Multilayered microresonators fabricated using microelectromechanical systems (MEMS) have potential applications such as mixer-filters [1,2] and detection of mass [3,4], prostate-specific antigens [5], proteins [6], virus [7,8], chemical species [9], gas [10,11], and magnetic fields [12–14]. These microresonators have several advantages such as small size, high sensitivity, high resolution, low power consumption, and low cost in mass production [15,16]. They can be excited in the first bending resonant frequency through different types of sources such as magnetic, electrostatic, thermal, and piezoelectric [17]. Their frequency shift and deflections can also be monitored using capacitive, piezoelectric, piezoresistive or optical sensing techniques [18,19].

The microresonators are extremely sensitive to surface processes due to their large surface area to mass ratios [20]. For chemical and biological sensing, the surface of microresonators can be covered with a sensitive coating to absorb the chemical and biological species [21,22]. Several microresonators also use piezoelectric multilayers due to their advantages such as compact structure, low driving voltage, self-actuation and self-sensing capability, and easy integration with electronic circuits [23].

An important issue in the mechanical design process of a multilayered microresonator with variable cross-section is the correct determination of its fundamental resonant frequency. Another important parameter is the estimation of its deflection under an excitation load, which can be used to study its mechanical behavior. Theoretical models can predict the resonant frequency shift of microresonators caused by variations of their geometrical variables and mechanical properties. Lobontiu et al. developed lumped and distributed-parameter models for predicting the bending or torsion resonant frequencies of microresonators with variable cross section and serially connected [24–28]. They also obtained analytically the deflection of this microresonator type; nevertheless, their models do not consider multilayered microresonators. Dufour and Fadel [29] reported analytical expressions in order to calculate the resonant frequencies of thin parallelepiped microcantilevers formed by a structural layer and a sensitive coating. They calculated the bending resonant frequency of microcantilevers using the deflection expression of its bending resonance mode. Their models are suited to microcantilevers with a rectangular plate and a sensitive coating. Ferguson et al. [30] presented an analytical model for predicting the fundamental bending resonant frequency of a composite free-free microbeam, which is connected to four torsional anchors. In addition, they determined the fundamental torsional frequency of the composite anchors. However, these models are limited to composite free-free microbeams and anchors with uniform cross section. Della et al. [31] employed an analytical model in order to study the free vibration of delaminated multilayered beams with two fixed ends. They analyzed the fundamental frequency shift of clamped-clamped multilayer beams with single and double delaminations in terms of the following parameters: normalized axial stiffness, normalized bending stiffness, and the relative slenderness ratio. This model is limited to delaminated multilayered beams with uniform width. Sampath et al. [32] determined the resonant frequency of a two-layer hybrid microcantilever with uniform cross section, which is integrated by an elastic base and viscoelastic layer. They reported that the molecular absorption process affects the resonant frequency of the microcantilever due to that modifies the mass load and the viscoelastic properties of the sensitive coating. Li et al. [33,34] studied the nonlinear transverse vibrations of a clamped-clamped composite microbeam using a refined integro-partial differential model. This model considers a composite microbeam with stepwise axially varying properties, rectangular cross section, and non-flat equilibrium position (i.e., a buckled beam). Based on this model, they estimated the first natural frequency of a clamped-clamped composite microbeam. For this, they employed a buckling factor that depends of the static compressive forces, stiffness, and geometric properties of the microbeam. Finally, they only reported the natural frequency of a multilayer microbeam with two clamped ends, which has a uniform width and a non-flat equilibrium position. Later, Li et al. [35] adopted a curved cross-section model to refine their previous buckling microbeam model [33,34]. This new model considers the cross-section curvature of the clamped-clamped composite microbeam induced by residual stress, which affects its structural stiffness and spatial response. Using this model, the analytical resonant frequencies of the clamped-clamped composite microbeam is closer to the experimental observations. However, it cannot be used for multilayered microresonators with variable width or serially connected multisegments. Pasini [36] developed a bending response model for multilayered microresonators considering their cross-section shape, symmetry and layers number, as well as their materials properties. However, his model does not consider different microbeam types connected in series. Edqvist et al. [37] established a general theoretical model for studying the quasi-static and dynamic electromechanical responses of piezoelectric multilayered microbeams with one fixed end. Their model was obtained through Euler-Bernoulli theory and only considers layers with uniform cross section. Herrera-May et al. [38,39] used theoretical models to estimate the first bending resonant frequency of microresonators integrated by an arrangement of microbeams with different cross-sections. All these models have important characteristics that can be useful in the design phase of some microresonators. However, several they have limitations to predict the resonant frequency of multilayered microresonators with variable cross-section. In order to overcome all these drawbacks, we present an analytical model for the first bending resonant frequency of multilayered microresonators with variable cross section and one fixed end. This model includes microresonators composed by an arrangement of different multilayered microbeam types. Furthermore, our model can be used to estimate the deflection expression of the proposed microresonators. The analytical model is based on the Rayleigh’s method, Macaulay’s functions, and Euler-Bernoulli beam theory. It does not consider the damping, residual stress and shear deformation of the microresonators.

