Advanced Rigid Body Mechanics in Three Dimensions

Name: Advanced Rigid Body Mechanics in Three Dimensions
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Mathematical intuition behind the fundamental equations in rigid body mechanics

Created byEmanuele Pesaresi

Last updated 3/2026

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What you'll learn

How to derive the fundamental equations on rigid body kinematics and dynamics
How to derive energy conservation from Newton's laws (Virtual Work Theorem)
Fundamental properties of the angular velocity of a rigid body

Course content

7 sections • 25 lectures • 5h 27m total length

Relation between velocities of points belonging to a rigid body15:05
Explain how a rigid body’s point velocity arises from a fixed absolute frame plus a body-fixed rotation, using a rotation matrix R and the angular velocity Omega.
Derivation of the angular velocity5:59
Derive the angular velocity vector Omega such that the velocity of point B equals Omega cross B. The lecture links this to a skew-symmetric matrix and determinant representation.
Angular velocity: application of the theory9:48
Demonstrates how a rotating disc produces tangential velocity at the circumference using angular velocity and v equals omega cross r. Connects 2d rotation matrices to the vector form of omega.
Angular velocity as a function of three angles3:49
Compute angular velocity as a function of three angles by composing rotations about the x, y, and z axes and differentiating the rotation matrix with respect to time.

Relative velocities and formula for accelerations12:26
Present the velocity of a point on a moving rigid body as v = v' + ω × r, and derive its acceleration including the Coriolis term.
Introduction to Mozzi (or Chasles) theorem4:53
Examine how a rigid body's instantaneous motion follows a motss axis parallel to the angular velocity, producing rotation about and translation along that axis—a helical motion.
Proof of Mozzi theorem11:53
Demonstrate the Mozzi theorem for a rigid body by showing a unique line, the Mozzi axis, along which all points share the same velocity, parameterized by lambda along omega.
Special cases of Mozzi theorem3:49
Examine special cases of rigid-body motion around the Mozzi axis, where zero along-axis velocity yields rotation, and vanishing angular velocity while velocities along the Mozzi axis persist yields translation.

First equation of motion11:53
Explore the dynamics of a general rigid body by linking distributed forces to the total external force, and derive that whole-body acceleration equals mass times the center of gravity's acceleration.
Highlighting the need for the equation of moments4:29
This lecture shows that in three-dimensional rigid body motion, the equation of moments supplies three equations and complements the center of mass position, velocity, and angular velocity descriptions.
Derivation of the equation of moments part 19:02
We derive the equation of moments by applying Newton's laws to a six-degree-of-freedom rigid body, summing infinitesimal forces and reactions, and using a cross product to relate angular momentum.
Derivation of the equation of moments part 27:05
Derive the equation of moments by integrating external moments over the system mass. Explore how reactions, center of gravity, and a six-equation, six-unknown framework influence the moment balance.
Euler rotation equations derivation part 13:14
Derive the three-dimensional rigid body moment equations by converting the mass integral to a volume integral, using center-of-mass coordinates and orientation angles with angular velocity concepts.
Euler rotation equations derivation part 212:20
Derive the external moment about the center of gravity using the inertia tensor (inertia matrix) and a cross product with angular velocity.

Derivation of the conservation of energy part 111:18
Explore how the rate of energy input relates to the kinetic energy of a rigid body, illustrating conservation of energy, moments, and the inertia matrix.
Derivation of the conservation of energy part 222:14
Derive torque and moments from forces using cross products and mixed products. Link power to torque and introduce the virtual work theorem for energy conservation.
Derivation of the conservation of energy part 310:29
Derive the virtual work theorem for a rigid body. Relate the sum of forces times velocity to the time derivative of kinetic energy and introduce the inertia matrix.
Kinetic energy of a rigid body20:33
Derives the kinetic energy of a rigid body in terms of angular velocity and the inertia matrix, then links it to the virtual work theorem and energy conservation.

Writing out the components of the equations of motion11:46
Derive the components of the rigid body equations of motion in three dimensions using a diagonal inertia tensor in principal axes, with two-dimensional simplifications and omega aligned with eigenvectors.
Stability analysis of the torque-free motion part 112:38
Explore torque-free motion of a rigid body with no external forces, analyzing stability of the angular velocity under various initial conditions and principal-axis alignments.
stability analysis of the torque-free motion part 218:39
Explore stability analysis of torque-free motion in three dimensions; derive elliptical relations for omega2 and omega3 under perturbations, and note that case three reveals instability along certain axes.

Mathematics and simulation of a four-bar linkage51:42
Analyze the motion of a four bar linkage, a one degree of freedom mechanism, using kinematic equations to express gamma and phi as functions of theta, illustrated with Matlab simulations.
Kinematically possible motion of a four-bar linkage0:14
Investigate the kinematically possible motion of a four-bar linkage within advanced rigid body mechanics in three dimensions.
Engineering problem on a four-bar linkage29:40
Analyze a four-bar parallelogram linkage with cranks and a connecting rod, and determine the torque needed to balance external force, gravity, and inertia using the virtual work principle.

Requirements

Newton laws of motion
Calculus, Multivariable Calculus (especially: derivatives, multiple integrals)
Vectors, dot products, cross products

Description

This course delves into the fundamental equations and concepts that govern the mechanics of rigid bodies. It provides a comprehensive and detailed derivation of all key equations, ensuring that students understand the underlying principles from first principles. To fully engage with the material, a solid understanding of the following mathematical concepts is required: vectors, dot and cross products, basic linear algebra (including matrices, determinants, eigenvectors, and eigenvalues), and essential calculus (with a focus on derivatives and volume integrals). On the physics side, the only prerequisite is familiarity with Newton's laws of motion, as they serve as the foundational framework for the entire course. Specifically, the laws governing point-particle dynamics (F = ma, where F is the total force acting on a particle, m is its mass, and a is its acceleration) are extended to construct the equations governing rigid body motion.

Throughout the course, we will derive the inertia matrix, which plays a crucial role in the equation of moments and in the expression of the kinetic energy of a rigid body. The concept of angular velocity will also be introduced, and its uniqueness will be demonstrated, setting the stage for a deeper understanding of rotational dynamics. Additionally, we will explore several important kinematic formulas that relate the velocities and accelerations of arbitrary points on a rigid body.

One of the key highlights of the course is the derivation of Chasles' theorem (also known as Mozzi–Chasles' theorem), which states that the most general displacement of a rigid body can be achieved by combining a translation along a line, known as the Mozzi axis, with a rotation about the same axis. This result has profound implications for the analysis of rigid body motion and serves as a cornerstone for further study in mechanics.

By the end of the course, students will gain a deep understanding of the mathematical and physical principles that govern the motion of rigid bodies, laying a strong foundation for advanced studies in mechanics, robotics, and engineering applications.

Who this course is for:

Students who would like to develop mathematical intution to tackle problems about rigid body systems.

Advanced Rigid Body Mechanics in Three Dimensions

What you'll learn

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Course content

Derivation of the angular velocity4 lectures • 35min

Formulae for: Velocities, Accelerations, and Mozzi's theorem4 lectures • 33min

Derivation of the equations of motion6 lectures • 48min

Derivation of the energy conservation (Virtual work theorem)4 lectures • 1hr 5min

Some important properties of the inertia matrix1 lecture • 22min

Stability analysis of the motion of a torque-free and force-free rigid body3 lectures • 43min

Four-bar linkage: important application in mechanical engineering3 lectures • 1hr 22min

Requirements

Description

Who this course is for: