What is the best way to introduce Laplace transforms for Engineers?

• matqkks
In summary, Laplace transforms can be introduced in order to motivate students to learn about the relationship between the time domain and the s-domain. They are used to solve linear differential equations, and have applications in engineering.
matqkks
Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function.

How can Laplace transforms be introduced so that students are motivated to learn? It needs to have an impact.

What are the applications of Laplace transforms?

The design of classical control laws is uses Laplace transforms.

S.G. Janssens
Analog EEs and MechEs that work with dynamic systems use Laplace Transforms all the time. We think in terms of Laplace Transforms and the s-domain. It's like asking if auto mechanics ever use wrenches. In the digital world of computer controls we work with it's discrete-time cousin the z transform.

vanhees71, S.G. Janssens, FactChecker and 2 others
matqkks said:
I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function.
In my experience, solving non-homogeneous DEs with forcing functions is much simpler when Laplace transforms are used.

vanhees71, S.G. Janssens and DaveE
matqkks said:
Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function.

How can Laplace transforms be introduced so that students are motivated to learn? It needs to have an impact.

What are the applications of Laplace transforms?
As far as introducing the Laplace transform, I'm not sure what else you could use for motivation other than it's another technique for solving linear differential equations. It's probably the simplest way for students to learn how the time domain and s-domain are related.

If you're looking for applications beyond just solving differential equations, I suggest you look through textbooks on linear systems analysis. But I think these applications, like analyzing the stability of a system, will assume that the student knows how the s-domain relates back to the time domain.

vanhees71 and matqkks
While I think it's good to introduce them with a bit of mathematical rigor (definitions, existence, integrals, etc.), engineers will soon forget all of that. I think it's important to show fairly early on how useful they are in practice, like solving a SHO problem for example, or their connection to frequency response plots. Show them the solution of a small network of passive analog electronics perhaps. The concept of electrical impedance expressed in the s domain is incredibly useful.

In practice, working engineers rarely find the transforms by solving integrals, we look them up and use them. We also rarely deal with obnoxious functions, it's always well behaved stuff that you can construct from a small list of standard stuff; like step, square, ramp, pulse, δ, sinusoids, exponential, etc.

PS: If you set s = jω (which we do all the time), you also get all of the benefits of the Fourier Transform too.

vanhees71, matqkks and berkeman
I would also stress the "algebra" theorems of the transforms: linearity, shifting, scaling, convolution, derivatives & integrals, repetition (periodic), etc. Also partial fraction expansion (even though everyone hates actually doing it). These are useful tools in constructing transforms from simple tables instead of brute force calculations.

PhDeezNutz and matqkks
When I took differential equations, a long, long time ago, in a place far, far away from where I live now, my very good professor (at a time when she was a newly hired assistant professor and I may have even sat in on her interview lectures, and also at a time when I was younger and thinner) treated it as a random tool box or Swiss Army knife of options course, without any deeper unity or method to the madness or motivation, sort of like collecting trading cards or fun meal toys.

Applications and practical relevance were introduced mostly through problem sets in the form of word problems. (Do you still call them "word problems" in upper division college math courses reserved for STEM majors who were good at calculus? But I'm spacing on an alternative name for them.)

While that could be pretty dry if done poorly, I recall it as being one of the most fun and entertaining math classes I took in college. She had so much enthusiasm for the topic as a newly hired professor that it was just infectious. It was magical. She left everyone feeling like:

Who wouldn't want to learn another sneaky trick to solve a class of problems (i.e. differential equations) that don't have a general solution that can be used in all cases to solve them analytically? The more you know, the smarter and more capable you are.

We all assumed, somewhat naively, that the problems we would solve with these tools were currently unknown and were waiting out there in the Platonic world of math and physics problems waiting to be discovered, and that we would conquer that world later, when we graduated, once we were well armed with tools our professors had given us, because we trusted our professors to give us tools that were good enough for anything we would encounter.

Having an attitude towards the material that it is exciting and worth knowing about, on an interpersonal and social interaction level, can make up for pretty thin gruel in the area of context and practical applications (even though both of those are out there to be had).

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matqkks
"Who wouldn't want to learn another sneaky trick to solve a class of problems (i.e. differential equations) that don't have a general solution that can be used in all cases to solve them analytically? The more you know, the smarter and more capable you are."
I have never been attracted to the "bag of tricks" approach to mathematics. I greatly prefer the techniques that hint at a deeper and profound truth that is widely applicable.

PhDeezNutz and S.G. Janssens
FactChecker said:
"Who wouldn't want to learn another sneaky trick to solve a class of problems (i.e. differential equations) that don't have a general solution that can be used in all cases to solve them analytically? The more you know, the smarter and more capable you are."
I have never been attracted to the "bag of tricks" approach to mathematics. I greatly prefer the techniques that hint at a deeper and profound truth that is widely applicable.
For a lot of mathematical disciplines, the deeper and profound truth thing works and I agree with you. For example, it is good for real and complex analysis, for fractals and chaos, for abstract algebra, and for plain old calculus. But other subfields, like differential equations and operations research, for example, just seem better suited to the bag of tricks approach. And the bag of tricks approach can be a pretty good one for engineering oriented students in contrast to pure math oriented students.

matqkks
ohwilleke said:
But other subfields, like differential equations and operations research, for example, just seem better suited to the bag of tricks approach.
I strongly disagree, as far as the subfield of differential equations is concerned.
I am not sufficiently family with OR to have an opinion about that.)

1. What is a Laplace transform and why is it important for engineers?

A Laplace transform is a mathematical operation that converts a function of time into a function of complex frequency. It is important for engineers because it allows them to analyze systems and signals in the frequency domain, which can provide valuable insights and simplify complex problems.

2. How are Laplace transforms used in engineering applications?

Laplace transforms are used in a wide range of engineering applications, such as control systems, circuit analysis, signal processing, and differential equations. They help engineers to model and analyze systems and signals in the frequency domain, which can provide valuable information about stability, response, and performance.

3. What are the advantages of using Laplace transforms over other mathematical tools?

Laplace transforms have several advantages over other mathematical tools, such as differential equations and Fourier transforms. They can simplify complex problems by converting them into algebraic equations, they can handle systems with discontinuous or non-periodic inputs, and they allow for easier analysis in the frequency domain.

4. What are the key concepts that engineers should understand when learning about Laplace transforms?

Engineers should have a strong understanding of basic calculus, complex numbers, and differential equations when learning about Laplace transforms. They should also be familiar with the properties of Laplace transforms, such as linearity, time-shifting, and differentiation, as well as the inverse Laplace transform.

5. How can engineers effectively incorporate Laplace transforms into their problem-solving process?

To effectively incorporate Laplace transforms into problem-solving, engineers should first understand the concepts and properties of Laplace transforms. Then, they should practice applying them to various engineering problems and systems. It is also helpful to use software tools, such as MATLAB, to assist with calculations and visualizations of Laplace transforms.

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