Size-Dependent Flexural Dynamics of Ribs-Connected Polymeric Micropanels

1Faculty of Engineering,Computing and Science,Swinburne University of Technology(Sarawak Campus),Jalan Simpang Tiga,Kuching 93350,Sarawak,Malaysia.

2Corresponding author.Tel:+6082260656;E-mail:kbmustapha@Swinburne.edu.my

1 Introduction

Microelectromechanical and nanoelectromechanical systems(MEMs and NEMS)occupy preeminent functional roles within the landscape of next generation devices.Often endowed with micro-sized structures,MEM s and NEMs have found unal-loyed advantage in portable field accelerometers,electrical filters,hydrophones,high-Q oscillators and inertia sensors[Nguyen(1995);Yazdi,Ayazi and Najafi(1998);Kun and Nguyen(1999);Mattila,Kiihamäki,Lamminmäki,Jaakkola,Rantakari,Oja,Seppä,Kattelus and Tittonen(2002)].A number of reasons shape the adoption of micro-systems across different emerging scientific fields:(i)prodigious capacity for fast response;(ii)ultra-high resonating frequency;and(iii)a high sensitivity to changes in stimulus.

The complexity of most microsystems depends on the intended functional purpose and operational requirements.In a simple form,a microsystem comprises cantilever arrays along with some other functional embodiment.In a more complex form,however,it may be composed of structural elements with different geometrical parameters for different functional purposes[Boisen,Dohn,Keller,Schmid and Tenje(2011)].Among the unique set of structural elements routinely employed in MEM s is the two-dimensional plate-like micro-scale structures(e.g.thin film,micro-scale panels,and orthotropic micron metallic web[Benkhelifa,Farnsworth,Tiwari and Bandi(2010)].These micro-structural elements have excellent magnetic,optical,mechanical and electrical properties that make them resourceful in Specific applications like thermal sealing and energy harvesting[Sakhaee-Pour,Ahmadian and Vafai(2008);Liu,Tu and Chung(2012);Manzaneque,Ruiz,Hernando-Garcia,Ababneh,Seidel and Sanchez-Rojas(2012)]Since the optimal design of MEMs and NEMs impels the understanding of their response under disparate mechanical loadings,the analyses of their constituent micro-structural elements has gained attention in the past few years[Younis(2011)].During the operational life of MEMs,some factors compromise the exhibition of their intended performance[Ardito,Baldasarre,Corigliano,De Masi,Frangi and Magagnin(2013)].Frequently,such factors are closely related to the interrelationship between the mechanical parameters of the underlying materials and the geometric parameters of the system’s architecture.This interplay imposes a strict constrain on the predictability of MEMs’mechanical responses.

Several studies relating to the prediction of MEMs’response have revealed the demands for the refinement of the mathematical models of the constituent elements of these miniature devices[Peddieson,Buchanan and M cNitt(2003);Reddy(2011);Mustapha(2014)].This is because,at small length scale,the vexing contribution of size-effect to the response of the structures starts to set in.Initial studies on the modeling of micro-scale structures through the framework of computational mechanics,employed the classical continuum theory(CCT).However,the underlying assumptions of the CCT do not account for the contribution of the latent sizedependency that manifests in micro and nano-scale structures[Fleck and Hutchinson(1993);Georgiadis and Velgaki(2003);Liew,Wong,He,Tan and Meguid(2004);Uchic,Dimiduk,Florando and Nix(2004);Volkert and Lilleodden 2006;Mahdavi,Farshidianfar,Tahani,Mahdavi and Dalir(2008);Chiroiu,Munteanu and Delsanto(2010)].It is this de ficiency of the CCT that has ignited the adoption of a number of enriched microstructure-dependent elasticity theories with higher-order constitutive laws[Farokhi,Ghayeshand Amabili(2013)].The models derived from these higher-order theories contain additional material constants,along with the well-known Lamé constants,to address the quantification of the phenomenological size-Specific property.

