Vibration analysis of mobile phone mast system by Rayleigh method

ARTICLE IN PRESS

JID: APM

[m3Gsc;October 21, 2016;20:45]

Applied Mathematical Modelling 0 0 0 (2016) 1–16

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Vibration analysis of mobile phone mast system by Rayleigh method Alexandre de M. Wahrhaftig a,∗, Reyolando M.L.R.F. Brasil b a Department of Construction and Structures, Polytechnic School, Federal University of Bahia (UFBa), Aristides Novís Street, nbr 02, 5th Floor, Federação, Salvador, BA 40210-910, Brazil b Department of Structural and Geotechnical Engineering, Polytechnic School, University of São Paulo (USP), Prof. Almeida Prado Av., tv. 2, nbr 83, University Campus, São Paulo, SP 05508-900, Brazil

a r t i c l e

i n f o

Article history: Received 24 December 2015 Revised 16 September 2016 Accepted 6 October 2016 Available online xxx Keywords: Actual Structures Fundamental Frequency Rayleigh Technique Finite Element Method Structural Dynamics Geometric Stiffness

a b s t r a c t To study the vibration of beams and columns, discretization techniques are required because such structures are continuous systems with infinite degrees of freedom. However, one can associate such systems to a system with a single degree of freedom, restricting the form to which the system will deform and describing their properties as a function of generalized coordinates. This technique is called the Rayleigh method. However, actual structures are more complex than simple beams and columns because their properties vary along their length. The objective of this work is to apply the technique recommended by Rayleigh to actual structures and find a single equation and correction factor that can be used to resolve practical problems in engineering. The structural elements selected for this study are metallic high-slenderness poles, for which the frequency of the first vibration mode were calculated analytically, as well as by finite element method-based computer modeling for comparative purposes. The results indicate that the analytical solution is 16% greater and 1% minor than the computational solution, and correction factors of 1.4 and 1.32 were found, respectively. © 2016 Elsevier Inc. All rights reserved.

1. Introduction The vibration response analysis of framed structures modeled as beams and columns has been studied by many researchers and continues to be treated extensively in the literature. Beams and columns constitute a continuous system with infinite degrees of freedom. To study the behavior of such systems, discretization techniques are required, wherein the structure is transformed into subsystems defined by points called joints. However, one can associate these subsystems to a system with a single degree of freedom, restricting the form to which the system will deform and describing their properties as a function of generalized coordinates. This technique was used by Rayleigh [1] in the study of the vibration of elastic systems, obtaining equations that were valid in the whole domain of the problem. However, actual structures are more complex than simple beams and columns because their properties vary along their length. In such cases, the Rayleigh method should be applied in parts and the integrals should be resolved within the limits established for each interval using the generalized properties calculated for each segment of the structure.

∗

Corresponding author. Fax: +5571 32839730. E-mail addresses: [email protected], [email protected] (A.d.M. Wahrhaftig), [email protected] (R.M.L.R.F. Brasil).

http://dx.doi.org/10.1016/j.apm.2016.10.020 0307-904X/© 2016 Elsevier Inc. All rights reserved.

Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

JID: APM 2

ARTICLE IN PRESS

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

It is important to note that axial compressive forces reduce the stiffness of the members of a structure. Geometric stiffness is a function of the normal force acting on a structural element and is due to the combined forces of the gravitational field and the self-weight of the structural element, as well as the devices attached to the element. Many well-known researchers in physics and mathematics have been drawn toward the investigation of the effect of the compression force on structural systems. In this sense, Timoshenko [2] pointed out that the first analytical model for understanding problems related to axially compressed bars could be attributed to Euler. More recently, Ratzersdorfer [3] presented a comprehensive study on the stability of compressed bars. Gambhir [4], who made important contributions to this field, stated that the studies of isolated bars are frequently related to the stability analysis of structural systems. With regard to the experimental analysis of the dynamic behavior of slender columns, Brasil et al. [5] performed an experiment to determine the parameters of reduction in the stiffness of unstressed sections of an RC pole and correctly calculate the displacements of all sections. The experiment was performed for a reinforced concrete tower used for telecommunications, which had a length of 30 m and a circular ring cross section of 50-cm diameter. The actual structures selected for this study were metallic high-slenderness poles, whose frequency of the first vibration mode was calculated by the Rayleigh method as described above, and one simplified equation was obtained. Computer models of these poles were developed using a geometric nonlinear solution by the finite element method (FEM) for comparative purposes. A relatively good agreement was obtained between the exact calculation and modeling results, with a difference of 16% and 1.1%, respectively for each case studied; hence, useful analytical solutions can be obtained independently, without the use of sophisticated computational programs, since a correction factor is applied to the simplified equation obtained from the analytical solution. The rest of this paper is organized as follows. Initially, a review of the mathematical aspects of the problem is presented, followed by the determination of both the analytical and FEM solutions, although in many cases, both the solutions are already known. Then, the simulations for each mathematical formulation are presented. Finally, conclusions based on the principal results of this study are outlined.

