Discrete the spring into n equal length segments, each small segment length ds, mass dm=ds, then the stiffness coefficient of each small segment spring is: k=l0dsk, when the oscillator displacement is x(t), the distance is fixed The elongation of the small segment at point s is: s=kx+Mg+l-slmgkl0ds, then the elastic potential energy of the small segment spring is: dEk=12kl0ds2s=12(kl0)kx+Mg+l-slmgkl02ds, the gravity of the small segment spring The increment of potential energy is: dEg=-dmgx(t)f(s), and the kinetic energy of the small segment spring is: dT=12(ds)t(s,t)2=12 2ds integral type, the kinetic energy of the system can be obtained as follows: T=12Mx2+12x2l0f2(s)ds(10) integral formula can be used to obtain the elastic potential energy of the system: Ek=EK0+12 The integral formula can be used to obtain the gravitational potential energy of the system: Eg=Eg0-Mgx-mgx1+m2m+4M. Because the resistance is not considered, the mechanical energy is conserved during the vibration process, so there are 12M+l0f2(s)dsx2+12kx2-mgxm +Mm+2M=E0-Ek0-Eg0, find the derivative of the above formula, and sort it out: M+l0f2(s)dsx+kx=m+Mm+2Mmg, let me=l0f2(s)ds,x+kM +mex=1M+mem+Mm+2Mmg, the vibration frequency of the vibrator can be obtained by the formula: =kM+me, and the equivalent mass of the spring can be obtained by the formula: me=l0f2(s)ds=m3+m2 (10M+6m) 30(2M+m)2, when mM, me=m3 when m=M, me=m3+16m270041m when m=2M, me=m3+11m1200425m when mM,me=m3+m5=815m In the transient process, the A1 and A2 lamps consume the same energy. We only discuss the A1 lamp. WA=0i2LRAdt=E2RA2(R0+RA)2L2(R+RA)1-1e2, the energy consumed by the A1 lamp in the steady state, the equal time, WA=E2(R+RA)2RA, WAWA=121-1e2 <1 Therefore, in such a configuration, when power is turned off, it is impossible for A1 and A2 to blink. If you want to be in the circuit, when the power is off, there will be a flashing phenomenon. In the case of the analogy, (R1+R)>1.32 (R0+R2), the A1 lamp can flash. The A2 lamp consumes less energy when it is transient, and therefore darkens.
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