1 Introduction

In a living cell, protein micro/nanotubules as an essential cytoskeletal component play an important role in making the cell shape and its mechanical characteristics. In other words, the protein micro/nanotubules are the most rigid of the cytoskeletal biopolymers, the bending stiffness of which is about hundred times greater than that of actin filaments. Zheng et al. [1] reported that through application of tension to sensory neurons creates new microtubule assembly concomitant. Omelchanko et al. [2] concluded that microtubules provide the necessary framework for polarization of fibroblasts and epitheliocytes. Gupton et al. [3] anticipated that drugs affect the rate of F-actin and microtubule convergence as well as microtubule buckling in a central cell region.

The structure of a living cell and its components may vibrate in various frequency ranges which causes to transfer mass, signals and energy between cells. Pokorny et al. [4] indicated that deterministic forces of biological polar molecules have the capability to transport particles and electrons with higher probability than forces of thermal origin only. They also analyzed vibration states in cells using numerical models. They found that the interaction forces between cancer cells may be lower than those between healthy cells [5]. Atanasov et al. [6] developed a physical model for vibration behavior of microtubules in living cell corresponding to the first four vibration modes relevant to transverse and longitudinal waves.

Due to the micron and submicron size of the microtubule dimensions, small-scale effects have significant influence on its mechanical behavior. In order to take these size effects into consideration, several unconventional continuum theories of elasticity have proposed and employed to predict size-dependent mechanical responses of micro/nanostructures [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. For instance, Gao and An [51] explored the buckling behavior of protein microtubules on the basis of a nonlocal anisotropic elastic shell model. Taj and Zhang [52] investigated the free vibration response of microtubules embedded in an elastic medium. Baninajjaryan and Tadi Beni [53] developed a size-dependent isotropic shell model based on the nonlocal elasticity theory for free vibration analysis of microtubules. Civalek and Demir [54] proposed a nonlocal finite element method for mechanical characteristics of protein microtubules. Tadi Beni et al. [55] predicted the size-dependent buckling behavior of protein microtubule under axial and radial compressions based on the couple stress theory of elasticity. Sahmani and Aghdam [56, 57] predicted the size-dependent nonlinear axial and radial instabilities of protein micro/nanotubules embedded in the cytoplasm of a living cell. They also anticipated the size-dependent nonlinear vibrations of axially loaded lipid protein micro/nanotubules within the prebuckling and postbuckling domains [58].

In accordance with reviewing the historical background, it is common that size effect in type of the stress nonlocality demonstrates softening influence caused a reduction in the stiffness, while the strain gradient small-scale effect plays a hardening role made an increment in the value of stiffness. It means that the previously proposed unconventional continuum theories have not the capability to cover the size dependencies in a perfect way. As a consequence, a new size-dependent theory of elasticity, namely as nonlocal strain gradient theory, was developed by Lim et al. [59] including simultaneously both softening and stiffening features of size effects. Thereafter, several investigations have been carried out using the newly proposed nonclassical continuum theory of elasticity. For instance, Li and Hu [60] employed the nonlocal strain gradient theory of elasticity to analyze the nonlinear buckling characteristics of Euler–Bernoulli nanobeams. Additionally, they predicted the nonlocal strain gradient frequency of wave motion on nanotubes conveying fluid [61]. Yang et al. [62] anticipated the critical nonlocal strain gradient voltages associated with the pull-in instability of functionally graded nanoactuators. Simsek [63] constructed a nonlocal strain gradient beam model for nonlinear vibrations of functionally graded Euler–Bernoulli nanobeams. Farajpour et al. [64] examined buckling behavior of orthotropic nonlocal strain gradient plates using a new size-dependent plate model. Tang et al. [65] analyzed the nonlocal strain gradient wave propagation in a viscoelastic nanotube. With the aid of the strain gradient elasticity theory, Sahmani and Aghdam [66] studied the linear and nonlinear vibrations of supramolecular lipid micro/nanotubules within both prebuckling and postbuckling domains. Li et al. [67] anticipated the size-dependent bending, buckling and free vibration characteristics of axially functionally graded nonlocal strain gradient beams. Lu et al. [68] explored the influences of nonlocality and strain gradient size dependency on the free vibration behavior of beams at nanoscale. Sahmani and Aghdam [69, 70] reported analytical expressions for the nonlocal strain gradient nonlinear buckling and postbuckling behavior of hydrostatic pressurized multilayer functionally graded nanoshells. Wang et al. [44] introduced a nonlocal strain gradient beam model for complex modal analysis for vibrational response of axially moving beams at nanoscale. Sahmani et al. [71,72,73] developed nonlocal strain gradient beam and plate models to analyze size-dependent nonlinear mechanical behaviors of functionally graded porous micro/nanostructures. Zhen et al. [74] employed the local adaptive differential quadrature method and the nonlocal strain gradient free vibrations of viscoelastic nanotubes.

In the present work, for the first time, the size-dependent nonlinear primary resonance of the protein micro/nanotubules under soft harmonic excitation and embedded in an elastic biomedium is investigated. To accomplish this purpose, the nonlocal strain gradient elasticity theory is employed within the framework of the refined hyperbolic shear deformable beam model. Via the Hamilton’s principle, the nonclassical governing differential equations of motion are constructed. After that, a numerical solution methodology based upon the generalized differential quadrature (GDQ) method in conjunction with the Galerkin technique is utilized to solve the nonlinear problem.

2 Mathematical formulations

As depicted in Fig. 1, a protein micro/nanotubule created by twisting of bilayer stripe protein molecules is considered. Due to the chirality, the protein molecules cannot pack parallel of themselves. The lipid micro/nanotubule is modeled as a beam-type structure with length L, thickness h and mid-radius R. Also, a coordinate system is attached to the lipid micro/nanotubule in such a way that its z-axis is along tubule thickness and its x-axis is along tubule length.

Fig. 1
figure 1

Schematic view of the a living cell including protein micro/nanotubules

Full size image

Based upon the hyperbolic shear deformation beam theory [75], the displacement field along different coordinate directions can be written as

$$u_{x} \left( {x,z,t} \right) = u\left( {x,t} \right) - zw_{,x} (x,t) + \left[ {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right]\psi \left( {x,t} \right)$$
(1a)
$$u_{z} \left( {x,z,t} \right) = w\left( {x,t} \right)$$
(1b)

where u and w in order are the displacement components of the biological lipid tubule along x- and z-axis. Additionally, \(\psi\) represents the rotation with respect to the cross section of the micro/nanotubule at neutral plane normal about y-axis.

