fundamentals of acoustics(2) 声学基础（英文版教学课件）

Chapter 1 Particles Vibrating Systems,IntroductionAcoustics may be defined as the generation, transmission, and reception of energy in the form of vibrational waves in matter.,This first chapter introduces the basic mathematics of simple vibrating systems.The treatment is limited to electromechanical devices because these are by far the most widely used and lend themselves to many purposes.According to function, we may group an electromechanical source into three sections:,The acoustical engineer is interested in the fidelity of reproduction of soundThe conversion of mechanical and electrical energy into acoustical energy,The design of acoustical transducers.The architect is more interested in the absorption and isolation of sound in buildings, and in controlled reverberation and echo prevention in auditoriums.The musician likes to know how to obtain rhythmic combinations of tones through vibrations of strings ,air columns.,Sound does not consist of air moving towards us in bulk; it travels through the air as a sound wave. As a vibrating body moves forward from its static equilibrium position, it pushes the air before it and compresses it . At the same time, a rarefaction occurs immediately behind the body, and air rushes in to fill this empty space left behind. In this way the compression of air is transferred to distant parts and air is set into a motion know as sound waves.,1. WAVES,Sound waves (in fluids) are longitudinal waves-the particles move in the direction of the wave motion.Propagation of sound waves involves the transfer of energy through space.While sound waves spread out in all directions from the source, they may be reflected and refracted, scattered and diffracted, interfered and absorbed.A medium is required for the propagation of sound wave, the speed of which depends on the density and temperature of the medium.,Sound pressure superimposed on atmospheric pressure,A sound may contain waves of only one frequency, in which case it is called a pure tone.,The cowboy will hear the train noise via the rails before he hears it through the air,2 SIMPLE HARMONIC MOTION (SHM),In studying vibrations it is advisable to begin with the simplest type: a one-dimensional vibration that has only a single frequency component ( a pure tone) If any object is displaced slightly from equilibrium it will oscillate about its equilibrium position in what is called simple harmonic motion (SHM).,Simple Harmonic Motion This fundamental vibrating system has been chosen as an instrument for the introduction of the basic concepts utilized in the description of sound.,Hookes Law: An elongation per unit length bears a linear relation to the force producing it.Let us assume that the restoring force f in newtons (N) can be expressed by the equation,where the equilibrium position is chosen to have x -coordinate x = 0 , x being positive when the spring is extended, and negative when the spring is shortened. and k is a constant that depends on the system under consideration. The units of D are:,f = - Dx (1-1),D=,Note: The negative sign in Hookes law ensures that the force is always opposite to the direction of the displacement.The constant k in Hookes law is traditionally called the spring constant for the system, even when the restoring force is not provided by a simple spring.,If the displacing force f is just sufficient to hold the mass stationary at the point x , the sum of forces on m is zero: f+ Dx =0 f = - Dx or the force exerted by the spring is opposing the displacing force.This condition is described by the linear second-order differential equation:,Both k and m are positive,so that we can define a constant,the equation may be written as,This is an important linear differential equation whose solution is well-Known and may be obtained by several methods.,One method is to assume a solution of the form,Differentiation and substitution into (1.3) shows that this is a solution if,It may similarly be shown that,is also a solution. The complete general solutions is the sum of these two solutions,Where A1 and A2 are arbitrary constants and the parameter,is the angular frequency in radians per second (rad/s) ,the frequency in hertz(Hz)is related to the angular frequency by,Another form of (1.4) may be written as :,Where A and,Are two new arbitrary constants.,A is the amplitude of the motion and,Is the initial phase angle of the motion.The value of A and,Are determined by the A1and A2,3 INITIAL CONDITIONS,If at time t=0 the mass has an initial displacement x0 and initial speed v0,then the arbitrary constants A1and A2 are fixed by these initial conditions, and the subsequent motion of the mass is completely determined.,(1-1-7),Substitution and simplification then gives,The value of A and,are,x,In these forms it is seen that the displacement lags 900 behind the speed and that the acceleration is 1800out of phase with the displacement.,4 ENERGY OF VIBRATION,The energy E of a system oscillating with simple harmonic motion of amplitude A and angular frequency is the sum of the systems potential energy EP and kinetic energy Ek.the potential energy Ep stored in the spring is,+,The kinetic energy possessed by the mass is,The total energy of the system is,+,The total energy can be rewritten in alternate forms,The total energy is a constant (independent of time) and is equal either to the maximum potential energy or to the maximum Kinetic energy.Since the system was assumed to be free of external forces and not subject to any frictional forces,it is not surprising that the total energy does not change with time.If all other quantities in the above equations are expressed in their MKS units, then Ep, Ek, and E will be in Joules(J),Problem: A 0.5 kg mass is hung on a vertical massless spring. The new equilibrium position of the spring is found to be 3 cm below the equilibrium position of the spring without the mass. a) What is the spring constant, D ? b) Show that the mass and spring system oscillates with simple harmonic motion about the new equilibrium position.,Definitions: Amplitude ( A ): The maximum distance that an object moves from its equilibrium position. A simple harmonic oscillator moves back and forth between the two positions of maximum displacement, at x = A and x = - A . Period ( T ): The time that it takes for an oscillator to execute one complete cycle of its motion. If it starts at t = 0 at x = A , then it gets back to x = A after one full period at t = T . Frequency ( f ): The number of cycles (or oscillations) the object completes per unit time.,1-2 DAMPED OSCILLATIONS,Whenever a real body is set into oscillation,frictional forces arise. These forces are of many types,depending on the particular oscillating system.but they will always result in a damping of the oscillations-a decrease in the amplitude of the free oscillations with time.,Solution: a) Since the mass/spring system is in equilibrium, the downward force of gravity must