Dynamic Simulation : Lagrangian Multipliers

1. 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.

2. Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 2 Lagrange’s Equation 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. Basic Problem in Multi-body Dynamics 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.

3. Non-conservative Forces Section 4 – Dynamic Simulation 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 non-conservative 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

4. Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 4 • The pendulum shown in the figure will be used as an example throughout this module. • 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. • Xcg,Ycgand q are the generalized coordinates. Simple Pendulum Simple Pendulum Y y x Ycg c.g. X Xcg

5. Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 5 The kinetic energy (T) and potential energy (V) of the pendulum are 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. Y y x Ycg Kinetic and Potential Energies c.g. X Xcg Unconstrained Body

6. Section 4 – Dynamic Simulation 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. • The constraint equations are X1,Y1 Y y x Ycg Constraint Equations c.g. X Xcg The c.g. lies on the y-axis halfway along the length .

7. Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 7 • The kinetic energy is augmented by adding the constraint equations multiplied by parameters called Lagrangian Multipliers. • 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. Y1 X1 Y Lagrangian Multipliers X What are the units of l1 and l2?

8. 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 ( ). Lagrange’s Equation Governing Equations Lagrangian

9. Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 9 Lagrange’s Equation Mathematical Steps Equation for 1st Generalized Coordinate Generalized Coordinates 1st Equation

10. Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 10 Lagrange’s Equation Mathematical Steps Equation for 2ndGeneralized Coordinate Generalized Coordinates 2nd Equation

11. Equation for 3rd Generalized Coordinate Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 11 Lagrange’s Equation Mathematical Steps Generalized Coordinates 3rd Equation

12. Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 12 Lagrange’s Equation Mathematical Steps Equation for 4th Generalized Coordinate Generalized Coordinates 4th Equation

13. Equation for 5th Generalized Coordinate Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 13 Lagrange’s Equation Mathematical Steps Generalized Coordinates 5th Equation

14. Section 4 – Dynamic Simulation 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). Summary of Equations

15. Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 15 Summation of Forces in the X-direction Summation of Forces in the Y-direction Free Body Diagram Approach Summation of Moments about the c.g. The application of Lagrange’s equation yields the same equations obtained by drawing a free-body diagram. Free Body Diagram with Inertial Forces

16. Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 16 Newton’s 2nd Law in x-direction Physical Significance of Lagrangian Multipliers 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).

17. Section 4 – Dynamic Simulation 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 • In this format, the conservative and non-conservative forces are lumped together on the right hand side of the equation. Matrix Format

18. Matrix Format Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 18 The kinetic energy augmented with Lagrangian Multipliers can be written in matrix format as Column array containing generalized coordinate velocities. Column array containing the constraint equations (refer to Module 3 in this section). Column array containing the Lagrangian multipliers. Matrix containing the mass and mass moments of inertia associated with each generalized coordinate. Inertia Matrix

19. Matrix Format Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 19 Lagrange’s equation for a mechanical system becomes Is the constraint equation Jacobian matrix introduced in Module 4 in this section. Column array containing both conservative and non-conservative forces.

20. Matrix Format Section 4 – Dynamic Simulation Module 7 – Lagrangian Multipliers Page 20 • Another equation for acceleration was obtained in Module 4 based on kinematics and the constraint equations. • Combining this equation with Lagrange’s equation from the previous slide yields: Matrix Form of Equations • 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.

21. 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. • AutodeskInventor’s Dynamic Simulation environment is an example of such software. Solution of Differential-Algebraic Equations (DAE)

22. Module Summary Section 4 – Dynamic Simulation 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.