#### Abstract

Vibration dynamics of elastic beams that are used in nanotechnology, such as atomic force microscope modeling and carbon nanotubes, are considered in terms of a fundamental response within a matrix framework. The modeling equations with piezoelectric and surface scale effects are written as a matrix differential equation subject to tip-sample general boundary conditions and to compatibility conditions for the case of multispan beams. We considered a quadratic and a cubic eigenvalue problem related to the inclusion of smart materials and surface effects. Simulations were performed for a two stepped beam with a piezoelectric patch subject to pulse forcing terms. Results with Timoshenko models that include surface effects are presented for micro- and nanoscale. It was observed that the effects are significant just in nanoscale. We also simulate the frequency effects of a double-span beam in which one segment includes rotatory inertia and shear deformation and the other one neglects both phenomena. The proposed analytical methodology can be useful in the design of micro- and nanoresonator structures that involve deformable flexural models for detecting and imaging of physical and biochemical quantities.

#### 1. Introduction

Atomic force microscopy (AFM) is a scanning probe microscopy (SPM) technique to obtain images of surface topography at the atomic scale, in a noninvasive manner, from a wide variety of samples on a scale from angstroms to 100 microns [1]. A typical AFM consists of a microcantilever with a sharp tip, a sample positioning system, a detection system, and a control system. The associated length scales are sufficiently small to call the applicability of classical continuum models into question [2]. In this work, we seek to develop a vibration dynamics framework for beams that include smart materials and subject to surface effects. This framework is also considered in the case of a two-span beam in which the first segment is governed by the Timoshenko model and the second segment is an Euler-Bernoulli beam model [3].

Recently, new generations of active microcantilevers have included piezoelectric materials locally attached at the microbeam with the role of sensors and/or actuators linear and nonlinear [4–9], among others. This has led to the study of multispan beams for AFM. The inclusion of smart materials layers will modify material properties between neighboring segments, producing discontinuities and fulfilling compatibility conditions for the continuity of the displacement and rotation of the beam and for the equilibrium of bending moment and shear at the discontinuity points.

Surface effects often play a significant role in the physical properties of micro- and nanosized materials and structures. Since the atoms within a very thin layer near surfaces experience a different local environment form that is experienced by atoms in bulk, the physical properties and mechanical response of surfaces will be distinct from those of bulk materials. Contrary to macroscopic structures, surface effects can strongly influence the stress and deformation properties of nanodevices. This latter is due to the increasing ratio between surface/interface area and volume. For instance, for a bulk deformation energy ( elastic stiffness) and ( surface energy per area), we shall have with distance . In [10], it has been shown that with a modified Timoshenko beam model the dynamics of nanoscaled tubes with surface effects deviate considerably from results obtained with classic theories.

In some models, considering effective properties () of nanosized beams when compared with standard continuum mechanics (), the relative error scale is found as , where is a geometrical constant depending upon the nanobeam considered, is a surface elastic constant, is a bulk elastic modulus, and is a length defining the size of the structural element [11]. Here we shall consider free vibration of a nanoscale continuum beam models which incorporates surface energy [12–14]. The search for time exponential solutions leads to a quadratic and cubic eigenvalue problems associated with a second-order modal differential system and a singular third-order differential system, respectively.

Simulations have been performed for triple span beams with and without a patch in the first segment and with loads in the first and last segment with tip-sample interaction. These loads included a uniform pulse load and its second derivative associated with the moment due to a piezoelectric patch. It is observed how the inclusion of piezoelectric materials absorbs vibrations when compared with classical multispan beams. Also, the robustness of the impulse response method with varying parameters and its influence in the behavior of the responses are observed.

The size dependence effects in Timoshenko microbeams with surface effects (TMB) have been simulated for the nondimensional natural frequency and compared with those of the classical Timoshenko beam model (TB). It has been observed that for beam length on the order of nanometer to microns, the difference between natural frequencies is apparent, and by increasing the length of the microbeam, the results tend to Timoshenko classical theory; that is, the surface effects are significant only in nanoscale. This same behavior was observed in [14] for a microbeam simply supported.

