What substitution do we make to find the modes of vibration given the equation of motion
W(x,t) = u(x) e^(iwt)
Remembering we only care about the real part of this expression
What is the form of the general solution after you have made the substitution into the eqn of motion
u = Acos kx +Bsin kx
Where k is found by auxiliary eqn method.
You can then plug boundary conditions in to find the constants A/B
Equation for coupled transfer function
Hc = (1/H1 + 1/H2 )^
.
How do you change the general solution for 4th order system?
Must include hyperbolic terms and also aux eqn gives k^4
What must we do to mode shapes when we want to use them in impulse/frequency responses
Mass normalise them
Involves integrating the mode shapes squared with respect to mass, and setting equal to 1.
Can change the dm to be in terms of the right variable for the mode shapes (eg x) by using the relation between the variables (eg dm = m dx)
When you are modelling a force of magnitude F and frequency ω what form do you use
Fe^iωt
What are the physical meaning of the zeros of a transfer function?
Zeros are the antiresonances of the bar, corresponding to the frequencies where a very high axial force creates no/little displacement
What’s a Q factor
A parameter that describes the resonance behaviour of a system.
Systems with higher Q factors resonate with greater amplitudes but have a smaller range of frequencies around which they resonate.
What system do peaks getting further apart indicate
Bending beam
What system do regularly spaced peaks indicate
A string or axial/torsional vibration of a bar
What does a peak at low frequencies on a velocity graph indicate
A rigid body mode
Things that can cause real data to differ from simulated
Peaks not regular- variation of properties along length, bending stiffness of string not accounted for
System may have modes of more than one type eg a beam with bending, torsional and axial modes
Measurements will have noise, antiresonances will not be as clean.
Peaks split in 2 due to vibration not happening precisely in one plane/different behaviour in different planes eg wood
How to find expression who’s roots give the natural frequencies for more complex series of simultaneous equations ie with hyperbolics/more than 2 terms
Put the two simultaneous equations into matrix form and since you have 2 variables and the right hand side equals zero due to boundary conditions, the non-trivial solution is found when the determinant of the matrix formed is zero
What are the physical meaning of the poles of the transfer function
They give the resonant frequencies
What does having poles on the y axis indicate
The system is marginally stable, so will vibrate indefinitely if perturbed.
What does the negative real component of a pole arise from
Damping