Consequently, the nonzero strain components are derived as

$$\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx} } \\ {\gamma_{xz} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}l} {u_{,x} + \left( {1/2} \right)w_{,x}^{2} - zw_{,xx} + \left[ {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right]\psi_{,x} } \hfill \\ {\left[ {\cosh \left( {1/2} \right) - \cosh \left( {z/h} \right)} \right]\psi } \hfill \\ \end{array} } \right\}$$

in which \(\varepsilon_{xx}\) and \(\gamma_{xz}\) denote, respectively, the normal and shear strains.

As it has been reported previously, the nonlocal elasticity theory and strain gradient elasticity theory do not consider size effect comprehensively. The nonlocal theory cannot take the higher-order stresses into account. On the other hand, the strain gradient theory has the capability to consider only local higher-order strain gradients. Motivated by this fact, Lim et al. [59] proposed a combination of these theories, namely as nonlocal strain gradient elasticity theory, which assess small-scale effects more reasonably. Accordingly, the total nonlocal strain gradient stress tensor Λ for a beam-type structure can be defined as below [59]

$$\varLambda_{xx} = \sigma_{xx} - \sigma_{xx,x}^{*}$$
(3a)
$$\varLambda_{xz} = \sigma_{xz} - \sigma_{xz,x}^{*}$$
(3b)

where \(\sigma\) and \(\sigma^{*}\) are the stress and higher-order stress tensors, respectively, which can be expressed as

$$\sigma_{ij} = \mathop \int \limits_{\varOmega } \left\{ {\varrho_{1} \left( {\left| {{\mathcal{X}}^{\prime} - {\mathcal{X}}} \right|} \right)C_{ijkl} \varepsilon_{kl} \left( {{\mathcal{X}}^{\prime} } \right)} \right\}{\text{d}}\varOmega$$
(4a)
$$\sigma_{ij}^{*} = l^{2} \mathop \int \limits_{\varOmega } \left\{ {\varrho_{2} \left( {\left| {{\mathcal{X}}^{\prime} - {\mathcal{X}}} \right|} \right)C_{ijkl} \varepsilon_{kl.x} \left( {{\mathcal{X}}^{\prime} } \right)} \right\}{\text{d}}\varOmega$$
(4b)

in which C denotes the elastic matrix, ϱ1 and ϱ2 in order represent the principal attenuation kernel function in the presence of the nonlocality and the additional kernel function related to the nonlocality effect of the first-order strain gradient field, \({\mathcal{X}}\) and \({\mathcal{X}}^{'}\) are, respectively, a point and any point else in the body, and l is the internal strain gradient length scale parameter. In accordance with the method of Eringen and assuming ϱ1 = ϱ2 = ϱ, the constitutive equation relevant to the total nonlocal strain gradient stress tensor of a beam-type structure is constructed as

$$\varLambda_{xx} - \mu^{2} \varLambda_{xx,xx} = C_{xxxx} \varepsilon_{xx} - l^{2} C_{xxxx} \varepsilon_{xx,xx}$$
(5a)
$$\varLambda_{xz} - \mu^{2} \varLambda_{xz,xx} = C_{xzxz} \varepsilon_{xz} - l^{2} C_{xzxz} \varepsilon_{xz,xx}$$
(5b)

where µ is the nonlocal parameter. Thereafter, the nonlocal strain gradient constitutive relations for a hyperbolic shear deformable micro/nanobeam made of nanoporous biomaterial can be expressed as

$$\begin{aligned} \varLambda_{xx} - \mu^{2} \varLambda_{xx,xx} & = \left[ {E/\left( {1 - \nu^{2} } \right)} \right]\left\{ {u_{,x} + \left( {1/2} \right)\left( {w_{,x} } \right)^{2} - zw_{,xx} + \left[ {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right]\psi_{,x} } \right. \\ & \quad \left. { - \,l^{2} \left( {u_{,xxx} + w_{,xxx} w_{,x} + w_{,xx}^{2} - zw_{,xxxx} + \left[ {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right]\psi_{,xxx} } \right)} \right\} \\ \varLambda_{xz} - \mu^{2} \varLambda_{xz,xx} & = \left[ {E/2\left( {1 + \nu } \right)} \right]\left\{ {\left[ {\cosh \left( {1/2} \right) - \cosh \left( {z/h} \right)} \right]\psi - l^{2} \left[ {\cosh \left( {1/2} \right) - \cosh \left( {z/h} \right)} \right]\psi_{,xx} } \right\} \\ \end{aligned}$$
(6)

Therefore, within the framework of the nonlocal strain gradient hyperbolic shear deformable beam model, the total strain energy of a biological lipid protein micro/nanotubule can be written as

$$\begin{aligned} \varPi_{s} & = \frac{1}{2}\mathop \int \limits_{0}^{L} \mathop \int \limits_{S} \left\{ {\sigma_{ij} \varepsilon_{ij} + \sigma_{ij}^{*} \nabla \varepsilon_{ij} } \right\}{\text{d}}S{\text{d}}x \\ & = \frac{1}{2}\mathop \int \limits_{0}^{L} \left\{ {N_{xx} \varepsilon_{xx}^{0} + M_{xx} \kappa_{xx}^{(0)} + R_{xx} \kappa_{xx}^{(2)} + Q_{x} \psi } \right\}{\text{d}}x \\ \end{aligned}$$
(7)

in which the stress resultants can be introduced as below

$$\begin{aligned} N_{xx} - \mu^{2} N_{xx,xx} & = A_{11}^{*} \left[ {u_{,x} + \left( {1/2} \right)w_{,x}^{2} - l^{2} \left( {u_{,xxx} + w_{,x} w_{,xxx} + w_{,xx}^{2} } \right)} \right] \\ M_{xx} - \mu^{2} M_{xx,xx} & = D_{11}^{*} \left( { - w_{,xx} + l^{2} w_{,xxxx} } \right) + F_{11}^{*} \left( {\psi_{,x} - l^{2} \psi_{,xxx} } \right) \\ R_{xx} - \mu^{2} R_{xx,xx} & = F_{11}^{*} \left( { - w_{,xx} + l^{2} w_{,xxxx} } \right) + H_{11}^{*} \left( {\psi_{,x} - l^{2} \psi_{,xxx} } \right) \\ Q_{x} - \mu^{2} Q_{x,xx} & = A_{44}^{*} \left( {\psi - l^{2} \psi_{,xx} } \right) \\ \end{aligned}$$
(8)

where

$$\begin{aligned} \left\{ {N_{xx} ,M_{xx} ,R_{xx} } \right\} & = 2\pi R\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \varLambda_{xx} \left\{ {1,z,z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right\}{\text{d}}z \\ Q_{x} & = 2\pi R\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \varLambda_{xz} \left\{ {\cosh \left( {1/2} \right) - \cosh \left( {z/h} \right)} \right\}{\text{d}}z \\ \end{aligned}$$
(9)

