# A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods

Authors: Tamer Ahmed El-Sayed; Said Hamed Farghaly
Journal: Shock and Vibration
 (2017)

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## Abstract

The exact solution for multistepped Timoshenko beam is derived using a set of fundamental solutions. This set of solutions is derived to normalize the solution at the origin of the coordinates. The start, end, and intermediate boundary conditions involve concentrated masses and linear and rotational elastic supports. The beam start, end, and intermediate equations are assembled using the present normalized transfer matrix (NTM). The advantage of this method is that it is quicker than the standard method because the size of the complete system coefficient matrix is 4 × 4. In addition, during the assembly of this matrix, there are no inverse matrix steps required. The validity of this method is tested by comparing the results of the current method with the literature. Then the validity of the exact stepped analysis is checked using experimental and FE(3D) methods. The experimental results for stepped beams with single step and two steps, for sixteen different test samples, are in excellent agreement with those of the three-dimensional finite element FE(3D). The comparison between the NTM method and the finite element method results shows that the modal percentage deviation is increased when a beam step location coincides with a peak point in the mode shape. Meanwhile, the deviation decreases when a beam step location coincides with a straight portion in the mode shape.


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## Body

## 1. Introduction

Stepped beam-like structure plays an important role in the construction of mechanical and civil engineering systems. Flexural vibrations were first investigated by Euler-Bernoulli in the eighteenth century. The rotary inertia effect was considered by Rayleigh [1]. Almost 95 years ago, Timoshenko introduced a correction for the beam theory to include the shear deformation effect [2]. The effect of rotary inertia and shear deformations on the beam natural frequencies is small at the lower normal modes and large at the higher normal modes. Cowper [3] derived formulae for the precise evaluation of the shear coefficient for rectangular and round cross sections as a function of Poisson’s ratio ν.The free vibrations of beams with discontinuities can be solved using either exact or approximate solution. The exact methods include the derivation of the transcendental eigenvalue equations in order to evaluate the beam natural frequencies. In the case of relatively simple problems closed form solutions for the eigenvalue problem were obtained [4, 5].For complicated problems, the beam eigenvalues are obtained by the decomposition of the domain. Numerical assembly technique (NAT) is one of the common methods used for the evaluation of the eigenvalue problem for beams with multiple discontinuity [6–8]. In this method, the size of the frequency equation determinant is 4n×4n for beam with n segments. Dynamic stiffness matrix is one of methods that is similar to the finite element method in assembling of the elements but with exact element rather than approximate element [9, 10]. The frequency equation determinant size for n segments is 2n+2×2n+2. The Laplace transformation method is used to obtain a solution for a Timoshenko beam mounted on elastic foundation with several combinations of discrete in-span attachments and with several combinations of attachments at the boundaries [11]. The attachments include translation and rotational springs, masses, and undamped single degree of freedom system. The characteristics equation for beam with n segments is 2n×2n. The transfer matrix method (TMM) is one of the favorable methods in the analysis of multispan beams. Many researchers used it to derive the frequency equation of a complicated beam system [12–14]. The advantage of this method is that, for n segments beam, the size of the frequency equation determinant is 4×4 which reduces the computational time. On the other hand, during the formulation of the beam frequency equation n-1 inverse matrix steps are required to form the final system transcendental eigenvalue problem. The use of linearly independent fundamental set of solution in solving the buckling and free vibrations of nonuniform rods was introduced by Li [15] and Li et al. [16]. This method enables obtaining the closed form solution of a multispan beam. A set of fundamental solutions which suit the analysis of single-span Timoshenko beams was introduced [17].On the other hand, there are numerous approximate methods to approximate the eigenvalue problem for the transverse vibrations of beams [21–23]. Finite element method is one of the most dominant methods for solving the free vibration of beams. The effect of step ratios and eccentricity on the free vibration of arbitrarily beam was investigated by Ju et al. [21]. The lowest three natural frequencies of a multistep up and down cantilever beam using a global Rayleigh-Ritz formulation, component modal analyses (CMA), ANSYS®, and experimental are evaluated [24]. Adomian decomposition method (ADM) was used to obtain the effect of step ratio and step location on the beam natural frequencies [25, 26]. The free and forced vibrations of beams with either single- or multiple-step changes using the composite element method (CEM) were introduced by Lu et al. [27]. The accuracy and convergence of CEM were compared with existing theoretical and experimental results. Differential transformation method (DTM) was applied in order to analyze the natural frequencies for different geometrically and material parameters stepped Bernoulli-Euler beam [28]. The differential quadrature element method (DQEM) was proposed to analyze the free vibration problem of beams with any discontinuities in cross-section [20]. Discrete singular convolution (DSC) was proposed for solving the free vibration analysis of stepped beams [29]. The solution for multistep Timoshenko beam using both numerical assembly technique and differential transformation method is proposed by Yesilce [19]. Experimental measurements were considered as a good tool for the validation of the analytical results.The exact free vibration of two-span Timoshenko stepped beams has been investigated by Gutierrez et al. [18] and Rossi et al. [32]. Farghaly [33] derived the exact solution for four-span Timoshenko beam with attachments. Yesilce [19] investigated the free vibration of stepped beams using exact numerical assembly technique (NAT) and using approximate differential transformation method. Recently, Farghaly and El-Sayed [6] drive the exact solution for the lateral vibration of Timoshenko beam with generalized start, end, and intermediate conditions using numerical assembly technique (NAT).The exact free vibration of a mechanical system composed of two elastic Timoshenko segments carried on an intermediate eccentric rigid body or on elastic supports was introduced by Farghaly and El-Sayed [34]. Their analysis was based on both analytical and experimental methods. They claimed a good agreement between the analytical and experimental results. Experimental setup using electromagnetic-acoustic transducer (EMAT) was introduced by Díaz-De-Anda et al. [35]. They compared the experimental results with those obtained theoretically using Timoshenko beam theory (TBT) with one and two shear coefficients. The flexural frequencies and amplitudes for cylindrical and rectangular Timoshenko beams were examined experimentally [35]. They found that the experimental results coincide very well with theoretical predictions. The transverse vibration of Bernoulli-Euler beams with discontinuous geometry and elastic support was investigated experimentally and analytically [36].During the last decades, many literatures were focused on the problem of free and forced vibration analysis of Timoshenko beam and the accuracy of the natural frequency predictions. To the authors’ knowledge, there is not enough research that has tackled the experimental modal frequencies of stepped thick beams, computationally and experimentally. Therefore, the main aim of this work is to investigate the results of the modal frequencies for such beams using analytical, experimental, and the three-dimensional finite element FE(3D). An analytical analysis is proposed which is based on the derivation of a set of fundamental solutions that suits the analysis of Timoshenko beams. This set of solutions is used to modify the TMM to include no inverse matrix procedure which may be called normalized transfer matrix method (NTM). The comparison between the experimental NTM and FE(3D) is done for selected single-step and two-step application models. The percentage deviations between NTM and FE(3D) are investigated. The results show that the finite element results are very close to the experimental results. The study includes the effect of increasing the step ratiod-, step location parameter (μ), and the length ratio. Finally, the capability of the present analysis to solve the free vibration of tapered beams has been investigated.


## 2. Mathematical Model

The mathematical model for beam with multiple-stepped sections is shown in Figure1. The total length of the beam is L. The beam model is divided into n segments. The beam has n+1 stations as shown in Figure 1. The station numbering corresponding to the start, intermediate, and end location is represented by 1,i+1,n+1, respectively. At each station, there are linear and rotational elastic supports and concentrated mass with mass moment of inertia. As shown in Figure 1, the beam segments are described by their material and cross-sectional properties ρi,Ei,Gi,Ai and Ii(i=1,2,…,n), which are the density, Young’s modulus of elasticity, rigidity modulus, cross-sectional area, and second moment of inertia, respectively. In this section, the frequency equation of the model is driven using the proposed normalized transfer matrix method. Since the current analysis is based on Timoshenko beam theory, the rotary inertia and shear deformations effects are considered. In the current analysis, the analytical solution is subject to the assumptions that the shear strain is assumed constant over the cross-section; therefore, a shear coefficient ki′ is used to compensate this assumption. In addition, the effect of stress concentration at the beam steps is neglected.Figure 1
Stepped multispan model.


