DOCSLIB.ORG

Appendix a the Laplace Transform

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix a the Laplace Transform Appendix A The Laplace Transform The Laplace transform was discovered originally by Leonhard Euler (1707–1783), the great 18th-century Swiss mathematician and physicist, but is named in honor of a French mathematician and astronomer Pierre-Simon Laplace (1749–1827), who used the transform in his work on probability theory. He was such a genius not only in mathematics that the great mathematician Simeon Poisson (1781–1840) labeled him the Isaac Newton of France, but also in politics that he could serve three regimes in revolutionary France – the republic, the empire of Napoleon, and the Bourbon restoration, having been bestowed a count from Napoleon and a marquis from Louis XVIII. The Laplace transform is a very useful tool in solving differential equations and much further, it plays an important role in dealing with linear time-invariant systems. A.1 Definition of the Laplace Transform The (unilateral or one-sided) Laplace transform is defined for a function x(t)ofa real variable t (often meaning the time) as ∞ X(s) = L{x(t)}= x(t)e−stdt (A.1) 0− where s is a complex variable, the lower limit, t−, of the integration interval is the instant just before t = 0, and x(t) is often assumed to be causal in the sense that it is zero for all t < 0. A.2 Examples of the Laplace Transform A.2.1 Laplace Transform of the Unit Step Function The unit step function is defined as 385 386 Appendix A δ lim 1 T T d (t ) ==lim rT (t) → (us(t + )–us (t – )) = us(t) T→0 T 0 T 2 2 dt T lim rT (t) T→0 us(t) 1 δ(t ) T t t t 00 0 (a) Unit step function (b) Rectangular pulse (c) Unit impulse function Fig. A.1 Unit step, rectangular pulse, and unit impulse functions 1fort ≥ 0 us (t) = (A.2) 0fort < 0 which is depicted in Fig. A.1(a). We can use Eq. (A.1) to obtain the Laplace transform of the unit step function as ∞ ∞ . L{ } (A=1) −st (A=2) −st (D=33) 1 −st∞ = 1 − = 1 us (t) us (t)e dt e dt e 0 (0 1) 0 0 −s −s s This Laplace transform pair is denoted by L 1 u (t) ←→ (A.3) s s A.2.2 Laplace Transform of the Unit Impulse Function The unit impulse function can be defined to be the limit of a rectangular pulse func- tion rT (t) with the pulsewidth T → 0 (converging to zero) or simply, the time derivative of the unit step function as 1 T T d δ(t) = lim rT (t) = lim us t + − us t − = us (t)(A.4) T →0 T →0 T 2 2 dt which is depicted in Fig. A.1(c). We can use Eq. (A.1) to obtain the Laplace transform of the unit impulse function as ∞ L{δ(t)}= δ(t)e−stdt = 1 0− This Laplace transform pair is denoted by L δ(t) ←→ 1(A.5) Appendix A 387 A.2.3 Laplace Transform of the Ramp Function The Laplace transform of the unit ramp function tus (t) is obtained as ∞ ∞ ∞ (A.1) (A.2) (D.36) t ∞ 1 (D.33) 1 L{tu (t)} = tu (t)e−stdt = te−stdt = e−st + e−stdt = s s 0 2 0 0 −s s 0 s This Laplace transform pair is denoted by L 1 tu (t) ←→ (A.6) s s2 A.2.4 Laplace Transform of the Exponential Function −at The Laplace transform of the exponential function e us (t) is obtained as ∞ ∞ ∞ . −at (A=1) −at −st (A=2) −at −st −(s+a)t (D=33) 1 L{e us (t)} e us (t)e dt e e dt = e dt 0 0 0 s + a This Laplace transform pair is denoted by L 1 e−atu (t) ←→ (A.7) s s + a A.2.5 Laplace Transform of the Complex Exponential Function Substituting σ + jω for a into (A.7) yields the Laplace transform of the complex exponential function as L −(σ+ jω)t (D.20) −σt e us (t) = e (cos ωt − j sin ωt)us (t) ←→ 1 s + σ ω = − j s + σ + jω (s + σ )2 + ω2 (s + σ )2 + ω2 L s + σ e−σt cos ωtu(t) ←→ (A.8) s (s + σ )2 + ω2 L ω e−σt sin ωtu(t) ←→ (A.9) s (s + σ )2 + ω2 A.3 Properties of the Laplace Transform Let the Laplace transforms of two functions x(t) and y(t)beX(s) and Y (s), respectively. 388 Appendix A A.3.1 Linearity The Laplace transform of a linear combination of x(t) and y(t) can be written as L α x(t) + βy(t) ←→ αX(s) + βY (s) (A.10) A.3.2 Time Differentiation The Laplace transform of the derivative of x(t) w.r.t. t can be written as L x (t) ←→ sX(s) − x(0) (A.11) Proof ∞ −st ∞ (A.1) dx (D.36) ∞ (A.1) L{ } = = −st − − −st = − x (t) e dt x(t)e 0 ( s) x(t)e dt sX(s) x(0) 0 dt 0 Repetitive application of this time differentiation property yields the Laplace trans- form of n th-order derivative of x(t) w.r.t. t as L x(n)(t) ←→ sn X(s) − sn−1x(0) − sn−2x (0) −···−x(n−1)(0) (A.12) A.3.3 Time Integration The Laplace transform of the integral of x(t) w.r.t. t can be written as t L 1 1 0 x(τ)dτ ←→ X(s) + x(τ)dτ (A.13) −∞ s s −∞ t This can be derived by substituting −∞ x(τ)dτ and x(t)forx(t) and x (t)into Eq. (A.11) as L t 0 x(t) ←→ X(s) = sL x(τ)dτ − x(τ)dτ ∞ −∞ Repetitive application of this time integration property yields the Laplace trans- form of n th-order integral of x(t) w.r.t. t as t t t 0 0 0 0 L ·· x(τ)dτ n ←→ s−n X(s) + s−n x(τ)dτ +···+ ·· x(τ)dτ n −∞ −∞ −∞ −∞ −∞ −∞ −∞ (A.14) Appendix A 389 A.3.4 Time Shifting – Real Translation The Laplace transform of a delayed function x(t) can be obtained as follows. ∞ . t1 (A=1) −st −st L {x(t − t1)} x(t − t1)e dτ + x(t − t1)e dτ −∞ t1 0 ∞ t−t1=τ − − τ − − τ = e st1 x(τ)e s dτ + e st1 x(τ)e s dτ; − t1 0 0 L if x(t)=0 ∀ t<0 −st1 −sτ −st1 x(t − t1), t1 > 0 ←→ e X(s) + x(τ)e dτ → e X(s) −t1 (A.15) τ τ τ = t τ = t t t 0 (a) (b) Fig. A.2 Switching the order of two integrations for double integral A.3.5 Frequency Shifting – Complex Translation L s1t e x(t) ←→ X(s − s1) (A.16) A.3.6 Real Convolution The (real) convolution of two (causal) functions g(t) and x(t) is defined as ∞ t if g(t)=0andx(t)=0 ∀ t<0 g(t) ∗ x(t) = g (τ) x (t − τ) dτ = g (τ) x (t − τ) dτ −∞ causality 0 (A.17) The Laplace transform of the convolution y(t) = g(t) ∗ x(t) turns out to be the product of the Laplace transforms of the two functions as L y(t) = g(t) ∗ x(t) ←→ Y (s) = G(s)X(s) (A.18) 390 Appendix A Proof ∞ ∞ t (A.1) (A.17) L {g(t) ∗ x(t)} = g(t) ∗ x(t)e−stdt = g(τ)x(t − τ)dτ e−stdt 0 0 0 ∞ ∞ Fig.A.2 = g(τ)e−sτ x(t − τ)e−s(t−τ)dt dτ 0 τ ∞ ∞ (t−τ=v) = g(τ)e−sτ x(v)e−svdv dτ 0 0 ∞ (A.1) (A.1) = g(τ)e−sτ dτ X(s) = G (s) X (s) 0 This property is used to describe the input-output relationship of a linear time- invariant (LTI) system which has the input x(t), the impulse function g(t), and the output y(t), where the Laplace transform of the impulse function, i.e., G(s) = L{g(t)}, is referred to as the system function or transfer function of the system. A.3.7 Partial Differentiation Given the Laplace transform of a function having a parameter a, that is L{x(t, a)}= X(s, a), the Laplace transform of the derivative of the parameterized function x(t, a) w.r.t. a equals the derivative of its Laplace transform X(s, a) w.r.t. a. ∂ L ∂ x(t, a) ←→ X(s, a) (A.19) ∂a ∂a A.3.8 Complex Differentiation The Laplace transform of −tx(t) equals the derivative of L{x(t)}=X(s) w.r.t. s. L d tx(t) ←→ − X(s) (A.20) ds This can be derived by differentiating Eq. (A.1) w.r.t. s. Example A.1 Applying Partial/Complex Differentiation Property −at To find the Laplace transform of te us (t), we differentiate Eq. (A.7) w.r.t. a and multiply the result by −1 to get L 1 te−atu (t) ←→ (A.21) s (s + a)2 Appendix A 391 Multiplying t by the left-hand side (LHS) and applying the complex differentia- tion property (A.20) repetitively, we get L 2! t2e−atu (t) ←→ (A.22) s (s + a)3 ..........................
Load more

Recommended publications
  • The Laplace Transform of the Psi Function
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 2, February 2010, Pages 593–603 S 0002-9939(09)10157-0 Article electronically published on September 25, 2009 THE LAPLACE TRANSFORM OF THE PSI FUNCTION ATUL DIXIT (Communicated by Peter A. Clarkson) Abstract. An expression for the Laplace transform of the psi function ∞ L(a):= e−atψ(t +1)dt 0 is derived using two different methods. It is then applied to evaluate the definite integral 4 ∞ x2 dx M(a)= , 2 2 −a π 0 x +ln (2e cos x) for a>ln 2 and to resolve a conjecture posed by Olivier Oloa. 1. Introduction Let ψ(x) denote the logarithmic derivative of the gamma function Γ(x), i.e., Γ (x) (1.1) ψ(x)= . Γ(x) The psi function has been studied extensively and still continues to receive attention from many mathematicians. Many of its properties are listed in [6, pp. 952–955]. Surprisingly, an explicit formula for the Laplace transform of the psi function, i.e., ∞ (1.2) L(a):= e−atψ(t +1)dt, 0 is absent from the literature. Recently in [5], the nature of the Laplace trans- form was studied by demonstrating the relationship between L(a) and the Glasser- Manna-Oloa integral 4 ∞ x2 dx (1.3) M(a):= , 2 2 −a π 0 x +ln (2e cos x) namely, that, for a>ln 2, γ (1.4) M(a)=L(a)+ , a where γ is the Euler’s constant. In [1], T. Amdeberhan, O. Espinosa and V. H. Moll obtained certain analytic expressions for M(a) in the complementary range 0 <a≤ ln 2.
