1 Introduction

Due to the high sensitivity and increased stability,micro/nanomechanical sensors or actuators have been widely applied to measure such physical parameters as mass,Young’s modulus,hardness and strength of micro-nanoscale materials.At this scale,surface effects become important because of the increasing ratio of surface/interface area to volume,which has attracted scientists and engineers towards the investigation of surface effects in buckling[1,2]or vibration analysis[3–5]of a micro/nanobeam.To this end,Gurtin et al.[6]presented a general theory of curved deformable interfaces in solids at equilibrium,and a continuum model of interface elastic constants was proposed.In addition,Wang and Feng[3]examined the impacts of surface elasticity and residual surface stress on the natural frequency of Euler–Bernoulli microbeams by assuming a sandwich beam model,which only holds true when the beam length is much larger in comparison with its thickness,e.g.,more than ten times.However,when the beam is not sufficiently long,the effects of shear deformation and rotatory inertia[7–9]must be considered.In other words,the Timoshenko beam model should be employed in this situation.Thus,a comprehensive Timoshenko beam with consideration of surface effects was presented to conduct the buckling and flexural vibration analysis of nanowires[1].Then,the corresponding vibration analysis was extended to nanotubes[5].

On the other hand,as the beam size reaches micro-or nanoscale,nonlocal elasticity theory[10]is widely accepted and applied to predict the static or dynamic behaviors[11–16]of a micro/nanostructure due to its scale dependence.The nonlocal elasticity theory assumes that the stress at a point is a function of the strains at all points of the body.As a contrast,classical continuum theory is referred to as local theory,because it states that the stress at a point is only dependent on the strain at this point.A comparison study on the flexural wave propagation in a carbon nanotube was carried out by Wang and Hu[17],and they noticed that the nonlocal theory can provide a better prediction for the decrease of phase velocity induced by the scale effect for a large wave number,while the local theory cannot.Also,Wang et al.[11]performed the free vibration analysis of micro/nanobeams based on nonlocal elasticity theory and Timoshenko beam theory,and exact vibration solutions were obtained to represent the vibration behaviour of short,stubby,micro/nanobeams.However,the effect of nonlocal elasticity on the constitutive relation of shear stress and strain is neglected.Vibrational modes of small scale Timoshenko beams were treated by Li and Wang[12]by incorporating the nonlocal effect into both normal and shear stresses.Then too,Reddy and El-Borgi[18]derived the governing equation of nonlocal micro/nanobeams by using modified von Kármán nonlinear strains to account for moderate rotations. The properties of two frequency spectra of nonlocal Timoshenko beams were considered in detail,and a reliable method to determine the nonlocal effect was also suggested by Zhang [19].

It is noticeable that the research work mentioned above considers either the nonlocal effect or surface effects.However,when the beam size shrinks to microns or nanometers,considerations of nonlocal effect and surface effects are both important inaccurately evaluating the natural frequency of the beam[20,21].Hence,in this study,a theoretical model of free vibration of a nonlocal Timoshenko beam with surface effects is established by considering three types of boundary conditions,i.e.,hinged–hinged,clamped–clamped and clamped–hinged ends.For the hinged–hinged beam,an exact and explicit natural frequency equation is derived,which links the natural frequency to the effects of nonlocal elasticity,surface elasticity,residual surface stress,shear deformation and rotatory inertia together.When one or several effects are ignored,the proposed frequency equation can become the equation derived by former researchers[3,11,19].However,for the clamped–clamped and clamped–hinged beams,the characteristic frequency equations obtained are both transcendental equations,which can only be solved by numerical computations.In fact,explicit frequency equations,even for approximate form,are much desired in engineering applications.Therefore,the Fredholm integral equation approach[22–24]in conjunction with a curve fitting method are employed to determine the approximate fundamental frequency equations due to the importance of the fundamental frequency in micro/nanobeams.

