Question 1

Question 1

Describe the process of finding the equation of motion for a multi degree of freedom system

Answer 1

1 - separate into free body diagrams
2 - find the force of the springs and dashpots in terms of the extension and velocity of the ends
3 - apply Newton’s laws (F=ma) for m1 and m2
4 - write as matrices in the form M(acceleration) +C(velocity) +K(displacement) = force

Question 2

describe the process of finding the natural frequencies for a multi degree of freedom system

Answer 2

1 - The natural frequencies are solutions to det[−𝜔n^2𝐌+𝐊] = 0
2 - find a quadratic
3 - solve for 𝜔n^2

Question 3

how do you convert from rad/s to Hz?

Answer 3

1 rad/s = 1/2pi Hz

Question 4

how do you find the mode shapes for the natural frequencies of an undamped multi degree of freedom system?

Answer 4

1 - The mode shape is given by substituting for the natural frequency into [−𝜔n^2𝐌+𝐊]𝐮 = 0
2 - we assume u11 and u21 = 1 so find u12 and u22 in terms of 𝜔n^2
3 - substitute the values calculated previously for 𝜔n^2 to find the mode shapes

Question 5

how do you find the damping ratios for the first 2 modes of a multi degree of freedom system?

Answer 5

1 - If we can find two scalars 𝛼 𝑎𝑛𝑑 𝛽,𝑚𝑎𝑘𝑖𝑛𝑔 𝑪=𝛼𝑴+𝛽𝑲 validated, the damping is proportional damping.
2 - write out full equation in terms of the matrices
3 - find beta then alpha
4 - confirm proportional damping
5 - use 2𝜉i𝜔n,i =𝛼+𝛽𝜔n,i^2 to find damping ratios

Question 6

how do you find the damped free response of both mases given initial conditions

Answer 6

1 - use equation given in data sheet for q(t)
2 - calculate 𝜔d =𝜔n sqrt(1−𝜉^2) for both natural frequencies
3 - Introduce the initial condition to q(0) eqtn and find A1 and A2
4 - differentiate equation for q’(0) to get q’(0)=(-𝜉1w1a1+b1wd1)u1+(-𝜉1w2a2+b2wd2)u2
5 - sub A into q’(0) eqtn to find B1 and B2
6 - now you can rewrite the q(t) eqtn in full with no unknowns

Question 7

how to find the mass normalised mode shape with damped free resonse

Answer 7

1 - use equation Uri^T M Uri = 1 where Ur1 = aU1 and Ur2 = bU2
2 - put your matrices into the formula to find a and b

Question 8

what are the steps for calculating the steady state response of harmonic motion

Answer 8

1. Calculate the mass normalised mode shape 𝒖𝑹𝒊
2. Form the matrix 𝑼=[𝒖𝑹𝟏 𝒖𝑹𝟐], and use UT to get decoupled excitation
3. Use single DOF knowledge to get the response in the decoupled coordinate
4. Use matrix U to convert decoupled response back to the original response

Question 9

how to form the matrix U and use U^T to get decoupled results

Answer 9

1 - U = [Ur1 Ur2]
2 - find U^T

Question 10

how to find the decoupled coordinate

Answer 10

decoupled coordinate = U^T*F
where F is given in question

Question 11

how to find the decoupled response

Answer 11

use p’‘1+2wn1𝜉1p’1 + wn1^2p1
where p = q/U

Question 12

how to find the result in the decoupled coordinate

Answer 12

1 - use p(t) = f0 Cs cos (wt-φs) for p1 and p2 in terms of t by finding Cs1,2, φs1,2, r1,2
2 - convert back to original response q = Up