2. Mathematical considerations based on Rayleigh method The Rayleigh method [1], combined with the principle of virtual work, composes the mathematical basis of this study. Bert [6] stated that this modified method was originally applied by Rayleigh in 1894 to the one-dimensional problem of determining the fundamental frequency of a stretched string undergoing small-amplitude vibrations. The essence of the method is that it does not use a specific trial function with an undetermined coefficient for the deflection, as is done in the ordinary Rayleigh method. The applications of the Rayleigh technique to mechanical systems with vibration problems are found in a wide range of scientific studies. Nikkhoo et al. [7] used the Rayleigh–Ritz method to obtain the natural frequencies and dynamic response of various beams under the excitation of a moving mass; in this method, trigonometric shape functions based on the end conditions of the beams were utilized. Moreover, they showed that a high level of accuracy could be obtained by utilizing a low number of shape functions, which had to be achieved to find the deformation field of various beams. Nguyen et al. [8] also used the Rayleigh–Ritz method to estimate the frequencies of poles and antenna masts while studying the aerodynamics of these structures. Along the same lines, Bhat [9] employed a method for calculating the natural frequencies of rectangular plates that have at least two parallel edges that are not simply supported and hence an exact solution cannot be obtained. Bhat inferred that the Rayleigh and Rayleigh–Ritz methods of analysis, when used with beam characteristic functions, could give good results in obtaining the natural frequencies of such plates. Chakraverty and Behera [10] investigated the free vibration of nonuniform Euler–Bernoulli nanobeams based on nonlocal elasticity theory. In this work, boundary characteristic orthogonal polynomials were implemented in the Rayleigh–Ritz method, which made the procedure computationally efficient because some elements of the mass and stiffness matrices of the generalized eigenvalue problem became either zero or one owing to the orthonormality of the assumed shape functions. It is important to note that the technique developed by Rayleigh, and presented in his first book, was aimed to calculate the fundamental frequency of continuous elastic systems, which were transformed in systems with a unique degree of freedom. The precision of this method depends on the functional form used to represent the free vibration mode. The choice of an appropriate functional form is described by El Bikri et al. [11]; they performed a theoretical investigation of the geometrically nonlinear free vibrations of a clamped–clamped beam containing an open crack. Their approach involves the use of a semi-analytical model based on the extension of the Rayleigh–Ritz method to nonlinear vibrations; the model is mainly affected by the choice of the admissible functions. Monterrubio and Ilanko [12] discussed the characteristics of sets of admissible functions to be used in the Rayleigh–Ritz method. Of particular interest were sets that could lead to converged results when penalty terms were added to the model constraints and elements were interconnected in vibration and buckling problems of beams as well as plates and shells of a rectangular planform. The discussion included the use of polynomials, trigonometric functions, and a combination of both. In the past, several sets of admissible functions have been used that have a limit on the number of terms that can be included in the solution without producing ill-conditioning. On the other hand, a combination of trigonometric and loworder polynomials have been found to produce accurate results without ill-conditioning for any number of terms and any number of penalty parameters that can be accommodated by the computer memory. Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

3

N(x) q(t) m0 = mn

e(t)

v(x,t) mi+1 mi

xn xi+1

xi

mi-1 xi-1

EI

x

m

Fig. 1. Parameters for developing the mathematical model.

Cheung and Zhou [13] studied the free vibration of thin orthotropic rectangular plates with intermediate line supports in one or two directions. Chiba and Sugimoto [14] used the Rayleigh–Ritz method to solve the problem of a cantilever plate attached to a “spring–mass system. Hu et al. [15] studied the problem of the vibration characteristics of shells subjected to axial forces such as centrifugal forces and used algebraic polynomial functions as the functional form. Laura et al. [16] used the Rayleigh–Ritz method to address the problem of vibrations in a circular plate. Kandasamy and Singh [17] analyzed the free vibration of isotropically skewed open circular cylindrical shells using a modified version of the Rayleigh–Ritz method. Wang [18] used a new displacement field applied to the Euler–Bernoulli theory for studying the free vibration and buckling problems of both thick and thin beams and plates. Zhou and Cheung [19] used the Rayleigh method to calculate the frequencies of a tapered Timoshenko beam under a Taylor series of static load; this method was applied to derive the eigenfrequency equation. To apply the Rayleigh method, the virtual-work principle must be described by adequately chosen generalized coordinates, and by a shape function that describes the mode of vibration. At the end of the calculation, the movement equation is written in terms of the generalized coordinates, from which one can extract the generalized properties of the system. Consider, as shown in Fig. 1, a system composed of a prismatic bar made from an elastic linear-hardening material; the bar ¯ and masses mi that are representatives is embedded at its base, bearing its own weight, represented by the parameter m of the bodies fixed along its length, which include one specially designed by m0 that is situated on the top. The movement of the system does not alter the orientation of the normal force N(x), which should be taken into consideration. A similar mathematical development can be found in Clough and Penzien [20] but without the inclusion of self-weight. Note that the inclusion of the self-weight was a difficulty found by Euler when he calculated the buckling force of a column. Assuming the well-known trigonometric function

