Clarification required for through and across variable. What is basis for declaring force as through variable and velocity as across variable?

Suppose there is elastic rubber band (showing ideal spring elastic behavior) fixed at one end and extended from the other end, from basics we know that other end will move with velocity v(t) and force(tension) inside rubber is f(t or x) at any point of time, while modeling using Simscape, using mechanical translational library blocks, spring component have f as through variable and v as across variable. From Simscape documentation Basic Principles of Modeling Physical Networks the explanation given about Variable Types is not clear. It goes like this

Physical Network approach supports two types of variables:

Through — Variables that are measured with a gauge connected in series to an element.

Across — Variables that are measured with a gauge connected in parallel to an element.

How we measure velocity and force using series and parallel gauge? may be this type of explanation is not valid for my problem, however exact definition of through variable also looks abstract sometime, Now take this: By definition Through variable is " their values have balance at a node, sum of all its values flowing into a branch point equals sum of all its values flowing out." As I am engineering grad, till now we have only come across mass conservation or energy conservation, where something really goes in and comes out, but here force is vector, what do you mean by force going in and coming out? Till now I have only learned that force is balanced on desired system taking into account final acceleration of system, Can you please elaborate this Simscape terminology. Please give me some insights. Thank you in advance.

Answer 1

[Sebastian Castro]

I like to think about it this way. Assuming no friction or other losses:

• If you push a block with force from your arm, all of the force will go towards accelerating that block. In a sense, force "flows" from your arm to the block and that is the only path it can take.
• Now, suppose you're using your arm to push two blocks. Some of that force will push block A and the rest will push block B. So again, force is analogous to current in that it flows and is divided among all the paths it could take. How that force is divided depends on all the component equations.

Then, velocity is the potential difference which you can see from the equations of a damper. The higher the speed across the damper, the larger the force. This is exactly the same as saying "the higher the voltage across a resistor, the larger the current".

- Sebastian

Answer 2

[David Blackwell]

I believe it is helpful to review the attached lecture notes from the MIT class on Dynamics and Control. The notes are part of the MIT OpenCourseWare project. The concept of through and across variables comes from a generalization of how to model physical systems. The MIT lecture notes provide the typical comparison of electrical and mechanical elements for power. Although the notes are limited in categorizing the many through & across variables (not mentioning those in thermal, fluid, & other systems), the notes are clear and direct. You should be able to extend these concepts to other variables you use.