## Abstract

Vibrations of thin and thick beams containing internal complexities are analyzed through generalized bases made of global piecewise-smooth functions (GPSFs). Such functional bases allow us to globally analyze multiple domains as if these latter were only one, such that a unified formulation can be used for different mechanical systems. Such bases were initially introduced to model a specific part of stress and displacement components through the thickness of multi-layered plates; subsequent extensions were introduced in the literature to allow the modeling of thin-walled beams and plates. However, in these latter cases, certain analytical difficulties were experienced when inner boundary conditions needed to be englobed into the GPSFs; in this work, such mentioned difficulties are successfully overcome through certain affine transformations which allow the analyses of vibrating complex beam systems through a straightforward analytical procedure. The complex mechanical components under investigation are Euler and/or Timoshenko models containing inner complexities (stepped beams, concentrated mass or stiffness, internal constraints, etc.). The ability of the models herein analyzed is shown through the comparison of the resulting solutions to exact counterparts, if existing, or to finite elements solutions.

## 1 Introduction

Within the frame of bent beams and plates, the theories describing the dynamic or static behavior of mechanical components originate from the ideas developed on two models of beam, i.e., the Euler–Bernoulli model (developed around the mid 18th century, let us say ∼1750) (e.g., Love [1] and its historical introduction) and Timoshenko [2,3]. The Euler–Bernoulli model along with its derived models is also generally recognized as associated with the so-called classical theory, whilst the Timoshenko model, along with its derived ones, is generally associated with the so-called uniform shear deformable theory. Higher order theories have also been developed in the 20th century but they are not relevant to the present context.

Despite the age of the classical beam theory, such a model is still appreciated today for its own simplicity and is involved in uncountable engineering contexts to predict static or dynamic behaviors in beam models. However, although such a classical theory can be considered of great importance within the engineering landscape, it is not always able to fulfill all the needs required. Indeed, its growing inability to accurately predict frequencies and modes as the wavelengths become increasingly comparable to the thickness of the beams is well recognized. To this end, the uniform shear deformable theory can generously assist to predict frequencies and modes of the real system with greater accuracy. Such an improvement, of uniform shear deformable theory with respect to the classical counterpart, has been revealed so much significant in the engineering landscape that some authors consider Timoshenko’s model one of the most remarkable events in the development of the structural dynamics of the 20th century [4]. During the 20th century, Timoshenko’s model has been longely debated, for example in order to identify a better shear coefficient, to conjecture a frequencies range for its usability, and last but not least, the existence of two series or double spectrum of frequencies (e.g., Refs. [57]). We can even find in the literature a recent and significant historical analysis carried out by Elishakoff [8] aimed at attempting to attribute the paternity of the ideas lying at the base of the mentioned uniform shear deformable theory; an analysis which finally led Elishakoff [8] to realize that the beam theory that incorporates both the rotary inertia and shear deformation as is known presently, with shear correction factor included, should be referred to as the Timoshenko–Ehrenfest beam theory rather than uniquely Timoshenko beam theory as universally recognized today. With regard to the aforementioned models, i.e., the Euler–Bernoulli and Timoshenko–Ehrenfest beam models, in the mechanical engineering landscape, this work intends to extend certain analytical methods to solve the equations related to both models in the case of multi-domains containing inner complexities. This need is essentially justified by engineering modeling reasons. Indeed, theories of simple single beams subjected solely to classical boundary conditions, are certainly an interesting subject for academic purposes but the modeling of systems both based on such theories and closer to real systems is an even more valuable subject of investigation. This latter is the case when inner and/or outer complexities are present on beam and plate systems. In this work, the focus is on beam-type systems.

When inner and/or outer complexities are present, exact methods, developed for simple beams (e.g., Ref. [9]), need to be substituted by valid analytical alternatives or even by numerical procedures. To this end, global piecewise-smooth functions (GPSFs) [10] are considered a valuable modeling base. The initial introduction of the GPSFs originally dates back to 2002 and was followed over the years by successful applications (e.g., Refs. [1115]). Subsequent extensions of GPSFs were introduced in the literature to both allow (i) a possible reduction of the computational effort [16] making the GPSFs adaptive (i.e., AGPSFs) and (ii) modeling thin-walled beams and plates [17,18]. In particular, in Ref. [17] the modelization of thin-walled beams and plates was carried out by adding certain supplementary analytical conditions whilst in Ref. [18] an integration procedure was used to substitute such mentioned conditions. Moreover, the way to also globally approximate discontinuous functions which can occur when certain types of inner complexities are mounted along beam systems was established in Ref. [18]. Although an intrinsic simplicity was researched in both cases [17,18], it is retained that a better procedure still needs to be identified in order to use AGPSFs both to (i) reduce the complexity related to creating the AGPSFs as functional bases and (ii) for modeling beams and plates containing different types of discontinuities along the beam system. In this work certain transformations, which have been called affine, have been identified and used to model thin-walled beams containing inner complexities more straightforwardly than has previously been done. In particular, such affine transformations can be seen as a variant of the procedure illustrated in the original work [10]; recursive modification at the ends of the local bases are now able to make the respective global functions continuous with their first derivative continuous; such a modification will provide a C1 continuity condition allowing us to model thin-walled beams, intended as models based on the Euler–Bernoulli beam theory. In this work, AGPSFs are also used to model Timoshenko–Ehernfest beam systems in order to complete earlier investigations [9] and to show the flexibility of AGPSFs in modeling systems based on different theories.

