2.003SC Recitation 9 Notes: Generalized Forces with Double Pendulum Example Generalized Forces Consider a system with n generalized coordinates, qj , acted upon by N non-conservative forces, where • F i = non-conservative forces acting on the system • ri = positions of the points at which the non-conservative forces act • δri = virtual displacements of those points when generalized coordinates are varied Z O Y X Recall that, when a force goes through a displacement, W = F · dr = |F ||dr|cosθ the work done is given by since only the component of the force in the direction of the motion ”counts”. So the virtual work done by all the non-conservative forces is δW nc = N N F i · δri = N N i=1 Fi · n N ∂r i=1 j=1 i ∂qj δqj = n N N n N N ∂r ( F i · i )δqj = Qj δqj ∂q j j=1 i=1 j=1 And the generalized forces are given by Qj = N N i=1 Fi · ∂ri ∂qj j = 1, 2, ..., n Generalized Forces on Double Pendulum - Problem Statement A double pendulum is shown below with a force F acting at point P. The generalized coordinates are q1 = θ1 and q2 = θ2 . Determine the generalized forces for the system. Intermediate steps: • Write an expression for rP , (the location of the point at which the force is applied). • Write expressions for ∂ri ∂θ1 and ∂ri ∂θ2 . • Write an expression for the generalized forces, Qθ1 and Qθ2 . • Write an expression for the virtual work force F does on the system. Recall that the generalized forces are deﬁned by the following Qj = N � i=1 fi · ∂ri ∂qj j = 1, 2, ..., n where • n = number generalized coordinates, generalized forces • N = number of forces acting on the system • f i = forces acting on the system • ri = location of points at which forces act • qj = the system’s generalized coordinates Generalized Forces on Double Pendulum - Solution rP , which describes the location of point P , can be expressed as rP = (l1 sinθ1 + l2 sinθ2 )Iˆ + (−l1 cosθ1 − l2 cosθ2 )Jˆ So the variations are given by ∂ri = l1 cosθ1 Iˆ + l1 sinθ1 Jˆ ∂θ1 ∂ri = l2 cosθ2 Iˆ + l2 sinθ2 Jˆ ∂θ2 The non-conservative force, F can be expressed as follows. F = Fx Iˆ + Fy Jˆ So the generalized forces are Qθ1 = F · ∂ri = Fx l1 cosθ1 + Fy l1 sinθ1 ∂θ1 Qθ2 = F · ∂ri = Fx l2 cosθ2 + Fy l2 sinθ2 ∂θ2 The virtual work done on the system is δW nc = Qθ1 δθ1 + Qθ2 δθ2 δW nc = (Fx l1 cosθ1 + Fy l1 sinθ1 )δθ1 + (Fx l2 cosθ2 + Fy l2 sinθ2 )δθ2