This work presents the development of analytical solutions regarding the symmetric free vibrations of pyramidal trusses, considering a general large strain measure. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. Illustrative examples considering the quadratic (Green–Lagrange) and logarithmic (Hencky) strain measures are used to exemplify the influence of the strain measure on the results. Pyramidal trusses are well-known bistable structures and in the unloaded configurations may present different analytical solutions depending on the initial conditions: small amplitude vibrations around each stable equilibrium configuration or large cross-well vibrations. These families of solution are separated by two homoclinic orbits, constituting the boundaries of the two potential wells, which are related to the saddle point initial conditions. In addition, closed-form time solutions for the undamped vibrations are derived and the nonlinear frequency–amplitude relations, which are a measure of the degree and type of nonlinearity (hardening or softening), are obtained. Finally, a semi-analytical procedure for the damped vibrations is also constructed.

这项工作提出了一种关于金字塔桁架对称自由振动的解析解决方案，并考虑了一般的大应变测量。在这种情况下，连续体的未变形和变形构型明显不同，因此需要对其进行明确区分。考虑了二次（格林-拉格朗日）和对数（亨克基）应变测量的说明性示例用于举例说明应变测量对结果的影响。金字塔形桁架是众所周知的双稳态结构，在空载配置下，可能会根据初始条件提供不同的分析解决方案：每个稳定的平衡配置周围的振幅较小的振动或跨井振动的较大。这些解决方案族由两个同斜轨道分开，构成两个势阱的边界，这与鞍点的初始条件有关。另外，推导了无阻尼振动的封闭形式的时间解，并获得了非线性频率-振幅关系，该关系是非线性程度（硬化或软化）的量度。最后，还建立了阻尼振动的半解析程序。