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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) = ______.
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The ______ is equivalent to the ODE. If we are given the ______ we can reconstruct the corresponding ODE.
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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.
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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).
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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.
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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.
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The model is ______ if any root has a positive real part.
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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.
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Model:
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Characteristic Equation:
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Natural Frequency:
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Damping Ratio:
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the natural frequency of oscillation of the free response will be ωn = ________/m
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the period of the oscillation is ____/ωn
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the frequency of oscillation of the free response is ____
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the damping ratio is defined as the ratio of the a______tual value of ______ to its ______riti______al value, ζ = ____/√(2mk)
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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.
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The desired transfer functions are given by equations (2) and (3).
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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.
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Which of the following statements is true about stable systems?
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What is the characteristic of an unstable system?
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What can be inferred about marginally stable systems?
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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.
