1、1,Chapter 18: Mechanical Vibrations,2,第十八章 机械振动基础,3,Vibrations are common phenomena in everyday life and in engineering practice. Such as the swinging of a clock pendulum, the bumping of a traveling car, the vibrations of working machine parts or motors, and the quake of buildings caused by an earth
2、quake.,3. The aim of the study of vibrations is the elimination or reduction of harmful oscillations, or the helpful use of them for special purposes.,Advantages and disadvantages of vibrations: Vibrating feeder, vibrating screen, vibrating pile driver extractor, for example, are advantageous applic
3、ations. But vibrations induce friction loss, affect intensity, make noise, have bad effects on working conditions, consume unnecessary energy and reduce the precision of a machine etc.,Vibrations are the back and forth movements of a system around its equilibrium position.,Dynamics,4,动力学,振动是日常生活和工程实
4、际中常见的现象。例如：钟摆的往复摆动,汽车行驶时的颠簸，电动机、机床等工作时的振动，以及地震时引起的建筑物的振动等。,利：振动给料机 弊：磨损，减少寿命，影响强度振动筛 引起噪声，影响劳动条件振动沉拔桩机等 消耗能量，降低精度等。,3. 研究振动的目的：消除或减小有害的振动，充分利用振动为人类服务。,2. 振动的利弊：,1. 所谓振动就是系统在平衡位置附近作往复运动。,5,The present chapter discusses mainly free vibrations and forced vibrations of systems with one degree of freedom
5、.,According to the cause of the vibrations they can be classified into,free vibrations,including undamped free vibrations and damped free vibrations,forced vibrations,including undamped forced vibrations and damped forced vibrations,self-excited forced vibrations,Dynamics,6,动力学,本章重点讨论单自由度系统的自由振动和强迫振
6、动。,7,181 Undamped free vibration of a system with one degree of freedom182 Methods of determination of the natural frequency of a system183 Damped free vibration of a system with one degree of freedom184 Undamped forced vibration of a system with one degree of freedom185 Damped forced vibration of a
7、 system with one degree of freedom186 The concepts of critical speed of rotation, vibration reduction andvibration isolation,Chapter 18: Mechanical vibrations,8,181 单自由度系统无阻尼自由振动182 求系统固有频率的方法183 单自由度系统的有阻尼自由振动184 单自由度系统的无阻尼强迫振动185 单自由度系统的有阻尼强迫振动186 临界转速 减振与隔振的概念,第十八章 机械振动基础,9,18-1 Undamped free vib
8、rationof a system with one degree of freedom,1.Concept of free vibration,Dynamics,10,动力学,18-1 单自由度系统无阻尼自由振动,一、自由振动的概念：,11,Equilibrium position x=0,Equilibrium position j=0,Dynamics,Equilibrium position j=0,12,动力学,13,A force acting on a vibrating body which is always directed towards its equilibrium
9、position is called a restoring force. After the action of an initial disturbance, the vibration of a system around its equilibrium position under the action of only a restoring force is called an undamped free vibration.,For a mass-spring system:For a simple pendulum:For a compound pendulum:,Dynamic
10、s,14,动力学,运动过程中，总指向物体平衡位置的力称为恢复力。物体受到初干扰后，仅在系统的恢复力作用下在其平衡位置附近的振动称为无阻尼自由振动。,质量弹簧系统： 单摆： 复摆：,15,2. Differential equation and its solution for an undamped free vibration of a system with one degree of freedom,the standard form of the differential equation of free vibration is,Its solution is：,Dynamics,1
11、6,二、单自由度系统无阻尼自由振动微分方程及其解,动力学,对于任何一个单自由度系统，以q 为广义坐标（从平衡位置开始量取 ），则自由振动的运动微分方程必将是：,a, c是与系统的物理参数有关的常数。令,则自由振动的微分方程的标准形式：,解为：,17,Assuming that at t = 0 ， we get,or：,Dynamics,18,动力学,设 t = 0 时， 则可求得：,或：,C1，C2由初始条件决定为,19,3.Properties of a free vibration without damping:AThe quantity ,which is the maximum d
12、istance of a vibrating body from the equilibrium position, is called the amplitude. n t + The quantity, which defines the position of the vibrating body at any given time, is called the phase of the vibration. The quantity, which defines the initial phase, at which the motion starts. T The time T du
