A transfer function designation for a mechanical system. The examined mechanical system is composed of a spring and a viscous damper which are connected in parallel. The system's input signal is the force which causes system to change its position; the input signal is marked as u(t). The output signal is shift y(t) i.e. it is the displacement of the system position as a result of the input signal u(t) (the force F). The shift y(t) is a difference between initial and actual positions of the examined system. The resistance force of the spring is given by formula Fs = k * y(t), where k is the spring's constant. The resistance force of the viscous damper is Fv = B1 * dy(t)/dt, where B1 is the viscous damper constant.
The motion equation of the examined mechanical system is given by a formula.
The motion equation, which have been written above, needs to transformed from the time domain to the domain of variable s i.e. the Laplace transformation needs to be performed. The Laplace form of the motion equation of considered mechanical system is following.
It should be noted that the input and the output signal in Laplace form are explicitly visible in the equation.
U(s) - represents the Laplace form of the input signal
Y(s) - represents the Laplace form of the output signal
The transfer function of the examined mechanical system will be marked as G(s) and
it equals to a division result of the Laplace transformation of output signal Y(s) and the Laplace transformation of the input signal U(s).
G(s) = Y(s)/U(s)
After rewriting the equation for the transfer function G(s) to a form easier to digest; it is given by following formula:
From the transfer function's equation above the time constant of the system can easily distinguished and the final transfer function G(s) may be rewritten into even more convenient form.