    [Show full text]
  • 7 Laplace Transform
    7 Laplace transform The Laplace transform is a generalised Fourier transform that can handle a larger class of signals. Instead of a real-valued frequency variable ω indexing the exponential component ejωt it uses a complex-valued variable s and the generalised exponential est. If all signals of interest are right-sided (zero for negative t) then a unilateral variant can be defined that is simple to use in practice. 7.1 Development The Fourier transform of a signal x(t) exists if it is absolutely integrable: ∞ x(t) dt < . | | ∞ Z−∞ While it’s possible that the transform might exist even if this condition isn’t satisfied, there are a whole class of signals of interest that do not have a Fourier transform. We still need to be able to work with them. Consider the signal x(t)= e2tu(t). For positive values of t this signal grows exponentially without bound, and the Fourier integral does not converge. However, we observe that the modified signal σt φ(t)= x(t)e− does have a Fourier transform if we choose σ > 2. Thus φ(t) can be expressed in terms of frequency components ejωt for <ω< . −∞ ∞ The bilateral Laplace transform of a signal x(t) is defined to be ∞ st X(s)= x(t)e− dt, Z−∞ where s is a complex variable. The set of values of s for which this transform exists is called the region of convergence, or ROC. Suppose the imaginary axis s = jω lies in the ROC. The values of the Laplace transform along this line are ∞ jωt X(jω)= x(t)e− dt, Z−∞ which are precisely the values of the Fourier transform.
    [Show full text]
  • SMU PHYSICS 1303: Introduction to Mechanics
    SMU PHYSICS 1303: Introduction to Mechanics Stephen Sekula1 1Southern Methodist University Dallas, TX, USA SPRING, 2019 S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 1 Outline Conservation of Energy S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 2 Conservation of Energy Conservation of Energy NASA, “Hipnos” by Molinos de Viento and available under Creative Commons from Flickr S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 3 Conservation of Energy Key Ideas The key ideas that we will explore in this section of the course are as follows: I We will come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed. I We will understand the mathematical description of energy conservation. I We will explore the implications of the conservation of energy. Jacques-Louis David. “Portrait of Monsieur de Lavoisier and his Wife, chemist Marie-Anne Pierrette Paulze”. Available under Creative Commons from Flickr. S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 4 Conservation of Energy Key Ideas The key ideas that we will explore in this section of the course are as follows: I We will come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed. I We will understand the mathematical description of energy conservation. I We will explore the implications of the conservation of energy. Jacques-Louis David. “Portrait of Monsieur de Lavoisier and his Wife, chemist Marie-Anne Pierrette Paulze”. Available under Creative Commons from Flickr. S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 4 Conservation of Energy Key Ideas The key ideas that we will explore in this section of the course are as follows: I We will come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed.
    [Show full text]
  • Subtleties of the Laplace Transform
    Troubles at the Origin: Consistent Usage and Properties of the Unilateral Laplace Transform Kent H. Lundberg, Haynes R. Miller, and David L. Trumper Massachusetts Institute of Technology The Laplace transform is a standard tool associated with the analysis of signals, models, and control systems, and is consequently taught in some form to almost all engineering students. The bilateral and unilateral forms of the Laplace transform are closely related, but have somewhat different domains of application. The bilateral transform is most frequently seen in the context of signal processing, whereas the unilateral transform is most often associated with the study of dynamic system response where the role of initial conditions takes on greater significance. In our teaching we have found some significant pitfalls associated with teaching our students to understand and apply the Laplace transform. These confusions extend to the presentation of this material in many of the available mathematics and engineering textbooks as well. The most significant confusion in much of the textbook literature is how to deal with the origin in the application of the unilateral Laplace transform. That is, many texts present the transform of a time function f(t) as Z ∞ L{f(t)} = f(t)e−st dt (1) 0 without properly specifiying the meaning of the lower limit of integration. Said informally, does the integral include the origin fully, partially, or not at all? This issue becomes significant as soon as singularity functions such as the unit impulse are introduced. While it is not possible to devote full attention to this issue within the context of a typical undergraduate course, this “skeleton in the closet” as Kailath [8] called it needs to be brought out fully into the light.