This paper is organized as follows: in Section 2, we describe the analytical model for the bending resonant frequency of a multilayered microresonator with variable cross section. Next, in Section 3, this model is applied to two multilayered microresonators with mass sensing applications reported in the literature. The results of our analytical model are compared with those obtained from a simple cantilever model, FEMs, and experimental data. Finally, we presented our conclusions and proposed future research in Section 4.

This section describes the analytical modeling to predict the first bending resonant frequency and deflection of multilayered microresonators with variable cross section. Rayleigh’s method can predict the first bending resonant frequency of structures with complex geometrical shapes considering energy conservation in the structures [40]. This method requires the deflection expression, bending stiffness, and mass per unit length of the structures.

The bending deflection response y(x,t) at a given point of a resonant structure with a harmonic motion can be determined as a product between a spatial function y(x) and a time-dependent one [27]. Thus, y(x,t) = y(x)sin(2πft), where t is the time and f is the frequency.

For out-of plane bending in single and double clamped beams with length L and cross-section area A(x), the maximum potential energy (U_m) and kinetic energy (K_m) are determined by [41]: (1) U m = 1 2 ∫ 0 L EI ( x ) ( ∂ 2 y ( x ) ∂ x 2 ) 2 dx (2) K m = ( 2 π f ) 2 2 ∫ 0 L ρ A ( x ) y 2 ( x ) dxwhere E, I, and ρ are the Young’s modulus, inertia moment, and density of the beam, respectively.

Based on Rayleigh’s method, the first bending resonant frequency (f_r) of single and double clamped beams can be obtained assuming energy conservation (U_m = K_m) in the beams. By substituting Equations (1) and (2) into U_m = K_m, the f_r is given by: (3) f r = 1 2 π ( ∫ 0 L EI ( x ) ( ∂ 2 y ( x ) ∂ x 2 ) 2 dx ∫ 0 L ρ A ( x ) y 2 ( x ) dx ) 1 / 2

In order to use the Rayleigh’s method in a multilayered microresonator with variable cross section, we need to know its elastic centroid, bending stiffness and mass per unit length. We propose a multilayered microresonator with variable cross section, as shown in Figure 1. This microresonator is formed by layers of different materials and geometric shapes, which are symmetrically distributed with respect to the xy–plane. The microresonator is divided into four different sections, which contain mth, nth, pth, and qth layers, respectively. We employed the nomenclature jth section in order to describe each one of the four sections of the microresonator. Figure 2 shows the geometrical nomenclature proposed for kth layer placed over any of the jth section. In this case, b_iSj, and t_iSj represent the width and thickness of the ith layer located on the jth section. In addition, the parameter h_iSj indicates the distance from the bottom plane of the first layer to the top plane of the ith layer placed over the jth section. Based on our multilayered microresonator, several microresonator designs used for detection of mass and gas can be obtained. Figure 3(a–d) illustrates the configurations of four different multilayered microresonators that are derived from our general-microresonator design. An analytical model for estimating the first bending resonant frequency of our general microresonator can help designers and researchers in the design phase of particular microresonators. For instance, an analytical model for a general multilayered microresonator can be used to know the effect of the materials and dimensions of the multilayers on the resonant frequency and deflection of particular microresonators.

