Nonlinear Dynamics Lec 1 and 2 (but actually 5) Flashcards
Why is important to consider none linearity
In reality everyting is none linearto some extent or other, we can only model linear behaviour mathematically so pretend everything is linear
How can we check an equation is linear simply
Check that it contains no powers of the variables of interest higher than the first (interest meaning input and outputs)
Why could you aruge all interesting systems are nonelinear
Nonelinearity is a signiture of interactions and that systems are important or interesting because they contain or express interactions
How is nonlinearity necessary for interactions
without nonelinearity there are no electromagnetic force
electrons would not bind to nuclei and atoms could not exist, no atms, no molecules, no life
How can we have interaction in dynamics through mass and springs and not have nonelinearity
As shown through modal analysis, through a change of coordinate system you can treat masses as SDOF systems, i.e. not interacting
Where no dynamics related scenarios do we see nonlinearity
interactions between particles
mathemtatical epidemology
Where does nonlinearity usually arise in structural dynamics
From geometrical sources (large deflections) or non hookian material behaviour where none linear force diplacement behaviour is seen
What level of deflection in encastre beams roughly causes nonlinearity
deformations greater than the thickness of the beam itself, due to large deflections changing the length of the beam generating axial strains
What is the simplest possible nonlinear dynamic equation
Duffings quation mY.. + k1Y + k3Y^3 = F
What ar the common types of nonlinearity in dynamics
cubic stiffness bilinear stiffness/damping nonlinear (quadratic) damping coulomb friction piecewise linear stiffness
What happens to an FRF plot with standard cubic stiffness nonlinearity as in duffings equation
the k3 y^3 term acts like an ever increasing stiffness with higher excitation stiffer the structure gets and the FRF peak move to the right
If a structure is known to be nonelinear what can be done to work out a true resonant frequency
If standard cubic stiffness nonlinearity, apply a small excitation will lead to a small y value thus y^3 becomes very small and the nonlinear term disappears. If a large excitation is applied without realising and use FRF to work back to physical properties get increased stiffness to true
What happens to nonlinear FRFs due to increased excitation
If polynomial nolinearity, low excitations linear terms dominate and FRF indistinguishable from underlying system, higher excitaion FRF will deviate or distort from linear term
When is the only time you wouldnt get sin(wt) in, sin(wt) out in a linear system
transient phase at start, but this quickly decays after short time
In nonlinear system what is the output dependent on
amplitude, velocity and frequency
What physical components often lead to nonlinearities
actuators, bearings, linkages or elastomeric elements
What is a bode plot
plot of the magnitude and phase separately against frequency
What is a Nyquist/Argand representation
Plots the real part against the imaginary part - a locus in the complex planes as frequency varies
What shape is given on a nyquist plot for a linear system
Nyquist plot for a mobility FRF is an ellipse
Why is the nyquist plot of a receptane FRF for a linear system not a circle
as FRF is equal to 1/(mw^2 + icw + k)
as frequency tends to zero FRF tends to 1/k
as frequency tends to infinity FRF tends to 0
Dont get back to 1/k
For velocity pure circle as tend to zero
Draw a sketch of a linear system bode and nyquist plot
See presentation
Whats the simplest system we can consider for a nonlinear system
Consider a SDOF oscialltor with separate nonlinear damping and stiffness term
my.. + fd(y.) + fs(y) = x(t)
In duffiings equation what does the nonlinear stiffness look like
k3*y^3
if k3 positive system gets stiffer as amplitude increases i.e hardening noe linearity
if k3 negative system gets softer
What does k3 control in duffings equation
Controls extent on nonlinearity if big none linear at small forces, if small need to push to see none linear
Draw a diagram of what happens to the FRF bode and nyquist plot with a positive cubic stiffness and where might you see this
See presentation
higher level of excitation restoring force greater than expected from the linear term alone
effect increases as the forcing level increases hardening
in clamped plates and beams
Draw a diagram of what happens to the FRF bode and nyquist plot with a negative cubic stiffness and where might you see this
See presentation
Effectives stiffness decreases as the level of excitation increases, referred to as softening
buckling beams
Why are truly negative cubic stiffness referred to as unphyiscal
Because they are unstable, they may have more than one equilibrium position
What is a bilinear stiffness
Stiffness has a change over point f = k1y @ y>0, f = k2y @ y<0
What is the most extreme example of a bilinear system
impact osciallator, infinite stiffess on one side and zero stiffness on the other
Example being a ball bouncing against a wall
Where might you find bilinear dampers
Shock abosrbers, slow down motion in both direction, want to stay in contact with the road -> road holding high damping
Comfort for user -> low damping
How do bilinear stiffnesses/damping distort FRFs
Dont distort as one rare nonlinear system which display homogeneity, but this is only true if the position of the change in stiffness is at the origin. If offset by any degree, dont get homogenity if level of excitation is sufficiently high
What is quadratic damping
Most common form of polynomial damping, occurs when fluid flows through an orifice
Draw a diagram of what happens to the FRF bode and nyquist plot with a quadratic damping
See presentation
Increasing excitation increases damping
What is coloumb damping
Constant resistance to veloicty, dependent only on direction, common in any situation with relative motion of interfaces in contact
Draw a diagram of what happens to the FRF bode and nyquist plot with a coloumb damping
See presentation
Friction unusual as most evident at low levels of excittion, where in extreme cases stick slip can occur
At high levels of excitation friction breaks out and system will behave monially linearly
Higher damping at lower excitations
What is piecewise linear stiffness
Have three distinct regions of stiffness that are linear
What are the two special cases of piecewise linear stiffness
saturation or limiter nonlinearity where central stiffness is constant and the outer areas are zero
clearence or backlash nolinearity has central stiffness that is zero
Why is backlash nonlinearity worse than central saturation nonlinearity
if keep going back the break point moves