Evidently,attempt on the theoretical modeling of micro structured materials stretches back to the work of the Cosserat brothers[Cosserat E.and F(1909);Chiroiu,Munteanu and Gliozzi(2010)].Consolidating on the work of the Cosserat brothers,a select list of advanced continuum theories that have emerged to be of relevant to the modeling of micro structured materials include the Eringen’s nonlocal elasticity theory[Eringen(1972);Eringen and Edelen(1972);Fotouhi,Firouz-Abadi and Haddadpour(2013)],the strain gradient theory[Aifantis(1992);Tang,Shen and Atluri(2003);Papacharalampopoulos,Karlis,Charalambopoulos and Polyzos(2010)],the couple stress theory[Toupin(1962);Yang,Chong,Lam and Tong(2002)]and the m icropolar elasticity[Toupin(1962);Xie and Long(2006);Marin,Agarwal and Othman(2014)].A concise summary of the distinction between these theories is well-stated in Aifantis[Aifantis(2011)].Meanwhile,a well-founded challenge with the new higher-order models is the dif ficulty associated with the determination of the microstructure-dependent material constants introduced by these theories[Lam,Yang,Chong,Wang and Tong(2003)].Consequently,models of the structural elements with fewer length scale parameters offer both experimental benefit and mathematical convenience.The Modified couple stress theory(MCST),which is adopted in the current study,is one such higher-order elasticity theories.With the MCST,the characterization of size-effect is done with just one material length scale parameter[Yang,Chong,Lam and Tong(2002);Papargyri-Beskou,Tsepoura,Polyzos and Beskos(2003)].The MCST has already set off a burst of research activities in the theoretical prediction of the deformational response of micro-scale structural elements.Initial application of the MCST focused on the analyses of micro-scale beams,rods and micro-scale pipes[Anthoine(2000);Park and Gao(2006);Güven(2011);Reddy(2011);Mustapha and Zhong(2012);Mustapha and Zhong(2012);Akgöz and Civalek(2013);Wang,Liu,Ni and Wu(2013)].However,a limited number of recent studies have also extended the work to the analyses of micro-scale plates[Yin,Qian,Wang and Xia(2010);Jomehzadeh,Nooriand Saidi(2011);Akgözand Civalek(2013);Gao,Huangand Reddy(2013)]An experimental validation of the MCST is recently presented in Romanoff and Reddy(2014).

Draw ing on the strength of the MCST,the present study investigates the size dependent flexural vibration of a rib-connected system of micro-panels.The system under investigation is part of an on-going design of a polymeric vibration isolation pad for electronic communicating systems.For the derivation,each micro-panel is treated as a micro-scale plate with an internal material length scale.The stiffness of the ribs between the micro-panel is derived by treating the ribs as a collection of clamped-clamped elastic beams.A hybrid numerical experiment based on the Rayleigh method of eigenvalue extraction and an applied statistical method of design of experiment(DOE)is adopted for parametric analysis of the derived model.The adopted hybrid approach helps to illuminate the pattern of the resonant frequency shifts under varying influences of the stiffness of the ribs.The effects of the material length scale of the micro-panel,the Poisson’s ratio and the rotary inertia are also assessed.The ensuing part of the paper proceeds with the presentation of the derivation of the size-dependent elastodynamics governing equation of the system in section 2.The solution procedure for the free vibration study is detailed in section 3.Reduced special cases of the derived model are highlighted in section 4.In section 5,numerical results and the discussion of the influence of the model’s parameters on the frequency shift of the system are presented.Basic conclusions from the analysis are given in section 6.

2 Variational Formulation

2.1 The higher-order elasticity theory

The system being considered is the elastically connected micro-panels shown in Fig.1.In what follows,the equations governing the flexural vibration of each micro-panel are derived from the Hamilton’s principle and the Modified couple stress theory(MCST).In deriving the governing equation,we treat each micropanel as an isotropic linear elastic body occupying a volume∀.In line with the theoretical framework of the MCST,the size-dependent strain energy of a deformed micro-scale structure is characterized by a quartet of tensors related as[Yang,Chong,Lam and Tong(2002);Reddy(2011)]:

In general,thevolume∀is taken to bean open setin R3with a well-behaved surface boundary(see Fig.1b for the unit normal to the boundaries’of a micro-panel).Accordingly, ΠU,εandσare the strain energy,the dilatation strain tensor and the Cauchy stress tensor,respectively.Furthermore,the tensorsmm mandχrefer to the deviatoric components of the couple stress and the symmetric curvature tensor,respectively.

Figure 1:Schematic of the rib-connected system of micro-panels:(a)a 3D view of the coupled system;and(b)the mid-plane of a micro-panel with the adopted coordinate system.