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2 Model development

2.1 Governing equations

In this section,the governing equations of a nonlocal Timoshenko beam with consideration of surface effects are derived.In order to consider the surface effects,it is assumed that the upper and lower surfaces of the beam have surface stiffness Es and residual surface stress[3].Due to the presence of the surface stiffness,the effective bending rigidity of the beam becomes

where E,b,h andare the Young’s modulus,width,thickness and second moment of area of the beam,respectively.The Laplace–Young equation[6]is employed to determine the influence of residual surface stress by relating the stress jumpacross a surface to the curvature tensor〈καβ〉of the surface,

Multiplying Eq.(5)by(−y d A)and integrating over the cross-sectional area A of the beam one obtains the following equations,

Similarly,the substitution of Eqs.(23)and(25)into Eq.(38)yields the characteristic equation,

whereεxx andγxy are the normal axial strain and shear strain,respectively,w is the transverse displacement,φ is the rotation angle of beam cross-section.Obviously,∂φ/∂x denotes the bending curvature of the beam.Hence,the transverse load on the surface of the beam obtained by the Laplace–Young equation[6]is given by

whereτu andτb are the residual surface tensions of the upper and lower surfaces of the beam,respectively.

which yields the following solutions,

where σxx and σxy are the normal and shear stresses,respectively,E eff is the effective Young’s modulus,G is the shear modulus,and lc is an internal characteristic length of nonlocal effect[25].It is worth noting that the nonlocal effect is considered both in the normal and shear constitutive relations.

where is the surface stress where α,β =1,2,ni is the unit normal vector of the surface(i=1,2,3).

where M and Q are the bending moment and shear force,respectively,and λ is the shear correction factor.

The equilibriums of the force and the moment are applied to produce the following two equations,

where ρ is the density of the beam.

Substituting Eq.(7)into Eq.(6)yields the following nonlocal bending moment and shear force,

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The governing equations of the nonlocal Timoshenko beam with surface effects is obtained by substituting Eq.(8)into Eq.(7),

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The following dimensionless quantities are introduced to have no dimensions for the above governing equations,

where L is the beam length.Thus,Eq.(9)becomes

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where

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Physically,A1 and A2 indicate the effects of surface elasticity and residual surface tension,respectively,A3 and A4 indicate the effects of shear deformation and rotatory inertia,respectively,and A5 is a dimensionless scale parameter characterizing the nonlocal effect.

The corresponding assumption expression for the clamped–hinged beam is the follow ing[33],

whereω is the dimensionless circular frequency of vibration.

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Substituting Eq.(13)into Eq.(11)obtains

The above coupled ordinary differential equations(ODEs)can be decoupled,and we obtain the following uncoupled ODE for Φ by eliminating Y,

where

Similarly,the uncoupled ODE for Y can also be determined by eliminating Φ,

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The bending moment and shear force in Eq.(8)are lacking dimensions and decoupled as follows

where

2.2 Solution procedures

In order to solve Eqs.(15)or(17),we set orand substitute them respectively into corresponding equations to lead to the following characteristic equation,

To incorporate the nonlocal effect in the governing equations,the normal and shear constitutive relations of the beam are deduced according to Eringen’s nonlocal elasticity theory[10],

being positive or negative should be estimated at first in order to determine the vibration modes of the nonlocal Timoshenko beam with surface effects.In Eq.(20),two asymptotic frequencies are obtained by setting k→∞,

These two frequencies are called cut-off frequencies,indicating that the vibration frequency of the beam cannot surpass either of them.Therefore,we conclude that c1 is always positive for all the possible vibration frequencies.Specially,when neglecting the surface effects(A1=A2=0),Eq.(22)becomeswhich are the cut-off frequencies derived by Lei et al.[26].For c3,three cases,i.e.,c3＜0,c3=0 and c3＞0,all exist due to the uncertainty of the sign.c3=0 is a special scenario of thickness shear vibration without transverse displacement[27],resulting in a critical frequency,Clearly,ωC divides the solution of Eq.(15)or(17)into two different forms,which is the so-called two frequency spectra[19,28–31]of a Timoshenko beam vibration.ω ＜ ωC,i.e.,c3＜ 0,is called as the first spectrum and ω ＞ ωC(c3 ＞ 0)corresponds to the second spectrum.Herein,we only consider lower-order vibration modes in this study,which means that the case of c3＜0 is examined.It is noted that c3＜0 leads toand in Eq.(21)irrespective of if c2 is positive or negative.Hence,the solution form for Φ is written as

where

and the constants Di(i=1,2,3,4)are determined from the boundary conditions.Substituting Eq.(23)into Eq.(14)obtains the following solution form for Y,

where

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3 Exact frequency equations

The exact natural frequency equations of a nonlocal Timoshenko beam with surface effects are derived by including three types of boundary conditions,i.e.,hinged–hinged,clamped–clamped and clamped–hinged ends.

