MAE488 Exam 2

A damper resists ________ but a spring resists ________.

Choose from the following:

acceleration, inertia, displacement, gravity, velocity, or mass

velocity

displacement

If the work done by force is path independent, we say the force is a non-conservative force.

T/F

False

The sum of kinetic and potential energy in a system is always a constant.

T/F

False

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.

T/F

True

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

equilibrium posistion

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

T = 1/2mv_cg^2 + 1/2I_cg w^2

The equivalent spring constant for multiple springs in a series is simply the sum of all of the spring constants.

T/F

False

The mass moment of inertia of a thin solid disk of mass m and radius r, about the center of the disk is 1/2mr^2. What is the mass moment of inertial about a point halfway from the center to the outer edge of the disk? Recall that I_a = I_s + md^2

Ia=(1/2)mr^2+md^2=(1/2)mr^2+m*(r/2)^2=3/4mr^2

A damper resist

velocity

If the work done by a force is path-independent, we say the force is a conservative force

True

What is Newtons second law for a translational system and a rotational system

Translational___________________ Rotational______________________

F=ma

Iw_dot=M

When an object is undergoing rolling motion the linear velocity is related to the angular velocity as v=rw

true

The equivalent spring constant to two springs in parallel is the sum of the spring constant

true

A spring resists

displacement

Draw a FBD for a mass m suspended by a spring k, damper c, and applied force f(t) acting down

A damper resist velocity whereas a spring resist displacement

true

The critical damping value for a mass spring damper system, c=2sqrt(mk) is the dividng line between an oscillatory response and a non-oscillatory resposne

true

The sum of the kinetic and potential energy in a system is a constant for conservative forces

true

The free response of an ideal mass-spring system will eventually approach a constant value

false

State your assumptions and neatly draw FBD. Write the equation of motion for m1

m1x_doubledot-k2x2+(k1+k2)x1=f

If a model has the inputs v1(t) and v2(t) and the outputs i1(t) and i2(t) what are the transfer functions

I1(s)/V1(s), I1(s)/V2(s), I2(s)/V1(s), I2(s)/V2(s)

The current in a resistor flows form the negative to the positive terminal

false

In an armature controlled DC motor the motor speed is controlled by varying the armature voltage

true

Impedance is the transfer function with the current output and voltage input

false

Dc motors are based on two concepts from physics:__________________ force and __________________________ law

Lorentz Faraday

The torque constant is the relationship between the motor torque and the voltage

false

State the two golden rules that govern the operational amplifier

No current through input terminals i1=i2=0

Zero voltage differential across input terminals, v_in=0

The equation for the power produced by or dissipated by an element is:

P=iV

Kirchoffs current law states that the sum of the currents entering a node is equal to

the sum of the currents exiting the node

Back emf is proportional to the motor

angular velocity or speed

Kirchoff's Voltage Law states that the signed sum of the voltages in a closed circuit or loop is equal to:

zero

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

parallel

Impedance is the transfer function with voltage output and current input

true

The Lorentz force equation is F=Ia*LxB. if the current is in the direction of the +x axis and the magnetic field is in the direction of the +z axis, what is the direction of the of the force?

-y axis

A differentiator can be constructed using only an operational amplifier and resistor and a capacitor

true

write the standard form of the state space model

x_dot=Ax+Bu

y=Cx+Du

What is Kirchoff's Current Law?

KCL says sum of the currents into a node = sum of the currents out of a node

The voltage drop across an inductor is L*di/dt

true

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

2

The state equation of a state space model is of the form: xdot=Ax+Bu

true

The equivalent resistance of two resistors in parallel is the sum of the resistances

false

A commutator changes the direction of the armature current

true

Operational amplifiers can be used to integrate and differentiate an input signal

true

The state variables in a RLC circuit are the capacitor________________ and the inductor _______________________

voltage

current

Capacitors store energy in the form of____

Charge

Inductors store energy in__________

A magnetic field

the _______ is proportional to the armature current.

Torque

Ohms law equation is

V=iR

A voltage divider is used to decrease the voltage at the output

True

Derive the equation for a voltage divider with V_in as the input R1 and R2 is the load resistor

Vo=(R2/(R2+R1))Vin

The state variables are the minimum set of variables required to describe the state of a system

True

Back EMF is the consequence of the Lorentz force law

False

Current is the derivative of charge with respect to time

True

A system has 2 inputs and 5 outputs. What is the size of the matrix transfer function

5x2

Eigenvectors are unique

false

The roots of the characteristic equation of a system are the same as the eigenvalues of the A matrix in the corresponding state space equations

True

The definition of the states in a state space model is not unique

True

A system has 2 inputs, 3 states, and 5 outputs. What are the sizes of the following matrices in the state space model?