### 2.1. Analytical Method and Frequency Equation

The objective of this section is to derive the system frequency equation which represents the model shown in Figure1. Timoshenko differential coupled equations of motion may be written here for ith span as follows:(1)k′GAiyi′′-ψi′xi,t-ρiAiy¨ixi,t=0,EIiψi′′xi,t-ρiIiψ¨ixi,t+k′GAiyi′-ψixi,t=0.Let(2)yi=Yiξiejωt,ψi=Ψiξiejωt,ξi=xiLi,where Yi is the normal function of yi, Ψi is the normal function of ψi, ξi is nondimensional length of each beam span i, and j=-1.Substituting (2) into (1) and omitting the factor ejωt, the following equations can be derived:(3)Yi′′ξi+λi4si2Yiξi-LiΨi′ξi=0,(4)si2LiΨi′′ξi+λi4ri2si2-1LiΨiξi+Yi′ξi=0,where(5a)λi4=ρiAiLi4ω2EiIi,(5b)si2=2ri21+νiki′,(5c)ri2=IiAiLi2.After decoupling the functions Yiξi and LiΨiξi in ((3)-(4)), the decoupled fourth-order differential equations in the nondimensional form can be written as(6)Yi′′′′ξi+αiYi′′ξi+βi2Yiξi=0,(7)LiΨi′′′′ξi+αiLiΨi′′ξi+βi2LiΨiξi=0,where7aαi=λi4ri2+si2,7bβi2=λi4λi4ri2si2-1.The general solution of (6) and (7) can be written, respectively, in the form(8)Yiξi=C1isin⁡aiξi+C2icos⁡aiξi+C3isinh⁡biξi+C4icoshbiξi,LiΨiξi=-δ1iaiC1icos⁡aiξi+δ1iaiC2isin⁡aiξi+δ2ibiC3icosh⁡biξi+δ2ibiC4isinh⁡biξi.Here,(9a)ai2=αi2+αi22-βi2,(9b)bi2=-αi2+αi22-βi2.One can derive the expressions of δ1i and δ2i using (8), together with (3) or (4) in the form(10a)δ1i=si2λi4-ai2,(10b)δ2i=si2λi4+bi2.Here i denotes the ith span, i=1,2,…,n in the case of multispan model.In order to introduce the current analysis, the linearly independent fundamental solutionsSY1iξ,SY2iξ,SY3iξ,SY4iξ and the corresponding SΨ1iξ,SΨ2iξ,SΨ3iξ,SΨ4iξ are derived. In order to simplify the solution of Timoshenko beam, the following dependent functions are defined:(11a)Γiξ=Yi′ξ-LiΨiξ,(11b)SΓ1iξ=SY1i′ξ-SΨ1iξ,(11c)SΓ2iξ=SY2i′ξ-SΨ2iξ,(11d)SΓ3iξ=SY3i′ξ-SΨ3iξ,(11e)SΓ4iξ=SY4i′ξ-SΨ4iξ.The Timoshenko solution will be normalized at the origin of coordinates as follows:(12)SY1i0SΨ1i0SΨ1i′0SΓ1i0SY2i0SΨ2i0SΨ2i′0SΓ2i0SY3i0SΨ3i0SΨ3i′0SΓ3i0SY4i0SΨ4i0SΨ4i′0SΓ4i0=1000010000100001.Substituting the general solution of (8) in each raw in (12) we get the following set of fundamental solutions:(13a)SY1iξi=1δ1i-δ2i-δ2icos⁡aiξi+δ1icosh⁡biξi,(13b)SY2iξi=1δ2iai2+δ1ibi2aiδ2i-bi2sin⁡aiξi+biδ1i+ai2sinh⁡biξi,(13c)SY3iξi=1δ1i-δ2icos⁡aiξi-cosh⁡biξi,(13d)SY4iξi=1δ2iai2+δ1ibi2δ2iaisin⁡aiξi+δ1ibisinh⁡biξi,(13e)SΨ1iξi=δ1iδ2iδ1i-δ2i-1aisin⁡aiξi+1bisinh⁡biξi,(13f)SΨ2iξi=1δ2iai2+δ1ibi2-δ1iδ2i-bi2cos⁡aiξi+δ2iδ1i+ai2cosh⁡biξi,(13g)SΨ3iξi=1δ1i-δ2iδ1iaisin⁡aiξi-δ2ibisinh⁡biξi,(13h)SΨ4iξi=δ1iδ2iδ2iai2+δ1ibi2-cos⁡aiξi+cosh⁡biξi.The general solution of the beam can be presented in terms of the set of fundamental solutions as(14)Yiξi=yi0SY1iξi+ψi0SY2iξi+ψi0′SY3iξi+γi0SY4iξi,LiΨiξi=yi0SΨ1iξi+ψi0SΨ2iξ+ψi0′SΨ3iξ+γi0SΨ4iξi,Γiξi=yi0SΓ1iξi+ψi0SΓ2iξi+ψi0′SΓ3iξi+γi0SΓ4iξi,where yi0=Yi0, ψi0=LiΨi0, ψi0′=LiΨi′0, and γi0=Γi0.The beam start boundary conditions at the point of attachment 1 can be presented in nondimensional form as(15)L1Ψ1′0+J-1λ14-Φ1L1Ψ10=0,Γ10+s12m-1λ14-Z1Y10=0,where(16a)J-1=J1ρ1A1L13,(16b)λ14=ρ1A1L14ω2E1I1,(16c)Z1=k1L13E1I1,(16d)Φ1=ϕ1L1E1I1,(16e)m-1=m1ρ1A1L1,where J1 is the mass moment of inertia at station 1, k1 and ϕ1 are the linear and rotational elastic supports at station 1, respectively, and m1 is the concentrated mass at station 1; see Figure 1 for details.Substituting the solutions presented in (13a), (13b), (13c), (13d), (13e), (13f), (13g), (13h)-(14) into (15), the following equations are obtained:(17)J-1λ14-Φ1ψ10+ψ10′=0,s12m-1λ14-Z1y10+γ10=0.The start boundary conditions in (17) can be presented in matrix form as(18)0J-1λ14-ϕ110s12m-1λ14-Z1001y10ψ10ψ10′γ10=0.This equation can be simply written as(19)UsΔ1=0,where(20)Δ1=y10ψ10ψ10′γ10t,where the superscript t indicates vector transpose.At stationn+1, the beam end boundary conditions can be written in the nondimensional form as(21)Ψn′1+Φn+1-λn4J-n+1Ψn1=0,Γn1+Zn+1-m-n+1λn4sn2Yn1=0,where(22a)J-n+1=Jn+1ρnAnLn3,(22b)λn4=ρnAnLn4ω2EnIn,(22c)Zn+1=kn+1Ln3EnIn,(22d)Φn+1=ϕn+1LnEnIn,(22e)m-n+1=mn+1ρnAnLn,where Jn+1 is the mass moment of inertia at station n+1, kn+1 and ϕn+1 are the linear and rotational elastic supports at station n+1, respectively, and mn+1 is the concentrated mass at station n+1.Substituting the solutions in (13a), (13b), (13c), (13d), (13e), (13f), (13g), (13h)-(14) into (21), the following equations are obtained:(23)yn0SΨ1n′1+Φn+1-λn4J-n+1SΨ1n+ψn0SΨ2n′1+Φn+1-λn4J-n+1SΨ2n+ψn0′SΨ3n′1+Φn+1-λn4J-n+1SΨ3n+γn0SΨ4n′1+Φn+1-λn4J-n+1SΨ4n=0,yn0SΓ1n1+sn2Zn+1-m-n+1λn4SY1n1+ψn0SΓ2n1+sn2Zn+1-m-n+1λn4SY2n1+ψn0′SΓ3n1+sn2Zn+1-m-n+1λn4SY3n1+γn0SΓ4n1+sn2Zn+1-m-n+1λn4SY4n1=0.Equation (23) can be written in the matrix form as(24)SΨ1n′1+Φn+1-λn4J-n+1SΨ1nSΨ2n′1+Φn+1-λn4J-n+1SΨ2nSΨ3n′1+Φn+1-λn4J-n+1SΨ3nSΨ4n′1+Φn+1-λn4J-n+1SΨ4nSΓ1n1+sn2Zn+1-m-n+1λn4SY1n1SΓ2n1+sn2Zn+1-m-n+1λn4SY2n1SΓ3n1+sn2Zn+1-m-n+1λn4SY3n1SΓ4n1+sn2Zn+1-m-n+1λn4SY4n1yn0ψn0ψn0′γn0=0.This equation can be simply written as(25)UE2×4Δn=0,where(26)Δn=yn0ψn0ψn0′γn0t.The beam intermediate continuity conditions can be presented in nondimensional form as(27)Yi1=Yi+10,Li+1iLiΨi1=Li+1Ψi+10,EIii+1∗Li+1i2Ψi′1+Φi+1-J-i+1λi4Ψi1=Li+1Ψi+1′0,k′GAii+1Li+1iΓi1+si2Zi+1-mi+1λi4Yi1=Γi+10,where(28a)J-i+1=Ji+1ρiAiLi3,(28b)λi4=ρiAiLi4ω2EiIi,(28c)Zi+1=ki+1Li3EiIi,(28d)Φi+1=ϕi+1LiEiIi,(28e)m-i+1=mi+1ρiAiLi,where Ji+1 is the mass moment of inertia at station i+1, ki+1 and ϕi+1 are the linear and rotational stiffness at station (i+1), respectively, and mi+1 is the concentrated mass at station i+1.Substituting the solutions in (13a), (13b), (13c), (13d), (13e), (13f), (13g), (13h)-(14) into (27), we get the following equations:(29)yi+10=yi0SY1i1+ψi0SY2i1+ψi0′SY3i1+γi0SY4i1,ψi+10=Li+1iyi0SΨ1i1+ψi0SΨ2i1+ψi0′SΨ3i1+γi0SΨ4i1,ψi+10′=EIii+1∗Li+1i2yi0SΨ1i′1+Φi+1-J-i+1λi4SΨ1i1+ψi0SΨ2i′1+Φi+1-J-i+1λi4SΨ2i1+ψi0′SΨ3i′1+Φi+1-J-i+1λi4SΨ3i1+γi0SΨ4i′1+Φi+1-J-i+1λi4SΨ4i1,γi+10=k′GAii+1Li+1iyi0SΓ1i1+si2Zi+1-mi+1λi4SY1i1+ψi0SΓ2i1+si2Zi+1-mi+1λi4SY2i1+ψi0′SΓ3i1+si2Zi+1-mi+1λi4SY3i1+γi0SΓ4i1+si2Zi+1-mi+1λi4SY4i1.Equation (29) can be written in matrix form as(30)SY1i1SY2i1SY3i1SY4i1Li+1iSΨ1i1Li+1iSΨ2i1Li+1iSΨ3i1Li+1iSΨ4i1W1iSΨ1i′1+Φi+1-J-i+1λi4SΨ1i1W1iSΨ2i′1+Φi+1-J-i+1λi4SΨ2i1W1iSΨ3i′1+Φi+1-J-i+1λi4SΨ3i1W1iSΨ4i′1+Φi+1-J-i+1λi4SΨ4i1W2iSΓ1i1+si2Zi+1-mi+1λi4SY1i1W2iSΓ2i1+si2Zi+1-mi+1λi4SY2i1W2iSΓ3i1+si2Zi+1-mi+1λi4SY3i1W2iSΓ4i1+si2Zi+1-mi+1λi4SY4i1yi0ψi0ψi0′γi0=yi+10ψi+10ψi+10′γi+10,where(31a)W1i=EIii+1∗Li+1i2,(31b)W2i=k′GAii+1Li+1i.Equation (30) can be presented as(32)Ti4×4∗Δi=Δi+1,where(33a)Δi=yi0ψi0ψi0′γi0t,(33b)Δi+1=yi+10ψi+10ψi+10′γi+10t.From (32), one can find that(34)Δn=Tn-1Δn-1=Tn-1Tn-2⋯T2Δ1.The intermediate spans transfer matrix can be presented as(35)T4×4=Tn-1Tn-2⋯T2;then (34) can be presented as(36)Δn=T4×4Δ1.Substituting (36) into the end condition of (25) results in the following equations:(37)UE2×4T4×4Δ1=0,(38)UIE2×4=UE2×4T4×4,(39)UIE2×4Δ1=0.The general beam equation can be presented using the start boundary condition in (19) and the beam intermediate and end condition in (39) as shown below(40)Utot4×4Δ1=0,where(41)Utot4×4=US2×4UIE2×4.Equating the determinant of Utot by zero results in the system frequency equation. In general, the TMM has advantages over the traditional methods in that the final frequency equation is 4×4 for any number of beam segments. The advantage of the current method NTM over the TMM is significant in the using of tailored solution that is normalized at the origin of coordinates. This type of solution enables the formulation of the system equations without the need to any inverse matrix procedures as shown previously. This reduces the computational time comparing with the TMM.


### 2.2. Finite Element Method

Among the numerical tools, finite element method is considered one of most efficient methods to perform the vibration analysis of mechanical and structural components. In this section, finite element is used to obtain the natural frequencies and mode shapes of uniform and stepped beams. ANSYS finite element commercial package is used to perform the finite element analysis. The analysis is done using three-dimensional (3D) solid element models and SOLID95 elements are used for meshing. Since all the experimentally investigated samples in the current work are round and stepped. The beam cross-section is free meshed using 87 SOLID95 elements for smaller cross-section and 171 elements for the larger cross-section. This mesh is then extruded using 40 elements along the length of the beam. The total number of the element is ranging from 3480 (40 × 87) to 6840 (40 × 171) elements based on the location of the step; see Figure2. Modal analysis module is used in this analysis and Block Lanczos method is used for the mode extraction method. The finite element model FE(3D) results are compared with those obtained experimentally and analytically.Figure 2
Finite element 3D mesh.


