16.61 Aerospace Dynamics Spring 2003 Simple example · Spring- mass system ass syster Linear spring Frictionless table m · Lagrangian L=T-V L=T-Ⅴ=Dmx · Lagranges equation d aLaL aq, a q e Do the derivatives aL d aL dL -x mmX dt( dq Put it all together d dL dL n元+kx=0 dt(d qi dc Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 5 Simple Example • Spring – mass system Spring mass system • Linear spring • Frictionless table m x k Spring mass system • Linear spring • Frictionless table m x k • Lagrangian L = T – V 1 1 2 2 L = T V 2 2 −= − mx kx • Lagrange’s Equation 0 i i dL L dt q q ∂ ∂ − = ∂ ∂ • Do the derivatives i L mx q ∂ = ∂ , i d L mx dt q ∂ = ∂ , i L kx q ∂ = − ∂ • Put it all together 0 i i dL L mx kx dt q q ∂ ∂ − = += ∂ ∂
16.61 Aerospace Dynamics Spring 2003 Consider the mgr problem with the mass oscillating between the two springs only i degree of freedom of interest here so take gi = R qi oIR+R 0|+|0 0 o(R+r 0 0 (1)(4)=(R2+02(R+R) =2-R2 L=T-V=m(R2+03(R+R)2)-kR dL tar dL mo(R+R)-2kR dR So the equations of motion are: mR-mo (r+r)+2kR=o 2k r+ or=Ro2 which is the same as on(3-4) Massachusetts Institute of Technology C How, Deyst 2003( Based on notes by Blair 2002
16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology © How, Deyst 2003 (Based on notes by Blair 2002) 6 Consider the MGR problem with the mass oscillating between the two springs. Only 1 degree of freedom of interest here so, take qi=R D D D ( ) (D ) (D ) D ( ) D ( ) D DD ( ) r R RR R R R T m r r m R RR V k R LTV m R R R kR d dt L R mR L R m R R kR M I o o M I T M I o o o = L N M M M O Q P P P + L N M M M O Q P P P L + N M M M O Q P P P = + L N M M M O Q P P P = =++ = =−= + + − ∂ ∂ F H G I K J = ∂ ∂ = +− × 0 0 0 0 0 0 0 2 2 2 2 2 2 22 2 2 22 2 2 2 ω ω ω ω ω c h c h So the equations of motion are: mR m R R kR R k m R R o o DD ( ) DD − ++ = + − F H I K = ω ω ω 2 2 2 2 0 2 or which is the same as on (3- 4)