1P1 Vibrations

Question 1

What does it mean for displacements to be compatible?

Answer 1

They are the same, i.e. a spring attached to a mass both have compatible displacements.

Question 2

Where do you find the time constant for first order systems?

Answer 2

Tyddot + y = x.
The coefficient of the second order term if the coefficient of the 0th order term is 1.

Question 3

How do you determine the time constant from a step response graph?

Answer 3

Draw a gradient at the origin and where this line meets the asymptote is at T.

Question 4

What distinguishes the response of a second and first order system?

Answer 4

Second order systems are continuous in both y and ydot at t=0. Whereas, first order systems can be neither.

Question 5

What order system is a mass and a dashpot?

Answer 5

First order as only features yddot and ydot, so can be integrated.

Question 6

How do you find the ramp response?

Answer 6

Integrate the step response.

Question 7

How do you find the harmonic response of a first order system?

Answer 7

Find the PI in terms of sin and cos, apply R, alpha method to remaining cos and sin.

Question 8

Max value of step response of a second order system.

Answer 8

2x equilibrium

Question 9

Define under damped, critically damped, and over damped.

Answer 9

Under damped zeta < 1, oscillatory response with a decreasing amplitude.

Critically damped, zeta = 1. (A+Bt)e^(-wt). Two repeated, real roots.

Over Damped, zeta > 1, exponential response.

Question 10

What is logarithmic decrement?

Answer 10

ln(y2/y1), formula given in databook. Can be used to estimate the damping coefficient.
Ln (y1/y(1+N)) = 2pi N zeta

Question 11

Case (a) in databook.

Answer 11

When base motion is zero. Harmonic exciting force, f

Question 12

What is Q factor?

Answer 12

(Y/X)max

Question 13

What is the effect of the complementary function for second order harmonic response?

Answer 13

Disturbs the initial reponse, harmonic at the damped natural frequency.

Question 14

Case (c) in the databook?

Answer 14

absolute motion relative to base motion.

Question 15

What are the different natural frequencies?

Answer 15

Natural frequency, wn, found from the original DE.

Damped natural frequency, wd, the frequency of response of the CF.

Resonant frequency, the frequency at which there is a maximum response for a harmonic oscillator. Very different for case a b and c.

Question 16

Case b in the databook.

Answer 16

Relative motion, to base excitation.

Question 17

What ways can the damping factor be determined from a Y/X against freqency diagram.

Answer 17

Q Factor:
(Y/X)max = 1/2zeta

Half power bandwidth:
at 1/sqrt(2) x (Y/X)max, w2 - w1 = 2 zeta x wn

Question 18

What is the transmitability of vibrations?

Answer 18

T = Y/X (where X is base excitation)

Question 19

How do you minimise transmitability?

Answer 19

T<1 for w/wn > sqrt(2) ~ 1.4

Low natural frequency, lower damping - lower oscillation. Danger if frequencies do approach resonance.

Question 20

How do you work out vibration transmission with a fixed base and oscillating mass?

Answer 20

fT = ky + lambda ydot

Can use complex numbers to determine the ratio of force transmitted, ends up being identical to case c.

Question 21

Conditions for seismic transducers to work.

Answer 21

w»wn
But with damping around 0.5, w/wn >0.85

Question 22

What case is applicable for an accelerometer?

Answer 22

Case a for w«wn

Question 23

Define the degrees of freedom of a system.

Answer 23

The number of coordinates required to describe the configuration of a system.

Question 24

Equation of motion of an undamped multi DoF system.

Answer 24

[m]yddot + [k] y = 0

Question 25

Eigenvalue formulation of multi dof vibration problem.

Answer 25

[m]^-1[k] Y = w^2 Y

Question 26

Equation to find natural frequencies of a multi dof system.

Answer 26

det ( - w^2 [m] + [k] ) = 0

Question 27

What is characterstic about the eigenvectors of a mutli dof system?

Answer 27

They are independent so any motion can be expressed as a linear combination of the eigen vectors.

Question 28

What is the rigid body mode of vibration?

Answer 28

When w = 0, so will not vibrate.

Question 29

How do you solve the resonant response of a multi dof problem?

Answer 29

Y = {-w^2[m] + [k]}^-1 F

Question 30

How can you expressed the determinant of the adjoint matrix for multi dof harmonic reponse?

Answer 30

(1- w^2/w1^2)(1-w^2/w2^2)…

Question 31

What does damping do to a tuned mass damper?

Answer 31

Increase the bandwith of low (Y/(f/k)), but at no point does it go to zero. Wider bandwidth of operation, but reduced performance at wn.

But at no damping, the response goes to zero at w. Tune w to be the original natural frequency of the mass system.