力学专业英语10

Lesson 10Vibrations of a single particleWhen a medium is disturbed by the passage of a wave through it, the particles comprising the medium are caused to vibrate. To take a simple example, corks floating on the surface of a pond will bob up and down owing to the influence of water waves. The bob of a simple pendulum and the weight hanging freely from the end of a spring are other examples of particles which may be set in vibration, and most readers will have a good mental picture of these vibrate.The motion of a vibrating particle is periodic, that is, after equal intervals of time the system finds itself in exactly the same situation. The bob of the pendulum, for example, is found to be at the same position, moving with the same velocity and acceleration as it was T seconds earlier, and these quantities will be the same T, 2T , 3T, etc. Seconds later. During the interval of one period, a vibrating system is said to go through a cycle of situation, and the frequency is defined as the number of cycles occurring in one second. Clearly, then ,f=1/T, the dimension of f is second-1 . This unit is termed the hertz.The simplest kind of periodic motion is that experienced by a point, moving along a straight line, whose acceleration is directed towards a fixed point on the line and is proportional to its distance from the fixed point. This is called simple harmonic motion. Suppose a point Pmoves along a straight line so that its position with respect to a fixed point o is completely specified by the single coordinate x. The acceleration of the moving point is d2x/dt2 ,which is directed towards the point o and proportional to he distance x. Thus where ωis a positive constant.Equation (1) shown above is a linear, second-order differential equation. The word linear is due to x and its derivative appear to the power one only, and second order because the highest derivative is d2x/dt2 . Equation (1) is referred to as the differential equation governing the motion. It is not the equation of motion. To find the equation of motion we have to solve Eq. (1) for x and obtain the general solution as follows. The quantity ωt+εis known as the phase of the motion and εthe phase constant or epoch. The constant a is the amplitude of the motion. It is the greatest possible value that x can have, sincethe maximum value of sin (ωt+ε) is unit. Thus the motion takes place entirely between the limits x=±a.The vibrating systems we considered above vibrated for ever with undiminished amplitude. Real physical systems do not, of course, vibrate indefinitely. For example, oscillation of a mass suspended on an elastic string eventually comes to rest. The reason for the decay is that the energy originally stored in the system is gradually transformed into heat energy by a number of processes. The suspended mass, as it moves,experiences a viscous force due to the air through which it passes. Work has to be done to overcome this force, and hence energy is lost. Even if the mass ere in vacuum, there would be hysteresis losses in stretching the elastic string, and also frictional losses where the fixed end of the string is clamped.It is an experimental fact that when a particle moves through a viscous fluid, such as a gas, there is a resistive force which is proportional to the velocity of the particle relative to the fluid, provided the velocity is not too great. Frictional forces can also be shown experimentally to be proportional to relative velocity. The resultant of all the forces which resist the motion may be regarded as a single damping force.Further, we will assume that this force at any instant is proportional to he particle velocity, i.e., the damping force is equal to D dx/dt , where D is the coefficient of damping. Applying Newton’s second law of motion to the particle of mass m gives. The actual equation of motion can be obtained by solving this differential equation. It is shown that the behavior of solution s as a function of t depends on the magnitude of hte damping constant.The vibrating systems treated in the preceding sections are sometimes referred to as free vibrations. The frequency of such vibrations is entirely determined by the system itself. There is no further interference by the outside world, in particular no further energy isintroduced. When a vibrating system is subjected to a continuous periodic disturbance of some kind, the resulting vibrations are known as forced vibrations which are quite different from free vibrations.Let us consider the simplest case of vibration where a mass m suspended from an elastic string of stiffness k is subjected to a varying force. The force has maximum value F O and varies sinusoidally with time. When we include the driving force in Eq. (3), we obtain, by a direct application of Newton’s second law of motion. This is the differential equation of damped forced vibration of a particle. The quantity p in Eq.(4) is called the circular frequency.From the theory of differential equation, the solution of Eq.(4) is x=x1+x2, where x1is the general solution of the equation without the right side, and x2 is a particular solution of the complete equation. The contribution of the general solution x1 diminishes with time, for which reason it is called the transient solution. On the other hand, the particular solution x2dose not vanish for large t and is known as the steady-state solution. Thus the transient and steady-state solutions can be obtained separately and then combined to obtain the complete solution.。