Could someone help me find the Laplace transform of the time funtion $f(t)={e}^{-K{t}^{n}}$ where K is real and positive and 0<n<1 ?

wurpenxd
2022-09-11
Answered

Could someone help me find the Laplace transform of the time funtion $f(t)={e}^{-K{t}^{n}}$ where K is real and positive and 0<n<1 ?

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Part II

29.[Poles] (a) For each of the pole diagrams below:

(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.

(ii) Write down two examples of such f(t) and F(s).

The diagrams are:$(1)1,i,-i.(2)-1+4i,-1-4i.(3)-1.(4)$ The empty
diagram.

(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by$w(t)=u(t){e}^{-\frac{t}{2}}\mathrm{sin}(\frac{3t}{2})$

(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?

(ii) Sketch the pole diagram of the system.

(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?

29.[Poles] (a) For each of the pole diagrams below:

(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.

(ii) Write down two examples of such f(t) and F(s).

The diagrams are:

(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by

(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?

(ii) Sketch the pole diagram of the system.

(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?

asked 2021-09-23

Let f(t) be a function on $[0,\mathrm{\infty}]$ . The Laplace transform of f is the function F defined by the integral $F\left(s\right)={\int}_{0}^{\mathrm{\infty}}{e}^{-st}f\left(t\right)dt$ . Use this definition to determine the Laplace transform of the following function.

$f\left(t\right)={e}^{-4t}\mathrm{sin}9t$

asked 2022-01-22

I am trying to show this kind of non-linear $y{}^{\u2057}={y}^{\prime}y\frac{{}^{\u2033}}{1+x}$ in normal form. For example here if $y={e}^{x}\to {y}^{\left(n\right)}={e}^{x}\to x=-1$ , where $y}^{\left(n\right)$ means n-th differential, then $x=-1$ , too weak idea. When I google with differential or anything like that, most of the material does not look the material that I need. I need to solve different type of problems such as this homework

$$\{\begin{array}{l}{u}^{\prime}=(u+v{)}^{2}\\ {v}^{\u2033}=x+{u}^{\prime}{v}^{\prime}\end{array}$$

I am not requesting you to solve them but I am requesting some material because I find my book quite hard-reading in this section. The earlier chapter begun that something is something, without much further ado really why?, and now the next advanced chapters are referring to the past chapters. The idea is a rush introduction to this topic in an engineering course so I think it explains quite a bit about the pedagogy.

I am not requesting you to solve them but I am requesting some material because I find my book quite hard-reading in this section. The earlier chapter begun that something is something, without much further ado really why?, and now the next advanced chapters are referring to the past chapters. The idea is a rush introduction to this topic in an engineering course so I think it explains quite a bit about the pedagogy.

asked 2021-12-26

For each of the following differential equations, determine the general or particular solution: $5x{e}^{-y}+2\mathrm{cos}\left(3x\right)]{y}^{\prime}+5{e}^{-y}-3\mathrm{sin}\left(3x\right)=0$