## 2.1. Analytical Method and Frequency Equation

The objective of this section is to derive the system frequency equation which represents the model shown in Figure1. Timoshenko differential coupled equations of motion may be written here for ith span as follows:(1)k′GAiyi′′-ψi′xi,t-ρiAiy¨ixi,t=0,EIiψi′′xi,t-ρiIiψ¨ixi,t+k′GAiyi′-ψixi,t=0.Let(2)yi=Yiξiejωt,ψi=Ψiξiejωt,ξi=xiLi,where Yi is the normal function of yi, Ψi is the normal function of ψi, ξi is nondimensional length of each beam span i, and j=-1.Substituting (2) into (1) and omitting the factor ejωt, the following equations can be derived:(3)Yi′′ξi+λi4si2Yiξi-LiΨi′ξi=0,(4)si2LiΨi′′ξi+λi4ri2si2-1LiΨiξi+Yi′ξi=0,where(5a)λi4=ρiAiLi4ω2EiIi,(5b)si2=2ri21+νiki′,(5c)ri2=IiAiLi2.After decoupling the functions Yiξi and LiΨiξi in ((3)-(4)), the decoupled fourth-order differential equations in the nondimensional form can be written as(6)Yi′′′′ξi+αiYi′′ξi+βi2Yiξi=0,(7)LiΨi′′′′ξi+αiLiΨi′′ξi+βi2LiΨiξi=0,where7aαi=λi4ri2+si2,7bβi2=λi4λi4ri2si2-1.The general solution of (6) and (7) can be written, respectively, in the form(8)Yiξi=C1isin⁡aiξi+C2icos⁡aiξi+C3isinh⁡biξi+C4icoshbiξi,LiΨiξi=-δ1iaiC1icos⁡aiξi+δ1iaiC2isin⁡aiξi+δ2ibiC3icosh⁡biξi+δ2ibiC4isinh⁡biξi.Here,(9a)ai2=αi2+αi22-βi2,(9b)bi2=-αi2+αi22-βi2.One can derive the expressions of δ1i and δ2i using (8), together with (3) or (4) in the form(10a)δ1i=si2λi4-ai2,(10b)δ2i=si2λi4+bi2.Here i denotes the ith span, i=1,2,…,n in the case of multispan model.In order to introduce the current analysis, the linearly independent fundamental solutionsSY1iξ,SY2iξ,SY3iξ,SY4iξ and the corresponding SΨ1iξ,SΨ2iξ,SΨ3iξ,SΨ4iξ are derived. In order to simplify the solution of Timoshenko beam, the following dependent functions are defined:(11a)Γiξ=Yi′ξ-LiΨiξ,(11b)SΓ1iξ=SY1i′ξ-SΨ1iξ,(11c)SΓ2iξ=SY2i′ξ-SΨ2iξ,(11d)SΓ3iξ=SY3i′ξ-SΨ3iξ,(11e)SΓ4iξ=SY4i′ξ-SΨ4iξ.The Timoshenko solution will be normalized at the origin of coordinates as follows:(12)SY1i0SΨ1i0SΨ1i′0SΓ1i0SY2i0SΨ2i0SΨ2i′0SΓ2i0SY3i0SΨ3i0SΨ3i′0SΓ3i0SY4i0SΨ4i0SΨ4i′0SΓ4i0=1000010000100001.Substituting the general solution of (8) in each raw in (12) we get the following set of fundamental solutions:(13a)SY1iξi=1δ1i-δ2i-δ2icos⁡aiξi+δ1icosh⁡biξi,(13b)SY2iξi=1δ2iai2+δ1ibi2aiδ2i-bi2sin⁡aiξi+biδ1i+ai2sinh⁡biξi,(13c)SY3iξi=1δ1i-δ2icos⁡aiξi-cosh⁡biξi,(13d)SY4iξi=1δ2iai2+δ1ibi2δ2iaisin⁡aiξi+δ1ibisinh⁡biξi,(13e)SΨ1iξi=δ1iδ2iδ1i-δ2i-1aisin⁡aiξi+1bisinh⁡biξi,(13f)SΨ2iξi=1δ2iai2+δ1ibi2-δ1iδ2i-bi2cos⁡aiξi+δ2iδ1i+ai2cosh⁡biξi,(13g)SΨ3iξi=1δ1i-δ2iδ1iaisin⁡aiξi-δ2ibisinh⁡biξi,(13h)SΨ4iξi=δ1iδ2iδ2iai2+δ1ibi2-cos⁡aiξi+cosh⁡biξi.The general solution of the beam can be presented in terms of the set of fundamental solutions as(14)Yiξi=yi0SY1iξi+ψi0SY2iξi+ψi0′SY3iξi+γi0SY4iξi,LiΨiξi=yi0SΨ1iξi+ψi0SΨ2iξ+ψi0′SΨ3iξ+γi0SΨ4iξi,Γiξi=yi0SΓ1iξi+ψi0SΓ2iξi+ψi0′SΓ3iξi+γi0SΓ4iξi,where yi0=Yi0, ψi0=LiΨi0, ψi0′=LiΨi′0, and γi0=Γi0.The beam start boundary conditions at the point of attachment 1 can be presented in nondimensional form as(15)L1Ψ1′0+J-1λ14-Φ1L1Ψ10=0,Γ10+s12m-1λ14-Z1Y10=0,where(16a)J-1=J1ρ1A1L13,(16b)λ14=ρ1A1L14ω2E1I1,(16c)Z1=k1L13E1I1,(16d)Φ1=ϕ1L1E1I1,(16e)m-1=m1ρ1A1L1,where J1 is the mass moment of inertia at station 1, k1 and ϕ1 are the linear and rotational elastic supports at station 1, respectively, and m1 is the concentrated mass at station 1; see Figure 1 for details.Substituting the solutions presented in (13a), (13b), (13c), (13d), (13e), (13f), (13g), (13h)-(14) into (15), the following equations are obtained:(17)J-1λ14-Φ1ψ10+ψ10′=0,s12m-1λ14-Z1y10+γ10=0.The start boundary conditions in (17) can be presented in matrix form as(18)0J-1λ14-ϕ110s12m-1λ14-Z1001y10ψ10ψ10′γ10=0.This equation can be simply written as(19)UsΔ1=0,where(20)Δ1=y10ψ10ψ10′γ10t,where the superscript t indicates vector transpose.At stationn+1, the beam end boundary conditions can be written in the nondimensional form as(21)Ψn′1+Φn+1-λn4J-n+1Ψn1=0,Γn1+Zn+1-m-n+1λn4sn2Yn1=0,where(22a)J-n+1=Jn+1ρnAnLn3,(22b)λn4=ρnAnLn4ω2EnIn,(22c)Zn+1=kn+1Ln3EnIn,(22d)Φn+1=ϕn+1LnEnIn,(22e)m-n+1=mn+1ρnAnLn,where Jn+1 is the mass moment of inertia at station n+1, kn+1 and ϕn+1 are the linear and rotational elastic supports at station n+1, respectively, and mn+1 is the concentrated mass at station n+1.Substituting the solutions in (13a), (13b), (13c), (13d), (13e), (13f), (13g), (13h)-(14) into (21), the following equations are obtained:(23)yn0SΨ1n′1+Φn+1-λn4J-n+1SΨ1n+ψn0SΨ2n′1+Φn+1-λn4J-n+1SΨ2n+ψn0′SΨ3n′1+Φn+1-λn4J-n+1SΨ3n+γn0SΨ4n′1+Φn+1-λn4J-n+1SΨ4n=0,yn0SΓ1n1+sn2Zn+1-m-n+1λn4SY1n1+ψn0SΓ2n1+sn2Zn+1-m-n+1λn4SY2n1+ψn0′SΓ3n1+sn2Zn+1-m-n+1λn4SY3n1+γn0SΓ4n1+sn2Zn+1-m-n+1λn4SY4n1=0.Equation (23) can be written in the matrix form as(24)SΨ1n′1+Φn+1-λn4J-n+1SΨ1nSΨ2n′1+Φn+1-λn4J-n+1SΨ2nSΨ3n′1+Φn+1-λn4J-n+1SΨ3nSΨ4n′1+Φn+1-λn4J-n+1SΨ4nSΓ1n1+sn2Zn+1-m-n+1λn4SY1n1SΓ2n1+sn2Zn+1-m-n+1λn4SY2n1SΓ3n1+sn2Zn+1-m-n+1λn4SY3n1SΓ4n1+sn2Zn+1-m-n+1λn4SY4n1yn0ψn0ψn0′γn0=0.This equation can be simply written as(25)UE2×4Δn=0,where(26)Δn=yn0ψn0ψn0′γn0t.The beam intermediate continuity conditions can be presented in nondimensional form as(27)Yi1=Yi+10,Li+1iLiΨi1=Li+1Ψi+10,EIii+1∗Li+1i2Ψi′1+Φi+1-J-i+1λi4Ψi1=Li+1Ψi+1′0,k′GAii+1Li+1iΓi1+si2Zi+1-mi+1λi4Yi1=Γi+10,where(28a)J-i+1=Ji+1ρiAiLi3,(28b)λi4=ρiAiLi4ω2EiIi,(28c)Zi+1=ki+1Li3EiIi,(28d)Φi+1=ϕi+1LiEiIi,(28e)m-i+1=mi+1ρiAiLi,where Ji+1 is the mass moment of inertia at station i+1, ki+1 and ϕi+1 are the linear and rotational stiffness at station (i+1), respectively, and mi+1 is the concentrated mass at station i+1.Substituting the solutions in (13a), (13b), (13c), (13d), (13e), (13f), (13g), (13h)-(14) into (27), we get the following equations:(29)yi+10=yi0SY1i1+ψi0SY2i1+ψi0′SY3i1+γi0SY4i1,ψi+10=Li+1iyi0SΨ1i1+ψi0SΨ2i1+ψi0′SΨ3i1+γi0SΨ4i1,ψi+10′=EIii+1∗Li+1i2yi0SΨ1i′1+Φi+1-J-i+1λi4SΨ1i1+ψi0SΨ2i′1+Φi+1-J-i+1λi4SΨ2i1+ψi0′SΨ3i′1+Φi+1-J-i+1λi4SΨ3i1+γi0SΨ4i′1+Φi+1-J-i+1λi4SΨ4i1,γi+10=k′GAii+1Li+1iyi0SΓ1i1+si2Zi+1-mi+1λi4SY1i1+ψi0SΓ2i1+si2Zi+1-mi+1λi4SY2i1+ψi0′SΓ3i1+si2Zi+1-mi+1λi4SY3i1+γi0SΓ4i1+si2Zi+1-mi+1λi4SY4i1.Equation (29) can be written in matrix form as(30)SY1i1SY2i1SY3i1SY4i1Li+1iSΨ1i1Li+1iSΨ2i1Li+1iSΨ3i1Li+1iSΨ4i1W1iSΨ1i′1+Φi+1-J-i+1λi4SΨ1i1W1iSΨ2i′1+Φi+1-J-i+1λi4SΨ2i1W1iSΨ3i′1+Φi+1-J-i+1λi4SΨ3i1W1iSΨ4i′1+Φi+1-J-i+1λi4SΨ4i1W2iSΓ1i1+si2Zi+1-mi+1λi4SY1i1W2iSΓ2i1+si2Zi+1-mi+1λi4SY2i1W2iSΓ3i1+si2Zi+1-mi+1λi4SY3i1W2iSΓ4i1+si2Zi+1-mi+1λi4SY4i1yi0ψi0ψi0′γi0=yi+10ψi+10ψi+10′γi+10,where(31a)W1i=EIii+1∗Li+1i2,(31b)W2i=k′GAii+1Li+1i.Equation (30) can be presented as(32)Ti4×4∗Δi=Δi+1,where(33a)Δi=yi0ψi0ψi0′γi0t,(33b)Δi+1=yi+10ψi+10ψi+10′γi+10t.From (32), one can find that(34)Δn=Tn-1Δn-1=Tn-1Tn-2⋯T2Δ1.The intermediate spans transfer matrix can be presented as(35)T4×4=Tn-1Tn-2⋯T2;then (34) can be presented as(36)Δn=T4×4Δ1.Substituting (36) into the end condition of (25) results in the following equations:(37)UE2×4T4×4Δ1=0,(38)UIE2×4=UE2×4T4×4,(39)UIE2×4Δ1=0.The general beam equation can be presented using the start boundary condition in (19) and the beam intermediate and end condition in (39) as shown below(40)Utot4×4Δ1=0,where(41)Utot4×4=US2×4UIE2×4.Equating the determinant of Utot by zero results in the system frequency equation. In general, the TMM has advantages over the traditional methods in that the final frequency equation is 4×4 for any number of beam segments. The advantage of the current method NTM over the TMM is significant in the using of tailored solution that is normalized at the origin of coordinates. This type of solution enables the formulation of the system equations without the need to any inverse matrix procedures as shown previously. This reduces the computational time comparing with the TMM.


## 2.2. Finite Element Method

Among the numerical tools, finite element method is considered one of most efficient methods to perform the vibration analysis of mechanical and structural components. In this section, finite element is used to obtain the natural frequencies and mode shapes of uniform and stepped beams. ANSYS finite element commercial package is used to perform the finite element analysis. The analysis is done using three-dimensional (3D) solid element models and SOLID95 elements are used for meshing. Since all the experimentally investigated samples in the current work are round and stepped. The beam cross-section is free meshed using 87 SOLID95 elements for smaller cross-section and 171 elements for the larger cross-section. This mesh is then extruded using 40 elements along the length of the beam. The total number of the element is ranging from 3480 (40 × 87) to 6840 (40 × 171) elements based on the location of the step; see Figure2. Modal analysis module is used in this analysis and Block Lanczos method is used for the mode extraction method. The finite element model FE(3D) results are compared with those obtained experimentally and analytically.Figure 2
Finite element 3D mesh.


## 3. Results and Discussion

### 3.1. Verification and Validation of NTM Results

#### 3.1.1. Verification Example 1

In this example, the first five nondimensional natural frequenciesλi∗4=ρ1A1L4ω2/E1I1 of stepped beam are compared with the exact solution presented by Gutierrez et al. [18]; see Figure 3. The model is solved at two different step locations and several values of b21 and h21 as shown in Table 1. Three different values of rotary inertia r12=vs,0.0036 and 0.01 are considered in order to validate the current model in case of Bernoulli-Euler and Timoshenko beams. The value of s12=3.12r12 is considered in order to evaluate the shear deformation [18]. The values of nondimensional linear and rotational elastic supports stiffness at start and the end are Z1=10,Φ1=vl, Z3=vs,Φ3=vl, respectively. The results of Table 1 show that the present NTM results are in good agreement with the exact solution presented by Gutierrez et al. [18].Table 1
The first five natural frequenciesλi∗2 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation ϕ1∗=vl and elastically restrained in translation Z1∗=10 in comparison with [18].
r
1
2
μ
b
21
h
21
λ
1
∗
2
λ
2
∗
2
λ
3
∗
2
λ
4
∗
2
λ
5
∗
2
10
-
7
[18]
0.25
1.0
0.8
3.010
9.696
34.010
74.430
132.341
Present NTM
3.0098
9.6956
34.0101
74.4300
132.3410
[18]
0.8
0.8
3.249
9.664
34.315
74.645
131.803
Present NTM
3.2490
9.6638
34.3151
74.6445
131.7507
[18]
0.8
0.6
3.385
8.117
28.630
60.997
103.439
Present NTM
3.3851
8.1171
28.6302
60.9947
103.4075
[18]
0.50
1.0
0.8
2.958
10.046
34.993
80.145
139.666
Present NTM
2.9579
10.0460
34.9931
80.1455
139.6667
[18]
0.8
0.8
3.124
10.165
34.688
80.570
139.175
Present NTM
3.1240
10.1655
34.6882
80.5707
139.1759
[18]
0.8
0.6
3.284
9.499
28.501
70.123
118.482
Present NTM
3.2841
9.4989
28.5006
70.1232
118.4830
0.0036
[18]
0.25
1.0
0.8
3.007
9.668
33.571
72.393
126.111
Present NTM
3.0066
9.6678
33.5711
72.3926
126.1107
[18]
0.8
0.8
3.245
9.635
33.862
72.607
125.617
Present NTM
3.2454
9.6354
33.8621
72.6068
125.6169
[18]
0.8
0.6
3.381
8.103
28.364
59.839
100.245
Present NTM
3.3812
8.1031
28.3643
59.8385
100.2447
[18]
0.50
1.0
0.8
2.946
9.919
33.205
71.469
116.725
Present NTM
2.9462
9.9192
33.2046
71.4693
116.7245
[18]
0.8
0.8
3.112
10.030
32.926
71.741
116.362
Present NTM
3.1120
10.0304
32.9259
71.7415
116.3618
[18]
0.8
0.6
3.272
9.391
27.449
64.020
102.697
Present NTM
3.2721
9.3908
27.4490
64.0200
102.6966
0.01
[18]
0.25
1.0
0.8
3.001
9.619
32.841
69.229
117.238
Present NTM
3.0009
9.6192
32.8407
69.2292
117.2380
[18]
0.8
0.8
3.239
9.586
33.109
69.444
116.798
Present NTM
3.2390
9.5858
33.1092
69.4435
116.7981
[18]
0.8
0.6
3.374
8.078
27.914
57.970
95.370
Present NTM
3.3744
8.0784
27.9139
57.9705
95.3698
[18]
0.50
1.0
0.8
2.926
9.710
30.711
61.768
95.623
Present NTM
2.9255
9.7104
30.7111
61.7680
95.6230
[18]
0.8
0.8
3.019
9.809
30.464
61.912
95.326
Present NTM
3.0908
9.8086
30.4643
61.9117
95.3257
[18]
0.8
0.6
3.251
9.211
25.893
56.578
86.647
Present NTM
3.2511
9.2111
25.8933
56.5776
86.6473Figure 3
Verification Example 1 [18], two-span stepped beam.


#### 3.1.2. Verification Example 2

In this example, Timoshenko beam with three-step round cross-section presented in [19] is investigated; see Figure 4. An intermediate lumped mass of m3∗=m3/ρ1A1L=1 is located at a distance of 750 mm from point 1. The input data for this example is listed in the caption of Figure 4. Table 2 shows the results of the first five natural frequencies ωi in (rad/sec) for pinned-pinned, free-clamped, clamped-free, clamped-pinned, and clamped-clamped configurations. The results of reference [19] are calculated using two methods. The first method is numerical assembly technique (NAT) and this method is the exact solution. The second method is differential transformation method and this method is approximate method. The pinned-pinned and free-clamped boundary conditions results are found in [19]. The results of the current analysis are in good agreement with the results presented in [19]. As can be seen from Table 2, the rotary inertia and shear deformation reduce the modal frequency especially for higher modes. Comparing the results of the clamped-free beam with the results of the free-clamped beam shows that the natural frequency results of the clamped-free beam are lower than that of free-clamped beam. This may be explained by the fact that fixing the beam from the thinner span results in lowering of the beam stiffness.Table 2
Comparison of the first five natural frequencies, in rad/sec, using the present NTM results with those obtained in [19] for the example shown in Figure 4.
ω
n  (rad/sec)
Method
Bernoulli-Eulerr1∗2=0
Timoshenkor1∗2≠0
Pinned-pinned(P-P)
ω
1
NAT [19]
319.4341
316.5288
NTM
319.4340
316.4855
ω
2
NAT [19]
1853.3864
1789.4207
NTM
1853.3864
1784.8723
ω
3
NAT [19]
4110.1341
3825.8438
NTM
4110.1341
3836.5347
ω
4
NAT [19]
7709.5714
6642.9094
NTM
7709.5712
6639.2170
ω
5
NAT [19]
11621.7699
9886.6243
NTM
11621.7698
9918.8328
Free-clamped(F-C)
ω
1
NAT [19]
371.5354
367.9440
NTM
371.5354
367.1487
ω
2
NAT [19]
1243.9063
1216.6220
NTM
1243.9065
1211.5531
ω
3
NAT [19]
3082.0846
2827.0352
NTM
3082.0845
2849.1465
ω
4
NAT [19]
5541.1410
4957.9240
NTM
5541.1409
5003.7433
ω
5
NAT [19]
9599.3810
7673.7578
NTM
9599.3810
7751.3336
Clamped-free(C-F)
ω
1
NTM
55.1257
55.0130
ω
2
NTM
616.6018
598.5917
ω
3
NTM
2702.3238
2532.1483
ω
4
NTM
5483.2379
4969.1659
ω
5
NTM
9570.9586
7956.5299
Clamped-pinned(C-P)
ω
1
NTM
480.8915
469.8444
ω
2
NTM
2098.0556
1991.9611
ω
3
NTM
5007.3315
4589.5676
ω
4
NTM
8366.8073
7049.5393
ω
5
NTM
13678.7112
11215.0349
Clamped-clamped(C-C)
ω
1
NTM
836.3527
807.0997
ω
2
NTM
3019.5886
2772.2732
ω
3
NTM
5583.6469
4980.5538
ω
4
NTM
9625.8943
7734.8319
ω
5
NTM
14403.3163
11538.1581Figure 4
Verification Example 2 [19], three-step beam having E=206.9GPa. ρ=7836.9kg/m3, G=79.5769GPa, and k′=0.75.