This work involves a methodology that may be adequately adapted to study recent advances in micro-/nanosized structures that are intensively used to design advanced micro-/nanosensors and molecular transportation systems devices for various engineering and medical applications. These sensors due to their ultrahigh resonant frequencies are important in sensitive sensing, molecular transportation, molecular separation, high-frequency signal processing, and biological imaging. Some deposited processes in the formation of thin-film composite membranes can be similar to types of multispan vibrating beams. However, the complex configuration of materials used in real world devices requires the study of material properties stemming from their atomically thin layered structures. Chemical deposited processes in the manufacturing of thin films on semiconductors can involve low temperatures with silicon rich nitride. Conventional structural analysis methods assume ideal structures (free of irregularities) but material and/or geometrical variations in a structure may result in drastic changes in its dynamical behaviors. Diffusion- and reaction-controlled interfacial polymerization is an important and practical topic that is beyond our scope [15–19].

This paper is organized as follows. In Section 2, we consider the Euler-Bernoulli multispan beam model, their compatibility conditions, and the boundary conditions in a matrix form. In Section 3, the dynamic response of the matrix Euler-Bernoulli multispan beam model subject to tip-sample interactions and external forcing is given in terms of the distributed matrix impulse response. The case of a microcantilever with a piezoelectric layer is discussed in Section 4. In Section 5 are presented the results of a double-span beam, obtained by the expansion of the first segment of the triple piezoelectric beam. In Sections 6 and 7 is considered the eigenanalysis of Timoshenko beam models with surface effects and their comparison with the Timoshenko classical model, for micro- and nanoscale. In Section 8, we consider the multispan Timoshenko beam model. AFM-based nanoscale processing with continuum surrounding media such as that found in biology and nanomachining applications [20–22] suggests observing frequency effects that arise with an academic two span beam model in which one segment includes rotatory inertia and shear deformation and the other one neglects both effects.

#### 2. Flexural Vibrations Using the Euler-Bernoulli Multispan Beam Model

We consider a multispan microcantilever of length composed of segments in which the displacement in the segment is governed by the Euler-Bernoulli modelwhere is the mass per unit length, is the cross section area, is the mass density, is the flexural stiffness, and is the transverse dynamic load at the station, . Here and .

##### 2.1. Matrix Formulation

The stepped Euler-Bernoulli model (1) can be written as a second-order differential equationwith diagonal matrix coefficients

whereHere the displacement and the dynamic load and , at the multispan beam, are given in vector form as

##### 2.2. Boundary and Internal Conditions for a Multispan Cantilever Beam

For a cantilever beam, the boundary conditions can be written in a compact matrix formwhere

At the points , , located between two consecutive segments, the compatibility conditions for the displacement, slope, bending moment, and shear lead to the compatibility matrix relationshipwithbeing for

#### 3. Forced Vibrations

The dynamic response of a forced multispan Euler-Bernoulli model given in (2), subject to homogeneous boundary conditions, could be described in terms of the diagonal matrix impulse response or initial-value Green function satisfying the initial-value problemwhere is the identity matrix and , are as given in (10), by using the i-th diagonal component of .

To work on the frequency domain, we need to introduce transfer function as being the Laplace transform of with respect to time (For inputs we seek outputs where with defined in (17)). Thus

It turns out that acts as an integrating factor in Lagrange’s adjoint method for the forced equation (2). Multiplying (2) by and integrating by parts, we obtain the dynamic response

where and are given initial conditions.

The procedure mentioned above is also related to the Riemann function method for integrating partial differential equations. Dynamic responses have been considered in the field of control of distributed systems and in elastodynamics in connection with vibrations and cracking problems [23–26].

In practice, when computing the convolution integral, which corresponds to the forced response , with null initial values , we actually have , where is a free vibration introduced by the system, whose initial values are a priori unknown, and is a particular response. When can be determined by other means, then those initial values can be supplied as and [27]. Thus, the induced system free response, due to a permanent response , is given by

##### 3.1. Frequency Response

Harmonic and piecewise linear forcing are of interest in frequency analysis. When seeking a response of the same type, the transfer function is introduced. Given the harmonic inputwe have the harmonic output responsewhere

In particular, for a concentrated force at a point , of spatial amplitude , we have the permanent responseWith the initial values , , the induced free response is given bywhereFor a pulse amplitudewhere is the unit step function, the permanent response turns out

As before, by substituting the initial values in (14), the induced free response will now bewith given as in (20).