and

$$\begin{aligned} \left\{ {A_{11}^{*} ,D_{11}^{*} } \right\} & = \left[ {2\pi RE/\left( {1 - \nu^{2} } \right)} \right]\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left\{ {1,z^{2} } \right\}{\text{d}}z \\ \left\{ {F_{11}^{*} ,H_{11}^{*} } \right\} & = \left[ {2\pi RE/\left( {1 - \nu^{2} } \right)} \right]\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left\{ {z^{2} \cosh \left( {1/2} \right) - zh\sinh \left( {z/h} \right),\left( {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right)^{2} } \right\}{\text{d}}z \\ \left\{ {A_{44}^{*} } \right\} & = \left[ {2\pi RE/2\left( {1 + \nu } \right)} \right]\mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left\{ {\cosh \left( {1/2} \right) - \cosh \left( {z/h} \right)} \right\}{\text{d}}z \\ \end{aligned}$$
(10)

Additionally, the kinetic energy for a biological lipid protein micro/nanotubule on the basis of the nonlocal strain gradient hyperbolic shear deformable beam model can be expressed as

$$\varPi_{T} = \frac{1}{2}\mathop \int \limits_{0}^{L} \mathop \int \limits_{S} \rho \left\{ {\left( {u_{x,t} } \right)^{2} + \left( {u_{z,t} } \right)^{2} } \right\}{\text{d}}S{\text{d}}x = \frac{1}{2}\mathop \int \limits_{0}^{L} \left\{ {I_{0} \left( {u_{,t} } \right)^{2} + I_{2} \left( {w_{,xt} } \right)^{2} + I_{3} w_{,xt} \psi_{,t} + I_{4} \left( {\psi_{,t} } \right)^{2} + I_{0} \left( {w_{,t} } \right)^{2} } \right\}{\text{d}}x$$

where

$$\left\{ {I_{0} , I_{2} , I_{3} , I_{4} } \right\} = 2\pi R\rho \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \left\{ {1,z^{2} ,z^{2} \cosh \left( {1/2} \right) - zh\sinh \left( {z/h} \right), \left( {z\cosh \left( {1/2} \right) - h\sinh \left( {z/h} \right)} \right)^{2} } \right\}{\text{d}}z$$
(11)

The external work performed by the elastic biomedium can be written as

$$\varPi_{P} = \mathop \int \limits_{0}^{L} \left( {k_{1} w^{2} + k_{2} w_{,x}^{2} } \right){\text{d}}x$$
(12)

in which \(k_{1}\) and \(k_{2}\) represent the normal and shear stiffness of the elastic biomedium, respectively.

Moreover, the performed work associated with the external transverse force q can be defined as below

$$\varPi_{w} = \mathop \int \limits_{0}^{L} q(x,t)w{\text{d}}x$$
(13)

Afterward, based upon the Hamilton’s principle, the governing differential equations of motion in terms of stress resultants are derived as

$$N_{xx,x} = I_{0} u_{,tt}$$
(14a)
$$M_{xx,xx} + \left( {N_{xx} w_{,x} } \right)_{,x} - k_{1} w + k_{2} w_{,xx} + q = I_{0} w_{,tt} - I_{2} w_{,xxtt} - I_{3} \psi_{,xtt}$$
(14b)
$$R_{xx,x} - Q_{x} = I_{3} w_{,xtt} + I_{4} \psi_{,tt}$$
(14c)

Consequently, through inserting Eq. (8) in Eq. (14), the size-dependent equations of motion can be rewritten as

$$A_{11}^{*} \left[ {u_{,xx} + w_{,x} w_{,xx} - l^{2} \left( {u_{,xxxx} + 3w_{,xx} w_{,xxx} + w_{,x} w_{,xxxx} } \right)} \right] = I_{0} \left( {u_{,tt} - \mu^{2} u_{,xxtt} } \right)$$
(15a)
$$\begin{aligned} D_{11}^{*} \left( { - w_{,xxxx} + l^{2} w_{,xxxxxx} } \right) & + \,F_{11}^{*} \left( {\psi_{,xxx} - l^{2} \psi_{,xxxxx} } \right) + A_{11}^{*} \varGamma_{1} - l^{2} A_{11}^{*} \varGamma_{2} - \mu^{2} A_{11}^{*} \varGamma_{3} + \mu^{2} l^{2} A_{11}^{*} \varGamma_{4} \\ & = \,I_{0} \left( {w_{,tt} - \mu^{2} w_{,xxtt} } \right) - I_{2} \left( {w_{,xxtt} - \mu^{2} w_{,xxxxtt} } \right) - I_{3} \left( {\psi_{,xtt} - \mu^{2} \psi_{,xxxtt} } \right) - q \\ & + \,\mu^{2} q_{,xx} + k_{1} \left( {w - \mu^{2} w_{,xx} } \right) - k_{2} \left( {w_{,xx} - \mu^{2} w_{,xxxx} } \right) \\ \end{aligned}$$
(15b)
$$\left( {F_{11}^{*} \left( { - w_{,xxx} + l^{2} w_{,xxxxx} } \right) + H_{11}^{*} \left( {\psi_{,xx} - l^{2} \psi_{,xxxx} } \right) - A_{44}^{*} \left( {\psi - l^{2} \psi_{,xx} } \right) = I_{3} \left( {w_{,xtt} - \mu^{2} w_{,xxxtt} } \right) + I_{4} \left( {\psi_{,tt} - \mu^{2} \psi_{,xxtt} } \right) } \right)$$
(15c)

where

$$\begin{aligned} \varGamma_{1} & = u_{,xx} w_{,x} + u_{,x} w_{,xx} + \left( {3/2} \right)w_{,xx} w_{,x}^{2} \\ \varGamma_{2} & = u_{,xxxx} w_{,x} + u_{,xxx} w_{,xx} + 4w_{,x} w_{,xx} w_{,xxx} + \left( {w_{,xx} + w_{,xxxx} } \right)w_{,x}^{2} \\ \varGamma_{3} & = u_{,xxxx} w_{,x} + 3u_{,xxx} w_{,xx} + 3u_{,xx} w_{,xxx} + u_{,x} w_{,xxxx} + 3w_{,xx}^{3} + 9w_{,x} w_{,xx} w_{,xxx} + \left( {3/2} \right)w_{,x}^{2} w_{,xxxx} \\ \varGamma_{4} & = u_{,xxxxxx} w_{,x} + 3u_{,xxxxx} w_{,xx} + 3u_{,xxxx} w_{,xxx} + u_{,xxx} w_{,xxxx} + 8w_{,xx} w_{,xxx}^{2} + 10w_{,xxxx} w_{,xx}^{2} \\ & \quad + \,4w_{,xx} w_{,xxx}^{2} + 14w_{,x} w_{,xxx} w_{,xxxx} + 8w_{,x} w_{,xx} w_{,xxxxx} + 6w_{,x} w_{,xx} w_{,xxx} \\ & \quad + \,\left( {w_{,xxxx} + w_{,xxxxx} } \right)w_{,x}^{2} + 2w_{,xx}^{3} \\ \end{aligned}$$
(16)

