AdvancedTopic 79 of 106Engineering Mechanics
Multi-degree-of-freedom systems
Have several natural frequencies and mode shapes found by eigenvalue analysis.
In depth
An n-DOF system has n natural frequencies and mode shapes from det(K-ω²M)=0. Each mode is an independent pattern in which all masses move in phase. Real structures have infinitely many modes but low ones dominate.
Key points
- Mass and stiffness matrices
- Orthogonality of modes
- Coupled equations diagonalised
ExampleA two-storey building has two modes: sway together and sway opposite.
What is Multi-degree-of-freedom systems?
Have several natural frequencies and mode shapes found by eigenvalue analysis.
Why is it important to understand multi-degree-of-freedom systems?
Real structures have many natural frequencies and mode shapes, so multi-degree-of-freedom models are needed for accurate vibration prediction.
What is a mode shape?
It is the pattern of relative motion for one natural frequency. Each mode has its own shape, and the modes are independent in a linear system.
Why does a multi-degree-of-freedom system have several frequencies?
Each degree of freedom can vibrate in a different pattern, so the number of natural frequencies equals the number of degrees of freedom.
How is Multi-degree-of-freedom systems used in practice?
Typical applications include predicting the vibration modes of a multi-storey building under earthquake load, modelling a vehicle with separate body and wheel masses and analysing the coupled vibration of two connected machines.
What should I learn before Multi-degree-of-freedom systems?
Study "Vibration isolation" first (topic 78 of 106 in Engineering Mechanics). It covers the groundwork this topic builds on.
What are the key points of Multi-degree-of-freedom systems?
Mass and stiffness matrices. Orthogonality of modes. Coupled equations diagonalised.
Can you give an example of Multi-degree-of-freedom systems?
A two-storey building has two modes: sway together and sway opposite.
What comes after Multi-degree-of-freedom systems?
The next topic is "Modal analysis", which builds directly on this one.
How difficult is Multi-degree-of-freedom systems?
It is classed as Advanced in Engineering Mechanics (topic 79 of 106). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mechanics.