System dynamics step/ramp formula

[kdinser]

HI all, I'm taking a basic system dynamics course and I'm either having a problem with a concept or just with the notation.

I think the easiest way explain the problem I'm having is with an example ( see jpeg).

If I have f(t) = 0 when t <= a and [(bt)/(a+b)] - b when t>a I would have the laplace transform that equals int[((bt)/(a+b) - b)* e^(-st) dt] I'm fine with integrating this by hand and coming up with the correct answer, but I'm not sure how to get the answer from the tables. The book I'm using ("system dynamics) by katsuhiko ogata) seems to be a very good book, but there is some notation in there regarding step and ramp functions that I'm not understanding.

mainly stuff like f(t-a)1(t-a). Can someone explain this notation to me and or how to use the laplace tables when dealing with this kind of function?

 

[Aaron Bex]

The notation you are referring to is called the Heaviside Step Function, which is used to represent a unit step or "on/off" type of behavior. The notation f(t-a)1(t-a) can be interpreted as follows: f(t-a) is a function of time (t) shifted by an amount of time (a). The 1(t-a) is the Heaviside Step Function which takes on the value of 0 for t<a and 1 for t>a. So the notation represents a function that is shifted by a certain amount of time and then is turned "on" at this time.

To use the Laplace tables to solve this type of problem, you will need to use the properties of the Heaviside Step Function. For example, if you want to solve for the Laplace transform of f(t-a)1(t-a), you can use the property that the Laplace transform of a unit step is 1/s. Using this property, you can rewrite the expression as f(t-a)*1/s, which can then be solved using the Laplace tables. Hope this helps!

 

[piercas]

I can provide a response to your question about the system dynamics step/ramp formula. The notation you are referring to, specifically f(t-a)1(t-a), is commonly used to represent a step function in system dynamics. The "1(t-a)" part of the notation indicates that the function is equal to 1 for t > a and 0 for t < a. This means that the function has a step at t=a, where it changes from 0 to 1.

In the case of your example, f(t) = 0 when t <= a and [(bt)/(a+b)] - b when t>a, the function can be rewritten as f(t) = [(bt)/(a+b)] - b1(t-a). This notation makes it easier to understand the behavior of the function, as it clearly indicates that there is a step at t=a.

When using Laplace tables to solve for the Laplace transform of a function, you can use the properties of the Laplace transform to simplify the integral. In this case, you can use the shifting property, which states that the Laplace transform of f(t-a) is e^(-as)F(s), where F(s) is the Laplace transform of f(t). Applying this property to your example, you can rewrite the integral as int[(bt)/(a+b) * e^(-(a+b)s) - b*e^(-as) dt]. From there, you can use the Laplace transform table to solve for the transform of each term separately.

I hope this helps clarify the notation and how to use Laplace tables in dealing with step and ramp functions in system dynamics. Keep practicing and don't hesitate to ask for help if you encounter any further difficulties. Good luck with your course!