13、ring which the vibrating body makes one complete vibration is called the period of vibration, f The quantity, which specifies the number of oscillations per second and is the inverse of the period, is called the frequency of the vibration, f = 1 / T. The quantity, which specifies the number of oscil
14、lations in 2 seconds, is called natural frequency of the vibration.It characterizes the dynamics of a given vibrating system and depends on the inherent parameters describing the system.,Dynamics,20,动力学,三、自由振动的特点：A物块离开平衡位置的最大位移，称为振幅。n t + 相位，决定振体在某瞬时 t 的位置 初相位，决定振体运动的起始位置。T 周期，每振动一次所经历的时间。f 频率，每秒钟振动
15、的次数， f = 1 / T 。 固有频率，振体在2秒内振动的次数。反映振动系统的动力学特性，只与系统本身的固有参数有关。,21,Properties of undamped free vibrations:,4. Other results 1. If a constant force is applied to the system in the direction of its vibration then this constant force does not affect the law of the vibration of the system. It only displac
16、es the equilibrium position O into the direction of this force. The amplitude, frequency and phase of the vibration do not change.,(2) The amplitude A and the initial phase depend on the initialconditions (initial displacement and initial velocity).,(1) It is a simple harmonic vibration.,(3)The peri
17、od T and the natural frequency depend only on the natural parameters of the system itself (m, k, I ,etc.).,Dynamics,22,动力学,无阻尼自由振动的特点是：,(2) 振幅A和初相位 取决于运动的初始条件(初位移和初速度)；,(1) 振动规律为简谐振动；,(3)周期T 和固有频率 仅决定于系统本身的固有参数(m,k,I )。,四、其它 1. 如果系统在振动方向上受到某个常力的作用，该常力只影响静平衡点O的位置，而不影响系统的振动规律，如振动频率、振幅和相位等。,23,2. Equiv
18、alent stiffness of series and parallel spring systems,For a parallel spring system,For a series spring system.,parallel,series,Dynamics,24,动力学,2. 弹簧并联系统和弹簧串联系统的等效刚度,并联,串联,25,18-2 Methods of determination of the natural frequency of a system,1. The differential equation of the vibration of a system i
19、n the standard form is,3. Energy Method:,From the equation Tmax=Umax we can determine .,Dynamics,26,动力学,18-2 求系统固有频率的方法,：集中质量在全部重力作用下的静变形,由Tmax=Umax , 求出,27,A system with undamped free vibration is a conservative system.The mechanical energy of system does not change during its motion.When the vibra
20、ting body moves to the maximum distance from the equilibrium position, its velocity becomes zero. Hence, the kinetic energy of the system is zero when its potential energy has its maximum value (choose the static equilibrium position of the system as the zero point of potential energy).When the vibr
21、ating body moves to the equilibrium position, the potential energy of the system is zero, while its kinetic energy is maximum.,Example:,Dynamics,28,动力学,无阻尼自由振动系统为保守系统，机械能守恒。当振体运动到距静平衡位置最远时，速度为零，即系统动能等于零，势能达到最大值（取系统的静平衡位置为零势能点）。当振体运动到静平衡位置时，系统的势能为零，动能达到最大值。,如：,29,The energy method is based on the con
22、servation law of the mechanical energy. It is a simple and convenient method in order to compute the natural frequency of a more complicated vibrational system.,Example 1 The system is shown in the figure.Assume that the system does not swing horizontally. The wheel, which is homogeneous of radius R
23、 and mass M, and the string do not slide one relative to the other. The load has mass M. Neglecting the mass of the string and of the spring,write down the differential equation of vibration of the system and determine its natural frequency.,Dynamics,we get,30,动力学,能量法是从机械能守恒定律出发，对于计算较复杂的振动系统的固有频率来得更
24、为简便的一种方法。,例1 图示系统。设轮子无侧向摆动，且轮子与绳子间无滑动，不计绳子和弹簧的质量，轮子是均质的，半径为R，质量为M，重物质量 m ，试列出系统微幅振动微分方程，求出其固有频率。,31,Solution 1: Let x be the generalized coordinate. The origin of the coordinate is at the position of static equilibrium of the system.,In static equilibrium we have,At an arbitrary position x we get,Dy
