1P1 Vibrations Flashcards
What does it mean for displacements to be compatible?
They are the same, i.e. a spring attached to a mass both have compatible displacements.
Where do you find the time constant for first order systems?
Tyddot + y = x.
The coefficient of the second order term if the coefficient of the 0th order term is 1.
How do you determine the time constant from a step response graph?
Draw a gradient at the origin and where this line meets the asymptote is at T.
What distinguishes the response of a second and first order system?
Second order systems are continuous in both y and ydot at t=0. Whereas, first order systems can be neither.
What order system is a mass and a dashpot?
First order as only features yddot and ydot, so can be integrated.
How do you find the ramp response?
Integrate the step response.
How do you find the harmonic response of a first order system?
Find the PI in terms of sin and cos, apply R, alpha method to remaining cos and sin.
Max value of step response of a second order system.
2x equilibrium
Define under damped, critically damped, and over damped.
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.
What is logarithmic decrement?
ln(y2/y1), formula given in databook. Can be used to estimate the damping coefficient.
Ln (y1/y(1+N)) = 2pi N zeta
Case (a) in databook.
When base motion is zero. Harmonic exciting force, f
What is Q factor?
(Y/X)max
What is the effect of the complementary function for second order harmonic response?
Disturbs the initial reponse, harmonic at the damped natural frequency.
Case (c) in the databook?
absolute motion relative to base motion.
What are the different natural frequencies?
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.
Case b in the databook.
Relative motion, to base excitation.
What ways can the damping factor be determined from a Y/X against freqency diagram.
Q Factor:
(Y/X)max = 1/2zeta
Half power bandwidth:
at 1/sqrt(2) x (Y/X)max, w2 - w1 = 2 zeta x wn
What is the transmitability of vibrations?
T = Y/X (where X is base excitation)
How do you minimise transmitability?
T<1 for w/wn > sqrt(2) ~ 1.4
Low natural frequency, lower damping - lower oscillation. Danger if frequencies do approach resonance.
How do you work out vibration transmission with a fixed base and oscillating mass?
fT = ky + lambda ydot
Can use complex numbers to determine the ratio of force transmitted, ends up being identical to case c.
Conditions for seismic transducers to work.
w»wn
But with damping around 0.5, w/wn >0.85
What case is applicable for an accelerometer?
Case a for w«wn
Define the degrees of freedom of a system.
The number of coordinates required to describe the configuration of a system.
Equation of motion of an undamped multi DoF system.
[m]yddot + [k] y = 0
Eigenvalue formulation of multi dof vibration problem.
[m]^-1[k] Y = w^2 Y
Equation to find natural frequencies of a multi dof system.
det ( - w^2 [m] + [k] ) = 0
What is characterstic about the eigenvectors of a mutli dof system?
They are independent so any motion can be expressed as a linear combination of the eigen vectors.
What is the rigid body mode of vibration?
When w = 0, so will not vibrate.
How do you solve the resonant response of a multi dof problem?
Y = {-w^2[m] + [k]}^-1 F
How can you expressed the determinant of the adjoint matrix for multi dof harmonic reponse?
(1- w^2/w1^2)(1-w^2/w2^2)…
What does damping do to a tuned mass damper?
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.
