线性化非线性摆韧带振荡微分方程课件.pdfVIP

18.03Class38,May10,2010

Linearization:Thenonlinearpendulumandphugoidoscillation

[1]Nonlinearpendulum

[2]Phugoidoscillation

[3]Bucklingbridge

[1]Thebobofapendulumisattachedtoarod,soitcanswingclear

aroundthepivot.Thissystemisdeterminedbythreeparameters:

Llengthofpendulum

mmassofbob

gaccelerationofgravity

Wewillassumethatthemotionisrestrictedtoaplane.

Todescribeitweneedadynamicalvariable.Wecoulduseahorizontal

displacement,butitturnsouttobeeasiertowritedowntheequation

controllingitifyouusetheangleofdisplacementfromstraightdown.

Writethetaforthatangle,measuredtotheright.

Hereisaforcediagram:

*

|\

|theta

|\

|\

mg|/\(thisissupposedtobearightangle!)

|\/

|/

|/mgsin(theta)

|/

|/

*

Writesforarclengthalongthecircle,withs=0

straightdown.Ofcourse,

s=Ltheta

Newtonslawsays

ms=F

Theforcehasthe-mgsin(theta)componentoftheforceofgravity

(andnoticethesign!),andalsoafrictionalforcewhichdependsupon

s=Ltheta

Makethesimplestmodelforfriction,-cs=-cLtheta.So:

mLtheta=-mgsin(theta)-cLtheta

DividethroughbymLandweget

theta+btheta+ksin(theta)=0

wherek=g/Landb=c/m.

Thisisanonlinearsecondorderequation.Itstillhasacompanion

firstordersystem,obtainedbysetting

x=theta,y=x

soy=theta=-ksin(theta)-bthetaor

x=y

y=-ksin(x)-by

Thisisanautonmoussystem.Letsstudyitsphaseportrait.

Equilibria:y=0,sin(x)=0;thatis,

x=0,+-pi,+-2pi,.