Figure 4 shows the reaction loads, bending moments, and uniformly distributed loads of the proposed microresonator.

For this case, R₀ and M₀ represent the total reaction load and bending moment at the fixed end of the microresonator. Furthermore, ω_Sj indicates the uniformly distributed weight of all the layers located in the jth section. The analytical model developed in this work is based on the following assumptions:

Multilayered microresonator is integrated by layers made of homogeneous and isotropic materials. The film mechanical properties of the different layers must be considered;

For our microresonator, the elastic centroid (a_Sj) in the jth section of the microresonator can be determined as [42]: (4) a S j = ( ES ) S j ( EA ) S j = ∬ A S j E S j y S j ( x ) dydz ∬ A S j E S j dydz = 1 2 ∑ i = 1 k E iS j b iS j t iS j ( h iS j + h ( i − 1 ) S j ) ∑ i = 1 m E iS j b iS j t iS jwhere t_iSj = h_iSj − h_(i−1)Sj, the subscript S_j is related to the jth section, the value k depends of the layers number located over each one of the four sections (i.e., k can take either values m, n, p or q), A_Sj symbolizes the domain occupied by the jth section, y_Sj (x) is the deflection along of the jth section, E_iSj is the Young’s modulus of the ith layer located in the jth section, h_(i−1)Sj is distance from the bottom plane of the first layer to the top plane of the (i-1)th layer placed over the jth section, (ES)_Sj is the elastic modulus weighted first moment of area and (EA)_Sj the elastic modulus weighted area. For the first section, b_iS1 considers the overall width of the ith layer (i.e., b_iS1 includes the sum of the width of the three layers located on the same distance h_iS1, which have the same material). In addition, the term h_0Sj = 0.

The effective bending stiffness (EI_z)_Sj in the jth section of the multilayered microresonator can be calculated as: (5) ( EI z ) S j = ∑ i k ( E i I zi ) S j = ∬ A S j E S j y S j ( x ) dy = 1 3 ∑ i = 1 k E iS j b iS j [ ( h iS j − a S j ) 3 − ( h ( i − 1 ) S j − a S j ) 3 ]

The maximum potential energy (U_mL) and kinetic energy (K_mL) of the multilayered microresonator can be obtained as: (6) U mL = 1 2 ( EI z ) S 1 ∫ 0 L S 1 ( ∂ 2 y S 1 ( x ) ∂ x 2 ) 2 dx + 1 2 ( EI z ) S 2 ∫ L S 1 L S 12 ( ∂ 2 y S 2 ( x ) ∂ x 2 ) 2 dx + 1 2 ( EI z ) S 3 ∫ L S 12 L S 123 ( ∂ 2 y S 3 ( x ) ∂ x 2 ) 2 dx + 1 2 ( EI z ) S 4 ∫ L S 123 L S 1234 ( ∂ 2 y S 4 ( x ) ∂ x 2 ) 2 dx (7) K mL λ 2 = 1 2 ( ∑ i = 1 m ρ iS 1 b iS 1 h iS 1 ) ∫ 0 L S 1 ( y S 1 ( x ) ) 2 dx + 1 2 ( ∑ i = 1 n ρ iS 2 b iS 2 h iS 2 ) ∫ L S 1 L S 12 ( y S 2 ( x ) ) 2 dx + 1 2 ( ∑ i = 1 p ρ iS 3 b iS 3 h iS 3 ) ∫ L S 12 L S 123 ( y S 3 ( x ) ) 2 dx + 1 2 ( ∑ i = 1 q ρ iS 4 b iS 4 h iS 4 ) ∫ L S 123 L S 1234 ( y S 4 ( x ) ) 2 dxwhere L_S12 = L_S1 + L_S2, L_S123 = L_S1 + L_S2 + L_S3, L_S1234 = L_S1 + L_S2 + L_S3+ L_S4, and λ = 2πf.