A_______________

B_______________

C_______________

D_______________

A=3x3

B=3x2

C=5x3

D=5x2

The eigenvalues of the A matrix are the same as the roots of the corresponding characteristic equation

true

Integration in the time domain corresponds to multiplication by s in the frequency domain

false

What is the size of the matrix of transfer functions for a system with 2 inputs, 4 states, 3 outputs

3x2

write the standard form of the state space model

x_dot=Ax+Bu

y=Cx+Du

A system has 2 inputs 4 states and 3 outputs. What is the size of each of the matrices in the state space model

A_______________

B_______________

C_______________

D_______________

A=4x4

B=4x2

C=4x4

D=3x2

Name two modeling techniques we have discussed other than Simulink and differential equations

Block Diagrams

Transfer Functions

A block diagram can be reduced to a single transfer function block

true

What is the equation used to convert a state space model to a transfer function

Y(s)/U(s)=C(sI-A)^-1B+D

The state variables are the minimum set of variables required to describe the state of a system

True

The eigenvalues of a system can be computed from the solutions of

| lambda*I-A|=0

true

Full state feedback is the configuration where all of the states are fed back and compared to the input. If full state feedback is used what must the C and D matricies in the input equation be?

C identity matrix

D zero matrix

The ____________________ is used to analyze input signals

transfer function

m is

number of inputs

p is

number of outputs

n is

the order of the system

integration in the time domain is equivalent to division by s in the frequency domain.

true

A=

n*n

B=

n*m

C=

p*n

D=

p*m

What is the general form of the free response of a 1st order system in time (t) and the time constant (r)?

x(t) = x(0)e^-t/T

The free response of a 2nd order system has three factors. What are they generally?

constant

exponential

sinusoid

Fill in the corresponding response types for each value or range of the damping ratio. Choose from: overdamped, underdamped, critically damped, unstable, sinusoidal

zeta less than zero

unstable

Fill in the corresponding response types for each value or range of the damping ratio. Choose from: overdamped, underdamped, critically damped, unstable, sinusoidal

zeta equal zero

sinusoidal

Fill in the corresponding response types for each value or range of the damping ratio. Choose from: overdamped, underdamped, critically damped, unstable, sinusoidal

zeta between 0 and 1

underdamped

Fill in the corresponding response types for each value or range of the damping ratio. Choose from: overdamped, underdamped, critically damped, unstable, sinusoidal

zeta equal 1

critically damped

Fill in the corresponding response types for each value or range of the damping ratio. Choose from: overdamped, underdamped, critically damped, unstable, sinusoidal

zeta greater than 1

overdamped

The response of a system with both dominant and non-dominant poles is almost the same as the response of the system with just the dominant poles.

T/F

true

Poles ____________ (closer to / further from) the origin will be dominant with respect to non-dominant poles.

closer to

When is omega_d = omega_n? Your answer should be in the form "When the parameter _________ has a value of ________."

When the parameter zeta or c has a value of zero.

The frequency of an un-damped or underdamped free response increases as a pole with positive imaginary part is moved _____ in the complex plane. Choose from: up, down, left, right

up

The free response is faster when a pole is moved ________ in the complex plane. Choose from: up, down, left, right.

left

What box is labeled 1 in terms of zeta and/or omega_n?

omega_n

What box is labeled 2 in terms of zeta and/or omega_n.

-zeta omega_n

What is the box labeled 3 in terms of t?

1/T

What is the box labeled 4 in terms of sigma?

cos sigma

what is j omega d equivalen to

jwnsqrt(1-zetta^2)

If the poles of a stable, 2nd order system are moved horizontally to the right, the free response of the system will be ___ compared to the original system.

slower

A 4th order system can be written in terms of two second order transfer functions. System 1 has a time constant of T=1.5 and system 2 has a time constant of T=0.15 which system has the dominant roots?

system 1

The roots of a 2nd order system result in a non-oscillatory free response. Therefore the imaginary part of the roots is zero

true

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 _____________ the damping ratio of system 2

greater than

The damping ratio of a second order system determines the type of response

true