Forced Oscillation 1

Equations of Motion Linear differential equation of order n=2 2

General solution: General solution = Complimentary + Particular solution 3

Particular solution:P(t) Complementary solution: C(t) 4

2 nd order linear inhomogeneous differential equation with constant coefficients General solution : Particular integral: obtained by special methods, solves the equation with f(t)  0; without any additional parameters A & B : obtained from initial conditions Complementary function 5

External Forcing SHO with an additional external force Why this particular type of force ? 6

For any arbitrary time varying force 7

Driving force: where 8

Equation of motion x=x r +ix i 9

Trial solution: x = Obtaining the particular integral Note: As the complementary solution has been discussed earlier, we shall ignore this term here. 10

Amplitude, Relative Phase 11

Amplitude and Phase 12 For the case

At resonance [  =  o ] 13 where We have

Low Frequency Response Stiffness Controlled Regime 14 Because

High Frequency Response Mass Controlled Regime 15

Undamped forced oscillation  Stiffness controlled regime (   )  Resonance (  0 )  Mass controlled regime (  >  0 ) 16

General solution: 17

0 18 Initial conditions:

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t Sin (1/2)Sin 2 20

Sin+ (1/2)Sin 2 21

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Fourier Series 25 A Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.

Fourier Series:f(t), << T t 0 A0 =A0 = f(t) A n / 2 = f(t)Cos n 26 For a periodic function f(t) that is integrable on [−π, π] or [0,T], the numbers A n and B n are called the Fourier coefficients of f. B m / 2 =f(t)Sin m f(t) = A 0 + A 1 Cos + A 2 Cos 2 + A 3 Cos 3 + A 4 Cos 4 + ……. + B 1 Sin + B 2 Sin 2 + B 3 Sin 3 +…

Examples 27

Sin (1/2)Sin 2 28

Sin+ (1/2)Sin 2 29

Sin + (1/2)Sin 2+ (1/3)Sin 3 + (1/4)Sin 4 30

6 terms of the series 31

10 terms of the series 32

20 terms of the series 33

Sin (1/3)Sin3 34

(1/3)Sin3Sin+ 35

Sin+(1/3)Sin3+(1/5)Sin5 +(1/7)Sin7 36

6 terms of the series 37

10 terms of the series 38

20 terms of the series 39

Cos+ (1/9)Cos 3 40

Cos+ (1/9)Cos 3+ (1/25)Cos 5 + (1/49)Cos 7 41