Given that the deformation of a material point of the micro-panel is described by the displacement fielduu uand a rotation vectorθ,then the tensorsεandχin Eq.(1)are known to satisfy the follow ing geometric relations:

where the operator∇is the 2D gradient notation.From kinematics consideration,the rotation vectorθis easily related to the displacement field in the form:

Additionally,the stress fields(σandmm m)are mapped to the displacement field through the enriched constitutive rules as:

In Eq.(5a)the stress field is related to the displacement field through the parametersλ(the bulk modulus)andG(the shear modulus)respectively.However,in Eq.(5b),the higher-order stress field(mm m)is related to the displacement field through an additional material constant(ζ)This new constant is associated with the material length scale parameter[Reddy(2011)].

For the purpose of characterizing the deformation of the micro-panel through the above higher-order constitutive rules,an infinitesimal bounded volume of the micro-panel is considered.The infinitesimal bounded volume of the micro-panel is treated as a differentiable manifold embedded in a Euclidean 3-space∀.Consequently,the following displacement trial field holds:

whereu1,u2,andu3are components of the displacement vector of an arbitrary material point of the micro-panel in thex,yandzdirections.Based on the small deflection theory of thin plates,which rests on the Kirchhoff assumptions,each of these components of the displacement field is defined as:

whereu,v,andware displacement components in thex,yandzdirections,respectively.Based on Eqs.(7a)–(7c),the non-zero components of the strain tensor are obtained as:

The components of the rotation vector,from Eq.(4),are obtained as:

With the help of Eq.(9)bearing in m ind Eq.(3),the components of the symmetric curvature tensor,are derived as:

Given the above derived kinematic variables,the first variation of the microstructuredependent strain energy is now written as:

From the displacement field defined in Eq.(7),the first variation of the kinetic energy,with the rotary inertia included,is defined as[Reddy(2002);Szilard(2004)]:

whereρis the mass density andhis the constant thickness of the micro-panel.In the same spirit,the first variation of the virtual work done by external loads is[Akgöz and Civalek(2013)]:

wherefi,ci,tiandsiare components of the body force,the body couple,the traction and the surface couple,respectively.With Eqs.(11)–(13),one invokes the variational statement of the Hamilton’s principle,which mathematically translates to:

where the integration in Eq.(14)is carried out between the time interval(0,T).Eqs.(11)–(13)are substituted in Eq.(14),and the fundamental lemma of variational calculus is invoked to retrieve the size-dependent governing equations of a single micro-panel as:

The terms that arise from the use of the MCST are underlined in Eqs.(15)–(17).In general,Eqs.(15)and(16)are adequate to predict the in-plane(extensional)vibration of a single micro-panel.On the other hand,Eq.(17)is suitable for the quantification of the transverse motion of a single micro-panel.Now,while Eq.(17)can be tackled alone,Eqs.(15)and(16)are coupled,and thus they cannot be solved independent of each other.Of interest in this study is the case of two inextensible micro-panels separated by evenly distributed ribs.On this premise,the focus of the current study is restricted to Eq(17)and its eigen-analysis.For the purpose of incorporating the influence of the ribs,the following assumptions are adopted:

1.the ribs are made of similar materials and have resistance to stretching and compression;

2.the ribs suffer negligibly marginal distortion of their positions with respect to their initial contact positions with the surface of each micro-panel;

3.no structural anisotropy is introduced by the ribs and;

4.the two micro-panels are made of the same material,length,width and density.

With the stated assumptions,the concentrated rigidities of the ribs can be replaced by the distributed continuous support of the Winkler’s type.Given this simplification,the contribution of the ribs’stiffness is now reflected in the equation governing the transverse vibration of individual micro-panel as:

Upper micro-panel:

Lower micro-panel:

It is pointed out that in Eqs.(18) – (19),the total stiffness(K)of the Winkler’s foundation is taken to be equivalent to the averaged stiffness of the overall ribs between the micro-panels.Besides,wuandwldenote the transverse motions of the upper and lower micro-panel,respectively.The two displacement field variables describing the response of each micro-panel are homogenized by subtracting Eq.(19)from(18)to get:

A further modification of the system’s response is sought by employing a change of variable as done in Murmu and Adhikari(2011).In this vein,the relative displacement(wu-wl)of the upper micro-panel with respect to the lower micro-panel is denoted bywa.The change of variable leads to the equations of the rib-connected micro-panels as:

Equations(21)and(22)describe the transverse motion of the connected Micropanels,whereIis the moment of inertia per unit area of a micro-panel.It deserves to be pointed out that if the underlined terms are eliminated,the classical size-independent governing equation of the Kirchhoff plate theory is retrieved.In succeeding sections,the dynamic behavior of the system is investigated under two idealized boundary conditions:(i)simple support on the four edges(that is,SSSS,where S stands for simply-supported);and(ii)a built-in support on the four edges(that is,CCCC,where C stands for clamped)as shown in Fig.2.