#### 3.1.3. Verification Example 3

The third verification example is shown in Figure5 with identical dimensions to that used in [5, 20]. The rotary inertia and shear deformation are considered. The shear coefficient k′ is calculated based on [3], keeping Poisson’s’ ratio ν = 0.33. Table 3 shows the first three nonzero free-free (F-F) eigenvalues in Hz and in comparison with [5, 20]. The present results are computed using both numerically (40) and FE(3D) methods. The results of Table 3 show that the present analysis results are very close to the experimental results. The percentage error between the present FE(3D) results and the experimental results of [5] is less than 0.302%. This represents the importance of including the effect of rotary inertia and shear deformation.Table 3
Comparison of the present NTM results with those obtained in [5, 20], for F-F beam shown in Figure 5.
Mode number
DQEM [20]
Exp. [5]
NTM (40)
FE(3D)
% error
I
II
III
IV
(II & IV)
1
292.440
291
290.316
290.120
0.302
2
1181.300
1165
1162.158
1167.100
−0.180
3
1804.100
1771
1767.321
1772.200
−0.068Figure 5
Verification Example 3 [5, 20], single-step (F-F) aluminum beam having, E=71.7GPa, ρ=2830Kg/m3, and ν=0.33.


#### 3.1.4. Verification Example 4

The fourth example is shown in Figure6. It is for a twenty-span uniform beam carrying 19 equally spaced concentrated masses with mi∗=0.1. Several beam start and end conditions are investigated as listed in Table 4. The results are evaluated using the present NTM method and previously published numerical assembly technique NAT [6] method. The computational time required to obtain the first three frequency parameters λ1∗,λ2∗, and λ3∗ using both NAT and NTM methods is calculated and listed in Table 4. Considerable reduction in computational time is observed in all the investigated cases as shown in Table 4.Table 4
Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated massesmi∗=0.1,(r1∗2=s1∗2=vs) in comparison with [6].
Method
Z
1
∗
,
Φ
1
∗
Z
21
∗
,
Φ
21
∗
λ
1
∗
λ
2
∗
λ
3
∗
Computational time (sec)
NAT [6]
v
s
,
v
s
v
s
,
v
s
3.7177
6.1716
8.6392
48.17
Present NTM
v
s
,
v
s
v
s
,
v
s
3.7177
6.1716
8.6392
13.53
NAT [6]
v
l
,
v
s
v
l
,
v
s
2.3871
4.7742
7.1612
34.67
Present NTM
v
l
,
v
s
v
l
,
v
s
2.3871
4.7742
7.1612
12.14
NAT [6]
v
l
,
v
l
v
l
,
v
l
3.5941
5.9671
8.35471
40.40
Present NTM
v
l
,
v
l
v
l
,
v
l
3.5941
5.9671
8.35471
13.47
NAT [6]
10,10
10,10
1.5989
2.5329
4.5969
22.13
Present NTM
10,10
10,10
1.5989
2.5329
4.5969
10.23Figure 6
Verification Example 4, twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general end flexibilities conditions.


### 3.2. Test Samples and Experimental Procedures

In order to measure the natural frequencies of the system under study, the free-free test samples were put in free oscillations by using an instrumented hammer model B  and  K 8202. An accelerometer model B  and  K 4366 is fixed to the shaft in order to capture the vibration signal. The output of the charge amplifier B  and  K 2635 is connected to NI 6216 data acquisition card. This card is connected to the PC and managed by Lab VIEW software. Figure10 shows a photo of the current experimental setup. The used card settings are sample frequency of 20 kHz, sampling time of 3, 5, 8 sec, and the size of samples block to read is 1 k. The time domain data is captured and transformed into frequency domain. The resonant frequencies were obtained by the average of the results of 10 impacts applied in three different locations of the sample. Figures 7–9 show two groups of different stepped test samples, S2 and S3, respectively. All the test samples are manufactured from steel rods of dimensions Ø 80 and Ø 40 mm. The common data for the steel test samples are ρ=7850kg/m3, E=206.8GPa, and k′=0.845.Figure 7
Schematic drawings for the experimental samples Group S2.Figure 8
Schematic drawings for the experimental samples Group S3.Figure 9
Real image for typical experimental test samples.Figure 10
Test setup.Two-span twelve test samples are shown in group S2, namely, S2-40/40, S2-40/100, and S2-40/140, shown in Figure7. The dimensions of these samples are presented in Table 5. The natural frequency results of these samples are presented in Table 7 using experimental, analytical, and FE(3D) methods. The captured signal for the free vibration response for selected case from Table 7 is shown in Figure 11. This case is italicized in the table. The results of Table 7 show that the three-dimensional FE solution is closer to the experimental results than the analytical results. The maximum percentage error between the experimental and the FE results is less than 0.5%.Table 5
Typical samples dimension of group S2 withd-=0.75 shown in Figure 7.
Group S2
(Single-step beam samples of steel rodØ 40, L=200 mm and d-=30/40)
L
L
1
μ
L
2
d
1
d
2
m
t (kg)
r
1
∗
2
S2-40/40
200
40
0.2
160
40
30
1.282
0.002
S2-40/100
200
100
0.5
100
40
30
1.375
0.002
S2-40/140
200
140
0.7
60
40
30
1.541
0.002Figure 11
Experiment first four natural frequencies for test sample S2-40/40.Group S3 consists of four different samples, namely, S3-80/250, S3-80/200, S3-80/150, and S3-80/100. More details about the geometrical dimensions and material properties for these samples are shown in Table6 and Figure 8. The results of Table 8 show that the deviation between the FE(3D) and the experimental is less than 1.16%. The free vibration signals of selected case italicized in Table 8 are plotted in Figure 12. In general, the results of the three-span samples reveal the same conclusion drawn from the two-span samples that the FE(3D) results are closer to the experimental results than the analytical results. The conclusion driven from the investigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction of the natural frequency of stepped beam.Table 6
The sample dimensions of group S3 as shown in Figure8.
Group S3
(Four cylindrical samples have two steps of steel rodØ 80, L=500 mm)
L
L
1
L
2
d
1
d
2
m
t (kg)
L
2
∗
r
1
∗
2
S3-80/250
500
200
250
20
80
10.480
0.5
0.000100
S3-80/200
500
225
200
20
80
7.398
0.4
0.000100
S3-80/150
500
250
150
20
80
6.781
0.3
0.000100
S3-80/100
500
275
100
20
80
4.932
0.2
0.000100Table 7
Percentage error between the computational and experimental results for single-step test samples (Group S2). Lowest four nonzero free-free modes.
d
-
L
1
μ
Method
Modal frequencies in Hz
1
2
3
4
20/40
40
0.2
(A) EXP
1848.940
5028.140
9854
16303
(B) NTM
1873.981
5194.229
10211.79
16776.400
(C) FE(3D)
1859.490
5056.749
9893.345
16380.170
(D) % error (A, B)
1.354
3.303
3.630
2.903
(E) % error (A, C)
0.570
0.568
0.399
0.473
100
0.5
(A) EXP
2153.900
7713.530
13701.800
19845.900
(B) NTM
2244.601
7923.127
13661.990
20230.700
(C) FE(3D)
2142.012
7700.169
13663.580
19793.360
(D) % error (A, B)
4.211
2.717
−0.290
1.938
(E) % error (A, C)
−0.551
−0.173
−0.278
−0.264
140
0.7
(A) EXP
3596.820
8103.960
15773.100
20733.800
(B) NTM
3813.426
8160.003
15739.160
21174.340
(C) FE(3D)
3581.521
8090.723
15705.070
20634.360
(D) % error (A, B)
6.022
0.691
−0.215
2.124
(E) % error (A, C)
−0.425
−0.163
−0.431
−0.479
25/40
40
0.2
(A) EXP (Figure11)
2430.240
6510.630
12186.700
19359.300
(B) NTM
2444.111
6638.069
12492.520
19588.580
(C) FE(3D)
2430.512
6518.094
12223.080
19314.540
(D) % error (A, B)
0.570
1.957
2.509
1.184
(E) % error (A, C)
0.011
0.114
0.298
−0.231
100
0.5
(A) EXP
2796.050
8693.500
14542.200
22134
(B) NTM
2912.321
8781.311
14633.780
22182.430
(C) FE(3D)
2806.782
8698.963
14551.070
22095.920
(D) % error (A, B)
4.158
1.010
0.629
0.218
(E) % error (A, C)
0.383
0.062
0.060
−0.172
140
0.7
(A) EXP
4015.620
8665.980
16238.9
22684
(B) NTM
4167.385
8797.248
16074.330
22814.460
(C) FE(3D)
4021.237
8655.476
16159.230
22610.690
(D) % error (A, B)
3.779
1.514
−1.013
0.575
(E) % error (A, C)
0.139
−0.121
−0.490
−0.323
30/40
40
0.2
(A) EXP
3023.690
7906
14381.400
21856.600
(B) NTM
3023.321
7946.457
14429.850
21806.140
(C) FE(3D)
3014.442
7878.443
14307.960
21781.750
(D) % error (A, B)
−0.012
0.511
0.336
−0.230
(E) % error (A, C)
−0.305
−0.348
−0.510
−0.342
100
0.5
(A) EXP
3401.940
9329.910
15681.400
23607.700
(B) NTM
3478.703
9293.555
15691.770
23326.460
(C) FE(3D)
3405.324
9313.349
15645.690
23513.650
(D) % error (A, B)
2.256
−0.389
0.066
−1.191
(E) % error (A, C)
0.099
−0.177
−0.227
−0.398
140
0.7
(A) EXP
4201.680
9322.630
16596.400
23967.700
(B) NTM
4268.272
9431.643
16437.830
23729.240
(C) FE(3D)
4211.100
9322.286
16530.200
23881.400
(D) % error (A, B)
1.584
1.169
−0.955
−0.994
(E) % error (A, C)
0.224
−0.003
−0.398
−0.360
35/40
40
0.2
(A) EXP
3608.280
9126.360
16128.600
23817.3
(B) NTM
3607.895
9104.530
16000.810
23500.960
(C) FE(3D)
3607.465
9106.544
16054.460
23720.110
(D) % error (A, B)
−0.010
−0.239
−0.792
−1.328
(E) % error (A, C)
−0.022
−0.217
−0.459
−0.408
100
0.5
(A) EXP
3885.030
9778.800
16719.900
24538.800
(B) NTM
3906.518
9705.047
16577.640
24125.780
(C) FE(3D)
3882.419
9759.382
16658.180
24417.41
(D) % error (A, B)
0.553
−0.754
−0.850
−1.683
(E) % error (A, C)
−0.067
−0.198
−0.369
−0.495
140
0.7
(A) EXP
4248.240
9888.540
17028
24738.400
(B) NTM
4253.660
9878.822
16849.850
24311.230
(C) FE(3D)
4247.271
9870.286
16963.900
24616.760
(D) % error (A, B)
0.127
−0.098
−1.046
−1.726
(E) % error (A, C)
−0.022
−0.184
−0.376
−0.491Table 8
Percentage error between the computational and experimental results for two-step samples (Group S3). Lowest four nonzero free-free modes.
Method
Modal frequencies in Hz
1
2
3
4
S3-80/250
(A) EXP
403.148
2126.460
4139.830
5153.500
(B) NTM
420.212
2203.984
4334.192
5605.556
(C) FE(3D)
404.180
2126.900
4153.700
5169.800
(D) % error (A, B)
−3.966
3.517
4.484
−8.428
(E) % error (A, C)
−0.255
−0.020
−0.333
−0.315
S3-80/200
(A) EXP (Figure12)
354.790
1727.900
2291.490
4531.380
(B) NTM
367.381
1788.472
2551.555
4660.983
(C) FE(3D)
355.230
1728.300
2318.400
4532.200
(D) % error (A, B)
−3.549
3.386
10.192
−2.860
(E) % error (A, C)
0.124
0.023
1.160
0.018
S3-80/150
(A) EXP
335.859
1360.050
1545.130
3778.200
(B) NTM
346.243
1434.113
1638.174
3889.033
(C) FE(3D)
336.440
1364.200
1548.500
3781.900
(D) % error (A, B)
−3.091
−5.445
−6.021
−2.933
(E) % error (A, C)
0.172
0.305
0.218
0.097
S3-80/100
(A) EXP
337.400
1018.240
1371.070
3221.200
(B) NTM
345.942
1074.574
1424.992
3303.837
(C) FE(3D)
337.480
1024
1372.900
3215
(D) % error (A, B)
−2.531
−5.532
−3.932
−2.565
(E) % error (A, C)
0.023
0.565
0.133
−0.192Figure 12
Experimental first four natural frequencies for test sample S3-80/200.