Using the Rayleigh’s method, the first bending resonant frequency of the multilayered microresonator is obtained as: (8) f r = 1 2 π ( U mL K mL / λ 2 ) 1 / 2

Equations (6–8) show a dependence of the bending resonant frequency with respect to the deflection y(x), mechanical properties (Young’s modulus and density) and geometrical parameters of the layers from each section of the resonant microestructure. Any variation of these factors will cause a change in its resonant frequency. For the case of mass and gas sensing, the particles deposited on the surface of a microresonator can affect its bending resonant frequency due to changes of its mass and stiffness. This frequency shift is used to detect the amount and type of particles deposited on the microresonator surface. For the case of small particles deposited on a concentrated region of the microresonator surface, the resonant frequency shift can be estimated only considering the kinetic energy variation. In order to determine this energy shift, the mass added by the particles must be included into Equation (7). For instance, if the mass of the particles is considered as a small concentrated mass load (m_add) located to a distance x_add of the microresonator’s fixed end, then the term m_addy²(x_add)/2 must be summed into Equation (7). For more information of this topic we recommended the work developed by Dong et al. [11]. We do not include the study of the resonant frequency shift of the microresonator.

The deflection y_Sj (x) in each one of the four sections of the microresonator are obtained using Euler-Bernoulli beam theory and Macaulay’s method [43]. This method is suitable to describe several load types acting on structures with variable cross section [44]. For this, it employs functions whose nomenclature considers a bracket notation 〈x − c〉^ε, since these functions have the zero value for x < c and (x − c)^ε for x ≥ c [45]. The value of superscript ε depends of the load type acting on structure. For instance, ε = 0 for a uniformly distributed load, ε = −1 for a concentrated load, and ε = −2 for a bending moment.

Thus, the deflection in each one of the four sections of proposed microresonator are determined from: (9a) ( EI z ) S 1 ∂ 2 y S 1 ( x ) ∂ 2 x = M S 1 ( x )     0 < x < L S 1 (9b) ( EI z ) S 2 ∂ 2 y S 2 ( x ) ∂ 2 x = M S 2 ( x )     L S 1 < x < L S 12 (9c) ( EI z ) S 3 ∂ 2 y S 3 ( x ) ∂ 2 x = M S 3 ( x )     L S 12 < x < L S 123 (9d) ( EI z ) S 4 ∂ 2 y S 4 ( x ) ∂ 2 x = M S 4 ( x )     L S 123 < x < L S 1234where M_Sj is the bending moment in the jth section of the multilayered microresonator, which can be determined through Macaulay’s functions.

The deflection y_Sj(x) of our microresonator must satisfy the following boundary conditions: (10) y S 1 ( 0 ) = 0 ∂ y S 1 ( 0 ) ∂ x = 0 y S 1 ( L S 1 ) = y S 2 ( L S 1 ) ∂ y S 1 ( L S 1 ) ∂ x = ∂ y S 2 ( L S 1 ) ∂ x y S 2 ( L S 12 ) = y S 3 ( L S 12 ) ∂ y S 2 ( L S 12 ) ∂ x = ∂ y S 3 ( L S 12 ) ∂ x y S 3 ( L S 123 ) = y S 4 ( L S 123 ) ∂ y S 3 ( L S 123 ) ∂ x = ∂ y S 4 ( L S 123 ) ∂ x

Using Macaulay’s method, we find the total load function F(x) of the microsensor. It can be described as [43]: (11) F ( x ) = − M 0 〈 x − 0 〉 − 2 + R 0 〈 x − 0 〉 − 1 − ω S 1 〈 x − 0 〉 0 + ω S 1 〈 x − L S 1 〉 0 − ω S 2 〈 x − L S 1 〉 0 + ω S 2 〈 x − L S 12 〉 0 − ω S 3 〈 x − L S 12 〉 0 + ω S 3 〈 x − L S 123 〉 0 − ω S 4 〈 x − L S 123 〉 0 + ω S 4 〈 x − L S 1234 〉 0with (12) R 0 = ∑ i = 1 4 ω S j L S j (13) M 0 = ∑ i = 1 4 ω S j L S j ( 1 2 L S j + L S j − 1 + L S j − 2 + L S j − 3 ) (14) ω S j = ∑ i = 1 k ρ iS j g b iS j t iS jwhere ω_Sj represents the weight per unit length in the jth section and g is the gravitational acceleration. For the first section, b_iS1 considers the overall width of the all layers located in h_iS1 (e.g., if there are three layers made with the same material located to a distance h_iS1, then b_iS1 includes the sum of each width of the layers).