##### 3.2. Closed Form of the Transfer Function for a Multispan Cantilever Beam

The Green function of a multispan Euler-Bernoulli can be determined by using an appropriate solution basis in each segment. From (12), is a diagonal matrix, with entries that satisfy the differential equationwhereThe functions constitute a solution basis of the equation . Hereis the solution of the initial-value problemBy denotingin each segment we havefor In order to determine , we need to find the initial-value vectorsBy using the compatibility conditionsgiven similarly to (9), and by deriving (30) with respect to .

The above process will be carried out for the case of a two stepped cantilever beam with the intermediate discontinuity point .

The boundary conditions and the initial values of lead to a simplification of in the first and last segments. We have two null initial values at , and we can use the boundary conditions at as initial values, instead of those at . This is possible by changing the solution basis at the last segment in (30) to the one generated by .

Thus, for

By applying the compatibility conditions

we get the algebraic system for determining from (30).

*Remark*. The case of axially loaded stepped beams has been investigated in [28] with nonclassical conditions. There, the Green function was obtained by working with a solution basis that considers hyperbolic and trigonometric functions in the general solution. These functions depend upon simple roots of a complete quadratic polynomial due to the inclusion of a second derivative spatial term in (24). When the axial load is removed, the solution basis reduces to the one considered for particular cases of vibrating beams. We observe that through a limit process the use of , which involves a division by root parameters, can also handle the case of repeated roots, which is of interest in the static case or critical frequencies.

#### 4. A Cantilever Beam with a Piezoelectric Layer in AFM

Active microcantilever beams due to their structural flexibility and sensitivity to atomic and molecular forces have received increased attention in a variety of nanoscale sensing and measuring applications, including atomic force microscopy, thermal scanning microscopy, and biomass sensing.

In [9], a microcantilever model was proposed for studying atomic force microscope. It was formulated as a three stepped beam with a piezoelectric layer patch in the first segment and the other two segments were simple beams with different cross-sectional areas according to Figure 1(a).

**(a)**

**(b)**

The governing equations can be written as a system of three second-order partial differential equations with constant coefficientssubject to cantilever boundary conditions (7) and compatibility conditions (9) at the discontinuity points . The parameters and for each segment of the beam are given by

The parameters , are the mass density of piezoelectric material and mass density of beam material; , , and are the piezoelectric layer width, the width of the second, and third beam segments, respectively; and are the piezoelectric layer thickness and beam thickness. and are the Young modulus of piezoelectric layer material and Young modulus of second and third segment material. , , , are the beam moment of inertia, layer moment of inertia, and layer-beam moment of inertia and are calculated as , , ; . And is the neutral axis of the beam on the composite portion expressed by

The values of these parameters [9] are presented in Table 1.

The distributed cross-sectional moment in the first segment induces a forcing term. Its time amplitude is given bywhere and are the coefficients of the converse piezoelectric effect and the applied voltage, respectively.

##### 4.1. Forced Responses

For homogeneous boundary conditions and null initial data, we have from (13) the forced responsewhere . ThusBy assuming a discrete complete spectrum of natural frequencies and the normal mode property, the Galerkin method can be used for determining approximate forced vibrations. We have the modal seriesand the spatial Green functionwhereHere is the spatial amplitude of exponential solutions of the unforced Euler-Bernoulli model that satisfy the boundary conditions (6), and the compatibility conditions (9) at the discontinuity points and .

The modal approximation

has been used in simulations with and without the inclusion of a piezoelectric layer. Expansions were truncated with a small number of terms, usually between 5 and 10 [30].

