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Almost periodic oscillations and waves PDF

308 Pages·2009·1.53 MB·English
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2 1 Introduction Ifweconsidertherectilinearmotionofamaterialpointofmassmwithout friction that is attracted by a fixed point on the line of motion with a force proportionaltothedistancetothisfixedpoint,thenNewton’slawofdynamics (ma=f) leads to the following second-order ordinary differential equation: mx¨(t)=−kx(t). (1.1) In equation (1.1), x(t) represents the abscissa of the point in motion, while k > 0 is the proportionality constant. We must assume k > 0 because the fixed point of attraction is taken as the origin on the line of motion, and only inthiscaseistheoriginattractingthepointofmassm.Ofcourse,x¨(t)stands for the acceleration a= dv, where the velocity v = dx =x˙(t). dt dt Let us now consider equation (1.1) and denote ω2 = k. Then equation m (1.1) becomes x¨(t)+ω2x(t)=0, (1.2) whichisahomogeneoussecond-orderdifferentialequationwithconstantcoef- ficients. It is well known that cosωt and sinωt are linearly independent so- lutions of equation (1.2). Therefore, the general solution of equation (1.2) is x(t) = c1cosωt+c2sinωt, with arbitrary constants c1,c2. If one de(cid:2)not(cid:3)es c1 =Acosα,c2 =−(cid:4)As(cid:5)inα,whichmeansA=(c21+c22)1/2,α=−arctan cc21 , c (cid:2)= 0, and α = − π signc , c = 0, these notations allow us to write the 1 2 2 1 general solution of equation (1.2) in the form x(t)=Acos(ωt+α). (1.3) Formula (1.3) shows that any motion of the harmonic oscillator is a simple harmonic of frequency 2 π. The positive number A is called the amplitude of ω the oscillatory motion described by equation (1.3). The number α is called thephaseofthose oscillations. BothAandα dependontheinitial conditions x(0)=x and v(0)=v . In terms of x and v , the amplitude and the phase 0 0 0 0 are expressed by A=(x20+v02ω−2)1/2, α=−arctan(vω0 x0). Equation (1.2) describes the motion of the material point without friction and free of any external force. When other forces are involved, we have to modifyequation(1.2)accordingly.Weareledinthismannertodealwiththe so-called forced oscillations, as opposed to the free oscillations described by equation (1.2). The new equation will have the form mx¨(t)=−kx(t)+F(t), with F(t) designating the external force acting on the point at the moment t. Denoting f(t)=m−1F(t), the equation of forced oscillations becomes x¨(t)+ω2x(t)=f(t), (1.4) in which f(t) is the density of the external force, or the force on a unit of mass of the moving point. Inmathematicalterms,equation(1.4)forforcedoscillationsofthematerial point is an inhomogeneous equation of the second order. Using the variation

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This text is well designed with respect to the exposition from the preliminary to the more advanced and the applications interwoven throughout. It provides the essential foundations for the theory as well as the basic facts relating to almost periodicity. In six structured and self-contained chapter
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