Unit 5 mechanical vibrations.doc

Unit 5 Mechanical Vibration 机械振动A mechanical vibration is an oscillatory, periodically repeated motion of a particle or body about a position of equilibrium. An engineer is frequently confronted with the problem of mechanical vibrations since they are encountered to some degree in almost all types of machinery and structures. Most vibrations are undesirable in machines and structures because they produce excessive stresses or repeated stresses; cause added wear; increase bearing loads; induce fatigue; create acute passenger discomfort in planes, ships, trains, buses, and automobiles; and absorb energy that could otherwise do useful work. The collapse of the Tacoma Narrows Bridge in 1940 is an example of structural failure due to excessive stresses produced by vibrations. The accuracy of precision instruments, tools, and machines may be impaired by excessive vibrations. Rotating machine parts require careful balancing in order to prevent damage from vibrations. When part of propeller of an airplane is shot off or breaks off in flight the propeller is no longer symmetrical, and the vibrations from the engine may tear the engine from the plane unless it can be stopped in time. The vibrations produced in an automobile by the engine or by driving on rough roads set up repeated stresses in certain parts that can eventually lead to failure of the members.机械振动是一个质点或物体以一个平衡位置为中心做往复周期的振荡运动。由于工程师几乎要面对所有的机械和结构中，因此机械振动就成了一个经常有遇要到的问题。在机器和结构中，大多数振动是不希望存在的，因为振动会产生附加应力或交变应力，引起额外磨损，增大轴承载荷，导致疲劳破坏，使飞机、船、火车及汽车上的乘客产生严重的不很舒服感，并且振动会吸收本可以做有用功的能量。1940年的Tacoma Narrows Bridge倒塌就是一个由振动引起附加应力而导致结构破坏的例子。附加振动可能损坏精密仪表、工具和机械的精度。为了防止来自振动的破坏，旋转机械需要很好的平衡。当飞机的螺旋桨在飞行过程中损坏或断裂时，螺旋桨就将不再对称，除非可以及时停止，否则发动机产生的振动就会把它从飞机上撕裂下来。汽车由发动机或在崎岖不平的路上行驶时产生的振动会在一定的部位产生交变应力，最终导致杆件疲劳失效。Vibrations are sometimes used to produce desirable effects. For example, vibrators are used to compact concrete in the forms and to separate the grain from the chaff in threshing machines. Instruments which function properly on an airplane with a conventional engine may tend to stick when used in gliders or jet-powered planes because of the lack of vibration. In such instances, a vibrator is sometimes installed on the instrument panel.有时候振动也能产生有益的效果。比如，振动可以压实物体，用脱粒机将谷粒与谷壳分离。那些在具有常规发动机的飞机上可正常运行的仪器，当用于滑翔机或喷气机时，因为缺乏振动，可能会趋于粘滞。在这种情况下，就需在仪器的仪表板上安装振动器。When a particle or body that is supported by a system of springs, a shaft, a beam, or any other elastic system is disturbed from its position of equilibrium by the application and sudden removal of an additional force，the particle or body will vibrate. Some common example are (1) the vertical vibratory motion of body A in Fig.1.19 (a) when it is displaced vertically from its equilibrium position on a helical spring and then released; (2) the vertical vibratory motion of body B of Fig.1.19(b) when it is displaced from its equilibrium position on a flexible springboard (cantilever beam) of negligible mass and released; and (3) the motion of the pendulum bob C in Fig.1.19(c) as it swings in a vertical plane when supported by a cord of negligible mass.当质点或物体是由弹簧系统支撑时，由于实际应用和额外力的移走，轴、梁或其他弹性系偏离平衡位置，这时质点或物体将开始振动。一些常见的例子有：图1.19（a）中，A物体在螺旋弹簧上被垂直拉离平衡位置时释放，它就会垂直振动；图1.19（b）中，当B物体在一个悬臂梁（忽略质量）上偏离平衡位置并释放时，它的垂直振动；图1.19（c）中，当摆锤C由细线（忽略质量）固定在竖直平面时的摇摆运动。Fig.1.20 (a) shows a body W suspended from a support by a spring and hanging at its equilibrium position. If the body is pulled from its equilibrium position by a force F and released, it will, in the absence of any frictional forces, continue to oscillate about the equilibrium position indefinitely. Fig.1.20 (b) shows a graph of the displacement, y, of the body W from its equilibrium position as a function of time. One of the fundamental properties of mechanical vibrations is that the motion repeats itself at definite intervals of time. The period, T, of the oscillation is the minimum amount of elapsed time before the motion starts to repeat itself. The motion completed in one period is a cycle. The frequency, f, of the oscillation is the number of cycles which occur in a given unit of time; the customary unit is cycles per second, cps, or hertz, Hz. It should be noted that the frequency is the reciprocal of the period, or in equation from, (1.28) 如图1.20（a）所示，小物块W通过弹簧悬挂在固定支撑上并处于平衡位置。