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Question 1

What will solving [M]-1[K]{X} = w^2{x} find?

Answer 1

Need to solve iteratively

Finds information on highest mode of vibration

Question 2

What will solving [A][M]{X} = 1/w^2{x} find?

Answer 2

Need to solve iteratively

Finds information on lowest mode of vibration

Question 3

What is virtual displacement

Answer 3

An instantaneous imaginary infinitesimal variation of the coordinate. It must be compatible with the restraints of the system

Question 4

What is virtual work

Answer 4

The work done b all the active forces

Question 5

Give D’Alembert’s Principle (virtual work)

Answer 5

𝛿W = Σ (Fi - mr(dotdot))𝛿r = 0

Question 6

Lagrange:
T =
U =

Answer 6

T = 1/2 m x(dot)^2
U = 1/2 k(xi-xi-1)^2

Question 7

Give Lagrange equation

Answer 7

d/dt(dT/dq(dot)i) - dT/dqi + dU/dqi = 0

Question 8

What is assumption in Rayleigh method

Answer 8

• Assumes 1 effective mass for entire system
• Approximate method that treats a continuous system as a discrete system
  (advantage is that doesn’t deal with partial differential equations)
  (deals with large number of ordinary differential equations which is great for fast computing machines)

Question 9

Write Rayleigh equation

Answer 9

.

Question 10

Assumption for Rayleigh-Ritzs

Answer 10

Based on premise that a closer approximation to exact natural mode can be obtained by:

1. Superposing a number of assumed functions than by using a single assumed function
2. Also provides values of the higher natural frequencies and the mode shapes
3. Number of frequencies = number of functions

Question 11

Describe viscous damping

Answer 11

• commonly used

- proportional to velocity

Question 12

Equation for critical damping

Answer 12

Cc = 2Sqrt(km)

Question 13

Equation for damping ratio

Answer 13

l = C / Cc

Question 14

Describe coulomb damping

Answer 14

• steady friction force that occurs in structures

- forces independent of amplitude and frequency

Question 15

Describe hysteretic damping

Answer 15

• internal dissipation of energy observed when a material is under cyclic stresses
• independent of frequency

Question 16

Describe complex stiffness

Answer 16

• most real structure not possible to distinguish stiffness and damping effects - therefore often considered together

Question 17

When would you use damping for troubleshooting

Answer 17

1. Issues with excessive vibration in a machine or structure
2. Can re-sit or change machinery causing vibration but vibration caused by external forces need to be controlled by damping
3. Vibration isolation
4. Vibration absorbers attached to machinery to alleviate vibration at resonance

Question 18

What happens to natural frequency of system after vibration absorbes added

Answer 18

• System now possesses two naturla frequencies that oscillate less at frequencies alongside original natural frequency

Question 19

What is the expansion theorem

Answer 19

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

Question 20

What is the importance of finding modal properties

Answer 20

1. Avoid resonance through knowledge of natural frequency
2. Provide info for calculating fatigue life of components
3. Understand system response to shock and random or periodic excitation
4. Predict where in the system vibration and noise might be an issue

Question 21

If response is (1 / 2 / 3), [H(w)] is made up of _____ FRFs

1. Displacement
2. Velocity
3. Acceleration

Answer 21

1. Receptance
2. Mobility
3. Accelerance

Question 22

Define no-linearity

Answer 22

A system where superposition does not hold true

Question 23

Define superposition

Answer 23

• consider system with inputs xi and outputs yi with initial conditions yo = y(dot)(o) = 0
• superposition holds if the system response to an input axi + Bxj is ayi + byj for all values of a, B, i and j

Question 24

Give duffing equation

Answer 24

my(dotdot) + cy(dot) + k1y + k3y^3 = Xsin(wt)

Question 25

What is reciprocity

Answer 25

For linear system when the frequency response is the same when excited at x1, measure at x2 and when excited at x2 measured at x1

Question 26

Give 5 types of common non-linearity

Answer 26

1. Cubic stiffness 2. Bilinear stiffness and damping 3. Piecewise linear stiffness 4. Non-linear damping - quadratic 5. Coulomb friction

Question 27

What does a phase portrait provide

Answer 27

- information about equilibrium and stability in a system without solving the porblem

Question 28

Difference of stable and unstable on a phase portrait

Answer 28

Stable: if the motion follows a closed loop after being displaced from equilibrium Unstable: if the motion followed is large after displaced from equilibrium

Question 29

Define chaos

Answer 29

when a system, given initial conditions, responds in what seems to be a completely random and unpredictable manner

Question 30

Understand jumping phenomena

Answer 30

Different from low --> high frequency and | high ---> low frequency

Question 31

When Θ is very small; sinΘ = cosΘ =

Answer 31

``` sinΘ = Θ = 0 cosΘ = 1-1/2Θ^2 ```

Question 32

Describe theoretical modal analysis

Answer 32

1. Description of structure (spatial model) 2. Vibration modes (modal model) 3. Response levels (response model)

Question 33

Describe experimental modal analysis

Answer 33

1. Response properties 2. Vibration modes 3. Structural model