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D'alembert's Principle Video 2.1a Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Introduction to Lagrangian Mechanics Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Analytical Mechanics • Aristotle • Euler • Galileo • Lagrange • Bernoulli • D’Alembert 1. Principle of Virtual Work: Static equilibrium of a particle, system of N particles, rigid bodies, system of rigid bodies 2. D’Alembert’s Principle: Incorporate inertial forces for dynamic analysis 3. Lagrange’s Equations of Motion Robo3x-1.3 3 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Generalized Coordinate(s) A minimal set of coordinates required to describe the configuration of a system No. of generalized coordinates = no. of degrees of freedom Robo3x-1.3 4 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Displacements Virtual displacements are small displacements consistent with the constraints g y A particle of mass m constrained to move vertically generalized coordinate y Robo3x-1.3 5 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Displacements Virtual displacements are small displacements consistent with the constraints l f q x The slider crank mechanism: a single degree of freedom linkage generalized coordinate x, q, or f Robo3x-1.3 6 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Displacements Virtual displacements are small displacements consistent with the constraints O q l P The pendulum: a single degree of freedom linkage generalized coordinate q Robo3x-1.3 7 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Work The work done by applied (external) forces through the virtual displacement, mg A particle of mass m constrained to move vertically Robo3x-1.3 8 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Work The work done by applied (external) forces through the virtual displacement, l f q F x The slider crank mechanism: a single degree of freedom linkage Robo3x-1.3 9 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Video 2.1b Vijay Kumar and Ani Hsieh Robo3x-1.3 10 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Work The work done by applied (external) forces through the virtual displacement, O q l P mg The pendulum: a single degree of freedom linkage Robo3x-1.3 11 Property of Penn Engineering, Vijay Kumar and Ani Hsieh The Principle of Virtual Work The virtual work done by all applied (external) forces through any virtual displacement is zero The system is in equilibrium Robo3x-1.3 12 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Static Equilibrium mg Robo3x-1.3 13 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Static Equilibrium l f q F x Robo3x-1.3 14 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Static Equilibrium O q l P mg Robo3x-1.3 15 Property of Penn Engineering, Vijay Kumar and Ani Hsieh D’Alembert’s Principle The virtual work done by all applied (external) forces through any virtual displacement is zero include inertial principle of virtual work forces The system is in static equilibrium Equations of motion for the system inertial force = - mass x acceleration Robo3x-1.3 16 Property of Penn Engineering, Vijay Kumar and Ani Hsieh D’Alembert’s Principle acceleration inertial force e2 mg e1 Robo3x-1.3 17 Property of Penn Engineering, Vijay Kumar and Ani Hsieh D’Alembert’s Principle acceleration O q inertial force l e P q mg er equation of motion Robo3x-1.3 18 Property of Penn Engineering, Vijay Kumar and Ani Hsieh D’Alembert’s principle with generalized coordinates generalized coordinate contribution contribution from from inertial external force(s) force(s) Particle in the vertical plane Simple pendulum Robo3x-1.3 19 Property of Penn Engineering, Vijay Kumar and Ani Hsieh The Key Idea The contribution from the inertial forces can be expressed as a function of the kinetic energy and its derivatives Kinetic Energy Robo3x-1.3 20 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Lagrange’s Equation of Motion Particle in the vertical plane Simple pendulum Robo3x-1.3 21 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Lagrange’s Equation of Motion for a Conservative System Conservative System There exists a scalar function such that all applied forces are given by the gradient of the potential function Scalar Function Gravitational force is conservative Scalar function is the potential energy Robo3x-1.3 22 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Standard Form of Lagrange’s Equation of Motion The Lagrangian Robo3x-1.3 23 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Lagrange’s Equation of Motion Particle in the vertical plane Simple pendulum Robo3x-1.3 24 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Video 2.2 Vijay Kumar and Ani Hsieh Robo3x-1.3 25 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Newton Euler Equations Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 26 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Analytical