratio of the system
Magnification factor ~ frequency ratio for different values of damping ratio.
Phase angle ~ frequency ratio for different values of damping ratio.
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