In Table 2 are showed the first six natural frequencies obtained in this work for a triple span beam with piezoelectric patch and without it. The first three frequencies are compared with those obtained experimentally in [9]. There is an agreement with the methodology of this work and it can be observed that the elimination of piezoelectric layers diminishes the frequency values. In Figure 2 are shown the corresponding vibration modes; in the first segment the modes of piezoelectric beam have smaller amplitude.

The effects of the inclusion of a piezoelectric patch are appreciated by applying a harmonic rectangular pulse load in the first segment and in the third segment for a three stepped beam with and without piezoelectric layer. It is observed that the spatial amplitude diminishes with the inclusion of a piezoelectric patch when the pulse is positioned in the piezoelectric layer segment; this effect does not appear when the pulse is positioned in the third segment (Figure 3).

In Figure 4 are compared the forced responses due to forcing terms only in the first segment. The first output tracks the forcing input . For the original forcing term proposed in [9] , we can observe that the effect of doublet due to derivation of Heaviside function is to concentrate the output at the point . Also, the amplitudes of forced response are smaller for three stepped beam with piezoelectric layer.

#### 5. Double-Span Microbeam with Material and Geometric Discontinuities

We consider a double-span beam of length , formed for two segments with different rectangular transversal section area, according to Figure 1(b). For simulation purposes, we consider the same parameters utilized in [9] for the three stepped beam with a piezoelectric patch in the first segment, and now for the double-span beam, obtained by removing the second segment of the first three stepped beam, which means extending the piezoelectric layer from to . In our methodology, this amounts to consider the limit case when tends to , so that the flexural stiffness and mass density will be valid on the first segment , since was extended to , and on the new second segment we will have and , described in (37) and (38).

The first six natural frequencies of this beam, with and without piezoelectric patch, were given in the third and fourth rows of Table 2 and were obtained by the modal method. We observed that the reduction of three segments for two segments, maintaining the others material and geometric properties, influences the magnitude of the natural frequencies. When the piezoelectric patch is removed, then the frequencies also decrease. The results were the same as those of the case of three stepped beam when removing the piezoelectric patch.

In Figure 5, by using the methodology described from (24) to (35), is presented a 3D graphic of the Green function of the two stepped piezoelectric beam, for the sixth natural frequency. The amplitude of this Green function has peaks at the same natural frequencies obtained by the modal method (Table 2, third row).

#### 6. Surface Elasticity and Residual Surface Tension in the Timoshenko Beam Model

The Timoshenko beam model has been modified with the inclusion of the residual tension and surface elastic modulus [14]. By using the energy method with the elastic strain energy of surface induced in the potential energy and residual surface tensions, expressed by Laplace-Young equations, it was obtained the modified Timoskenko modelfor the flexural deflection of the beam and the rotation angle of the cross section of the beam. Here denotes a transverse dynamic load and a moment load. The valuesare the effective curvature effect and flexural rigidity, respectively. Here and denote the upper and lower surfaces residual tensions and is a surface elastic modulus. Without these later parameters, the model (48) that includes surface effects becomes the classical Timoshenko model. As usual, Young’s modulus, moment of inertia of the cross section area, shear modulus, mass density area, and shear deformation factor of the beam are denoted as , , , , and , respectively. It is assumed that the beam has length and area , where , are the width and thickness of the beam and . The boundary conditions are those of a cantilever beam or subject to balance of the moment and shear at the free end [31].

##### 6.1. Matrix Formulation

The coupled Timoshenko model (48) can be written as a second-order differential equation with matrix coefficientswherewithFor a microcantilever beam of length , we have the boundary conditions in a matrix formulationor in a more compact form

##### 6.2. Eigenanalysis

The search of exponential solutionsof the unforced Timoshenko modelsubject to general separated homogeneous boundary conditions (54), results in that is the general solution of a second-order matrix differential equation and is given byfor constant vectors and . Here is the matrix solution of the initial-value problemwhere denotes the null matrix and the identity matrix. The matrix coefficients are

By using the initial values of** h**(x) in (57) and the clamped boundary condition , b, it turns out that . Thus we have to determine so thatsatisfies the boundary condition at the free end . By assuming homogeneous boundary conditions, we have the nonlinear eigenmatrix problemFrom this, it turns out the characteristic equation

We thus have the reduced systemwithwhereis the solution of the initial-value problemwithHereFrom (62), the natural frequencies will satisfy

The fundamental matrix response is obtained in a similar way to the classical case [32]

We observe from Figure 6 that for frequencies not above the critical frequency, the fundamental response has lower amplitude when considering surface effects, while for higher frequencies the oscillatory behavior of follows the classical one.