Mechanics 1. Principle of Virtual Work: Static equilibrium of a particle, system of N particles, rigid bodies, system of rigid bodies 2. D’Alembert’s Principle: Incorporate inertial forces for dynamic analysis 3. Lagrange’s Equations of Motion Robo3x-1.3 27 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Newton Euler Equations Recall Newton’s 2nd law of motion m net external force = inertia x acceleration Robo3x-1.3 28 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Newton Euler Equations Newton’s equations of motion for a translating rigid body Euler’s equations of motion for a rotating rigid body Robo3x-1.3 29 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Motion of Systems of Particles fj Center of Mass y Pi m i C Pi fj rOP mj i x O z Newton’s equations of motion net external force = total mass x Robo3x-1.3 30 accelerationProperty of Penn Engineering, of Vijay Kumarcenter and Ani Hsieh of mass Newton’s Second Law for a System of Particles The center of mass for a system of particles, S, accelerates in an inertial frame, A, as if it were a single particle with mass m (equal to the total mass of the system) acted upon by a force equal to the net external force. Linear momentum Rate of change of linear momentum in an inertial frame is equal to the net external force acting on the system. Also true for a rigid body Robo3x-1.3 31 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Equations of Motion for a Rotating Rigid Body • Rigid body with a point O fixed in an inertial frame • Rigid body with the center of mass C Angular momentum O C y P x Robo3x-1.3 32 Property of Pennmoment Engineering, Vijay of Kumar inertia and Ani Hsieh Equations of Motion for a Rotating Rigid Body • Rigid body with a point O fixed in an inertial frame • Rigid body with the center of mass C The rate of change of angular momentum of a rigid body (or a system of rigidly connected particles) in an inertial frame with O or C as an origin is equal to the net external moment acting (with the same origin) on the body. Robo3x-1.3 33 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Example O q l e2 P mg e1 equation of motion Robo3x-1.3 34 Property of Penn Engineering, Vijay Kumar and Ani Hsieh 67 KB Moment of Inertia of Planar Objects e2 m 2b m 2a C e O 1 m m Robo3x-1.3 35 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Moment of Inertia of Planar Objects dm r C y x Robo3x-1.3 36 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Video 2.3 Vijay Kumar and Ani Hsieh Robo3x-1.3 37 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Dynamics and Control Acceleration Analysis: Revisited Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 38 Property of Penn Engineering, Vijay Kumar and Ani Hsieh b2 Position Vectors b1 • Reference frame Q b3 P • Origin B • Basis vectors • Position Vectors • Position vectors for P and Q in A a a2 1 O • Position vector of Q in B A a3 Robo3x-1.3 39 Property of Penn Engineering, Vijay Kumar and Ani Hsieh General Approach to Analyzing Multi-Body System 2. Pair of points fixed to the same 1. Points common to body adjacent bodies 3. Pair of points on adjacent bodies whose relative motions can be easily described 4. Write equations relating pairs of points either on same or adjacent bodies. Robo3x-1.3 40 Property of Penn Engineering, Vijay Kumar and Ani Hsieh b2 Position Analysis b1 Q b3 P B a a2 1 O A a3 Robo3x-1.3 41 Property of Penn Engineering, Vijay Kumar and Ani Hsieh b Velocity Analysis 2 b1 Q b3 P B 3x3 skew symmetric matrix a a2 1 O A a3 Robo3x-1.3 42 Propertyangular of Penn Engineering, velocity Vijay Kumar and of Ani HsiehB in A Acceleration Analysis • Acceleration of P and Q in A Q P B a a2 1 O A a3 Robo3x-1.3 43 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Angular Acceleration The angular acceleration of B in Q A, is defined as the derivative of P the angular velocity of B in A: B a a2 1 O A a3 Robo3x-1.3 44 Property of Penn Engineering, Vijay Kumar and Ani Hsieh b Acceleration Analysis 2 b1 Q b3 P B a a1 tangential 2 acceleration O A a3 centripetal (normal) acceleration Robo3x-1.3 45 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Serial Chain of Rigid Bodies OD C OC D B OB A OA Robo3x-1.3 46 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Serial Chain of Rigid Bodies OD C OC D B OB A OA Robo3x-1.3 47 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Video 2.4 Vijay Kumar and Ani Hsieh Robo3x-1.3 48 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Newton Euler Equations (continued) Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 49 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Newton Euler Equations Newton’s equations of motion A rigid body B accelerates in an inertial frame A as if it were a single particle with the same mass m (equal to the total mass of the system) acted upon by a force equal to the net external force.
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