The shear load function V(x) is obtained by integrating Equation (11) with respect to x and using the integration rules of the Macaulay’s functions [43]. The function V(x) is given by: (15) V ( x ) = − M 0 〈 x − 0 〉 − 1 + R 0 〈 x − 0 〉 0 − ω S 1 〈 x − 0 〉 1 + ω S 1 〈 x − L S 1 〉 1 − ω S 2 〈 x − L S 1 〉 1 + ω S 2 〈 x − L S 12 〉 1 − ω S 3 〈 x − L S 12 〉 1 + ω S 3 〈 x − L S 123 〉 1 − ω S 4 〈 x − L S 123 〉 1 + ω S 4 〈 x − L S 1234 〉 1 + C 1

Integrating Equation (15) with respect to x, the bending moment function M(x) is determined as: (16) M ( x ) = − M 0 〈 x − 0 〉 0 + R 0 〈 x − 0 〉 1 − 1 2 ω S 1 〈 x − 0 〉 2 + 1 2 ω S 1 〈 x − L S 1 〉 2 − 1 2 ω S 2 〈 x − L S 1 〉 2 + 1 2 ω S 2 〈 x − L S 12 〉 2 − 1 2 ω S 3 〈 x − L S 12 〉 2 + 1 2 ω S 3 〈 x − L S 123 〉 2 − 1 2 ω S 4 〈 x − L S 123 〉 2 + 1 2 ω S 4 〈 x − L S 1234 〉 2 + C 1 x + C 2where C₁ and C₂ are constants calculated through boundary conditions (V(0) = R₀ and M(0) = M₀) of the shear load and bending moment functions. Substituting these conditions into Equations (15) and (16), we obtain C₁ = C₂ = 0.

The bending moment functions for each one of the four sections of the multilayered microresonator are determined from Equation (16):

For 0 < x < L_S1 (17a) M S 1 ( x ) = − M 0 〈 x − 0 〉 0 + R 0 〈 x − 0 〉 1 − 1 2 ω S 1 〈 x − 0 〉 2For L_S1 < x < L_S12 (17b) M S 2 ( x ) = − M 0 〈 x − 0 〉 0 + R 0 〈 x − 0 〉 1 − 1 2 ω S 1 〈 x − 0 〉 2 + 1 2 ω S 1 〈 x − L S 1 〉 2 − 1 2 ω S 2 〈 x − L S 1 〉 2For L_S12 < x < L_S123 (17c) M S 3 ( x ) = − M 0 〈 x − 0 〉 0 + R 0 〈 x − 0 〉 1 − 1 2 ω S 1 〈 x − 0 〉 2 + 1 2 ω S 1 〈 x − L S 1 〉 2 − 1 2 ω S 2 〈 x − L S 1 〉 2 + 1 2 ω S 2 〈 x − L S 12 〉 2 − 1 2 ω S 3 〈 x − L S 12 〉 2For L_S123 < x < L_S1234 (17d) M S 4 ( x ) = − M 0 〈 x − 0 〉 0 + R 0 〈 x − 0 〉 1 − 1 2 ω S 1 〈 x − 0 〉 2 + 1 2 ω S 1 〈 x − L S 1 〉 2 − 1 2 ω S 2 〈 x − L S 1 〉 2 + 1 2 ω S 2 〈 x − L S 12 〉 2 − 1 2 ω S 3 〈 x − L S 12 〉 2 + 1 2 ω S 3 〈 x − L S 123 〉 2 − 1 2 ω S 4 〈 x − L S 123 〉 2

Then, the deflection y_Sj(x) in each section of the multilayered microresonator is determined by substituting Equation (17) into Equation (9) and considering the boundary conditions expressed by Equation (10), as well as the integration rules of the Macaulay’s functions. Thus, the deflections y_Sj(x) can be determined as:

