Engineering Mechanics – Dynamics Unit 5 Review: Rigid Body Kinematics

Key Concepts and Definitions

• Rigid body a solid object where the distance between any two points remains constant, regardless of external forces applied
• Kinematics the study of motion without considering the forces causing the motion
• Degrees of freedom (DOF) the number of independent parameters needed to fully describe the motion of a rigid body
  • In 3D space, a rigid body has 6 DOF (3 translational and 3 rotational)
• Translation motion where all points in a rigid body move along parallel paths
• Rotation motion where a rigid body turns about a fixed axis or point
• Angular displacement $(\theta)$ the angle through which a rigid body rotates, measured in radians or degrees
• Angular velocity $(\omega)$ the rate of change of angular displacement with respect to time, typically expressed in rad/s or deg/s
• Angular acceleration $(\alpha)$ the rate of change of angular velocity with respect to time, usually measured in rad/s² or deg/s²

Types of Motion in Rigid Bodies

• Pure translation all points in the rigid body undergo the same displacement, velocity, and acceleration
• Pure rotation the rigid body rotates about a fixed axis, with no translational motion
• General plane motion a combination of translation and rotation in a single plane
  • Can be described using 3 DOF (2 translational and 1 rotational)
• General spatial motion a combination of translation and rotation in 3D space
  • Requires 6 DOF (3 translational and 3 rotational) for complete description
• Rolling motion a special case of general plane motion where the rigid body rolls without slipping
• Sliding motion a special case of general plane motion where the rigid body slides without rotation
• Screw motion a combination of rotation about an axis and translation along the same axis

Reference Frames and Coordinate Systems

• Reference frame a set of axes used to describe the position, orientation, and motion of a rigid body
• Fixed (inertial) reference frame a non-accelerating frame, typically attached to the Earth or a stationary object
• Moving (non-inertial) reference frame a frame that translates or rotates relative to a fixed frame
• Cartesian coordinate system a rectangular coordinate system with mutually perpendicular axes (x, y, z)
• Cylindrical coordinate system a 3D coordinate system using radius (r), angle (θ), and height (z)
• Spherical coordinate system a 3D coordinate system using radius (r), polar angle (θ), and azimuthal angle (φ)
• Body-fixed coordinate system a coordinate system attached to and moving with the rigid body
• Space-fixed coordinate system a coordinate system that remains stationary relative to the fixed reference frame

Angular Velocity and Acceleration

• Angular velocity vector $(\vec{\omega})$ a vector representing the instantaneous angular velocity of a rigid body
  • Direction of $\vec{\omega}$ is along the axis of rotation, following the right-hand rule
  • Magnitude of $\vec{\omega}$ is the angular speed, $|\vec{\omega}| = \sqrt{\omega_x^2 + \omega_y^2 + \omega_z^2}$
• Angular acceleration vector $(\vec{\alpha})$ a vector representing the instantaneous angular acceleration of a rigid body
  • Direction of $\vec{\alpha}$ indicates the axis about which the angular velocity is changing
  • Magnitude of $\vec{\alpha}$ is the rate of change of angular speed, $|\vec{\alpha}| = \sqrt{\alpha_x^2 + \alpha_y^2 + \alpha_z^2}$
• Euler's equations of motion relate the angular velocity, angular acceleration, and moments acting on a rigid body
• Rotational kinetic energy $(\text{KE}_\text{rot}) = \frac{1}{2} I \omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular speed
• Angular momentum $(\vec{L}) = I \vec{\omega}$, a vector quantity describing the rotational motion of a rigid body

Relative Motion Analysis

• Relative position the position of one point with respect to another point or reference frame
• Relative velocity the velocity of one point with respect to another point or reference frame
  • $\vec{v}_{B/A} = \vec{v}_B - \vec{v}_A$, where $\vec{v}_{B/A}$ is the velocity of B relative to A
• Relative acceleration the acceleration of one point with respect to another point or reference frame
  • $\vec{a}_{B/A} = \vec{a}_B - \vec{a}_A$, where $\vec{a}_{B/A}$ is the acceleration of B relative to A
• Coriolis acceleration $(\vec{a}_C) = 2 \vec{\omega} \times \vec{v}_{rel}$, an apparent acceleration experienced by an object moving in a rotating reference frame
• Centripetal acceleration $(\vec{a}_{cp}) = \omega^2 \vec{r}$, the acceleration directed towards the center of rotation
• Tangential acceleration $(\vec{a}_t) = \alpha \vec{r}$, the acceleration tangent to the circular path of rotation

Instantaneous Center of Zero Velocity

• Instantaneous center of zero velocity (IC) a point in a rigid body or its extension that has zero velocity at a given instant
• Kennedy's theorem states that the IC of a rigid body in general plane motion lies at the intersection of lines drawn perpendicular to the velocities of any two points in the body
• The IC can be used to simplify the analysis of rigid body motion by treating the motion as pure rotation about the IC
• The location of the IC changes with time as the motion of the rigid body changes
• The concept of IC is particularly useful in analyzing rolling motion and gears

Applications in Engineering

• Vehicle dynamics analyzing the motion of cars, trucks, and other vehicles
  • Stability, handling, and ride comfort are important considerations
• Robotics designing and controlling the motion of robotic arms and manipulators
  • Inverse kinematics and trajectory planning rely on rigid body kinematics principles
• Aerospace engineering studying the motion of aircraft, spacecraft, and satellites
  • Attitude control and orbital mechanics involve rigid body kinematics concepts
• Biomechanics analyzing the motion of human body segments and joints
  • Gait analysis and prosthetic design benefit from understanding rigid body kinematics
• Machine design creating mechanisms and machines that perform desired motions
  • Cam-follower systems and linkages are examples of rigid body kinematics applications

Problem-Solving Strategies

• Identify the type of motion (pure translation, pure rotation, general plane motion, or general spatial motion)
• Choose an appropriate reference frame and coordinate system
• Draw free-body diagrams and kinetic diagrams to visualize forces and motions
• Use vector algebra and calculus to express position, velocity, and acceleration relationships
• Apply the relevant kinematic equations and constraints
  • Position: $\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2} \vec{a} t^2$ (translational) or $\theta(t) = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$ (rotational)
  • Velocity: $\vec{v}(t) = \vec{v}_0 + \vec{a} t$ (translational) or $\omega(t) = \omega_0 + \alpha t$ (rotational)
• Use relative motion analysis when multiple reference frames are involved
• Consider the instantaneous center of zero velocity for general plane motion problems
• Double-check units and perform dimensional analysis to ensure consistency