MAE 488 Exam 2 (with Quizzes 4-7)

What happens to the free response of a system when it's stable poles (roots of the characteristic equation) are moved to the left on the complex plane?

It decays faster

What happens to the free response of a system when it's stable poles (roots of the characteristic equation) are moved downward (i.e. towards the real axis) on the complex plane?

The frequency decreases

What type of free response will a system have if it has poles on the jw axis?

Oscillatory

(T/F) The magnitude of the response of a system at resonance depends on the damping ratio ζ.

True

Label these response types from L to R, top to bottom:

x(t) = x0e^(-at) + b/a(1 - e^(-at))

x(t) = b/a + (x0 - b/a)*e^(-at)

free, forced, steady-state, transient

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

free response or initial condition response

The ___________________is the part of response due to the forcing function (i.e. input). If the input is zero, there is no forced response.

forced response

The ____________________ is the part of the response that remains with time

steady-state response

The _______________ is the part of the response that disappears (decays to zero) with time.

transient response

The characteristic equation of a 2nd order response is:

ms^2 + cs + k = 0

The roots of the characteristic equation of a 2nd order response are:

a +/- jb, where a = -c/2m and b = sqrt(k/m - (c/2m)^2) = wd

The damping coefficient is represented by the letter _________ , which reduces the frequency of oscillation from the __________

c; natural frequency

____________ is the natural frequency

wn

If c = 0, the system doesn't have any damping and is equivalent to a _____________

mass-spring system

_____________ is the damped frequency

wd

If wd is zero or imaginary, the system will NOT _________

oscillate, and the roots will be real and equal

The value of c for which wd will be zero is ________. This is known as the __________ value

c = 2*sqrt(mk); critical damping value

An oscillatory response has _________ roots, while a non-oscillatory response has _______ roots.

complex conjugate; real

ζ = c/(2*sqrt(mk)) is known as the _________

damping ratio/factor, or the ratio of c to the critical damping value.

The standard form is _______________

Y(s) = wn^2/(s^2 + 2ζwn*s + wn^2) (laplace table #21)

wd = _________

wn*sqrt(1 - ζ^2)

A 4th order system can be written in terms of two transfer functions. Their time constants are T1 > T2. Which system has the dominant roots?

T1

(T/F) The roots of a 2nd order system result in an undamped oscillatory free response. Therefore the imaginary part of the roots is zero.

False

What is the locus of all possible stable roots with a constant natural frequency?

A half circle

The free response of two 2nd order systems with the same natural frequency are being compared. The free response of system 1 decays faster than the free response of system 2. Therefore the damping ratio of system 1 is ________________ than the damping ratio of system 2.

greater than

Dominate roots are roots that ____________

dominate the response

Steps to obtain a steady state response from a model:

Laplace both sides, solve for output, then inverse Laplace both sides.

f(t) = Asin(wt + phi); What is the amplitude and frequency?

A and w respectively

Kirchhoff's voltage law states that the signed sum of the voltages in a closed circuit or loop is equal to:

0

Kirchhoff's current law states that the sum of the currents entering a node is equal to:

The sum of the currents exiting the node.

Resistors in series are combined into an equivalent resistor in the same way that springs in _________ are combined into an equivalent spring.

parallel

The state variables in an RLC Circuit are the capacitor ____________ and the inductor ______________

voltage; current

(T/F) The current in a resistor flows from the negative to the positive terminal.

False

What is the voltage drop across an inductor?

L* di/dt

Current is the flow of___________

electrons

Current, i, equals (derivative) _______

dQ/dt

The unit of charge is__________

coulomb (C)

The current flowing in a voltage supply moves from the ______ terminal to the __________

negative; positive

P = ________

i*V

V = __________

i*R

Voltage and current are inverse. Current is ________ in series resistors and Voltage is _________ in parallel resistors.

equal, equal

The capacitance equation is:

v = Q/C

Capacitance is a measure of how much charge can be stored for a given ___________

voltage

The inductance equation is:

v = L*di/dt

The capacitance energy formula is:

1/2Cv^2

The inductance energy formula is:

1/2Li^2

Capacitors deal directly with _________, while resistors and inductors deal with ___________

voltage, current

state definition

condition at a specific time

state variable definition

minimum set of variables needed to describe the state of a system

Capacitors and inductors have the state variables of _________ and _________ respectively

voltage, current

A circuit has a voltage source in series with a resistor, capacitor, and an inductor. What is the minimum number of state variables required?

