1 Mechanical Systems Translation  Point mass concept  P  P(t) = F(t)*v(t)  Newton’s Laws & Free-body diagrams Rotation  Rigid body concept  P  P(t) = T(t)*w(t)  Newton’s laws & Free-body diagrams Transducer devices and effects

2 Mechanical rotation Newton’s Laws (applied to rotation)  Every body persists in a state of uniform (angular) motion, except insofar as it may be compelled by torque to change that state.  The time rate of change of angular momentum is equal to the torque producing it.  To every action there is an equal and opposite reaction. (Principia Philosophiae, 1686, Isaac Newton)

3 Quantities and SI Units “F-L-T” system  Define F: force [N]  Define L: length [m]  Define T: time [s] Derive  T: torque (moment) [N-m]  M: mass [kg]  w: angular velocity [rad/s]  J: mass moment of inertia [kg-m^2]

4 Physical effects and engineered components Inertia effect - rigid body with mass in rotation Compliance (torsional stiffness) effect – torsional spring Dissipation (rotational friction) effect – torsional damper System boundary conditions: u motion conditions – angular velocity specified u torque conditions - drivers and loads

5 Rotational inertia Physical effect:  r^2*  *dV Engineered device: rigid body “mass” Standard schematic icon (stylized picture) Standard multiport representation Standard icon equations

6 Rigid body in fixed-axis rotation: standard form J T1T2 w 1 I:J w T1T2

7 Compliance (torsional stiffness) Physical effects:  =E*  Engineered devices: torsional spring Standard schematic icon Standard multiport representation Standard icon equations

8 Torsional compliance 0 C w1 w2 T

9 Dissipation (torsional resistance) Physical effects Engineered devices: torsional damper Standard schematic icons Standard multiport representation Standard icon equations

10 Torsional resistance 0 R w1 w2 T

11 Free-body diagrams Purpose: Develop a systematic method for generating the equations of a mechanical system. Setup method: Separate the mechanical schematic into standard components and effects (icons); generate the equation(s) for each icon. Standard form of equations: the composite of all component equations is the initial system set; select a reduced set of key variables (generalized coordinates); reduce the initial equation set to a set in these variables.

12 Multiport modeling of mechanical translation Multiport representations of the standard icons: focus on power ports Equations for the standard icons Multiport modeling using the free-body approach

13 Multiport modeling of fixed-axis rotation based on free-body diagrams Identify each rotating rigid body. Define an inertial angular velocity for each. Use a standard multiport component to represent each rotating rigid body (with or without mass). Write the standard equation(s) for each component.

14 Example 1: torsional system