φ (x ) = 1 − cos

πx 2L

,

(1)

the elastic stiffness can be determined as

 K0 =

L 0



d 2 φ (x ) EI d x2

2

dx,

(2)

where E is the elasticity modulus and I is the inertia of the section. The geometric stiffness is given by

 Kg =

L 0

 N (x )

d φ (x ) dx

2 dx,

(3)

¯ (L − x )]g. The total generalized mass is where N(x) is a function of the normal force N (x ) = [mR + m

M = mR + m,

(4)

where mR is the design generalized mass lumped, generated by the mass mi at the element joint xi obtained by

mR =



mi φ ( xi )2 ,

(5)

and m is the generalized mass originated by the self-weight of the structural element given by

 m=

0

L

¯ (φ (x ) ) , m 2

(6)

¯ is the mass per unit length, obtained by multiplying the density of the material by the cross-sectional area. It is of where m interest to mention that the shape function given by Eq. (1) is considered to be valid throughout the entire domain of the Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM 4

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

structure. Studies to evaluate Eq. (1) and the shape of vibration of the real cases have been performed by Wahrhaftig [21]. Finally, the natural cyclical frequency is calculated as



ω=

K , M

(7)

taking into account

K = K0 − Kg .

(8)

The final expression to calculate the first undamped resonant frequency, in hertz, is given by

1 f = 2π





π 4 EI + π 2 2mR +m¯ L 32 L3 16 L



 12

¯ g − 14 m

¯ mR + 3π2π−8 Lm

,

(9)

considering the normal force of compression to be positive. In Eq. (9), the conventional parcel of the stiffness of the column to the left side of the negative signal in the numerator, and the geometric parcel of the stiffness to the right side of the same signal, can be seen. The first one is function of the modulus of elasticity of the material and the second of the normal force. Here, it is possible to take into account the lumped mass at the free extremity of the column through to the mass mR , plus its self-weight, by considerating the parameter ¯ , which is the distributed mass per length unit, both of which multiply the acceleration of gravity, as can be seen. The m generalized mass of the system is observed at the denominator. With the use of Eq. (9) it is possible to perform a geometric nonlinear analysis in a single mathematical operation, by considering the effect of the normal force, dispensing complex computational tools, or interactive calculations. It is opportune to mention that Eq. (9) has been evaluated by computational modeling using the finite element method (FEM), and experimentally in a physical laboratory by Wahrhaftig et al. [22], who found it to be a very good approximation, and have provided more detail about the previous mathematical development. 3. Geometric nonlinear solution based on fem Structural dynamics can be employed to obtain solutions to homogeneous differential equations, the shape of which represents vibration modes that exist in the coordinate system at the same frequency range, and occur harmonically in time. The equation describes the vibration of the system, according to a normal mode of vibration, and corresponds to the frequency. The mathematical solution to the dynamic problem is a polynomial equation of degree n that contains the variable ω2 , and is commonly known as the frequency equation. The n solutions for ωi are real and positive, and are considered the natural frequencies of the system. The smallest frequency is typically denoted as ω1 , while the largest frequency is denoted as ωn . Thus, n modes of vibration can be determined and collected in a modal n x n matrix, which contains columns representing the n modes of undampened, normalized free vibration. Each pair of eigenvalues and eigenvectors corresponds to a frequency and mode of vibration for the system. The previous description represents a classical method of performing dynamic analysis of structures – the so-called modal analysis – in which enough information about systems or structures is obtained to reproduce their dynamics. In classic modal analysis, this information is related to the natural frequencies of the system (eigenvalues) and the modes of vibration (eigenvectors). The formulation corresponding to the FEM in relation to the mathematical procedure is the geometrically nonlinear formulation, through which geometric stiffness is introduced into the stiffness matrix of the structure. The modal analysis by eigenvalues is resolved as



det [K]−ω2 [M] = 0,

(10)

