MAE488 Quiz 2

Transfer Function

A transfer function is the Laplace transform of the output divided by the laplace transform of the input, assuming the initial conditions are equal to zero.

order

highest derivative

response of a system

output/s of a system in response to an initial condition or input

free response

part of the response that is due to the initial condition. If the initial conditions are zero, there is no free response

Forced response

the part of the response due to the forcing function (input). If the input is zero there is no forced response

steady state response

the part of the response that remains with time

transient response

part of the response that disappears with time

complete response

the sum of the free and forced response. also the sum of the steady state and transient response

damping ratio

the ratio between damping in a system and the critical damping value of the system

critically damped

damping value equal to the critical damping value

fastest possible without exceeding the final steady state value

zeta = 1

over damped

damping value greater than the critical damping value and will not oscillate

underdamped

damping alue that is less than the critical damping value and will oscillate

Stability

characteristic of a system that has a bounded output for any bounded input. The free response of a stable system approaches zero

equilibrium

a state of the system that will not change except if disturbed from the state by an input

dominant roots (of a stable system)

those that result in the longest lasting terms in the transient responsse

impulse

a mathematical function designed to represent an input that is applied for an infinitesimal time

ration function

any function that can be defined as a ratio of polynomials

proper transfer function

is a transfer function in which the degree of the numerator is less thatn or equal to the degree of the denominator

strictly proper transfer function

a transfer function in which the degree (order) of the numerator is strictly less than the degree of the denominator

normal form of a transfer function

the form in which the coefficient of the highest power of s in the denominator is one

the characteristic polynomial

the denominator of the transfer function

characteristic equation

results from setting the characteristic polynomial to zero

homogeneous v nonhomogeneous

right hand side is equal to 0 is homogeneous

four ways of solving odes

direct integration, separation of variables, trial-solution, laplace transform

Final Value Theorem

limit of x(t) as t approaches infinity = limit of sX(s) as s approaches 0

FVT tells us the final steady state value

only valid if both xt and dxdt have laplace transforms and xt approaches a s constant value as t approaches infinity

laplace transform steps

1. Take the laplace transform of both sides of the ode

2. solve for the dependent variable as a function of s

3. take the inverse laplace transform of the result

natural frequency of a system

w_n = sqrt(k/m)

k = spring constant

m = mass

unit: rad/sec

damped natural frequency

c=0

no damping

c >= 2sqrt(mk)

wd = zero or imaginary

critical damping value

c >= 2sqrt(mk)

damping ratio/factor

zeta can be used to determine wheter or not a system is stable

globally stable

stable for any initial condition

locally stable

only stable for some initial conditions

marginally stable

will oscillate about an equilibrium point

characteristic equation

mx’’ + cx’ + kx =0

ms²+cs+k=0

pure sinosoid

damping ratio, zeta, = 0

unstable at what value

damping ratio is less than 0

damped sinusoid

zeta between 0 and 1