For the hinged–hinged beam,zero transverse displacements and bending moments at two ends yield the following four expressions,

Substituting Eqs.(23)and(25)into Eq.(27)obtains a series of algebraic equations with respect to Di(i=1,2,3,4),

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By setting the determinant of the coefficients matrix to be zero,the characteristic equation is given as follows

It is noted from Eqs.(24)and(26)thatHence,we have two choices,sin(β1)=0 or sinh(β2)=0.sinh(β2)=0 leads to β2=0,and further yields c=0,which is the case of thickness shear vibration.Therefore,The remaining one is sin(β1)=0,which leads to β1=nπ (n=1,2,3,···).According to Eq.(24a),we obtain the following equation,

Substituting Eq.(16)into Eq.(30)and conducting a squaring operation produces the following two roots,

where

where

Equation(33)is the dimensionless exact and explicit natural frequency equation of the nonlocal hinged–hinged Timoshenko beam with surface effects.Clearly,the effects of nonlocal elasticity,surface elasticity,residual surface stress,shear deformation and rotatory inertia are all incorporated in the equation.For a nonlocal Rayleigh beam with surface effects,i.e.,A3−＞∞,Eq.(33)be comes the following form,

For a local Euler–Bernoulli beam with consideration of surface effects,i.e.,A3−＞∞and A4=A5=0,Eq.(33)reduces to[3].For a local Timoshenko beam,i.e.,A1=A2=A5=0,Eq.(33)gives the following result[19],

For a nonlocal Euler–Bernoulli beam,i.e.,A1=A2=A4= 0 and A3− ＞ ∞,Eq.(33)becomes ωn=[32].When A1=A2=A4=A5=0 and A3−＞∞,Eq.(33)recovers the classical equation of ωn=n2π2 for a local Euler–Bernoulli beam.

It is worth noting that Eq.(33)is applicable for either of the two frequency spectra of Timoshenko beam vibration even though it is derived based on the first frequency spectrum.This is because the same characteristic equation is obtained regardless of c3＜0 or c3＞0,which is also observed by Levinson and Cooke[30]and Zhang[19].However,this conclusion only holds true for the hinged–hinged beam. For other types of boundary conditions such as clamped–clamped and clamped–hinged ones,different frequency spectrum leads to different frequency equations.

For the clamped–clamped beam,the boundary conditions are given as follows

By substituting Eqs.(23)and(25)into Eq.(36)and setting the determinant of the matrix equation to be zero,we arrive at the following characteristic equation,

For the clamped–hinged beam,we have the four boundary conditions,

In this study,x and y denote the coordinates along the beam length and thickness,which are measured from the left end and the mid-plane of the beam,respectively.Based on the established coordinate system and Timoshenko beam theory,the strain-displacement relations of the beam are written as

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4 Approximate fundamental frequency equations

In Sect.3,the exact and explicit natural frequency equation of the nonlocal hinged–hinged Timoshenko beam with surface effects is derived.However,for the beam with clamped–clamped and clamped–hinged ends,the frequency equations shown in Eqs.(37)and(39)are both transcendental equations so that natural frequencies must be numerically calculated,which is rather inconvenient for engineering applications.In practice,explicit frequency equations are much desired.Hence,an alternative Fredholm integral equation approach[22–24]is formulated to derive the required frequency equations.However,for the present Timoshenko beam model,the Fredholm integral equation obtained is too complicated to solve it effectively.In this study,the corresponding equation for a nonlocal Rayleigh beam with surface effects is determined at first,and then the proposed equation is extended to the case of a nonlocal Timoshenko beam by using analogy analysis.For all the computations,A1 and A2 characterizing the surface effects are both ranging from−0.2 to 0.2[3],and A5 indicating the nonlocal effect varies from 0.0 to 0.1[19].For a rectangular cross-section beam,the shear correction factor is taken as λ =5(1+ υ)/(6+5υ)with 0≤ υ≤ 0.5.In view of G=E/[2(1+υ)]and h/L∈(0,1/4]adopted in this study,A3 and A4 always hold in the range of(A3≥56.4706)and(0＜A4≤0.0052),respectively,which covers a wide array of slenderness ratios of the beam.

