ME451 Kinematics and Dynamics of Machine Systems Generalized Forces 6.2 October 16, 2013 Radu Serban University of Wisconsin-Madison

2 Before we get started… Last Time: Discussed Principle of Virtual Work and D’Alembert’s Principle Introduced centroidal reference frames Derived the Newton-Euler EOM for a single rigid body Today: Inertia properties Generalized forces Constrained variational EOM (variational EOM for a planar mechanism) Miscellaneous: Matlab 5 and Adams 3 – due tonight, (11:59pm) Homework 7 – (6.1.1, 6.1.2, 6.1.3, 6.1.4) – due Friday, October 18 (12:00pm) Matlab 6 and Adams 4 – due October 23, (11:59pm) Solutions to Midterm Problems – available on the course page Student feedback responses – available on the SBEL course page Monday (October 21) lecture – simEngine2D discussion – in EH 2261

3 Roadmap: Check Progress What have we done so far? Derived the variational and differential EOM for a single rigid body These equations are general but they must include all forces applied on the body These equations assume their simplest form in a centroidal RF What is left? Properties of the polar moment of inertia Define a general strategy for including external forces Treatment of constraint forces Derive the variational and differential EOM for systems of constrained bodies

Properties of the Centroid and Polar Moment of Inertia Inertial Properties of Composite Bodies 6.1.4, 6.1.5

5 Location of the Center of Mass (1/2)

6 Location of the Center of Mass (2/2) For a rigid body, the COM is fixed with respect to the body If the body has constant density, the COM coincides with the centroid of the body shape If the rigid body has a line of symmetry, then the COM is somewhere along that axis Notes: Here, symmetry axis means that both mass and geometry are symmetric with respect to that axis If the rigid body has two axes of symmetry, the centroid is on each of them, and therefore is at their intersection

7 Polar Moment of Inertia Parallel Axis Theorem Jakob Steiner (1796– 1863)

8 Masses, centroid locations, and PMI for rigid bodies with constant density and of simple shapes can be easily calculated Question: how do we calculate these quantities for bodies made up of rigidly attached subcomponents? Step 1: Calculate the total body mass Step 2: Compute the centroid location of the composite body Step 3: For each subcomponent, apply the parallel axis theorem to include the PMI of that subcomponent with respect to the newly computed centroid, to obtain the PMI of the composite body Note: if holes are present in the composite body, it is ok to add and subtract material (this translates into positive and negative mass) Inertial Properties of Composite Bodies

9 Roadmap: Check Progress

Virtual Work and Generalized Force 6.2

11 Including Concentrated Forces (1/3)

12 Including Concentrated Forces (2/3)

13 Including Concentrated Forces (3/3)

14 Including a Point Force (1/2)

15 Including a Point Force (2/2)

16 Including a Torque

17 Tractor Model [Example 6.1.1]

18 (TSDA) Translational Spring-Damper-Actuator (1/2)

19 (TSDA) Translational Spring-Damper-Actuator (2/2) Note: tension defined as positive Hence the negative sign in the virtual work

20 (RSDA) Rotational Spring-Damper-Actuator (1/2)

21 (RSDA) Rotational Spring-Damper-Actuator (2/2) Note: tension defined as positive Hence the negative sign in the virtual work

22 Generalized Forces: Summary Integral manipulations (use rigid-body assumptions) Redefine in terms of generalized forces and virtual displacements Explicitly identify virtual work of generalized forces Virtual work of generalized external forces Virtual work of generalized inertial forces D’Alembert’s Principle effectively says that, upon including a new external force, the body’s generalized accelerations must change to preserve the balance of virtual work. As such, to include a new force (or torque), we are interested in the contribution of this force on the virtual work balance.

23 Roadmap: Check Progress What have we done so far? Derived the variational and differential EOM for a single rigid body Defined how to calculate inertial properties Defined a general strategy for including external forces Concentrated (point) forces Forces from compliant elements (TSDA and RSDA) What is left? Treatment of constraint forces Derive the variational and differential EOM for systems of constrained bodies

Variational Equations of Motion for Planar Systems 6.3.1

25 Variational and Differential EOM for a Single Rigid Body

26 Matrix Form of the EOM for a Single Body Generalized Virtual Displacement (arbitrary) Generalized Mass Matrix Generalized Accelerations

27 [Side Trip] A Vector-Vector Multiplication Trick

28 Variational EOM for the Entire System (1/2)

29 Variational EOM for the Entire System (2/2) Generalized Force Generalized Virtual Displacement Generalized Mass Matrix Generalized Accelerations

30 A Closer Look at Generalized Forces

31 Constraint Forces Forces that develop in the physical joints present in the system: (revolute, translational, distance constraint, etc.) They are the forces that ensure the satisfaction of the constraints (they are such that the motion stays compatible with the kinematic constraints) KEY OBSERVATION: The net virtual work produced by the constraint forces present in the system as a result of a set of consistent virtual displacements is zero Note that we have to account for the work of all reaction forces present in the system This is the same observation we used to eliminate the internal interaction forces when deriving the EOM for a single rigid body Therefore provided  q is a consistent virtual displacement

32 Consistent Virtual Displacements

33 Constrained Variational EOM We can eliminate the (unknown) constraint forces if we compromise to only consider virtual displacements that are consistent with the constraint equations Arbitrary Consistent Constrained Variational Equations of Motion Condition for consistent virtual displacements