16.61 Aerospace Dynamics Spring 2003 Lecture #9 rtual Work And the Derivation of Lagrange's Equations Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Lecture #9 Virtual Work And the Derivation of Lagrangeís Equations Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 1
16.61 Aerospace Dynamics Spring 2003 Derivation of lagrangian equations Basic Concept: Virtual Work Consider system of N particles located at(, x2, x,,.x3N )with 3 forces per particle(f. f, f..fn). each in the positive direction 5 4 F (x4,x5,x6) 2 N-2 FN. N-2 Assume system given small, arbitrary displacements in all directions Called virtual displacements No passage of time Applied forces remain constant Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Derivation of Lagrangian Equations Basic Concept: Virtual Work Consider system of N particles located at ( x x 1 2 , , x3,…x3N ) ) with 3 forces per particle ( 1 2 3 3 , , , F F F …F N , each in the positive direction. F3 F2 F1 (x1, x2, x3) F5 F6 F4 (x4, x5, x6) xi xj xk FN FN-1 FN-2 (xN-2, xN-1, xN) F3 F2 F1 (x1, x2, x3) F5 F6 F4 (x4, x5, x6) xi xj xk FN FN-1 FN-2 (xN-2, xN-1, xN) Assume system given small, arbitrary displacements in all directions. Called virtual displacements - No passage of time - Applied forces remain constant Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 2
16.61 Aerospace Dynamics Spring 2003 The work done by the forces is termed virtual work W=∑F6x Note use of &x and not dx Note There is no passage of time e The forces remain constant In vector form 6W=∑F·or Virtual displacements MUSt satisfy all constraint relationships d Constraint forces do no work Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 The work done by the forces is termed Virtual Work. 3 1 N j j j δW F δ x = = ∑ Note use of δx and not dx. Note: • There is no passage of time • The forces remain constant. In vector form: 3 1 i i i δW δ = = ∑F r • Virtual displacements MUST satisfy all constraint relationships, ! Constraint forces do no work. Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 3
16.61 Aerospace Dynamics Spring 2003 Example: Two masses connected by a rod ○ R Constraint forces R1=-R2=-R2 Now assume virtual displacements Or, and Sr-but the displacement components along the rigid rod must be equal, so there is a constraint equation of the form Yi Ⅴ irtual work: W=R1·or1+R2·r2 R2e e·r1+R2e·r2 (R2-R2)·or So the virtual work done of the constraint forces is zero Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 Example: Two masses connected by a rod l R1 R2 àr e l R1 R2 àr e Constraint forces: 1 2 à R R = − = −R e2 r Now assume virtual displacements δ 1r , and δ 2 r - but the displacement components along the rigid rod must be equal, so there is a constraint equation of the form 1 2 e r e r r •δ = r •δ Virtual Work: ( ) 1 1 2 2 2 1 2 2 2 1 à à à 0 R r R r r r r r r r W R e R e R R e = • + • = − • + • = − • = 2 δ δ δ δ δ δ So the virtual work done of the constraint forces is zero Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 4
16.61 Aerospace Dynamics Spring 2003 This analysis extends to rigid body case Rigid body is a collection of masses e Masses held at a fixed distance i Virtual work for the internal constraints of a rigid bod displacement is zero Example: Body sliding on rigid surface without friction Since the surface is rigid and fixed, Or=0, >w=0 For the body, SW=R.Or, but the direction of the virtual displacement that satisfies the constraints is perpendicular to the constraint force. Thus sW=0 Massachusetts Institute of Technology C How, Deyst 2003 (Based on Notes by Blair 2002)
16.61 Aerospace Dynamics Spring 2003 This analysis extends to rigid body case • Rigid body is a collection of masses • Masses held at a fixed distance. ! Virtual work for the internal constraints of a rigid body displacement is zero. Example: Body sliding on rigid surface without friction Since the surface is rigid and fixed, 0, 0 s δ r W = → = δ For the body, δW = • R1 δ 1 W r , but the direction of the virtual displacement that satisfies the constraints is perpendicular to the constraint force. Thus δ = 0. Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002) 5