where [K]is the total stiffness matrix, with the same notation as that in Eq. (8), and [M] is the mass matrix, representing modal analysis, it with geometric nonlinear characteristics based on the geometric stiffness, as presented by Filho [23]. Geometric stiffness has been introduced in several analyses in the FEM environment while considering nonlinear effects or geometric nonlinearity (GNL). Levy et al. [24] used this consideration to study problems with membranes by using symbolic algebra. They affirmed that the geometric stiffness matrix plays an important role in the nonlinear analysis of membrane shells undergoing large rotations and small strains. Another work that demonstrates the importance of the geometric stiffness is the analysis by Spillers [25], who investigated the effect of this parameter on the checking of space frames; he found that the omission of terms in the geometric stiffness matrix implies the omission of some buckling mechanisms in physical systems, which can be important in some physical situations. It is important to mention that the interpolation functions used in the FEM formulations to determine the full stiffness matrix are third-degree polynomials, e.g., Filho [23] and Wilson and Bathe [26]. Lin and Trethewey [27] employed cubic expressions as interpolation functions when using the FEM for the dynamic analysis of elastic beams subjected to dynamic loads induced by the arbitrary movement of a spring-mass-damper system. Ormarsson and Dahlblom [28] also employed third-degree polynomials as shape functions to introduce the viscoelastic behavior of a wood material when formulating mathematical procedures within the principles of the FEM. Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

JID: APM

ARTICLE IN PRESS A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

[m3Gsc;October 21, 2016;20:45] 5

Continuing the discussion on the problems of structural dynamics, Likins [29] indicated that nonlinear terms were incorrectly omitted before his development work (Likins [30]) and showed how these nonlinearities can, in some cases, contribute to the geometric stiffness matrix of the finite element model. Carrion et al. [31] stated that a well-known concept used in the FEM is the stiffness matrix of an element, which is used to relate the external forces applied at the nodes of the structural element to its nodal displacements. El-Absy and Shabana [32] conducted analytical studies based on the virtual work formulation to examine the effect of geometric stiffness forces on the stability of elastic and rigid body modes. A simple rotating beam model was used to demonstrate the effect of axial forces and dynamic coupling between the modes of displacement on the rigid body motion. Finding the numerical solution of large nonlinear structural dynamics problems using the FEM and direct time integration techniques can be a computationally intensive task for any computer, and very often, this task exceeds the available computational capabilities. Some research efforts have been directed toward the development of techniques to reduce the computational cost for these structural problems, as discussed by Idelsohn and Cardona [33]. With regard to FE analysis, Wilson and Habibullah [34] stated that further consideration of the normal force in structural dynamics is a viable technique for calculating the second-order effects because the effects are linearized and the solution to the problem is obtained directly and accurately without interactions. Lou et al. [35] developed an FE model to simulate the short-term full-range response and long-term service-load behavior of slender prestressed concrete columns subjected to uniaxial eccentric loads. They analyzed the coupling of axial and flexural fields and their interaction with P-delta, when P-delta effects are considered by introducing a geometric stiffness matrix. Avilés et al. [36] discussed the kinematic analysis of mechanisms and affirmed that the main advantages of using the geometric matrix are that it gives a general method that is valid for any linkage with any kinematic configuration and number of rigid-body degrees of freedom and that it provides complete information on the kinematic properties, including positions, velocities, accelerations, jerks, and singular positions. They also stated that the approach based on a reduced form of the stiffness matrix referred to as the geometric stiffness matrix, or simply as the geometric matrix, offers a number of major advantages, especially with respect to simplicity and generality. The computational cost is also very low because of the simplicity of the numerical calculations and the reduced dimensions of the matrices involved. Li et al. [37] used the concept of geometric stiffness to realize a nonlinear seismic analysis for RC framed structures considering full-range factors, including stiffness and strength degradation and structural member failure; the analysis was established based on the fundamental concept of the force analogy method. According to them, the proposed analysis can evaluate the exact response of RC structures because the procedure is performed on an RC framed structure to simulate the full-range nonlinear response, which consists of material nonlinearity and GNL, when subjected to earthquake motions. Kalkan and Laefer [38] explained that the accurate modeling of long poles is of particular importance because of the significant P-delta effects they exhibit. These second-order moment effects intensify as the height of the pole and the number of antennas and platforms increase and as the amplitude of motion intensifies. They presented many techniques readily available in the literature for evaluating the second-order behavior. The application of the geometric stiffness matrix is a general approach to include these effects during the analysis of all types of structural systems. They found that while modeling RC and steel poles by the FEM, the first two modes in each direction were characterized by extremely low frequencies in the range of 0.4–1.1 Hz. Jingbo and Zhenyu [39] carried out the seismic analysis of an RC structure characterized by abrupt changes in the horizontal connection stiffness using the FEM and found the first natural frequency of the structure to be 0.675 Hz; the big tube of the main building was 59.4-m high. In the structure model, the floor slabs and shear walls were simulated by discrete plate elements, whereas the beams and columns were simulated by beam elements. Nguyen et al. [8] studied the aeroelastic effects over poles and antenna masts that had similar properties to the mast considered in this study; they found that the first mode of vibration was located at 0.77 Hz. However, they did not take into account the geometric effects on the dynamics of the problem. Yuana et al. [40] applied the exact dynamic stiffness method, using an approximation for the vibration of Bernoulli–Euler members, to the flexural free vibration of nonuniform Timoshenko beams with gradual or stepwise nonuniformity of geometric and/or material properties and to the Euler buckling of similarly nonuniform columns. The formulation of the governing ordinary differential equations (ODEs) for dynamic stiffnesses and their derivatives were used, allotting one part of the equations to the geometric stiffness matrix. The solution of the ODE problem was obtained by standard ODE solvers; unlike the FEM, the efficiency, accuracy, and reliability of the proposed method were exact and not approximate. The effect of the external force on a structure was studied by Kovacic et al. [41]. Their study was concerned with the way in which a static force could dramatically change the dynamic behavior of a harmonically excited quasi-zero stiffness SDOF system. They also presented reduced order modeling techniques for geometrically nonlinear structures, more specifically, techniques that are applicable to structural models constructed using commercial finite element software. They depicted the geometric matrix as an alternative to the Jacobian matrices traditionally employed in the kinematic analysis of mechanisms. By using this matrix, which is a type of stiffness matrix based on a very simple mathematical model in which the mechanism is represented by its primary and secondary basic nodes, the analysis of velocities, accelerations, and jerks, as well as the solution to position problems of mechanisms with any kinematical configuration, can be carried out in a simple and general manner. The main advantage of using the geometric matrix is that it gives a general method that is valid for any linkage with any kinematic configuration and number of rigid-body degrees of freedom. More details about specific GNL formulation by the FEM in a similar context to this work are available in the study by Wahrhaftig and Brasil [42]. Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM

400

40.64 t = 0.48

600

65 t = 0.80

600

70 t = 0.80

700

80 t = 0.80

1400

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

90 t = 0.80

1100

4800

6

[m3Gsc;October 21, 2016;20:45]

(a) Slender metallic pole

173.11 t = 0.80

(b) Geometric details

(c) Discretized of FEM model

Fig. 2. 48 m-high slender metallic pole and its geometric details.

Table 1 Structural properties and discretization of the FEM model.

φ ext

φ ext

φ ext

φ ext

Height (m)

(cm)

t (cm)

Height (m)

(cm)

t (cm)

Height (m)

(cm)

t (cm)

Height (m)

(cm)

t (cm)

48.00 46.00 44.00 42.00 40.00 38.00 36.00 34.00 32.00 31.00

40.64 40.64 40.64 65.00 65.00 65.00 70.00 70.00 70.00 80.00

0.48 0.48 0.48 0.80 0.80 0.80 0.80 0.80 0.80 0.80

30.00 29.00 28.00 27.00 26.00 25.00 24.00 23.00 22.00 21.00

80.00 80.00 80.00 80.00 80.00 80.00 90.00 90.00 90.00 90.00

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

20.00 19.00 18.00 17.00 16.00 15.00 14.00 13.00 12.00 11.00

90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

10.00 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00

97.56 105.11 112.67 120.22 127.78 135.33 142.89 150.44 158.00 165.56 173.11

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

The results obtained by computer modeling in this study are consistent with those found by Silva and Brasil [43], who studied the dynamic behavior of a post under linearity and GNL; the post had similar characteristics to those considered in this study, although the post was made of reinforced concrete. Silva et al. [44] conducted studies on the dynamic behavior of 90 telecommunication poles under wind forces by means of a simplified procedure; they obtained consistent results for more complex analysis.

4. Simulations and discussion 4.1. Case 1 – 48-m high pole – actual structure characteristics The actual structure selected for this study is a metallic high-slenderness Fig. 2(a). The geometric details are shown in Fig. 2(b), where t is the thickness of the wall of each segment of the structure. Fig. 2(c) shows the finite element model and the discretization of the structure with 40 bar elements. Table 1 lists the structure properties and model discretization values. Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

7

Table 2. Devices and weights on the structure. Device

Height

Weight and distributed weight

Pole Stairs Cables Antenna and supports

from 0 to 48 m from 0 to 48 m from 0 to 48 m 48 m

7850 kN m−3 0.15 kN m−1 0.25 kN m−1 3.36 kN

Table 3 Parameters of analytical procedure. Parameter

Value

Elastic modulus of steel Density of steel Lumped mass at the top Distributed mass per unit height Gravitational acceleration

