P(r)P振子的自由振动频率规律性研究

Regularity research on free vibration frequency of P(r)P oscillators

将恢复力为ε·sign（x）·｜x｜^r的振子命名为具有单纯r次幂型恢复力的振子,记为P（r）P振子,给出了其自由振动频率的精确计算公式。提出了频率幂次常数和当量刚度等概念,将P（r）P振子自由振动频率归纳为频率幂次常数与当量刚度平方根的乘积。提出了P（r）P振子的分类方法：把幂次r在0和1之间的称为渐柔型P（r）P振子,振幅愈大,其自由振动频率愈小;幂次r大于1的称为渐刚型P（r）P振子,其自由振动频率随振幅的增大而增大。

Oscillators with restoring forceε·sign（x）·｜x｜^r are named oscillators with pure r-th power type restoring force,abbreviated as P（r）Poscillators,and an exact free vibration frequency formula of this group of oscillators is presented.The concepts of frequency-power constant and equivalent stiffness are proposed,and the free vibration frequency of P（r）P oscillators is explained as a product of frequency-power constant and square root of equivalent stiffness consequently.The free vibration frequencies of P（r）P oscillators are related with the amplitude except linear oscillator.The exponent ris the key parameter of P（r）Poscillators,which influences the free vibration frequencies remarkably and determines frequency-power constants particularly.Frequency-power constant monotone decreases as exponent rincreases.P（r）Poscillators can be divided into two groups.The softening P（r）P oscillators have exponent r in the range of 0to 1,whose free vibration frequencies decrease with an increase in amplitude while the hardening P（r）P oscillators have exponent r bigger than 1,whose free vibration frequencies increase with an increase in amplitude.