### 3.3. Percentage Modal Deviation between FE(3D) and NTM Method

The results of the previous section show the accuracy of the FE(3D) model in evaluating the natural frequencies of stepped beam. Therefore, in this section, a free-free two-span model is deeply investigated using (40) and FE(3D) methods to justify the validity of the analytical solution in predicting the stepped beam results. Two categories of samples are considered as shown in Figure 13. The first category includes 600 mm length samples with r1∗2=2.78×10-4 and the second category includes 200 mm length samples with r1∗2=205×10-3. The effects of changing μ and d- are investigated, μ varies from 0 to 1, and meanwhile, only three values are listed in Table 9. Four different values of d- are investigated 0.5, 0.625, 0.75, and 0.875. The study in this section focuses only on the first three nonzero free-free modes. The percentage modal deviation in analytical NTM solution prediction in reference to the FE(3D) solution is presented in Figure 13 for long samples and in Figure 14 for short samples. This percentage deviation is calculated using the following formula:(42)Devi=fNTMi-fFEifFEi∗100,whereDevi is the percentage deviation in the ith modal frequency,f(FE)i is the ith mode natural frequency using three-dimensional finite element,f(NTM)i is the ith mode natural frequency using analytical NTM method.Table 9
Typical relative deviation between present NTM and finite element (3D) results. Lowest three nonzero modes for two categories of single-step F-F beam; see Figure13.
d
-
Method
μ
Modal frequencies in Hz
L
=
600 mm
L
=
200 mm
1
2
3
1
2
3
0.5
NTM
0.1
217.034
635.604
1267.590
1878.302
5163.860
9645.650
0.4
232.036
797.543
1834.110
1995.497
6510.740
13831.40
0.7
447.629
1037.270
2253.940
3813.426
8160
15739.200
FE(3D)
0.1
216.987
634.850
1263.259
1876.267
5123.790
9448.138
0.4
229.554
785.722
1819.305
1930.166
6271.645
13642.314
0.7
437.591
1033.915
2245.125
3583.014
8091.330
15706.439
% dev.
0.1
−0.022
−0.119
−0.343
−0.108
−0.782
−2.090
0.4
−1.082
−1.504
−0.814
−3.385
−3.812
−1.386
0.7
−2.294
−0.324
−0.392
−6.431
−0.849
−0.208
0.625
NTM
0.1
283.930
812.357
1608.220
2446.347
6532.940
11979.500
0.4
308.616
991.955
2074.890
2639.638
7837.580
15025.700
0.7
495.792
1128.820
2333.290
4167.385
8797.250
16074.300
FE(3D)
0.1
283.898
811.795
1604.956
2445.458
6509.821
11855.709
0.4
305.955
981.347
2072.554
2570.408
7647.979
15037.839
0.7
489.234
1121.452
2336.177
4021.301
8655.042
16159.248
% dev.
0.1
−0.011
−0.069
−0.203
−0.036
−0.355
−1.044
0.4
−0.870
−1.081
−0.113
−2.693
−2.479
0.080
0.7
−1.341
−0.657
0.123
−3.633
−1.643
0.526
0.750
NTM
0.1
355.027
993.279
1944.090
3030.222
7818.400
14004
0.4
381.639
1149.720
2248.110
3236.367
8841.600
15842.900
0.7
511.227
1230.890
2406.190
4268.272
9431.640
16437.800
FE(3D)
0.1
355.036
993.179
1943.016
3031.657
7819.722
13978.716
0.4
379.583
1144.159
2251.351
3181.291
8759.634
15934.664
0.7
508.439
1224.396
2410.039
4205.930
9312.425
16529.197
% dev.
0.1
0.002
−0.010
−0.055
0.047
0.017
−0.181
0.4
−0.542
−0.486
0.144
−1.731
−0.936
0.576
0.7
−0.548
−0.530
0.160
−1.482
−1.280
0.553
0.875
NTM
0.1
429.273
1177.820
2274.860
3616.212
9013.570
15741.600
0.4
448.134
1269.840
2419.690
3759.295
9566.420
16569.700
0.7
511.380
1312.660
2498.460
4253.660
9878.820
16849.800
FE(3D)
0.1
429.347
1178.446
2277.013
3621.275
9047.347
15829.045
0.4
447.258
1269.273
2423.695
3739.834
9595.934
16695.492
0.7
510.778
1310.738
2502.153
4247.181
9870.045
16963.843
% dev.
0.1
0.017
0.053
0.094
0.140
0.373
0.552
0.4
−0.196
−0.045
0.165
−0.520
0.308
0.753
0.7
−0.118
−0.146
0.148
−0.153
−0.089
0.672Figure 13
Long and short stepped samples at three values ofμ=0.1,0.4,  and  0.7 and four values of d-=h21=0.5,0.625,0.75,  and  0.875.Figure 14
Variation of the percentage modal deviation between FE(3D) and NTM results as a function ofμ and d-, for free-free long samples L = 600 mm: (a) 1st mode, (b) 2nd mode, and (c) 3rd mode.
(a)
(b)
(c)The results of Table9 show that, for the investigated examples with step ratio smaller than one, the increase in μ and/or d- increases the modal frequencies. The percentage deviations in the analytical NTM results for the short samples are higher than those for the longer samples. The percentage deviations in the analytical natural frequency prediction are plotted for the long and short examples in Figures 14 and 15, respectively. Figures 14(a) and 15(a) present a plot for Dev1, Figures 14(b) and 15(b) present a plot for Dev2, and Figures 14(c) and 15(c) present a plot for Dev3. Figure 16(a) presents the first mode shape at the conditions of the peak point in Figure 15(a), Figure 16(b) presents the second mode shape at the conditions of the peak point in Figure 15(b), and Figure 16(c) presents the third mode shape at the conditions of the peak point in Figure 15(c).Figure 15
Percentage modal deviation between FE(3D) and NTM as a function ofμ and d-, for free-free short samples L = 200 mm: (a) 1st mode, (b) 2nd mode, and (c) 3rd mode.
(a)
(b)
(c)Figure 16
Typical first three modal shapes at the peak points shown in Figure15 for free-free short sample and d- = 0.5.
(a)
First mode
(b)
Second mode
(c)
Third modeFigures14(a) and 15(a) show that, for long and short stepped samples, the Dev1 attains the peak when μ lies between 0.5 and 0.7; that is, the step in the beam is located around the peak of the first free-free mode shape; see Figure 16(a). Figures 14(b) and 15(b) show that there are two peaks in Dev2 prediction. The location of these peaks is found to be near the position of the peaks of the second free-free mode shape; see Figure 16(b). In addition, the value of Dev2 approaches zero when the step location lies in the semistraight line between the two peaks in the second F-F mode shape. The same trend is repeated in the third mode shape as shown in Figures 14(c), 15(c), and 16(c). In addition, when the location of diametric step in shaft coincides with a peak point in i mode shape, the value of Devi is increased. Meanwhile, when the step lies in a straight portion of the mode shape, the Devi approaches zero.


### 3.4. Tapered Beam Approach

Due to the importance of tapered or conical beams in many engineering applications, the current section is devoted to show how to use the present analysis to solve the problem of taper beam. The current analysis is based on uniform beams, while the partial differential equation which represents the lateral vibration of tapered or conical beams is fourth-order Bessel equation [30, 31, 37, 38]. To simulate nonuniform beam using the current analysis, the beam is divided into multiple equal length spans as shown in Figure 17. The height and/or width of these spans are varying linearly between the start and the end to simulate the tapered or conical beam. The height and/or width ratio of i span can be calculated from the following formula:(43)hi1=hn1+1-hn1-xiL+Li12n,bi1=bn1+1-bn1-xiL+Li12n,where(44)hi1=hih1,hn1=hnh1,bi1=bib1,bn1=bnb1,Li1=LiL1.hi is the height of i beam segment and bi is the width of i beam segment.Figure 17
Stepped tapered beam with varying depth and width.To verify the suitability of the current model to represent conical beams, the results of the current model are compared with the exact solution for cantilevered (C-F) conical beam with variable taper ratiohn1=bn1=0.2,0.5, and 0.7 as shown in Table 10. The model was investigated using the present NTM and using several number of spans n=20,100,200, and 1000. The first three eigenvalues are evaluated for each taper ratio and number of spans. The time used for computing the first three natural frequencies is evaluated. The model results are also evaluated using NAT previously published in [6] at n=20 in order to compare the time saving when using the present method.Table 10
The first three eigenvalues for C-F conical beam using NTM in comparison with those of the exact solution presented in [30] and NAT results using the program of [6].
h
n
1
=
b
n
1
[30]
NAT [6]  n=20
NTM n=20
NTM n=100
NTM n=200
NTM n=1000
Time (s)
Time (s)
Time (s)
Time (s)
Time (s)
0.2
λ
1
∗
2
6.1964
6.1683
6
6.1706
1.8
6.1954
5.3
6.1962
10.5
6.1963
32.6
λ
2
∗
2
18.3855
18.2513
18.2513
18.3801
18.3840
18.3853
λ
3
∗
2
39.8336
39.4809
39.4809
39.8194
39.8300
39.8336
0.5
λ
1
∗
2
4.6252
4.6155
6.2
4.6175
1.9
4.6249
5.4
4.6250
10.9
4.6251
34
λ
2
∗
2
19.5476
19.5074
19.5074
19.5460
19.5472
19.5477
λ
3
∗
2
48.5789
48.4725
48.4725
48.5746
48.5779
48.5788
0.7
λ
1
∗
2
4.0669
4.0615
6.2
4.0635
1.9
4.0668
5.6
4.0669
11.1
4.0670
35
λ
2
∗
2
20.5554
20.5360
20.5361
20.5547
20.5553
20.5554
λ
3
∗
2
—
53.9625
53.9625
54.0131
54.0147
54.0152The results of Table10 show that increasing the number of spans results in increasing the accuracy of the evaluated beam eigenvalues in comparison with the exact solution in [30]. On the other hand, the results show that increasing the number of spans results in increasing the computational time. The comparison between the computational time using the present NTM and previously published NAT shows that the present NTM method is quicker than the NAT method [6] for the same number of spans.To investigate the capability of the present model to evaluate the natural frequencies of taper or conical beam with variable boundary conditions, some cases of nonuniform beams are selected form [31]. The analysis in [31] is numerical and based on solving the taper beam partial differential equation using Runge-Kutta method. The boundary conditions and the taper ratio are listed in Table 11. The number of spans used in evaluating the current example is n=1000. The eigenvalue results using the present NTM method are in good agreement with numerical results of [31].Table 11
The first four eigenvalues for taper or conical beam subjected to variable end conditions using the present NTM and [31].
Beam parameters
Frequency parameterλ1∗2
[31]
Present NTM
h
n
1
=
1  Z1=Zn+1=108  Φ1=108,Φn+1=1000
21.94
21.9518
60.44
60.5461
118.7
118.7589
196.4
196.4162
b
n
1
=
0.5  Z1=Zn+1=10  Φ1=Φn+1=102
4.98
5.0292
12.50
12.4289
40.13
39.8154
88.28
88.0372
b
n
1
=
1
,
h
n
1
=
1.5
1
/
3  Z1=102,Zn+1=2×102  Φ1=3×102,Φn+1=4×102
13.63
13.6709
24.59
24.9270
48.37
48.7915
97.28
97.5762
b
n
1
=
h
n
1
=
3
1
/
4  Z1=Zn+1=108  Φ1=Φn+1=108
25.83
25.8327
71.00
71.1461
139.3
139.4223
230.3
230.4320


## 3.1. Verification and Validation of NTM Results

### 3.1.1. Verification Example 1

In this example, the first five nondimensional natural frequenciesλi∗4=ρ1A1L4ω2/E1I1 of stepped beam are compared with the exact solution presented by Gutierrez et al. [18]; see Figure 3. The model is solved at two different step locations and several values of b21 and h21 as shown in Table 1. Three different values of rotary inertia r12=vs,0.0036 and 0.01 are considered in order to validate the current model in case of Bernoulli-Euler and Timoshenko beams. The value of s12=3.12r12 is considered in order to evaluate the shear deformation [18]. The values of nondimensional linear and rotational elastic supports stiffness at start and the end are Z1=10,Φ1=vl, Z3=vs,Φ3=vl, respectively. The results of Table 1 show that the present NTM results are in good agreement with the exact solution presented by Gutierrez et al. [18].Table 1
The first five natural frequenciesλi∗2 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation ϕ1∗=vl and elastically restrained in translation Z1∗=10 in comparison with [18].
r
1
2
μ
b
21
h
21
λ
1
∗
2
λ
2
∗
2
λ
3
∗
2
λ
4
∗
2
λ
5
∗
2
10
-
7
[18]
0.25
1.0
0.8
3.010
9.696
34.010
74.430
132.341
Present NTM
3.0098
9.6956
34.0101
74.4300
132.3410
[18]
0.8
0.8
3.249
9.664
34.315
74.645
131.803
Present NTM
3.2490
9.6638
34.3151
74.6445
131.7507
[18]
0.8
0.6
3.385
8.117
28.630
60.997
103.439
Present NTM
3.3851
8.1171
28.6302
60.9947
103.4075
[18]
0.50
1.0
0.8
2.958
10.046
34.993
80.145
139.666
Present NTM
2.9579
10.0460
34.9931
80.1455
139.6667
[18]
0.8
0.8
3.124
10.165
34.688
80.570
139.175
Present NTM
3.1240
10.1655
34.6882
80.5707
139.1759
[18]
0.8
0.6
3.284
9.499
28.501
70.123
118.482
Present NTM
3.2841
9.4989
28.5006
70.1232
118.4830
0.0036
[18]
0.25
1.0
0.8
3.007
9.668
33.571
72.393
126.111
Present NTM
3.0066
9.6678
33.5711
72.3926
126.1107
[18]
0.8
0.8
3.245
9.635
33.862
72.607
125.617
Present NTM
3.2454
9.6354
33.8621
72.6068
125.6169
[18]
0.8
0.6
3.381
8.103
28.364
59.839
100.245
Present NTM
3.3812
8.1031
28.3643
59.8385
100.2447
[18]
0.50
1.0
0.8
2.946
9.919
33.205
71.469
116.725
Present NTM
2.9462
9.9192
33.2046
71.4693
116.7245
[18]
0.8
0.8
3.112
10.030
32.926
71.741
116.362
Present NTM
3.1120
10.0304
32.9259
71.7415
116.3618
[18]
0.8
0.6
3.272
9.391
27.449
64.020
102.697
Present NTM
3.2721
9.3908
27.4490
64.0200
102.6966
0.01
[18]
0.25
1.0
0.8
3.001
9.619
32.841
69.229
117.238
Present NTM
3.0009
9.6192
32.8407
69.2292
117.2380
[18]
0.8
0.8
3.239
9.586
33.109
69.444
116.798
Present NTM
3.2390
9.5858
33.1092
69.4435
116.7981
[18]
0.8
0.6
3.374
8.078
27.914
57.970
95.370
Present NTM
3.3744
8.0784
27.9139
57.9705
95.3698
[18]
0.50
1.0
0.8
2.926
9.710
30.711
61.768
95.623
Present NTM
2.9255
9.7104
30.7111
61.7680
95.6230
[18]
0.8
0.8
3.019
9.809
30.464
61.912
95.326
Present NTM
3.0908
9.8086
30.4643
61.9117
95.3257
[18]
0.8
0.6
3.251
9.211
25.893
56.578
86.647
Present NTM
3.2511
9.2111
25.8933
56.5776
86.6473Figure 3
Verification Example 1 [18], two-span stepped beam.


