До розрахунку вільних коливань дисипативного осцилятора з нелінійною пружністю

An approximate method for calculating the amplitudes of free damping oscillations of an oscillator with a nonlinear power term in terms of the elastic characteristic under the force of linear viscous resistance is presented. The method does not require the construction of a solution of a nonlinear differential equation of motion and is based on the well-known assumption that the envelope graph of the free damping oscillations of a dissipative Duffing oscillator is approximately described by an exponential function, as in a linear oscillator. Based on this position, the calculation of the damping oscillations of the oscillator with power nonlinearity in the expression of elasticity, where there is also a linear term, is reduced to recurrent relations. In the case of a rigid power characteristic, the two-digit Lambert function of the negative argument, where its first branch is used, is included in the ratio. For a soft power characteristic, the recurrence relation has a Lambert function of a positive argument, which is unambiguous. In order to simplify the numerical implementation of the analytical solutions obtained, it is recommended to use known tables of this special function, and in the case of small modular values of the argument, the proposed approximation by its elementary functions. In order to provide information on the actual errors of the proposed calculation method, examples are given where the results to which it results are compared with the results of numerical computer integration of the differential equation of motion for quadratic and cubic nonlinearities. The satisfactory consistency of the comparison results confirmed the suitability of using the proposed method in engineering calculations. Calculations showed that the influence of the nonlinear component of elasticity weakens during oscillation damping and, at small amplitudes, they are close to the terms of the geometric progression, as in a linear oscillator.