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Active vibration control of the multilayered smart nanobeams: velocity feedback gain effects on the system’s behavior

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Abstract

The primary focus of this study is to analyze the vibrational properties of a nanobeam with a porous structure, which is composed of a functionally graded material. Additionally, the nanobeam comprises a magnetostrictive substance known as Terfenol-D. The nanobeam is postulated to include three distinct layers, whereby the central layer is comprised of a magnetostrictive material, while the outside layers are built of functionally graded material. On the other hand, the use of higher-order parabolic shear deformation beam theory is utilized to get the kinematic relations. Moreover, Eringen’s nonlocal theory is used to include the impact of small-scale phenomena. The governing equations are derived by the application of Hamilton’s principle and then solved using analytical techniques. This study presents a thorough examination and elucidation of the influence exerted by several elements, such as aspect ratio, feedback gain, and gradient index, on the system being studied. Based on extant research, empirical evidence indicates that increasing the feedback gain results in a decrease in the natural frequencies. In order to establish the accuracy and dependability of our present study, we have undertaken a comparison examination of our results in relation to the existing body of the literature. The results of the present study have the potential to advance the understanding and progress in the field of nano-systems, namely nano-sensors and nano-actuators.

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Acknowledgements

The authors would like to thank the reviewers for their comments and suggestions to improve this article’s clarity.

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  1. Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran

    Farzad Ebrahimi & Mehrdad Farajzadeh Ahari

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  1. Farzad Ebrahimi
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  2. Mehrdad Farajzadeh Ahari
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Correspondence to Farzad Ebrahimi.

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The authors declare that there is no conflict of interest. Professor F. Ebrahimy defined the subject and guided authors regarding limitations and errors and revised the manuscript. M. F. Ahari analyzed the data and drafted the manuscript.