It is herein stressed that this work is not intended only to analyze beam-type systems; rather, the aim of this work also offers too the perspective to adopt the named GPSFs of class C−1, C0, C1 (where C−1 means functions containing finite jumps, C0 means continuous functions having only discontinuous derivatives and, finally, C1 refers to continuous functions with first derivatives continuous) for plates where such types of discontinuities need to be modeled. Moreover, in order to show the flexibility of such bases (AGPSFs) here the vibrations of beams, containing a combination of the discontinuities C−1, C0, C1 differently located on the same multi-beam systems, are also investigated.

This research is organized as follows. Section 2 presents the mathematical details regarding the new bases and the affine transformations leading from one base to another. In this section, discussions are provided in the context of Ritz-type analysis derived through the theorem of virtual displacements along with global bases (i.e., AGPSFs) fulfilling the inner and outer essential boundary conditions. Section 3 presents the mechanical systems containing inner complexities under analyses and the related variational models based on Euler–Bernoulli and Timoshenko–Ehrenfest theories. Section 4 presents and discusses the results of the numerical simulations and in Sec. 5 relevant conclusions are drawn.

## 2 The GPSFs of C0 and C1 Class Through Affine Transformation

The functional bases of GPSFs were originally introduced in 2002 [10] with the intent to model piecewise physical entities (stress and displacements). In 2015 [16] GPSFs became A-GPSFs when a way to make them adapt to the complexities of the functions, and for certain aspects to reduce computational effort, was clarified. The aforementioned studies were initially thought to model piecewise entities through the thickness of freely vibrating components. Subsequent development of A-GPSFs followed in the middle plane of thin-walled beams and plates to study the free vibrations of the same components [17,18]. The adoption of A-GPSFs within the middle plane of a structural element needed a variant for using these bases. Indeed, thin-walled beams and plates when used along with the theorem of virtual displacement can require the continuity of the first derivative in order to fulfill essential boundary conditions, whilst, GPSFs were initially designed only to be piecewise continuous without requiring any higher order of smoothness.

Particular attention should be paid to the aforementioned requirements. In particular, classical GPSFs and A-GPSFs show the natural ability to model smooth functions in particular cases of unsmooth ones, but this would happen within a purely mathematical process of fitting (e.g., least square approximation, etc.). When the mathematical process of fitting is involved in a resolution process aimed at solving differential equations through a Ritz procedure, essential conditions need to be fulfilled under the penalty of divergence. Therefore, what we generally need to study vibrations of thin-walled beams and plates are functional components of A-GPSFs occasionally equipped with a higher order of smoothness. This was the main reason for the introduction of certain supplementary analytical conditions added to A-GPSFs [17] or the introduction of an integration process [18].

In this paper, we show how certain affine transformations, recursively applied to local functions, can achieve global functional bases having different degrees of smoothness. This provides a unified approach to solving different beam systems. For illustrative purposes, let us initially take into account Fig. 1 where a global domain with x ∈ [−1, 1] contains NL = 3 local domains with $x∈[−1,−1/3]∪[−1/3,1/3]∪[1/3,1]$.

Fig. 1
Fig. 1
Close modal
Each local domain in Fig. 1 contains three functional local functions (N = 3) for each subdomain corresponding to the first three Legendre polynomials which can be obtained through the following well known expressions in [−1, 1]:
$L0(x)=1,L1(x)=x,L2(x)=(3x2−1)/2,…,Lq(x)=12qq!⋅dqdxq[(x2−1)q]$
(1)
where q in Eq. (1) corresponds to the degree of the ith = (q + 1)th Legendre polynomial generated through Rodrigues’ formula. Once the ith Legendre polynomial (i = 1, …, N) has been generated in [−1, 1], the same polynomial can be moved into any local domain [a, b] by simply substituting the variable x with (2(xa)/(ba) − 1), which is the procedure carried out to produce Fig. 1.

Figure 1 is the starting point to produce the GPSFs of any class (C0 or C1). Indeed, based on Fig. 1, GPSFs are obtained through the assembling scheme of Fig. 2. A-GPSFs can be constructed by using the same scheme as in Fig. 2 followed by the eventual elimination of repeated paths [16].

Fig. 2
Fig. 2
Close modal

Figure 2 represents the local functions of Fig. 1 through black dots whilst, their junctions are represented through paths. In any case, two consecutive levels (i and i + 1) always provide a number of GPSFs equal to the number of local domains (NL = 3 in Fig. 2). The closing path joining all the local functions in the last level of expansion (N) is added only in the last two levels.

The leading idea consists in leaving the responsibility to create global functions of class C0 or C1 to the path. Therefore, by addressing the attention to an arbitrarily established path (e.g., the continuous or the dashed one in Fig. 2), when two adjacent local functions follow from left to right, in the established path, the local downstream polynomial should undergo an affine transformation, i.e., its degree is kept invariant but in the interface between the adjacent sub-domains, the local functions should assume the same value, when a GPSFs of C0 class is required, or both the same value and the same first derivative, when a GPSFs of class C1 is required. In particular, with respect to Fig. 3, where f(x) represents the antecedent function whilst φ(x) is the following function, the continuity conditions in xo are established through Eq. (2).
${Co:g(x)=f(xo)φ(xo)φ(x)C1:g(x)=f′(xo)φ′(xo)φ(x)+φ′(xo)f(xo)−φ(xo)f′(xo)φ′(xo)$
(2)
Fig. 3
Fig. 3
Close modal

