Rigid-Body Dynamics

Below are selected topics from rigid-body dynamics, a subtopic of classical mechanics involving the use of Newton's laws of motion to solve for the motion of rigid bodies moving in 1D, 2D, or 3D space.^B.11 We may think of a rigid body as a distributed mass, that is, a mass that has length, area, and/or volume rather than occupying only a single point in space. Rigid body models have application in stiff strings (modeling them as disks of mass interconnect by ideal springs), rigid bridges, resonator braces, and so on.

We have already used Newton's to formulate mathematical dynamic models for the ideal point-mass (§B.1.1), spring (§B.1.3), and a simple mass-spring system (§B.1.4). Since many physical systems can be modeled as assemblies of masses and (normally damped) springs, we are pretty far along already. However, when the springs interconnecting our point-masses are very stiff, we may approximate them as rigid to simplify our simulations. Thus, rigid bodies can be considered mass-spring systems in which the springs are so stiff that they can be treated as rigid massless rods (infinite spring-constants , in the notation of §B.1.3).

So, what is new about distributed masses, as opposed to the point-masses considered previously? As we will see, the main new ingredient is rotational dynamics. The total momentum of a rigid body (distributed mass) moving through space will be described as the sum of the linear momentum of its center of mass (§B.4.1 below) plus the angular momentum about its center of mass (§B.4.13 below).

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Subsections

• Center of Mass
  • Linear Momentum of the Center of Mass
  • Whoops, No Angular Momentum!
  
  
• Translational Kinetic Energy
• Rotational Kinetic Energy
• Mass Moment of Inertia
  • Circular Disk Rotating in Its Own Plane
  • Circular Disk Rotating About Its Diameter
  
  
• Perpendicular Axis Theorem
• Parallel Axis Theorem
• Stretch Rule
• Area Moment of Inertia
• Radius of Gyration
  • Rectangular Cross-Section
  • Circular Cross-Section
  
  
• Two Masses Connected by a Rod
  • Striking the Rod in the Middle
  • Striking One of the Masses
  
  
• Angular Velocity Vector
• Vector Cross Product
  • Cross-Product Magnitude
  • Mass Moment of Inertia as a Cross Product
  • Tangential Velocity as a Cross Product
  
  
• Angular Momentum
  • Relation of Angular to Linear Momentum
  
  
• Angular Momentum Vector
  • Angular Momentum Vector in Matrix Form
  
  
• Mass Moment of Inertia Tensor
  • Simple Example
  • Example with Coupled Rotations
  • Off-Diagonal Terms in Moment of Inertia Tensor
  
  
• Principal Axes of Rotation
  • Positive Definiteness of the Moment of Inertia Tensor
  
  
• Rotational Kinetic Energy Revisited
• Torque
• Newton's Second Law for Rotations
• Equations of Motion for Rigid Bodies
  • Body-Fixed and Space-Fixed Frames of Reference
  • Angular Motion in the Space-Fixed Frame
  • Euler's Equations for Rotations in the Body-Fixed Frame
  • Examples

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Previous: Conservation of Momentum
Next: Center of Mass

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.