Analytical Dynamics

Taught by: Dr. Riccardo Bevilacqua
Taken: FS 2015
Text: Dynamics of Particles and Rigid Bodies: A Systematic Approach
by: Anil Rao

Table of Contents

Kinematics

Kinetics of Particles

Kinematics and Kinetics of Rigid Bodies/Systems of Particles

Analytical Dynamics

Scalars

Vectors

Reference Frames

Coordinate System

Vector Derivatives

Transport Theorem

Transport Theorem Derivation

Cylindrical Coordinate Systems

Cylindrical Coordinate System

Spherical Coordinate Systems

Spherical Coordinate System

Euler Angles

Euler Angle First Rotation
Euler Angle Second Rotation
Euler Angle Third Rotation

Limitations of Euler Angles

Intrinsic Coordinates

Motion Between Two Points in the Same Reference Frame

Rolling and Slipping

Inertial Reference Frames

Three Laws of Mechanics

  1. The first law of mechanics or the inertia law states that an object at rest tends to remain at rest and an object in motion tends to remain in motion
  2. The second law of mechanics relates forces to momentum as it's most common form shows (\underline{F} = m ^{N}\underline{a} where ^{N}\underline{a} is the acceleration in an inertial reference frame)
  3. The last of the three laws of mechanics states that for every action there is an equal and opposite reaction
    • If the resultant force lies on the same line of action as the initial force (sliding vectors) then it is considered the strong form of the 3rd law
    • The weak form of the third law is when the resultant force and the initial force are not on the same line of action

Angular Momentum

Relative Velocity

Friction Force Models

Coulomb Friction

Viscous Friction

Linear Spring Force Model

Center of Mass

Tensors

Tensor Product

Angular Momentum of Rigid Bodies/Systems of Particles

Pure Torque

Euler's Laws

First Law

Second Law

Parameterize a Problem

Lagrange Equations

Kinetic Energy of a Particle

Kinetic Energy for a System of Particles

Kinetic Energy for a Rigid Body (Koenig Decomposition)