25、namics,.,32,动力学,解：以 x 为广义坐标（静平衡位置为 坐标原点）,则任意位置x 时：,静平衡时：,33,Applying the theorem of kinetic energy we have：,From: we get,The differential equation of vibration is：Therefore, the natural frequency is,Dynamics,34,动力学,应用动量矩定理：,由 ， 有,振动微分方程：固有频率：,35,Solution 2: Let x be the generalized coordinate,the or
26、igin is at the position of static equilibriumof the system. Applying the conservation law of mechanical energy we have,Choose the position of equilibrium as the zero point of potential energy. When the displacement of the center of the wheel is x, the elongation of the spring is 2x.,In equilibrium,
27、we have,Dynamics,36,动力学,解2 ： 用机械能守恒定律以x为广义坐标（取静平衡位置为原点）,以平衡位置为计算势能的零位置，并注意轮心位移x时，弹簧伸长2x,因平衡时,37,From T+U= we get,Differentiation with respect to time t and elimination of the common multiplier results in,Dynamics,38,动力学,由 T+U= 有：,对时间 t 求导，再消去公因子 ，得,39,Solution: choose the distance x of C from the eq
28、uilibrium position as the generalized coordinate. The maximum kinetic energy of the system is,Example 2 A tab wheel, whose mass is M and whose gyroradius with respect to the center of the wheel is , rotates without sliding on a horizontal plane. The radii of the big and the small wheels are R and r.
29、 the stiffness of the two springs are and , the load is m. Neglecting the masses of the wheel and the springs, and assuming the string is not extensible, determine the natural frequency of the system in small oscillation.,Dynamics,40,动力学,例2 鼓轮：质量M，对轮心回转半径，在水平面上只滚不滑，大轮半径R，小轮半径 r ，弹簧刚度 ，重物质量为m, 不计轮D和弹
30、簧质量，且绳索不可伸长。求系统微振动的固有频率。,解：取静平衡位置O为坐标原点，取C偏离平衡位置x为广义坐标。系统的最大动能为：,41,The maximum potential energy of the system is：,Dynamics,42,动力学,系统的最大势能为：,43,Writing we have,From Tmax=Umax we then get,Dynamics,44,动力学,设 则有,根据Tmax=Umax , 解得,45,18-3 Damped free vibration of a system with one degree of freedom,1. Con
31、cept of Resistance：Resistance: Resistance is the force acting on an oscillation system restoring its equilibrium. Viscous Damping：In many cases, the resistance caused by the surrounding medium is proportional to the first power of the velocity. Such a resistance is called viscous damping.,Its projec
32、tion form is,where C is called the viscous resistance coefficient, or coefficient of damping.,Dynamics,46,动力学,18-3 单自由度系统的有阻尼自由振动,一、阻尼的概念：阻尼：振动过程中，系统所受的阻力。粘性阻尼：在很多情况下，振体速度不大时，由于介质粘性引起的阻尼认为阻力与速度的一次方成正比，这种阻尼称为粘性阻尼。,投影式：,c 粘性阻尼系数，简称阻尼系数。,47,2.Differential equation of a damped free vibration and its sol
33、ution.For a spring system with viscous damping we get,2,and,2,n,=,=,m,c,n,m,k,w,we write,Using the abbreviations,This is the differential equation of the damped free vibration in standard form.,Dynamics,48,动力学,二、有阻尼自由振动微分方程及其解：质量弹簧系统存在粘性阻尼：,有阻尼自由振动微分方程的标准形式。,49,The general solution is discussed for
34、three cases: 1. In the case of small resistance( or ) the general solution is,Where is the circular frequency of the damped free vibration. Assuming at and , we get,Dynamics,50,动力学,其通解分三种情况讨论：1、小阻尼情形,有阻尼自由振动的圆频率,51,Properties of a damped vibration: The period of the vibration increases, the frequenc
35、y decreases., ratio of resistance,Hence,If then and we get approximately,Dynamics,52,动力学,衰减振动的特点： (1) 振动周期变大，频率减小。,阻尼比,有阻尼自由振动：,当 时， 可以认为,53,(2) The amplitude of the vibration decreases in geometric progression.,The logarithmic decrement is defined as,The ratio of two adjacent amplitude is,2、In the
