MAE 488 Final Conceptual Terms

Calculate the roots of 𝑠^2+2*z*wn*s+wn^2 Simplify your answer. There is more space on page 2 if needed but write your answer below.

(s1, s2) = -z*wn+-j*wn*sqrt(1-z^2)

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

1) Constant 2) Exponential 3) Sinusoid

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

1) I1(s)/V1(s) 2) I1(s)/V2(s) 3) I2(s)/V1(s) 4) I2(s)/V2(s)

State the two golden rules that govern the operational amplifier

1) Input terminals have no current 2) Op Amp maintains 0 differential input voltage.

Consider the following transfer function that relates the output Y(s) and the input X(s). H(s) = 25 / s^2 + 3s + 36 1) Is the transfer function in standard form? 2) What is the time constant? 3) Suppose the input is X(s) = 2/s, what is the Laplace transform of the output, Y(s)?

1) No 2) 2/3 3) Y(s) = 50 / s^3 + 3s^2 + 36s

How many variables fully describe the state of a 2nd order system?

2

The input to a stable, linear, time-invariant system is the sinusoid x(t) = 5sin(2pit + pi/2). What is the frequency of the output?

2pi rad/sec

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

5 x 2

The displacement of a system in meters is described by y(t) = 7sin(2*pi*t) where t is in seconds. Determine the amplitude, frequency in rad/s, frequency in Hz., the period, and the phase.

A = 7 w = 2pi rad/s phase = 0 T = 1 s f = 1 Hz

A sinusoidal function is described as f(t)=4sin(2t+pi/4) in. A) Circular Frequency? B) Period? C) Amplitude? D) Phase Angle in Degrees?

A) 2 rad/sec B) pi sec C) 4 in D) 45 degrees

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

A) 3 x 3 B) 3 x 2 C) 5 x 3 D) 5 x 2

Consider the following differential equation for some system: 𝑤̈(𝑡)+3𝑤̇(𝑡)+2𝑤(𝑡)=3e^(-at)u(t)+5cos(5t+pi)u(t). A) Input? B) Output? C) Order? D) Linear? E) Independent variable? F) Dependent variable?

A) 3e^(-at)u(t)+5cos(5t+pi)u(t) B) w(t) C) 2nd order D) yes E) t F) w

Fill in the corresponding response types for each value or range of the damping ratio(z). A) Zeta less than zero B) Zeta equal to zero C) Zeta between 0 and 1 D) Zeta equal to 1 E) Zeta greater than 1

A) Unstable B) Sinusoidal C) Underdamped D) Critically damped E) Overdamped

Match the value of the damping ratio, 𝜁, to the type of free response for a 2nd order system. Enter the letter corresponding to the response next to each value(s) of the damping ratio. A) 𝜁 <0 B) 𝜁 =0 C) 0<𝜁 <1 D) 𝜁 =1 E) 𝜁 >1

A) Unstable B) Sinusoidal with constant amplitude C) Underdamped D) Critically damped E) Overdamped

Which of these two exponential functions decays faster? A) f(t)=2e^(-7t) B) f(t)=2e^(-2t)

A) f(t)=2e^(-7t)

The solution to the ODE given below 𝑥̇(𝑡)+𝑎𝑥(𝑡)=𝑏𝑢(𝑡), 𝑥(0)= 𝑥0 can be obtained by taking the inverse Laplace transform of: 𝑋(𝑠)= 𝑥0/s+a + b/a(1/s - 1/s+a). Write the expression for: A) Free response B) Forced response C) Steady - state response D) Transient response

A) x(t)free = x0a^(-at) u(t) B) x(t)forced = b/a(1-e^(-at) u(t) C) x(t)ss = b/a u(t) D) x(t)transient = (x0 - b/a) e^(-at) u(t)

You are given the magnitude M(w) and phase @(w) of the frequency response for a given system. Suppose the Input is A1 sin(wt + @1). Write an expression for the output in terms of the given information.

A1*M(w)*sin[w(hat) + @1 + @(2)]

Care must be taken when using a differentiator due to

Amplifying noise

Which of the following is NOT one of the reasons we said why control is used. A) Modify the dynamic response B) Stabilize an unstable plant C) Compensate for a disturbance D) Change the input

D) Change the input

Consider the step response of a 2nd order system with complex conjugate roots. If the damping ratio, 𝜁, increases what happens to the time constant, 𝜏 ?

