Analysis of the Spatial Vibration of Nonprismatic Arches by Means of Recurrence Relations for the Coefficients of the Chebyshev Series Expansion of the Solution
The problem of spatial vibrations, both aperiodically forced and free vibrations, of an arch with an arbitrary distribution of material and geometric parameters is considered. Approximation with Chebyshev series was used to solve a conjugated system of partial differential equations describing the problem. The system of differential equations was solved using an algorithm generating a recursive infinite system of equations, developed by S. Paszkowski in “Numerical applications of Chebyshev polynomials” (in Polish), Warsaw PWN, 1975. Since the coefficients of the obtained system of equations are defined by closed analytical formulas they can be directly used to solve any nonprismatic arch, without it being necessary to solve again the considered problem. The algorithm is highly accurate; i.e., already at a small approximation base it yields results agreeing with exact analytical solutions (obviously for problems in the case of which such solutions can be derived). In order to demonstrate this the eigenfrequencies and eigenvectors obtained for a circular prismatic arch were compared with their precise values determined from the exact analytical solutions. The results yielded by the proposed method were also compared with the results obtained by other methods and by other authors. As an illustration, the proposed method was used to solve a more complex problem, i.e., the problem of the free and aperiodically forced vibrations of a nonprismatic arch with its axis described by a catenary curve. In the example the effect of the lack of cross-sectional symmetry of the arch on the form of the system’s spatial free and forced vibrations was analysed.
Since curved beams are often used in civil, mechanical, and aerospace engineering applications, the analysis of their vibrations is of great practical importance. Being described by a conjugated system of partial differential equations with four or six (when the effect of shearing forces is taken into account) unknown functions, the spatial arch vibration problem is quite complicated. It is particularly difficult to solve when all the system parameters, such as curvature and geometric and material characteristics, are variable.
The arch vibration problem has been investigated by many researchers, as evidenced by the abundant literature on this subject. Most of this literature concerns planar systems in which only in-plane or out-of-plane vibrations occur or spatial systems with separated in-plane and out-of-plane vibrations (vibrations are separated when the “generalized” cross-sectional moment of deviation equals zero). The problem of the free vibration of prismatic arches was examined by, among others, Chidamparam and Leissa  and Lee et al. . In Chidamparam and Leissa’s paper  the problem was solved analytically with and without axial deformability taken into account. Also the effect of a static axial force on the eigenproblem solution by the Galerkin method was examined there. Circular arches with different opening angles were considered, but only to determine the natural frequencies. In paper  by Lee et al. the natural vibrations of a rotating curved beam with elastically restrained root were analysed. The differential equations describing the problem were derived using the Hamilton principle. The fundamental solution of the system of the differential equations was obtained using the power series method.
The problem of the free vibration of arches with stepped cross sections was solved, using different methods, by, among others, Huang at al. , Kawakami et al. , Liu and Wu , Shin et al. , and Tong et al. . Huang et al.  analysed arches with any curvature and any cross section and employed the Frobenius method combined with the dynamic stiffness method to solve the problem. The same method was used by Huang et al. in  to solve the free vibration-and-stability problem with shape deformation taken into account. Kawakami et al. in paper  solved the eigenproblem using the discrete Green function. Liu and Wu  applied the generalized differential quadrature rule (GDQR) to solve the eigenproblem, assuming the arch axis to be inextensible. The numerical examples presented in  concerned circular arches with different opening angles and with a constant and stepwise or linearly varying cross section. The obtained nondimensional natural frequencies were compared with the ones determined by other methods by other researchers. The generalized differential quadrature method (GDQM) and the differential transformation method (DTM) were employed by Shin et al. . In paper  by Tong et al. a circular arch was analysed and a solution for a prismatic arch was derived, which was then used to solve an arch (considered as a physical approximation) with stepped cross sections. Besides the eigenproblem also the harmonically forced vibration problem was solved there. Karami and Malekzadeh in paper  used the differential quadrature method (DQM) to analyse the free vibrations of a circular arch with stepped cross sections. Nieh et al.  analytically determined (by the power series method) a stiffness matrix for a prismatic arch, which was then used to solve the free vibration-and-stability problem for an elliptical arch. Eftekhari  analysed the vibration of a circular arch with a varying cross-sectional height, applying DQM and Fourier series approximation to solve problems of the free and forced vibrations induced by a moving force.
