Structural Dynamics / module 3

Single-degree-of-freedom systems

The equation of motion, derived three ways, and the four forces that balance at every instant of a structure's movement.

What you will understand

Derive m x″ + c x′ + k x = F(t) from Newton's second law and from D'Alembert's principle
Identify the inertia, damping, stiffness and applied forces in a physical problem
State the units of every term and check an equation dimensionally
Distinguish static from dynamic displacement, and relative from absolute motion
Explain what the damping term does and does not represent
Set up the equation of motion for a real single-degree-of-freedom idealisation

Lessons

The equation of motion
Four forces, one equation, and three routes to it — Newton, D'Alembert and energy — which must agree because they describe the same thing.

Module summary

m x″ + c x′ + k x = F(t) is a statement of equilibrium, not of F = ma
The four forces balance at every instant, including instants when F(t) is zero
Inertia opposes acceleration, damping opposes velocity, stiffness opposes displacement
Each term has units of force, and checking that catches most setup errors
The viscous damping term is a convenience, not a description of what dissipates energy
Under base excitation the equation is written in RELATIVE displacement
Mass in kg, stiffness in N/m, damping in N·s/m — consistent SI or nothing
Every later method in this course solves this one equation differently

Teaching basis and review status

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

Check your understanding

Practice

A 15 tonne mass sits on columns of total lateral stiffness 6.0 MN/m with 4% damping. What is the damping coefficient c, in kN·s/m?

Check yourself

Why does gravity not appear in the equation of motion for a vertically hanging mass?