Post on 26-Oct-2021
transcript
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