Research Article | Open Access
A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods
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.
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 . Almost 95 years ago, Timoshenko introduced a correction for the beam theory to include the shear deformation effect . 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  derived formulae for the precise evaluation of the shear coefficient for rectangular and round cross sections as a function of Poisson’s ratio .
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 for beam with 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 segments is . 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 . The attachments include translation and rotational springs, masses, and undamped single degree of freedom system. The characteristics equation for beam with segments is . 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 segments beam, the size of the frequency equation determinant is which reduces the computational time. On the other hand, during the formulation of the beam frequency equation 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  and Li et al. . 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 .
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. . 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 . 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. . 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 . The differential quadrature element method (DQEM) was proposed to analyze the free vibration problem of beams with any discontinuities in cross-section . Discrete singular convolution (DSC) was proposed for solving the free vibration analysis of stepped beams . The solution for multistep Timoshenko beam using both numerical assembly technique and differential transformation method is proposed by Yesilce . 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.  and Rossi et al. . Farghaly  derived the exact solution for four-span Timoshenko beam with attachments. Yesilce  investigated the free vibration of stepped beams using exact numerical assembly technique (NAT) and using approximate differential transformation method. Recently, Farghaly and El-Sayed  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 . 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. . 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 . 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 .
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 ratio , step location parameter , 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 Figure 1. The total length of the beam is . The beam model is divided into segments. The beam has stations as shown in Figure 1. The station numbering corresponding to the start, intermediate, and end location is represented by , 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 and , 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 is used to compensate this assumption. In addition, the effect of stress concentration at the beam steps is neglected.
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 Figure 1. Timoshenko differential coupled equations of motion may be written here for th span as follows:Letwhere is the normal function of , is the normal function of , is nondimensional length of each beam span , and .
Substituting (2) into (1) and omitting the factor , the following equations can be derived:whereAfter decoupling the functions and in ((3)-(4)), the decoupled fourth-order differential equations in the nondimensional form can be written aswhereThe general solution of (6) and (7) can be written, respectively, in the formHere,One can derive the expressions of and using (8), together with (3) or (4) in the formHere denotes the th span, in the case of multispan model.
In order to introduce the current analysis, the linearly independent fundamental solutions and the corresponding are derived. In order to simplify the solution of Timoshenko beam, the following dependent functions are defined:The Timoshenko solution will be normalized at the origin of coordinates as follows:Substituting the general solution of (8) in each raw in (12) we get the following set of fundamental solutions:The general solution of the beam can be presented in terms of the set of fundamental solutions aswhere , , , and .
The beam start boundary conditions at the point of attachment 1 can be presented in nondimensional form aswherewhere is the mass moment of inertia at station 1, and are the linear and rotational elastic supports at station 1, respectively, and 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:The start boundary conditions in (17) can be presented in matrix form asThis equation can be simply written aswherewhere the superscript indicates vector transpose.
At station , the beam end boundary conditions can be written in the nondimensional form aswherewhere is the mass moment of inertia at station , and are the linear and rotational elastic supports at station , respectively, and is the concentrated mass at station .
Substituting the solutions in (13a), (13b), (13c), (13d), (13e), (13f), (13g), (13h)-(14) into (21), the following equations are obtained:Equation (23) can be written in the matrix form asThis equation can be simply written aswhereThe beam intermediate continuity conditions can be presented in nondimensional form aswherewhere is the mass moment of inertia at station , and are the linear and rotational stiffness at station , respectively, and is the concentrated mass at station .
Equation (29) can be written in matrix form aswhereEquation (30) can be presented aswhereFrom (32), one can find thatThe intermediate spans transfer matrix can be presented asthen (34) can be presented asSubstituting (36) into the end condition of (25) results in the following equations: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 belowwhereEquating the determinant of 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 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 Figure 2. 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.
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 frequencies of stepped beam are compared with the exact solution presented by Gutierrez et al. ; see Figure 3. The model is solved at two different step locations and several values of and as shown in Table 1. Three different values of rotary inertia and are considered in order to validate the current model in case of Bernoulli-Euler and Timoshenko beams. The value of is considered in order to evaluate the shear deformation . The values of nondimensional linear and rotational elastic supports stiffness at start and the end are , , 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. .
3.1.2. Verification Example 2
In this example, Timoshenko beam with three-step round cross-section presented in  is investigated; see Figure 4. An intermediate lumped mass of 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 in (rad/sec) for pinned-pinned, free-clamped, clamped-free, clamped-pinned, and clamped-clamped configurations. The results of reference  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 . The results of the current analysis are in good agreement with the results presented in . 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.
3.1.3. Verification Example 3
The third verification example is shown in Figure 5 with identical dimensions to that used in [5, 20]. The rotary inertia and shear deformation are considered. The shear coefficient is calculated based on , keeping Poisson’s’ ratio = 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  is less than 0.302%. This represents the importance of including the effect of rotary inertia and shear deformation.
3.1.4. Verification Example 4
The fourth example is shown in Figure 6. It is for a twenty-span uniform beam carrying 19 equally spaced concentrated masses with . 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  method. The computational time required to obtain the first three frequency parameters , and 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.
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. Figure 10 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 80 and 40 mm. The common data for the steel test samples are , , and .
Two-span twelve test samples are shown in group S2, namely, S2-40/40, S2-40/100, and S2-40/140, shown in Figure 7. 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%.
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 Table 6 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.
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 and the second category includes 200 mm length samples with . The effects of changing and are investigated, varies from 0 to 1, and meanwhile, only three values are listed in Table 9. Four different values of 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:where is the percentage deviation in the th modal frequency, is the th mode natural frequency using three-dimensional finite element, is the th mode natural frequency using analytical NTM method.
The results of Table 9 show that, for the investigated examples with step ratio smaller than one, the increase in and/or 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 , Figures 14(b) and 15(b) present a plot for , and Figures 14(c) and 15(c) present a plot for . 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).
(a) First mode
(b) Second mode
(c) Third mode
Figures 14(a) and 15(a) show that, for long and short stepped samples, the attains the peak when 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 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 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 mode shape, the value of is increased. Meanwhile, when the step lies in a straight portion of the mode shape, the 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 span can be calculated from the following formula:where is the height of beam segment and is the width of beam segment.
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 ratio , and as shown in Table 10. The model was investigated using the present NTM and using several number of spans , and . 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  at in order to compare the time saving when using the present method.
The results of Table 10 show that increasing the number of spans results in increasing the accuracy of the evaluated beam eigenvalues in comparison with the exact solution in . 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  for the same number of spans.
To investigate the capability of the present model to evaluate the natural frequencies of taper or conical bea