Examples

Examples Content

Kinematics

Kinetics

Kinetics for a System of Particles

Kinetics for Rigid Bodies

Transport Theorem Examples

  • Given a disk (D) rotating about its centerpoint (O), find the velocity and acceleration of point P as seen by the disk (D) and the ground (G)
Example 1, Diagram 1
  • Start by defining reference frames and then coordinate systems within them
  • For this problem two reference frames will be used: the disk (D) and the ground (G)
  • Coordinate system fixed in the disk reference frame (D)
    • Origin at point O
    • \underline{e}_{r} = along the line \underline{OP}
    • \underline{e}_{z} = out of the page (positive with theta)
    • \underline{e}_{\theta} = \underline{e}_{z} \times  \underline{e}_{r}
Disk Reference Frame
  • Coordinate system fixed in the ground reference frame (G)
    • Origin at point O
    • \underline{E}_{z} = \underline{e}_{z}
    • \underline{E}_{x} = along the line \underline{OP} @ \theta = 0
    • \underline{E}_{y} = \underline{E}_{z} \time \underline{E}_{x}
Ground Reference Frame
  • Now to find the velocity and acceleration of point P in the disk reference frame we take the time derivative of the vector from the origin O to point P (\underline{r} = \underline{OP} = r\underline{e}_{r})
    • ^{D}\underline{V}_{P} = \frac{^{D}d\underline{r}}{dt} =  \frac{^{D}d}{dt}(r\underline{e}_{r}) =  \frac{dr}{dt}\underline{e}_{r} + r\frac{^{D}d\underline{e}_{r}}{dt}
    • Neither r nor \underline{e}_{r} are changing in the disk reference frame therefore ^{D}\underline{V}_{P} = 0
  • Next the acceleration of point P in the disk reference frame is to be found by taking a time derivative of the velocity of point P in the disk reference frame
    • ^{D}\underline{a}_{P} = \frac{^{D}d}{dt}(^{D}\underline{V}_{P}) = 0
  • Now the velocity needs to be found for point P in the ground reference frame and two different approaches will be shown
  • First approach involves writing all of the equations with respect to the coordinate system fixed in reference frame G
    • \underline{r} = r\underline{e}_{r} = rcos(\theta)\underline{E}_{x} +  rsin(\theta)\underline{E}_{y}
    • ^{G}\underline{V}_{P} = \frac{^{G}d\underline{r}}{dt} =  -r\dot{\theta}sin(\theta)\underline{E}_{x} +  r\dot{\theta}cos(\theta)\underline{E}_{y}
  • The second approach involves leaving the equations with terms referenced in the disk reference frame and uses the transport theorem
    • ^{G}\underline{V}_{P} = (^{D}\underline{V}_{P}) +  ^{G}\underline{\omega}^{D} \times \underline{r}
      • ^{D}\underline{V}_{P} = 0
      • ^{G}\underline{\omega}^{D} =\dot{\theta}\underline{e}_{z}
      • \underline{r} = r\underline{e}_{r}
    • ^{G}\underline{V}_{P} = \dot{\theta}\underline{e}_{z} \times  r\underline{e}_{r} = \dot{\theta}r\underline{e}_{\theta}
  • The second approach leaves a much cleaner equation for further derivatives and so this is the form we'll use when finding the acceleration
    • ^{G}a_{P} = \frac{^{G}d}{dt}(^{G}\underline{V}_{P}) =  \frac{^{D}d}{dt}(^{G}\underline{V}_{P}) + ^{G}\underline{\omega}^{D}  \times ^{G}\underline{V}_{P} = \ddot{\theta}r\underline{e}_{\theta}  + \dot{\theta}\underline{e}_{z} \times  \dot{\theta}r\underline{e}_{\theta} =  \ddot{\theta}r\underline{e}_{\theta} -  \dot{\theta}^{2}r\underline{e}_{r}
  • This example is the same as the previous one execept it has an additional rotation \beta around its base
  • It's good practice to have an additional reference frame for each rotation and so in this example we'll use three reference frames: disk (D), disk at \theta = 0 (C) and the ground (G)
Example 2, Diagram 1
  • Always start by defining the reference frames and then the coordinate systems within them
  • Coordinate system fixed in the disk reference frame (D)
    • Origin at point O
    • \underline{u}_{r} = along \underline{OP}
    • \underline{u}_{z} = perpendicular to the disk, positive with \theta
    • \underline{u}_{\theta} = \underline{u}_{z} \times  \underline{u}_{r}
Disk Reference Frame 2
  • Coordinate system fixed in the ground reference frame (G)
    • Origin at point O
    • \underline{E}_{z} = Out of the page
    • \underline{E}_{x} = along the line \underline{OP} @ \theta = 0, \beta = 0
    • \underline{E}_{y} = \underline{E}_{z} \times \underline{E}_{x}
Ground Reference Frame
  • Coordinate system fixed in the disk at \theta = 0 reference frame (C) (rotates with \beta)
    • Origin at point O
    • \underline{e}_{\theta} = \underline{E}_{y}
    • \underline{e}_{r} = along \underline{OP} at \theta = 0
    • \underline{e}_{z} = \underline{u}_{z}
Disk Reference Frame Fixed with Theta = 0
  • Just as in the previous example the velocity and acceleration in the disk reference frame (D) will equal zero
    • ^{D}\underline{V}_{P} = (^{D}\underline{a}_{P}) = 0
  • Now to find the velocity of point P in the ground reference frame we are going to take advantage of the transport theorem again
    • \underline{r} = r\underline{u}_{r}
    • ^{G}\underline{\omega}^{C} = \dot{\beta}\underline{e}_{\theta}
    • ^{C}\underline{\omega}^{D} = \dot{\theta}\underline{u}_{z}
    • ^{G}\underline{V}_{P} = \frac{^{G}d\underline{r}}{dt} =  \frac{^{D}d\underline{r}}{dt} + ^{G}\underline{\omega}^{D} \times  \underline{r}
      • ^{G}\underline{V}_{P} = ^{D}\underline{V}_{P} +  (^{G}\underline{\omega}^{C} + ^{C}\underline{\omega}^{D}) \times  \underline{r}
      • ^{G}\underline{V}_{P} = 0 + (\dot{\beta}\underline{e}_{\theta} +  \dot{\theta}\underline{u}_{z}) \times (r\underline{u}_{r})
      • ^{G}\underline{V}_{P} =  [\dot{\beta}(cos(\theta)\underline{u}_{\theta} +  sin(\theta)\underline{u}_{r}) + \dot{\theta}\underline{u}_{z}]  \times (r\underline{u}_{r})
      • ^{G}\underline{V}_{P} =  -\dot{\beta}cos(\theta)r\underline{u}_{z} +  \dot{\theta}r\underline{u}_{\theta}

Cylindrical Coordinate System Example

Cylindrical Coordinate System
Cylindrical Coordinate System Example

Spherical Coordinate System Example

Spherical Coordinate System
Cylindrical Coordinate System Example
Spherical Coordinate System

Intrinsic Coordinate System Example

Intrinsic Coordinate System

Rolling Motion

Roll and Slip Example
Roll and Slip Example Ground Reference Frame

Basic Equation of Motion

Basic Equations of Motion Example
Basic Equations of Motion Example Free Body Diagram

Angular Momentum

Angular Momentum Example
Angular Momentum Example Free Body Diagram

Incomplete

Friction Force Models

Friction Force Models Example

Incomplete

Linear Spring Force Models

Linear Spring Model Example

Incomplete

Basic System of Particles

Basic System of Particles Example
Wedge FBD
Basic System of Particles Wedge FBD
Block FBD
Basic System of Particles Block FBD
System FBD
Basic System of Particles System FBD

Angular Momentum

Basic Rigid Body Problem

Gyroscope Example

Pendulum Example