In this paper, a simplified model is established based on the principle of Coulomb torsion scale experiment, and the stability problem of the dynamic system model is analyzed and discussed. In order to make the problem more intuitive and concise, first establish a simplified model of the system: a stiffness coefficient of k, a spring that can be freely vibrated on a smooth horizontal surface is fixed at one end, and the other end is a small ball with a mass of m and a charge of q1. Fix a small ball 2 with a different electric charge and a power of q2 not far away, and make the vibration direction of the spring coincide with the connection of the two small balls, as shown in Fig. 1. For example, if the position of the small ball 1 is the coordinate origin O when the spring is not extended, and the distance from the small ball 2 to the origin is 1, the dynamic equation of the system is: md2xdt2=-kx+q1q24PE0#1(1-x)2 (1) Let X20=km, B2=q1q24PE0m(1) can be written as x+X20x-B21(lx)2=0(2), and the restoring force of the system is f(x)=X20x-B21(lx)2 (3), the potential energy of the origin of the coordinate is zero, and the potential energy function of the system is u(x)=12X20x2-B2xl(lx)(4).
Since the system's equilibrium point and phase trajectory curve are conservative, there is an energy integral E(x, y)=12¤x2+12X20x2-B2xl(lx)=hO(5), so that the phase trajectory equation is ¤ X2+X20x2-2B2xl(lx)=2h0(6), take l=1, get a cubic equation x3-2x2+x-B2X20=0(7) from f(x)=0, let y=x-23L =B2X20 Equation (7) can be transformed into y3-13y-(L-227)=0(8). It can be known from the relationship between the root of the cubic equation and the coefficient that when L>4P27, equation (8) has only one real root. Since this root is not in the range of <0,1>, the system does not have a balance point, and the system cannot reach the equilibrium state.
When L = 4P27, equation (8) has only one real root in the range <0, 1>, and the system has only one unstable equilibrium point.
When L < 4P27, equation (8) has two real roots in the range <0, 1>, that is, the system has two equilibrium points.
The stability analysis of the system shifts the coordinates, with x1 as the new coordinate origin. The displacement of the mass is still represented by x. The linear restoring force of the system can be expressed as -k(x1+x), and the Coulomb gravity can be expressed as q1q2P4PE0#1P ( L-x1-x)2, let a=l-x1, the dynamic equation of the system can be written as d2xdt2=-X20(x1+x)+B2#1(ax)2(9). Considering the micro-vibration, the second term on the right side of the equation is developed by McLaughlin. The first three terms are obtained. The dynamic equation of the system is: d2xdt2=-(X20-2B2a3)x+3B2a4x2(10), and the approximate linear equations are ¤x=y¤y=-(X20-2a3#B2)x(11). Its characteristic equation is -K,1c, -K=K2+pK+q=0(12) where p=0q=-c=(X20-2a3B2) since f(x1)=X20x1-B2(1-x1)2 =0x1X0 and 2x11-x1<1, so q=X20-2a3B2=1x1(X20x1-B2(1-x1)22x11-x1)>0(13), thereby determining that the singular point x1 is the center point and is stable. point. a=l-x2, whose characteristic equation is also of the formula (12). Since 2x21-x2>1q=1x2
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