16.61 Aerospace Dynamics Spring 2003 Lecture #10 Friction in Lagrange's Formulation Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
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 ∑m(8x+16y+22)=∑(Fx+F+F。=) Define virtual displacements sx Substitute in and noting the independence of the 8q,, for each DOf we get one Lagrange equation oa +F +F Applying lots of calculus on LHS and noting independence of the Sq, for each dof we get a Lagrange equation d a ar ∑ +F +F Further, we moved the conservative forces(those derivable from a potential function to the lhs aLaL ∑ az +F +F Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Generalized Forces Revisited • Derived Lagrange’s Equation from D’Alembert’s equation: ( ) ( ) 1 1 i i i p p i i i i i i i x i y i z i i i m x δ x y δ δ y z z F δ x F δ y F δ z = = ∑ ∑ && + + && && = + + • Define virtual displacements 1 N i i j j j x x q = q ∂ = ∂ δ ∑ δ • Substitute in and noting the independence of the j δ q , for each DOF we get one Lagrange equation: 1 1 i i i p p i i i i i i i i i r x y z i i r r r r r r x y z x y z m x y z q F F F q = = q q q q q q ∂ ∂ ∂ ∂ ∂ ∂ + + = + + ∂ ∂ ∂ ∂ ∂ ∂ ∑ ∑ && && && r δ δ • Applying lots of calculus on LHS and noting independence of the i δ q , for each DOF we get a Lagrange equation: 1 i i i p i i x y z r r i r r x i r d T T y z F F F dt q q = q q q ∂ ∂ ∂ ∂ ∂ − = + + ∂ ∂ ∂ ∂ ∂ & ∑ • Further, we “moved” the conservative forces (those derivable from a potential function to the LHS: 1 i i i p i i x y z r r i r r x i r d L L y z F F F 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 Fyi oqr +F e Recall that the rhs was derived from the virtual work SW qr og 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 DOF=3-2 g k Constraint equations: y=2=0 q1) Generalized coordinate: q Kinetic Energy: T=mq Potential Energy: V=kq'-mggsin 8 Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 • Define Generalized Force: 1 r i i i p i i q x y z i r r x i r y z Q F F F = q q 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 g c k m q(t) µ θ • DOF = 3 – 2 = 1. • Constraint equations: y = z = 0. • Generalized coordinate: q • Kinetic Energy: 1 2 2 T m = q& • Potential Energy: 1 2 sin 2 V k = − q mgq θ Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 3
16.61 Aerospace Dynamics Spring 2003 · Lagrangian:L=T- na ka mgg sin e Derivatives OL dl aL aL ng -kg +mg sin 6 Lagrange's Equation aL aL mq+kq-mg sin=o 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 o Look at the free body diagram 7g o Since body in motion at the time of the virtual displacement, use F the d'alembert principle and include the inertia forces as well N as the real external forces o Sum forces perpendicular to the motion: N=mg cose Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 • Lagrangian: 1 1 2 2 sin 2 2 L T = −V = mq& − kq + mgq θ • Derivatives: , , L d L L mq mq kq mg q dt q q ∂ ∂ ∂ = = = − + ∂ ∂ ∂ & && & & sinθ • Lagrange’s Equation: sin r q d L L mq kq mg Q dt q q ∂ ∂ − = + − = ∂ ∂ && & θ • To handle friction force in the generalized force term, need to know the normal force Æ Lagrange approach does not indicate the value of this force. Fs mg Fd N Ff 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: N m= g cosθ Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 4
16.61 Aerospace Dynamics S 2003 Recall SW=F Ss. Two nonconservative components, look at each component in turn o Damper: W=-cqoq o Friction force δW=-sgn(q)NSq sgn(qmg cos eoq TotalⅤ irtual work 8W=(cq-sgn( q)umg cos0)&q The generalized force is thus W (cq-sgn( q)umg cos 0) O e And the eom is mq+kg -mg sin 0=-cq-sgn(qumg cos 6 =mq+cq+kg=mg(sin 8-sgn(q)ucos 0) Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 • Recall δW = ⋅ F δ s q . Two nonconservative components, look at each component in turn: o Damper: δW c = − q&δ o Friction Force: sgn( ) sgn( ) cos W q N q q mg q = − = − δ µ δ µ θδ • Total Virtual Work: δW c = −( q& − sgn(q)µ θ mg cos )δ q • The generalized force is thus: ( ) sgn( ) cos r q r W Q cq q mg 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 µ θ ⇒ θ µ θ 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 ox +F o For example the gravity force mg, y=-qsin 8 sin e O.=mg 0 Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 • Note: Could have found the generalized forces using the coordinate system mapping: 1 r i i i p i i q x y z i r r x i r y z Q F F F = q q q ∂ ∂ ∂ = + + ∂ ∂ ∂ ∑ o o For example, the gravity force: , sin , sin i r i y i q y F mg y q q Q mg sin ∂ = − = ∂ − = = θ θ θ − ⇒ Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 6
16.61 Aerospace Dynamics Spring 2003 Ravleigh's Dissipation Function For systems with conservative and non-conservative forces we developed the general form of Lagrange's equation d OL aL Q with l=t-v and x Qa=Fo-+F +F qr For non-conservative forces that are a function of q, there is an alternative approach. Consider generalized forces QN=-∑cn(q where the c, are the damping coefficients, which are dissipative in nature, result in a loss of energy Now define the rayleigh dissipation function F=22∑ Ci g, g Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
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 r r d L L Q dt q q ∂ ∂ − = ∂ ∂ & with L=T-V and r N q x y z r r x r y z Q F F F q q q ∂ ∂ ∂ = + + ∂ ∂ ∂ • For non-conservative forces that are a function of , there is an alternative approach. Consider generalized forces q& 1 ( , ) n N i ij j Q c q = = −∑ &j t q where the are the damping coefficients, which are dissipative in nature Î result in a loss of energy ij c • Now define the Rayleigh dissipation function 1 1 1 2 n n ij i j i j F c = = = ∑∑ q& &q Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 7
16.61 Aerospace Dynamics Spring 2003 Then we can show that aF ∑ qry q So that we can rewrite Lagrange's equations in the slightly cleaner form (aL OL aF 0 In the example of the block moving on the wedge d/ aL OL aF +o.=mq+kg-mg sin 8+cq dt a where O' now only accounts for the friction force 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) 8 • Then we can show that 1 r r n N q j j q r j F c q Q q = ∂ = = − ∂ ∑ & & • So that we can rewrite Lagrange's equations in the slightly cleaner form 0 r r r d L L F dt q q q ∂ ∂ ∂ − + = ∂ ∂ ∂ & & • In the example of the block moving on the wedge, 1 2 2 F = cq& sin r q d L L F mq kq mg cq Q dt q q q ∂ ∂ ∂ − + = + − + = ′ ∂ ∂ ∂ && & & & θ where r Qq ′ now only accounts for the friction force
