MDOF Systems

Question 1

What is a MDOF system

Answer 1

• Multi-degree of freedom system
• for each degree of freedom, there is an equation of motion
• Each Wn has it’s own mode shape

Question 2

Give three ways of finding equations of motion for MDOF

Answer 2

• N2L of motion
• Influence coefficients
• Lagrange’s equations

Question 3

Describe response for free vibrations

Answer 3

response is a combination of different modes

Question 4

Describe response for forced vibrations

Answer 4

excitation around a natural frequency will result in the structure vibrating in the associated mode shape

Question 5

List procedure for finding EoM using N2L

Answer 5

• 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

Question 6

What do eigenvalues mean

Answer 6

eigenvalues are the natural frequencies of the system

Question 7

What do eigenvectors mean

Answer 7

eigenvectors are the mode shapes of the system

Question 8

When does orthoganlity hold true

Answer 8

orthoganality properties only hold if the damping matrix [c] is proportional to [m]/[k]

Question 9

Why is understanding dynamic response important

Answer 9

• avoid resonance at Wn
• calculate fatigue life
• understand system response to shock
• predict where noise might be an issue

Question 10

What is experimental modal analysis

Answer 10

• 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)

Question 11

What happens in experimental modal analysis if data is taken at a node

Answer 11

a natural frequency will be lost in the FRF

Question 12

What happens to phase when you pass through a resonance and an antiresonance

Answer 12

resonance - lose 180°

antiresonance - gain 180°

Question 13

How is damping ratio found in experimental modal anaylsis

Answer 13

using 3-dB method

Question 14

What does orthogonal mean

Answer 14

orthogonal - of or involving right angles

Question 15

What is the expansion theorem

Answer 15

for free vibration, the vibration of a system is a superposition of the normal modes

Question 16

What is Kij

Answer 16

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)

Question 17

How to get influence coefficients

Answer 17

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

Question 18

What is flexibility

Answer 18

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 ]

Question 19

What is the flexibility coefficient aij

Answer 19

aij is the deflection at point i due to a unit force at point j

Question 20

How to find flexibility coefficient aij

Answer 20

Exact same method as to finding influence coefficients,

however instead of K11 and F1

Question 21

What are inertia influence coefficients

Answer 21

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

Question 22

How to calculate inertia influence coefficients

Answer 22

• 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