Structural Dynamics / module 9
Two-degree-of-freedom systems
Use a two-degree-of-freedom system to study coupling, natural frequencies and mode shapes with calculations that can be followed by hand.
What you will understand
- 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
Coupled motion and the matrices
Two masses that cannot move independently, the matrices that describe them, and what every entry physically means.
The eigenvalue problem, solved by hand
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.
Module summary
- Two degrees of freedom means two natural frequencies and two mode shapes
- K[i][j] is the force at i needed to hold everything still while j moves one unit
- Coupling lives in the off-diagonal terms — here in K, not in M
- det(K − ω²M) = 0 gives a quadratic in ω² and hence two roots
- A mode shape is a RATIO; its absolute magnitude means nothing
- Mode 1 has both floors moving together; mode 2 has them opposed
- Change one storey and BOTH frequencies and BOTH shapes move
- φ₁ᵀMφ₂ = 0 — orthogonality, which is what makes modal analysis possible
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 notesCheck your understanding
Practice
A two-storey shear building has k₁ = 120 MN/m and k₂ = 70 MN/m. What is K[0][0], in MN/m?
Practice
A 2DOF system has m₁ = m₂ = 2 000 kg and k₁ = k₂ = 300 kN/m. What is the second natural frequency, in rad/s?
Check yourself
A modal analysis of a two-storey building returns a first mode shape of {1, −0.8}. What does this indicate?
Practice
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.
Check yourself
What is orthogonality of mode shapes actually FOR?