E = 205 GPa ρ = 7850 kg m−3 m0 = 342.40 kg ¯ = 40 kg m−1 m g = 9.806650 m/s2

Table 2 lists the structural parameters and the existing devices on the structure. This metal pole is used for supporting the transmission system of mobile telephony signals. The structure is 48-m high and has a hollow circular section with variable external diameter (φ ext ) and thickness (t); the slenderness of the pole is 310. According to Nguyen et al. [8], lighting poles and antenna masts may appear as simple structures that can be modeled easily by slender vertical cantilever beams with one or more concentrated masses usually located at their top; engineering calculations are often based on such models. However, a deeper reflection on this type of structures indicates that they have totally different characteristics. Such structures are being built in such large numbers that they represent a relevant economic problem in spite of their low single cost. They are characterized by increasing height, lightness, and slenderness, and have complicated shapes, making them extremely sensitive to complex aeroelastic phenomena and wind-excited vibrations. Therefore, sophisticated analyses are required to capture their physical behavior. 4.1.1. Analytical procedure 4.1.1.1. Geometric definitions and parameters adopted. The parameters used for the analytical procedure using the Rayleigh method are as follows (Table 3): The corresponding ordinates of the heights in the structure and the geometric properties of the cross sections of the respective segments are given as follows 2 π ( D 4 − d 4 ). For the base (x= 0):D1 = 173.11 cm, t1 = 0.80 cm, d1 =D1 − 2t1 , A1 = π4 (D1 2 − d1 ), I1 = 64 1 1 For the following segment with variable properties: D(x ) =

π

4

64 (D (x )

4

− d ( x ) ).

D2 −D1 L1 x +

2 2 D1 , d(x) = D(x) − 2t1 ,A(x ) = π4 (D(x ) − d (x ) ), I (x ) =

2 π (D 4 − d 4 ). For L = For the ordinate L1 = 11 m:D2 = 90.00 cm, t2 = 0.80 cm, d2 = D2 − 2t2 , A2 = π4 (D2 2 − d2 ), I2 = 64 2 2 2

2 π (D 4 − d 4 ). For L = 32.00 m: D = 70.00 cm, 25.00 m: D3 = 80.00 cm, t3 = 0.80 cm, d3 = D3 − 2t3 , A3 = π4 (D3 2 − d3 ), I3 = 64 3 3 3 4

2 π (D 4 − d 4 ). ForL = 38.00 m: D = 65.00 cm, t = 0.80 cm, d = D − 2t , t4 = 0.80 cm, d4 = D4 − 2t4 , A4 = π4 (D4 2 − d4 ), I4 = 64 4 4 4 5 5 5 5 5 2 π (D 4 − d 4 ). For L = 44.00 m and L = 48.00 m: D = 40.64 cm, t = 0.48 cm, d = D − 2t , A = A5 = π4 (D5 2 − d5 ), I5 = 64 5 5 6 5 6 6 6 6 6 6

π ( D 2 − d 2 ), I = π ( D 4 − d 4 ). 6 6 6 5 5 4 64

4.1.1.2. Calculation of generalized mass. The generalized mass was determined using the following integral arrangement. For the first segment,



m1 =

L1

0

¯. mI (x )φ (x ) dx, with mI (x ) = A(x )ρ + m 2

(11)

For the second segment,



m2 =

L2

L1

¯. mII φ (x ) dx, with mII = A2 ρ + m 2

(12)

Similarly, for the other segments, the general form can be expressed as



mi =

Li Li−1

¯ , where being i = 1, 2 . . . 6 and i = I, II . . . V I. mι φ (x ) dx, with mι = Ai ρ + m 2

(13)

The generalized distributed mass is obtained as

m=

6 

mi ,

(14)

i=1

Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM 8

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

and the total generalized mass is obtained as

M = m0 + m.

(15)

The total generalized mass on the structure is 1960.54 kg. 4.1.1.3. Calculation of generalized geometric stiffness. The calculation of the generalized geometric stiffness is necessary to determine the normal forces acting on the parts defined in the structure geometry. From the top to the bottom, the axial forces acting on the structure are

 F0 = m0 g, F6 =

L

L5

 mV I gdx, F5 =

L5 L4

mV gdx,

(16)

and so on. This can be generalized as

 Fi =

Li

mι gdx.

Li−1

(17)

The axial force on the first segment, which is linearly variable, is obtained by the following expression:

 F1 =

L1

mI (x )gdx,

0

(18)

where g is the acceleration due to gravity. Therefore, the generalized axial force F is

F=

6 

Fi ,

(19)

i=0

and the geometric stiffness is calculated using the following expressions:

 Kg6 =

L



L5

 Kg5 =

L5 L4

 Kg4 =

L4 L3

 Kg3 =

L3 L2

 Kg2 =

L2 L1

 Kg1 =

L1

2 



d F0 + mV I (L6 − x )g φ (x ) dx



(a )

,

2 



d F0 + F6 + mV (L5 − x )g φ (x ) dx



(b )

,

2 



d F0 + F6 + F5 + mIV (L4 − x )g φ (x ) dx



(c )

,

2



d F0 + F6 + F5 + F4 + mIII (L3 − x )g φ (x ) dx



2 



d F0 + F6 + F5 + F4 + F3 + mII (L2 − x )g φ (x ) dx



(d )

,

(e )

,

2 



d F0 + F6 + F5 + F4 + F3 + F2 + mI (x )(L1 − x )g φ (x ) dx

0

.