Figure 2:The two types of boundary supports investigated for the rib-connected micro-panels.

In order to specify the expressions for the stipulated boundary conditions the follow ing stress resultants are defined:

whereandCyzare the couple moments that need to be mapped to the displacement field through Eq.(10).A lso,andare the moments(related to the Cauchy stress tensor)that are mapped to the displacement field through Eq.(8).Under the SSSS boundary condition,the following constraints are imposed around the periphery of the coupled micro-panels:

Analogously,under the CCCC boundary condition the expressions stated in Eqs.(25)–(28),are supplemented with the follow ing additional constraints around the periphery of coupled micro-panels:

3 Solution Procedure

The Rayleigh method is adopted to evaluate the dynamic response of the system.The energy functional Π from the governing equation of the coupled is defined as:

With the non-dimensional parameters,the explicit expressions for the maximum energy terms for the overall system become:

The Rayleigh solution procedure requires that the mid-plane deflections of the micro-panel be expressed in terms of an assumed mode shape function in the form of a double Fourier series as:

such thatpmnis the amplitude of the function,whileφm(ξ)andβn(η)are the mode functions that satisfy the Dirichlet boundary conditions of the micro-panel.The following comparison functions are used for the two boundary conditions considered in this study:

wheremandnare the nodal lines in theξandηdirections that determine the wave modes of the micro-panels.For the frequency values to be determined,the Rayleigh method minimizes the energy functional(by ensuring that∂Π/∂pmn=0)[Liu(2011)].The minimization procedure results in a system of algebraic equations whose secular determinant yields the natural frequencies of the coupled system.In the next section,detailed numerical results are provided to evaluate the influence of different parameters of the model on the dynamic response of the coupled systems.

4 Special cases

Three different mechanistic cases of practical interests can be derived from the governing equation.These cases,shown in Fig.3,are briefly highlighted below.For convenience,the following additional non-dimensional terms have been introduced in what follows:

4.1 Asynchronous motion of the coupled system

The case of the asynchronous motion of the coupled system,which is also referred to as the out-of-phase vibration,occurs when the two micro-panels move in different directions[Murmu and Adhikari(2011)].In such case the condition of motion stipulates thatUnder this condition,the Rayleigh method yields an upper estimate of the frequency of the coupled micro-panels with the SSSS edge conditions as:

In Eq.(45),γnow represents the influence of the size-effect(that is,the material length scale of the micro-panel).This Specific parameter quantifies the size dependent vibration Dynamics of the micro-panel with respect to the micro-panel’s thickness.A lso,αeffis the non-dimensional averaged stiffness of the elastic connections,while µ and φ represent the thickness-to-length ratio and the aspect ratio of the micro-panel,respectively

4.2 Synchronous motion of the coupled system

Under a synchronous motion,the movements of the two micro-panels are in sync.Hence,this case is also referred to as the in-phase vibration Here,the condition of motion during the in-phase vibration requires thatwa=0.With this,the Rayleigh method again yields an upper estimate of the frequency of the coupled micro-panels with SSSS edge conditions as:

4.3 Motion of the upper-panel supported by the ribs

The third special case relates to the situation when the lower micro-panel is a rigid base.Such situations do arise in the development of a mass sensing device[Agache,Blanco-Gomez,Cochet and Caillat(2011)].When this occurs,thenwl=0.Consequently,the frequency of the remnant systems is governed by the geometric and material properties as well as the boundary conditions of the upper micro-panel and the ribs.The upper estimate of the frequency of the remnant system with a SSSS edge conditions for the upper micro-panel then takes the simplified form:

Figure 3:Special cases of the rib-connected micro-panels:(a)out-of-phase motion;(b)in-phase motion;and(c)motion of a single rib-supported micro-panel.