The function g(x) replaces φ(x) in the established path according to the class of continuity we are interested in achieving in the GPSFs. Of course, Eq. (2) must be recursively applied along the whole established path in order to achieve one GPSF per path. At this stage, some observations should be made. By looking at Eq. (2), it can be deduced that the function g(x) is still a polynomial having the same degree of the original one in the same domain. This preserves the completeness of the local bases and, consequently, the completeness of the global bases (i.e., of the GPSFs). Moreover, it is interesting to observe that Eq. (2) is well posed as there is no possibility that both φ(xo) and φ′(xo) can become zero at the ends of local polynomials; in fact, local polynomials are orthogonal polynomials, in particular of Legendre, and, as such, they never cancel out at the ends. This last condition would seem to fail for constant components along with the requirement to get C1 continuity; however, it should be noticed that in the case of C1 class of continuity the first two levels of expansions always lead to only two global functions: the constant and the linear term. This is justified for geometric reasons and by the fact that the number of class C1 GPSFs when N = 2 always reduces to two functions, whatever the number of local domains (NL), because NLN-2(NL−1) = 2 for any NL.

Based on the aforementioned illustrated analysis, a functional base of of class C0 or C1 GPSFs can be produced through a subroutine that automatically implements Eqs. (1) and (2) along with the assembly scheme of Fig. 2. Therefore, based on Fig. 1, Figs. 4 and 5 illustrate GPSFs of class C0 and C1 respectively. In these figures, the dashed line corresponds to the zero line, the first column is the functional base of GPSFs whilst the second and third columns illustrate the first and second derivatives of GPSFs respectively.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

When the problem consists of approximating trends or functions, for example in the sense of minimum least squares, a set of functional bases of GPSFs (i.e., the first column of Figs. 4 and 5) suffices the scope. Class C0 or C1 GPSFs can be used depending on the features of the function we need to approximate, on the convergence speed we would like to get, and, finally and on the accuracy required. When the problem consists only of approximating trends or functions, the worst result we could achieve is an approximation of low accuracy; this could be due, for example, to an attempt to approximate C0 functions through GPSFs of class C1. When the GPSFs have to be used in a Ritz-type analysis, as is the case herein dealt with for both the Euler–Bernoulli and Timoshenko–Ehrenfest beam model, the functional set of GPSFs needs to be complemented by their respective derivatives (i.e., the second and third columns in Figs. 4 and 5). Every single functional component of the set of GPSFs possibly along with its derivatives involved in its relevant analysis must fulfill the essential (or geometric) boundary conditions under penalty of divergence from the exact results. In particular, when the Euler–Bernoulli model is analyzed, the relevant boundary value problem involves an unknown displacement function through a 4th order (in the space) partial differential equation; in this case, the essential boundary conditions regard both the GPSFs and their first derivatives. The fulfillment of the natural boundary conditions is entrusted to the variational statement leading to the Ritz-type analysis and involving the second derivatives. When the Timoshenko–Ehrenfest model is analyzed, the relevant boundary value problem involves two unknown generalized displacement functions through a 2th order (in the space) system of two partial differential equations; in this case, the essential boundary conditions regard the GPSFs only. The fulfillment of the natural boundary conditions is entrusted to the variational statement leading to the Ritz-type analysis and involving the first derivatives. Therefore, except for special cases that may derive from specific and/or atypical complexities present in the Euler–Bernoulli beam systems, a Ritz-type analysis generally requires the use of class C1 GPSFs (Fig. 5); while, Timoshenko–Ehrenfest beam systems generally require the use of lower class GPSFs (i.e., C0 as in Fig. 4).

## 3 Free Vibrations of Mechanical Components With Internal Complexities Through a Unified Consistent Variational Approach Along With A-GPSFs

In this section, let us focus our attention on beam-type systems including internal complexities. The system has a total length L and could even have constant piecewise thickness h or stepwise material properties. The in-plane and normal coordinate length parameters are denoted with x and z respectively, and u and w represent the corresponding displacement components.

The free vibrations of the beam-type system illustrated in Fig. 6 are analyzed through both the classical Euler–Bernoulli and the Timoshenko–Ehrenfest beam theory. In this regard, relevant GPSFs are used in a unique formulation that will be developed through functions in a single global domain [0, L].

Fig. 6
Fig. 6
Close modal
Therefore, based on the assumed displacement field Eq. (3) which is the characteristics of the Euler–Bernoulli beam theory,
$u(x,z;t)=−zw(x;t),x$
(3)
where (),x = ∂()/∂x and the time part is intended as w(x;t) = w(x)cos(ω · t), the theorem of virtual displacements can be written as in the following Eq. (4).
$∫VolσxδεxdVol+∫Volρw¨δwdVol+mw¨m⋅δwm+kwk⋅δwk=0$
(4)
Here, ρ is the volumetric density, δ is the virtual operator whilst, σx and ɛx are the longitudinal stress and strain respectively; these latter are related through Young’s modulus (σx = x). In Eq. (4), the letteral indexes m and k refer to the specific section where the respective quantities are collocated on the beams of Fig. 6. Therefore, based on Eq. (4) an eigenproblem, aimed at extracting the natural frequencies and associated eigenfunctions, can be achieved once an expansion series of w(x) has been set as in Eq. (5).
$w(x)=∑i=1naiW(x)i=aTW$
(5)

Let us notice that the expansion in series, W(x)i with i = 1,…,n, has been set as global in the whole domain [0, L] of the beam without caring about the internal complexities; the proper modeling of these latter are entrusted to the choice of the base which can be one among the GPSFs depending on the type of internal complexity in the mechanical component. At this stage, an aspect of the resolution procedure deserves to be noted: we are developing the model exactly as we would have done on a simple beam, i.e., using a series expansion and ignoring the involvement in the structure of internal complexities. However, even though, we are carrying out such a development a convergence to the exact results is expected for merit which belongs only to the GPSFs constructed through a simple affine transformation.

