The paper deals with the dynamics of a lumped mass mechanical system containing two nonlinear springs connected in series. The external harmonic excitation, linear and nonlinear damping are included into considerations. The mathematical model contains both differential and algebraic equations, so it belongs to the class of dynamical systems governed by the differential–algebraic system of equations (DAEs). An approximate analytical approach is used to solve the initial value problem for the DAEs. We adopt the multiple scales method (MSM) that allows one to obtain the sufficiently correct approximate solutions both far from the resonance and at the resonance conditions. The steady and non-steady resonant vibrations are analyzed by employing the modulation equations of the amplitudes and phases which are yielded by the MSM procedure.

中文翻译：

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具有串联弹簧的集总系统的非线性振动

该论文涉及包含两个串联连接的非线性弹簧的集总质量机械系统的动力学。考虑了外部谐波激励、线性和非线性阻尼。该数学模型包含微分方程和代数方程，因此它属于由微分代数方程组 (DAE) 控制的动力系统类。近似分析方法用于解决 DAE 的初始值问题。我们采用多尺度方法 (MSM)，它允许人们在远离共振的情况下和在共振条件下获得足够正确的近似解。通过使用由 MSM 程序产生的振幅和相位的调制方程来分析稳态和非稳态共振。