For 0 < x < L_S1 (18a) y S 1 ( x ) = 1 ( E I z ) S 1 [ − 1 2 M 0 〈 x − 0 〉 2 + 1 6 R 0 〈 x − 0 〉 3 − 1 24 ω S 1 〈 x − 0 〉 4 ]For L_S1 < x < L_S12 (18b) y S 2 ( x ) = 1 ( E I z ) S 2 [ − 1 2 M 0 〈 x − 0 〉 2 + 1 6 R 0 〈 x − 0 〉 3 − 1 24 ω S 1 〈 x − 0 〉 4 + 1 24 ω S 1 〈 x − L S 1 〉 4 − 1 24 ω S 2 〈 x − L S 1 〉 4 − 1 2 M 0 L S 1 2 + 1 6 R 0 L S 1 3 − 1 8 ω S 1 L S 1 4 + ( M 0 L S 1 − 1 2 R 0 L S 1 2 + 1 6 ω S 1 L S 1 3 ) x ] + 1 ( EI z ) S 1 [ ( M 0 L S 1 + 1 2 R 0 L S 1 2 − 1 6 ω S 1 L S 1 3 ) x + 1 2 M 0 L S 1 2 − 1 3 R 0 L S 1 3 + 1 8 ω S 1 L S 1 4 ]For L_S12 < x < L_S123 (18c) y S 3 ( x ) = 1 ( E I z ) S 3 [ − 1 2 M 0 〈 x − 0 〉 2 + 1 6 R 0 〈 x − 0 〉 3 − 1 24 ω S 1 〈 x − 0 〉 4 + 1 24 ω S 1 〈 x − L S 1 〉 4     − 1 24 ω S 2 〈 x − L S 1 〉 4 + 1 24 ω S 4 〈 x − L S 12 〉 4 − 1 24 ω S 3 〈 x − L S 12 〉 4 + C 3 x + C 4 ]     C 3 = ( EI z ) S 3 ( EI z ) S 2 [ − M 0 L S 2 + 1 2 R 0 L S 2 ( 2 L S 1 + L S 2 ) − 1 2 ω S 1 L S 1 L S 2 L S 12 − 1 6 ω S 2 L S 2 3 ]         + ( EI z ) S 3 ( EI z ) S 1 [ − M 0 L S 1 + 1 2 R 0 L S 1 − 1 6 ω S 1 L S 1 3 ]         + M 0 L S 12 − 1 2 R 0 L S 12 2 + 1 6 ω S 1 L S 1 ( L S 1 2 + 3 L S 1 L S 2 + 3 L S 2 2 ) + 1 6 ω S 2 L S 2 3 C 4 = ( EI z ) S 3 ( EI z ) S 2 [ 1 2 M 0 L S 2 ( 2 L S 1 + L S 2 ) − 1 3 R 0 L S 2 [ 3 L S 1 L S 12 + L S 2 2 ] + 1 12 ω S 1 L S 1 L S 2 [ 3 L S 1 ( 2 L S 1 + 3 L S 2 ) + 4 L S 2 2 ]     + 1 24 ω S 2 L S 2 3 ( 4 L S 1 + 3 L S 2 ) ] + ( EI z ) S 3 ( EI z ) S 1 [ 1 2 M 0 L S 1 2 − 1 3 R 0 L S 1 3 + 1 8 ω S 1 L S 1 4 ] − 1 2 M 0 L S 12 2 + 1 3 R 0 L S 12 3     − 1 24 ω S 1 L S 1 [ 3 L S 1 ( L S 1 2 + 4 L S 1 L S 2 + 6 L S 2 2 ) + 8 L S 2 3 ] − 1 24 ω S 2 L S 2 3 ( 4 L S 1 + 3 L S 2 ) .For L_S123 < x < L_S1234 (18d) y S 4 ( x ) = 1 ( EI z ) S 4 [ − 1 2 M 0 〈 x − 0 〉 2 + 1 6 R 0 〈 x − 0 〉 3 − 1 24 ω S 1 〈 x − 0 〉 4 + 1 24 ω S 1 〈 x − L S 1 〉 4 − 1 24 ω S 2 〈 x − L S 1 〉 4       + 1 24 ω S 2 〈 x − L S 12 〉 4 − 