Upload
yin-ka
View
9
Download
0
Embed Size (px)
DESCRIPTION
Engineering Math II Lecture 11School Notes
Citation preview
CIV210Engineering Mathematics II
Lecture 11
1
XU Dong-sheng
Email: [email protected]
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)
2
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
3
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
4
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
5
Example 2: Example 2: Application of Laplace transform Application of Laplace transform
6
Example 2: Example 2: Application of Laplace transform Application of Laplace transform
7
Example 2: Example 2: Application of Laplace transform Application of Laplace transform
8
Apply the inverse transform to obtain
Laplace Transform and Inverse TransformLaplace Transform and Inverse Transform
Original function (given function) : f(t)
9
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
10
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
11
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
10
1 tL F s f d
s
ntL e f t F s n Shifting theorem 1:
12
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:
13
Example type 2:Example type 2:Initial Value Problem: The Basic Laplace Steps
14
Summary diagram for the approachSummary diagram for the approach
15
Example:Example:
17
cosn x
dx