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Coupled mode parametric resonance in a vibrating screen model

机译：振动筛模型中的耦合模式参数共振

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摘要

We consider a simple dynamic model of the vibrating screen operating in the parametric resonance (PR) mode. This model was used in the course of designing and setting of such a screen in LPMC. The PR-based screen compares favorably with conventional types of such machines, where the transverse oscillations are excited directly. It is characterized by larger values of the amplitude and by insensitivity to damping in a rather wide range. The model represents an initially strained system of two equal masses connected by a linearly elastic string. Self-equilibrated, longitudinal, harmonic forces act on the masses. Under certain conditions this results in transverse, finite-amplitude oscillations of the string. The problem is reduced to a system of two ordinary differential equations coupled by the geometric nonlinearity. Damping in both the transverse and longitudinal oscillations is taken into account. Free and forced oscillations of this mass-string system are examined analytically and numerically. The energy exchange between the longitudinal and transverse modes of free oscillations is demonstrated. An exact analytical solution is found for the forced oscillations, where the coupling plays the role of a stabilizer. In a more general case, the harmonic analysis is used with neglect of the higher harmonics. Explicit expressions for all parameters of the steady nonlinear oscillations are determined. The domains are found where the analytically obtained steady oscillation regimes are stable. Over the frequency ranges, where the steady oscillations exist, a perfect correspondence is found between the amplitudes obtained analytically and numerically. Illustrations based on the analytical and numerical simulations are presented.
机译：我们考虑在参数共振（PR）模式下运行的振动筛的简单动态模型。在LPMC中设计和设置此类屏幕的过程中使用了此模型。基于PR的筛网可与传统类型的此类机器相媲美，在传统机器中，此类横向振动直接被激发。它的特点是幅度值较大，并且在相当宽的范围内对阻尼不敏感。该模型表示由线性弹性线连接的两个相等质量的初始应变系统。自平衡的纵向谐波力作用在质量上。在某些条件下，这会导致弦的横向有限振幅振荡。该问题简化为由几何非线性耦合的两个常微分方程组。考虑了横向和纵向振动的阻尼。对该质量弦系统的自由振动和强迫振动进行了分析和数值分析。演示了自由振荡的纵向和横向模式之间的能量交换。对于强迫振荡，找到了一种精确的解析解，其中耦合起着稳定器的作用。在更一般的情况下，谐波分析在忽略较高谐波的情况下使用。确定了稳态非线性振荡所有参数的显式表达式。在解析获得的稳定振荡状态稳定的地方找到域。在存在稳定振荡的频率范围内，通过分析和数值获得的振幅之间找到了完美的对应关系。提出了基于解析和数值模拟的插图。

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