### 3.1.2. Verification Example 2

In this example, Timoshenko beam with three-step round cross-section presented in [19] is investigated; see Figure 4. An intermediate lumped mass of m3∗=m3/ρ1A1L=1 is located at a distance of 750 mm from point 1. The input data for this example is listed in the caption of Figure 4. Table 2 shows the results of the first five natural frequencies ωi in (rad/sec) for pinned-pinned, free-clamped, clamped-free, clamped-pinned, and clamped-clamped configurations. The results of reference [19] are calculated using two methods. The first method is numerical assembly technique (NAT) and this method is the exact solution. The second method is differential transformation method and this method is approximate method. The pinned-pinned and free-clamped boundary conditions results are found in [19]. The results of the current analysis are in good agreement with the results presented in [19]. As can be seen from Table 2, the rotary inertia and shear deformation reduce the modal frequency especially for higher modes. Comparing the results of the clamped-free beam with the results of the free-clamped beam shows that the natural frequency results of the clamped-free beam are lower than that of free-clamped beam. This may be explained by the fact that fixing the beam from the thinner span results in lowering of the beam stiffness.Table 2
Comparison of the first five natural frequencies, in rad/sec, using the present NTM results with those obtained in [19] for the example shown in Figure 4.
ω
n  (rad/sec)
Method
Bernoulli-Eulerr1∗2=0
Timoshenkor1∗2≠0
Pinned-pinned(P-P)
ω
1
NAT [19]
319.4341
316.5288
NTM
319.4340
316.4855
ω
2
NAT [19]
1853.3864
1789.4207
NTM
1853.3864
1784.8723
ω
3
NAT [19]
4110.1341
3825.8438
NTM
4110.1341
3836.5347
ω
4
NAT [19]
7709.5714
6642.9094
NTM
7709.5712
6639.2170
ω
5
NAT [19]
11621.7699
9886.6243
NTM
11621.7698
9918.8328
Free-clamped(F-C)
ω
1
NAT [19]
371.5354
367.9440
NTM
371.5354
367.1487
ω
2
NAT [19]
1243.9063
1216.6220
NTM
1243.9065
1211.5531
ω
3
NAT [19]
3082.0846
2827.0352
NTM
3082.0845
2849.1465
ω
4
NAT [19]
5541.1410
4957.9240
NTM
5541.1409
5003.7433
ω
5
NAT [19]
9599.3810
7673.7578
NTM
9599.3810
7751.3336
Clamped-free(C-F)
ω
1
NTM
55.1257
55.0130
ω
2
NTM
616.6018
598.5917
ω
3
NTM
2702.3238
2532.1483
ω
4
NTM
5483.2379
4969.1659
ω
5
NTM
9570.9586
7956.5299
Clamped-pinned(C-P)
ω
1
NTM
480.8915
469.8444
ω
2
NTM
2098.0556
1991.9611
ω
3
NTM
5007.3315
4589.5676
ω
4
NTM
8366.8073
7049.5393
ω
5
NTM
13678.7112
11215.0349
Clamped-clamped(C-C)
ω
1
NTM
836.3527
807.0997
ω
2
NTM
3019.5886
2772.2732
ω
3
NTM
5583.6469
4980.5538
ω
4
NTM
9625.8943
7734.8319
ω
5
NTM
14403.3163
11538.1581Figure 4
Verification Example 2 [19], three-step beam having E=206.9GPa. ρ=7836.9kg/m3, G=79.5769GPa, and k′=0.75.


### 3.1.3. Verification Example 3

The third verification example is shown in Figure5 with identical dimensions to that used in [5, 20]. The rotary inertia and shear deformation are considered. The shear coefficient k′ is calculated based on [3], keeping Poisson’s’ ratio ν = 0.33. Table 3 shows the first three nonzero free-free (F-F) eigenvalues in Hz and in comparison with [5, 20]. The present results are computed using both numerically (40) and FE(3D) methods. The results of Table 3 show that the present analysis results are very close to the experimental results. The percentage error between the present FE(3D) results and the experimental results of [5] is less than 0.302%. This represents the importance of including the effect of rotary inertia and shear deformation.Table 3
Comparison of the present NTM results with those obtained in [5, 20], for F-F beam shown in Figure 5.
Mode number
DQEM [20]
Exp. [5]
NTM (40)
FE(3D)
% error
I
II
III
IV
(II & IV)
1
292.440
291
290.316
290.120
0.302
2
1181.300
1165
1162.158
1167.100
−0.180
3
1804.100
1771
1767.321
1772.200
−0.068Figure 5
Verification Example 3 [5, 20], single-step (F-F) aluminum beam having, E=71.7GPa, ρ=2830Kg/m3, and ν=0.33.


### 3.1.4. Verification Example 4

The fourth example is shown in Figure6. It is for a twenty-span uniform beam carrying 19 equally spaced concentrated masses with mi∗=0.1. Several beam start and end conditions are investigated as listed in Table 4. The results are evaluated using the present NTM method and previously published numerical assembly technique NAT [6] method. The computational time required to obtain the first three frequency parameters λ1∗,λ2∗, and λ3∗ using both NAT and NTM methods is calculated and listed in Table 4. Considerable reduction in computational time is observed in all the investigated cases as shown in Table 4.Table 4
Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated massesmi∗=0.1,(r1∗2=s1∗2=vs) in comparison with [6].
Method
Z
1
∗
,
Φ
1
∗
Z
21
∗
,
Φ
21
∗
λ
1
∗
λ
2
∗
λ
3
∗
Computational time (sec)
NAT [6]
v
s
,
v
s
v
s
,
v
s
3.7177
6.1716
8.6392
48.17
Present NTM
v
s
,
v
s
v
s
,
v
s
3.7177
6.1716
8.6392
13.53
NAT [6]
v
l
,
v
s
v
l
,
v
s
2.3871
4.7742
7.1612
34.67
Present NTM
v
l
,
v
s
v
l
,
v
s
2.3871
4.7742
7.1612
12.14
NAT [6]
v
l
,
v
l
v
l
,
v
l
3.5941
5.9671
8.35471
40.40
Present NTM
v
l
,
v
l
v
l
,
v
l
3.5941
5.9671
8.35471
13.47
NAT [6]
10,10
10,10
1.5989
2.5329
4.5969
22.13
Present NTM
10,10
10,10
1.5989
2.5329
4.5969
10.23Figure 6
Verification Example 4, twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general end flexibilities conditions.


## 3.1.1. Verification Example 1

In this example, the first five nondimensional natural frequenciesλi∗4=ρ1A1L4ω2/E1I1 of stepped beam are compared with the exact solution presented by Gutierrez et al. [18]; see Figure 3. The model is solved at two different step locations and several values of b21 and h21 as shown in Table 1. Three different values of rotary inertia r12=vs,0.0036 and 0.01 are considered in order to validate the current model in case of Bernoulli-Euler and Timoshenko beams. The value of s12=3.12r12 is considered in order to evaluate the shear deformation [18]. The values of nondimensional linear and rotational elastic supports stiffness at start and the end are Z1=10,Φ1=vl, Z3=vs,Φ3=vl, respectively. The results of Table 1 show that the present NTM results are in good agreement with the exact solution presented by Gutierrez et al. [18].Table 1
The first five natural frequenciesλi∗2 for stepped Timoshenko beam in the case where the beam is rigidly restrained against rotation ϕ1∗=vl and elastically restrained in translation Z1∗=10 in comparison with [18].
r
1
2
μ
b
21
h
21
λ
1
∗
2
λ
2
∗
2
λ
3
∗
2
λ
4
∗
2
λ
5
∗
2
10
-
7
[18]
0.25
1.0
0.8
3.010
9.696
34.010
74.430
132.341
Present NTM
3.0098
9.6956
34.0101
74.4300
132.3410
[18]
0.8
0.8
3.249
9.664
34.315
74.645
131.803
Present NTM
3.2490
9.6638
34.3151
74.6445
131.7507
[18]
0.8
0.6
3.385
8.117
28.630
60.997
103.439
Present NTM
3.3851
8.1171
28.6302
60.9947
103.4075
[18]
0.50
1.0
0.8
2.958
10.046
34.993
80.145
139.666
Present NTM
2.9579
10.0460
34.9931
80.1455
139.6667
[18]
0.8
0.8
3.124
10.165
34.688
80.570
139.175
Present NTM
3.1240
10.1655
34.6882
80.5707
139.1759
[18]
0.8
0.6
3.284
9.499
28.501
70.123
118.482
Present NTM
3.2841
9.4989
28.5006
70.1232
118.4830
0.0036
[18]
0.25
1.0
0.8
3.007
9.668
33.571
72.393
126.111
Present NTM
3.0066
9.6678
33.5711
72.3926
126.1107
[18]
0.8
0.8
3.245
9.635
33.862
72.607
125.617
Present NTM
3.2454
9.6354
33.8621
72.6068
125.6169
[18]
0.8
0.6
3.381
8.103
28.364
59.839
100.245
Present NTM
3.3812
8.1031
28.3643
59.8385
100.2447
[18]
0.50
1.0
0.8
2.946
9.919
33.205
71.469
116.725
Present NTM
2.9462
9.9192
33.2046
71.4693
116.7245
[18]
0.8
0.8
3.112
10.030
32.926
71.741
116.362
Present NTM
3.1120
10.0304
32.9259
71.7415
116.3618
[18]
0.8
0.6
3.272
9.391
27.449
64.020
102.697
Present NTM
3.2721
9.3908
27.4490
64.0200
102.6966
0.01
[18]
0.25
1.0
0.8
3.001
9.619
32.841
69.229
117.238
Present NTM
3.0009
9.6192
32.8407
69.2292
117.2380
[18]
0.8
0.8
3.239
9.586
33.109
69.444
116.798
Present NTM
3.2390
9.5858
33.1092
69.4435
116.7981
[18]
0.8
0.6
3.374
8.078
27.914
57.970
95.370
Present NTM
3.3744
8.0784
27.9139
57.9705
95.3698
[18]
0.50
1.0
0.8
2.926
9.710
30.711
61.768
95.623
Present NTM
2.9255
9.7104
30.7111
61.7680
95.6230
[18]
0.8
0.8
3.019
9.809
30.464
61.912
95.326
Present NTM
3.0908
9.8086
30.4643
61.9117
95.3257
[18]
0.8
0.6
3.251
9.211
25.893
56.578
86.647
Present NTM
3.2511
9.2111
25.8933
56.5776
86.6473Figure 3
Verification Example 1 [18], two-span stepped beam.


