The differential equations of motion for unidirectional composite non-cylindrical helical springs with rectangular cross-section including warping deformation of wire cross-section were derived using the naturally curved and twisted anisotropic beam theory. They consisted of 14 first-order partial differential equations with variable coefficients. An explicit analytical expression for warping function of Saint-Venant's torsion of unidirectional composite bars with rectangular cross-section was also obtained. The natural frequencies and vibration modal shapes of the springs were determined by using improved Riccati transfer matrix method. The element transfer matrix used in the solution was calculated using the Scaling and Squaring method and Pad'e approximations. Numerical results showed that the warping deformation has a significant influence on the natural frequencies of such springs, it should be considered in their free vibration analysis. Finally, the effects of various parameters on the natural frequencies of unidirectional composite conical springs with rectangular cross-section were investigated.%以各向异性自然弯扭梁理论为基础,首次导出了考虑簧丝截面翘曲变形的单向复合材料矩形截面非圆柱(锥形、双曲形和桶形)螺旋弹簧的运动微分方程,它们由14个变系数的一阶偏微分方程组成.同时得到了单向复合材料矩形截面杆件扭转翘曲函数的显式表达式.弹簧的固有频率和振动模态可以使用改进的Riccati传递矩阵法确定,单元传递矩阵则采用Scaling-Squaring方法以及Pad'e逼近表达式进行计算.数值结果表明,对于单向复合材料矩形截面的非圆柱螺旋弹簧,翘曲变形对其固有频率有着重大的影响,在自由振动分析中必须加以考虑.最后研究了各种设计参数对单向复合材料矩形截面锥形弹簧固有频率的影响.
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