MEAM 535 Principle of Virtual Work Aristotle Galileo (1594) Bernoulli (1717) Lagrange (1788) 1. Start with static equilibrium of holonomic system of N particles 2. Extend to rigid bodies 3. Incorporate inertial forces for dynamic analysis 4. Apply to nonholonomic systems University of Pennsylvania 25 MEAM 535 D’Alembert’s principle D’Alembert’s principle for a single particle of mass m subject to a resultant external force F with the acceleration a is written in the form: F - m a = 0 (-ma) is regarded as a fictitious external force called the inertial force The inertial force can be regarded as an applied force in the principle of Virtual Work University of Pennsylvania 26 MEAM 535 Recap: Principle of Virtual Work A system of N particles (P1, P2,…, PN) is in static equilibrium if and only if the virtual work done by all the applied (active) forces though any (arbitrary) virtual displacement is zero. A holonomic system of N particles is in static equilibrium if and only if all the generalized (active) forces are zero. Only “applied” or “active” forces contribute to the generalized force The jth generalized force is given by University of Pennsylvania 27 MEAM 535 Lagrange-D’Alembert Equations for Holonomic Systems Apply the Principle of Virtual Work to a system of particles Fi - mi ai = 0 (-miai), the inertial force, is an applied force acting on the ith particle N (a) ∑(Fi − mia i )⋅δri = 0 i=1 Holonomic active generalized forces € Holonomic generalized active inertial forces N * Pi Q j = ∑(−mia i )⋅v j , j =1, 2,…, n i=1 University of Pennsylvania 28 € MEAM 535 Kane’s equations for holonomic systems The motion of a holonomic system with N particles and n speeds, is governed by n equations of motion given by: Advantages Principle of virtual work (Galileo, Bernouli) Ignore reaction forces and focus on the forces and moments that do work D’Alembert’s principle Now incorporate inertial forces Kane’s equation: One equation of motion for each degree of freedom University of Pennsylvania 29 MEAM 535 Example 1 Generalized speed: l G=2τ/r B u=dθ/dt Q r φ θ m F Generalized Active Forces P No friction, gravity Only the piston has mass x F = -Fa1 F = − Kinematic Analysis € Need Partial Velocities University of Pennsylvania 30 MEAM 535 Example 2 Find the equation(s) of motion for the system. r φ θ y l linear elastic P springs, constant k, free length, d m c a2 a1 University of Pennsylvania 31 MEAM 535 Example 3 Find the equation(s) of motion for the three degree of freedom arm in the horizontal plane if the links are massless but the motors have masses mi. Neglect frictional forces. There are m2 no external forces acting on m the arm. 3 m1 University of Pennsylvania 32 .