Mechanical Systems

1
Mechanical Systems

• Translation
• Point mass concept
• P(t) F(t)v(t)
• Newtons Laws Free-body diagrams
• Rotation
• Rigid body concept
• P(t) T(t)w(t)
• Newtons laws Free-body diagrams
• Transducer devices and effects

2
Mechanical rotation

• Newtons 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-m2

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
• motion conditions angular velocity specified
• torque conditions - drivers and loads

5
Rotational inertia

• Physical effect ?r2?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
T1
T2
7
Compliance (torsional stiffness)

• Physical effects ?E?
• Engineered devices torsional spring
• Standard schematic icon
• Standard multiport representation
• Standard icon equations

8
Torsional compliance
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
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.

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Example 1 torsional system