Decreases

A marginally stable system will always return to and settle at its equilibrium position.

False

Eigenvectors are unique.

False

Global stability is a characteristic of a system that depends on the initial conditions.

False

If the initial conditions are all zero, the forced response is zero.

False

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

False

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

False

Numerator dynamics do not affect the response.

False

The Bode plot is a plot of the frequency response.

False

The ODE: z^(..)+4z^(.)z-5t+14z=0 is homogeneous.

False

The characteristic polynomial is the numerator of the transfer function.

False

The current in a resistor flows from the negative to positive terminal.

False

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

False

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

False

The gain M(w) is the ratio of the input amplitude to the output amplitude.

False

The initial conditions of a system are always equal to the value of the response an infinitesimal time after t=0

False

The poles of a transfer function are the zeros of the equation obtained by setting the numerator polynomial of the transfer function to zero.

False

The resonant frequency is the same as the natural frequency.

False

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

False

The system with transfer function 5s+5/2s^2+18S+40 is unstable.

False

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

False

The transfer function can be used to calculate the free response.

False

The transient response is the response due to initial conditions.

False

The mass moment of inertia of a thin solid disk of mass 𝑚 and radius 𝑟, about the center of the disk is ½ 𝑚𝑟ଶ. What is the mass moment of inertial about a point halfway from the center to the outer edge of the disk? Recall that Ia=Iz+md^2.

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

Define control

Influencing or forcing a desired behavior of a system.

A proportional (P) controller results in a response with too much steady-state error. Which of the following will help reduce the error?

Integrator

DC Motors are based on two concepts from physics:

Lorentz force and Faraday law

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

P = iv

What is the output of an ideal closed loop control system if the input is a unit ramp?

Same as the input (if they have the same units)

If the plots 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

If the current output of an element or system depends ONLY on the current input, it is

Static and causal

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

Static equilibrium

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

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

Define transfer function.

The transfer function is the Laplace Transform of the Outputs over the Laplace Transform of the Inputs assuming zero initial conditions.

Any continuous periodic signal can be represented as the sum of the constant and an infinite sum of the sinusoids.

True

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

True

Pole placement is a control design technique based on equating the desired closed loop characteristic polynomial and the actual closed loop characteristic polynomial which is expressed in terms of the unknown gain.

True

The damping ratio is the ratio of the damping in a system to the critical damping value.

True

The definitions of the state space model is not unique.

True

The frequency transfer function is given by T(jw) = T(s)|z=jw

True

The impulse function is the derivative of the step function.

True

The magnitude of the output of a 2nd order linear time-invariant systemin response to a sinusoidal input is a function of the input frequency w and the damping ratio z.

True

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

The product of a complex number and its complex conjugate is a real number (no imaginary part).

True

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.

True

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 time constant is a measure of how fast a decaying exponential decays.

True

When wd = wn? Your answer should be in the form "When the parameter _________ has a value of _______."

When the parameter (zeta) has a value of (zero).

A PI controller results in a response with too much overshoot. Will a derivative term help?

Yes

What are the two components of the superposition principle?

additive property and homogeneity

Back emf is proportional to the motor

angular velocity

All physically realizable elements or systems are

causal

Poles __________ the origin will be dominant with respect to non-dominant poles.

closer to

The frequency transfer function is a _______ function of the ____________ of the __________.

complex frequency input

If the current output of an element or system depends on the current and past input, it is

dynamic and causal

The complete response is the sum of the ___________ response and the _____________ response. It is also the sum of the ________________ response and the ________________ response.

free and forced transient and steady-state

If a system is linear and a sinusoidal input is applied to the system, the steady-state output will have the same

frequency

The free response is faster when a pole is moved ________ in the complex plane.

left

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

parallel

Kirchoff's Current law states that the sum of the currents entering a node is equal to

sum of currents exiting the node

A linear system obeys what principal?

superposition

What is the time constant of the response x(t)=4e^(-0.2t)?

tc = 1/.2 = 5

The frequency of an un-damped or underdamped free response increases as a pole with positive imaginary part is moved _______ in the complex plane.

up

A damper resists ___________________ but a spring resists __________________. Choose from the following: acceleration, inertia , displacement, gravity, velocity, mass.

velocity displacement

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

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

Kirchoff's Voltage law states that the signed sum of the voltage in a closed circuit or loop is equal to

zero