Dynamic discussion on multi-elastic oscillation system
The same N spring oscillators are placed in series on a smooth horizontal surface with one end fixed and the other end free to move. Let the displacement of the ball at any time be x 1, x 2, ..., xn, ..., x N, respectively. The kinetic energy of the system is T = 6 N n = 1 1 2 m x2 n The potential energy of the system is V = 6 N n = 1 1 2 k(xn - xn - 1)2 Therefore, the Lagrangian function is L = T - V = 6 N n = 1 1 2 m x2 n - 1 2 k(xn - xn - 1)2 Substituting it into the Lagrangian equation of motion, mxn - k(xn +1 - 2 xn + xn - 1 =0, (n =1, 2, ..., N - 1) mx N + k(x N - x N - 1) =0 so xn - Ω2(xn +1 - 2 xn + xn - 1) =0 , (n =1,2,...,N - 1)x N +Ω2(x N - x N - 1) =0(1)(2) is the dynamic equation of the system, where Ω2 = km, x 0 =0.2 Analytical solution of system motion When the system is making small vibrations, according to the literature < 3 >, < 4 >, the test solution of equation (1) can be set as xn = A 0 sin n Substituting it into equation (3), the equation of motion of the system is xn(t) = 6 N i =1 sin n < i 4 2 N +1 6 N n =1 xn(0) sin n < i cosωi t + 4(2 N +1)ωi 6 N n =1 xn(0) sin n < i sinωi t(4)3 The motion of the system 3. The motion curve and phase trajectory of the system are visible from equation (4), the system The motion condition will be determined by the initial conditions and the number N of spring oscillators. We choose the initial condition xn(0) = A 0, xn(0) = 0 (n = 1, 2,...,N ), which is easy to implement and easy to discuss. In order to comprehensively and intuitively analyze the motion of the system, we have made small according to formula (4), when the initial conditions are selected and N is different, with Ωt as the horizontal axis and xn(t) / A 0 as the vertical axis. The motion curve of the ball, and the phase trajectory of each ball is made with xn(t) / A 0 as the horizontal axis xn(t) / (A 0Ω) as the vertical axis. When N = 1, the system is a spring oscillator, which is described in detail in many textbooks as a simple harmonic vibration model. The small ball is a simple harmonic vibration whose displacement is determined by the equation x(t) = A 0 cosωt = A 0 cosΩt , and the trajectory is an ellipse in the phase plane. When N = 2, the system is a double-spring system, and the equation of motion of each ball can be written by equation (4) as x 1(t) ≈ A 0 <0. 7236cos(0. 618Ωt) +0. 2764cos(1. 618Ωt) > x 2(t)≈A 0 <1. 1708cos(0. 618Ωt) - 0. 1708cos(1. 618Ωt) >When N =3, N =5 and N =20, use Matlab The software made the motion curve of each small ball and took Ωt = 200 to make the phase trajectory of some small balls as shown. summary Comparing and analyzing the motion curves and phase trajectories of each ball, it is easy to conclude that the motion of the system under the condition that the initial displacement of each ball is the same and the initial velocity is zero: 1) as shown by 5 and 8. The motion curve, combined with phases, 4, 6, 7, 9, 10, 11 shows that each ball performs a non-periodic motion. 2) As the number of spring oscillators in series increases, that is, N becomes larger, and the difference between some small balls is obvious. 3) With the increase of N, the phase trajectory of the small ball looks similar to the chaotic motion, and the phase trajectories always circulate in a certain area, but they are not chaotic motion. Because an important feature of chaotic motion is the sensitive dependence of the system on the initial conditions, that is, a slight change in the initial conditions, will lead to a huge difference in the behavior of the system at a later time, for which the initial conditions within the elastic limit of the spring When making small changes, the phase trajectory does not change much. Baby Bathtub Baby Bathtub,clawfoot tub feet types,Baby Bathtubs,modern freestanding tub,Walk-in Bathtubs Guangzhou Aijingsi Sanitary Products Co.,Ltd , https://www.infinityedgehottub.com