Nonlinear Dynamics Lec 1 and 2 (but actually 5)

Question 1

Why is important to consider none linearity

Answer 1

In reality everyting is none linearto some extent or other, we can only model linear behaviour mathematically so pretend everything is linear

Question 2

How can we check an equation is linear simply

Answer 2

Check that it contains no powers of the variables of interest higher than the first (interest meaning input and outputs)

Question 3

Why could you aruge all interesting systems are nonelinear

Answer 3

Nonelinearity is a signiture of interactions and that systems are important or interesting because they contain or express interactions

Question 4

How is nonlinearity necessary for interactions

Answer 4

without nonelinearity there are no electromagnetic force

electrons would not bind to nuclei and atoms could not exist, no atms, no molecules, no life

Question 5

How can we have interaction in dynamics through mass and springs and not have nonelinearity

Answer 5

As shown through modal analysis, through a change of coordinate system you can treat masses as SDOF systems, i.e. not interacting

Question 6

Where no dynamics related scenarios do we see nonlinearity

Answer 6

interactions between particles

mathemtatical epidemology

Question 7

Where does nonlinearity usually arise in structural dynamics

Answer 7

From geometrical sources (large deflections) or non hookian material behaviour where none linear force diplacement behaviour is seen

Question 8

What level of deflection in encastre beams roughly causes nonlinearity

Answer 8

deformations greater than the thickness of the beam itself, due to large deflections changing the length of the beam generating axial strains

Question 9

What is the simplest possible nonlinear dynamic equation

Answer 9

Duffings quation mY.. + k1Y + k3Y^3 = F

Question 10

What ar the common types of nonlinearity in dynamics

Answer 10

cubic stiffness
bilinear stiffness/damping
nonlinear (quadratic) damping
coulomb friction
piecewise linear stiffness

Question 11

What happens to an FRF plot with standard cubic stiffness nonlinearity as in duffings equation

Answer 11

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

Question 12

If a structure is known to be nonelinear what can be done to work out a true resonant frequency

Answer 12

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

Question 13

What happens to nonlinear FRFs due to increased excitation

Answer 13

If polynomial nolinearity, low excitations linear terms dominate and FRF indistinguishable from underlying system, higher excitaion FRF will deviate or distort from linear term

Question 14

When is the only time you wouldnt get sin(wt) in, sin(wt) out in a linear system

Answer 14

transient phase at start, but this quickly decays after short time

Question 15

In nonlinear system what is the output dependent on

Answer 15

amplitude, velocity and frequency

Question 16

What physical components often lead to nonlinearities

Answer 16

actuators, bearings, linkages or elastomeric elements

Question 17

What is a bode plot

Answer 17

plot of the magnitude and phase separately against frequency

Question 18

What is a Nyquist/Argand representation

Answer 18

Plots the real part against the imaginary part - a locus in the complex planes as frequency varies

Question 19

What shape is given on a nyquist plot for a linear system

Answer 19

Nyquist plot for a mobility FRF is an ellipse

Question 20

Why is the nyquist plot of a receptane FRF for a linear system not a circle

Answer 20

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

Question 21

Draw a sketch of a linear system bode and nyquist plot

Answer 21

See presentation

Question 22

Whats the simplest system we can consider for a nonlinear system

Answer 22

Consider a SDOF oscialltor with separate nonlinear damping and stiffness term
my.. + fd(y.) + fs(y) = x(t)

Question 23

In duffiings equation what does the nonlinear stiffness look like

Answer 23

k3*y^3
if k3 positive system gets stiffer as amplitude increases i.e hardening noe linearity
if k3 negative system gets softer

Question 24

What does k3 control in duffings equation

Answer 24

Controls extent on nonlinearity if big none linear at small forces, if small need to push to see none linear

Question 25

Draw a diagram of what happens to the FRF bode and nyquist plot with a positive cubic stiffness and where might you see this

Answer 25

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

Question 26

Draw a diagram of what happens to the FRF bode and nyquist plot with a negative cubic stiffness and where might you see this

Answer 26

See presentation
Effectives stiffness decreases as the level of excitation increases, referred to as softening
buckling beams

Question 27

Why are truly negative cubic stiffness referred to as unphyiscal

Answer 27

Because they are unstable, they may have more than one equilibrium position

Question 28

What is a bilinear stiffness

Answer 28

Stiffness has a change over point f = k1y @ y>0, f = k2y @ y<0

Question 29

What is the most extreme example of a bilinear system

Answer 29

impact osciallator, infinite stiffess on one side and zero stiffness on the other
Example being a ball bouncing against a wall

Question 30

Where might you find bilinear dampers

Answer 30

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

Question 31

How do bilinear stiffnesses/damping distort FRFs

Answer 31

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

Question 32

What is quadratic damping

Answer 32

Most common form of polynomial damping, occurs when fluid flows through an orifice

Question 33

Draw a diagram of what happens to the FRF bode and nyquist plot with a quadratic damping

Answer 33

See presentation

Increasing excitation increases damping

Question 34

What is coloumb damping

Answer 34

Constant resistance to veloicty, dependent only on direction, common in any situation with relative motion of interfaces in contact

Question 35

Draw a diagram of what happens to the FRF bode and nyquist plot with a coloumb damping

Answer 35

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

Question 36

What is piecewise linear stiffness

Answer 36

Have three distinct regions of stiffness that are linear

Question 37

What are the two special cases of piecewise linear stiffness

Answer 37

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

Question 38

Why is backlash nonlinearity worse than central saturation nonlinearity

Answer 38

if keep going back the break point moves