Newtonian Dynamics Flashcards
Give the basic system equation
mx.. + kx + cx. = F
Natural frequnecy is defined as
frequency at which a system would osciallte if it were unforced and undamped
Unique natural frequency associated with
unique mode shapes
Why do we use simple models in vibration analysis
When modelling we take the approach that we would like the simplest possible model that provides the information that we need
What is a multi degree of freedom system
One that requires two or more coordinates to describe its motion
For each degree of freedom there is
an equation of motion
Why can we ignore deflection caused by mass weight
We take deflection from static equilibrium where mg = delta K, therefore mg and delta k cancel out
Derive the matrix equations of motions for a 3 DOF chain system
See powerpoint
For a linear system forced at a certain frequency how will it respond
Will respond at the same frequency
What type of equation is the force vibration equation of motion
Non homogenous differential equation
Why does it matter that the forced vibration equation is non homohenous differential equation
As it has two parts to the solution, a transient (complementary function) and a steady state response (particular integral)
What is the steady state response of a system
Solution of the system where frequency of the response is equal to the frequency of the forcing, only thing that can vary for each degree of freedom is the amplitude
What is the transient response
Homogenous solution, dies away quickly, solution for forcing equal to zero, all modeshapes are present
What is the solution to the nonhomogenous differential equation ax.. + bx. + cx = F
x(t) = xo(t) + x(t)
xo(t) homogenous solution (transient)
x(t) steady state response
For an unforced system what will be the reponse
Combination of all modeshapes. e.g. for 2 DOF
x1 = sin(w1t) + sin(w2t)
x2 = sin(w1t) + sin(w2t)
What do we assume when solving equations of motion
if forced that focing is harmonic F1 = Fsinvt, thus solution is harmonic x = Xsin(vt - phi)
if unforced that solution is harmonic
To determine the natural frequencies how would you go from the equations of motion to the eigenvalue problem
m x.. + kx = 0 Assume harmonic response - w^2 mX + kX = 0 kX = w^2 mX m^-1 k X = w^2 X Take determinant of dynamic matrix A (=m^-1 k) - lambda I and equate it to 0
How are modeshapes related
orthoganal, multiply them together and will get 0
What are modeshapes orthoganal with
other modeshapes, mass and stiffness matrix, and damping if proportional damping (i.e damping proportional to mass or stiffmess matrix)
What does the superscript and subscript refer to on X ^(1) sub 2
superscript is the natural frequency number ie first natural frequency, and the subscript is the mass
Why should you assume a solution including phase ie the form x1 = X1 cos (wt + phi)
Allows you to include damping, the phase cancels out so not an issue and assume damping but means we have a full solution with initial conditions
When doing ratio of modeshapes which one should always be on the bottom
Can write either way mathematically either way but as engineers should have first mass modeshape on bottom
When might you get a negative natural frequency and what do we do with it
-ve frequency doesnt make any sense so disregard at all times, occurs in fourier transform and when squarerooting the eigenvalues to give natural frequency
How would you write the full response of a system
first response
x1(t) = X1^(1) cos (w1t+phi1) +X1^(2) cos (w2t+phi2)
x2(t) = X2^(1) cos (w1t+phi1) +X2^(2) cos (w2t+phi2)
How would you work out full response of system
Eliminate some variable using modeshape ratios, then use initial conditions to determine X1 and X2 as well as phase values
Using vectors formulate FRF magnitude and phase
See powerpoint
what is the natural frequency of a SDOF system equal to
sqrt (k/m)
What is critical damping equal to in an SDOF system
2 m wn
What is damping factor equal to in an SDOF system
c/ critical damping or c / 2mwn
Nondemensionalise the FRF and phase for an SDOF system
See powerpoint
What happens to FRF magnitude as damping increases
The peak gets shorter and wider
Take the inverse of a 2x2 matrix [a b; c d]
1/dert * [d -b, -c a]
What is damping
energy dissipated in a structure
What structures have damping
All structures dissipate energy to some extent, when the amount of energy dissipated is small we can attempt an analysis presuming the structure is undamped
What are the type of damping
viscous damping
coloumb damping
hyteretic damping
What is vsicour damping
Commonly used, viscous force proportional to velocity
critical damping given by sqrt 4mk, less than critical damping get exponential decay
What is coloumb dampingq
steady friction force (dry friction) that occurs in many strucutres, forces are independent of amplitude and frequency, always oppose motion and magnitude may be approximated as constant
What is hysteretic damping
Internal dissipation of energy (material damping) obseerved what a material is subjected to cyclic stress, independent of freqency of vibration, cant use vsicous, need to divide damping term by frequency of osciallation
How is hysteretic damping simplified in equation of motion
As an increased stiffness value k* = k(1+i*eta) where eta is a loss factor for a material
How might excessive vibration in a machine or structure be resolved
Add or increase damping
Resiting or changing machinery that generate vibration if not an external source (wind or tuburlence causing)
