The lateral vibration of the spring oscillator is shown in the general configuration of the lateral vibration for the spring oscillator system shown. Let the length of each spring be l, then the force of each spring on the object M is T=K(l-a0), the direction is along the axial direction of the spring, so the restoring force provided by each spring is TsinH The direction is negative along the y axis. From Newton's second law and sinH=y/l, Md2ydt2=Fy=-2TsinH=-2K(l-a0)yl=-2Ky(1-a0l)(5) Use uT to indicate that the spring oscillator leaves the equilibrium position for lateral vibration. The displacement, there is d2uLdt2+X2LuT(1-a0l)=0(6) Since l=a2+u2L, that is, l is a function of uT, the above equation is not a simple harmonic equation, which is because the restoration on M The force is not strictly proportional to the displacement uT of the object leaving the equilibrium position. Since the equation is difficult to obtain an analytical solution, we will solve this equation by numerically solving the differential equation and express the result as a graphical representation.
If the parameters X2L=4s-2, A=4m, U=0, a0=5m, a=10m, the equation is solved by computer numerical values ​​and the results are plotted as a function of displacement and velocity as a function of time. Compared with the longitudinal harmonic vibration of the spring oscillator, it is seen that the macroscopic vibration frequency (note that this is not a simple harmonic motion) is significantly smaller.
The lateral vibration of the toy spring approximation toy spring approximation means that Hooke's law is still satisfied when a0/a<<1, so when a0/a<<1, we can ignore a0/l in (6), thus The solution d2uTdt2+X2LuT=0(7) is uT=Acos(XLt+U), which indicates that under this approximation, the lateral vibration is a simple harmonic vibration, and the vibration frequency is the same as the longitudinal vibration, ie XT=XL. There is no limit to the amplitude A.
To illustrate the difference between this approximation and the rigorous solution, we take X2L=4s-2, A=4m, U=0, a0=011m, a=10 (satisfying a0/a<<1), and solving the equation by computer numerical value ( 6) and (7), the relationship between displacement and velocity as a function of time, and the 2-point line and the solid line represent the solutions of equations (6) and (7), respectively.
From the comparison between the solid line and the dotted line of the two graphs, it can be found that the two meet at the beginning of the period, but the difference between them gradually increases with time. This shows that the toy spring approximation is only established within a certain period of time even if a0/a<<1 is satisfied. As the frequency between the two is not exactly the same, the images of the two solutions are gradually separated.
Small vibration approximation of lateral vibration If a0 cannot be ignored with respect to a, the toy spring approximation is no longer applicable. However, if the displacement uT is much smaller than the length a, that is, the small vibration condition, 1lU1a1-12u2Ta2+ is obtained from l=a2+u2T, and (8) is substituted into the formula (6) to obtain d2uTdt2+X2LuT1-a0a1-(12u2Ta2). ), + (9) If the quadratic term and the higher term are omitted, d2uTdt2+X2TuT=0(10) where X2T=X2L(1-a0a), where the lateral vibration can be approximated as a simple harmonic vibration However, its vibration frequency is less than the longitudinal vibration frequency.
Take X2L=4s-2, A=1m, U=0, a0=5m, a=10m (approximating the small vibration condition, but not satisfying a0/a<<1), and solving the equation (6) and equation by computer numerical value. (10), and plot the results of displacement versus velocity versus time, and 0, where the dotted and solid lines correspond to the solutions of equations (6) and (10), respectively.
It can be seen from the comparison between the solid line and the dotted line of the two figures that the two are consistent in a certain period of time, but the difference between them increases with time. This shows that the small vibration approximation is only established for a period of time at the beginning, and as the time increases because the frequencies between the two are not exactly the same, the images of the two solutions are gradually separated.
Conclusion We discuss the solution of the lateral vibration of the spring oscillator in general, and give the solution under two approximate cases. It can be concluded that the lateral vibration is no longer a simple harmonic motion, and its macro frequency is always less than The frequency of longitudinal vibration; under the approximation of the toy spring, the lateral vibration can be approximated as a simple harmonic vibration whose frequency is equal to the frequency of longitudinal vibration; under the small vibration approximation, the lateral vibration can also be approximated as a simple harmonic motion, but its frequency is less than The frequency of longitudinal vibration; the satisfaction of the approximation condition is related to the length of time we care about, that is, the approximate condition is relatively more severe when the time is long, and if the time of study is shorter, the approximation condition can be appropriately relaxed.
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