In order to perform the numerical solving process in a more general form, the following dimensionless parameters are taken into consideration

$$X = \frac{x}{L} ,\;U = \frac{u}{h},\; W = \frac{w}{h},\; \varPsi = \psi ,\;\eta_{1} = \frac{l}{L} ,\;\eta_{2} = \frac{\mu }{L} ,\; \beta = \frac{h}{L}$$
$$T = \frac{t}{L}\sqrt {\frac{{A_{11}^{*} }}{{I_{0} }}} ,\;\left\{ {a_{11}^{*} ,a_{44}^{*} ,d_{11}^{*} ,f_{11}^{*} ,h_{11}^{*} } \right\} = \left\{ {\frac{{A_{11}^{*} }}{{A_{11}^{*} }},\frac{{A_{44}^{*} }}{{A_{11}^{*} }},\frac{{D_{11}^{*} }}{{A_{11}^{*} h^{2} }},\frac{{F_{11}^{*} }}{{A_{11}^{*} h^{2} }},\frac{{H_{11}^{*} }}{{A_{11}^{*} h^{2} }}} \right\}$$
(17)
$$\left\{ {I_{0}^{*} , I_{2}^{*} , I_{3}^{*} , I_{4}^{*} } \right\} = \left\{ {\frac{{I_{0} }}{{I_{0} }},\frac{{I_{2} }}{{I_{0} h^{2} }},\frac{{I_{3} }}{{I_{0} h^{2} }},\frac{{I_{4} }}{{I_{0} h^{2} }}} \right\} ,\; {\mathcal{Q}} = \frac{{qL^{2} }}{{A_{11}^{*} h}}, \; \left\{ {K_{1} ,K_{2} } \right\} = \left\{ {\frac{{L^{4} k_{1} }}{{A_{11}^{*} h^{2} }} , \frac{{L^{2} k_{2} }}{{A_{11}^{*} h^{2} }}} \right\}$$

As a result, the dimensionless form of the size-dependent governing differential equations of motion can be constructed as

$$a_{11}^{*} \left[ {U_{,XX} + \beta W_{,X} W_{,XX} - \eta_{1}^{2} \left( {U_{,XXXX} + 3\beta W_{,XX} W_{,XXX} + \beta W_{,X} W_{,XXXX} } \right)} \right] = I_{0}^{*} \left( {U_{,TT} - \eta_{2}^{2} U_{,XXTT} } \right)$$
(18a)
$$\begin{aligned} d_{11}^{*} \left( {W_{,XXXX} - \eta_{1}^{2} W_{,XXXXXX} } \right) & - \,f_{11}^{*} \left( {\varPsi_{,XXX} - \eta_{1}^{2} \varPsi_{,XXXXX} } \right) + K_{1} \left( {W - \eta_{2}^{2} W_{,XX} } \right) \\ & - \,K_{2} \left( {W_{,XX} - \eta_{2}^{2} W_{,XXXX} } \right) - a_{11}^{*} \tilde{\varGamma }_{1} + \eta_{1}^{2} a_{11}^{*} \tilde{\varGamma }_{2} + \eta_{2}^{2} a_{11}^{*} \tilde{\varGamma }_{3} - \eta_{1}^{2} \eta_{2}^{2} a_{11}^{*} \tilde{\varGamma }_{4} \\ & = \,{\mathcal{Q}} - \eta_{2}^{2} {\mathcal{Q}}_{,XX} - I_{0}^{*} \left( {W_{,TT} - \eta_{2}^{2} W_{,XXTT} } \right) + I_{2}^{*} \left( {W_{,XXTT} - \eta_{2}^{2} W_{,XXXXTT} } \right) \\ & + \,I_{3}^{*} \left( {\varPsi_{,XTT} - \eta_{2}^{2} \varPsi_{,XXXTT} } \right) \\ \end{aligned}$$
(18b)
$$\begin{aligned} f_{11}^{*} \left( {W_{,XXX} - \eta_{1}^{2} W_{,XXXXX} } \right) & - \,h_{11}^{*} \left( {\varPsi_{,XX} - \eta_{1}^{2} \varPsi_{,XXXX} } \right) + a_{44}^{*} \left( {\varPsi - \eta_{1}^{2} \varPsi_{,XX} } \right) \\ & = \, - \,I_{3}^{*} \left( {W_{,XTT} - \eta_{2}^{2} W_{,XXXTT} } \right) - I_{4}^{*} \left( {\varPsi_{,TT} - \eta_{2}^{2} \varPsi_{,XXTT} } \right) \\ \end{aligned}$$
(18c)

in which

$$\begin{aligned} \tilde{\varGamma }_{1} & = \beta U_{,XX} W_{,X} + \beta U_{,X} W_{,XX} + \left( {3/2} \right)\beta^{2} W_{,XX} W_{,X}^{2} \\ \tilde{\varGamma }_{2} & = \beta U_{,XXXX} W_{,X} + \beta U_{,XXX} W_{,XX} + 4\beta^{2} W_{,X} W_{,XX} W_{,XXX} + \beta^{2} \left( {W_{,XX} + W_{,XXXX} } \right)W_{,X}^{2} \\ \tilde{\varGamma }_{3} & = \beta \left[ {U_{,XXXX} W_{,X} + 3U_{,XXX} W_{,XX} + 3U_{,XX} W_{,XXX} + U_{,X} W_{,XXXX} } \right] \\ & \quad + \,\beta^{2} \left[ {3W_{,XX}^{3} + 9W_{,X} W_{,XX} W_{,XXX} + \left( {3/2} \right)W_{,X}^{2} W_{,XXXX} } \right] \\ \tilde{\varGamma }_{4} & = \beta \left[ {U_{,XXXXXX} W_{,X} + 3U_{,XXXXX} W_{,XX} + 3U_{,XXXX} W_{,XXX} + U_{,XXX} W_{,XXXX} } \right] \\ & \quad + \,\beta^{2} \left[ {10W_{,XXXX} W_{,XX}^{2} + 12W_{,XX} W_{,XXX}^{2} + 14W_{,X} W_{,XXX} W_{,XXXX} + 8W_{,X} W_{,XX} W_{,XXXXX} } \right. \\ & \quad \left. { + \,6W_{,X} W_{,XX} W_{,XXX} + \left( {W_{,XXXX} + W_{,XXXXX} } \right)W_{,X}^{2} + 2W_{,XX}^{3} } \right] \\ \end{aligned}$$
(19)