In many papers the finite element method has been used to analyse vibrations. Krishnan et al.  using three different types of finite elements (differing in their number of degrees of freedom) solved the eigenproblem, but for only the first natural frequency. In Raveendranath et al. , various types of elements, differing in the degrees of the power polynomials used to approximate the shape function, were developed. The elements were used to solve the vibration problem for a circular prismatic arch. Yang et al. in  developed two types of elements, corresponding to two arch models. One of the models took into account the effect of shape deformability and rotary inertia, whereas the other did not, assuming that the arch axis did not deform axially. The shape functions were approximated with power polynomials. The defined finite elements were used to solve many numerical examples (including arches with variable parameters) in which eigenfrequencies and eigenforms were determined. Öztürk et al. , using the assumption about arch axial nondeformability and assuming the shape functions in the form of a combination of trigonometric functions and classical polynomials, developed a finite element and employed it to solve the free vibration-and-dynamic stability problem for the arch. An element developed using exact (for the static problem) shape functions (Litewka and Rakowski ) was employed in Litewka and Rakowski . The functions, being combinations of trigonometric functions, were further transformed by replacing the trigonometric functions with their expansions into power series and rejecting the higher-order terms. The finite element was used to solve the free vibration problem. Also Zhu and Meguid  worked on developing a finite element for the curved beam. A three-node element was defined and used to analyse free vibrations. The results were compared with experimental results. An interesting approach to the solution of the free vibration problem for an arch with discontinuities (additional elastic constraints, mass elements, and stepwise change in curvature), based on wave propagation techniques was presented in Kang et al.  and Riedel and Kang . In the latter paper only the discontinuities arising from additional elastic constraints were taken into account. Using the coupled displacement field methodology and coupled shape functions derived from the static equilibrium consideration Ishaquddin et al.  developed a curvilinear beam finite element for the Euler-Bernoulli beam and the Timoshenko beam. The designed element is resistant to shear, flexure, and torsion locking. The element was used in an analysis of the circular arch eigenproblem. The computed out-of-plane free vibration frequencies were compared with the analytical solution.
Curved laminated composite beams with constant curvature were considered by Jafari-Taookolaei et al. , who analytically solved the eigenproblem, taking delamination into account, and developed a curvilinear finite element. Li et Geudes Soares  developed a spectral finite element model based on shear deformation theory. The exact solution of the governing homogeneous differential equations was used as the shape functions. The eigenproblem of a circular laminated arch with a constant cross section was analysed and natural frequencies and eigenforms were determined. Laminated composite and sandwich beams with constant and varying (circular, elliptical, and parabolic) curvature were analysed by Ye, Jin, and Su . A spectral-sampling surface method was developed and applied to the eigenproblem of the curved laminated beam. Sadeghpour et al.  analysed a debonded curved sandwich beam. Relevant equations of motion were derived from the Lagrange equation. The Rayleigh-Ritz method was used to discretize the system. The eigenproblem was solved using the Lanczos algorithm. The determined natural frequencies and eigenforms were compared with the results yielded by the finite element method.
Most of the above works on laminated beams focused on the analysis of the effect of lamination on the system’s response. The considerations were limited to arches with constant curvature and a cross section invariable along their length.
The work by Yu et al.  is an example of an analysis of more complex models, including curved spatial systems. A spatial beam model based on Washizu’s static model was analysed. The model was used to investigate the in-plane and out-of-plane vibrations of a circular arch with a triangular cross section. Also the spatial problem was solved and the eigenfrequencies of a helical cylindrical spring were determined. The spatial free vibrations of circular planar arches were analysed in work  by Caliò, assuming the differential equations describing the in-plane and out-of-plane vibrations of the arch to be separated. Caliò introduced dynamic stiffness matrices for an arch and a curved-in-plane girder. He solved the eigenproblem using the Wittrick and Williams method. The results obtained as part of a numerical example illustrating the analysis of the vibrations of a spatial system consisting of arch elements were compared with the results yielded by FEM.
The above survey of literature covers papers published in the last ten-twenty years, and so it is obviously incomplete. A wide survey of the earlier papers can be found in, e.g., the review study by Auciello and De Rosa . Nevertheless, even this brief survey of literature shows that most of the research on the vibrations of arches has been limited to solving the eigenproblem and only a few of the papers are devoted to the problem of (usually harmonically) forced vibrations.