Based on Eqs. (3)(5) along with the relationship between strain and displacement (ɛx = u,x) the following eigenproblem Eq. (6), aimed at extracting the natural frequencies and associated analytical eigenfunctions, can be achieved.
$Ka=ω2MaK=∫0LEJW″⋅W″Tdx+kWkWkTM=∫0LρAW⋅WTdx+mWmWmT$
(6)
In Eq. (6), J and A represent the inertia and area of the transversal section respectively, Wk and Wm correspond to the vector W(x) evaluated at xk and xm respectively.
An eigenproblem similar to Eq. (6) can be formulated with respect to the Timoshenko–Ehrenfest theory. Here, the assumed displacement field corresponds to Eq. (7),
$u(x,z;t)=−zψ(x;t)$
(7)
where ψ(x; t) represents the rotation of the transversal section of the beam as independent from the transversal displacement w(x; t), thus that two unknown functions can be set into the model, such as $w(x;t)=w(x)cos(ω⋅t)$ and $ψ(x;t)=ψ(x)cos(ω⋅t)$. The theorem of virtual displacements is taken into account in the following Eq. (8).
$∫Vol(σxδεx+τzxδγzx)dVol+∫Volρw¨δwdVol+∫Volρz2ψ¨δψdVol+mw¨m⋅δwm+kwk⋅δwk=0$
(8)
Therefore, based on Eq. (8) an eigenproblem, aimed at extracting the natural frequencies and associated eigenfunctions, can be achieved once an expansion series of both w(x) and ψ(x) have been set as in Eq. (9).
${ψ(x)=∑i=1naiΨ(x)i=aTΨw(x)=∑i=1lbiW(x)i=bTW$
(9)
where the formulation is still constructed regardless of the fact that the beam may be homogeneous or inhomogeneous with steps and/or containing inner complexities located in certain sections. Based on Eqs. (7)(9) along with the relationship between strain and displacement (ɛx = u, x and γzx = u, z + w, x) the following eigenproblem Eq. (10), aimed at extracting the natural frequencies and associated analytical eigenfunctions, can be achieved.
$[K11K12K12TK22](ab)=ω2[M1100M22](ab);K11=∫0L(EJΨ′⋅Ψ′T+χAGΨ⋅ΨT)dx;K22=∫0LχAGW′⋅W′Tdx+kWkWkT;K12=−∫0LχAGΨ⋅W′Tdx;M11=∫0LρJΨ⋅ΨTdx;M22=∫0LρAW⋅WTdx+mWmWmT$
(10)

In Eq. (10), χ represents the shear correction factor and G is the transverse shear modulus. In both models, Euler–Bernoulli and Timoshenko–Ehrenfest, the transversal section is assumed rectangular with thickness h and width b.

Before closing this section, let us still notice that the expansion in series, Ψ(x)i with i = 1, …, n, W(x)i with i = 1, …, l have been set as global in the whole domain [0, L] of the beam without caring about the internal complexities; the proper modeling of these latter is entrusted to the choice of the base which can be one among the GPSFs depending on the type of internal complexity in the mechanical component. Still, we should notice that we have developed the model exactly as we would have done on a simple beam, i.e., using a series expansion and ignoring the involvement in the structure of internal complexities. However, even though, we are carrying out such a development, a convergence to the exact results is expected for merit which belongs only to the GPSFs constructed through a simple affine transformation.

## 4 Numerical Simulations and Discussions

In this part of the manuscript, the analytical model presented in Sec. 3 is analyzed in conjunction with the GPSFs presented in Sec. 2 through the new affine transformation.

In the first simulation, a simple unique beam simply supported at the ends is taken into account. This test illustrates how the GPSFs can get the relevant results of a simple beam as a particular case. Indeed, a simple beam can be seen as if it were of an arbitrary number of sub-domains. A subdivision of NL sub-domains having equal lengths along with an equal number of local functional components (N) has been adopted in each subdomain to build all the GPSFs. For this specific initial test only, the Timoshenko–Ehrenfest model has been taken into account and compared with existing exact results [9].

To compare the current results with those published in Ref. [9], simply supported boundary conditions have been simulated; therefore, once the GPSFs were obtained, all the functional components of the set were multiplied by the function x(x-L) in order to make able GPSFs able to geometrical fulfill the simply supported boundary conditions. The results reported in Table 1 were obtained by using the following geometrical-material characteristics: L = 1 m, h = 25 cm, E = 210 GPa, ρ = 7850 kg/m3, ν = 0.3, and χ = 5 (1 + ν)/(6 + 5ν) with a beam having unitary width b.