    [Show full text]
  • A Note on Laplace Transforms of Some Particular Function Types
    A Note on Laplace Transforms of Some Particular Function Types Henrik Stenlund∗ Visilab Signal Technologies Oy, Finland 9th February, 2014 Abstract This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform or its inverse when there is a specific type of an implicit function involved.1 0.1 Keywords Laplace transform, inverse Laplace transform 0.2 Mathematical Classification MSC: 44A10 1 Introduction 1.1 General The following two Laplace transforms have appeared in numerous editions and prints of handbooks and textbooks, for decades. arXiv:1402.2876v1 [math.GM] 9 Feb 2014 1 ∞ 3 s2 2 2 4u Lt[F (t )],s = u− e− f(u)du (F ALSE) (1) 2√π Z0 u 1 f(ln(s)) ∞ t f(u)du L− [ ],t = (F ALSE) (2) s s ln(s) Z Γ(u + 1) · 0 They have mostly been removed from the latest editions and omitted from other new handbooks. However, one fresh edition still carries them [1]. Obviously, many people have tried to apply them. No wonder that they are not accepted ∗The author is obliged to Visilab Signal Technologies for supporting this work. 1Visilab Report #2014-02 1 anymore since they are false. The actual reasons for the errors are not known to the author; possibly it is a misprint inherited from one print to another and then transported to other books, believed to be true. The author tried to apply these transforms, stumbling to a serious conflict.
    [Show full text]
  • Sliding and Rolling: the Physics of a Rolling Ball J Hierrezuelo Secondary School I B Reyes Catdicos (Vdez- Mdaga),Spain and C Carnero University of Malaga, Spain
    Sliding and rolling: the physics of a rolling ball J Hierrezuelo Secondary School I B Reyes Catdicos (Vdez- Mdaga),Spain and C Carnero University of Malaga, Spain We present an approach that provides a simple and there is an extra difficulty: most students think that it adequate procedure for introducing the concept of is not possible for a body to roll without slipping rolling friction. In addition, we discuss some unless there is a frictional force involved. In fact, aspects related to rolling motion that are the when we ask students, 'why do rolling bodies come to source of students' misconceptions. Several rest?, in most cases the answer is, 'because the didactic suggestions are given. frictional force acting on the body provides a negative acceleration decreasing the speed of the Rolling motion plays an important role in many body'. In order to gain a good understanding of familiar situations and in a number of technical rolling motion, which is bound to be useful in further applications, so this kind of motion is the subject of advanced courses. these aspects should be properly considerable attention in most introductory darified. mechanics courses in science and engineering. The outline of this article is as follows. Firstly, we However, we often find that students make errors describe the motion of a rigid sphere on a rigid when they try to interpret certain situations related horizontal plane. In this section, we compare two to this motion. situations: (1) rolling and slipping, and (2) rolling It must be recognized that a correct analysis of rolling without slipping.
    [Show full text]
  • Coupling Effect of Van Der Waals, Centrifugal, and Frictional Forces On
    PCCP View Article Online PAPER View Journal | View Issue Coupling effect of van der Waals, centrifugal, and frictional forces on a GHz rotation–translation Cite this: Phys. Chem. Chem. Phys., 2019, 21,359 nano-convertor† Bo Song,a Kun Cai, *ab Jiao Shi, ad Yi Min Xieb and Qinghua Qin c A nano rotation–translation convertor with a deformable rotor is presented, and the dynamic responses of the system are investigated considering the coupling among the van der Waals (vdW), centrifugal and frictional forces. When an input rotational frequency (o) is applied at one end of the rotor, the other end exhibits a translational motion, which is an output of the system and depends on both the geometry of the system and the forces applied on the deformable part (DP) of the rotor. When centrifugal force is Received 25th September 2018, stronger than vdW force, the DP deforms by accompanying the translation of the rotor. It is found that Accepted 26th November 2018 the translational displacement is stable and controllable on the condition that o is in an interval. If o DOI: 10.1039/c8cp06013d exceeds an allowable value, the rotor exhibits unstable eccentric rotation. The system may collapse with the rotor escaping from the stators due to the strong centrifugal force in eccentric rotation. In a practical rsc.li/pccp design, the interval of o can be found for a system with controllable output translation. 1 Introduction components.18–22 Hertal et al.23 investigated the significant deformation of carbon nanotubes (CNTs) by surface vdW forces With the rapid development in nanotechnology, miniaturization that were generated between the nanotube and the substrate.