The problem was solved using the method previously applied to solve the free vibration problem for the Euler beam in the author’s papers [29, 30] and for the Timoshenko beam in . The method is based on the way of approximately solving ordinary linear differential equations by means of Chebyshev series, presented in the monograph by Paszkowski . The way referred to makes it possible to reduce the differential equations to a system of algebraic equations. Unfortunately, the coefficients of the latter equations are combinations of the linear coefficients in the expansions of the initial functions (occurring in the differential equations) and their higher derivatives. An effective algorithm can be developed by transforming the coefficients so that they will contain only the terms of the expansions of the initial functions and not their derivatives. Because of their length and the limited confines of this paper the transformations are not included here. It should be noted that the final solution for the given form of differential equations has a general character and enables ones to solve a system with any geometric and material parameters. Moreover, as demonstrated in this paper and in the earlier papers by the author, the proposed method makes it possible (within the adopted model) to obtain highly accurate solutions.
The use of Chebyshev series to solve structural mechanics problems (and other problems) is a known fact. The series are used owing to, among other things, their good approximation properties. However, to the present authors’ knowledge there have been no other works in which the recursive algorithm (see Paszkowski ) employed in this paper was used to generate equations enabling the direct determination of the expansion coefficients of the sought functions.
In order to verify the derived formulas, the algorithm was applied to solve two examples. The first one was taken from papers by other authors and used to compare the obtained results with the ones determined by other methods. The eigenproblem was solved in this example. In the second example the problem of the aperiodically forced vibrations of an arch with a varying cross section was analysed. Since in this example the eigentransformation method was employed to solve the aperiodically forced vibration problem, the eigenproblem was solved in the first step. A rectangular pulse and a nonstationary harmonic excitation were assumed as the aperiodic excitation.
2. Problem Formulation
A nonprismatic arch, described in accordance with the Bernoulli-Euler theory, is analysed.
The arch is subjected to time-variable force loads and moment loads (Figure 1). The axis of the arch is a plane curve having length , defined by parametric equation , where .
If the nondimensional variables and functions , , , , , , , and are introduced, the following system of fourth-order partial differential equations describing the arch vibrations is obtained:and the nondimensional internal forces are defined by the following formulas:
(i) the axial forces
(ii) the bending moments
(iii) the shearing forces
(iv) the torsional momentwhere is a nondimensional displacement tangent to the arch axis, and are nondimensional displacements perpendicular to the arch axis, is the angle of twist, is the nondimensional arch curvature, , , , , are nondimensional external forces, and the nondimensional material and geometric characteristics are , density per unit length, , axial stiffness, , , flexural stiffness, , Young’s modulus, , the cross-sectional area, and , the generalized moments of inertia, expressed by the formulas:The constants in formulas ((1a), (1b), (1c), and (1d))–(7) arewhere parameters are reference quantities.
3. Problem Solution
The solution of differential equations system ((1a), (1b), (1c), and (1d)) is sought in the form of Chebyshev series of the first kindwhere is the -th Chebyshev polynomial of the 1st kind and , , , and are unknown coefficients of the expansion of displacement functions , , , into Chebyshev series, hereafter denoted as , , , .
The method presented in Appendix A and in the author’s papers [29–31] will be used to solve the problem. In this method the system of ordinary differential equations is reduced to a recursive infinite system of algebraic equations. If the initial equations are partial differential equations, a system of ordinary differential equation is obtained.
The characteristic feature of this method is that the coefficients of the obtained infinite system of equations are in an analytical form whereby the equations derived in this paper can be used to solve any arch, without it being necessary to transform the initial equations. The high accuracy of the results yielded by proposed method, demonstrated in this paper, is comparable with that of the exact closed analytical solutions.
In order to apply this method to the considered problem let us reduce the system of equations to the matrix formwhere matrix functions and vector are expressed by the formulas (a simplified form of derivative is used to shorten the notation)Having determined matrix functions and ( is calculated similarly as by substituting functions for functions in formula (A.10)) from formula (A.10) and expanded them and functions (13) into Chebyshev series and substituted the calculated series coefficients into (A.9), one gets the infinite system of ordinary differential equations
At this stage of the solution the , elements in system of (14) are a linear combination of the coefficients of expansion of the “input functions” and the coefficients of expansion of their first and second derivatives. The term “input functions” applies to the products of the functions described in formula (B.4) (see Appendix B). After complex transformations involving the use of relation , where and , coefficients become a linear combination of only the coefficients of expansion of the “input functions”. A detailed description of the transformations can be found in the author’s paper  in which the Euler beam vibration problem was analysed. The ultimate form of coefficients , , and is presented in Appendix B.
