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- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/11%3A_Mechanical_Systems_with_Rigid-Body_Plane_Translation_and_Rotation/11.03%3A_ExamplesExamples of equations of motion for rigid bodies in plane motion
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/06%3A_General_Time_Response_of_First_Order_Systems_by_Application_of_the_Convolution_Integral/6.06%3A_Chapter_6_HomeworkEvaluate in all detail the convolution integral in Equation 6.2.4 to show that the exact response solution is \(x(t)=b c \tau_{1}\left[\left(t_{z}+\tau_{1}\right)\left(1-e^{-t / \tau_{1}}\right)-t\rig...Evaluate in all detail the convolution integral in Equation 6.2.4 to show that the exact response solution is \(x(t)=b c \tau_{1}\left[\left(t_{z}+\tau_{1}\right)\left(1-e^{-t / \tau_{1}}\right)-t\right]=b c \tau_{1}\left[t_{z}+\tau_{1}-t-\left(t_{z}+\tau_{1}\right) e^{-t / \tau_{1}}\right]\).
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/08%3A_Pulse_Inputs_Dirac_Delta_Function_Impulse_Response_Initial_Value_Theorem_Convolution_SumIn this chapter, we will continue analysis of pulse excitation and subsequent dynamic response. Most importantly, we will extend the analysis to the concept of the impulse, which in its simplest form ...In this chapter, we will continue analysis of pulse excitation and subsequent dynamic response. Most importantly, we will extend the analysis to the concept of the impulse, which in its simplest form is the product of excitation and the duration of excitation. In particular, the Dirac delta or ideal unit-impulse function is essential in the theory of linear systems and also useful in practical applications.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/17%3A_Introduction_to_System_Stability-_Frequency-Response_CriteriaTo examine the stability of a closed-loop control system, we examine in particular the frequency response of the associated open-loop system, for which the transfer function \(O \operatorname{LTF}(s)=...To examine the stability of a closed-loop control system, we examine in particular the frequency response of the associated open-loop system, for which the transfer function \(O \operatorname{LTF}(s)=G(s) \times H(s)\) was described in Section 16.6. Recall that root-locus analysis is based on the characteristic equation \(1+G(s) \times H(s)=0\), or \(G(s) \times H(s)=-1\); obviously, the right-hand-side number −1 is important to the closed-loop poles and to closed-loop system stability.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/00%3A_Front_Matter
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/18%3A_Appendix_A-_Table_and_Derivations_of_Laplace_Transform_Pairs/18.01%3A_A.1-_Table_of_Laplace_Transform_Pairs\(\frac{\omega_{n}^{2}}{\omega_{d}} e^{-\zeta \omega_{n} t} \sin \omega_{d} t\) \(\int_{t=0}^{t=\infty} e^{-s t} f(t) d t, \text { or } \int_{t=0^{-}}^{t=\infty} e^{-s t} f(t) d t\) In these examples,...\(\frac{\omega_{n}^{2}}{\omega_{d}} e^{-\zeta \omega_{n} t} \sin \omega_{d} t\) \(\int_{t=0}^{t=\infty} e^{-s t} f(t) d t, \text { or } \int_{t=0^{-}}^{t=\infty} e^{-s t} f(t) d t\) In these examples, MATLAB finds the forward transform \(L[\sinh (a t)]= a /\left(s^{2}-a^{2}\right)\), and the inverse transform \(L^{-1}\left[(s+a) /\left[(s+a)^{2}+\omega^{2}\right]\right]=e^{-a t} \cos \omega t\).
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/17%3A_Introduction_to_System_Stability-_Frequency-Response_Criteria/17.04%3A_The_Nyquist_Stability_CriterionWhich, if either, of the values calculated from that reading, \(\mathrm{GM}=(1 / \mathrm{GM})^{-1}\) is a legitimate metric of closed-loop stability? \(\PageIndex{4}\) includes the Nyquist plots for b...Which, if either, of the values calculated from that reading, \(\mathrm{GM}=(1 / \mathrm{GM})^{-1}\) is a legitimate metric of closed-loop stability? \(\PageIndex{4}\) includes the Nyquist plots for both \(\Lambda=0.7\) and \(\Lambda =\Lambda_{n s 1}\), the latter of which by definition crosses the negative \(\operatorname{Re}[O L F R F]\) axis at the point \(-1+j 0\), not far to the left of where the \(\Lambda=0.7\) plot crosses at about \(-0.73+j 0\); therefore, it might be that the appropria…
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/16%3A_Introduction_to_System_Stability_-_Time-Response_Criteria/16.01%3A_General_Time-Response_Stability_Criterion_for_LTI_SystemsIn this derivation, we shall consider explicitly only the type of input function whose Laplace transform is a fraction of polynomials in the Laplace variable \(s\), with the numerator polynomial being...In this derivation, we shall consider explicitly only the type of input function whose Laplace transform is a fraction of polynomials in the Laplace variable \(s\), with the numerator polynomial being of lower degree than the denominator polynomial. (However, the same stability criterion that we derive for this special type of input function can be derived for any physically realizable input function.) The following are simple examples of this type of input function, which are defined to be non…
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/07%3A_Undamped_Second_Order_SystemsThe ideal 2 nd order mass-damper-spring system was introduced in Section 1.9, and a theoretical response solution for an undamped system by elementary ODE methods was demonstrated in Section 1.10. In ...The ideal 2 nd order mass-damper-spring system was introduced in Section 1.9, and a theoretical response solution for an undamped system by elementary ODE methods was demonstrated in Section 1.10. In this chapter, we explore ideal undamped 2 nd order systems in greater detail, deriving theoretical response solutions by means of Laplace transformation with application of the inverse convolution transform from Chapter 6.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/15%3A_Input-Error_Operationsproportional, integral, and derivative types of control
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Introduction_to_Linear_Time-Invariant_Dynamic_Systems_for_Students_of_Engineering_(Hallauer)/05%3A_Basic_Electrical_Components_and_Circuits/5.01%3A_IntroductionBut electrical variables (voltage, current, charge) and circuits are quite different physically than mechanical variables and systems, and the physical laws and methods for deriving ordinary different...But electrical variables (voltage, current, charge) and circuits are quite different physically than mechanical variables and systems, and the physical laws and methods for deriving ordinary differential equations that describe behaviors of electrical circuits also are entirely different. This chapter is an introduction to the theory of electrical circuits and basic analog electronics, and to some common but simple practical applications of the theory.