Table 1

First ten natural frequencies (Hz) through GPSFs of class C1; b.c.: SS; L = 1 m, h = 25 cm

No.Exact plane stressa [9]Exact Timoshenko [9]N(NL)
18(1)6(6)8(4)10(3)12(2)
1535.523535.468535.468535.468535.468535.468535.468
21774.261772.911772.911772.911772.911772.911772.91
33275.883270.163270.163270.163270.163270.163270.16
44858.324845.474845.474845.474845.474845.474845.47
56415.326585.446585.446585.446437.486585.446585.44
66458.756437.486437.486437.486585.446437.486437.48
77015.207211.007211.007211.007211.007211.007211.00
88055.328025.848025.848025.858025.848025.848025.84
98435.818711.708711.708711.708711.708711.708711.70
109640.719604.469604.469604.499604.479604.469604.46
Size[K]b3652525244
No.Exact plane stressa [9]Exact Timoshenko [9]N(NL)
18(1)6(6)8(4)10(3)12(2)
1535.523535.468535.468535.468535.468535.468535.468
21774.261772.911772.911772.911772.911772.911772.91
33275.883270.163270.163270.163270.163270.163270.16
44858.324845.474845.474845.474845.474845.474845.47
56415.326585.446585.446585.446437.486585.446585.44
66458.756437.486437.486437.486585.446437.486437.48
77015.207211.007211.007211.007211.007211.007211.00
88055.328025.848025.848025.858025.848025.848025.84
98435.818711.708711.708711.708711.708711.708711.70
109640.719604.469604.469604.499604.479604.469604.46
Size[K]b3652525244
a

Symmetrical/extensional modes intentionally excluded.

b

2 N NL–4(NL−1).

Table 1 lists the first ten natural frequencies obtained through the minimum number of functional components. For example, 8(4) means that once the expansion level 8 has been fixed for each domain (N = 8) the values in Table 1 correspond to the minimum number of sub-domains required to get that frequencies (NL = 4: {8, 8, 8, 8}). Thus, Table 1 clarifies that the most advantageous procedure (from the point of view of the size of the eigenproblem) would seem associated with a beam modeled as it is, i.e., through a unique domain. However, in this case, the classical Legendre polynomials used to require up to the 17th degree to get the exact results on the first six significant figures. Any further increasing number of domains provides a slightly higher size of the eigenproblem, even if through a constant size, but with the advantage of reducing the maximum polynomial degrees involved. In any case, during the convergence analysis, the analytical model based on class C1 GPSFs generated through an affine transformation has shown an excellent monotonic convergence to the exact results listed in Table 1. Finally, we should note, similarly to Ref. [9], that the analytical procedures herein dealt with naturally provide frequencies (in Table 1) falling into one single set regardless of the existence of one or two spectra.

In order to further stress the ability of the method to converge to exact results, Table 2 compares these exact results produced by Eisenberger [19] to the results achieved through A-GPSFs.

Table 2

First five natural frequencies, $ω^=ωL2ρA/EJ$, b.c.: CS with internal slide at x = xc [19]

ξ = xc/L
0.250.50.750.9
No.Ref. [19]
13.39215.48005.95614.3332
232.323318.826521.165026.6222
368.344769.882850.802171.0179
4105.0211109.5340120.8773123.4203
5199.3862214.9602220.8016182.2637
No.A-GPSFs
13.39225.48005.95624.3332
232.323418.826421.165126.6222
368.344869.882850.802271.0179
4105.0212109.5340120.8773123.4203
5199.3862214.9602220.8016182.2637
{N1, N2}{6, 14}{9, 10}{12, 7}{15, 5}
Size[K]19181819
ξ = xc/L
0.250.50.750.9
No.Ref. [19]
13.39215.48005.95614.3332
232.323318.826521.165026.6222
368.344769.882850.802171.0179
4105.0211109.5340120.8773123.4203
5199.3862214.9602220.8016182.2637
No.A-GPSFs
13.39225.48005.95624.3332
232.323418.826421.165126.6222
368.344869.882850.802271.0179
4105.0212109.5340120.8773123.4203
5199.3862214.9602220.8016182.2637
{N1, N2}{6, 14}{9, 10}{12, 7}{15, 5}
Size[K]19181819

The results listed in Table 2 are those obtained through the minimum number of functional components. A careful perusal of the results in Table 2 reveals an excellent agreement between the results obtained using this method and the exact results listed by Eisenberger [19, Table 4]; extremely slight discrepancies can be observed but these, whenever existing, seem mainly due to different roundings used. In Table 2, it is also interesting to note the flexibility of the A-GPSFs which allow convergence in a cost-effective manner by increasing the order of the polynomial components only in the largest domains thus keeping the computational effort contained and balanced.

The next system herein analyzed, as aimed at testing the performance of the resolution procedure based on Ritz-type analysis along with GPSFs, is the system depicted in Fig. 7 which is based on the following geometrical-material characteristics kept fixed during the simulation process: L = 60 cm, xm = 15 cm, xc = 30 cm, xk = 45 cm, width uniform of 2 cm, m = 100 g, k = 100 3EJ/(L/4)3, E = 206 GPa, ρ = 7800 kg/m3, ν = 0.3, and χ = 5 (1 + ν)/(6 + 5ν).

Fig. 7
Fig. 7
Close modal

The system illustrated in Fig. 7 presents such unusual inner complexities that a resolution aimed at treating the system as if it were a unique domain, without recurring to GPSFs, would be unthinkable; as it will be clear in this set of simulations the new GPSFs perform the related dynamical analysis efficiently based on both the Euler–Bernoulli and the Timoshenko–Ehrenfest models.