The first blocks of (14) – (i.e., sixteen equations) are satisfied identitywise (0=0). The equations are replaced with the twelve boundary conditions which have not been used so far. The number of the conditions follows from the order of the equations in system ((1a), (1b), (1c), and (1d)). In the formulation of the equations stemming from the boundary conditions at the arch’s ends () one uses the expansions of displacement functions (6), formulas (2)-(7) for internal forces, and the following formulas for calculating the Chebyshev polynomials at points Exemplary equations for the two main ways of fixing the arch are expressed by the formulas:(i)clamped end (respectively on the left and right end of the arch)(ii)hinged endthe first five equations are identical as (16)1,2,3,4,6, whereas the sixth equation stems from the condition and has the form
In order to solve the modified system of (14) it was reduced to the finite system of equations . This is tantamount to the assumption that each of the displacement functions in formula (10) is approximated by a finite series with terms. When the order of the terms is changed, the system is expressed by the formulawhere , a , , , and .
System of (18) is further transformed. After left multiplying (18) by and using the theorem presented in Appendix C, matrix was reduced to the Jordan form. For the analysed system the solution of the eigenproblemleads to single eigenvalues , the transformation matrix has the form , where is an eigenmatrix obtained by solving eigenproblem (19), and the matrix becomes a diagonal matrix.
After substituting , introducing nondimensional time , adding a component describing the dumping, and making some simple transformations, system (18) was reduced to the system of separated equations
In order to reduce the computing time and eliminate the vibration forms encumbered with a large error (the eigenforms corresponding to the higher complex vibrational frequencies), the incorrect (inessential) eigenforms should be rejected. In this case, eigenforms are considered and transformation matrices and then become rectangular matrices with, respectively, and dimensions (Kleiber et al. , p.128), and vector .
The solution of the single equation from system of separated equations (20), when the function describing the load has the form (the Green function), is described by the formulawhere . In the case of the load described by any function, the solution was obtained by calculating the convolution of the load and the Green function.After all the components of vector had been calculated, the sought vector was determined.
4. Numerical Examples
In order to check the correctness and effectiveness of the proposed algorithm two examples are considered. In the first of the examples, systems taken from other authors’ papers are solved with an aim of comparing the results obtained by the proposed method with the ones obtained by other methods. Example 2 shows how the method can be used to solve a more complex system, i.e., one with an arbitrarily variable curvature and cross section, subjected to any aperiodic load. In the examples the parameter defining the size of the approximation base amounts to m=20. The computations were performed using the Wolfram Mathematica® 7 software .
Example 1. A circular arch with length , where is the radius and is the opening angle (), and a constant square () cross section was analysed.
In the case of in-plane vibrations, two variants of arch clamping, stiff clamping at both ends (C-C) and hinged fixing (H-H), are assumed. In the case of out-of-plane vibrations, stiff clamping at both ends (C-C) is considered. The determined natural frequencies are nondimensional and defined by the formula , where is the arch curvature radius. The calculations were performed for different values of parameter , assuming and . It is apparent from the definition of parameter that it indirectly defines the ratio of the beam’s flexural stiffness to its axial stiffness.
The following methods were used to solve the eigenproblem:(i)the method presented in this paper,(ii)the analytical method, a close solution was obtained using Mathematica® ,(iii)the finite element method, where 3D nonprismatic (tapered) beam elements with six degrees of freedom in each node were used (Cosmos/M),(iv)the approximation method based on classic power polynomials (Taylor series), where the sought displacements and rotation angles are described by the formulas
In the case of numerical approximation method calculations, each of the functions was approximated with 30 terms of the series, whereas for FEM the system was divided into 20 finite elements. Such a number of finite elements (20) were selected that the number of parameters in FEM and the number of parameters in the approximation methods were equal or almost equal. In the case of the approximation methods, 120 parameters were defined (four functions x 30 terms of the series), while in FEM there were 126 generalized coordinates (21 nodes x 6 coordinates) minus the coordinates eliminated from the calculations due to the assumed boundary conditions. The reason why “only” 20 elements were used was to compare the effectiveness of the two types of approximation (the approximation presented in this paper and the FEM approximation) as applied to the considered problem. In order to verify the convergence of the solutions obtained by FEM the calculations were also performed for a much denser finite element grid – by dividing the arch into respectively 100 and 500 finite elements. To ensure that the comparisons were made for the same systems the values of all the geometric characteristics defining the arch’s cross section were directly input into the Cosmos/M program.
