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Objectives
Objectives
By the end of this section, you should be able to
Recall what the Laplace transform does and why it is useful for solving linear ordinary differential equations in time.
Use partial fraction expansion to rewrite rational functions so that they match common Laplace transform table entries.
Write and solve simultaneous linear equations in matrix form.
Recognize and interpret basic electronic components and their standard symbols in simple circuits.
These concepts typically come from earlier coursework in
Differential equations and Laplace transforms
Calculus and algebra techniques for partial fractions
Linear algebra and matrix methods
Introductory electrical circuits and physics
This page supports the early part of EGR 306 Measurement and Dynamic Response and also serves as a focused review for engineering students who want to refresh key tools for system dynamics, mechanical vibrations, and control systems.
The topics here are standard in many differential equations, linear algebra, and introductory circuits courses. The goal is to get you ready to use them confidently when modeling and analyzing dynamic systems.
Laplace Transforms
Laplace Transforms
Laplace transforms are a central tool in system dynamics, mechanical vibrations, signals, and control engineering. The transform turns differential equations in the time domain into algebraic equations in the Laplace domain, which are often easier to solve and interpret.
Partial Fraction Expansion
Partial Fraction Expansion
Partial fraction expansion lets you decompose a rational function into simpler terms that match entries in a Laplace transform table. This step is often what makes it possible to invert a Laplace transform and recover the time-domain solution for a dynamic system.
Matrix/Linear Algebra
Matrix/Linear Algebra
Matrix and linear algebra tools appear throughout system dynamics and control, especially in state-space modeling and multi-degree-of-freedom mechanical systems.
Electrical Components
Electrical Components
Many measurement, actuation, and control problems involve basic circuits. Understanding how standard components behave is essential for building realistic dynamic models.
Before you move on, you should comfortably be able to:
Explain, in a sentence or two, how the Laplace transform helps you solve linear ordinary differential equations.
Take a simple rational function and perform a partial fraction expansion.
Set up and solve a 2 x 2 or 3 x 3 linear system in matrix form.
Identify the main passive electrical components by symbol and describe how they relate voltage and current qualitatively.
If any of these feel uncertain, spend more time with the slides and your notes from your earlier math and circuits courses before starting the next section.
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