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Dynamic Simulation: Lagrangian Multipliers Objective The objective of this module is to introduce Lagrangian multipliers that are used with Lagrange’s equation to find the equations that control the motion of mechanical systems having constraints. The matrix form of the equations used by computer programs such as Autodesk Inventor’s Dynamic Simulation are also presented. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Basic Problem in Multi-body Dynamics Module 7 – Lagrangian Multipliers Page 2 In the previous module (Module 6) we developed Lagrange’s equation and showed how it could be used to determine the equations of simple motion systems. Lagrange’s Equation d L L 0 dt qi qi The examples we considered were for systems in which there were no constraints between the generalized coordinates. The basic problem of multi-body dynamics is to systematically find and solve the equations of motion when there are constraints that bodies in the system must satisfy. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Non-conservative Forces Module 7 – Lagrangian Multipliers Page 3 The derivation of Lagrange’s equation in the previous module (Module 6) considered only processes that store and release potential energy. These processes are called conservative because they conserve energy. Lagrange’s equation must be modified to accommodate nonconservative processes that dissipate energy (i.e. friction, damping, and external forces). A non-conservative force or moment acting on generalized coordinate qi is denoted as Qi. The more general form of Lagrange’s equation is d L L Qi dt qi qi © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Simple Pendulum Module 7 – Lagrangian Multipliers Page 4 The pendulum shown in the figure will be used as an example throughout this module. Simple Pendulum Y The position of the pendulum is known at any instance of time if the coordinates of the c.g., Xcg,Ycg, and the angle q are known. y Ycg x θ c.g. X Xcg,Ycg and q are the generalized coordinates. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Xcg www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Kinetic and Potential Energies Page 5 The kinetic energy (T) and potential energy (V) of the pendulum are 1 2 1 2 1 2 T Iq mX cg mYcg 2 2 2 V mgYcg These equations also give the kinetic and potential energy of the unconstrained body flying through the air. There needs to be a way to include the constraints to differentiate between the two systems. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Y y Ycg x θ c.g. X Xcg Unconstrained Body www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Constraint Equations Module 7 – Lagrangian Multipliers Page 6 In addition to satisfying Lagrange’s equations of motion, the pendulum must satisfy the constraints that the displacements at X1 and Y1 are zero. y Ycg The constraint equations are X cg sin q X 1 0 2 Ycg cos q Y1 0 2 © 2011 Autodesk X1,Y1 Y Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. x θ c.g. X Xcg The c.g. lies on the yaxis halfway along the length . www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Lagrangian Multipliers Module 7 – Lagrangian Multipliers Page 7 The kinetic energy is augmented by adding the constraint equations multiplied by parameters called Lagrangian Multipliers. Y1 X1 Y Note that since the constraint equations are equal to zero, we have not changed the magnitude of the kinetic energy. The Lagrangian multipliers are treated like unknown generalized coordinates. What are the units of l1 and l2? © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. θ X 1 2 1 2 1 2 Iq mX cg mYcg 2 2 2 l1 X cg sin q X 1 2 l2 Ycg cos q y1 2 T www.autodesk.com/edcommunity Education Community Governing Equations Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 8 In the following slides, Lagrange’s equation will be used in a systematic manner to determine the equations of motion for the pendulum. The governing equations that will be used are shown here. There are no non-conservative forces acting on the system ( Qi 0 ). Lagrange’s Equation d L dt qi L Qi q i Lagrangian n L Ti Vi i 1 1 2 1 2 1 2 L Iq mX cg mYcg l1 X cg sin q X 1 l2 Ycg cos q Y1 mgYcg 2 2 2 2 2 © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Equation for 1st Section 4 – Dynamic Simulation Generalized Coordinate Module 7 – Lagrangian Multipliers Page 9 1 2 1 2 1 2 L Iq mX cg mYcg l1 X cg sin q X 1 l2 Ycg cosq Y1 m gYcg 2 2 2 2 2 Lagrange’s Equation d L dt qi L Qi q i Mathematical Steps L mX cg q1 Generalized Coordinates q1 X cg q2 Ycg q3 q q4 l1 d L mXcg dt q1 L l1 q1 1st Equation mXcg l1 0 q5 l2 © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Equation for 2nd Generalized Coordinate Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 10 1 2 1 2 1 2 L Iq mX cg mYcg l1 X cg sin q X 1 l2 Ycg cosq Y1 m gYcg 2 2 2 2 2 Lagrange’s Equation d L dt qi L Qi q i Mathematical Steps L mYcg q 2 d L mYcg dt q 2 Generalized Coordinates L l2 mg q2 q1 X cg q2 Ycg q3 q q4 l1 2nd Equation mYcg l2 mg 0 q5 l2 © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Equation for 3rd Generalized Coordinate Module 7 – Lagrangian Multipliers Page 11 1 2 1 2 1 2 L Iq mX cg mYcg l1 X cg sin q X 1 l2 Ycg cosq Y1 m gYcg 2 2 2 2 2 Lagrange’s Equation d L dt qi L Qi q i Generalized Coordinates q1 X cg q2 Ycg q3 q q4 l1 q5 l2 © 2011 Autodesk Mathematical Steps L Iq q3 d L Iq dt q3 L l1 cos q l2 sin q q3 2 2 3rd Equation Iq l1 cos q l2 sin q 0 2 2 Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Equation for 4th Generalized Coordinate Module 7 – Lagrangian Multipliers Page 12 1 2 1 2 1 2 L Iq mX cg