Three-Dimensional Kinematics of Rigid Bodies

Learning Objectives

  • Classify the different types of rigid body motion in three dimensions.
  • Understand and apply Euler's Rotation Theorem.
  • Describe 3D orientation using Euler Angles (Precession, Nutation, and Spin).
  • Formulate the angular velocity and angular acceleration as 3D spatial vectors.
  • Apply relative motion equations for velocity and acceleration in 3D.
  • Calculate acceleration in a rotating frame of reference, including the Coriolis acceleration.

The kinematics of rigid bodies in three dimensions extends the principles of two-dimensional (planar) motion. While planar motion is restricted to a single plane, 3D motion allows for rotation about any axis in space and translation in any direction. This requires the use of spatial vectors and a more general formulation of angular velocity and angular acceleration.

Civil Engineering Applications

Although many civil engineering dynamics problems can be simplified to 2D, 3D kinematics is required for complex spatial analysis:

  • Heavy Crane Operations: Modeling the translation, boom rotation, and payload swing simultaneously in three dimensions.
  • Seismic Structural Analysis: Evaluating multidirectional twisting and torsional vibrations of asymmetric buildings during an earthquake.
  • Wind Turbine Dynamics: Analyzing the complex spatial rotation of blades under varying wind loads.

Types of Rigid Body Motion in 3D

Translation

Motion where all points of the body have the same velocity and acceleration. The body's orientation remains constant in space.

Rotation about a Fixed Axis

Motion where the body rotates about a line that is fixed in space. The angular velocity vector is directed along this axis.

Rotation about a Fixed Point

Motion where the body rotates about a single stationary point. The instantaneous axis of rotation passes through this point but can change direction over time.

General Motion

A combination of translation and rotation. Any displacement can be modeled as a translation of a base point followed by a rotation about an axis passing through that point.

Euler's Theorem

A foundational principle for understanding 3D rotation is Euler's theorem, which describes the general displacement of a rigid body with one point fixed.

Euler's Rotation Theorem

Euler's Theorem states that any displacement of a rigid body such that a single point OO on the body remains fixed in space is equivalent to a single rotation about some axis that passes through the fixed point OO.

This means that instead of thinking of a sequence of complex rotations (e.g., pitch, roll, yaw), we can mathematically describe the final orientation as one distinct rotation through an angle θ\theta about a single spatial axis n\mathbf{n}. This unique axis is called the axis of rotation, and the corresponding vector ω\mathbf{\omega} is always aligned with it at any given instant.

Euler Angles

When a rigid body undergoes general 3D rotation, its orientation in space is often described using three successive rotations about different axes, known as Euler angles. These angles provide a systematic way to define the rotational position of a body-fixed frame (x,y,z)(x,y,z) relative to a stationary, fixed reference frame (X,Y,Z)(X,Y,Z).

Euler Angles (Z-X'-Z'' Sequence)

A common convention to define successive rotations: Precession (ϕ\phi), Nutation (θ\theta), and Spin (ψ\psi).

Successive Rotations Components

  • Precession (ϕ\phi): First rotation about the fixed ZZ-axis. The angular velocity component is ϕ˙K\dot{\phi} \mathbf{K}.
  • Nutation (θ\theta): Second rotation about the intermediate (new) x′x'-axis (often called the line of nodes). The angular velocity component is θ˙i′\dot{\theta} \mathbf{i}'.
  • Spin (ψ\psi): Third rotation about the final (body-fixed) zz-axis. The angular velocity component is ψ˙k\dot{\psi} \mathbf{k}.

Total Angular Velocity from Euler Angles

The total angular velocity of the rigid body is the vector sum of the three angular velocity components.

