© 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. Education Community 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. Education Community Calculus of Variations  Problems in dynamics can be formulated in such a way that it is necessary to find the stationary value of a definite integral.  Lagrange ( ) 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 ( ) 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. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 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. Education Community Euler and Lagrange Leonhard Euler Joseph-Louis Lagrange Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 3

© 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. Education Community Hamilton’s Principle 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. 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. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 4

© 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. Education Community Principle of Least Action 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. Action Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 5

© 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. Education Community Stationary Value of an Integral  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. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 6

© 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. Education Community Stationary Value of a Function  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.  In this example, the function has a stationary point at x=x 1. At this point, its first derivative is equal to zero. x y y=f(x) x1x1 y1y1 Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 7

© 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. Education Community 3D Stationary Points 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. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 8

© 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. Education Community Variation of a Function x y y=f(x) dy dx yy a b x x+dx   (x) is an arbitrary function that satisfies the boundary conditions at a and b.  g(x) can be made infinitely close to f(x) by making the parameter  infinitesimally small. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 9 Actual Path Candidate Path

© 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. Education Community Meaning of  y  The Calculus of Variations considers a virtual infinitesimal change of function y = f(x).  The variation  y 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. x y y=f(x) dy dx yy a b x x+dx Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 10

© 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. Education Community Variation of a Derivative In the calculus of variations, the derivative of the variation and the variation of the derivative are equal. Derivative of the VariationVariation of the Derivative The order of operation is interchangeable. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 11

© 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. Education Community Variation of a Definite Integral Variation of an Integral Integral of a Variation In the calculus of variations, the variation of a definite integral is equal to the integral of the variation. The order of operation is interchangeable. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 12

© 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. Education Community Specific Definite Integral 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 The actual path that the system will follow will be the one that makes the definite integral stationary. q i are the generalized coordinates used to define the position and orientation of each component in the system. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 13

© 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. Education Community Euler-Lagrange Equation Derivation A first order Taylor’s Series was used in the last step. The stationary value of an integral is found by setting its variation equal to zero. For an arbitrary value of , Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 14

© 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. Education Community Euler-Lagrange Equation Derivation Integration by Parts Substitutions The second integral is integrated by parts.  is equal to zero at t 1 and t 2. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 15

© 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. Education Community Euler-Lagrange Equation Derivation Lagrange’s equation The only way that this definite integral can be zero for arbitrary values of   is for the partial differential equation in parentheses to be zero at all values of x in the interval t 1 to t 2. or Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 16

© 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. Education Community Euler-Lagrange Summary 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 Makes this integral stationary Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 17

© 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. Education Community Examples  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. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 18

© 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. Education Community Vibrating Spring Mass Example Governing Equations Equation of Motion m k y Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 19 y is measured from the static position. Mathematical Operations

© 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. Education Community Falling Mass Example Governing Equations m y g x Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 20 Mathematical Operations Equation of Motion

© 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. Education Community Module Summary  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 “Euler- Lagrange 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. Section 4 – Dynamic Simulation Module 6 – Lagrange’s Equation Page 21