Mathematical Models in Electrical and Mechanical Systems

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