MDOF

Question 1

What is the damped and undamped MDOF system matrix equation of motion?

Answer 1

Damped: [M]{x..(t)} + [C]{x.(t)} + [K]{x(t)} = {f(t)}
Undamped: [M]{x..(t)} + [K]*{x(t)} = {f(t)}
[M] = Mass matrix
{x..(t)} = Acceleration
[C] = Damping Matrix
{x.(t)} = Velocity
[K] = Stiffness matrix
{x(t)} = Dispalcement
{f(t)} = Prescribed time-variting Loads

Question 2

Descibe the equation below in words:
[M]{x..(t)} + [C]{x.(t)} + [K]*{x(t)} = {f(t)}

Answer 2

Equilibrium of inertia, damping, elastic (or stiffness, or
restoring) forces and external forces, acting in the direction
of each DOF, is satisfied at each instant of time.

Question 3

Why can’t the matrix equation for a damped MDOF system be solved with normal partial differential equations?

Answer 3

It can’t be solved by normal PDEs because there are coupling terms. This means that there are terms for each floor/node in both equations needing to be solved. So the movement at each node effects the other nodes.

Question 4

What are the two groups of methods that find general solutions to the equation of motion and which is better for linear dynamic problems?

Answer 4

1) Mode superposition methods
2) Direct time integration methods

Mode superposition methods are better for linear dynamic problems as:
- it provides a physical insight into the vibration behaviour of the structure modelled.
- it is considerably more efficient computationally.

Question 5

what does the so called expansion theorem state?

Answer 5

Any N-dimensional vector can be presented as a linear combination of N-dimensional vectors which are orthogonal and, therefore, linearly independent (i.e. none of them can be expressed as a linear combination of others).

Question 6

If all nodes vibrate with the same frequency how would you find the displacement and amplitude at each level?

Answer 6

{x(t)} = {X} * sin(wt + θ)
{x..(t)} = -w²{X} * sin(wt + θ)

{X} = {Φr} * Ar
{Φr} = Mode shape r
Ar = Amplitude factor of mode r

Question 7

How would you find the angular frequency of the system (w) if it was an undamped system?

Answer 7

det[ -w² *[M] + [K] ] = 0

(Once expanded this is called the characteristic equation or frequency equation)

det[ a,b ; c,d ] = ad - bc

Question 8

What is the standard eigenvalue problem for an undamped system?

Answer 8

wr² [M]{Φr} = [K]{Φr}
or
( [K] - wr^2 * [M]) * {Φr} = 0

r = mode shape number
wr = angular frequency of mode r

Question 9

what is the generalised coordinate?

Answer 9

It is the response of the equivalent SDOF system corresponding to mode r.
qr(0) = initial displacement of mode r
q.r(0) = Initial velocity of mode r

Question 10

How do you calculate the mass and stiffness of each mode?

Answer 10

Modal mass = mr = {Φr} T * [M] * {Φr}
Modal stiffness = kr = {Φr} T * [K] * {Φr}

Question 11

How can you find the lowest natural frequency of a system?

Answer 11

You find it by minimising the strain energy and maximising the kinetic energy. I.e. the easiest movement mode 1

Question 12

How do you get the final solution for the undamped free vibration of a MDOF system?

Answer 12

x(t) = (N, r=1) Sum( {phir} * qr(t) )

Use the equation for qr transient but make wrd = wr and all damping terms = 0.

Question 13

What is the spectral matrix and where is it used?

Answer 13

The spectral matrix [Ω^2] is defined as an NxN diagonal matrix containing squared natural frequencies on its diagonal.

[K] * [φ] = [Μ] * [φ] * [Ω^2]

φ = Node Shape

Question 14

What is Classical, Proportional damping or Rayleigh damping and when is it possible?

Answer 14

Systems where diagonalisation of the damping matrix is possible are called Classical, Proportional damping or Rayleigh damping.

It is achieved when the matrix is proportional to the mass and stiffness matrices in the form:
[C] = α[M] + β[K]
Where α & β are constant numbers

Question 15

For a MDOF system under harmonic load, what affects the amplitude and response?

Answer 15

Amplitude is frequency-dependent and increases around all resonant frequencies.

The response will be at the same frequency as the excitation.

The transient response is a superposition of a lower frequency decaying sinusoid having frequency which is lower than the frequency of the constant steady-state sinusoid superimposed on it. This lower frequency corresponds to the natural frequency of the system

Question 16

What is the solution to the motion of a damped MDOF under arbitrary loading

Answer 16

Formual sheet {x(t)}

Question 17

How to find the natural frequencies of continuous systems?

Answer 17

wm = (α*L)^2 / L^2 * √(EI/m)
m = Mass/length
α = coefficient that depends on the boundary conditions of the beam.

Question 18

What does sinh(α) and cosh(α) = ?

Answer 18

sinh(α) = (e^α - e^-α) / 2
cosh(α) = (e^α + e^-α) / 2

Question 19

For a simply supported beam with uniform properties and using Unity-scaled mode shapes what does modal mass equal?

Answer 19

0.5 * Mtotal

Question 20

Outline the general method for solving MDOF systems

Answer 20

1. Develop equation of motion
   [M]acceleration + [C]Velocity + [K]displacement = force
2. Calculate mode shapes
   formula sheet Mr. and kr
3. Apply mode superposition, modal properties must be known or assumed
   (genralised coordinates)
4. Formulate generalised equation of motion
   Pr(t) = sum(mode shape ij * forcej)
5. By summing mode shapes sclaed by the corresponding genralise coordinates
   formula sheet {x(t)}

Question 21

How do you find the first and secton mode shape vectors?

Answer 21

Find the mass and stiffness matrix and use one of the the eigenvector equations by subbing in the first mode shape angular frequency. Then solve to find the normalised mode shape. Repeat for all other mode shapes using the same equation.

Question 22

Describe how you would find the static external load ( q(x) ) of a continuous system?

Answer 22

Find the equilibrium equations in terms of shear, bending and uniform external forces. Know that the external load is equal to the second derivative of the bending moment and the bending moment is equal to EI times the second derivative of deflection.

q(x) = EI * d^4v/dx^4

EI = Stiffness
v = vertical deflection
x = location

Question 23

How do you find the undamped and damped equation of motion for a continuous system?

Answer 23

Undamped:
Using the static external load (q(x) = EI * d^4v/dx^4) and inertia force ( m*d^2v/dt^2 ) you can show:

m*d^2v/dt^2 + EI * d^4v/dx^4 = P(x,t)

Damped
md^2v/dt^2 + cdv/dt + EI * d^4v/dx^4 = P(x,t)

Question 24

For a simply supported beam what is teh ration between the second and first modal angular frequencies?

Answer 24

w2/w1 = (2pi)^2/(pi)2 = 4

Question 25

Describe the general way to find vertical displacements for a distributed system.

Answer 25

1. Set up the equation of motion
2. Find modal mass
   (0.5 * Total)
3. Find the euqtaion for teh solution by separation of variabkles
4. Find modal force (Qn(t))
5. Find vertical displacement.