36、case of critical damping,critical viscous resistance coefficient,Dynamics,the general solution is,54,动力学,(2) 振幅按几何级数衰减,对数减缩率,相邻两次振幅之比,55,We can see that the body tends to move to the position of equilibrium with increasing time exponentially slowly.,Substituting the initial conditions,(at,) we get,a
37、nd,0,0,x,x,x,x,&,&,=,=,3、In the case of overdamping the general solution is,The motion is not a periodic one, With increasing time the quantity x goes exponentially to zero, the body does not vibrate.,Dynamics,56,动力学,可见，物体的运动随时间的增长而无限地趋向平衡位置，不再具备振动的特性。,代入初始条件,3、过阻尼（大阻尼）情形,57,Example 3 In a spring sy
38、stem W=150N, st=1cm, A1=0.8cm, A21=0.16cm. Determine the viscous resistance coefficient c.,Solution：,Because is very small we can write approximatelyand get,Dynamics,58,动力学,例3 质量弹簧系统，W=150N，st=1cm , A1=0.8cm, A21=0.16cm。 求阻尼系数c 。,解：,由于 很小，,59,18-4 Undamped forced vibrationof a system with one degree
39、 of freedom,1. Concept of forced vibration. A forced vibration is the oscillation of a body under the action of an external disturbing force (except the restoring force). A simple harmonic disturbing force is given by , where H is the amplitude of the force, is its circular frequency and is its init
40、ial phase.,This is the differential equation of an undamped forced vibration in standard form. It is a linear inhomogeneous differential equation of second order. its solution is,Dynamics,60,动力学,18-4 单自由度系统的无阻尼强迫振动,一、强迫振动的概念强迫振动：在外加激振力作用下的振动。简谐激振力：H力幅； 激振力的圆频率 ； 激振力的初相位。,无阻尼强迫振动微分方程的标准形式，二阶常系数非齐次线性微
41、分方程。,二、无阻尼强迫振动微分方程及其解,61,The total solution is：,x2 is a stable state force vibration.,3. Main properties of stable state force vibration.,1)It is the additional vibration of a system in one dimension, caused by a simple harmonic disturbing fore. 2)The frequency of this part is equal to the frequency
42、 of the simple harmonic disturbing force, it does not depend on the mass and the stiffness of the system. 3)The amplitude of it depends on the natural frequency of the system and of the frequency and the amplitude of the disturbing force, but not the initial conditions.,the general solution of the c
43、orresponding homogeneous differential equation and,a particular solution of the complete equation.,Dynamics,62,动力学,为对应齐次方程的通解 为特解,3、强迫振动的振幅大小与运动初始条件无关，而与振动系统的固有频率、激振力的频率及激振力的力幅有关。,三、稳态强迫振动的主要特性：,1、在简谐激振力下，单自由度系统强迫振动亦为简谐振动。,2、强迫振动的频率等于简谐激振力的频率，与振动系统的 质量及刚度系数无关。,63,is the amplitude ratio (or the dynam
44、ic coefficient). is the frequency ratio. This is the - curve, the dependence of the amplitude on the frequency.,1,(1) At =0,(3) At ，the phase of the forced vibrations is inverse to the phase of the disturbing force,i.e.,the phase shift is,(2)At ，the amplitude b of this vibration increase with w . an
45、d goes to infinity with w wn .,where b decreases with increasing :,If w goes to , b goes to b0. If w goes to , b goes to zero.,Dynamics,64,动力学,(1) =0时,(2) 时，振幅b随 增大而增大；当 时，,(3) 时，振动相位与激振力相位反相，相差 。,b 随 增大而减小；,65,At resonance we get,4. Resonance,then,b is the infinite( ), this phenomenon is called res
46、onance.,If,Dynamics,66,动力学,4、共振现象,，这种现象称为共振。,此时，,67,18-5 Damped forced vibration of a system with one degree of freedom,1. Differential equation of a damped forced vibration and its solution.,This equation is the differential equation of a damped forced vibration in standard form. It is a linear inh
47、omogeneous differential equation of second order. Its solution is,Dividing both sides of the last equation by m and introducing the notations,we obtain,Dynamics,68,动力学,18-5 单自由度系统的有阻尼强迫振动,一、有阻尼强迫振动微分方程及其解,将上式两端除以m ，并令,有阻尼强迫振动微分方程的标准形式，二阶常系数非齐次微分方程。,69, the amplitude of the forced vibration.,e is the phase shift of the forced vibration, the difference to the phase of the disturbing force.,