(f )

(20)

Therefore, the generalized geometric stiffness Kg of the structure is obtained as

Kg =

6 

Kgi .

(21)

i=1

The generalized geometric stiffness on the structure is 0.653 kN m−1 . 4.1.1.4. Calculation of generalized conventional stiffness. The conventional stiffness segments can be represented as

 K01 =

L1

0



EI (x )

2

d2 φ (x ) d x2

 dx, K02 =



L2

E I2 L1

2

d2 φ (x ) d x2

dx.

(22)

In the same way, the other segments can be expressed as

 K0i =



Li

Li−1

E Ii

2

d2 φ (x ) d x2

dx,

(23)

Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

9

Table 4 Normal force on the structure. L

Analytical procedure

FEM

Difference

(m)

(kN)

(kN)

Absolute

(%)

48.00 44.00 38.00 32.00 25.00 11.00 0.00

3.355520 6.786842 16.585633 26.964392 40.426203 70.056344 102.174047

3.355520 6.786842 16.585633 26.964392 40.426204 70.056345 102.174049

0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0

0.0 0 0 0 0 0 −0.0 0 0 0 01 −0.0 0 0 0 01 −0.0 0 0 0 01 −0.0 0 0 0 02 −0.0 0 0 0 02 −0.0 0 0 0 02

and the generalized conventional stiffness is given by the sum of the segments

K0 =

6 

K0i .

(24)

i=1

The generalized conventional stiffness is 25.768 kN m−1 . 4.1.1.5. Calculation of frequency. The frequency for the first mode of vibration calculated using Eq. (7) and converted to hertz is 0.569646 Hz. To employ Eq. (8) directly, it is necessary to adopt a criterion for weighting the geometric properties of the structure. One such approach is presented below:

A=

( A1 +2 A2 )L1 + A2 (L2 − L1 ) + A3 (L3 − L2 ) + A4 (L4 − L3 ) + A5 (L5 − L4 ) + A6 (L − L5 ) L

.

(25)

Thus, the weighted area is A = 0.022 m2

I=

( I1 +2 I2 )L1 + I2 (L2 − L1 ) + I3 (L3 − L2 ) + I4 (L4 − L3 ) + I5 (L5 − L4 ) + I6 (L − L5 ) L

.

(26)

Thus, the inertia weighted is I = 0.003 m4 . The frequencies calculated using the weighting parameters are 0.413132 Hz (linear) and 0.403193 Hz (nonlinear). 4.2. Computational modeling To compare the development results obtained using the previous mathematical procedure and the proposed procedure, a computational model was developed by the FEM and a modal analysis was realized for the model. The structure analyzed in the study was modeled using bar elements with constant and variable cross sections, as appropriate. The forces listed in Table 4 were applied to the model, along with the corresponding masses. The frequencies for the first mode of vibration by the FEM were: without geometric nonlinearity (Linear), 0.499057 Hz; and with geometric nonlinearity (GNL), 0.489339 Hz. The frequency obtained by the FEM was calculated by taking into account the effect of the normal forces, using the formulation corresponding to the FEM. The formulation is realized by considering the geometric stiffness in the structure stiffness matrix, and can be used to take into account of second-order effects, which are particularly important in slender structures. A comparison, assessing the quality of the analytical results of the mathematical development presented in this work, shows an increase in the normal force on each segment along the structure, as listed in Table 4. The results show no differences between the analytical and computational solutions. 4.3. Case 2 – 30-m high pole – actual structure characteristics The investigated structure is a truncated cone metallic pole with 52 and 82 cm top and bottom diameters, respectively. It is intended to sustain a mobile phone broadcasting system. It is a 30 m high, hollow section. The external diameter (φ ext ) and thickness (t) vary along the height. The structure data were acquired in the field. The diameters were measured with a metallic tape measure and the thickness with ultrasound equipment. For a given vertical line, several thickness measurements were carried out to obtain a relative average of the band. The union of the pole segments is formed by successive fittings, by placing and screw-fastening the metallic parts. Each superpositioning band has 20 cm length. In these joint areas, the thickness of the transverse section corresponds to the sum of the measures of the superpositioning bands, conform is indicated in Fig. 3. Table 5 presents the properties and the discretization used to model the structure. The assessed slenderness of the structure is 256. The metallic pole sustains two working platforms, one situated at 20 m height and the other at the superior extremity. There is still a set of antennas located at 27 m from the base and attached to the body of the pole through metallic devices. The supporting devices and the platforms follow the composition presented in Tables 6 and 7, where φ designates the Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

2nd Platform and antennas Support and antennas

20

52 t = 0.60

Var t = 0.6

Var t = 1.2

580 20

590

10

[m3Gsc;October 21, 2016;20:45]

1st Platform and antennas

Var

20

580

3000

t = 1.2

(b) General photographic views.