5 Discussion

The centerpiece of this section is to detail the qualitative and quantitative results relating to the free vibration properties of the system,based on the formulation presented in the foregoing sections.Principally,the section details the variation of the natural frequencies of the system with respect to changes in the aspect ratio(ϕ),the material length scale parameter(γ),the Poisson’s ratio(ν),the thicknessto-span ratio(µ),and the stiffness of the ribs(αeff).For the reported analyses,the effective material properties of the polymeric micro-panel are assumed to be:E=1.44GPa;ρ=1220kg/m3;andν=0.38.Furthermore,based on the experimental work of Lam,Yang,Chong,Wang and Tong(2003),the material length scale parameter for the epoxy micro-plate is restricted to a conservative value ofζ=17.6µm.Meanwhile,for polymeric structural elements with underlying polymer chains of finite stiffness,the material length scale parameter(ζ)is reported to be directly related to the effective averaged Frank elastic constant[Gao,Huang and Reddy(2013)].For the analyses that follow,the thickness values considered is in the range 20µm≤h≤176µm.Besides,each rib is idealized as a fixed- fixed bar.Thus for each rib,the equivalent stiffness is estimated from,whereandLrare the cross-sectional area,the moment of inertia,the Young’s modulus and length of each rib.

5.1 Validation

For the purpose of validation,the governing equation of the system is Modified by eliminating the parametersµ,αeffandγfrom the mathematical model.With this elimination,the governing equation is reduced to that of the Kirchhoff’s plate theory[Szilard(2004)].Tab.1 reveals the comparison of the frequency values,based on the reduced form of the current model,with the classic work of Leissa(1973).Tab.1 contains selected numerical results for a panel(treated as a plate)under the SSSS edge conditions and the CCCC edge conditions.

Tab.2 provides the validation of the non-dimensional frequencies of a micro-plate with simply-supported edges.The governing equation employed for the results in Tab.2 containsγ,but it does not containαeffandµ.The validation involves comparison with the closed-form expression presented in Yin,Qian,Wang and Xia(2010).It is noticed from Tab.2 that a good agreement is achieved between the predicted non-dimensional frequency from the current method and the closed-form solution.

For brevity sake,the discussion is restricted to the two lowest symmetrical modes of the flexural vibration(whose non-dimensional frequency parameters areλ11andλ22)in the next subsections.For completeness,the mode shapes corresponding to bothλ11andλ22are shown in Figs 4 and 5,respectively.In these figures,the contour plot corresponding to each of the two vibration modes are also provided under each of the mode shape.From the plots,it is noticed that increasing values of the small-scale parameter stiffen the response of the micro-panel.Besides,as seen from the contour plots,higher values of the small-scale parameter generate more equilibrium positions during the deformation of the system.

5.2 Frequency distribution and frequency shift

To evaluate the frequency shift induced by the small-scale parameter,a percentage change in the computed natural frequencies based on the MCST and the classical(size-independent)theory is defined as:

where(λnm)MCSTis the natural frequency of the system from the MCST,while(λnm)CCTis the corresponding natural frequency from the CCT.Besides,λnmis the non-dimensional natural frequency parameter defined as:

Table 1:Validation of the non-dimensional frequencies of a classical plate.

Table 2:Validation of the non-dimensional frequencies of a micro-plate with simply-supported edges.

Figure 4:The variation in the mode shape along with the corresponding contour plots of the lowest symmetrical mode(λ11):(a)γ=0;(b)γ=0.2;and(c)γ=0.8.

Summarized in Figs 68 are the distributions of the frequency variations for the three mechanistic cases of the coupled systems highlighted in section 4.Precisely,illustrated in Figs 6(a)and 6(b)are the distributions of the natural frequencies for the first symmetric vibration mode,under the SSSS and the CCCC boundary conditions,respectively.In each of the plot,the dummy variablesC1,C2 andC3 refer to case 1,case 2 and case 3,respectively.It is recalled that case 1 relates to the out-of phase motion of the coupled micro-panels,case 2 refers to the in-phase vibration of the coupled micro-panels,and case 3 is the motion of a single rib-supported micropanel(case 3).Fig.7 is indicative of the distribution of the natural frequencies for the second symmetric vibration modes of the three mechanistic cases.It is noticed from Fig.7 that the frequency response increases with the aspect ratio(similar to Fig.6).While a clear pattern of skew-symmetric distribution of the frequency values is easily noticed in Figs 6 and 7,a number of additional subtle observations can be made from the plots.First,in Fig.6a,the lowest non-dimensional natural frequency value for the out-of-phase motion(C1)is more than the highest nondimensional natural frequency value for the other two cases for the aspect ratio of 0.2.However,under an increased value of the aspect ratio,the difference in the natural frequency values for the three cases approaches each other,althoughC1 still maintains the lead.Furthermore,in Fig.6(b),the difference in the degrees of influence of the aspect ratio and the small-scale parameter on the pattern of the frequency increase is found to be comparatively similar across the three cases.

