ME451 Kinematics and Dynamics of Machine Systems Initial Conditions for Dynamic Analysis Constraint Reaction Forces October 23, 2013 Radu Serban University of Wisconsin-Madison

2 Before we get started… Last Time: Derived the variational EOM for a planar mechanism Introduced Lagrange multipliers Formed the mixed differential-algebraic EOM Today Slider-crank example – derivation of the EOM Initial conditions for dynamics Recovering constraint reaction forces Assignments: Homework 8 – – due today Matlab 6 and Adams 4 – due today, (11:59pm) Miscellaneous No lecture on Friday (Undergraduate Advising Day) Draft proposals for the Final Project due on Friday, November 1

3 Lagrange Multiplier Form of the EOM Equations of Motion Position Constraint Equations Velocity Constraint Equations Acceleration Constraint Equations Most Important Slide in ME451

4 Mixed Differential-Algebraic EOM

5 Slider-Crank Example (1/5)

6 Slider-Crank Example (2/5)

7 Slider-Crank Example (3/5) Constrained Variational Equations of Motion Condition for consistent virtual displacements

8 Slider-Crank Example (4/5) Lagrange Multiplier Form of the EOM Constraint EquationsAcceleration EquationVelocity Equation

9 Slider-Crank Example (5/5) Mixed Differential-Algebraic Equations of Motion Constraint EquationsVelocity Equation

Initial Conditions 6.3.4

11 The Need for Initial Conditions Informally, consider an ordinary differential equation with 2 states The differential equation specifies a “velocity” field in 2D An IC specifies a starting point in 2D Solving the IVP simply means finding a curve in 2D that starts at the specified IC and is always tangent to the local velocity field

12 Another example

13 ICs for the EOM of Constrained Planar Systems

14 Specifying Position ICs (1/2)

15 Specifying Position ICs (2/2)

16 Specifying Velocity ICs (1/2)

17 Specifying Velocity ICs (2/2)

18 Specifying ICs in simEngine2D

19 Initial Conditions: Conclusions

20 ICs for a Simple Pendulum [handout]

Constraint Reaction Forces 6.6

22 Reaction Forces Remember that we jumped through some hoops to get rid of the reaction forces that develop in joints Now, we want to go back and recover them, since they are important: Durability analysis Stress/Strain analysis Selecting bearings in a mechanism Etc. The key ingredient needed to compute the reaction forces in all joints is the set of Lagrange multipliers

23 Reaction Forces: The Basic Idea Recall the partitioning of the total force acting on the mechanical system Applying a variational approach (principle of virtual work) we ended up with this equation of motion After jumping through hoops, we ended up with this: It’s easy to see that

24 Reaction Forces: Important Observation

25 Reaction Forces: Framework

26 Reaction Forces: Main Result

27 Reaction Forces: Comments

28 Reaction Forces: Summary A joint (constraint) in the system requires a (set of) Lagrange multiplier(s) The Lagrange multiplier(s) result in the following reaction force and torque An alternative expression for the reaction torque is

29 Reaction force in a Revolute Joint [Example 6.6.1]