    [Show full text]
  • Lectures on the Local Semicircle Law for Wigner Matrices
    Lectures on the local semicircle law for Wigner matrices Florent Benaych-Georges∗ Antti Knowlesy September 11, 2018 These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero expectation and constant variance. We state and prove the local semicircle law, which says that the eigenvalue distribution of a Wigner matrix is close to Wigner's semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. This local semicircle law is formulated using the Green function, whose individual entries are controlled by large deviation bounds. We then discuss three applications of the local semicircle law: first, complete delocalization of the eigenvectors, stating that with high probability the eigenvectors are approximately flat; second, rigidity of the eigenvalues, giving large deviation bounds on the locations of the individual eigenvalues; third, a comparison argument for the local eigenvalue statistics in the bulk spectrum, showing that the local eigenvalue statistics of two Wigner matrices coincide provided the first four moments of their entries coincide. We also sketch further applications to eigenvalues near the spectral edge, and to the distribution of eigenvectors. arXiv:1601.04055v4 [math.PR] 10 Sep 2018 ∗Universit´eParis Descartes, MAP5. Email: [email protected]. yETH Z¨urich, Departement Mathematik. Email: [email protected].
    [Show full text]
  • Semi-Parametric Likelihood Functions for Bivariate Survival Data
    Old Dominion University ODU Digital Commons Mathematics & Statistics Theses & Dissertations Mathematics & Statistics Summer 2010 Semi-Parametric Likelihood Functions for Bivariate Survival Data S. H. Sathish Indika Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/mathstat_etds Part of the Mathematics Commons, and the Statistics and Probability Commons Recommended Citation Indika, S. H. S.. "Semi-Parametric Likelihood Functions for Bivariate Survival Data" (2010). Doctor of Philosophy (PhD), Dissertation, Mathematics & Statistics, Old Dominion University, DOI: 10.25777/ jgbf-4g75 https://digitalcommons.odu.edu/mathstat_etds/30 This Dissertation is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion in Mathematics & Statistics Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. SEMI-PARAMETRIC LIKELIHOOD FUNCTIONS FOR BIVARIATE SURVIVAL DATA by S. H. Sathish Indika BS Mathematics, 1997, University of Colombo MS Mathematics-Statistics, 2002, New Mexico Institute of Mining and Technology MS Mathematical Sciences-Operations Research, 2004, Clemson University MS Computer Science, 2006, College of William and Mary A Dissertation Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the Requirement for the Degree of DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS AND STATISTICS OLD DOMINION UNIVERSITY August 2010 Approved by: DayanandiN; Naik Larry D. Le ia M. Jones ABSTRACT SEMI-PARAMETRIC LIKELIHOOD FUNCTIONS FOR BIVARIATE SURVIVAL DATA S. H. Sathish Indika Old Dominion University, 2010 Director: Dr. Norou Diawara Because of the numerous applications, characterization of multivariate survival dis­ tributions is still a growing area of research.