1 24 ω S 3 〈 x − L S 12 〉 4 + 1 24 ω S 3 〈 x − L S 123 〉 4 − 1 24 ω S 4 〈 x − L S 123 〉 4 + C 5 x + C 6 ] C 5 = ( E I z ) S 4 ( EI z ) S 3 [ − M 0 L S 3 + 1 2 R 0 L S 3 ( 2 L S 12 + L S 3 ) − 1 2 ω S 1 L S 1 L S 3 ( L S 1 + 2 L S 2 + L S 3 ) − 1 2 ω S 2 L S 2 L S 3 ( L S 2 + L S 3 ) ]         + ( EI z ) S 4 ( EI z ) S 2 [ − M 0 L S 2 + 1 2 R 0 L S 2 ( 2 L S 1 + L S 2 ) − 1 2 ω S 1 L S 1 L S 2 L S 12 − 1 6 ω S 2 L S 2 3 ]         + ( EI z ) S 4 ( EI z ) S 1 [ − M 0 L S 1 + 1 2 R 0 L S 1 2 − 1 6 ω S 1 L S 1 3 ] + M 0 L S 123 − 1 2 R 0 L S 123 2         + 1 6 ω S 1 L S 1 [ L S 1 ( L S 1 + 3 L S 2 + 3 L S 3 ) + 3 L S 2 ( L S 2 + 2 L S 3 ) + 3 L S 3 2 ]         + 1 6 ω S 2 L S 2 [ L S 2 ( L S 2 + 3 L S 3 ) + 3 L S 3 2 ] + 1 6 ω S 3 L S 3 2 C 6 = ( EI z ) S 4 ( EI z ) S 3 [ − M 0 L S 3 ( 2 L S 12 + L S 3 ) − 1 3 R 0 L S 3 [ 3 L S 1 ( L S 1 + 2 L S 2 + L S 3 ) + 3 L S 2 ( L S 2 + L S 3 ) + L S 3 2 ]         + 1 12 ω S 1 L S 1 L S 3 [ 3 L S 1 ( 2 L S 1 + 6 L S 2 + 3 L S 3 ) + 12 L S 2 ( L S 2 + L S 3 ) + 4 L S 3 2 ] − 1 24 ω S 3 L S 3 4         + 1 12 ω S 2 L S 2 L S 3 [ 6 L S 1 ( L S 2 + L S 3 ) + 3 L S 2 ( 2 L S 2 + 3 L S 3 ) + 4 L S 3 2 ] ] + ( E I z ) S 4 ( EI z ) S 2 [ 1 2 M 0 L S 2 ( 2 L S 1 + L S 2 )         − 1 3 R 0 L S 2 ( 3 L S 1 L S 12 + L S 2 2 ) + 1 12 ω S 1 L S 1 L S 2 [ 3 L S 1 ( 2 L S 1 + 3 L S 2 ) ] + 1 24 ω S 2 L S 2 3 ( 4 L S 1 + 3 L S 2 ) ]         + ( EI z ) S 4 ( EI z ) S 1 [ 1 2 M 0 L S 1 2 − 1 3 R 0 L S 1 3 + 1 8 ω S 1 L S 1 4 ] − 1 2 M 0 L S 123 2 + 1 3 R 0 L S 123 3 − 1 24 ω S 3 L S 3 3 ( 4 L S 12 + 3 L S 3 )         − 1 24 ω S 1 L S 1 [ 3 L S 1 ( L S 1 2 + 4 L S 1 L S 2 + 4 L S 1 L S 3 + 12 L S 2 L S 3 + 6 L S 2 2 + 6 L S 3 2 ) + 8 L S 2 ( L S 2 2 + 3 L S 2 L S 3 + 3 L S 3 2 ) + 8 L S 3 3 ]         − 1 24 ω S 2 L S 2 [ 4 L S 1 ( L S 2 2 + 3 L S 2 L S 3 + 3 L S 3 2 ) + 3 L S 2 ( L S 2 2 + 4 L S 2 L S 3 + 6 L S 3 2 ) + 8 L S 3 3 ] .