ω=ϕ˙K+θ˙i′+ψ˙k\mathbf{\omega} = \dot{\phi} \mathbf{K} + \dot{\theta} \mathbf{i}' + \dot{\psi} \mathbf{k}

Variables

SymbolDescriptionUnit
ω\mathbf{\omega}Total angular velocity vectorrad/s
ϕ˙K\dot{\phi} \mathbf{K}Precession angular velocity componentrad/s
θ˙i′\dot{\theta} \mathbf{i}'Nutation angular velocity componentrad/s
ψ˙k\dot{\psi} \mathbf{k}Spin angular velocity componentrad/s

Angular Velocity and Angular Acceleration Vectors

In 3D kinematics, the angular velocity ω\mathbf{\omega} and angular acceleration α\mathbf{\alpha} are spatial vectors representing the rotation of the rigid body.

Angular Velocity Vector

Representation of angular velocity in Cartesian coordinates.

ω=ωxi+ωyj+ωzk\mathbf{\omega} = \omega_x \mathbf{i} + \omega_y \mathbf{j} + \omega_z \mathbf{k}

Variables

SymbolDescriptionUnit
ω\mathbf{\omega}Angular velocity vectorrad/s
ωx,ωy,ωz\omega_x, \omega_y, \omega_zScalar components of angular velocityrad/s
i,j,k\mathbf{i}, \mathbf{j}, \mathbf{k}Unit vectors in x, y, and z directions-

Angular Acceleration Vector

Representation of angular acceleration as the time derivative of angular velocity.

α=dωdt=ω˙xi+ω˙yj+ω˙zk\mathbf{\alpha} = \frac{d\mathbf{\omega}}{dt} = \dot{\omega}_x \mathbf{i} + \dot{\omega}_y \mathbf{j} + \dot{\omega}_z \mathbf{k}

Variables

SymbolDescriptionUnit
α\mathbf{\alpha}Angular acceleration vectorrad/s²
ω\mathbf{\omega}Angular velocity vectorrad/s
ω˙x,ω˙y,ω˙z\dot{\omega}_x, \dot{\omega}_y, \dot{\omega}_zTime derivatives of angular velocity componentsrad/s²

Relative Motion Analysis

When analyzing the general motion of a rigid body, a translating frame of reference attached to a base point A is commonly used. The velocity and acceleration of any other point B on the body can be expressed relative to A.

Relative Velocity Equation

Relates the velocity of point B to the velocity of point A.

vB=vA+ω×rB/A\mathbf{v}_B = \mathbf{v}_A + \mathbf{\omega} \times \mathbf{r}_{B/A}

Variables

SymbolDescriptionUnit
vB\mathbf{v}_BVelocity of point Bm/s
vA\mathbf{v}_AVelocity of base point Am/s
ω\mathbf{\omega}Angular velocity of the bodyrad/s
rB/A\mathbf{r}_{B/A}Position vector of B with respect to Am

Relative Acceleration Equation

Relates the acceleration of point B to the acceleration of point A in 3D.

aB=aA+α×rB/A+ω×(ω×rB/A)\mathbf{a}_B = \mathbf{a}_A + \mathbf{\alpha} \times \mathbf{r}_{B/A} + \mathbf{\omega} \times (\mathbf{\omega} \times \mathbf{r}_{B/A})

Variables

SymbolDescriptionUnit
aB\mathbf{a}_BAcceleration of point Bm/s²
aA\mathbf{a}_AAcceleration of base point Am/s²
α\mathbf{\alpha}Angular acceleration of the bodyrad/s²
ω\mathbf{\omega}Angular velocity of the bodyrad/s
rB/A\mathbf{r}_{B/A}Position vector of B with respect to Am

Cross Products in 3D

Unlike planar motion where ω×(ω×rB/A)\mathbf{\omega} \times (\mathbf{\omega} \times \mathbf{r}_{B/A}) reduces to −ω2rB/A-\omega^2 \mathbf{r}_{B/A}, in 3D motion this term must be evaluated strictly using the vector cross product, as ω\mathbf{\omega} and rB/A\mathbf{r}_{B/A} are not necessarily perpendicular.

Rotating Frames of Reference and the Coriolis Effect

Sometimes it is convenient to use a coordinate system that is rotating with respect to a fixed (Newtonian) frame. For example, analyzing the motion of an object relative to the rotating Earth.

