Theoretical Mechanics

Theoretical mechanics is a fundamental engineering discipline that studies the general laws of mechanical motion and equilibrium of material bodies. It provides a mathematical foundation for further study of mechanics of materials, theory of mechanisms and machines, machine dynamics, structural mechanics, and other engineering subjects.

What theoretical mechanics studies

Real bodies and mechanical systems are represented by idealized models such as a particle, a rigid body, or a system of particles. These models make it possible to isolate the essential laws of motion and equilibrium and describe them mathematically.

The course consists of three main branches: statics, kinematics, and dynamics. They are studied in a natural sequence — from forces and equilibrium, through the description of motion, to the causes of that motion.

Statics

Statics studies force systems and the conditions of equilibrium of material bodies. Topics include forces and their projections, moments of forces, couples, reduction of force systems, constraints and reactions, equilibrium of two- and three-dimensional systems, distributed loads, centers of gravity, and friction.

The main practical goal is to learn how to construct a free-body diagram, identify external forces and support reactions correctly, and write independent equilibrium equations.

Kinematics

Kinematics describes mechanical motion without considering the forces that cause it. Its principal quantities include position, trajectory, velocity, acceleration, angular velocity, and angular acceleration.

The section covers particle kinematics, translation and fixed-axis rotation of rigid bodies, plane motion, relative motion of a particle, Coriolis acceleration, spherical motion, and general motion of a free rigid body.

Dynamics

Dynamics establishes the relationship between motion and the forces that produce it. It uses Newton's laws, differential equations of motion, momentum, angular-momentum and work-energy theorems, and conservation laws.

For mechanical systems, the course covers center-of-mass motion, general theorems of dynamics, D'Alembert's principle, virtual displacements, the general equation of dynamics, and Lagrange's equations of the second kind. The course concludes with an introduction to small oscillations of a one-degree-of-freedom system.

How to study the course

A recommended sequence is Statics → Kinematics → Dynamics. For each topic, first understand the physical meaning of the concepts and the assumptions of the model, then study the governing equations and the procedure for applying them. After that, move to problems, where the key skill is selecting an appropriate mechanical model and solution method.

Notation and units

The formulas use standard vector and scalar notation of engineering mechanics. Unless stated otherwise, quantities should be expressed in a consistent system of units, preferably SI: length in meters, time in seconds, mass in kilograms, force in newtons, and moment of force in N·m.