3 Numerical solution methodology

The nonlocal strain gradient governing equations of motion are solved numerically with the aid of a solution methodology based upon the GDQ method together with the Galerkin technique [76,77,78]. Consequently, in order to discretize \(X\) domain, the shifted Chebyshev–Gauss–Lobatto grid point is put to use as below

$$X_{i} = \left( {1/2} \right)\left[ {1 - \cos \left( {\pi \left( {i - 1} \right)/\left( {n - 1} \right)} \right)} \right],\quad i = 1,2,3, \ldots , n$$
(20)

Thereby, the discretized governing equations of motion can be written in terms of mass matrix, damping matrix and stiffness matrix as below

$${\mathcal{M}}{\ddot{\mathbf{p}}} + {\mathcal{C}}{\dot{\mathbf{p}}} + {\mathcal{K}}_{\varvec{L}} {\mathbf{p}} + {\mathcal{K}}_{\varvec{N}} = {\mathcal{Q}}\cos \left( {\varOmega T} \right)$$
(21)

where

$$\begin{aligned} {\mathbf{p}} & = \left[ {\begin{array}{*{20}c} {{\mathbf{U}}^{{\mathbf{T}}} } \\ {{\mathbf{W}}^{\text{T}} } \\ {{\varvec{\Psi}}^{\text{T}} } \\ \end{array} } \right]_{3 \times n} \\ {\mathcal{M}} & = \left[ {\begin{array}{*{20}c} { - \,I_{0}^{*} \left( {{\mathcal{D}}_{X}^{(0)} - \eta_{2}^{2} {\mathcal{D}}_{X}^{(2)} } \right)} & 0 & 0 \\ 0 & {I_{0}^{*} \left( {{\mathcal{D}}_{X}^{\left( 0 \right)} - \eta_{2}^{2} {\mathcal{D}}_{X}^{\left( 2 \right)} } \right) - I_{2}^{*} \left( {{\mathcal{D}}_{X}^{(2)} - \eta_{2}^{2} {\mathcal{D}}_{X}^{(4)} } \right)} & { - \,I_{3}^{*} \left( {{\mathcal{D}}_{X}^{(1)} - \eta_{2}^{2} {\mathcal{D}}_{X}^{(3)} } \right)} \\ 0 & {I_{3}^{*} \left( {{\mathcal{D}}_{X}^{(1)} - \eta_{2}^{2} {\mathcal{D}}_{X}^{(3)} } \right)} & {I_{4}^{*} \left( {{\mathcal{D}}_{X}^{(0)} - \eta_{2}^{2} {\mathcal{D}}_{X}^{(2)} } \right)} \\ \end{array} } \right] \\ {\mathcal{K}}_{\varvec{L}} & = \left[ {\begin{array}{*{20}c} {a_{11}^{*} \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} - \eta_{1}^{2} {\mathcal{D}}_{X}^{\left( 4 \right)} } \right)} & 0 & 0 \\ 0 & {d_{11}^{*} \left( {{\mathcal{D}}_{X}^{\left( 4 \right)} - \eta_{1}^{2} {\mathcal{D}}_{X}^{\left( 6 \right)} } \right) + K_{1} \left( {1 - \eta_{2}^{2} {\mathcal{D}}_{X}^{\left( 2 \right)} } \right) - K_{2} \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} - \eta_{2}^{2} {\mathcal{D}}_{X}^{\left( 4 \right)} } \right)} & { - \,f_{11}^{*} \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} - \eta_{1}^{2} {\mathcal{D}}_{X}^{\left( 5 \right)} } \right)} \\ 0 & {f_{11}^{*} \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} - \eta_{1}^{2} {\mathcal{D}}_{X}^{\left( 5 \right)} } \right)} & {a_{44}^{*} \left( {{\mathcal{D}}_{X}^{\left( 0 \right)} - \eta_{1}^{2} {\mathcal{D}}_{X}^{\left( 2 \right)} } \right) - h_{11}^{*} \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} - \eta_{1}^{2} {\mathcal{D}}_{X}^{\left( 4 \right)} } \right)} \\ \end{array} } \right] \\ {\mathcal{K}}_{\varvec{N}} & = \left[ {\begin{array}{*{20}c} {k_{N}^{u} } \\ {k_{N}^{w} } \\ 0 \\ \end{array} } \right]_{3 \times n} , \quad {\mathcal{Q}}^{\text{T}} = \left[ {\begin{array}{*{20}c} 0 &\quad q &\quad 0 \\ 0 & \quad q & \quad 0 \\ 0 & \quad q & \quad 0 \\ \vdots & \quad \vdots & \quad \vdots \\ 0 &\quad q & \quad 0 \\ \end{array} } \right]_{n \times 3} \\ {\mathcal{C}} & = \left( {2c/\omega_{L} } \right){\mathcal{K}}_{\varvec{L}} \varvec{ } \\ \end{aligned}$$
(22)