如果将物体在力F的作用下偏离平衡位置然后释放，在不考虑任何摩擦力的情况下，物体将关于平衡位置做无休止的振荡运动。图1.20（b）所示的是物体W偏离平衡位置的位移y与时间的函数关系图。振动的一个基本性质是以一定的时间间隔重复运动。振动的周期T是重复运动的最小时间间隔。完成一个周期的运动是一个循环。振动频率f是在给定的单位时间内完成循环的个数；常用的单位是次每秒，cps，或赫兹，Hz。需要注意的是频率是周期的倒数，用公式表示为：。The amplitude, A, of the oscillation is the maximum displacement, rather linear or angular, of the body from its equilibrium position.不管是线性的还是成角度的，振动的振幅是物体偏离平衡位置的最大位移。 Mechanical vibrations which are maintained by elastic , and sometimes gravity，forces are called free vibrations. Once started, a free vibration, frequently called a natural vibration, continues to oscillate at its natural frequency. A forced vibration is produced and maintained by a periodic exciting force external to the system, and it takes place at the frequency of the exciting force. If the system is nearly frictionless and contains no dissipative elements, the amplitude of the forced oscillation will become very large when the frequency of the exciting force is close to the natural frequency of the system. Thus, although the frequency of a forced vibration is independent of the natural frequency of the elastic system, the resulting amplitude is affected by both frequencies. Vibrations are also classified as damped or undamped. When friction, air resistance，viscous damping, and all other resisting forces are negligible the vibration is undamped. When any of these effects is appreciable the body has damped vibrations. In practice there is always a frictional damping force which will eventually stop a free vibration, even though it may be neglected for many purposes.机械振动是通过弹力保持的，有时是重力，这些力被称为自由振动。自由振动通常称为固有振动，一旦开始，就将以固有频率持续振动。强迫振动由系统外的周期性激荡力产生并保持，它以激荡力的频率发生。如果系统近乎无摩擦和无耗散，那么，当激荡力的固有频率与系统固有频率接近时，强迫振动的振幅就会变得非常大。因此，尽管强迫振动的频率与弹性系统的固有频率无关，但是，所产生的振幅会受到两个频率的影响。振动也被分为阻尼振动和无阻尼振动。当摩擦力，空气阻力，粘性阻尼和其他所有阻力都忽略时，振动就是无阻尼的。当考虑这些因素时，物体就发生阻尼振动。在实际应用中，尽管由于各种原因，摩擦阻尼力可能被忽略，但是，它最终会阻碍自由振动。If the motion of a particle or body is constrained so that its position can be completely specified by one coordinate, it is said to have a single degree of freedom. If the system can vibrate in two directions or is composed of two articles that can each vibrate independently in one direction, it is said to have two degree of freedom because two coordinates are required to specify the position of the system at any instant. For example, a system composed of a single particle, supported by a spring as indicated in Fig.1.21 (a), which vibrates in a vertical direction only, has one degree of freedom. A system composed of two particles which move vertically and are supported as indicated in Fig.1.21 (b) has two degrees of freedom because two coordinates are required to locate the particles at any instant. A system composed of one particle supported by four springs and constrained to move in a vertical plane as indicated in Fig.1.21 (c) has two degrees of freedom because two coordinates are required to locate the particle in the vertical plane at any instant. A single rigid body has, in general, six degrees of freedom, since it may have translation in three coordinate directions and it may rotate about coordinate axes.当一个质点或物体的运动被限制时，它的位置用一个坐标就能完全表示，这时，我们就说它有一个自由度。当一个系统可以在两个方向振动或由两个可以在同一方向独立振动的物体组成时，我们说它有两个自由度，这时，质点任意时刻的位置需要两个坐标来确定。例如，一个单质点系统，通过弹簧固定，如图1.21（a）所示，只能在竖直方向振动，有一个自由度。一个可沿铅垂方向振动的双质点系统，所受约束如图1.21（b）所示，有两个自由度，这是因为要确定任意时刻质点的位置需要两个坐标。一个由四个弹簧固定的单质点系统，如图1.21（c）所示，在竖直面内被限制运动，有两个自由度，这是因为在竖直面内要确定任意时刻质点的位