Coriolis Acceleration

When a particle moves within a rotating frame of reference, it experiences an apparent acceleration from the perspective of an observer in the fixed frame, known as the Coriolis acceleration. It is responsible for the rotation of weather systems and the deflection of long-range projectiles on Earth.

Acceleration in a Rotating Frame

Calculates the absolute acceleration of a particle P observed from a rotating frame.

aP=aA+Ω˙×rP/A+Ω×(Ω×rP/A)+2Ω×(vP/A)xyz+(aP/A)xyz\mathbf{a}_P = \mathbf{a}_A + \dot{\mathbf{\Omega}} \times \mathbf{r}_{P/A} + \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r}_{P/A}) + 2\mathbf{\Omega} \times (\mathbf{v}_{P/A})_{xyz} + (\mathbf{a}_{P/A})_{xyz}

Variables

SymbolDescriptionUnit
aP\mathbf{a}_PAbsolute acceleration of point Pm/s²
aA\mathbf{a}_AAbsolute acceleration of the origin of the rotating framem/s²
Ω\mathbf{\Omega}Angular velocity of the rotating framerad/s
Ω˙\dot{\mathbf{\Omega}}Angular acceleration of the rotating framerad/s²
rP/A\mathbf{r}_{P/A}Position of P relative to origin Am
(vP/A)xyz(\mathbf{v}_{P/A})_{xyz}Velocity of P relative to the rotating framem/s
(aP/A)xyz(\mathbf{a}_{P/A})_{xyz}Acceleration of P relative to the rotating framem/s²
2Ω×(vP/A)xyz2\mathbf{\Omega} \times (\mathbf{v}_{P/A})_{xyz}Coriolis acceleration componentm/s²

Coriolis Effect Simulation

Interactive Simulation

Interact with the simulation below to explore the Coriolis effect.

Coriolis Effect — Rotating Reference Frame

Controls

CCW (−)CW (+)
Coriolis Deflection Preview
Trajectory previewFixedRotating

Coriolis Equations

Rotating Frame Acceleration:
aabs=arel+Ω˙×r+Ω×(Ω×r)+2Ω×vrel\mathbf{a}_{\text{abs}} = \mathbf{a}_{\text{rel}} + \dot{\boldsymbol{\Omega}} \times \mathbf{r} + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r}) + 2\boldsymbol{\Omega} \times \mathbf{v}_{\text{rel}}
Coriolis Term:
aCor=2Ω×vrel\mathbf{a}_{\text{Cor}} = 2\boldsymbol{\Omega} \times \mathbf{v}_{\text{rel}}
∣aCor∣=2×1.5×2.0=6.00 m/s2|\mathbf{a}_{\text{Cor}}| = 2 \times 1.5 \times 2.0 = 6.00\,\text{m/s}^2
Fixed observer: particle moves straight
Rotating observer: particle curves
🌍 Fixed (Inertial) Observer

Fixed: Particle travels in straight line; disc rotates under it.

Rotating: Particle appears to deflect — this is the Coriolis pseudo-force.

Drag to orbit view.

Key Takeaways
  • Euler's Theorem: Any displacement with a fixed point can be defined by a single rotation around a single axis passing through that point.
  • Angular Velocity Vector: ω=ωxi+ωyj+ωzk\mathbf{\omega} = \omega_x\mathbf{i} + \omega_y\mathbf{j} + \omega_z\mathbf{k} is required to describe 3D rotation.
  • Acceleration Equation: Requires evaluating ω×(ω×r)\mathbf{\omega} \times (\mathbf{\omega} \times \mathbf{r}) sequentially using cross products, unlike planar 2D assumptions.
  • Euler Angles (Precession, Nutation, Spin) are a common way to specify orientation.
  • Rotating frames introduce the Coriolis acceleration, 2Ω×vrel2\mathbf{\Omega} \times \mathbf{v}_{rel}, which depends on the velocity of the particle relative to the rotating frame.
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