For multilayered microresonators with two or three different sections, only the first two or three deflections equations must be used. Next, y_Sj(x) is used into Equations (6) and (7) to find the maximum potential and kinetic energy of the microresonator. Finally, by substituting these energies into Equation (8), the first bending resonant frequency of the multilayered microresonator can be estimated.

We applied our analytical model to estimate the first bending resonant frequency and deflection of two multilayered microresonators reported in the literature. In addition, we used finite element models (FEMs) to study the bending vibration of these microresonators. Our analytical results agree with the FEMs and experimental results of the two microresonators.

We considered the geometrical configuration of two multilayered microresonators for mass sensing applications developed by Lu et al. [46,47]. Figure 5 shows a SEM image of the first multilayered microresonator of Lu et al. [46], which we called microresonator type-A. In addition, we named their second multilayered microresonator microresonator type-B (see Figure 6).

These microresonators have self-actuation and self-sensing capability. The details of their fabrication process can be found elsewhere [46,47]. Figure 7 shows the geometrical configuration of the layers and load types on the microresonator type-A. It uses small electrodes of piezoelectric lead zirconate titanate (PZT) for microresonator excitation and contains a piezoresistive gauge for detection of its resonant frequency (see Figure 8). It can be divided in four different sections of layers. The first two sections contain nine layers composed by the following materials: silicon (Si), silicon dioxide (SiO₂), titanium (Ti), platinum (Pt), PZT, chromium (Cr), and gold (Au). The order of the nine layers on the first and second section is the following: Si/SiO₂/Ti/Pt/PZT/Cr/Au/Cr/SiO₂. Only the first section has two holes to decrease initial stress due to PZT deformation, which improve the signal to noise of the piezoresistive gauge contained in the first section. The third only contains a Si layer and the fourth section has three layers of Si/Cr/Au.

Table 1 indicates the order of the layers in each one of the four sections for the microresonator type-A. Figure 9 depicts the main dimensions of the microresonator type-A. In addition, Table 2 indicates the geometrical parameters of all layers used in the microresonator type-A. For the first section, b_iS1 include the overall width of the layers located to a same distance h_iS1.

Figure 10 shows the geometrical configuration of the layers and load types on the microresonator type-B, which contains four different sections of layers. Its first section is integrated by nine layers that are fabricated with the same materials used in the first section of the microresonator type A. The second section is composed by two layers (Si/SiO₂) and the third section is only formed by Si (see Figure 11). The fourth section contains three layers that are made of Si/Cr/Au. The order of the layers in each one of the four sections for the microresonator type-B is showed in Table 3. Table 4 indicates the geometrical parameters of all layers employed in the microresonator type-B.

Both microresonator type-A and type-B have four sections with different numbers of layers in each one of them. In addition, their layers have a symmetrical configuration with respect to the xy−plane. Therefore, our analytical model for a general multilayered microresonator can be used to estimate the first bending resonant frequency of both microresonator type-A and type-B. Table 5 shows the mechanical properties of all layers used in the two microresonators, which were obtained from the literature [48–51]. Based on the mechanical properties and values of the geometrical parameters of the two microresonators, we first find the weight by unit length, reaction load, bending moment, and effective stiffness for both microresonators (see Table 6). Using these values, we substitute the deflections Equation (18) into Equations (6) and (7) in order to determine the maximum potential and kinetic energies. Finally, these energies were applied into Equation (8) to estimate the first bending resonant frequency (f_r) of the microresonators. Thus, our analytical results for the microresonator type-A and type-B are 13.18 kHz and 186.51 kHz, respectively. These results agree well with the experimental results of both microresonators obtained by Lu et al. [46,47], which are 12.81 kHz and 186.0 kHz, respectively. Our analytical results present a small relative difference of 2.9% and 0.3% with respect to experimental data. Also, we compared our analytical results with those obtained through two FEMs and a simple cantilever model (SCM).

For a simple cantilever model where the origin of its coordinate system is located at its fixed end and the xz–plane is considered its neutral plane, the first bending resonant frequency (f_rc) can be calculated as [52]: (19) f rc = 0.162 h c L c 2 E c ρ cwhere h_c and L_c are the thickness and length of the cantilever, respectively; E_c and ρ_c are the Young’s modulus and density of the cantilever material.