in which

$${\mathbf{U}} = \left[ {\begin{array}{*{20}c} {U_{1} } \\ {U_{2} } \\ {U_{3} } \\ \vdots \\ {U_{n} } \\ \end{array} } \right], \;{\mathbf{W}} = \left[ {\begin{array}{*{20}c} {W_{1} } \\ {W_{2} } \\ {W_{3} } \\ \vdots \\ {W_{n} } \\ \end{array} } \right],\;{\varvec{\Psi}} = \left[ {\begin{array}{*{20}c} {\varPsi_{1} } \\ {\varPsi_{2} } \\ {\varPsi_{3} } \\ \vdots \\ {\varPsi_{n} } \\ \end{array} } \right]$$
(23)
$$\begin{aligned} k_{N}^{u} & = a_{11}^{*} \beta \left( {{\mathcal{D}}_{X}^{(1)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) - 3a_{11}^{*} \eta_{1}^{2} \beta \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{W}}} \right) - a_{11}^{*} \eta_{1}^{2} \beta \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{W}}} \right) \\ k_{N}^{w} & = - \,a_{11}^{*} \beta \left[ {\left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) + \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right)} \right] - \left( {3/2} \right)a_{11}^{*} \beta^{2} \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \\ & \quad \circ \,\left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) + a_{11}^{*} \eta_{1}^{2} \beta \left[ {\left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) + 3\left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right)} \right] \\ & \quad + \,a_{11}^{*} \eta_{1}^{2} \beta^{2} \left[ {4\left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{W}}} \right) + \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right)} \right. \\ & \quad \left. { + \,\left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right)} \right] \\ & \quad + \,a_{11}^{*} \eta_{2}^{2} \beta \left[ {\left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) + 3\left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) + 3\left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{W}}} \right)} \right. \\ & \quad \left. { + \,\left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{W}}} \right)} \right] \\ & \quad + \,a_{11}^{*} \eta_{2}^{2} \beta^{2} \left[ {3\left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) + 9\left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{W}}} \right)} \right. \\ & \quad \left. { + \,\left( {3/2} \right)\left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{W}}} \right)} \right] \\ & \quad - \,a_{11}^{*} \eta_{1}^{2} \eta_{2}^{2} \beta \left[ {\left( {{\mathcal{D}}_{X}^{\left( 6 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) + 3\left( {{\mathcal{D}}_{X}^{\left( 5 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) + 3\left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{W}}} \right) + \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{U}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{W}}} \right)} \right] \\ & \quad - \,a_{11}^{*} \eta_{1}^{2} \eta_{2}^{2} \beta^{2} \left[ {10\left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) + 12\left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{W}}} \right)} \right. \\ & \quad \circ \,\left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{W}}} \right) + 14\left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{W}}} \right) + 8\left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \\ & \quad \circ \,\left( {{\mathcal{D}}_{X}^{\left( 5 \right)} {\mathbf{W}}} \right) + 6\left( {{\mathcal{D}}_{X}^{\left( 1 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 3 \right)} {\mathbf{W}}} \right) + \left( {{\mathcal{D}}_{X}^{\left( 4 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \\ & \quad \left. { + \,\left( {{\mathcal{D}}_{X}^{\left( 5 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) + 2\left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right) \circ \left( {{\mathcal{D}}_{X}^{\left( 2 \right)} {\mathbf{W}}} \right)} \right] \\ \end{aligned}$$

where \(c\) is the damping parameter, ∘ denotes the Hadamard product, and the derivative operators corresponding to each order can be introduced as

$$\left[ {{\mathcal{D}}_{X}^{\left( i \right)} } \right]_{kj} = \left\{ {\begin{array}{*{20}l} {\left[ {{\text{Identity}}\;{\text{matrix}}} \right]_{n \times n} } \hfill & {\quad i = 0} \hfill \\ {\frac{{\mathop \prod \nolimits_{j = 1,j \ne k}^{n} \left( {X_{k} - X_{j} } \right)}}{{\left( {X_{k} - X_{j} } \right)\mathop \prod \nolimits_{k = 1,k \ne j}^{n} \left( {X_{j} - X_{k} } \right)}}} \hfill & {\quad i = 1,\;j,k = 1,2,3, \ldots ,n} \hfill \\ {i\left( {\left[ {{\mathcal{D}}_{X}^{\left( 1 \right)} } \right]_{kj} \left[ {{\mathcal{D}}_{X}^{{\left( {i - 1} \right)}} } \right]_{kk} - \frac{{\left[ {{\mathcal{D}}_{X}^{{\left( {i - 1} \right)}} } \right]_{kj} }}{{\left( {X_{k} - X_{j} } \right)}}} \right)} \hfill & {\quad i = 2,3,4, \ldots ,n - 1 ,\;j,k = 1,2,3, \ldots ,n} \hfill \\ { - \sum\nolimits_{k = 1,k = j}^{n} {\left[ {{\mathcal{D}}_{X}^{\left( i \right)} } \right]_{kj} } } \hfill & { \quad i = 1,2,3, \ldots ,n - 1,\;j,k = 1,2,3, \ldots ,n} \hfill \\ \end{array} } \right.$$
(24)

In order to continue the solution methodology, the variable matrix \({\mathbf{p}}\) is expressed separately as below

$${\mathbf{p}}\left( {X,T} \right) = {\varvec{\upvarphi}}\left( X \right)g\left( T \right)$$
(25)

By inserting Eq. (25) in Eq. (21), one will have

$${\mathcal{M}}{\varvec{\upvarphi}}{\ddot{\mathbf{g}}} + {{\mathcal{C}}}{\varvec{\upvarphi}}{\dot{\mathbf{g}}} + {{\mathcal{K}}}_{\varvec{L}} {\varvec{\upvarphi}}{\mathbf{g}} + {\mathcal{K}}_{\varvec{N}} = {{\mathcal{Q}}}\cos \left( {\varOmega T} \right)$$
(26)

Thereafter, with the aid of the Galerkin technique, the Duffing-type equation of motion relevant to the forced oscillations of the lipid protein micro/nanotubule can be extracted as

$${\hat{\mathcal{M}}}{\ddot{\mathbf{g}}} + {\hat{\mathcal{C}}}{\dot{\mathbf{g}}} + {\hat{\mathcal{K}}}_{\varvec{L}} {\mathbf{g}} + {\hat{\mathcal{K}}}_{\varvec{N}} \left( {{\mathbf{g}}^{3} } \right) = {\hat{\mathcal{Q}}}\cos \left( {\varOmega T} \right)$$
(27)

in which

$$\begin{aligned} {\hat{\mathcal{M}}} & = {\varvec{\upvarphi}}_{\text{diagonal}}^{{\mathbf{T}}} {{\mathcal{M}}}{\varvec{\upvarphi}} ,\; {\hat{\mathcal{C}}} = {\varvec{\upvarphi}}_{\text{diagonal}}^{{\mathbf{T}}} {{\mathcal{C}}}{\varvec{\upvarphi}},\; {\hat{\mathcal{K}}}_{\varvec{L}} = {\varvec{\upvarphi}}_{\text{diagonal}}^{{\mathbf{T}}} {{\mathcal{K}}}_{\varvec{L}} {\varvec{\upvarphi}} \\ {\hat{\mathcal{K}}}_{\varvec{N}} & = {\varvec{\upvarphi}}_{\text{diagonal}}^{{\mathbf{T}}} {{\mathcal{K}}}_{\varvec{N}} ,\varvec{ }\;\varvec{ }{\hat{\mathcal{Q}}} = {\varvec{\upvarphi}}_{\text{diagonal}}^{{\mathbf{T}}} {{\mathcal{Q} }} \\ \end{aligned}$$
(28)

It should be noted that the expressions for \({\mathbf{\varphi }}\left( X \right)\) corresponding to different boundary conditions are actually represent the associated linear vibrational mode shapes (eigenvectors) which can be obtained numerically for each type of boundary conditions.

