Damped Simple Harmonic Motion—Critical Damping

Author

Eric W. Weisstein
May 19, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/DampedSimpleHarmonicMotionCriticalDamping.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DampedSimpleHarmonicMotionCriticalDamping.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

SHO[{beta_,w_,t_},C_List:{1,0}]:=Module[{D=beta^2-4w^2,r1,r2,gamma},Which[D>0,r1=(-beta+Sqrt[beta^2-4 w^2])/2;
r2=(-beta-Sqrt[beta^2-4 w^2])/2;
C[[1]] Exp[r2 t]+C[[2]] Exp[r1 t],D<0,gamma=Sqrt[4 w^2-beta^2]/2;
Exp[-beta t/2](C[[1]] Cos[gamma t]+C[[2]] Sin[gamma t]),D==0,(C[[1]]+C[[2]]t)Exp[-w t]]]

SHOType[{beta_,w_,t_},C_List:{1,0},periods_:2]:=Module[{D=beta^2-4w^2,r1,r2,gamma},Which[D>0,Print["Overdamped, D=",D,">0"],D<0,Print["Underdamped, D=",D,"<0"],D==0,Print["Critically damped, D=",D];
];
Print[SHO[{beta,w,t}]];
]

SHOForced[{beta_,w_,w0_,t_},A_:1]:=Module[{D=beta^2-4w^2,r1,r2,gamma,delta},Which[D>0,delta=ArcTan[beta w/(w^2-w0^2)];
A/Sqrt[(w^2-w0^2)^2+beta^2 w^2] Cos[w t+delta],D<0,delta=ArcTan[beta w/(w^2-w0^2)];
A Sqrt[(w0^2-w^2)^2+beta^2 w^2]/((w0^2-w^2)^2-w^2(4w0^2-beta^2)) Cos[w t+delta],D==0,delta=ArcTan[2 w w0/(w^2-w0^2)];
A/(w^2+w0^2)Cos[w t+delta]]]

Critically Damped

SHOType[{.3,.3/2,t}]

Plot[{SHO[{.3,.15,t},{1.,0.}],SHO[{.3,.15,t},{0.,1.}],SHO[{.3,.15,t},{.5,.5}]},{t,0,20},PlotRange->All,PlotStyle->{{},RGBColor[1,0,0],RGBColor[0,0,1]}]

Solution

DSolve[x''[t]+β x'[t]+((ω0)^2)x[t]==0/.β->2ω0,x[t],t]

w=y1 [t]y2'[t]-y1'[t]y2[t]/.{
y1->Function[t,Exp[-(Subscript[ω, 0]) t]],
y2->Function[t,t Exp[-(Subscript[ω, 0]) t]]
}//FullSimplify

DSolve[{
v1'[t]y1'[t]+v2'[t]y2'[t]==C Cos[ω t],
v1'[t]y1[t]+v2'[t]y2[t]==0}/.{
y1->Function[t,Exp[-(Subscript[ω, 0]) t]],
y2->Function[t,t Exp[-(Subscript[ω, 0]) t]]
},{v1[t],v2[t]},t]//FullSimplify

{v1,v2}={-Integrate[((Exp[-(Subscript[ω, 0]) t]C Cos[ω t])/w),t],Integrate[((t Exp[-(Subscript[ω, 0]) t]C Cos[ω t])/w),t]}//FullSimplify

xstar=v1 y1[t]+v2 y2[t]/.{
y1->Function[t,Exp[-(Subscript[ω, 0]) t]],
y2->Function[t,t Exp[-(Subscript[ω, 0]) t]]
}

Together[xstar]

{a,b}=.

{a,b}=Reverse[List@@Collect[Together[xstar],{_Cos,_Sin}]]/{Cos[ω t],Sin[ω t]}//Simplify

FullSimplify[(Sqrt[(a^2)+(b^2)])/.{α->β/2,γ->(Sqrt[((Subscript[ω, 0])^2)-(β^2)/4])},C>0&&ω>0&&(Subscript[ω, 0])>0&&β∈Reals]

Factor[(ω^4)-2 (ω^2) (Subscript[ω, 0]^2)+(Subscript[ω, 0]^4)]

Simplify[DSolve[x''[t]+β x'[t]+((ω0)^2)x[t]==C Cos[ω t]/.β->2ω0,x[t],t]]