【摘要】 以质量-弹簧系统为研究对象，系统地探讨了线性与非线性振动模型的数学建模方法，重点分析了平衡点的稳定性、微振动与大振幅振动的动力学行为，并揭示了一种反常规现象。用拉格朗日方程建立了1个质量-弹簧系统的非线性振动数学模型，求出了系统平衡点的个数和位置。微振动分3种类型处理。对于常规情况，即仅有1个稳定平衡点，且势能有二次项的情况，采用近似方法得到线性振动方程；对于有多个稳定平衡点的情况，在每个平衡点附近用近似方法得到多个线性振动方程；对于稳定平衡位置，没有势能二次项的本质非线性，仅保留势能的四次项，得到纯三次位移的近似微振动方程。采用谐波-能量平衡法求出了纯三次位移微振动方程的解析解，对微振动情况将近似振动方程的解析解与精确非线性振动方程的数值解进行了对比，求解结果相似性高。对于大幅振动采用数值仿真分析方法，对不同参数、不同初始条件，采用时程曲线、相图、快速傅里叶变换（Fast Fourier Transform, FFT）分析相结合的方式进行了对比分析，验证了模型的合理性和方法的有效性。同时，发现了一种反常规现象：小位移初始条件引起了大振幅振动，大位移初始条件却引起小振幅振动，并给出了反常规现象的合理解释。更多还原

【Abstract】 Taking the mass-spring system as the research object, the mathematical modeling methods for linear and nonlinear vibration models were systematically explored. Emphasis was placed on analyzing the stability of equilibrium points, the dynamic behaviors of micro-vibrations and large-amplitude vibrations,and an unconventional phenomenon was revealed. A nonlinear vibration mathematical model of the mass-spring system was established using Lagrange’s equations. The number and positions of the system’s equilibrium points were solved, and micro-vibrations were treated in three categories. For the conventional case(with only one stable equilibrium point and a quadratic term in potential energy), a linear vibration equation was obtained via an approximate method. For the case with multiple stable equilibrium points, multiple linear vibration equations were derived using the approximate method near each equilibrium point. For the stable equilibrium position with essential nonlinearity(without a quadratic term in potential energy), only the quartic term of potential energy was retained, leading to the acquisition of an approximate micro-vibration equation with pure cubic displacement. The analytical solution of the approximate micro-vibration equation with pure cubic displacement was solved using the harmonic-energy balance method.For micro-vibrations, the analytical solution of the approximate vibration equation was compared with the numerical solution of the exact nonlinear vibration equation, and the solutions exhibited high similarity. For large-amplitude vibrations, numerical simulation was adopted; comparative analyses were conducted under different parameters and initial conditions by integrating time-history curves, phase portraits, and FFT analysis, which verified the rationality of the model and the effectiveness of the method. Meanwhile, an unconventional phenomenon was discovered: small-displacement initial conditions induced largeamplitude vibrations, whereas large-displacement initial conditions induced small-amplitude vibrations. A reasonable explanation for this unconventional phenomenon was also provided.更多还原