Vibration isolation may be used to isolate machinery generation vibration from it surronding
Vibration absorbers may be attatched to machinery to alleviate excessive vibration at resonance
What are vibration absorbers
Simple mass spring system added to a structure which is designed to greatly reduce the amplitude of vibration at a particular frequency
When might you use a vibration absorber
If a particular resonance behaviour is an issue (particular frequency/mode)
How does a vibration absorber work
If tuned correctly at wanted frequency, input force into main body will equal input force into main body from vibration absorber, meaning at that frequency amplitude of vibration will = 0
What is often used when insufficient damping on a structure
viscoelastic material (polymers with such properties) widely used to provide additional damping
Whats the issue/benefit of viscoelastic material
Both frequency and temperature have a large influence on the effectiveness of the material for damping
why is constrained layer damping often used
polymers material that possess high damping properties often lack rigidity and cannot be used for structural purposes on their own, instead bonded to more rigid material (metals) to add damping, often in the form of sandwhich panels. Shear effects in damping layers dissipate energy
What is free layer damping
Thermally sprayed coating such as ceramic deposited using thermal spray process, provide damping through complex microstructure, useful for gently curved surfaces
What are particle dampers
Containers filled with particles attatched to the structure as discrete point damper, damping proportional to mass of particles, dissipate energy through inter particle friction
What are the benefits and negatives of particles dampers
not temperature dependent, behaviour is amplitude and frequency dependent
Where do the generalised mass and stiffness matricies come from
transpose (modeshape i) * mass matrix * modeshape i = Mii
transpose (modeshape i) * stiffness matrix * modeshape i = Kii
What does the orthoganality of modeshapes mean
transpose (modeshape i) * mass matrix * modeshape j = 0
transpose (modeshape i) * stiffness matrix * modeshape j = 0
When would you get orhoganality with damping matrix and a generalised damping quantity
When you have proportional damping
What is the modal matrix psi
all normal modes assembled into one matrix
If we use the modal matrix on the mass/stiffness matrix what can we get
Generalised mass and stiffness matrix
What can we do with the generalised mass, stiffness and modal matrix
As they are all diagonal matricies we can solve each of the n equation of motion separately
How do we create the othonormal modes
Divide the modeshapes by the square root of the generalised mass
What can you do with the othonormal modes
othonormal when multiplied with the mass matrix give 1 and the stiffness matrix give the natural frequencies
With proportional damping matrix you would get 2damping ratiosqrt(natural frequency)
Derive the orthognoality properties
See powerpoint
What are the different types of FRFs
receptence when displacement is measured
mobility when velocity is measured
accelerance when acceleration is measured
What is an FRF
Ratio of outputs/inputs
What does a H12 FRF mean
Force applied at point 2 and response measured at point 1
For a linear system what does Hij equal
Hji
What other form of response can we assume for damped system
Xe^iwt
Where are the natural frequencies on an FRF plot
Frequency at the peaks from any FRF as long as not measuring at a node
What happens to the peak on an FRF is there is no damping
Goes to infinity
What happens to the peak on an FRF is there is critical damping
Peak goes flat
What happens to the peak of an FRF is the structure becomes less stiff
Peak moves to the left
What can peaks moving be used to do an FRF
Use to detect damage, peak moving to the left means structure getting less stiff, peak moving to the right means structure getting more stiff. From this can determine damage
How do you go from time to frequency domain
Using the fourier transform
What is the FRF
Frequency response function ratio of output over input
What is needed from an FRF to evulate modeshapes
complete row or coloumn, take the imaginary domain first peak in each FRF for first mode, second for second mode etc.
Why does Hij equal Hji
As in linear systems we have reciprocity
Why might younot see a peak in an FRF
If there is a node
When might you see antiresonances
Will always see for drive point measurements i.e. Hii in magnitude plot see between the peaks, in phase plot will gain 180 degrees over a resonance and lose 180 degrees over antiresonance
Whys it bad to take a measurement on a node
Will lose peak in FRF and so lose alot of information
What are the difficulties with extracting FRFs from a bridge
Big structure dificult to excite, if background are things like traffic and wing strong enough to excite damage, difficult to work out damping, environmental facors like temperature impact stiffness and damping thus properties changes and therefore FRFs change
In experimental modal analaysis what is often done to make the process easier
Single degree of freedom curve fit, look at each peak individually
How can damping be estimated from an FRF
Using 3dB method, based on the width of the peaks
What is the equation for damped natural frequency
wd = wn * sqrt(1 - zeta^2)
What is the equation for resonant natural frequency
wr = wn * sqrt(1 - 2*zeta^2)
