Mathematical Models in Electrical and Mechanical Systems

Study Notes

Mathematical Models

Electrical and mechanical systems share a common mathematical representation and can be modeled and analyzed similarly using equations of motion, Newton's laws, KCL, KVL, and Ohm's Law.
Force always acts opposite to displacement in mechanical systems.

Analogies

The force-current analogy relates through and across variables, helping in creating a digital twin to predict system behavior.
Analogies can provide additional insight into systems.

Linear Systems

Linear systems satisfy the principles of superposition and homogeneity.
Most linear systems are approximately linear in a small range around an operating point.

Modeling Systems

Mechanical systems can be modeled by analyzing forces acting on masses using free-body diagrams (FBD).
Electrical systems can be modeled by examining current through or voltage across components.
Differential equations are a common approach to modeling system dynamics.

Laplace Transforms

Laplace transforms are used to convert complex operations into simpler ones.
They are used to convert differential equations into algebraic ones, particularly in engineering for control systems.
Laplace transforms involve changing from the time domain to the Laplace domain (s).

Laplace Transform Properties

The Laplace transform pair consists of the Laplace transform and inverse Laplace transform.
The frequency shift theorem involves a translation in the Laplace domain.
The time shift theorem involves multiplication in the Laplace domain.
The derivative theorem involves the difference between s times the Laplace transform and the initial value.
The integration theorem involves the Laplace transform divided by s.
The final value theorem gives the value of a function as time approaches infinity.
The initial value theorem gives the value of a function at time 0.

Laplace Transforms of Functions

The Laplace transform of an impulse function is 1.
The Laplace transform of a step function is 1/s.
The Laplace transform of a ramp function is 1/s^2.

Transfer Functions

A transfer function is defined as the Laplace transform of the output divided by the Laplace transform of the input.
Transfer functions can only be used for linear systems.
They provide an input-output description of system behavior.
One way to find the transfer function of a system is to use an impulse and record the response.
The transfer function can also be found by finding the step response and dividing by s.

Finding Transfer Functions

For a mass-spring-damper system, the transfer function can be found by:
- Creating a free-body diagram (FBD)
- Summing forces on each mass
- Deriving motion equations
- Taking the Laplace transform of the motion equations
- Rearranging and substituting (if more than one mass) to find the input-output relationship

Description

This quiz covers the mathematical representation of electrical and mechanical systems, including equations of motion, Newton's laws, KCL, KVL, and Ohm's Law. It also explores the force-current analogy and its application in building digital twins.