MAE 2030 Lecture 7 2 2421Today Forced damped oscillator

MidpointmethodMultiple masses

Forced damped harmonic oscillatora famous often used model very simple covers

lots of phenomena like things that vibrate

K Lomm

i

d GM F to sin wot

Ci Forcing

I Lo I

FBI T Mlm Ts Kx

Td 1J Td CK

Ifa M Ii tosin wot

LMB IF moi

I Ts Td t Fosin wot MId tki Cri

Math know

Form M t CX c Kx to sin wot this

mffurter I Eosin wot kx cyt.fmI

w

X v

1storder form j fFosin wot Kx wm

I f titL z F

Lots of special cases

lots of phenomenawith e9n reagdouifbool

vibrations

ex Xg X ht Xp

ex Long term behavior

Assume that a 0

mightbe verysmall

Xh t OTt a

Xp It steady slate soin

Xp Acos wot t Bsin wotXa

c tAFT

X'ta

resonance

Fodriving frequency

I 7

KfWo

Im Wn natural frequency

ex overdamped I underdampedXh

c O

µunderdamped wecriticallydamped s

c is

Coverdamphedgge

Read book check phenomena w computer

simulation

Midpoint Method for numerical Solin of ODES

ex z zaHZ

errorwith

midpoint

gerror

withEuler

to sten tenth

1

one Eulerstep

Algorithmif f It it

r h S function

given ten Z n h halfa tiffeep

Ey f th t Ih 2 n t In

2htt I Zn th Eyz

adds v2 lines of code

ODE2 keep ODE I

Multi Dof many degrees of freedom

ex n C Cz

EmTIEmEDm FLoK OO LozK2 00

Goal ODEs we can solve

FBDs

Mass ITd1 m TokTsf Tsz

mass 2 Tok j Mz FTsz

system Ide ft m 7pm Mr F1st

k1

M X Tat Tsz Ta Tsz

mass 2 Mz iz Td Ts t F

need to figure out tensions

spring 1 Ts K L Lo be carefulof signs

dashpot 1 Td Ct Ctx

spring 2 Tsz K2 L2 Loa

x z X

dashpot 2 Td Cz iz Cz iz ni