The results yielded by the approximate methods were compared with the exact analytical solution.
The first twelve eigenfrequencies were compared for each of the considered cases, where at the adopted approximation base numerical computations yielded complex values already for the third eigenfrequency or higher eigenfrequencies (depending on the case). The calculated frequencies (units rad/s) are shown in Figures 2, 3, 4, 5, 6, and 7.
The frequencies connected with “the same” eigenform are given in the same row in the tables. Some eigenforms determined using FEM are “transposed”. For example, to the eigenform connected with eigenfrequency ω11=440.240 rad/s (the analytical solution) corresponds frequency 375.266 rad/s in the FEM solution, which is frequency no. 10 in the latter. To the eigenform connected with eigenfrequency ω10=416.206 rad/s (the analytical solution) corresponds frequency 417.517 rad/s in FEM, which is frequency no. 11 in the latter. In such cases, the frequencies corresponding to “the same” eigenform are connected with an arrow.
For comparison, also the natural frequencies obtained by other authors are included there. The results taken from Liu and Wu  (Figures 4 and 5) are for , i.e., for a model with neglected arch axial deformability.
In order to compare the eigenforms relative error functions were calculated from the formulawhere is the close analytical solution, is the approximate solution, and .
The analytically determined eigenforms were normalized so that the maximum value of displacement () was equal to one. Considering that the eigenforms were determined with an accuracy to a constant multiplier, the other solutions were normalized so that the mean-square error was minimumGraphs of relative error functions (24) on the logarithmic scale, for selected eigenforms 1, 4, 8, and 12, are shown in Figures 8–10.
The relative error (relative to the exact analytical solution) of the first 12 eigenfrequencies determined using the proposed method is below for arch in-plane vibrations and below for arch out-of-plane vibrations. Although the classic series approximation yielded slightly more accurate results (the relative error of ) for the first two eigenfrequencies, the next eigenfrequencies are encumbered with an increasingly larger error (ranging from to in the case of arch = 0.01 and from to in the case of arch = 0.1), whereas the complex numbers obtained for eigenfrequencies 3-5 in the case of the hinge-supported arch and for eigenfrequencies 6-8 in the case of the stiff clamping were incommensurable. In the analysed interval of eigenfrequencies FEM yields results whose accuracy ranges from to for the H-H arch and from to for the C-C arch in the case of the eigenfrequencies describing the arch in-plane vibrations and from to ( in isolated cases) in the case of the eigenfrequencies describing the arch out-of-plane vibrations.
An analysis of the obtained eigenvalues (Figures 2, 3, 4, 5, 6, 7) and the graphs of the relative errors (Figures 8–10) shows that the results yielded by the proposed method are considerably more precise than the ones obtained using the other methods.
In the case of eigenfrequencies, the relative error is smaller by several orders of magnitude than the errors of the other solution methods. Only the power series method yielded similarly accurate results for the first eigenforms. No further eigenforms were analysed because of the complex eigenfrequency values corresponding to them.
For the considered cases also the relative errors of the eigenforms were analysed at h/R = 0.1. The differences between the analytically determined eigenforms and the ones calculated using the proposed method for h/R = 0.01 were by 2-3 orders of magnitude smaller than the results discussed above for h/R = 0.1.
If one analyses the results presented in Figures 2, 3, 4, 5, 6, and 7, it becomes apparent that the “transposition” of the eigenforms in the case of the solutions obtained using FEM occurs mostly for out-of-plane vibration. Another observed regularity is the more frequent occurrence of “transpositions” for systems with h/R = 0.1 than for systems with h/R = 0.01 (“transpositions” occur more often in “stocky” systems). “Transpositions” are particularly numerous in Figure 6 (out-of-plane vibration and h/R = 0.1). All the “transpositions” occurring in Figures 6 and 7 concern “nearly pure” torsional modes, i.e., eigenforms in which the angle of cross section rotation is 2-3 orders higher than displacement .
A comparison of the eigenfrequencies yielded by the proposed method (at the approximation base of 30 series terms) with the analytically determined eigenfrequencies shows that they are practically identical (in the adopted range of accuracy) or only slightly different.