What is requested in order to analyze the system in Fig. 7 is a proper choice of GPSFs (which here, for the sake of brevity, we will assume always having the same level of expansion for any sub-domain).

In this latter regard, Fig. 8 can assist our choices. In particular, Fig. 8 shows the continuity conditions the GPSFs bases should have in order to converge to the expected exact results.

Fig. 8
Fig. 8
Close modal

First, from Fig. 8 we can immediately notice that the class of GPSFs is not the same, neither for the different sections nor for the theory we use.

As far as the Timoshenko–Ehrenfest beam theory is concerned, the model generally requires class C0 GPSFs (for both the functions W and Ψ). Of course, this different choice is due to the fact that all the components of a base of GPSFs must fulfill the relevant inner and outer essential boundary conditions. This is the reason why, in xc, W must be discontinuous (through a jump [18]) whilst Ψ must preserve the rotation, and thus it needs to be of C0 class. The jump in W at xc is ensured by adding a constant term for x ∈ [0, xc] to the set of GPSFs for W and nil elsewhere. The essential boundary conditions at the ends are ensured by multiplying all the functional components of W and Ψ by x(x0.6) and x respectively. Therefore, the resultant set of GPSFs used in the Timoshenko–Ehrenfest model for N = 3 in NL = 4 sub-domains (i.e., {3,3,3,3}) is illustrated in Fig. 9.

Fig. 9
Fig. 9
Close modal

As far as the Euler–Bernoulli beam theory is concerned, the model generally requires class C1 GPSFs (for W) in order to let the base of GPSFs converge at the exact results; still such a requirement is related to the need for all the functional components of the set to fulfill the relevant inner and outer essential boundary conditions. This is the reason why in xc the first derivative of W (the rotation of the transversal section) must be locally continuous while leaving W to be discontinuous through a jump. The jump in W at xc is ensured by adding a constant term for x ∈ [0, xc] to the set of GPSFs and nil elsewhere. The essential boundary conditions at the ends and at inner points are ensured by multiplying the first functional component W1 and the remaining functional components Wi for i > 1, by (100/9)(4x2–40x3/3 + 100x4/9) and x2(x–0.6) respectively. The set of GPSFs used in the Euler–Bernoulli model for N = 3 in NL = 4 sub-domains (i.e., {3,3,3,3}) is illustrated in Fig. 10.

Fig. 10
Fig. 10
Close modal

Based on the aforementioned analytical considerations, numerical simulations were carried out in order to produce the results. The results have been assessed by using the theories herein dealt with (through the mentioned GPSFs) and compared to finite element (FE) solutions based on Euler–Bernoulli and Timoshenko's elements described in Petyt [20]. Aside from the mention regarding the finite elements having equal length and distributed through the beam-type system of Fig. 8, here any further detail aimed at illustrating the simulation of the FE system is not mentioned both for the sake of brevity and because the relevant modeling is retained known.

Tables 3 and 4 report the frequencies of the system of Fig. 8 when the thickness is uniform and equal to 3 mm; therefore, we are considering a system having a length/thickness ratio of 200 which is quite slender in the range of the first ten frequencies; for this, it is not surprising that both theories (Euler–Bernoulli and Timoshenko–Ehrenfest) give similar results.

Table 3

First ten natural frequencies (Hz) through GPSFs, refer to Fig. 8; hi = {3,3,3,3} mm; theory: Euler–Bernoulli

GPSFsFE solutions (140 elements)
No.{4,4,4,4}{5,5,5,5}{6,6,6,6}{8,8,8,8}{10,10,10,10}
128.892328.764228.759328.759028.759028.7590
277.718377.254777.250977.248577.248577.2485
3149.045148.067147.946147.928147.928147.928
4227.637226.501226.390226.387226.387226.387
5437.768431.592430.235430.151430.151430.151
6560.551550.649549.703549.688549.687549.687
7912.214819.523796.815796.130796.123796.123
81136.956978.380949.622946.735946.727946.727
91910.601545.931481.001457.881457.801457.80
102289.961717.941702.181675.131674.701674.70
Size[K]1115192735280
GPSFsFE solutions (140 elements)
No.{4,4,4,4}{5,5,5,5}{6,6,6,6}{8,8,8,8}{10,10,10,10}
128.892328.764228.759328.759028.759028.7590
277.718377.254777.250977.248577.248577.2485
3149.045148.067147.946147.928147.928147.928
4227.637226.501226.390226.387226.387226.387
5437.768431.592430.235430.151430.151430.151
6560.551550.649549.703549.688549.687549.687
7912.214819.523796.815796.130796.123796.123
81136.956978.380949.622946.735946.727946.727
91910.601545.931481.001457.881457.801457.80
102289.961717.941702.181675.131674.701674.70
Size[K]1115192735280
Table 4

First ten natural frequencies (Hz) through GPSFs, refer to Fig. 8; hi = {3,3,3,3} mm; theory: Timoshenko–Ehrenfest