    [Show full text]
  • SEISMIC ANALYSIS of SLIDING STRUCTURES BROCHARD D.- GANTENBEIN F. CEA Centre D'etudes Nucléaires De Saclay, 91
    n 9 COMMISSARIAT A L'ENERGIE ATOMIQUE CENTRE D1ETUDES NUCLEAIRES DE 5ACLAY CEA-CONF —9990 Service de Documentation F9II9I GIF SUR YVETTE CEDEX Rl SEISMIC ANALYSIS OF SLIDING STRUCTURES BROCHARD D.- GANTENBEIN F. CEA Centre d'Etudes Nucléaires de Saclay, 91 - Gif-sur-Yvette (FR). Dept. d'Etudes Mécaniques et Thermiques Communication présentée à : SMIRT 10.' International Conference on Structural Mechanics in Reactor Technology Anaheim, CA (US) 14-18 Aug 1989 SEISHIC ANALYSIS OF SLIDING STRUCTURES D. Brochard, F. Gantenbein C.E.A.-C.E.N. Saclay - DEHT/SMTS/EHSI 91191 Gif sur Yvette Cedex 1. INTRODUCTION To lirait the seism effects, structures may be base isolated. A sliding system located between the structure and the support allows differential motion between them. The aim of this paper is the presentation of the method to calculate the res- ponse of the structure when the structure is represented by its elgenmodes, and the sliding phenomenon by the Coulomb friction model. Finally, an application to a simple structure shows the influence on the response of the main parameters (friction coefficient, stiffness,...). 2. COULOMB FRICTION HODEL Let us consider a stiff mass, layed on an horizontal support and submitted to an external force Fe (parallel to the support). When Fg is smaller than a limit force ? p there is no differential motion between the support and the mass and the friction force balances the external force. The limit force is written: Fj1 = \x Fn where \i is the friction coefficient and Fn the modulus of the normal force applied by the mass to the support (in this case, Fn is equal to the weight of the mass).
    [Show full text]
  • Delta Functions and Distributions
    When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes Created October 2010, updated March 8, 2017. Abstract x = 0. That is, one would like the function δ(x) = 0 for all x 6= 0, but with R δ(x)dx = 1 for any in- These notes give a brief introduction to the mo- tegration region that includes x = 0; this concept tivations, concepts, and properties of distributions, is called a “Dirac delta function” or simply a “delta which generalize the notion of functions f(x) to al- function.” δ(x) is usually the simplest right-hand- low derivatives of discontinuities, “delta” functions, side for which to solve differential equations, yielding and other nice things. This generalization is in- a Green’s function. It is also the simplest way to creasingly important the more you work with linear consider physical effects that are concentrated within PDEs, as we do in 18.303. For example, Green’s func- very small volumes or times, for which you don’t ac- tions are extremely cumbersome if one does not al- tually want to worry about the microscopic details low delta functions. Moreover, solving PDEs with in this volume—for example, think of the concepts of functions that are not classically differentiable is of a “point charge,” a “point mass,” a force plucking a great practical importance (e.g. a plucked string with string at “one point,” a “kick” that “suddenly” imparts a triangle shape is not twice differentiable, making some momentum to an object, and so on.
    [Show full text]
  • An Analytic Exact Form of the Unit Step Function
    Mathematics and Statistics 2(7): 235-237, 2014 http://www.hrpub.org DOI: 10.13189/ms.2014.020702 An Analytic Exact Form of the Unit Step Function J. Venetis Section of Mechanics, Faculty of Applied Mathematics and Physical Sciences, National Technical University of Athens *Corresponding Author: [email protected] Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract In this paper, the author obtains an analytic Meanwhile, there are many smooth analytic exact form of the unit step function, which is also known as approximations to the unit step function as it can be seen in Heaviside function and constitutes a fundamental concept of the literature [4,5,6]. Besides, Sullivan et al [7] obtained a the Operational Calculus. Particularly, this function is linear algebraic approximation to this function by means of a equivalently expressed in a closed form as the summation of linear combination of exponential functions. two inverse trigonometric functions. The novelty of this However, the majority of all these approaches lead to work is that the exact representation which is proposed here closed – form representations consisting of non - elementary is not performed in terms of non – elementary special special functions, e.g. Logistic function, Hyperfunction, or functions, e.g. Dirac delta function or Error function and Error function and also most of its algebraic exact forms are also is neither the limit of a function, nor the limit of a expressed in terms generalized integrals or infinitesimal sequence of functions with point wise or uniform terms, something that complicates the related computational convergence. Therefore it may be much more appropriate in procedures.
    [Show full text]