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Appendix 1

Appendix 1

$$\begin{gathered} K_{11} = - A_{11} \left( {o^{2} } \right) \hfill \\ K_{12} = - B_{11} \left( {o^{2} } \right) + \alpha S_{11} \left( {o^{2} } \right) \hfill \\ K_{13} = \alpha S_{11} \left( {o^{3} } \right) \hfill \\ K_{21} = - B_{11} \left( {o^{2} } \right) + \alpha S_{11} \left( {o^{2} } \right) \hfill \\ K_{22} = - D_{11} \left( {o^{2} } \right) + 2\alpha M_{11} \left( {o^{2} } \right) - \alpha^{2} L_{11} \left( {o^{2} } \right) - A_{55} + 2\beta D_{55} - \beta^{2} M_{55} \hfill \\ K_{23} = \alpha M_{11} \left( {o^{3} } \right) - \alpha^{2} L_{11} \left( {o^{3} } \right) + 2\beta D_{55} \left( o \right) - \beta^{2} M_{55} \left( o \right) - A_{55} \left( o \right) \hfill \\ K_{31} = \alpha S_{11} \left( {o^{3} } \right) \hfill \\ K_{32} = \alpha M_{11} \left( {o^{3} } \right) - \alpha^{2} L_{11} \left( {o^{3} } \right) + 2\beta D_{55} \left( o \right) - \beta^{2} M_{55} \left( o \right) - A_{55} \left( o \right) \hfill \\ K_{33} = - \alpha^{2} L_{11} \left( {o^{4} } \right) - \beta^{2} M_{55} \left( {o^{2} } \right) + 2\beta D_{55} \left( {o^{2} } \right) - A_{55} \left( {o^{2} } \right) - K_{w} \hfill \\ \quad \quad \qquad\; - K_{g} \left( {o^{2} } \right) - K_{w} \left( {\mu^{2} o^{2} } \right) - K_{g} \left( {\mu^{2} o^{4} } \right)K_{33} = - \alpha^{2} L_{11} \left( {o^{4} } \right) - \beta^{2} M_{55} \left( {o^{2} } \right) \hfill \\ \quad \qquad \quad \; + 2\beta D_{55} \left( {o^{2} } \right) - A_{55} \left( {o^{2} } \right) - K_{w} - K_{g} \left( {o^{2} } \right) - K_{w} \left( {\mu^{2} o^{2} } \right) - K_{g} \left( {\mu^{2} o^{4} } \right) \hfill \\ \end{gathered}$$
$$\begin{gathered} M_{11} = I_{0} \left( {1 + \mu^{2} o^{2} } \right) \hfill \\ M_{12} = - \left( {I_{1} - I_{3} \alpha + \mu^{2} I_{1} o^{2} + I_{3} \alpha \mu^{2} o^{2} } \right) \hfill \\ M_{13} = - \left( { - I_{3} \alpha \left( o \right) - \mu^{2} I_{3} \alpha \left( {o^{3} } \right)} \right) \hfill \\ M_{21} = - \left( {I_{1} - I_{3} \alpha + \mu^{2} I_{1} o^{2} + I_{3} \alpha \mu^{2} o^{2} } \right) \hfill \\ M_{22} = - \left( {I_{2} - 2I_{4} \alpha + I_{6} \alpha^{2} + I_{2} \mu^{2} o^{2} - 2I_{4} \alpha o^{2} + I_{6} \mu^{2} \alpha^{2} o^{2} } \right) \hfill \\ M_{23} = - \left( {I_{6} \alpha^{2} o - I_{4} \alpha \left( o \right) + I_{6} \alpha^{2} \mu^{2} o^{3} - I_{4} \alpha \mu^{2} o^{3} } \right) \hfill \\ M_{31} = - \left( { - I_{3} \alpha \left( o \right) - \mu^{2} I_{3} \alpha \left( {o^{3} } \right)} \right) \hfill \\ M_{32} = - \left( {I_{6} \alpha^{2} o - I_{4} \alpha \left( o \right) + I_{6} \alpha^{2} \mu^{2} o^{3} - I_{4} \alpha \mu^{2} o^{3} } \right) \hfill \\ M_{33} = - \left( {I_{0} + I_{6} o^{2} \alpha^{2} + I_{0} \mu^{2} o^{2} + I_{6} \mu^{2} o^{4} \alpha^{2} } \right) \hfill \\ \end{gathered}$$
$$\begin{aligned} A_{{11}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{{11}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} C_{{11}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{{11}}^{f} {\text{d}}z \\ A_{{12}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{{12}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} C_{{12}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{{12}}^{f} {\text{d}}z \\ A_{{22}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{{22}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} C_{{22}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{{22}}^{f} {\text{d}}z \\ A_{{44}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{{44}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} C_{{44}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{{44}}^{f} {\text{d}}z \\ A_{{55}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{{55}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} C_{{55}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{{55}}^{f} {\text{d}}z \\ A_{{66}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{{66}}^{c} dz + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} C_{{66}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{{66}}^{f} {\text{d}}z \end{aligned}$$
$$\begin{gathered} B_{{11}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{{11}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} zC_{{11}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{{11}}^{f} {\text{d}}z \hfill \\ B_{{12}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{{12}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} zC_{{12}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{{12}}^{f} {\text{d}}z \hfill \\ B_{{22}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{{22}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} zC_{{22}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{{22}}^{f} {\text{d}}z \hfill \\ B_{{44}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{{44}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} zC_{{44}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{{44}}^{f} {\text{d}}z \hfill \\ B_{{55}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{{55}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} zC_{{22}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{{55}}^{f} {\text{d}}z \hfill \\ B_{{66}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{{66}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} zC_{{66}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{{66}}^{f} {\text{d}}z \hfill \\ \end{gathered}$$
$$\begin{gathered} D_{{11}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{{11}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{2} C_{{11}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{{11}}^{f} {\text{d}}z \hfill \\ D_{{12}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{{12}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{2} C_{{12}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{{12}}^{f} {\text{d}}z \hfill \\ D_{{22}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{{22}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{2} C_{{22}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{{22}}^{f} {\text{d}}z \hfill \\ D_{{44}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{{44}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{2} C_{{44}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{{44}}^{f} {\text{d}}z \hfill \\ D_{{55}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{{55}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{2} C_{{55}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{{55}}^{f} {\text{d}}z \hfill \\ D_{{66}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{{66}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{2} C_{{66}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{{66}}^{f} {\text{d}}z \hfill \\ \end{gathered}$$
$$\begin{aligned} S_{{11}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{3} C_{{11}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{3} C_{{11}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{3} C_{{11}}^{f} {\text{d}}z \\ S_{{12}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{3} C_{{12}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{3} C_{{12}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{3} C_{{12}}^{f} {\text{d}}z \\ S_{{22}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{3} C_{{22}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{3} C_{{22}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{3} C_{{22}}^{f} {\text{d}}z \\ S_{{44}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{3} C_{{44}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{3} C_{{44}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{3} C_{{44}}^{f} {\text{d}}z \\ S_{{55}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{3} C_{{55}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{3} C_{{55}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{3} C_{{55}}^{f} {\text{d}}z \\ S_{{66}} &= \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{3} C_{{66}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{3} C_{{66}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{3} C_{{66}}^{f} {\text{d}}z \\ \end{aligned}$$
$$\begin{gathered} M_{{11}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{4} C_{{11}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{4} C_{{11}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{4} C_{{11}}^{f} {\text{d}}z \hfill \\ M_{{12}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{4} C_{{12}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{4} C_{{12}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{4} C_{{12}}^{f} {\text{d}}z \hfill \\ M_{{22}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{4} C_{{22}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{4} C_{{22}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{4} C_{{22}}^{f} {\text{d}}z \hfill \\ M_{{44}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{4} C_{{44}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{4} C_{{44}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{4} C_{{44}}^{f} {\text{d}}z \hfill \\ M_{{55}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{4} C_{{55}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{4} C_{{22}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{4} C_{{55}}^{f} {\text{d}}z \hfill \\ M_{{66}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{4} C_{{66}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{4} C_{{66}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{4} C_{{66}}^{f} {\text{d}}z \hfill \\ \end{gathered}$$
$$\begin{gathered} L_{{11}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{6} C_{{11}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{6} C_{{11}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{6} C_{{11}}^{f} {\text{d}}z \hfill \\ L_{{12}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{6} C_{{12}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{6} C_{{12}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{6} C_{{12}}^{f} {\text{d}}z \hfill \\ L_{{22}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{6} C_{{22}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{6} C_{{22}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{6} C_{{22}}^{f} {\text{d}}z \hfill \\ L_{{44}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{6} C_{{44}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{6} C_{{44}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{6} C_{{44}}^{f} {\text{d}}z \hfill \\ L_{{55}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{6} C_{{55}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{6} C_{{55}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{6} C_{{55}}^{f} {\text{d}}z \hfill \\ L_{{66}} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{6} C_{{66}}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{{ - hc}}{2}}} z^{6} C_{{66}}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{6} C_{{66}}^{f} {\text{d}}z \hfill \\ \end{gathered}$$

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Ebrahimi, F., Ahari, M.F. Active vibration control of the multilayered smart nanobeams: velocity feedback gain effects on the system’s behavior. Acta Mech 235, 493–510 (2024). https://doi.org/10.1007/s00707-023-03769-y

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