Transfer Functions and Equations Quiz
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Questions and Answers

An equilibrium is a state of no ______.

change

The condition states that the second-order system whose characteristic polynomial is ms^2 + cs + k is stable if and only if m, c, and k have the same ______.

sign

For zero initial conditions the free response is ______, and the complete response is the same as the forced response.

zero

Consider the model ẋ + ax = f (t) and assume that x(0) = ______.

<p>0</p> Signup and view all the answers

The ______ is equivalent to the ODE. If we are given the ______ we can reconstruct the corresponding ODE.

<p>transfer function</p> Signup and view all the answers

The ______ of the transfer function is the characteristic polynomial, and thus the transfer function tells us something about the intrinsic behavior of the model, apart from the effects of the input and specific values of the initial conditions.

<p>denominator</p> Signup and view all the answers

If a model has more than one input, a particular transfer function is the ratio of the ______ over the input transform, with all the remaining inputs ignored (set to zero temporarily).

<p>output transform</p> Signup and view all the answers

Stability Test for Linear Constant-Coefficient Models: A constant-coefficient linear model is ______ if and only if all of its characteristic roots have negative real parts.

<p>stable</p> Signup and view all the answers

The model is ______ if one or more roots have a zero real part with no roots on the imaginary axis of multiplicity 2 or greater, and the remaining roots have negative real parts.

<p>neutrally stable</p> Signup and view all the answers

The model is ______ if any root has a positive real part.

<p>unstable</p> Signup and view all the answers

If a linear model is , then it not possible to find a set of initial conditions for which the free response approaches ∞ as t → ∞. However, if the model is un, there might still be certain initial conditions that result in a response that disappears in time.

<p>stable</p> Signup and view all the answers

Model:

<p>ẍ + 6ẋ + 25x = f (t)</p> Signup and view all the answers

Characteristic Equation:

<p>$s^2 + 6s + 25 = 0$</p> Signup and view all the answers

Natural Frequency:

<p>$ωn = \sqrt{\frac{k}{m}}$</p> Signup and view all the answers

Damping Ratio:

<p>$ζ = \frac{c}{2\sqrt{mk}}$</p> Signup and view all the answers

the natural frequency of oscillation of the free response will be ωn = ________/m

<p>k</p> Signup and view all the answers

the period of the oscillation is ____/ωn

<p>2π</p> Signup and view all the answers

the frequency of oscillation of the free response is ____

<p>b</p> Signup and view all the answers

the damping ratio is defined as the ratio of the a______tual value of ______ to its ______riti______al value, ζ = ____/√(2mk)

<p>c</p> Signup and view all the answers

The transfer functions X (s)/V (s) and Y (s)/V (s) are obtained by transforming both sides of each equation, assuming zero initial conditions.

<p>True</p> Signup and view all the answers

The desired transfer functions are given by equations (2) and (3).

<p>True</p> Signup and view all the answers

The forced response for x(t) and y(t) if the input is v(t) = 5u s (t) is 6 15 − 3e−5t + e−7t.

<p>True</p> Signup and view all the answers

Which of the following statements is true about stable systems?

<p>Stable systems have all poles with negative real parts.</p> Signup and view all the answers

What is the characteristic of an unstable system?

<p>At least one pole has a positive real part.</p> Signup and view all the answers

What can be inferred about marginally stable systems?

<p>They exhibit oscillatory behavior without exponential growth or decay.</p> Signup and view all the answers

Study Notes

Equilibrium and Stability

  • An equilibrium is a state of no change.
  • A system is stable if and only if m, c, and k have the same sign.
  • A system is stable if all its characteristic roots have negative real parts.

System Response

  • For zero initial conditions, the free response is zero, and the complete response is the same as the forced response.
  • The transfer function of a system tells us about its intrinsic behavior, apart from the effects of the input and specific values of the initial conditions.

Transfer Function

  • The transfer function is the ratio of the output transform over the input transform, with all the remaining inputs ignored.
  • The characteristic polynomial of the transfer function is equivalent to the ODE.

Model Classification

  • A model is marginally stable if one or more roots have a zero real part with no roots on the imaginary axis of multiplicity 2 or greater, and the remaining roots have negative real parts.
  • A model is unstable if any root has a positive real part.
  • A stable model does not have a response that approaches infinity as t → ∞.

Model Characteristics

  • The natural frequency of oscillation of the free response is ωn = √(k/m).
  • The period of the oscillation is 2π/ωn.
  • The frequency of oscillation of the free response is ωn.
  • The damping ratio is defined as ζ = a/√(2mk).

Forced Response

  • The forced response is obtained by transforming both sides of each equation, assuming zero initial conditions.
  • The forced response for x(t) and y(t) if the input is v(t) = 5u(t) is 6 15 − 3e−5t + e−7t.

System Properties

  • In a stable system, it is not possible to find a set of initial conditions for which the free response approaches infinity as t → ∞.
  • An unstable system has a response that approaches infinity as t → ∞.
  • Marginally stable systems have certain initial conditions that result in a response that disappears in time.

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Description

Test your understanding of transfer functions and their role in solving equations in this quiz. Explore how transforming both sides of an equation can help solve for the ratio X(s)/F(s) and learn about the concept of transfer functions. Discover how transfer functions can be used as multipliers to obtain the forced response transform from the input transform.

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