16.61 Aerospace Dynamics Spring 2003 Lecture #10 Friction in Lagrange’s Formulation Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 1 16.61 Aerospace Dynamics Spring 2003 Generalized Forces Revisited ? Derived Lagrange’s Equation from D’Alembert’s equation: () () 11 iii pp iii i i ii x i y i z i mxx yy zz F x F y F zδ δδ δ δ δ == ++= + + ∑∑ && && && ? Define virtual displacements 1 N i ij j j x xq q = ?? ? = ?? ? ?? ∑ δ δ ? Substitute in and noting the independence of the j qδ , for each DOF we get one Lagrange equation: 11 iii pp iii i i i iiir x y z rrr r r r xyz x y z mx y z q F F F q qqq q q q == ??? ??? ? ? ? ++ = + + ??? ??? ? ? ? ??? ∑∑ && && && r ? ? ? δ δ ? Applying lots of calculus on LHS and noting independence of the i qδ , for each DOF we get a Lagrange equation: 1 iii p ii xyz i rr r r x i r y zdT T FFF dt q q q q q = ?? ? ? ? ???? ?= + + ?? ? ?? ? ? ? ?? ? ∑ & ? ? ? Further, we “moved” the conservative forces (those derivable from a potential function to the LHS: 1 iii p ii xyz i rr r r x i r y zdL L FFF dt q q q q q = ?? ? ? ? ???? ?= + + ?? ? ?? ? ? ? ?? ? ∑ & ? ? Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 2 16.61 Aerospace Dynamics Spring 2003 ? Define Generalized Force: 1 riii p ii qxyz i rr x i r y z QFFF qq = ?? q ? ?? =++ ?? ??? ?? ∑ ? Recall that the RHS was derived from the virtual work: r q r W Q q = δ δ ? Note, we can also find the effect of conservative forces using virtual work techniques as well. Example ? Mass suspended from linear spring and velocity proportional damper slides on a plane with friction. ? Find the equation of motion of the mass. g c k m q(t) μ θ ? DOF = 3 – 2 = 1. ? Constraint equations: y = z = 0. ? Generalized coordinate: q ? Kinetic Energy: 2 1 2 = &Tmq ? Potential Energy: 2 1 sin 2 qmgq=?Vk θ Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 3 16.61 Aerospace Dynamics Spring 2003 ? Lagrangian: 22 11 sin 22 LTV mq kq mgq=?= ? +& θ ? Derivatives: ,, LdL L mq mq kq mg qdtq q ????? ===?+ ?? ?? &&& && sinθ ? Lagrange’s Equation: sin r q dL L mq kq mg Q dt q q ???? ?=+? = ?? ?? ?? && & θ ? To handle friction force in the generalized force term, need to know the normal force ? Lagrange approach does not indicate the value of this force. mg F s F d F f N mq&& o Look at the free body diagram. o Since body in motion at the time of the virtual displacement, use the d’Alembert principle and include the inertia forces as well as the real external forces o Sum forces perpendicular to the motion: cosNmg θ= Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 4 16.61 Aerospace Dynamics Spring 2003 ? Recall Wδ δ=?Fs q . Two nonconservative components, look at each component in turn: o Damper: Wcqδ δ=? & o Friction Force: sgn( ) sgn( ) cos WqNq qmg q =? =? δ μδ μ θδ ? Total Virtual Work: ( ) sgn( ) cosWcq qmg q=? ? & δ μθδ ? The generalized force is thus: () sgn( ) cos r q r W Qcqqmg q ==?? & δ μ θ δ ? And the EOM is: () sin sgn( ) cos sin sgn( ) cos mq kq mg cq q mg mq cq kq mg q +? =?? ++= ? && & && & θ μθ θ μθ ? Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 5 16.61 Aerospace Dynamics Spring 2003 ? Note: Could have found the generalized forces using the coordinate system mapping: 1 riii p ii qxyz i rr x i r y z QFFF qq = ?? q ? ?? =++ ?? ??? ?? ∑ o o For example, the gravity force: ,sin, sin i r i yi q y Fmgyq q Qmg sin ? =? = ? ?= = θ θ θ ? ? Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 6 16.61 Aerospace Dynamics Spring 2003 Rayleigh's Dissipation Function ? For systems with conservative and non-conservative forces, we developed the general form of Lagrange's equation N qr rr dL L Q dt q q ?? ?? ?= ?? ?? ?? & with L=T-V and r N qx y z rr x r y z QF F F qqq ? ?? =++ ? ?? ? For non-conservative forces that are a function of , there is an alternative approach. Consider generalized forces q& 1 (,) n N iij j Qcq = =? ∑ & j tq where the are the damping coefficients, which are dissipative in nature ? result in a loss of energy ij c ? Now define the Rayleigh dissipation function 11 1 2 nn ij i j ij Fc == = ∑∑ && q Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 7 16.61 Aerospace Dynamics Spring 2003 Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 8 ? Then we can show that 1 rr n N qj j q j r F cq Q q = ? ==? ? ∑ & & ? So that we can rewrite Lagrange's equations in the slightly cleaner form 0 rrr dL L F dt q q q ?? ??? ? += ?? ??? ?? && ? In the example of the block moving on the wedge, 2 1 2 F cq= & sin r q dL LF mq kq mg cq Q dt q q q ????? ′ ?+=+? += ?? ??? ?? && & && θ where r q Q now only accounts for the friction force. ′