## 3.1.2. Verification Example 2

In this example, Timoshenko beam with three-step round cross-section presented in [19] is investigated; see Figure 4. An intermediate lumped mass of m3∗=m3/ρ1A1L=1 is located at a distance of 750 mm from point 1. The input data for this example is listed in the caption of Figure 4. Table 2 shows the results of the first five natural frequencies ωi in (rad/sec) for pinned-pinned, free-clamped, clamped-free, clamped-pinned, and clamped-clamped configurations. The results of reference [19] are calculated using two methods. The first method is numerical assembly technique (NAT) and this method is the exact solution. The second method is differential transformation method and this method is approximate method. The pinned-pinned and free-clamped boundary conditions results are found in [19]. The results of the current analysis are in good agreement with the results presented in [19]. As can be seen from Table 2, the rotary inertia and shear deformation reduce the modal frequency especially for higher modes. Comparing the results of the clamped-free beam with the results of the free-clamped beam shows that the natural frequency results of the clamped-free beam are lower than that of free-clamped beam. This may be explained by the fact that fixing the beam from the thinner span results in lowering of the beam stiffness.Table 2
Comparison of the first five natural frequencies, in rad/sec, using the present NTM results with those obtained in [19] for the example shown in Figure 4.
ω
n  (rad/sec)
Method
Bernoulli-Eulerr1∗2=0
Timoshenkor1∗2≠0
Pinned-pinned(P-P)
ω
1
NAT [19]
319.4341
316.5288
NTM
319.4340
316.4855
ω
2
NAT [19]
1853.3864
1789.4207
NTM
1853.3864
1784.8723
ω
3
NAT [19]
4110.1341
3825.8438
NTM
4110.1341
3836.5347
ω
4
NAT [19]
7709.5714
6642.9094
NTM
7709.5712
6639.2170
ω
5
NAT [19]
11621.7699
9886.6243
NTM
11621.7698
9918.8328
Free-clamped(F-C)
ω
1
NAT [19]
371.5354
367.9440
NTM
371.5354
367.1487
ω
2
NAT [19]
1243.9063
1216.6220
NTM
1243.9065
1211.5531
ω
3
NAT [19]
3082.0846
2827.0352
NTM
3082.0845
2849.1465
ω
4
NAT [19]
5541.1410
4957.9240
NTM
5541.1409
5003.7433
ω
5
NAT [19]
9599.3810
7673.7578
NTM
9599.3810
7751.3336
Clamped-free(C-F)
ω
1
NTM
55.1257
55.0130
ω
2
NTM
616.6018
598.5917
ω
3
NTM
2702.3238
2532.1483
ω
4
NTM
5483.2379
4969.1659
ω
5
NTM
9570.9586
7956.5299
Clamped-pinned(C-P)
ω
1
NTM
480.8915
469.8444
ω
2
NTM
2098.0556
1991.9611
ω
3
NTM
5007.3315
4589.5676
ω
4
NTM
8366.8073
7049.5393
ω
5
NTM
13678.7112
11215.0349
Clamped-clamped(C-C)
ω
1
NTM
836.3527
807.0997
ω
2
NTM
3019.5886
2772.2732
ω
3
NTM
5583.6469
4980.5538
ω
4
NTM
9625.8943
7734.8319
ω
5
NTM
14403.3163
11538.1581Figure 4
Verification Example 2 [19], three-step beam having E=206.9GPa. ρ=7836.9kg/m3, G=79.5769GPa, and k′=0.75.


## 3.1.3. Verification Example 3

The third verification example is shown in Figure5 with identical dimensions to that used in [5, 20]. The rotary inertia and shear deformation are considered. The shear coefficient k′ is calculated based on [3], keeping Poisson’s’ ratio ν = 0.33. Table 3 shows the first three nonzero free-free (F-F) eigenvalues in Hz and in comparison with [5, 20]. The present results are computed using both numerically (40) and FE(3D) methods. The results of Table 3 show that the present analysis results are very close to the experimental results. The percentage error between the present FE(3D) results and the experimental results of [5] is less than 0.302%. This represents the importance of including the effect of rotary inertia and shear deformation.Table 3
Comparison of the present NTM results with those obtained in [5, 20], for F-F beam shown in Figure 5.
Mode number
DQEM [20]
Exp. [5]
NTM (40)
FE(3D)
% error
I
II
III
IV
(II & IV)
1
292.440
291
290.316
290.120
0.302
2
1181.300
1165
1162.158
1167.100
−0.180
3
1804.100
1771
1767.321
1772.200
−0.068Figure 5
Verification Example 3 [5, 20], single-step (F-F) aluminum beam having, E=71.7GPa, ρ=2830Kg/m3, and ν=0.33.


## 3.1.4. Verification Example 4

The fourth example is shown in Figure6. It is for a twenty-span uniform beam carrying 19 equally spaced concentrated masses with mi∗=0.1. Several beam start and end conditions are investigated as listed in Table 4. The results are evaluated using the present NTM method and previously published numerical assembly technique NAT [6] method. The computational time required to obtain the first three frequency parameters λ1∗,λ2∗, and λ3∗ using both NAT and NTM methods is calculated and listed in Table 4. Considerable reduction in computational time is observed in all the investigated cases as shown in Table 4.Table 4
Validation of the present NTM using 20-span uniform beam with 19 equally spaced concentrated massesmi∗=0.1,(r1∗2=s1∗2=vs) in comparison with [6].
Method
Z
1
∗
,
Φ
1
∗
Z
21
∗
,
Φ
21
∗
λ
1
∗
λ
2
∗
λ
3
∗
Computational time (sec)
NAT [6]
v
s
,
v
s
v
s
,
v
s
3.7177
6.1716
8.6392
48.17
Present NTM
v
s
,
v
s
v
s
,
v
s
3.7177
6.1716
8.6392
13.53
NAT [6]
v
l
,
v
s
v
l
,
v
s
2.3871
4.7742
7.1612
34.67
Present NTM
v
l
,
v
s
v
l
,
v
s
2.3871
4.7742
7.1612
12.14
NAT [6]
v
l
,
v
l
v
l
,
v
l
3.5941
5.9671
8.35471
40.40
Present NTM
v
l
,
v
l
v
l
,
v
l
3.5941
5.9671
8.35471
13.47
NAT [6]
10,10
10,10
1.5989
2.5329
4.5969
22.13
Present NTM
10,10
10,10
1.5989
2.5329
4.5969
10.23Figure 6
Verification Example 4, twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general end flexibilities conditions.


## 3.2. Test Samples and Experimental Procedures

In order to measure the natural frequencies of the system under study, the free-free test samples were put in free oscillations by using an instrumented hammer model B  and  K 8202. An accelerometer model B  and  K 4366 is fixed to the shaft in order to capture the vibration signal. The output of the charge amplifier B  and  K 2635 is connected to NI 6216 data acquisition card. This card is connected to the PC and managed by Lab VIEW software. Figure10 shows a photo of the current experimental setup. The used card settings are sample frequency of 20 kHz, sampling time of 3, 5, 8 sec, and the size of samples block to read is 1 k. The time domain data is captured and transformed into frequency domain. The resonant frequencies were obtained by the average of the results of 10 impacts applied in three different locations of the sample. Figures 7–9 show two groups of different stepped test samples, S2 and S3, respectively. All the test samples are manufactured from steel rods of dimensions Ø 80 and Ø 40 mm. The common data for the steel test samples are ρ=7850kg/m3, E=206.8GPa, and k′=0.845.Figure 7
Schematic drawings for the experimental samples Group S2.Figure 8
Schematic drawings for the experimental samples Group S3.Figure 9
Real image for typical experimental test samples.Figure 10
Test setup.Two-span twelve test samples are shown in group S2, namely, S2-40/40, S2-40/100, and S2-40/140, shown in Figure7. The dimensions of these samples are presented in Table 5. The natural frequency results of these samples are presented in Table 7 using experimental, analytical, and FE(3D) methods. The captured signal for the free vibration response for selected case from Table 7 is shown in Figure 11. This case is italicized in the table. The results of Table 7 show that the three-dimensional FE solution is closer to the experimental results than the analytical results. The maximum percentage error between the experimental and the FE results is less than 0.5%.Table 5
Typical samples dimension of group S2 withd-=0.75 shown in Figure 7.
Group S2
(Single-step beam samples of steel rodØ 40, L=200 mm and d-=30/40)
L
L
1
μ
L
2
d
1
d
2
m
t (kg)
r
1
∗
2
S2-40/40
200
40
0.2
160
40
30
1.282
0.002
S2-40/100
200
100
0.5
100
40
30
1.375
0.002
S2-40/140
200
140
0.7
60
40
30
1.541
0.002Figure 11
Experiment first four natural frequencies for test sample S2-40/40.Group S3 consists of four different samples, namely, S3-80/250, S3-80/200, S3-80/150, and S3-80/100. More details about the geometrical dimensions and material properties for these samples are shown in Table6 and Figure 8. The results of Table 8 show that the deviation between the FE(3D) and the experimental is less than 1.16%. The free vibration signals of selected case italicized in Table 8 are plotted in Figure 12. In general, the results of the three-span samples reveal the same conclusion drawn from the two-span samples that the FE(3D) results are closer to the experimental results than the analytical results. The conclusion driven from the investigated two- and three-span samples is that the three-dimensional finite element can be trusted in the prediction of the natural frequency of stepped beam.Table 6
The sample dimensions of group S3 as shown in Figure8.
Group S3
(Four cylindrical samples have two steps of steel rodØ 80, L=500 mm)
L
L
1
L
2
d
1
d
2
m
t (kg)
L
2
∗
r
1
∗
2
S3-80/250
500
200
250
20
80
10.480
0.5
0.000100
S3-80/200
500
225
200
20
80
7.398
0.4
0.000100
S3-80/150
500
250
150
20
80
6.781
0.3
0.000100
S3-80/100
500
275
100
20
80
4.932
0.2
0.000100Table 7
Percentage error between the computational and experimental results for single-step test samples (Group S2). Lowest four nonzero free-free modes.
d
-
L
1
μ
Method
Modal frequencies in Hz
1
2
3
4
20/40
40
0.2
(A) EXP
1848.940
5028.140
9854
16303
(B) NTM
1873.981
5194.229
10211.79
16776.400
(C) FE(3D)
1859.490
5056.749
9893.345
16380.170
(D) % error (A, B)
1.354
3.303
3.630
2.903
(E) % error (A, C)
0.570
0.568
0.399
0.473
100
0.5
(A) EXP
2153.900
7713.530
13701.800
19845.900
(B) NTM
2244.601
7923.127
13661.990
20230.700
(C) FE(3D)
2142.012
7700.169
13663.580
19793.360
(D) % error (A, B)
4.211
2.717
−0.290
1.938
(E) % error (A, C)
−0.551
−0.173
−0.278
−0.264
140
0.7
(A) EXP
3596.820
8103.960
15773.100
20733.800
(B) NTM
3813.426
8160.003
15739.160
21174.340
(C) FE(3D)
3581.521
8090.723
15705.070
20634.360
(D) % error (A, B)
6.022
0.691
−0.215
2.124
(E) % error (A, C)
−0.425
−0.163
−0.431
−0.479
25/40
40
0.2
(A) EXP (Figure11)
2430.240
6510.630
12186.700
19359.300
(B) NTM
2444.111
6638.069
12492.520
19588.580
(C) FE(3D)
2430.512
6518.094
12223.080
19314.540
(D) % error (A, B)
0.570
1.957
2.509
1.184
(E) % error (A, C)
0.011
0.114
0.298
−0.231
100
0.5
(A) EXP
2796.050
8693.500
14542.200
22134
(B) NTM
2912.321
8781.311
14633.780
22182.430
(C) FE(3D)
2806.782
8698.963
14551.070
22095.920
(D) % error (A, B)
4.158
1.010
0.629
0.218
(E) % error (A, C)
0.383
0.062
0.060
−0.172
140
0.7
(A) EXP
4015.620
8665.980
16238.9
22684
(B) NTM
4167.385
8797.248
16074.330
22814.460
(C) FE(3D)
4021.237
8655.476
16159.230
22610.690
(D) % error (A, B)
3.779
1.514
−1.013
0.575
(E) % error (A, C)
0.139
−0.121
−0.490
−0.323
30/40
40
0.2
(A) EXP
3023.690
7906
14381.400
21856.600
(B) NTM
3023.321
7946.457
14429.850
21806.140
(C) FE(3D)
3014.442
7878.443
14307.960
21781.750
(D) % error (A, B)
−0.012
0.511
0.336
−0.230
(E) % error (A, C)
−0.305
−0.348
−0.510
−0.342
100
0.5
(A) EXP
3401.940
9329.910
15681.400
23607.700
(B) NTM
3478.703
9293.555
15691.770
23326.460
(C) FE(3D)
3405.324
9313.349
15645.690
23513.650
(D) % error (A, B)
2.256
−0.389
0.066
−1.191
(E) % error (A, C)
0.099
−0.177
−0.227
−0.398
140
0.7
(A) EXP
4201.680
9322.630
16596.400
23967.700
(B) NTM
4268.272
9431.643
16437.830
23729.240
(C) FE(3D)
4211.100
9322.286
16530.200
23881.400
(D) % error (A, B)
1.584
1.169
−0.955
−0.994
(E) % error (A, C)
0.224
−0.003
−0.398
−0.360
35/40
40
0.2
(A) EXP
3608.280
9126.360
16128.600
23817.3
(B) NTM
3607.895
9104.530
16000.810
23500.960
(C) FE(3D)
3607.465
9106.544
16054.460
23720.110
(D) % error (A, B)
−0.010
−0.239
−0.792
−1.328
(E) % error (A, C)
−0.022
−0.217
−0.459
−0.408
100
0.5
(A) EXP
3885.030
9778.800
16719.900
24538.800
(B) NTM
3906.518
9705.047
16577.640
24125.780
(C) FE(3D)
3882.419
9759.382
16658.180
24417.41
(D) % error (A, B)
0.553
−0.754
−0.850
−1.683
(E) % error (A, C)
−0.067
−0.198
−0.369
−0.495
140
0.7
(A) EXP
4248.240
9888.540
17028
24738.400
(B) NTM
4253.660
9878.822
16849.850
24311.230
(C) FE(3D)
4247.271
9870.286
16963.900
24616.760
(D) % error (A, B)
0.127
−0.098
−1.046
−1.726
(E) % error (A, C)
−0.022
−0.184
−0.376
−0.491Table 8
Percentage error between the computational and experimental results for two-step samples (Group S3). Lowest four nonzero free-free modes.
Method
Modal frequencies in Hz
1
2
3
4
S3-80/250
(A) EXP
403.148
2126.460
4139.830
5153.500
(B) NTM
420.212
2203.984
4334.192
5605.556
(C) FE(3D)
404.180
2126.900
4153.700
5169.800
(D) % error (A, B)
−3.966
3.517
4.484
−8.428
(E) % error (A, C)
−0.255
−0.020
−0.333
−0.315
S3-80/200
(A) EXP (Figure12)
354.790
1727.900
2291.490
4531.380
(B) NTM
367.381
1788.472
2551.555
4660.983
(C) FE(3D)
355.230
1728.300
2318.400
4532.200
(D) % error (A, B)
−3.549
3.386
10.192
−2.860
(E) % error (A, C)
0.124
0.023
1.160
0.018
S3-80/150
(A) EXP
335.859
1360.050
1545.130
3778.200
(B) NTM
346.243
1434.113
1638.174
3889.033
(C) FE(3D)
336.440
1364.200
1548.500
3781.900
(D) % error (A, B)
−3.091
−5.445
−6.021
−2.933
(E) % error (A, C)
0.172
0.305
0.218
0.097
S3-80/100
(A) EXP
337.400
1018.240
1371.070
3221.200
(B) NTM
345.942
1074.574
1424.992
3303.837
(C) FE(3D)
337.480
1024
1372.900
3215
(D) % error (A, B)
−2.531
−5.532
−3.932
−2.565
(E) % error (A, C)
0.023
0.565
0.133
−0.192Figure 12
Experimental first four natural frequencies for test sample S3-80/200.


