双摆振动系统的同相振动性

Double pendulum system with the phase oscillation

采用MIRONENKO的反射函数法研究了双摆振动系统x′=A(t)x与y′=B(t)y的同相振动性,其中A(t)=(aij(t))2×2,B(t)=(bij(t))2×2.假设F(t),G(t)分别为x′=A(t)x,y′=B(t)y的反射矩阵,当A(t+2ω)=A(t),B(t+2ω)=B(t)时,矩阵F(-ω),G(-ω)分别相似于x′=A(t)x,y′=B(t)y的根本矩阵.若特征方程|λE-F(-ω)|=0与|μE-G(-ω)|=0具有相同的特征根,则x′=A(t)x与y′=B(t)y的稳定性相同.文中给出了特征方程|λE-F(-ω)|=0与|μE-G(-ω)|=0具有相同特征根的充分条件.

Using the method of reflective function of Mironenko,this paper studies the nature of the synchronous vibration of the double pendulum system,x′=A（t）x and y′=B（t）y,where A（t）=（aij（t））2×2 and B（t）=（bij（t））2×2 are continuous on R.Suppose that F（t） and G（t） are the reflective matrix of system x′=A（t）x and y′=B（t）y respectively.If A（t＋2ω）=A（t）,B（t＋2ω）=B（t）,then F（-ω） and G（-ω） are similar to the monodromy matrix of system x′=A（t）x and y′=B（t）y respectively.If the roots of the characteristic equations ｜λE-F（-ω）｜=0 and ｜μE-G（-ω）｜=0 are equal,then the stability of null solution of x′=A（t）x and y′=B（t）y are the same.The sufficient conditions for the existence of the same roots of equations ｜λE-F（-ω）｜=0 and ｜μE-G（-ω）｜=0 are also given.