4.1 Fredholm integral equation approach

In this subsection,the explicit fundamental frequency equation for a nonlocal Rayleigh beam with surface effects is derived by means of the Fredholm integral equation approach[22–24].As A3 approaches infinity,Eq.(17)reduces to the governing equation for the Rayleigh beam,which is expressed in the following,

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Integrating Eq.(40)four times with respect to ξ from 0 to ξ obtains the following four expressions,

where Ki(i=1,2,3,4)are the coefficients determined from the boundary conditions as shown in Eqs.(36)or(38).In conjunction with Eqs.(36)and(41),the expression forms of Ki for the clamped–clamped beam are calculated as follows

In conjunction with Eqs.(38)and(41),the expression forms of Ki for the clamped–hinged beam are the following,where

By substituting Eqs.(42)or(43)into Eq.(41d),the resulting Fredholm integral equation of the Rayleigh beam for these two types of boundary conditions is given as

In Eq.(31a),when A4− ＞ 0,which is unphysical and should be ignored.Hence, is kept for the root of Eq.(30)and it is rewritten as follows

Hence,the problem solving the fundamental frequency of the beam becomes the eigenvalue problem of the above Fredholm integral equation.In order to achieve this,an assumption expression of Y(ξ)satisfying the boundary conditions in Eqs.(36)or(38)needs to be proposed to characterize the first mode of the beam.In this study,the first mode of classical Euler–Bernoulli beam is adopted as the assumption expression.It should be noted that the approach is only valid for the first mode.For the clamped–clamped beam,the proposed assumption expression of Y(ξ)is given as[33]

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For the quantities,W and φ,we take the following forms,

Then,by substituting Eqs.(47)and(48)into Eq.(45)and integrating both sides with respect to ξ from 0 to 1,respectively,the approximate fundamental frequency equations of the clamped–clamped and clamped–hinged beams are determined.For the former,the approximate frequency equation obtained is the following,

The corresponding frequency equation for the latter is

The approximate fundamental frequency values given by Eqs.(49)and(50)are compared with the exact ones calculated numerically to verify the validity of these two equations.Excellent agreement between them is observed by considering the largest relative errors being 0.71%and 0.32%for the clamped–clamped and clamped–hinged beams,respectively.By comparing Eq.(34)with Eqs.(49)and(50),it is noted that the generalized expression forms of these frequency equations for the Rayleigh beam with three types of boundary conditions are nearly the same except the difference in the coefficient terms.Thus,it is reasonable to extend this analogy to the case of a nonlocal Timoshenko beam with surface effects.Since the exact and explicit natural frequency equation of the hinged–hinged Timoshenko beam has been derived in Eq.(33),we assume that the approximate fundamental frequency equations for the clamped–clamped and clamped–hinged beams have the following forms,respectively.

For the clamped–clamped beam,

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Fig.1 Comparison of fundamental frequency obtained from Eqs.(37)and(51)for a clamped–clamped Timoshenko beam with A3=56.4706 and A4=0.0052

where

For the clamped–hinged beam,

where

4.2 Curve fitting method

The approximate fundamental frequency equations of the Timoshenko beam with clamped–clamped and clamped–hinged ends have been presented in Eqs.(51)and(53),respectively.In order to demonstrate the validity of the equations,the approximate frequency values obtained are compared with the exact ones by numerically solving Eqs.(37)and(39).It is observed from Figs.1and 2 that even though the approximate frequency equations both present overestimated results,they can follow the accurate variation trend of the natural frequency with the change of these dimensionless parameters(A1,A2,A3,A4,A5).We conclude that the proposed equations are able to predict the fundamental frequency if they are appropriately amended.Thus,a correction factor fC is introduced to reduce the difference between one another,which is defined as the ratio of Eq.(51)to Eq.(37)or Eq.(53)to Eq.(39).Since Eqs.(51)and(53)become Eqs.(49)and(50),respectively,as A3 approaches infinity,which means that fC−＞1with A3−＞∞,fC is expressed in the following form,

Due to a variety of parameters included in Eq.(55),the effects of these parameters on fC are examined to simplify the complex expression.It is noted from Fig.3 that A2 has no impact on fC for the clamped–clamped beam,and the same goes for the clamped–hinged beam.In Fig.4,the A4 values varying from 0.0052(h/L=1/4)to 0.0005(h/L=1/13)are investigated,and an insignificant difference is observed by considering the largest percentage difference being 1.3%.Therefore,A2 and A4 are not incorporated in Eq.(55)for the clamped–clamped and clamped–hinged beams,and the expression form of fC becomes

A curve fitting method is employed to construct the explicit expression of fC.Eventually,the approximate fundamental frequency equations of the nonlocal Timoshenko beam with surface effects are determined.For the clamped–clamped beam,the proposed frequency equation is the following,

Fig.2 Comparison of fundamental frequency obtained from Eqs.(39)and(53)for a clamped–hinged Timoshenko beam with A3=56.4706 and A4=0.0052

Fig.3 Effect of A2 on the correction factor for a clamped–clamped Timoshenko beam with A3=56.4706 and A4=0.0052

