Randon and Non-Linear

Question 1

How is \rho defined when it comes to joint probability distributions?

Answer 1

E[X1 X2] = \rho sigma_x1 sigma_x2

Therefore, Y = X1 + X2

E[Y^2] = E[X1] + E[X2] + 2 * E[X1 X2]

Where the last term = 2 \rho sigma_x1 sigma_x2

Question 2

What is a typical FoS when using Miner’s Rule?

Answer 2

4 or 5 typical.

Question 3

How would an iterative solution be applied to:

x’’ + p^2 x + e * x^3 = a cos(w t)

Answer 3

To find x0, ignore the C.F. and e = 0 to get:

x0 = a/(p^2-w^2) cos(wt)
= a* cos(wt)

Then move x0 = a* cos(wt) to rhs with e and get:

x1’’ + p^2 x1 = a cos(wt) - e * (a*)^3 * cos^3(wt)

Question 4

Define stationary.

Answer 4

The temporal average is independent of time.

Question 5

Define ergodic.

Answer 5

The ensemble average is equal to the temporal average.

Question 6

Does ergodic -> stationary?

Answer 6

Yes. The temporal average is equal to the ensemble average (which doesn’t change with time). Therefore, the temporal average doesn’t change with time and is stationary.

Question 7

Does stationary -> ergodic.

Answer 7

No. The temporal average depends on the particular value chosen for that member of the ensemble. The classic example is x = a cos(t) where a is random. The ensemble average depends on the distribution of a, but the temporal average only depends on the particular instance of a.

Question 8

What shape is the amplitude - frequency graph for spring hardening?

Answer 8

It leans to the right.

Question 9

What shape is the amplitude - frequency graph for spring softening?

Answer 9

It leans to the left.

Question 10

How can you draw the x’ , x trajectories from knowing that

V = x^2

?

Answer 10

T + V = E along a trajectory,

so 1/2 (x’)2 + x^2 = E, which are easy to plot in x’ , x space.

Question 11

Define ergodic in terms of algebra only.

Answer 11

< F{ x(t) } > = E[ F{ x(t) } ]

Question 12

Why is an ergodic process useful?

Answer 12

The temporal average can be used to estimate the ensemble average. This makes calculating the ensemble average as simple as reording a single process over a long time.

Question 13

Define the cross-correlation function.

Answer 13

Rxy (T) = E[ x(t) * y(t+T) ]

Question 14

Express Rxy(T) in terms of Ryx(T)

Answer 14

Rxy(T) = E[ x(t) * y(t+T) ]
= E[ x(t+T) * y(t+T)]

ie they are the same.

Question 15

Express Sxy (w) in terms of Syx(w)

Answer 15

Sxy(w) = Syx(w) *

i.e. the complex conjugate. Datasheet expressions help to derive this.

Question 16

A non-linear equation is driven like this:

x’’ + p^2 x + e x^3 = a cos(wt)

How would you go about solving it for [10%]?

Answer 16

Set x = b cos(wt), then x^3 = 3b^3/4 cos(wt) + b^3/4 cos(3wt)

Neglect the 3wt term and get:

p^2 - w^2 + 3b^3 / 4 = a

Question 17

We have decided that x = A cos(wt). What is x^3?

Answer 17

A^3 * (3/4 cos(wt) + 1/4 cos(3wt) )

Don’t forget it’s A^3 out the front!

Question 18

How would you deal with a thing connected to a dashpot and spring in series?

Answer 18

Define a variable for displacement of the OBJECT and of the CONNECTION. Then can find two equations:

equilibrium of object

equilibrium of connection

Question 19

How would you deal with a thing connected to a dashpot and spring in parallel?

Answer 19

The sum of the dashpot and spring (and any D’Alambert forces) is equal to the forcing term

Question 20

I have calculated x(w). I have then found the integral of x’‘(w) = |(iw)|^2 * x(w).

This integral tends to infinity. What has gone wrong?

Answer 20

The question has likely asked for a white noise approximation. This approximation is non-sensical if the velocity is of interest. It’s the questions fault - not mine!