No.GPSFsFE solutions (520 elements)
{4,4,4,4}{5,5,5,5}{6,6,6,6}{8,8,8,8}{10,10,10,10}
128.792128.756428.755228.755228.755228.7552
277.717177.237877.231277.229877.229877.2298
3149.065147.887147.858147.855147.855147.855
4227.222226.489226.322226.316226.316226.316
5438.496432.002429.806429.757429.757429.757
6561.531551.987548.975548.895548.895548.895
7951.881803.987796.174794.439794.435794.435
81199.08969.184947.643944.728944.717944.718
92105.371536.041482.531453.431453.241453.24
103250.681742.031707.281668.311667.801667.80
Size[K]27354359751040
No.GPSFsFE solutions (520 elements)
{4,4,4,4}{5,5,5,5}{6,6,6,6}{8,8,8,8}{10,10,10,10}
128.792128.756428.755228.755228.755228.7552
277.717177.237877.231277.229877.229877.2298
3149.065147.887147.858147.855147.855147.855
4227.222226.489226.322226.316226.316226.316
5438.496432.002429.806429.757429.757429.757
6561.531551.987548.975548.895548.895548.895
7951.881803.987796.174794.439794.435794.435
81199.08969.184947.643944.728944.717944.718
92105.371536.041482.531453.431453.241453.24
103250.681742.031707.281668.311667.801667.80
Size[K]27354359751040

Tables 3 and 4, also show how both theories, based on GPSFs, provide results in excellent agreement with the finite element solutions. The ability of the present analytical models, based on GPSFs, to get converged frequencies with a size far less than the FE eigenproblem can also be noticed.

Finally, we should report that the finite element solutions in the case of the Euler–Bernoulli beam element correspond to the most convergent frequencies through the minimum number of degrees-of-freedom (280) and before of ill-conditioning warnings appeared. On the other hand, the finite element solutions in the case of the Timoshenko beam element correspond to the most convergent values through the minor number of degrees-of-freedom (1040).

Tables 5 and 6 regard a different simulation when compared to Tables 3 and 4. Here, with respect to the inner guided constraint, the left side of the beam keeps a uniform thickness of 3 mm, whilst the thickness on the right side is increased to 5 cm.

Table 5

First ten natural frequencies (Hz) through GPSFs, refer to Fig. 8; hi = {3, 3, 50, 50} mm; theory: Euler–Bernoulli

 No. GPSFs FE solutions (284 elements) {6,6,6,6} {8,8,8,8} {10,10,10,10} {12,12,12,12} {14,14,14,14} 1 28.7485 28.6727 28.6726 28.6726 28.6726 28.6726 2 35.8937 35.8882 35.8882 35.8882 35.8882 35.8882 3 173.171 173.144 173.144 173.144 173.144 173.144 4 566.099 566.079 566.078 566.078 566.078 566.078 5 919.294 918.661 918.652 918.652 918.652 918.652 6 1723.04 1706.15 1705.80 1705.80 1705.80 1705.80 7 2023.02 2022.80 2022.80 2022.80 2022.80 2022.80 8 2555.54 2274.29 2269.98 2269.94 2269.94 2269.94 9 3659.09 3474.59 3467.97 3467.86 3467.86 3467.86 10 5834.90 4301.04 4240.84 4239.52 4239.51 4239.51 Size[K] 19 27 35 43 51 568
 No. GPSFs FE solutions (284 elements) {6,6,6,6} {8,8,8,8} {10,10,10,10} {12,12,12,12} {14,14,14,14} 1 28.7485 28.6727 28.6726 28.6726 28.6726 28.6726 2 35.8937 35.8882 35.8882 35.8882 35.8882 35.8882 3 173.171 173.144 173.144 173.144 173.144 173.144 4 566.099 566.079 566.078 566.078 566.078 566.078 5 919.294 918.661 918.652 918.652 918.652 918.652 6 1723.04 1706.15 1705.80 1705.80 1705.80 1705.80 7 2023.02 2022.80 2022.80 2022.80 2022.80 2022.80 8 2555.54 2274.29 2269.98 2269.94 2269.94 2269.94 9 3659.09 3474.59 3467.97 3467.86 3467.86 3467.86 10 5834.90 4301.04 4240.84 4239.52 4239.51 4239.51 Size[K] 19 27 35 43 51 568
Table 6

First ten natural frequencies (Hz) through GPSFs, refer to Fig. 8; hi = {3, 3, 50, 50} mm; theory: Timoshenko–Ehrenfest

No.GPSFsFE solutions (768 elements)
{6,6,6,6}{8,8,8,8}{10,10,10,10}{12,12,12,12}{14,14,14,14}
128.580528.580328.580328.580328.580328.5803
235.872235.872135.872135.872135.872135.8721
3173.042173.039173.039173.039173.039173.039
4565.223565.156565.156565.156565.156565.156
5918.476916.343916.329916.329916.329916.330
61733.791698.711698.171698.171698.171698.17
71895.211895.121895.121895.121895.121895.12
82431.342260.012256.612256.582256.582256.58
93687.603451.573438.603438.443438.443438.45
105471.004340.144200.814196.174196.134196.13
Size[K]435975911071536
No.GPSFsFE solutions (768 elements)
{6,6,6,6}{8,8,8,8}{10,10,10,10}{12,12,12,12}{14,14,14,14}
128.580528.580328.580328.580328.580328.5803
235.872235.872135.872135.872135.872135.8721
3173.042173.039173.039173.039173.039173.039
4565.223565.156565.156565.156565.156565.156
5918.476916.343916.329916.329916.329916.330
61733.791698.711698.171698.171698.171698.17
71895.211895.121895.121895.121895.121895.12
82431.342260.012256.612256.582256.582256.58
93687.603451.573438.603438.443438.443438.45
105471.004340.144200.814196.174196.134196.13
Size[K]435975911071536

The finite element solutions require an increasing computational effort; in particular, by using a similar convergence criterion as mentioned in Tables 3 and 4, the finite element solutions for the Euler–Bernoulli beam element require a number of 568 degrees-of-freedom, whilst when the Timoshenko beam element is taken into account the convergence requires at least 1536 degrees-of-freedom. The analytical model based on GPSFs still converges without experiencing any particular analytical difficulties and with a size of the eigenproblem significantly lower than the finite element solution.

