CIV210Engineering Mathematics II

Lecture 11

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XU Dong-sheng

Email: xuds@chuhai.edu.hk

Tutorial

Application example 1: ODEApplication example 1: ODE

Modeling: Forced Oscillations. Resonance

we considered vertical motions of a massspring system (vibration of a mass m on an elastic spring, as shown in figure, and modeled it by the homogeneous linear ODE.

0my cy ky (1)

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Mass on a spring

0my cy ky

Here y(t) as a function of time t is the displacement of the body of mass m from rest.

The massspring system exhibited only free motion. This means no external forces (outside forces) but only internal forces controlled the motion. The internal forces are forces within the system. They are the force of inertia my, the damping force cy (if c>0), and the spring force ky, a restoring force.

(1)

Application example 1: ODEApplication example 1: ODE

Modeling: Forced Oscillations. Resonance

We now extend our model by including an additional force, that is, the external force r(x) on the right. Then we have:

my cy ky r x Mechanically this means that at each instant t the resultant of the internal forces is in equilibrium with r(t). The resulting motion is called a

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internal forces is in equilibrium with r(t). The resulting motion is called a forced motion with forcing function r(t), which is also known as input or driving force, and the solution y(t) to be obtained is called the output or the response of the system to the driving force.

The special interest are periodic external forces and we shall consider a driving force of the form 0 cosr x F tThen we have the nonhomogeneous ODE

0 cosmy cy ky F t (2)

Application example 1: ODEApplication example 1: ODE

Solving the Nonhomogeneous ODE (2)

From previous lecture, we know that a general solution of (2) is the sum of a general solution yh of the homogeneous ODE(1) plus any solution yp of (2). To find yp, we use the method of undetermined coefficients,

cos sinpy t a t b t

sin cosy t a t b t

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Substituting yp, yp and yp into (2) and collecting the cosine and the sine terms, we get

2 2 0cos sin cosk m a cb t ca k m b t F t So that:

2 0ca k m b

sin cospy t a t b t

2 2cos sinpy t a t b t

2 0k m a cb F

Application example 1: ODEApplication example 1: ODE

Solving the Nonhomogeneous ODE (2)

To eliminate b, obtaining

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Example 2: Example 2: Application of Laplace transform Application of Laplace transform

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Example 2: Example 2: Application of Laplace transform Application of Laplace transform

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Example 2: Example 2: Application of Laplace transform Application of Laplace transform

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Apply the inverse transform to obtain

Laplace Transform and Inverse TransformLaplace Transform and Inverse Transform

Original function (given function) : f(t)

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Important Important FormularsFormulars

0L f t sL f t f

2 0 0L f t s L f t sf f

3 2 0 0 0L f t s L f t s f sf f

1 2 1 2L c f t c g t c L f t c L g t

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3 2 0 0 0L f t s L f t s f sf f

1 2 10 0 0n n n n nL f t s L f t s f s f f

Some functions and their Some functions and their Laplace transformLaplace transformssFunction f(t) Laplace Transform

1 1/s, s>0

2 2/s, s>0

n (n=1,2,3,) n/s, s>0

t 1/s2

t2 2!/s3

tn n!/sn+1

t-1/2 (/s)1/2

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t-1/2 (/s)1/2

ent 1/(s-n), (s-n)>0

e-nt 1/(s+n), (s+n)>0

te-nt 1/(s+n)2, (s+n)>0

cost s/(s2+2)

sint /(s2+2)

cosht s/(s2-2)

sinht /(s2-2)

tcost (s2-2)/(s2+2)2

Laplace Transformation of IntegrationLaplace Transformation of Integration

01t

L f d L f ts

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1 tL F s f d

s

ntL e f t F s n Shifting theorem 1:

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L f t F s ntL e f t F s n

1 ntL F s n e f t

L f t F s nsnL f t n g t e F s

1 ns nL e F s f t n g t

If Then

Shifting theorem 2:

If Then

Example type 1:Example type 1:

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Example type 2:Example type 2:Initial Value Problem: The Basic Laplace Steps

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Summary diagram for the approachSummary diagram for the approach

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Fourier Series of all even function Fourier Series of all even function over over R

Example:Example:

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cosn x

dx