MDOF Systems Flashcards
What is a MDOF system
- Multi-degree of freedom system
- for each degree of freedom, there is an equation of motion
- Each Wn has it’s own mode shape
Give three ways of finding equations of motion for MDOF
- N2L of motion
- Influence coefficients
- Lagrange’s equations
Describe response for free vibrations
response is a combination of different modes
Describe response for forced vibrations
excitation around a natural frequency will result in the structure vibrating in the associated mode shape
List procedure for finding EoM using N2L
- set-up coordinate system
- adopt +ve directions for displacement, vel, acc
- determine static equilibrium position/configuration
- draw free body diagram (with spring, damping and external forces)
- apply N2L
What do eigenvalues mean
eigenvalues are the natural frequencies of the system
What do eigenvectors mean
eigenvectors are the mode shapes of the system
When does orthoganlity hold true
orthoganality properties only hold if the damping matrix [c] is proportional to [m]/[k]
Why is understanding dynamic response important
- avoid resonance at Wn
- calculate fatigue life
- understand system response to shock
- predict where noise might be an issue
What is experimental modal analysis
- to take measurements of the structure and then infer its modal properties
- measured responses can be:
displacement (receptance FRF)
velocity (mobility FRF)
acceleration (accelerance FRF)
What happens in experimental modal analysis if data is taken at a node
a natural frequency will be lost in the FRF
What happens to phase when you pass through a resonance and an antiresonance
resonance - lose 180°
antiresonance - gain 180°
How is damping ratio found in experimental modal anaylsis
using 3-dB method
What does orthogonal mean
orthogonal - of or involving right angles
What is the expansion theorem
for free vibration, the vibration of a system is a superposition of the normal modes
What is Kij
Kij is the force at the point i due to a unit displacement at point j, when all other points are fixed
Kij = Kji ( Maxwell’s reciprocity theorem)
How to get influence coefficients
Method:
- set the first mass to move with a unit step 1
- the rest of the masses have 0 displacement
- usual force is denoted as K11 (for first mass) K21, k31 …
- carry out N2L like equations and subsitute values for x
- values for K11, K21, K31 (first column) found
- now set second mass to move with unit step 1
- calculate values K12, K22, K32
- now set third mass to move with unit step 1
- calculate values K13, K23, K33
What is flexibility
Flexibility is the inverse of stiffness and denoted with [A]
[A] = [K]^-1
[K] = a b
c d
[K]^-1 = 1/(ad - bc) x [ d - b ]
[ -c a ]
What is the flexibility coefficient aij
aij is the deflection at point i due to a unit force at point j
How to find flexibility coefficient aij
Exact same method as to finding influence coefficients,
however instead of K11 and F1
What are inertia influence coefficients
defined as the set of impulses applied at each mass or point to produce a unit velocity at mass j, with zero velocity at all other points
How to calculate inertia influence coefficients
- assume set of impulses Iij applied at all points so as to produce a zero velocity at all but the jth point which will have a unit velocity
- repeat for j = 1,2, …, n
