July 2011

Table of Laplace Transforms of Elementary Functions

Below are some functions   $f(t)$   and their Laplace transforms   $F(s)$.
 

$f(t)$ $F(s) = \mathcal{L} \left\{f(t)\right\}$
$1$ $\dfrac{1}{s}$
$t$ $\dfrac{1}{s^2}$
$t^2$ $\dfrac{2}{s^3}$
... ...

 

Equilibrium of Force System

The body is said to be in equilibrium if the resultant of all forces acting on it is zero. There are two major types of static equilibrium, namely, translational equilibrium and rotational equilibrium.
 

Formulas
Concurrent force system
$\Sigma F_x = 0$

$\Sigma F_y = 0$
 

Parallel Force System
$\Sigma F = 0$

$\Sigma M_O = 0$
 

Non-Concurrent Non-Parallel Force System
$\Sigma F_x = 0$

$\Sigma F_y = 0$

$\Sigma M_O = 0$
 

Properties of Laplace Transform

Constant Multiple
If   $a$   is a constant and   $f(t)$   is a function of   $t$,   then
 

$\mathcal{L} \left\{ a \, f(t) \right\} = a \, \mathcal{L} \left\{ f(t) \right\}$

 

Linearity Property | Laplace Transform

Linearity Property
If   $a$   and   $b$   are constants while   $f(t)$   and   $g(t)$   are functions of   $t$   whose Laplace transform exists, then
 

$\mathcal{L} \left\{ a \, f(t) + b \, g(t) \right\} = a \, \mathcal{L} \left\{ f(t) \right\} + b \, \mathcal{L} \left\{ g(t) \right\}$

 

Proof of Linearity Property
$\displaystyle \mathcal{L} \left\{ a \, f(t) + b \, g(t) \right\} = \int_0^\infty e^{-st}\left[ a \, f(t) + b \, g(t) \right] \, dt$

Problem 01 | Linearity Property of Laplace Transform

Problem 01
Find the Laplace transform of   $f(t) = 5t - 2$.
 

Solution 01

Problem 02 | Linearity Property of Laplace Transform

Problem 02
By using the linearity property, show that

$\mathcal{L}(\cosh at) = \dfrac{s}{s^2 - a^2}$

 

Solution 02
$f(t) = \cosh at$

$\displaystyle \mathcal{L}\left\{ f(t) \right\} = \int_0^\infty e^{st} f(t) \, dt$

$\displaystyle \mathcal{L}(\cosh at) = \int_0^\infty e^{st} \cosh at \, dt$
 

But
$\cosh at = \dfrac{e^{at} + e^{-at}}{2}$
 

Thus,
$\displaystyle \mathcal{L}(\cosh at) = \int_0^\infty e^{st} \left( \dfrac{e^{at} + e^{-at}}{2} \right) \, dt$

First Shifting Property | Laplace Transform

First Shifting Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   when   $s > a$   then,
 

$\mathcal{L} \left\{ e^{at} \, f(t) \right\} = F(s - a)$

 

In words, the substitution   $s - a$   for   $s$   in the transform corresponds to the multiplication of the original function by   $e^{at}$.
 

Problem 356 | Equilibrium of Non-Concurrent Force System

Problem 356
The cantilever truss shown in Fig. P-356 is supported by a hinge at A and a strut BC. Determine the reactions at A and B.
 

Cantilever truss with equal loads at top joints

 

Problem 02 | First Shifting Property of Laplace Transform

Problem 02
Find the Laplace transform of   $f(t) = e^{-5t} \sin 3t$.
 

Solution 02

Problem 03 | First Shifting Property of Laplace Transform

Problem 03
Find the Laplace transform of   $f(t) = e^{-3t} \cos t$.
 

Solution 03

Problem 04 | First Shifting Property of Laplace Transform

Problem 04
Find the Laplace transform of   $f(t) = e^t \sinh 2t$.
 

Solution 04

Problem 357 | Equilibrium of Non-Concurrent Force System

Problem 357
The uniform rod in Fig. P-357 weighs 420 lb and has its center of gravity at G. Determine the tension in the cable and the reactions at the smooth surfaces at A and B.
 

Uniform rod supported by a cable

 

Problem 358 | Equilibrium of Non-Concurrent Force System

Problem 358
A bar AE is in equilibrium under the action of the five forces shown in Fig. P-358. Determine P, R, and T.
 

Bar in Equilibrium

 

Problem 359 | Equilibrium of Non-Concurrent Force System

Problem 359
A 4-m bar of negligible weight rests in a horizontal position on the smooth planes shown in Fig. P-359. Compute the distance x at which load T = 10 kN should be placed from point B to keep the bar horizontal.
 

Rigid bar resting on smooth inclined surface

 

Problem 360 | Equilibrium of Non-Concurrent Force System

Problem 360
Referring to Problem 359, what value of T acting at x = 1 m from B will keep the bar horizontal.
 

Rigid bar resting on smooth inclined surface

Problem 361 | Equilibrium of Non-Concurrent Force System

Problem 361
Referring to Problem 359, if T = 30 kN and x = 1 m, determine the angle θ at which the bar will be inclined to the horizontal when it is in a position of equilibrium.
 

Rigid bar resting on smooth inclined surface

 

Second Shifting Property | Laplace Transform

Second Shifting Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   and   $g(t)
= \begin{cases} f(t - a) & t \gt a \\ 0 & t \lt a \end{cases}$
 

then,

$\mathcal{L} \left\{ g(t) \right\} = e^{-as} F(s)$

 

Problem 01 | Second Shifting Property of Laplace Transform

Problem 01
Find the Laplace transform of   $g(t) = \begin{cases} f(t - 1)^2 & t \gt 1 \\ 0 & t \lt 1 \end{cases}$
 

Problem 02 | Second Shifting Property of Laplace Transform

Problem 01
Find the Laplace transform of   $g(t) = \begin{cases} f(t - 2)^3 & t \gt 2 \\ 0 & t \lt 2 \end{cases}$
 

Change of Scale Property | Laplace Transform

Change of Scale Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then,
 

$\mathcal{L} \left\{ f(at) \right\} = \dfrac{1}{a} F \left( \dfrac{s}{a} \right)$

 

Problem 01 | Change of Scale Property of Laplace Transform

Problem 01
Find the Laplace transform of   $f(t) = \cos 4t$   using the change of scale property.
 

Solution 01
$\mathcal{L} \left\{ f(at) \right\} = \dfrac{1}{a} F \left( \dfrac{s}{a} \right)$
 

$\mathcal{L} (\cos t) = \dfrac{s}{s^2 + 1}$
 

Thus,
$\mathcal{L} (\cos 4t) = \dfrac{1}{4} \left[ \dfrac{\dfrac{s}{4}}{\left( \dfrac{s}{4} \right)^2 + 1} \right]$

$\mathcal{L} (\cos 4t) = \dfrac{1}{4} \left[ \dfrac{\dfrac{s}{4}}{\dfrac{s^2}{16} + 1} \right]$

Problem 02 | Change of Scale Property of Laplace Transform

Problem 02
Given that   $\mathcal{L} \left( \dfrac{\sin t}{t} \right) = \arctan \left( \dfrac{1}{s} \right)$,   find   $\mathcal{L} \left( \dfrac{\sin 3t}{t} \right)$.
 

Solution 02
$\mathcal{L} \left\{ f(at) \right\} = \dfrac{1}{a} F \left( \dfrac{s}{a} \right)$
 

$\mathcal{L} \left( \dfrac{\sin 3t}{t} \right) = 3\mathcal{L} \left( \dfrac{\sin 3t}{3t} \right)$,   thus,   $a = 3$

Problem 03 | Change of Scale Property of Laplace Transform

Problem 03
Supposed that the Laplace transform of a certain function   $f(t)$   is   $\dfrac{s^2 - s + 1}{(2s + 1)^2 (s - 1)}$,   find the Laplace transform of   $f(2t)$.
 

Solution 03
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then by change of scale property,   $\mathcal{L} \left\{ f(at) \right\} = \dfrac{1}{a} F \left( \dfrac{s}{a} \right)$
 

$\mathcal{L} \left\{ f(t) \right\} = \dfrac{s^2 - s + 1}{(2s + 1)^2 (s - 1)}$

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