Finally, it is interesting to notice that, aside from the 7th mode, the discrepancy between the frequencies assessed through the Euler–Bernoulli theory and the frequencies of the Timoshenko–Ehrenfest theory are quite similar. This is because the 7th mode involves a significant displacement field on the right side when compared to the left side, thus involving the importance of the shear for the thicker part of the model on the right side.

The last numerical Tables 7 and 8, take into account uniformly thicker beams. The thickness is increased from 3 mm to 21 mm. The FE solution always requires a higher size of the eigenproblem with respect to the size of the analytical procedure. Significantly higher becomes the size of the eigenproblem when the Timoshenko beam element is taken into account. Here, still the present analytical method provides excellent agreement with the expected results.

Table 7

First ten natural frequencies (Hz) through GPSFs, refer to Fig. 8; hi = {21, 21, 21, 21} mm; theory: Euler–Bernoulli

No.GPSFsFE solutions (128 elements)
{4,4,4,4}{5,5,5,5}{6,6,6,6}{8,8,8,8}{10,10,10,10}
1232.085230.864230.829230.826230.826230.826
2564.597561.952561.947561.935561.935561.935
31290.881286.691286.301286.281286.281286.28
41618.711610.701610.201610.191610.191610.19
53071.793029.463021.123020.553020.553020.55
63960.023904.553891.453891.363891.363891.36
76878.176071.516016.576002.736002.716002.71
88241.517212.186975.616964.686964.626964.62
913,529.010,821.510,374.910,219.010,218.410,218.4
1017,432.412,182.712,099.811,808.911,805.411,805.4
Size[K]1115192735256
No.GPSFsFE solutions (128 elements)
{4,4,4,4}{5,5,5,5}{6,6,6,6}{8,8,8,8}{10,10,10,10}
1232.085230.864230.829230.826230.826230.826
2564.597561.952561.947561.935561.935561.935
31290.881286.691286.301286.281286.281286.28
41618.711610.701610.201610.191610.191610.19
53071.793029.463021.123020.553020.553020.55
63960.023904.553891.453891.363891.363891.36
76878.176071.516016.576002.736002.716002.71
88241.517212.186975.616964.686964.626964.62
913,529.010,821.510,374.910,219.010,218.410,218.4
1017,432.412,182.712,099.811,808.911,805.411,805.4
Size[K]1115192735256
Table 8

First ten natural frequencies (Hz) through GPSFs, refer to Fig. 8; hi = {21, 21, 21, 21} mm; Theory: Timoshenko–Ehrenfest

GPSFsFE solutions (800 elements)
No.{4,4,4,4}{5,5,5,5}{6,6,6,6}{8,8,8,8}{10,10,10,10}
1229.638229.500229.498229.498229.498229.498
2557.271555.589555.569555.566555.566555.566
31258.961256.121255.921255.921255.921255.92
41586.581583.531583.151583.141583.141583.14
52929.842900.672893.572893.462893.462893.46
63710.203669.443656.203655.993655.993656.00
76157.305550.395523.925516.585516.575516.59
87342.196388.356309.826299.296299.276299.29
911,749.49297.219061.638972.028971.558971.62
1015,650.110,410.510,185.310,058.010,057.010,057.1
Size[K]27354359751600
GPSFsFE solutions (800 elements)
No.{4,4,4,4}{5,5,5,5}{6,6,6,6}{8,8,8,8}{10,10,10,10}
1229.638229.500229.498229.498229.498229.498
2557.271555.589555.569555.566555.566555.566
31258.961256.121255.921255.921255.921255.92
41586.581583.531583.151583.141583.141583.14
52929.842900.672893.572893.462893.462893.46
63710.203669.443656.203655.993655.993656.00
76157.305550.395523.925516.585516.575516.59
87342.196388.356309.826299.296299.276299.29
911,749.49297.219061.638972.028971.558971.62
1015,650.110,410.510,185.310,058.010,057.010,057.1
Size[K]27354359751600

## 5 Conclusions

Within the frame of freely vibrating continuous systems, expansions in series based on GPSFs [10,17,18] have been complemented with an affine transformation in order to model both thin and thick elements. Such an affine transformation has significantly simplified the implementation of the GPSFs. The GPSFs built through affine transformations have allowed analytical models to efficiently implement a Ritz-type analysis based on both Euler–Bernoulli and Timoshenko–Ehrenfest beam theories. Excellent analytical stability has been experimented in both cases along with the ability of the GPSFs to converge to the exact results. The analytical methods presented in this work are an extension of the exact analytical models earlier published [9]; the present analytical model along with this latter reference, are able the show, for several systems, the reason why the generous Timoshenko–Ehrenfest model still today represents, after one century, a reference point in mechanical and/or structural engineering.

Although the present author believes that GPSFs can be efficiently improved for investigating systems containing different types of inner complexities and for different theories, it is retained that a significant step forward to future investigations on the dynamics of beams and plates containing inner complexities has been presented in the present work.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

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