2

In an electrical circuit, the number of state variables are equal to the number of ____________

storage elements (inductors and capacitors)

dv/dt = _________

1/C*i

Capacitor Laplace:

V(s) = (1/sC)*I(s)

Inductor Laplace:

V(s) = (sL)*I(s)

Resistor Laplace:

V(s) = (R)*I(s)

Impedance laplace:

Z(s) = V(s)/I(s)

An Operational Amp is a voltage amplifier with a very large ____________

gain, G > 10^5

The two op-amp rules are:

1. Input terminals have no current (i1 = i2 = 0)

2. The op amp will try to maintain a zero differential input voltage (vin = 0)

capacitor equation

i = c*dv/dt

inductor equation

v = L*di/dt

___________ is the only thing that breaks the traditional parallel and series laws

capacitor

capacitor impedance

1/sC

inductor impedance

sL

resistor impedance

R

Types of DC motors

Armature and field controlled

Lorentz Force

Force generated on a current carrying conductor in the presence of a magnetic field.

F =

(i*L) x B

Rotor

rotating part

stator

non-rotating part

armature

wire coil the force is applied to (it rotates)

field

magnetic field generated by magnets/circuit

commutator

changes the direction of the armature current so the force is always in the same direction

Torque, T =

2Fr

Newton's 2nd law for translational and rotational system

f = ma; T = Iw

(T/F) A voltage divider is used to decrease the voltage at the output

true

For an armature controlled DC motor with a given geometry and field, the torque is proportional to

the armature current, ia

3 inputs and 4 outputs means:

4x3 matrix

(T/F) The commutator changes the direction of the armature current so the force is always in the same direction

True

If the current is in the direction of the +x axis and the magnetic field the +z axis, what is the force direction?

- y axis

(T/F) Impedence is a transfer function with the current output and the voltage input

False

(T/F) Back EMF is a consequence of the Lorentz force law

False

(T/F) In a field controlled DC motor, the motor torque is modified by changing the field.

True

Torque (2) =

Kt*ia

Kt is the _______

torque constant

(T/F) EMF is another name for voltage

True, electro motive force

Faraday's law

an emf is set up in a closed circuit located in a magnetic field whenever the total magnetic flux linking the circuit is changed

Back EMF definition

The change in flux due to the moving coil of wire, producing a voltage (back EMF) that opposed the applied voltage.

vb = (faraday's law)

BLv = Kb*w

Kt amd Kb are _____

equal (torque constant and back emf constant)

Difference b/w a field controlled DC motor and an armature

field circuit has adjustable voltage source and the armature current is fixed.

field controllers adjust the ________ field by changing the voltage

magnetic

A damper resists __________ but a spring resists _______________

velocity, displacement

(T/F) If the work done by a force is path independent, we say the force is a non-conservative force.

False

(T/F) The sum of the kinetic and potential energy in a system is always a constant.

False

(T/F) The parallel axis theorem allows one to calculate the moment of inertia about an arbitrary axis in terms of a known moment of inertia about a parallel axis.

True

In a vertical mass-spring damper system, gravity can be ignored if the displacement is defined with respect to ________

Equilibrium

The kinetic energy of a translating and rotating body with mass m, velocity v, moment of inertia I, and angular velocity w is:

KE = 1/2mv^2 + 1/2Iw^2

Time constant = ________________

-1/r = 1/(ζ*wn)

The dominate roots produces the largest ____________ value

damping ratio, ζ, therefore decays slower, which dominates the response