## 3.3. Percentage Modal Deviation between FE(3D) and NTM Method

The results of the previous section show the accuracy of the FE(3D) model in evaluating the natural frequencies of stepped beam. Therefore, in this section, a free-free two-span model is deeply investigated using (40) and FE(3D) methods to justify the validity of the analytical solution in predicting the stepped beam results. Two categories of samples are considered as shown in Figure 13. The first category includes 600 mm length samples with r1∗2=2.78×10-4 and the second category includes 200 mm length samples with r1∗2=205×10-3. The effects of changing μ and d- are investigated, μ varies from 0 to 1, and meanwhile, only three values are listed in Table 9. Four different values of d- are investigated 0.5, 0.625, 0.75, and 0.875. The study in this section focuses only on the first three nonzero free-free modes. The percentage modal deviation in analytical NTM solution prediction in reference to the FE(3D) solution is presented in Figure 13 for long samples and in Figure 14 for short samples. This percentage deviation is calculated using the following formula:(42)Devi=fNTMi-fFEifFEi∗100,whereDevi is the percentage deviation in the ith modal frequency,f(FE)i is the ith mode natural frequency using three-dimensional finite element,f(NTM)i is the ith mode natural frequency using analytical NTM method.Table 9
Typical relative deviation between present NTM and finite element (3D) results. Lowest three nonzero modes for two categories of single-step F-F beam; see Figure13.
d
-
Method
μ
Modal frequencies in Hz
L
=
600 mm
L
=
200 mm
1
2
3
1
2
3
0.5
NTM
0.1
217.034
635.604
1267.590
1878.302
5163.860
9645.650
0.4
232.036
797.543
1834.110
1995.497
6510.740
13831.40
0.7
447.629
1037.270
2253.940
3813.426
8160
15739.200
FE(3D)
0.1
216.987
634.850
1263.259
1876.267
5123.790
9448.138
0.4
229.554
785.722
1819.305
1930.166
6271.645
13642.314
0.7
437.591
1033.915
2245.125
3583.014
8091.330
15706.439
% dev.
0.1
−0.022
−0.119
−0.343
−0.108
−0.782
−2.090
0.4
−1.082
−1.504
−0.814
−3.385
−3.812
−1.386
0.7
−2.294
−0.324
−0.392
−6.431
−0.849
−0.208
0.625
NTM
0.1
283.930
812.357
1608.220
2446.347
6532.940
11979.500
0.4
308.616
991.955
2074.890
2639.638
7837.580
15025.700
0.7
495.792
1128.820
2333.290
4167.385
8797.250
16074.300
FE(3D)
0.1
283.898
811.795
1604.956
2445.458
6509.821
11855.709
0.4
305.955
981.347
2072.554
2570.408
7647.979
15037.839
0.7
489.234
1121.452
2336.177
4021.301
8655.042
16159.248
% dev.
0.1
−0.011
−0.069
−0.203
−0.036
−0.355
−1.044
0.4
−0.870
−1.081
−0.113
−2.693
−2.479
0.080
0.7
−1.341
−0.657
0.123
−3.633
−1.643
0.526
0.750
NTM
0.1
355.027
993.279
1944.090
3030.222
7818.400
14004
0.4
381.639
1149.720
2248.110
3236.367
8841.600
15842.900
0.7
511.227
1230.890
2406.190
4268.272
9431.640
16437.800
FE(3D)
0.1
355.036
993.179
1943.016
3031.657
7819.722
13978.716
0.4
379.583
1144.159
2251.351
3181.291
8759.634
15934.664
0.7
508.439
1224.396
2410.039
4205.930
9312.425
16529.197
% dev.
0.1
0.002
−0.010
−0.055
0.047
0.017
−0.181
0.4
−0.542
−0.486
0.144
−1.731
−0.936
0.576
0.7
−0.548
−0.530
0.160
−1.482
−1.280
0.553
0.875
NTM
0.1
429.273
1177.820
2274.860
3616.212
9013.570
15741.600
0.4
448.134
1269.840
2419.690
3759.295
9566.420
16569.700
0.7
511.380
1312.660
2498.460
4253.660
9878.820
16849.800
FE(3D)
0.1
429.347
1178.446
2277.013
3621.275
9047.347
15829.045
0.4
447.258
1269.273
2423.695
3739.834
9595.934
16695.492
0.7
510.778
1310.738
2502.153
4247.181
9870.045
16963.843
% dev.
0.1
0.017
0.053
0.094
0.140
0.373
0.552
0.4
−0.196
−0.045
0.165
−0.520
0.308
0.753
0.7
−0.118
−0.146
0.148
−0.153
−0.089
0.672Figure 13
Long and short stepped samples at three values ofμ=0.1,0.4,  and  0.7 and four values of d-=h21=0.5,0.625,0.75,  and  0.875.Figure 14
Variation of the percentage modal deviation between FE(3D) and NTM results as a function ofμ and d-, for free-free long samples L = 600 mm: (a) 1st mode, (b) 2nd mode, and (c) 3rd mode.
(a)
(b)
(c)The results of Table9 show that, for the investigated examples with step ratio smaller than one, the increase in μ and/or d- increases the modal frequencies. The percentage deviations in the analytical NTM results for the short samples are higher than those for the longer samples. The percentage deviations in the analytical natural frequency prediction are plotted for the long and short examples in Figures 14 and 15, respectively. Figures 14(a) and 15(a) present a plot for Dev1, Figures 14(b) and 15(b) present a plot for Dev2, and Figures 14(c) and 15(c) present a plot for Dev3. Figure 16(a) presents the first mode shape at the conditions of the peak point in Figure 15(a), Figure 16(b) presents the second mode shape at the conditions of the peak point in Figure 15(b), and Figure 16(c) presents the third mode shape at the conditions of the peak point in Figure 15(c).Figure 15
Percentage modal deviation between FE(3D) and NTM as a function ofμ and d-, for free-free short samples L = 200 mm: (a) 1st mode, (b) 2nd mode, and (c) 3rd mode.
(a)
(b)
(c)Figure 16
Typical first three modal shapes at the peak points shown in Figure15 for free-free short sample and d- = 0.5.
(a)
First mode
(b)
Second mode
(c)
Third modeFigures14(a) and 15(a) show that, for long and short stepped samples, the Dev1 attains the peak when μ lies between 0.5 and 0.7; that is, the step in the beam is located around the peak of the first free-free mode shape; see Figure 16(a). Figures 14(b) and 15(b) show that there are two peaks in Dev2 prediction. The location of these peaks is found to be near the position of the peaks of the second free-free mode shape; see Figure 16(b). In addition, the value of Dev2 approaches zero when the step location lies in the semistraight line between the two peaks in the second F-F mode shape. The same trend is repeated in the third mode shape as shown in Figures 14(c), 15(c), and 16(c). In addition, when the location of diametric step in shaft coincides with a peak point in i mode shape, the value of Devi is increased. Meanwhile, when the step lies in a straight portion of the mode shape, the Devi approaches zero.


## 3.4. Tapered Beam Approach

Due to the importance of tapered or conical beams in many engineering applications, the current section is devoted to show how to use the present analysis to solve the problem of taper beam. The current analysis is based on uniform beams, while the partial differential equation which represents the lateral vibration of tapered or conical beams is fourth-order Bessel equation [30, 31, 37, 38]. To simulate nonuniform beam using the current analysis, the beam is divided into multiple equal length spans as shown in Figure 17. The height and/or width of these spans are varying linearly between the start and the end to simulate the tapered or conical beam. The height and/or width ratio of i span can be calculated from the following formula:(43)hi1=hn1+1-hn1-xiL+Li12n,bi1=bn1+1-bn1-xiL+Li12n,where(44)hi1=hih1,hn1=hnh1,bi1=bib1,bn1=bnb1,Li1=LiL1.hi is the height of i beam segment and bi is the width of i beam segment.Figure 17
Stepped tapered beam with varying depth and width.To verify the suitability of the current model to represent conical beams, the results of the current model are compared with the exact solution for cantilevered (C-F) conical beam with variable taper ratiohn1=bn1=0.2,0.5, and 0.7 as shown in Table 10. The model was investigated using the present NTM and using several number of spans n=20,100,200, and 1000. The first three eigenvalues are evaluated for each taper ratio and number of spans. The time used for computing the first three natural frequencies is evaluated. The model results are also evaluated using NAT previously published in [6] at n=20 in order to compare the time saving when using the present method.Table 10
The first three eigenvalues for C-F conical beam using NTM in comparison with those of the exact solution presented in [30] and NAT results using the program of [6].
h
n
1
=
b
n
1
[30]
NAT [6]  n=20
NTM n=20
NTM n=100
NTM n=200
NTM n=1000
Time (s)
Time (s)
Time (s)
Time (s)
Time (s)
0.2
λ
1
∗
2
6.1964
6.1683
6
6.1706
1.8
6.1954
5.3
6.1962
10.5
6.1963
32.6
λ
2
∗
2
18.3855
18.2513
18.2513
18.3801
18.3840
18.3853
λ
3
∗
2
39.8336
39.4809
39.4809
39.8194
39.8300
39.8336
0.5
λ
1
∗
2
4.6252
4.6155
6.2
4.6175
1.9
4.6249
5.4
4.6250
10.9
4.6251
34
λ
2
∗
2
19.5476
19.5074
19.5074
19.5460
19.5472
19.5477
λ
3
∗
2
48.5789
48.4725
48.4725
48.5746
48.5779
48.5788
0.7
λ
1
∗
2
4.0669
4.0615
6.2
4.0635
1.9
4.0668
5.6
4.0669
11.1
4.0670
35
λ
2
∗
2
20.5554
20.5360
20.5361
20.5547
20.5553
20.5554
λ
3
∗
2
—
53.9625
53.9625
54.0131
54.0147
54.0152The results of Table10 show that increasing the number of spans results in increasing the accuracy of the evaluated beam eigenvalues in comparison with the exact solution in [30]. On the other hand, the results show that increasing the number of spans results in increasing the computational time. The comparison between the computational time using the present NTM and previously published NAT shows that the present NTM method is quicker than the NAT method [6] for the same number of spans.To investigate the capability of the present model to evaluate the natural frequencies of taper or conical beam with variable boundary conditions, some cases of nonuniform beams are selected form [31]. The analysis in [31] is numerical and based on solving the taper beam partial differential equation using Runge-Kutta method. The boundary conditions and the taper ratio are listed in Table 11. The number of spans used in evaluating the current example is n=1000. The eigenvalue results using the present NTM method are in good agreement with numerical results of [31].Table 11
The first four eigenvalues for taper or conical beam subjected to variable end conditions using the present NTM and [31].
Beam parameters
Frequency parameterλ1∗2
[31]
Present NTM
h
n
1
=
1  Z1=Zn+1=108  Φ1=108,Φn+1=1000
21.94
21.9518
60.44
60.5461
118.7
118.7589
196.4
196.4162
b
n
1
=
0.5  Z1=Zn+1=10  Φ1=Φn+1=102
4.98
5.0292
12.50
12.4289
40.13
39.8154
88.28
88.0372
b
n
1
=
1
,
h
n
1
=
1.5
1
/
3  Z1=102,Zn+1=2×102  Φ1=3×102,Φn+1=4×102
13.63
13.6709
24.59
24.9270
48.37
48.7915
97.28
97.5762
b
n
1
=
h
n
1
=
3
1
/
4  Z1=Zn+1=108  Φ1=Φn+1=108
25.83
25.8327
71.00
71.1461
139.3
139.4223
230.3
230.4320


## 4. Conclusion

A new proposed normalized transfer matrix NTM uses new set of fundamental solution in combination with the transfer matrix method. This method has the advantage of the TMM in that the determinant of the frequency equation is 4 × 4 forn number of spans. In addition, the formation of the system frequency equation determinant is not included in any inverse matrix steps which reduces the computational time.The current work introduces a comparison between the experimental and analytical NTM and three-dimensional finite element analyses for stepped thick beams. Different system parameters such as the step diameters ratiod-, the step location parameter μ, and the elastic segment length ratio L2∗ are considered. The following conclusions can be drawn from the present work:(1)
An excellent agreement between the present experimental results, using several test samples, and those of FE(3D) predictions has been recorded.(2)
The results show that the three-dimensional finite element can be trusted in the prediction of the modal frequencies of stepped thick beams in structural and mechanical system.(3)
An interesting percentage of modal deviations between the FE(3D) results and those obtained using the present frequency equation (40) has been illustrated. An increase in the step down location parameter (μ) and/or (d-) increases the modal deviations. The percentage deviations are higher for shorter beam than for longer one. The maximum deviation value in a specific normal mode is obtained when the step location is adjacent to a peak location in this normal mode. Moreover, the minimum percentage deviation in a specific mode shape is obtained when the step location lies in a straight portion in the mode shape. This may be explained by the smaller effect of stress concentration at the step under such conditions.(4)
Typical example shows that the computational time using normalized transfer matrix (NTM) is greatly reduced in comparison with numerical assembly technique (NAT).In addition, to the above importance of conclusion, one can obtain accurate results for any combination of classical and elastic end conditions.(5)
Based on the stepped beam, the current analysis can be used to evaluate the results of tapered beam. The present NTM show a very good stability at very large number of spans which enables the accurate evaluation of the tapered beam results using stepped beam analysis.



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