The internal resonances between the longitudinal and transversal oscillations of a forced Timoshenko beam with an axial end spring are studied in depth. In the linear regime, the loci of occurrence of 1 : ir, \(ir \in \mathbb {N}\), internal resonances in the parameters space are identified. Then, by means of the multiple time scales method, the 1 : 2 case is investigated in the nonlinear regime, and the frequency response functions and backbone curves are obtained analytically, and investigated thoroughly. They are also compared with finite element numerical simulations, to prove their reliability. Attention is paid to the system response obtained by varying the stiffness of the end spring, and it is shown that the nonlinear behaviour instantaneously jumps from hardening to softening by crossing the exact internal resonance value, in contrast to the singular (i.e. tending to infinity) behaviour of the nonlinear correction coefficient previously observed (without properly taking the internal resonance into account).

深入研究了带有轴向端部弹簧的受力Timoshenko梁的纵向和横向振动之间的内部共振。在线性区域中，出现的位置为1：  ir，\（ir \ in \ mathbb {N} \）识别参数空间中的内部共振。然后，通过多时间尺度方法，在非线性状态下研究了1：2的情况，并分析了频率响应函数和主干曲线，并进行了深入研究。还将它们与有限元数值模拟进行比较，以证明其可靠性。注意通过改变端部弹簧的刚度获得的系统响应，并且表明，与奇异值（即趋于无穷大）相比，非线性行为通过与确切的内部共振值相交而从硬化状态立即变为软化状态。先前观察到的非线性校正系数的行为（没有适当考虑内部共振）。