Through definition of \(\tilde{T} = \varOmega T\), Eq. (28) can be rewritten as

$$\varOmega^{2} {\hat{\mathcal{M}}}{\ddot{\mathbf{g}}} + \varOmega {\hat{\mathcal{C}}}{\dot{\mathbf{g}}} + {\hat{\mathcal{K}}}_{\varvec{L}} {\mathbf{g}} + {\hat{\mathcal{K}}}_{\varvec{N}} \left( {{\mathbf{g}}^{3} } \right) = {\hat{\mathcal{Q}}}\cos \left( {\tilde{T}} \right)$$
(29)

Now, in order to discretize the time domain, it is assumed that

$$\tilde{T}_{i} = i/n_{t} \quad i = 1,2,3, \ldots ,n_{t}$$
(30)

where \(n_{t}\) denotes the number of discrete points on the time domain and is an even number.

Consequently, it yields

$$\varOmega^{2} {\hat{\mathcal{M}}}\left( {{\mathcal{D}}_{T}^{\left( 2 \right)} } \right)^{\text{T}} {\mathbf{G}} + \varOmega {\hat{\mathcal{C}}}\left( {{\mathcal{D}}_{T}^{\left( 1 \right)} } \right)^{\text{T}} {\mathbf{G}} + {\hat{\mathcal{K}}}_{\varvec{L}} {\mathbf{G}} + {\hat{\mathcal{K}}}_{\varvec{N}} \left( {{\mathbf{G}}^{3} } \right) = {\hat{\mathcal{Q}}}{\mathbf{S}}$$
(31)

in which \({\mathbf{G}}\) includes the first \(m\) discretized mode shapes (eigenvectors) relevant to the Galerkin technique as

$${\mathbf{G}} = \left[ {\begin{array}{*{20}c} {\left[ {q_{1}^{u} } \right]_{{1 \times n_{t} }} } & {\left[ {q_{2}^{u} } \right]_{{1 \times n_{t} }} } & \cdots & {\left[ {q_{m}^{u} } \right]_{{1 \times n_{t} }} } & {\left[ {q_{1}^{w} } \right]_{{1 \times n_{t} }} } & {\left[ {q_{2}^{w} } \right]_{{1 \times n_{t} }} } & \cdots & {\left[ {q_{m}^{w} } \right]_{{1 \times n_{t} }} } & {\left[ {q_{1}^{\psi } } \right]_{{1 \times n_{t} }} } & {\left[ {q_{2}^{\psi } } \right]_{{1 \times n_{t} }} } & \cdots & {\left[ {q_{m}^{\psi } } \right]_{{1 \times n_{t} }} } \\ \end{array} } \right]$$

and

$${\mathbf{S}} = \left[ {\begin{array}{*{20}c} {\cos \left( {\tilde{T}_{1} } \right)} & {\cos \left( {\tilde{T}_{2} } \right)} & \cdots & {\cos \left( {\tilde{T}_{{n_{t} }} } \right)} \\ \end{array} } \right]$$
(32)

Also, the time derivative operator corresponding to each order can be introduced explicitly in the following matrix forms

$${\mathcal{D}}_{T}^{\left( 1 \right)} = {\mathcal{G}}_{ij}^{\left( 1 \right)} \;where\;\left\{ {\begin{array}{*{20}l} {{\mathcal{G}}_{11}^{(1)} = 0} \hfill \\ {{\mathcal{G}}_{i1}^{(1)} = \left( { - 1} \right)^{i - 1} \cot \left( {\frac{{\pi \left( {i - 1} \right)}}{{n_{t} }}} \right)} \hfill \\ {{\mathcal{G}}_{ij}^{(1)} = \left( { - 1} \right)^{{n_{t} + 1 - j}} \cot \left( {\frac{{\pi \left( {n_{t} + 1 - j} \right)}}{{n_{t} }}} \right)} \hfill \\ {{\mathcal{G}}_{{\left( {i + 1} \right)\left( {j + 1} \right)}}^{(1)} = {\mathcal{G}}_{ij}^{(1)} } \hfill \\ \end{array} } \right.\quad i,j = 2,3, \ldots ,n_{t}$$
(33)
$${\mathcal{D}}_{T}^{\left( 2 \right)} = {\mathcal{G}}_{ij}^{\left( 2 \right)} \;{\text{where}} \;\left\{ {\begin{array}{*{20}l} {{\mathcal{G}}_{11}^{(2)} = - \left( {\frac{1}{6} + \frac{{n_{t}^{2} }}{12}} \right)} \hfill \\ {{\mathcal{G}}_{i1}^{(2)} = \left( { - 1} \right)^{i - 1} /\left[ {2\sin^{2} \left( {\frac{{\pi \left( {i - 1} \right)}}{{n_{t} }}} \right)} \right]} \hfill \\ {{\mathcal{G}}_{ij}^{(2)} = \left( { - 1} \right)^{{n_{t} + 1 - j}} /\left[ {2\sin^{2} \left( {\frac{{\pi \left( {n_{t} + 1 - j} \right)}}{{n_{t} }}} \right)} \right]} \hfill \\ {{\mathcal{G}}_{{\left( {i + 1} \right)\left( {j + 1} \right)}}^{(2)} = {\mathcal{G}}_{ij}^{(2)} } \hfill \\ \end{array} } \right.\quad i,j = 2,3, \ldots ,n_{t}$$
(34)

Now, by vectorization of matrices \({\mathbf{G}}\) and \({\hat{\mathcal{K}}}_{\varvec{N}}\) [71,72,73], and using Kronecker product, the vectorized definition of Eq. (32) can be expressed as

$$\left( {\varOmega^{2} \left( {{\mathcal{D}}_{T}^{\left( 2 \right)} \otimes {{\hat{\mathcal{M}}}}} \right) + \varOmega \left( {{\mathcal{D}}_{T}^{\left( 1 \right)} \otimes {{\hat{\mathcal{C}}}}} \right) + \left( {\varvec{I}_{\varvec{t}} \otimes {{\hat{\mathcal{K}}}}_{\varvec{L}} } \right)} \right)vec\left( {\mathbf{G}} \right) + vec\left( {{{\hat{\mathcal{K}}}}_{\varvec{N}} \left( {{\mathbf{G}}^{3} } \right)} \right) - {\mathbf{S}}\left( {\varvec{I}_{\varvec{t}} \otimes {{\hat{\mathcal{Q}}}}} \right) = 0$$
(35)

where \(\varvec{I}_{\varvec{t}}\) represents the identity matrix (zero order of the time derivative).

Finally, the pseudo-arc-length continuation method [79] is utilized to solve Eq. (35) as a set of nonlinear equations.