Example 2. The algorithm is used here to solve the eigenproblem and the aperiodically forced vibration problem of a catenary nonprismatic arch. Two kinds of arch fixing: clamped-clamped (C-C) and hinged-hinged (H-H) are assumed. Static schemes of these arches are shown in Figures 11 and 12.
The parametric equations for the arch, as a function of its length, are as follows:where is the height and is the span of the arch. Parameter in formula (26) is a function of the assumed ratio. The curvature of the arch is expressed by the formula is assumed. Hence parameter in formulas (26) and (27) is equal to .
The cross section’s height is described by the formulas
(i) for the clamped-clamped (C-C) arch
(ii) for the hinged-hinged (H-H) archwhere .
Two cross sections, a T cross section symmetric to the axis and an asymmetric L cross section, were considered (see Figure 13) in each of the static schemes. Such shapes and dimensions of the cross sections were selected that the latter’s areas and moments of inertia relative to the axis were equal in both arch versions. The cross sections only differed in the moments of inertia relative to the axis and in the moment of deviation (equal to zero for the T cross section and different from zero for the L cross section).
The eigenproblem was solved using the proposed method. Four cases, an arch with the symmetric T cross section and an arch with the asymmetric L cross section for each of the two arch clamping versions, were considered. Twenty terms of the series were used for approximation ( = 19). The same problem was solved using the FE method and forty finite elements with six degrees of freedom and a linearly variable cross section.
A comparison of the results obtained in this example shows high agreement between the natural frequencies calculated by the proposed method and the values yielded by FEM. In the case of the C-C arch, the maximum relative error between FEM and the proposed method for the first ten natural frequencies amounts to 3.709% and 1.461% for the T cross section and the L cross section, respectively. In the case of the H-H arch with the L cross section, the relative errors do not exceed 0.761%, except for the frequency, for which the error amounts to 6.246%. Slightly better results were obtained in the case of the H-H arch with the T cross section, where as regards the first ten frequencies the maximum difference between the FEM results and the results yielded by the proposed method amounted to 0.402%, except for the frequency, for which the relative error was 5.356%.
The first ten eigenforms were also compared for each of the arch clamping versions. The eigenforms nos. 1, 2, and 3 obtained by, respectively, the proposed method and the FE method are shown in Figures 14 and 15 (others eigenforms are not presented here due to the confines of this paper).
Figures 14 and 15 show good agreement between the eigenforms determined using the proposed method and the ones yielded by FEM. Up to eigenform 7 for the stiff clamping and up to eigenform 8 for the hinged fixing the graphs are practically identical. Single divergences appear at higher eigenforms (the latter are not presented here due to the confines of this paper).
By comparing the eigenfrequencies obtained for the systems with symmetric cross section T and asymmetric cross section L, the maximum differences between the eigenfrequencies were determined. The differences amount to 12.2% and 6.5% for, respectively, the C-C system and the H-H system.
Tables 3 and 4 show that the frequencies determined by the proposed method are convergent. It is also apparent that in the considered problems (20)-(30) terms of the series need to be used for the approximation in order to obtain sufficiently accurate results.
As regards the aperiodically forced vibration problem, the vibrations of the hinged-hinged arch loaded in the plane of its axis were analysed. The problem was solved for two cases differing in the spatial distribution of the load. In the first case, the load with intensity was uniformly distributed as shown in Figure 16. In the second case the arch was loaded with concentrated force , where , perpendicular to the arch (Figure 16). The force values were , . Calculations were performed assuming the following arch geometry parameters: b=0.01 m (Figure 13), (Figure 16). The parameters of the material properties were , , and .
Also two different cases of the time distribution of the load were considered. In the first case, the time distribution of the load had the form of a rectangular pulse and the load defining functions are expressed bywhere is the Heaviside function and is the pulse duration. In the second case, the time distribution had the form of the harmonic functionThe following functions of harmonic excitation were assumed:where is the first natural frequency of the arch with the L cross section; is the first natural frequency of the arch with the T cross section. The out-of-plane vibrations correspond with frequency , whereas the coupled spatial vibrations, where displacements and cross section rotation are one order of magnitude higher than displacements , , correspond with frequency .
