In this paper primary and secondary resonance of carbon nanotube conveying magnetic nanofluid and subjected to a longitudinal magnetic field resting on viscoelastic foundation with different boundary conditions is investigated. To investigate the small scale effects, stress driven nonlocal integral model has been used and to show the more correctness of stress driven nonlocal integral model response, in studying the behavior of carbon nanotube with different boundary conditions, its results are compared with strain gradient model. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Kármán strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The nonlinear natural frequencies are obtained from the perturbation method and the divergence and flutter instability due to the increase in nanofluid velocity is investigated. Then the frequency response for primary, subharmonic and superharmonic resonance is obtained using the method of multiple scales.

Finally, the effects of length small scale parameters, longitudinal magnetic field, magnetic nanofluid and boundary conditions on nonlinear free and forced vibration of carbon nanotube are investigated. As the most important results, as the intensity of the magnetic field increases, the critical flow velocity increases and divergence and flutter occur later. But the critical flow velocity decreases with increasing the intensity of the magnetic field for a carbon nanotube conveying magnetic nanofluid. In forced vibration, increasing the intensity of the magnetic field increases the amplitude of the response for all boundary conditions in primary and secondary resonance.

本文研究了输送磁性纳米流体的碳纳米管在具有不同边界条件的粘弹性基础上的纵向磁场作用下的初级和次级共振。为了研究小尺度效应，使用应力驱动的非局部积分模型并显示应力驱动的非局部积分模型响应的更正确性，在研究不同边界条件下碳纳米管的行为时，将其结果与应变梯度模型进行了比较。控制偏微分方程源自伯努利-欧拉梁理论，利用 von Kármán 应变-位移关系。使用伽辽金方法，将控制方程简化为非线性常微分方程。通过微扰法获得非线性固有频率，并研究了由于纳米流体速度增加引起的发散和颤振不稳定性。然后使用多尺度方法获得一次、次谐波和超谐波共振的频率响应。

最后，研究了长度小尺度参数、纵向磁场、磁性纳米流体和边界条件对碳纳米管非线性自由振动和受迫振动的影响。作为最重要的结果，随着磁场强度的增加，临界流速增加，发散和颤振发生较晚。但是，对于输送磁性纳米流体的碳纳米管而言，临界流速随着磁场强度的增加而降低。在强迫振动中，增加磁场强度会增加初级和次级共振中所有边界条件的响应幅度。