Module 09
Two-degree-of-freedom systems
The smallest system that has mode shapes — small enough that the eigenvalue problem can be solved by hand, and large enough that everything important about many degrees of freedom is already present.
What this module covers
- Assemble the mass and stiffness matrices of a two-storey shear building by inspection
- Explain what coupling means and which matrix carries it
- Derive and solve the eigenvalue problem for a 2DOF system
- Find mode shapes and explain why their magnitude is arbitrary
- Demonstrate orthogonality and say what it is for
- Decompose an arbitrary initial displacement into modal contributions
Lessons
Two masses that cannot move independently, the matrices that describe them, and what every entry physically means.
Start lesson →Is there a deformed shape that, once set moving, keeps its shape? For two degrees of freedom the question is a quadratic, and the answer is two shapes.
Start lesson →
Module checkpoint
Check what you have taken in
5 questions
Question 1
A two-storey shear building has k₁ = 120 MN/m and k₂ = 70 MN/m. What is K[0][0], in MN/m?
Question 2
A 2DOF system has m₁ = m₂ = 2 000 kg and k₁ = k₂ = 300 kN/m. What is the second natural frequency, in rad/s?
Question 3
A modal analysis of a two-storey building returns a first mode shape of {1, −0.8}. What does this indicate?
Question 4
Mode shapes φ₁ = {1, 2} and φ₂ = {1, −0.5} are claimed for a system with m₁ = 5 000 kg and m₂ = 2 500 kg. Compute φ₁ᵀMφ₂, in kg, to test orthogonality.
Question 5
What is orthogonality of mode shapes actually FOR?