Forced, undamped vibration, continued...

Suppose the forcing function is sinusoidal = Fsinet

The defining ODE for displacement, y(t), is:

my + ky = Fsinet

y(t) = C₁sinwt + C₂coswt + Y

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General Solution

ENGG 614 2023

Ninth Class 10-31-23

Revised of 10-30-23, 8:30pm

Substitute Y in the ODE and obtain:

Let Y = Asinet; don't need Bcoset since both y and y", the only dependent variable terms,

P

are both sines.

- A0²sinet + Aw²sinet = ( )sinwt#

(40)

A

1 = (-)( ²¹ =)#(41)

y(t) = C₁sinwt + C₂coswt + ((sinot# (42)

m

(30

Page 15 of 22/n-A0'sinet + Aw sinet

y(t) = C₁sinwt + C₂coswt + (=) ( ²1 )sinot#(42)

m

msinwt# (40)

^ = ( ) ( )* (41)

Let y = y = 0, system at rest at t = 0 when F is applied

For y(0) = 0: 0 = C₁0 + C₂·1+(.).0; C₂ = = 0

2

So the solution for y(t) is:

For y (0) = 0: _y(t) = wc₁coswt — wc,sinwt + () 1¹ecoset

0 = wC₁ • 1 - W. 0.0 +

C₁ = ( =)))*(¹4)

wsin Ot-Osinut

L'Hopital's Rule, 0 [The resonance condition]

y(t) = ( ) (

mw

W

FO

2

m(w²-0²)

y(t) = ( − 2)(~)( ¹²² )sinwt + (_)(_ ¹ )sinot

m

ய

= (-) (¹) - Osinw tw sinet]

y(t) = ((

wtcos@t-sino t

-20

#(45)

²), 0→w

. 1