My professor told us that it is possible to see the friction in a mass-friction-spring as the contribution of a closed loop control system. He wrote the following formulas:

Transfer function: $$G(S)=\frac s {s^2+\frac k M}$$ gain: $$\rho=h/M$$ where k=spring constant, h=friction constant, M=mass

Can you recognize these formulas and give me some indication about this method? I think it would be useful to understand what is the meaning of the transfer function is this case?


2 Answers 2


Based on the information you've given, I believe your professor is suggesting that a friction term can be represented as shown in the following block diagram.

A block diagram showing G(s) as the plant and rho in the feedback path.

The transfer function $G(s)$ relates force ($F(s)$, the input in the diagram) to velocity ($sX(s)$, the output in the diagram) for a mass-spring system. The damping ($\rho$) is represented in the feedback loop as a proportional gain acting on the velocity. We can use the feedback rule to create a single, closed loop transfer function:

\begin{align} G_{cl}(s) &= \frac{G(s)}{1+\rho G(s)},\\ &= \frac{\frac{s}{s^2+k/M}}{1 + \left(\frac{h}{M} \right)\frac{s}{s^2+k/M}},\\ &= \frac{s}{s^2 + \frac{h}{M}s + \frac{k}{M}}. \end{align}

As you can see, the closed loop transfer function is the same as the general mass-spring-damper transfer function.


You can see how to obtain the transfer function of the mass spring system in many well documented links, e.g.:

Having said that I couldn't understand what you meant by "it is possible to see the the friction in a mass-friction-spring as the contribute of a closed loop control system".

One possible explanation, is that it is possible to adjust the damping properties in order to obtain a certain responce from the mechanical system. To my knowledge though, this is not very usual when talking about the mass-damper-spring system.

Regarding what is the transfer function: is a way to represent a linear, time-invariant system in a more algebraic way (instead of a set of differential equation). What it connects is the relationship of input and outpus. In your particular transfer function, the output is the position of the mass spring system, while the input would (normally) be an excitation.


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