# Does a Mechanical equivalent of an Integrator exist

In electrical domain, we have resistors, capacitors, and inductors as fundamental constitutive elements. Similarly, in mechanical domain, we have masses, springs and dampers.

In most of the literature, we can immediately see that they have the one-to-one relation, perse i.e., we can go from one form of representation to other, or even mix them up (mechatronics). However, the problem I see is, in the electrical domain, we have the inductor which is an energy storage element, representing an integrator, collectively speaking. In contrast, when we look it up in the mechanical domain, all we have are mass-spring-damper equivalent, and there is no explicit representation for an inductor.

So, my question is this, does there exists a fundamental equivalent for an inductor in mechanical domain i.e., a mechanical equivalent of an integrator with a physical meaning in itself. If so, how it is called?

I have tried a lot of literature to find one but failed in the quest. So, all your help and suggestions are welcome.

• Well, an inductor in series or a capacitor in parallel is pretty much an integrator. – Carl Witthoft Mar 31 '17 at 14:48
• Any vehicle is an integrator of velocity wrt time. – user_1818839 Apr 1 '17 at 9:01
• @CarlWitthoft I think you mistook the question :) – Raaja_is_at_topanswers.xyz Apr 1 '17 at 12:36
• @BrianDrummond How do you say that ''any vehicle is a velocity of time''? – Raaja_is_at_topanswers.xyz Apr 1 '17 at 12:39
• That's not what I said. – user_1818839 Apr 1 '17 at 14:15

Consider an analogy where voltage and force are "efforts" and current and velocity are "flows". We can choose either the effort or flow as the input, and the other as the output.

A spring is a mechanical integrator in the sense that it integrates its velocity to generate a proportional force. (It integrates a flow input to generate an effort output). Alternatively, a mass can be considered a mechanical integrator in the sense that it integrates the applied force to generate a change in velocity. (It integrates an effort input to generate a flow output).

The analogy to the electrical domain should be clear.

If we consider "flow" as the input and "effort" as the output, then a capacitor integrates electrical flow (current) to generate an electrical effort (voltage), i.e., $e = \frac{1}{C}\int i\,dt$. Similarly, a spring integrates mechanical flow (velocity) to generate mechanical effort (force), i.e., $F = k\int v\,dt$.

Alternatively, if we consider the "effort" as the input and the "flow" as the output, then an inductor integrates electrical effort (voltage) to generate a change in electrical flow (current), i.e., $i = \frac{1}{L} \int e\,dt$. Similarly, a mass integrates mechanical effort (force) to generate a change in mechanical flow (velocity), i.e., $v = \frac{1}{m} \int F\,dt$.

• I think this answer does a better job answering the question from a systems and controls perspective (which seems to be the original intent of the question) than the accepted answer (which is about mechanical computing). Springs and masses in mechanical systems are analogous to capacitors and inductors in electrical systems. For what it's worth, +1. – ConjuringFrictionForces Apr 21 '17 at 12:42
• @ConjuringFrictionForces As you mentioned the original intent of this question lies in systems and control perspective!! +1 for the intuitive analogies. – Raaja_is_at_topanswers.xyz Apr 21 '17 at 12:48
• If you are interested in this perspective, I would recommend the text "Engineering System Dynamics" by Forbes Brown. There's an approach to system dynamics that lets you model nearly everything using extensions of these effort-flow analogies and thinking about power transmission through systems. For further reading you could also consider the lecture notes for this course. – Max Apr 21 '17 at 12:51
• @Max Good text! A similar text would be "System Dynamics: An Introduction" by Rowell and Wormley. That's what I used for my systems courses in my undergrad. – ConjuringFrictionForces Apr 21 '17 at 16:04
• @ConjuringFrictionForces and Max Fantastic literature!! Thanks. – Raaja_is_at_topanswers.xyz Apr 21 '17 at 18:04

If position is what is being manipulated then it's pretty simple, convert the input position to speed using a friction wheel on a spinning disk where the input position decides where the wheel is in relation to the axis of the spinning disk.

This video with a description a mechanical computer's components includes a description of an integrator at 30:52

• That's a nice video (+1). Having said this in the above-mentioned video, how do we call such an integrator, i.e., a generalised name with a mechanical analogue. – Raaja_is_at_topanswers.xyz Mar 31 '17 at 8:33
• @RaajaG there are many such elements we call them based on what they do. There are far more variance in the mechanical realm than in electrical realms. We have more than one way to integrate. Anyway surely you have read of simulation analogs – joojaa Mar 31 '17 at 14:52