Report

Dynamic Simulation: Lagrange’s Equation Objective The objective of this module is to derive Lagrange’s equation, which along with constraint equations provide a systematic method for solving multi-body dynamics problems. © 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 Calculus of Variations Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 2 Problems in dynamics can be formulated in such a way that it is necessary to find the stationary value of a definite integral. Lagrange (1736-1813) created the Calculus of Variations as a method for finding the stationary value of a definite integral. He was a self taught mathematician who did this when he was nineteen. Euler (1707-1783) used a less rigorous but completely independent method to do the same thing at about the same time. They were both trying to solve a problem with constraints in the field of dynamics. © 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 Euler and Lagrange Module 6 – Lagrange’s Equation Page 3 Leonhard Euler Joseph-Louis Lagrange 1707-1783 1736-1813 http://en.wikipedia.org/wiki/Leonhard_Euler http://en.wikipedia.org/wiki/Lagrange © 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 Hamilton’s Principle Module 6 – Lagrange’s Equation Page 4 Hamilton’s Principle states that the path followed by a mechanical system during some time interval is the path that makes the integral of the difference between the kinetic and the potential energy stationary. t2 A Ldt t1 L=T-V is the Lagrangian of the system. T and V are respectively the kinetic and potential energy of the system. The integral, A, is called the action of the system. © 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 Principle of Least Action Module 6 – Lagrange’s Equation Page 5 Hamilton’s Principle is also called the “Principle of Least Action” since the paths taken by components in a mechanical system are those that make the Action stationary. t2 A Ldt Action t1 © 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 Stationary Value of an Integral Module 6 – Lagrange’s Equation Page 6 The application of Hamilton’s Principle requires that we be able to find the stationary value of a definite integral. We will see that finding the stationary value of an integral requires finding the solution to a differential equation known as the Lagrange equation. We will begin our derivation by looking at the stationary value of a function, and then extend these concepts to finding the stationary value of an integral. © 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 Stationary Value of a Function Module 6 – Lagrange’s Equation Page 7 A function is said to have a “stationary value” at a certain point if the rate of change of the function in every possible direction from that point vanishes. y y=f(x) In this example, the function has a stationary y1 point at x=x1. At this point, its first derivative is equal to zero. x1 © 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 x Education Community 3D Stationary Points Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 8 In 3D the rate of change of the function in any direction is zero at a stationary point. Note that the stationary point is not necessarily a maximum or a minimum. © 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 Variation of a Function Module 6 – Lagrange’s Equation Page 9 g x f x x (x) is an arbitrary function that satisfies the boundary conditions at a and b. y y g x Candidate Path g(x) can be made infinitely close to f(x) by making the parameter infinitesimally small. dy y=f(x) dx Actual Path a © 2011 Autodesk dy 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 x+dx www.autodesk.com/edcommunity b x Education Community Meaning of dy Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 10 The Calculus of Variations considers a virtual infinitesimal change of function y = f(x). y y g x dy dy The variation dy refers to an arbitrary infinitesimal change of the value of y at the point x. The independent variable x does not participate in the process of variation. © 2011 Autodesk y=f(x) dx a 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 x+dx www.autodesk.com/edcommunity b x Education Community Section 4 – Dynamic Simulation Variation of a Derivative Module 6 – Lagrange’s Equation Page 11 In the calculus of variations, the derivative of the variation and the variation of the derivative are equal. Derivative of the Variation d d y dx d dx d dx d g x f x x dy dx d x dx dx d dy dx dg x d x dx d d y © 2011 Autodesk Variation of the Derivative dx df x dx d x dx The order of operation is interchangeable. 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 Variation of a Definite Integral Module 6 – Lagrange’s Equation Page 12 In the calculus of variations, the variation of a definite integral is equal to the integral of the variation. Integral of a Variation Variation of an Integral d b b b a a f x dx g x dx f x dx b b d f x dx g x f x dx a a a b b g x f x dx x dx a x dx a d © 2011 Autodesk b a b b a a f x dx d f x dx 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. The order of operation is interchangeable. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Specific Definite Integral Module 6 – Lagrange’s Equation Page 13 The specific definite integral that we want to find the stationary value of is the Action from Hamilton’s Principle. It can be written in functional form as t2 A t1 L q i , q i , t n L T q t V q t i i i 1 qi are the generalized coordinates used to define the position and orientation of each component in the system. The actual path that the system will follow will be the one that makes the definite integral stationary. © 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 Euler-Lagrange Equation Derivation Module 6 – Lagrange’s Equation Page 14 The stationary value of an integral is found by setting its variation equal to zero. L L d L q i , q i , t L q i i , q i i , t L q i , q i , t i i q i qi A first order Taylor’s Series was used in the last step. t2 dA d t1 t2 t2 L L Ldt d Ldt i i dt 0 q i qi t1 t1 t2 For an arbitrary value of , © 2011 Autodesk L L q i q i dt 0 i i t1 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 Euler-Lagrange Equation Derivation Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 15 The second integral is integrated by parts. Integration by Parts Substitutions t2 L L q i q i dt 0 i i t1 t2 t1 t2 L L i dt i q i q i t1 t2 t1 d L dt q i d uv udv vdu i dt is equal to zero at t1 and t2. L d L q dt q i i t1 t2 © 2011 Autodesk udv d uv vdu u L q i v i dt 0 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 Euler-Lagrange Equation Derivation Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 16 L d L q dt q i i t1 t2 d L q i dt q i L i dt 0 The only way that this definite integral can be zero for arbitrary values of i is for the partial differential equation in parentheses to be zero at all values of x in the interval t1 to t2. 0 or d L dt q i © 2011 Autodesk L 0 q i Lagrange’s equation 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 Euler-Lagrange Summary Module 6 – Lagrange’s Equation Page 17 Finding the stationary value of the Action, A, for a mechanical system involves solving the set of differential equations known as Lagrange’s equation. Solving these equations d L dt q i L q 0 i n L T q t V q t i i i 1 Makes this integral stationary t2 A L q , q , t i i t1 © 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 Examples Module 6 – Lagrange’s Equation Page 18 Although the derivation of Lagrange’s equation that provides a solution to Hamilton’s Principle of Least Action, seems abstract, its application is straight forward. Using Lagrange’s equation to derive the equations of motion for a couple of problems that you are familiar with will help to introduce their application. © 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 Vibrating Spring Mass Example Module 6 – Lagrange’s Equation Page 19 Governing Equations Mathematical Operations d L dt q i L q 0 i L T V T 1 m y y y m V ky k 2 y is measured from the static position. © 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 y 1 ky 2 2 m y d L dt y 2 2 m y 2 2 L 2 1 L 1 m y ky Equation of Motion m y ky 0 www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Falling Mass Example Module 6 – Lagrange’s Equation Page 20 Governing Equations d L dt q i Mathematical Operations L L q 0 i L y T 1 m y 2 y g 2 m y d L dt y L 2 V mgy m y mgy 2 m L T V 1 y m y mg x Equation of Motion m y mg 0 © 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 6 – Lagrange’s Equation Page 21 Lagrange’s equation has been derived from Hamilton’s Principle of Least Action. Finding the stationary value of a definite integral requires the solution of a differential equation. The differential equation is called “Lagrange’s equation” or the “EulerLagrange equation” or “Lagrange’s equation of motion.” Lagrange’s equation will be used in the next module (Module 7) to establish a systematic method for finding the equations that control the motion of mechanical 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. www.autodesk.com/edcommunity Education Community