mYcg l1 X cg sin q X 1 l2 Ycg cosq Y1 m gYcg 2 2 2 2 2 Lagrange’s Equation d L dt qi L Qi q i Generalized Coordinates q1 X cg q2 Ycg q3 q q4 l1 q5 l2 © 2011 Autodesk Mathematical Steps L 0 q 4 d L 0 dt q 4 L X cg sin q X 1 q4 2 4th Equation X cg sin q X 1 0 2 Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Equation for 5th Generalized Coordinate Module 7 – Lagrangian Multipliers Page 13 1 2 1 2 1 2 L Iq mX cg mYcg l1 X cg sin q X 1 l2 Ycg cosq Y1 m gYcg 2 2 2 2 2 Lagrange’s Equation d L dt qi L Qi q i Generalized Coordinates q1 X cg q2 Ycg Mathematical Steps L 0 q5 d L 0 dt q5 L Ycg cos q Y1 q5 2 5th Equation q3 q q4 l1 q5 l2 © 2011 Autodesk Ycg cos q Y1 0 2 Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Summary of Equations Module 7 – Lagrangian Multipliers Page 14 There are five unknown generalized coordinates including the two Lagrangian Multipliers. There are also five equations. Three of the equations are differential equations. Two of the equations are algebraic equations. Combined, they are a system of differential-algebraic equations (DAE). © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. l 0 mX cg 1 mYcg l2 mg 0 Iq l1 cos q l2 sin q 0 2 2 X cg sin q X 1 0 2 Ycg cos q Y1 0 2 www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Free Body Diagram Approach Module 7 – Lagrangian Multipliers Page 15 Summation of Forces in the X-direction l 0 mX cg 1 λ2 Summation of Forces in the Y-direction λ1 mYcg l2 mg 0 Summation of Moments about the c.g. cos q 2 Iq l1 cos q l2 sin q 0 2 2 The application of Lagrange’s equation yields the same equations obtained by drawing a free-body diagram. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. q mYcg 2 sin q 2 mXcg Iθcg mg Free Body Diagram with Inertial Forces www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Physical Significance of Lagrangian Multipliers Module 7 – Lagrangian Multipliers Page 16 Newton’s 2nd Law in x-direction mXcg l1 0 Force required to impose the constraint that X1 is a constant. Lagrangian Multipliers are simply the forces (moments) required to enforce the constraints. In general, the Lagrangian Multipliers are a function of time, because the forces (moments) required to enforce the constraints vary with time (i.e. depend on the position of the pendulum). © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Matrix Format Module 7 – Lagrangian Multipliers Page 17 The computer implementation of Lagrange’s equation is facilitated by writing the equations in matrix format. Separating the Lagrangian into kinetic and potential energy terms enables Lagrange’s equation to be written as d T T V Qi dt qi qi qi In this format, the conservative and non-conservative forces are lumped together on the right hand side of the equation. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Matrix Format Module 7 – Lagrangian Multipliers Page 18 The kinetic energy augmented with Lagrangian Multipliers can be written in matrix format as 1 T T T q M q l q, t 2 q Column array containing generalized coordinate velocities. q, t Column array containing the constraint equations (refer to Module 3 in this section). l M © 2011 Autodesk Column array containing the Lagrangian multipliers. Inertia Matrix m A M 0 0 0 0 mA 0 0 0 I cgA 0 0 0 0 0 Matrix containing the mass and mass moments of inertia associated with each generalized coordinate. Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Matrix Format Module 7 – Lagrangian Multipliers Page 19 Lagrange’s equation for a mechanical system becomes i M q l Q q j T i Is the constraint equation Jacobian matrix introduced in Module 4 q j in this section. Q © 2011 Autodesk Column array containing both conservative and non-conservative forces. Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Matrix Format Module 7 – Lagrangian Multipliers Page 20 Another equation for acceleration was obtained in Module 4 based on kinematics and the constraint equations. i q q j Combining this equation with Lagrange’s equation from the previous slide yields: © 2011 Autodesk Matrix Form of Equations M i q j i q j q Q 0 l This equation can be solved to find the accelerations and constraint forces at an instant in time. The accelerations must then be integrated to find the velocities and positions. Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Solution of Differential-Algebraic Equations (DAE) Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 21 The solution of even the simplest system of DAE requires computer programs that employ predictor-corrector type numerical integrators. The Adams-Moulton method is an example of the type of numerical method used. Significant research has led to the development of efficient and robust integrators that are found in commercial computer programs that generate, assemble, and solve these equations. Autodesk Inventor’s Dynamic Simulation environment is an example of such software. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Module Summary Module 7 – Lagrangian Multipliers Page 22 This module showed how Lagrangian Multipliers are used in conjunction with Lagrange’s equation to obtain the equations that control the motion of mechanical systems. The method presented provides a systematic method that forms the basis of mechanical simulation programs such as Autodesk Inventor’s Dynamic Simulation environment. The matrix format of the equations were presented to provide insight into the computations performed by computer software. The Jacobian and constraint kinematics developed in Module 4 of this section are an important part of the matrix formulation. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community