Fig.4 Effect of A4 on the correction factor for a clamped–clamped Timoshenko beam with A3=56.4706

where

The corresponding frequency equation for the clamped–hinged beam is given as

where

5 Validity of approximate fundamental frequency

The aim of this section is to confirm the accuracy of the approximate fundamental frequency equations specified in Eqs.(57)and(59).At first,the exact frequency values are calculated by numerically solving Eqs.(37)and(39).Then,the approximate frequencies obtained from Eqs.(57)and(59)are compared with the exact ones,respectively,and the relative error is employed to illustrate the difference.By comparison study,it is observed that the approximate values correlate very well with the exact ones with the largest relative error being 0.72%and 0.32%,respectively,by considering all possible combinations of A1,A2,A3,A4 and A5.Some comparison results for the clamped–clamped and clamped–hinged beams are tabulated in Tables 1 and 2.Figures 5–7 are plotted to further display the high goodness of fit between one another,and then to investigate the variation of the fundamental frequency as a function of the above parameters.In Fig.5,ω1 increases as A3 increases for the clamped–clamped and clamped–hinged beams.The reason is that the shear deformation effect decreases the system stiffness.A larger A3 leads to a smaller shear deformation,inducing the increase of ω1.When A3 reaches 104,the shear deformation effect can be nearly neglected,and ω1 becomes stable.Figure 6 shows the impact of rotatory inertia on the fundamental frequency.It is seen that ω1 slowly declines as A4 increases.This is because the rotatory inertia effect increases the system mass,resulting in the decrease of ω1.By comparison with Fig.5,we also notice that the rotatory inertia effect on the fundamental frequency is much smaller than that of shear deformation. The nonlocal effect on the fundamental frequency is presented in Fig.7.A larger A5 causes a significant drop of ω1,which is due to the nonlocal effect reducing the system stiffness.Figures5–7 all indicate that the surface effects have a remarkable influence on the frequency.Positive A1 and A2 will enlarge the system frequency,and the frequency will be reduced by negative A1 and A2.Therefore,when the beam size shrinks to microns or nanometers,thenonlocal effect and surface effects should both be taken into consideration to obtain accurate prediction of the natural frequency.Generally,the changes of the natural frequency with A1,A2,A3,A4 or A5 are determined by numerically solving characteristic frequency equations.However,in this study,the explicit frequency equations as presented in Eqs.(33),(57)and(59)are proposed to reveal the dependence of the frequency on above parameters(A1,A2,A3,A4,A5),avoiding complicated numerical computations.

Table1 Comparison of dimensionless fundamental frequencies obtained from Eqs.(37)and(57)for clamped–clamped beam with A4=0.0052

A3 (A1,A2) A5=0.0 A5=0.05 A5=0.1 Eq.(37) Eq.(57) Relative error(%)Eq.(37) Eq.(57) Relative error(%)Eq.(37) Eq.(57) Relative error(%)56.4706 (−0.2,−0.2) 15.1023 15.0931 −0.0609 11.9201 11.9175 −0.0218 10.0892 10.0660 −0.2299(−0.2,0.2) 15.1893 15.1875 −0.0119 12.1481 12.1452 −0.0239 10.4181 10.3891 −0.2784(0.2,−0.2) 16.8986 16.8294 −0.4095 13.4783 13.4532 −0.1862 11.4797 11.4266 −0.4626(0.2,0.2) 16.9618 16.9001 −0.3638 13.6488 13.6247 −0.1766 11.7278 11.6705 −0.4886 100 (−0.2,−0.2) 16.5998 16.6431 0.2608 12.9543 12.9942 0.3080 10.9114 10.9333 0.2007(−0.2,0.2) 16.6980 16.7461 0.2881 13.2023 13.2410 0.2931 11.2655 11.2825 0.1509(0.2,−0.2) 19.1045 19.1639 0.3109 15.0613 15.1188 0.3818 12.7573 12.7743 0.1333(0.2,0.2) 19.1784 19.2434 0.3389 15.2528 15.3104 0.3776 13.0327 13.0458 0.1005 100000 (−0.2,−0.2) 19.3362 19.3326 −0.0186 14.7048 14.7625 0.3924 12.2723 12.3550 0.6739(−0.2,0.2) 19.4557 19.4504 −0.0272 14.9833 15.0393 0.3737 12.6625 12.7451 0.6523(0.2,−0.2) 23.7043 23.6966 −0.0325 18.0657 18.1349 0.3830 15.1102 15.2108 0.6658(0.2,0.2) 23.8019 23.7928 −0.0382 18.2930 18.3609 0.3712 15.4287 15.5293 0.6520