An analysis of the above results shows that when the system is loaded with a concentrated force, the system’s responses (the three displacements and the rotation) in the investigated point are of the same order of magnitude. The exception are the vibrations induced by load pulses, where displacements are by one order of magnitude smaller than displacements . Moreover, in the case of the nonstationary vibrations induced by the harmonic load, displacements are several times smaller than displacements and . In the case of the vibrations induced by the load applied at a constant rate, all the displacements and the rotation are of the same order of magnitude, irrespective of the type of excitation (pulse or harmonic). In the case of the nonstationary vibrations of the L system, induced by the harmonic load, vibration amplifications (increases in vibration over time) are visible. Besides displacements , , also displacement and rotation are amplified, despite the fact that the load acts in the plane of the arch. The occurrence of displacements characteristic of out-of-plane arch vibrations is due to the appearance of conjugating elements , where , in system of (14). The amplification of the values of functions and is due to the amplification of vibrations in the resonance zone since excitation frequency is the eigenfrequency of the out-of-plane arch vibrations. Because of the separation of the equations describing the arch in-plane and out-of-plane vibrations (, where ) and the direction of the load (in the plane of the arch), no displacements characteristic of arch out-of-plane vibrations appears: .
The following general conclusions emerge from an analysis of the results obtained in the examples:(i)The derived final equations (enabling one to directly calculate the coefficients of expansion of the sought functions) have a general character stemming from the analytical form of the coefficients of the equations. Owing to this the vibration problem can be solved for any arch, described in accordance with the Bernoulli-Euler theory, without it being necessary to derive the equations again.(ii)The results obtained using the solution method proposed in this paper agree with the results obtained by other authors.(iii)The comparison of the results obtained by the different methods (FEM, the approximation method based on the classic power series, and the proposed method) with the analytical results has shown that results yielded by the proposed method are much more accurate than the ones obtained by the other methods.(iv)The method is highly accurate: already 30-power series element approximation yields results agreeing with the analytical solution results.(v)The examples show that the proposed method is fast convergent (Tables 3 and 4). When the size of the approximation base is increased to , 40, and 50, the accuracy of the results increases only slightly in comparison with the accuracy of the results obtained when .
The following detailed conclusions can be drawn from the analysis of the effect of cross section asymmetry on the form the spatial vibrations of the arch:(i)In the case of asymmetric cross sections, coupled vibrations are generated (arch in-plane vibrations are accompanied by out-of-plane vibrations and vice versa). When displacements connected with out-of-plane vibrations dominate, the accompanying coupled in-plane vibrations are relatively small. Whereas when displacements connected with in-plane vibrations dominate, the coupled out-of-plane vibrations arising in the system are of the same order of magnitude as the dominant vibrations. This regularity is observed in the case of free vibrations (Figures 14 and 15).(ii)The coupling of vibrations also occurs in the case of forced vibrations, especially when the vibrations are forced by a harmonically variable load. In the examples, even though the harmonic load acts in the plane of the arch, the out-of-plane vibrations are of the same order of magnitude as the in-plane vibrations. Since the excitation frequencies assume the resonant value or values close to those of the resonant frequencies, the characteristic increment in vibrations over time is visible in the vibration diagram.
A. Approximation Method of Solving Differential Equations with Variable Coefficients
In this paper, in order to solve the system of differential equations a generalization of the following theorem concerning ordinary differential equations ( p. 231) is used.
Theorem A.1. If function satisfies a linear equation with order where and functions have determinable coefficients of the Chebyshev series, then for each integer the following identity holds:where are polynomials of integer and is the -th coefficient of expansion of function into a Chebyshev series relative to Chebyshev polynomials of the 1st kind (the proof of this theorem can be found in  pp. 231-234).
The generalization of the theorem consists in the transference of the differential equation approximate solution method (described by the theorem) onto systems of linear differential equations (see  p. 323). In such a case, system of equations can be presented in this matrix formwhere coefficients are square matrices of degree and and are -element vectors. The differentiation of the vector means the differentiation of each of its components. If vector function satisfies system of (A.6) and the theorem’s assumptions hold good, then for each integer the following identity is true:Functions in the formula are matrix equivalents of the functions defined by formula (A.2)and stands for a vector whose elements are the -th coefficients of the Chebyshev expansion of the components of vector .
In a special case, when system of (A.6) is a 4th-order () system, the sought coefficients of the Chebyshev expansion of vector function satisfy the following infinite system of algebraic equations:
B. The Elements of Matrix Equation of Motion
The elements of matrix equation of motion (14) are expressed as follows: