Discussion on the role of spring loop swing

The simulation and analysis of the ball trajectory take the parameters k=100N/m, H0=5b, and the simulated motion results are shown in 4. It can be seen from the trajectory of the small ball in the Cartesian coordinates (t=7.8s) that the ball is aperiodic, but relatively close to the periodic motion. Further convert the Cartesian coordinates into polar coordinates, and divide the motion of the ball into radial and lateral motions, as shown.

It can be seen from the motion trajectory in polar coordinates (t=7.8s) that although the small ball does not perform periodic motion in the two-dimensional space, it is quasi-periodic motion in the radial direction and quasi-periodic in the lateral direction. Exercise, make rt, Ht relationship, as shown.

As can be seen, the spring pendulum performs quasi-periodic motion in both the lateral and radial directions. The following theoretical explanations are given for the calculation of simulation results. The equation of motion of a small sphere in polar coordinates is r-rH2=-kmr-L0+gcosHrH+2rH=-gsinH(1). It is known from the literature <2> that a series solution of the equation of motion is r=L+EAcosX1t+A1- 3E2LB22n2-4cos2X2t+2A2+, H=EBcosX2t+A2-E22n+1ABnn+2LcosX1+X2t+A1+A2+, (2) where L=L0+mg/k, which is the spring length of the equilibrium position, X1X2 is radial and lateral respectively The natural frequency of the vibration, n = X1/X2, EAA1 and EBA2 are determined by the initial conditions. It is known from the literature <2> that r is mainly controlled by EAcosX1t+A1, and H is mainly controlled by EBcosX2t+A2, so theoretically it is concluded that the spring pendulum performs quasi-periodic motion in both the lateral and radial directions. This proves that the lateral and radial motions of the spring pendulum under computational simulations are consistent with the theory.

By using the origin software to fit the sine function pair, it is concluded that the motion period in these two directions is TH=2.018s and Tr=0.199s; the motion curves in both directions are less than 10-- compared with the standard sine function. 4. The cycle of the pure ball and the period of the independent spring vibration are respectively determined as TH0=2Pl/g=2.007s and Tr0=2Pm/k=0.199s, which can be analyzed and calculated under the given conditions. The motion and radial motion have little effect on each other's motion, that is, the influence of mutual coupling is small, which is consistent with the theoretical results. It can be seen from equation (2) that the coupling effect between the lateral motion and the radial motion of the spring pendulum is mainly determined by E22n+1ABnn+2Lcos and 3E2LB22n2-4cos(2X2t+2A2), and the lateral and radial motions of the spring pendulum are respectively Mainly controlled by EBcos (X2t+A2) and EAcos (X1t+A1). As discussed in the literature <2>, the main control items of the lateral and radial motions of the spring pendulum are much larger than the coupling term, so the lateral direction of the spring pendulum The coupling between motion and radial motion is very weak.

The influence of the initial swing angle on the motion of the ball keeps the stiffness coefficient of the spring constant. Here, k=100N/m is still set, and the different swing angles are changed to obtain the lateral motion curves under different swing angles, as shown. The lateral motion curves at different initial swing angles can be seen from the fact that the lateral motion of the ball under different initial swing angles has a sinusoidal periodic variation, if a sine function H=H0 is used.

sin2PTt+U fits the above curve and the result is as follows.

The initial swing angle H0/(b) transverse motion period T/s fitting error under different initial swing angles G32.01721.81.41.71.83.17.0@10-5 can be seen, the swing angle becomes larger (such as 60b75b ), the small ball is still in the sinusoidal cycle motion in the horizontal direction, but the cycle is obviously larger than the relatively small swing angle. The single pendulum period formula Tc=T0 (1+0.062H20+0.0088H40) given by the literature optimization can be fitted and compared (T0 is the lateral motion period when H0=5b), and the result is as shown.

It can be seen from the comparison results of the period under different initial swing angles that when the swing angle of the spring swing changes, the transverse direction of the small sphere still moves in a sinusoidal period, the swing angle increases, and the period becomes longer, which is close to the result obtained under the pure swing, so The change of the swing angle hardly affects the lateral motion of the ball. From the theoretical conclusion, the same conclusion can be drawn. From equation (2), it can be seen that the change of the initial swing angle, that is, the change of EB, mainly affects the amplitude of the lateral motion. The larger the initial swing angle, the larger the amplitude (consistent). It does not affect the lateral motion of the ball.

It can be seen from the trend of 0 that when the initial swing angle is small, there is almost no influence on the radial motion, and when the initial swing angle becomes large, the amplitude of the radial motion increases, and the oscillation is intensified, so The large swing angle has a direct effect on the radial periodic vibration of the ball. The following theoretical explanations are given.

From equation (2) and document <2>, it is known that in the case of a small swing angle (ie, when EB is small), r is mainly controlled by EAcos (X1t+A1), so the change of the swing angle has almost no radial motion. Influence; but in the case of a large swing angle (ie, when EB is large), since the coupling term 3E2LB22n2-4cos(2X2t+2A2) of r is large, r is subject to EAcos(X1t+A1) and 3E2LB22n2-4cos(2X2t+). 2A2) The control of the two, so when the swing angle is large, the change of the swing angle has a direct influence on the radial motion of the ball, the swing angle is increased, the amplitude of the radial motion is increased, and the oscillation is intensified, as shown by 0. .

The influence of the stiffness coefficient on the small ball motion The initial swing angle of the small ball is kept 5b, and the different stiffness coefficients are changed. The lateral motion curve is calculated by the program, as shown in 1.

It can be seen from 1 that in the case where the stiffness coefficient k is small, the lateral motion is a non-periodic motion, and the larger k is, the more the periodic motion is sinusoidal. It can be approximated by calculation. When k\10N/m, the small ball is connected in the horizontal direction - close to the sinusoidal cycle, and the period of the small ball is consistent with the pure pendulum, so the small stiffness coefficient It will affect the lateral periodic motion law of the lateral motion curve under the different stiffness coefficients of the ball. The greater the stiffness coefficient, the smaller the effect on lateral motion and the closer the lateral motion is to pure swing.

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