4 Numerical results and discussion

In this section, the nonlinear primary resonance of a lipid protein nanotubule with thickness of 1.6 nm, mid-radius of 10.7 nm and length of L = 20 h is considered. Also, the material properties are considered as: E = 0.8 GPa, \(\nu = 0.3,\) and \(\rho = 1042\) kg/m3 [80]. Three different boundary conditions, namely as simply supported–simply supported (SS–SS), simply supported–clamped (SS–C) and clamped–clamped (C–C), are considered for the two ends of the micro/nanotubule.

Figures 2 and 3 illustrate the size-dependent frequency response curves associated with the nonlinear primary resonance corresponding to various nonlocal parameters and strain gradient parameters, respectively. It is revealed that by taking the nonlocality into account, the peak of jump phenomenon for the vibration amplitude increases and it is shifted to higher excitation frequency. In other words, it means that the nonlocal size effect leads to increase the geometrical nonlinearity regarding to the nonlinear primary resonance of nanotubules. However, by changing the end supports from simply supported to clamped one, the significance of this pattern decreases. On the other hand, the strain gradient size dependency leads to reduce the peak of jump phenomenon for the vibration amplitude and it is shifted to lower excitation frequency. In other words, it means that the strain gradient size effect causes to decrease the geometrical nonlinearity regarding to the nonlinear primary resonance of nanotubules. It can be seen again that through changing the boundary conditions from SS–SS to C–C, the influence of the strain gradient size effect decreases.

Fig. 2
figure 2

Size-dependent frequency response of the protein micro/nanotubules corresponding to different nonlocal parameters

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Fig. 3
figure 3

Size-dependent frequency response of the protein micro/nanotubules corresponding to different strain gradient parameters

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In Figs. 4 and 5, the size-dependent amplitude response curves related to the nonlinear primary resonance of lipid nanotubule are depicted corresponding to various values of nonlocal and strain gradient parameters, respectively. It is observed that by increasing the excitation amplitude, the vibrational response amplitude increases up to the first bifurcation point. Thereafter, increase in the vibrational response amplitude continues through reduction in the excitation amplitude up to the second bifurcation point. It is found that the nonlocal size effect causes to decrease the excitation amplitudes associated with both of the bifurcation points, but its effect on the first one is more considerable. However, the strain gradient size dependency has an opposite influence and leads to increase them. Moreover, it is seen that by changing the end supports from the simply supported to the clamped one, the influence of the nonlocality on the excitation amplitude associated with the bifurcation points increases, but the influence of the strain gradient size dependency decreases.

Fig. 4
figure 4

Size-dependent amplitude response of the protein micro/nanotubules corresponding to different nonlocal parameters

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Fig. 5
figure 5

Size-dependent amplitude response of the protein micro/nanotubules corresponding to different strain gradient parameters

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In Table 1, the dimensionless natural frequencies of the nanotubule with different boundary conditions are tabulated corresponding to various values of the small-scale parameters. It is indicated that the nonlocal size effect leads to decrease the natural frequency, but the strain gradient size effect causes to increase it. However, the increment caused by the strain gradient size dependency is more than the reduction caused by the nonlocality.

Table 1 Dimensionless natural frequencies of a nanotubule corresponding to various values of the small-scale parameters and different boundary conditions
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Figure 6 represents the size-dependent amplitude response of the lipid nanotubule under different frequency ratios. It is revealed that by increasing the value of the excitation frequency, the excitation amplitudes associated with the bifurcation points increase. Also, it leads to increase the difference between the excitation amplitudes of the two bifurcation points, and this pattern becomes more significant by changing the boundary conditions from SS–SS to C–C. This observation may be related to this point that by changing the boundary conditions from SS–SS to C–C, the deflection of the excited nanotubule reduces.

Fig. 6
figure 6

Size-dependent amplitude response of the protein micro/nanotubules corresponding to different excitation frequencies

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In Fig. 7, the size-dependent frequency response of the lipid nanotubule subjected to the soft excitations with various amplitudes is shown. It is indicated that through enhancement of the excitation amplitude, the peak of the jump phenomenon associated with the frequency response of nanotubule increases. This pattern is more significant for simply supported end conditions than clamped one, which may be related to higher deflection of a nanotubule with simply supported boundary conditions.

Fig. 7
figure 7

Size-dependent frequency response of the protein micro/nanotubules corresponding to different excitation amplitudes

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Figures 8 and 9 demonstrate, respectively, the frequency response and amplitude response of the lipid nanotubule embedded on a different biomedium. It is obvious that by taking the elastic biomedium into consideration, the peak of the jump phenomenon related to the frequency response decreases, especially for SS–SS boundary conditions, due to this fact that the elastic foundation causes to reduce the deflection of the excited nanotubule. However, it is seen that the excitation amplitudes associated with the bifurcation points increase. Moreover, it is displayed that this pattern is more significant for Pasternak type of biomedium including shear stiffness than the Winkler one.

Fig. 8
figure 8

Influence of the elastic biomedium on the size-dependent frequency response of the protein micro/nanotubules

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Fig. 9
figure 9

Influence of the elastic biomedium on the size-dependent amplitude response of the protein micro/nanotubules

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5 Concluding remarks

The prime objective of the current study was to predict the nonlinear primary resonance of a lipid protein micro/nanotubule in the presence of the nonlocality and strain gradient size dependency. To accomplish this purpose, the nonlocal strain gradient elasticity theory was utilized within the framework of the refined hyperbolic shear deformation beam theory. Through the numerical solving process, the size-dependent frequency response and amplitude response of the lipid micro/nanotubule were obtained corresponding to different small-scale parameters.

It was observed that by taking the nonlocality into account, the peak of jump phenomenon for the vibration amplitude increases and it is shifted to higher excitation frequency. However, the strain gradient size dependency leads to reduce the peak of jump phenomenon for the vibration amplitude and it is shifted to lower excitation frequency.

It was indicated that by changing the boundary conditions from SS–SS to C–C, the influence of the size effects on the frequency response of lipid protein micro/nanotubule reduces. In addition, it was seen that by increasing the excitation amplitude, the vibrational response amplitude increases up to the first bifurcation point. Thereafter, increase in the vibrational response amplitude continues through reduction in the excitation amplitude up to the second bifurcation point.

It was seen that the nonlocal size effect causes to decrease the excitation amplitudes associated with both bifurcation points, but its effect on the first one is more considerable. However, the strain gradient size dependency has an opposite influence and leads to increase them.

It was displayed that by increasing the value of the excitation frequency, the excitation amplitudes associated with the bifurcation points increase. It was also demonstrated that by taking the elastic biomedium into consideration, the peak of the jump phenomenon related to the frequency response decreases, especially for SS–SS boundary conditions, but the excitation amplitudes associated with the bifurcation points increase.