Figure 5:The variation in the mode shape and the contour plots of the second symmetrical mode(λ22):(a)γ=0;(b)γ=0.2;and(c)γ=0.8.

Figure 6:Distribution of the natural frequencies for the first symmetric vibration modes of the systems:(a)under four simply supported edges;(b)under fully clamped edges.

Fig.8 represents a visual display of the boundary condition under which the effect of the small-scale parameter exerts a greater influence.From this plot,it is seen that the CCCC edge condition is more heavily affected by the size-effect than the SSSS edge condition for the two vibration modes.It is worth pointing out that,since the CCCC edge conditions do naturally yield greater frequency response,normalized frequency values are employed for Fig.8.Still,it is noticed that the effect of the small-scale parameter is more pronounced on the frequency of the system under the CCCC edge condition.

Figure 7:Distribution of the natural frequencies for the second symmetric vibration modes(αeff=100):(a)micro-panels with four simply supported edges;(b)micropanels with fully clamped edges.

Additional plots to examine the effects of the geometric and small-scale parameters are provided in Figs 9-12,all of which are related to the system with the SSSS edge conditions.It is observed from Fig.9(a)that as the micro-panel’s thickness reduces,up to 20%increase in the frequency value is attainable for an aspect ratio of 1(basically a square micro-panel).On the other hand,Fig.9(b)illustrates the vanishing influence of the size-effect on the natural frequency value of the panels as the thickness increases.

Figure 8:A comparison of the effect of the small-scale parameter on the out-ofphase motion of the coupled system.

Fig.10 shows the differential trend of the influence of the small-scale effect on the responses of the three mechanistic cases of the coupled system.From this plot,one is able to conclude that the out-of-phase motion of the coupled system experiences the most softening effect arising from the decreasing value of the Micropanels’thickness.Demonstrated in Fig.11 is the inverse relationship between the frequency response of the out-of-phase motion of the coupled system with the thickness-to-span ratio(µ).

In Fig.12,the influence of the ribs’effective stiffness on the three special cases is shown.Specifically,Fig.12(a)reveals the trend of the frequency increase when the ribs’effective stiffness parameter approaches zero,while Fig.12(b)shows difference in the trend for the three cases when the ribs’effective stiffness parameter is maintained at a modest value of 100.

5.3 Identification of the order of influence of the model’s parameters

Figure 9:The quantification of the percentage change in frequency values:(a)the change in frequency against the small-scale effect;(b)the change in frequency against the micro-panels’thickness.

The results presented under subsection 5.3 are generated by the traditional method of analysis,where one varies a single parameter of the model while keeping all others constant.Essentially,this method is analogous to the so calledone factorat-a-time(OFAT)analysis[Karray and Silva(2004)].One of the drawbacks of the OFAT methodology in the traditional computational mechanics is the inherent underestimation of the possibility of interaction effects between the variables that influence the response of the system under consideration.In the present study for instance,we recognized five continuous numerical dimensionless parameters that could influence the response of the system as seen in Fig.13.These five continuous numerical parameters(or factors)are the aspect ratio(ϕ),the small-scale parameter(γ),the ribs’effective stiffness(αeff),the Poisson’s ratio(ν)and the thickness to-span ratio(µ).In what follows,the OFAT analysis is complemented with the applied statistical method of design of experiment(DOE)[Dean and Voss(1999);Mustapha and Zhong(2012)]to examine the order of influence of these parameters.

Figure 10:Variation of frequency values with varying value of the material length scale for the special motion types of the system.

Figure 11:The effect of rotary inertia on the two lowest symmetric vibration modes for the out-of-phase motion of the coupled system.

Figs.14-16 are indicative of the outcome of the analyses based on the use of the DOE.Two levels of each of the five parameters are considered for the numerical DOE.Based on the established procedure of DOE,an experimental design involving five input factors at two levels(low and high)becomes a problem of 2pnumber of experimental runs,where the superscriptpis the number of factors.The capability of the statistical software M initab[Lesik(2010)]is used to run a randomized design of the experiment.It is pointed out that the term experiment is used in the sense of numerical analysis experiment in the current setting.The values of the factors at each run of the experiment are presented in Tab.3.

