University is skillful! Vibration of spring vibrator under constant dry friction damping Zhuo Shichuang Dong Shenxing 21. School of Technical Education, Xuzhou Normal University, Xuzhou 2210, China; 1 Department of Physics, Suzhou University, Suzhou, Jiangsu 2150061 Introduction The vibration system encountered many forms of damping. For example, when the vibrating body vibrates in the medium, the damping caused by the viscosity of the medium is called viscous damping. As another example, the damping caused by the friction between the contact surfaces is called dry friction damping, also known as Coulomb damping. In addition, internal damping or the like is caused by deformation of the vibration system material. There are many vibration theories about viscous damping in various literatures, but there are few vibrational theories about dry frictional resistance. The direction of actual movement is opposite. The position of the small ball when the spring is naturally extended is used as the origin to establish a dimensional coordinate system. Similarly, the vibration body is set to vibrate from the right end of the maximum displacement 1 = office, and the initial velocity is 0. The differential equation of the vibration of the vibrating body is the vibration of the vibrating body from 1 left. Obey Equation 1; the vibrating body vibrates left and right, obeying Equation 1. Above, in some cases, dry friction damping plays a leading role. For example, in the vibration system composed of the small ball 7 and the lightweight spring worn on the horizontal thin rod, the air resistance can be neglected when the vibration body speed is not large, and the dry friction between the rod and the small ball is guided. Call. 1 Face some discussion about its vibration. 2 The differential equation of vibration and its general solution In order to highlight the main question, only the influence of dry frictional resistance is considered here. The size is constant, and the directions are always written in standard form. They are linear non-homogeneous differential equations with order coefficients. The corresponding homogeneous equation, +; =, is the general solution because the free term 1 of equation 2 is constant, so set it, + mountain 2., = order, the solution date is received, 200, 608; repair date 2001 Shen 023 has a specific physical meaning, which is discussed later. Therefore, the general solution of equation (2) is the same as the general solution of equation 2, which is the sum of two integral constants, which is determined by the initial conditions. What is redundant is to emphasize that the vibrating body reciprocates. Equations 1 and 1 and Equations 3 and 3 are alternately active. The final state of the previous phase is the initial condition for each half of the cycle. Therefore, the sum of the equations 3 and 3 is different at different stages. , the following are determined separately. The movement moves to the maximum displacement at the left end. When the initial 糸1 is cut = 0, especially = 1. and =0 are substituted into the equation 3, respectively, and the solution can be solved = the heart 1 = so the vibration equation of the first stage is the final state value. =兀. 1 =, 33 = 25 = 25, which is the initial condition of the second stage. The large displacement moves to the right end from the large displacement. When the initial condition is called =兀, 1=1.23 and =, respectively, substituted into Equation 3, which can be solved, = 1.38, 2 = 兀. The solution of the vibration equation of the second stage is adjusted to have V=X0, and the final state value is at the position=2, 1=meaning, 4=0, which is the initial condition of the third stage. Similarly, the vibration equation of the third stage is obtained as its final state value = ash, 1 =, 6 8, = the vibration equation of the fourth stage is the final state value of the output = buckle, 8 = 0. The final state of co4 is time = redundant time, especially = the general form of 2 solution, the formula is a natural number, and its value is determined by the sIfl YJ1=inty1 suffix 1 station to 51 language 3 rounding function, it returns An integer that does not work as an independent variable. 3 The basic law of vibration and the basic law of vibration 3.1 Vibration 3.1.1 Dynamic equilibrium position Reciprocating transition on both sides of the vibration center The dynamic equilibrium position is the force and balance point when the vibrating ball of the yellow vibrator is in the motion state of the corpse. At this. The ball is subjected to zero force and the maximum speed. Due to the presence of friction, when the vibrating body reaches the origin, the resultant force is zero, but the resultant force is not zero. Therefore, the origin is not the dynamic balance position of the vibrating body. However, since the vibrating body is always reciprocating on both sides of the origin. Therefore, the point is called the vibration center. There are two real dynamic balance positions of the vibrating body, which are located at the origin and on both sides. 2 When the vibrating body vibrates to the left, the dynamic balance point is located at the right side of the point. =, that is, 1 to +3 when the vibrating body vibrates to the right. The balance point is located at the left side of the point =, that is, 1 = ization = 8. Since the vibrating body reciprocates, the dynamic balance position reciprocates on both sides of the zero point. 3.1.2 The isochronism of vibration is strictly speaking. Muscle vibration is not a periodic motion. Because the vibration body by the attenuation of the Yin amplitude cannot completely repeat its own path. But it is still reciprocating on both sides of the vibration center, its sturdy nature. Therefore, the time taken for the vibrating body to reciprocate is also referred to as a period. It can be seen from Equation 4 that at different stages, although the vibrating body is obtained and the circular frequencies are the same, the time used in each stage is equal. If you use the two adjacent stages as a cycle, you will still be a factory friend. Can be determined by the frictional resistance. The vibration is still isochronous. 3.1.3 The amplitude of each stage of the vibration is in the order of the law of the difference of the number of stages. The amplitude of the half-cycle is in turn, 3, 5, 5, 2, 15 can be, the vibration is in each stage, the amplitude is decreasing by the law of the arithmetic series. The tolerance is the large displacement of the amplitude for the dynamic equilibrium position. Rather than the maximum displacement of a thousand vibration centers. 3.1.4 In each stage of vibration, the amount of energy attenuation is caused by the friction of the arithmetic series. The attenuation along the energy of each stage is equal to the work done by the friction at this stage. Knowing + can be used to vibrate the energy of each stage of the energy in accordance with the difference of the number of rules of the servant; into the decreasing tolerance of 4 for 3.2 vibration like the board to make vibrations. ,1. From the images in 3, not only the obvious changes such as the change with time, but also the isochronism of the decreasing vibration of the amplitude can be seen. It is also clearly staged along the cycle. When =7,3727, there is a continuous but unguided point on the image, and there is a breakpoint on the image. These points are the turning points between the stages, resulting from a sudden change in the direction of the friction. In the half cycle of each phase of the vibration, the axis of symmetry of the vibrating phase segment elliptic curve is different. When the vibrating body vibrates, the axis of symmetry of the semi-elliptic curve is at the point, the right side = 3; when the vibrating body vibrates to the right, the axis of symmetry of the semi-elliptic curve is discussed in the original 4 V.OH.=3j! I3v has extreme value. By the elimination of Equations 4 and 4, the general form of the semi-elliptic curve equation of the image can be obtained as a function of the relationship between production, human and ugly = total energy with time, 4, 0. It can be seen from 4 that during the vibration process In the middle, the kinetic energy and the potential energy are transformed, and the total energy of the system is continuously decreasing. In addition, it can be seen from 4 that the energy residual potential energy of the vibration final state system is +. The meaning of the existence of 3 is the amount of shift of the dynamic equilibrium position relative to the center of vibration. In addition, it is known from 3.1.3 that the amplitude of the amplitude of each adjacent half cycle is 2, and 8 is 7 again, the amount of amplitude attenuation due to friction loss. 4.2 The number of half-cycles of vibration is determined by the formula. The amplitude of the vibration at each stage is =, the attenuation is the most 23, and the vibration can be maintained. The half-cycle, that is, the solution is also a natural number, so there is 2.J, its vibration body The number of half cycles that can sustain vibration. 4.3 The conditions for forming vibration are given by Equation 5. The vibrator cannot vibrate at this time. Along, that is, 0 red. At this time, the vibrating body can only move from the point of the maximum displacement on the right side to the left by the right balance position at 5 o'clock to the left maximum displacement point and stop, 5 then the vibrating body can reciprocate and stop, 5. The condition of the vibrating body can form the secondary vibration of the vibrating foot. The foot of the vibrating milk is satisfied. The final state of the vibrating body is 4.4.1. The half-cycle number of the vibrating body can sustain vibration. After the determination, according to the final state edge value = and Equation 4, the final state position of the vibration system can be determined only. = 1 =, 2 on the right side of the center; 0, if it is an odd number, the vibration body 7 is on the left side of the vibration center = 2,. The + interval on the special axis is the interval in which the vibrating body can stably stay at the end of the vibration, which is called the static balance interval. 4.4.2 The residual potential energy of the vibration system vibrates the final state. If the vibrating body does not stop at the vibration center, it stops at other positions within the static 1 balance interval. Then the residual potential energy of the system is + zero. According to the final position of the vibrating body. Can calculate the residual potential energy of the system, = + do 2, 32. Dry, Huang Yu Yu Yunqiang and other translations. Beijing Science Press, 1979.294299. 2 Suandronov entered the Witts Ha Yijin. 0. Theory of vibration. Translation of vibration mechanics translation group. Beijing Science Press, 1973.156163. Paint An Shen, Du Yuying. The basis of mechanics. Beijing Higher Education Press, 1982.425429. 4 Department of Theoretical Mechanics, Harbin Institute of Technology. Theoretical mechanics is called. Beijing Higher Education Press, 1982.308, 314. Simple multi-range micro-current source digital metering instrument old type 4 appendix 13 type micro-meter panel meaning and internal main structure principle meaning 6 and 7 is not difficult to see that type 8 micrometer can be used as a range of 1.99 people DC digital current head with resistance of 0. 21 pages We are a specialized manufacturer integrating the development, production and distribution of various Safety Shoes and labor protection shoes. What`s more,we applied for and got the CE,ISO and LA certification to keep to international safety principle. The main functions include: Anti-slip, Anti-smash, Anti-puncture, Antistatic, Insulation, Oil resistant, Acid and Alkali resistant and High temperature resistant. Safety Shoes Safety Shoe, Safety Shoes For Men, Work Safety Shoes, Comfortable Safety Shoes, Industrial Safety Shoes, Construction Safety Shoes QINGDAO SANJET NEW POWER MACHINERY CO., LTD. , https://www.sanjet.net