Table2 Comparison of dimensionless fundamental frequencies obtained from Eqs.(39)and(59)for clamped–hinged beam with A4=0.0052

A3 (A1,A2) A5=0.0 A5=0.05 A5=0.1 Eq.(39) Eq.(59) Relative error(%)Eq.(39) Eq.(59) Relative error(%)Eq.(39) Eq.(59) Relative error(%)56.4706 (−0.2,−0.2) 11.4378 11.4310 −0.0595 9.1067 9.1017 −0.0549 7.7518 7.7634 0.1496(−0.2,0.2) 11.5789 11.5714 −0.0648 9.3395 9.3306 −0.0953 8.0527 8.0598 0.0882(0.2,−0.2) 13.1868 13.1751 −0.0887 10.5714 10.5552 −0.1532 9.0385 9.0439 0.0597(0.2,0.2) 13.2959 13.2832 −0.0955 10.7519 10.7319 −0.1860 9.2722 9.2733 0.0119 100 (−0.2,−0.2) 12.1585 12.1645 0.0493 9.6281 9.6371 0.0935 8.1763 8.1985 0.2715(−0.2,0.2) 12.3076 12.3133 0.0463 9.8725 9.8786 0.0618 8.4914 8.5104 0.2238(0.2,−0.2) 14.3308 14.3562 0.1772 11.4157 11.4291 0.1174 9.7316 9.7584 0.2754(0.2,0.2) 14.4482 14.4734 0.1744 11.6090 11.6197 0.0922 9.9812 10.0049 0.2374 100000 (−0.2,−0.2) 13.3121 13.2948 −0.1300 10.4375 10.4503 0.1226 8.8291 8.8573 0.3194(−0.2,0.2) 13.4746 13.4562 −0.1366 10.6993 10.7103 0.1028 9.1649 9.1920 0.2957(0.2,−0.2) 16.3365 16.3145 −0.1347 12.8368 12.8518 0.1169 10.8825 10.9167 0.3143(0.2,0.2) 16.4692 16.4463 −0.1390 13.0504 13.0640 0.1042 11.1567 11.1899 0.2976

Fig.5 Comparison of fundamental frequency obtained from approximate and exact equations for a Timoshenko beam with A4=0.0052 and A5=0.05

6 Conclusions

In this study,the governing equation of a nonlocal Timoshenko beam with surface effects is derived by considering both the nonlocal effect and surface effects.Three types of boundary conditions,i.e.,hinged–hinged,clamped–clamped and clamped–hinged ends,are examined.Based on the established governing equation,an exact and explicit natural frequency equation for the hinged–hinged beam is determined,which directly illustrates the relationship between the natural frequency and various parameters.By neglecting the effects of shear deformation,rotatory inertia and nonlocal elasticity,the proposed frequency equation becomes the equation for a local Euler–Bernoulli beam with surface effects[3].By neglecting the nonlocal effect and surface effects,the equation reduces to the one for a local Timoshenko beam[19].Furthermore by neglecting the effects of shear deformation and rotatory inertia,the equation recovers the classical one for a local Euler–Bernoulli beam.For the clamped–clamped and clamped–hinged beams,the characteristic frequency equations cannot be solved analytically due to their transcendental nature.Hence,the Fredholm integral equation approach[22–24]in conjunction with a curve fitting method is adopted to determine the approximate fundamental frequency equations.It is noted that the approximate equations can accurately predict the fundamental frequency for a wide range of parameters in view of the largest relative error being 0.72%and 0.32%,respectively,when compared with the exact values.In sum,explicit frequency equations for hinged–hinged,clamped–clamped and clamped–hinged beams are put forward to directly characterize the dependence of the natural frequency on various parameters,providing a more convenient means relative to numerical computation.By some extension,the proposed equations in this study can be used for many other materials and structures(e.g.,nanotubes,nanowires and nanorods).

Fig.6 Comparison of fundamental frequency obtained from approximate and exact equations for a Timoshenko beam with A3=56.4706 and A5=0.05

Fig.7 Comparison of fundamental frequency obtained from approximate and exact equations for a Timoshenko beam with A3=56.4706 and A4=0.0052

Acknowledgements The authors would like to thank the School of Civil and Environmental Engineering at Nanyang Technological University,Singapore for kindly supporting this research topic.

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