Module 03
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 this module covers
- 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
Four forces, one equation, and three routes to it — Newton, D'Alembert and energy — which must agree because they describe the same thing.
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Module checkpoint
Check what you have taken in
2 questions
Question 1
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?
Question 2
Why does gravity not appear in the equation of motion for a vertically hanging mass?