For a microcantilever beam described by the Timoshenko model with surface effects, the size dependence in the natural frequency of Timoshenko classical model and Timoshenko model including surface effects is illustrated in Figure 7. The solutions based on classical Timoshenko beam theory and Timoshenko beam theory including surface effects are denoted by TB and TMB, respectively. The fundamental natural frequencies are normalized to fundamental frequency of cantilever Euler-Bernoulli beam. In this figure are considered the parameters utilized in [14] for the same purposes. The parameters of surface elasticity and residual surface tension can be determined by molecular dynamics simulations or experiments. Residual surface stresses can be either positive or negative, depending on the crystallographic structure [33]. For an anodic alumina (Young’s modulus , Poisson’s ratio and ) are considered two types of crystallographic direction

In Figure 7, the size dependence effects in the nondimensional natural frequency of TMB microbeams in comparison to the classical TB are illustrated. We can observe that for beam length on the order of nanometer to microns, the difference between natural frequencies is apparent and, by increasing the length of the microbeam, the results tend to Timoshenko classical theory; that is, the surface effects are significant only in nanoscale. This same behavior was observed in [14] for a microbeam simply supported.

#### 7. Timoshenko Model with Surface Energy in Thick Nanobeams

In [13], the surface effects are included in the Timoshenko beam model following Gurtin-Murdoch continuum theory [34], in which is considered an elastic surface with zero thickness fully bonded to its bulk material. This elastic surface adds a set of specific constitutive equations relative to distinct material properties of the surface and surface energy effects. Let us denote an isotropic material beam, with rectangular transversal section, length , width , thickness , moment of inertia , and which is the transversal sectional area. The equations for the bulk are the same constitutive equations of classical theory of elasticity. The government equations of transverse and angular displacement, and , respectively, are then given by

In a compact notation we have

where

and is a sum of matrix differential operatorswith .

The material parameters of bulk are , , and , which represent Young’s modulus, Poisson’s ratio, and mass density, respectively. Also, we have beam surface parameters , , , , Lamé constants, surface residual stress, and mass density of surface. is the moment of inertia relative to the surface , and for rectangular transversal section . is the shear correction coefficient.

Rewriting (74), we have

Considering exponential solutions in the form , then

where

In order to guarantee nonzero, we must have . Thus we obtain the characteristic equation

##### 7.1. Singular Eigenvalue Problem

By assuming exponential solutions in (74), it follows the eigenvalue problemwhich can be written as a singular third-order differential matrix equationwithWe observe that the nature of the matrix characterizes the problem (83) as being singular. Following the spectral methodology, we seek exponential solutions , , of (83). It follows thatwhereFor each , non-zero solutions can be obtained whenever is a root of the characteristic polynomialThe characteristic equationwithhas the roots and , where

The case of simple roots can be characterized from the condition . This excludes the possibility of to be a simple root. Moreover, we can determine the eigenvectors satisfying (85). We summarize these results in the following.

Lemma 1. * is a simple root of if and only if and . For simple roots, the eigenvectors satisfying (85) are given bywhere is an arbitrary constant.**For the case of distinct roots , the general solutionand can be also presented in the formwhere , , , and*

The above representation of in terms of hyperbolic and trigonometric functions follows by introducing in (92) the basic change

###### 7.1.1. Repeated Roots Analysis

From the given characterization of simple roots, it is clear that the repeated roots can be only and . The conditions and exclude the possibility of having nonzero third roots. In such a case, will be a quadruple root. More precisely, we have the following.

Lemma 2. * If is a root of , then it is a double root if , or quadruple if .*

If and are roots, with , then they will be double roots.

The characteristic polynomial