The ordered ranking of the model’s factors(also refers to as main effects)as they influence the response of the system are presented in the form of the Pareto charts depicted in Figs 14-16.In these charts the termsA,B,C,DandEare dummy variables that symbolize the ribs effective stiffness,the aspect ratio,the size-effect,the thickness-to-span ratio and the Poisson’s ratio,respectively.The combinations of these dummy variables(such asAB,ACandABCetc.,)are visible in these charts,and they are called the interaction effects.Specifically,Figs.14,15 and 16 represent the Pareto chart of case 1(the asynchronous motion),case 2(the synchronous motion)and case 3(the motion of a rib-connected micro-panel),respectively.

Figure 12:The effect of the ribs’stiffness on the three special cases:(a)a negligible ribs’effective stiffness parameter;(b)ribs’effective stiffness maintained at 100.

Figure 13:The relation between the inputs and the output of the coupled Micropanels system.

It is observed from Figs 14-16 that the factor that affects the natural frequency of the motion of the three cases the most is the interaction factorBC.This interaction factor is created by the interaction between the aspect ratio and the size-effect parameter.Furthermore,it is observed from the Pareto charts that the Poisson’s ratio has a somewhat negligible effect on the motion of the three systems.However,the interaction effect created between the Poisson’s ratio,the aspect ratio and size effect(BCE)is the fourth most Significant factor that influences the behavior of the synchronous motion.An additional conclusion that can be drawn from these charts is the fact the size-effect is the second most Significant factor that alters the natural frequency of the system for case 1 and case 2.It is also the third most Significant factor for case 3.Besides,the thickness-to-span ratio is discovered not to Significantly affect the response of any of the three mechanistic cases.

Figure 14:The Pareto chart of the ordered ranking of the model’s factors for the asynchronous motion of coupled system.

Table 3:Randomized numerical experimental runs.

Figure 15:The Pareto chart of the ordered ranking of the model’s factors for the synchronous motion of coupled system.

Figure 16:The Pareto chart for the motion of ribs-supported single micro-panel.

6 Conclusion

On the basis of the extended Hamilton’s principle and the Modified couple stress theory,the governing equations of a rib-connected system of coupled micro-panels are derived.Starting from a displacement trial field,the derivation considers the deformation of a single micro-panel embedded in a Euclidean 3-space∀.The derivation procedure yields three field equations.Two of the field equations represent the extensional vibration mode of the micro-panel,while the third equation relates to the flexural vibration mode.Given the focus of the current study,the third field equation,which accounts for the flexural vibration of the micro-panel and is uncoupled from the extensional modes is adopted for comprehensive response analyses.In analyzing the behavior of the coupled micro-scale panels,three forms of special cases of motion of the system were assessed:(i)the out-of-phase motion;(ii)the in-phase motion;and(iii)the motion of a single micro-panel with rib connections and a rigid base.The vibration characteristics under each of these motion types were studied systematically for the determination of the frequency shift and the pattern of the frequency distribution.The study reaffirms the expected substantial increase in natural frequency value for a lower range of the micro-panel’s thickness.A skew-symmetric pattern of distribution of the shift in natural frequencies values is discovered for the first and second symmetric modes of the coupled system under the two edge conditions studied.Furthermore,it is noticed that,for lower values of the aspect ratio,the lowest non-dimensional natural frequency value for the out of-phase motion is more than the highest non-dimensional natural frequency value for both the in-phase motion and the motion of a single micro-panel with rib connection.It is also found that the effect of the material length scale on the frequency of the out-of-phase motion is higher when the edges are fully clamped than when simply-supported.Through Pareto analysis,notable interaction effects were discovered between the models’parameters.

Although extrapolating theoretical conclusions to experimental studies do come with certain challenges,preliminary theoretical investigations such as the one carried out in this study strives to provide a cogent basis for close scrutiny of future experimental results.The observations from this study are expected to pave the way for the recognition of key variables(main effects or interactions effects)that might be employed in the future optimization of the system’s geometry and response.

Acknowledgement:The author is greatly indebted to the anonymous reviewer for the insightful suggestions that lead to the improvement of the manuscript.The support of the management of Swinburne University(Sarawak Campus)is also acknowledged.

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