Var t = 0.6

Var

Var

20

580

t = 1.52 t = 0.76

Var

590

t = 1.52

Var t = 0.76

82

t = 0.76

(c) Base detail.

(a) Geometry ( in centimeters) and FEM discretization Fig. 3. 30 m-high slender metallic pole and its geometric details. Table 5 Structural properties and discretization of the FEM model.

φ ext

φ ext

φ ext

Height (m)

(cm)

t (cm)

Height (m)

(cm)

t (cm)

Height (m)

(cm)

t (cm)

30.00 29.00 28.00 27.00 26.00 25.00 24.10 23.90 23.00 22.00 21.00

52.00 53.00 54.00 55.00 56.00 57.00 57.90 58.10 59.00 60.00 61.00

0.60 0.60 0.60 0.60 0,60 0.60 0.60 0.60 0.60 0.60 0.60

20.00 19.00 18.10 17.90 17.00 16.00 15.00 14.00 13.00 12.10 11.90

62.00 63.00 63.90 64.10 65.00 66.00 67.00 68.00 69.00 69.90 70.10

0.60 0,60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.76

10.00 9.00 8.00 7.00 6.10 5.90 5.00 4.00 3.00 2.00 1.00 0.00

72.00 73.00 74.00 75.00 75.90 76.10 77.00 78.00 79.00 80.00 81.00 82.00

0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76

diameter of the platform. The local assessment revealed the presence of microwave (MW) antennas and of radio frequency (RF), which are listed with the rest of the structure accessories in Table 8. The data related to the antennas were obtained from the catalog of the manufacturer. All the aforementioned devices represent additional masses and concentrated forces on the structure, as shown in Table 9, which presents the structural parameters and the parameters of the existing devices, the specific weight adopted for the material of the structure, and the localized and distributed axial load. The geometry of the structure and the existing devices are schematically represented in Fig. 3, as well as photographic images of the pole and details of the base of the structure. 4.3.1. Analytical procedure 4.3.1.1. Geometric definitions and parameters adopted. The external diameter of the sections varies linearly with their height D−D according to the expression D(x ) = L 1 x + D1 , where D is the diameter of the top and D1 is the diameter of the base of the structure, with x varying from 0 to L (L is length or height of the structure). The parameters of the analytical procedure are listed in Table 10: Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

11

Table 6 Composition of the support. Support set for antenna

Mass (kg)

´ Pipe φ = 1(25.4 mm) Angle (L) 203 × 152 × 19 mm Staples U (φ =1)´ Top plate Total =

6 50 1 1 58

Table 7 Composition of the platform. Platform φ = 2.5 m

Mass (kg)

Floor sheet Lateral floor sheet Channel (U) 150 × 12.2 mm – Banister Angle (L) 102 × 76 × 6.4 mm – Banister Angle (L) 102 × 76 × 6.4 mm – Banister Angle (L) 102 × 76 × 6.4 mm – Floor support Platform lower ring Joints Banister bolts Angle (L) 152 × 102 × 9.5 mm – Platform lower support Total =

116 46 96 68 77 43 14 3 5 33 500

Table 8 Composition of the localized nodal masses. Mass Device Antenna RF 2.6 m Antenna RF 1.23 m Antenna MW Platform Support for antennas ´ mm) (Guide) Pipe φ = 1(25.4 ´ mm) (LC) Pipe φ = 3/4(19 Total (kg) =

1st Plat (20 m)

Support (27 m)

2nd Plat (30 m)

(kg/unit)

Quant.

(kg)

Quant.

(kg)

Quant.

(kg)

19 4 19 500 58 6 6

2 1 2 1 6 0 0

37 4 38 500 345 0 0 924

3 0 0 0 3 0 0

56 0 0 0 173 0 0 228

1 1 0 1 6 1 1

19 4 0 500 345 6 6 880

(LC = lightning conductor, MW = microwave, RF = radio frequency, Plat = platform) Table 9 Localized axial load and characteristics of the devices. Device

Frontal area

Height

Weight, distributed weight

Pole Ladder Cables 1st Platform Antenna of the 1st platform Intermediate antennas Intermediate supports 2nd Platform Antennas of the 2st platform

Variable 0.05 m2 /m 0.15 m2 /m 2.60 m2 1.99 m2 2.11 m2 0.56 m2 2.36 m2 0.90

0–30 m 0–30 m 0–30 m 20 m

77 kNm−3 0.15 kN m−3 0.25 kN m−3 9.06 kN

27 m

2.24 kN

30 m m2

8.63 kN

Table 10 Parameters of analytical procedure. Parameter

Value

Elastic modulus of steel Density of steel Lumped mass at the top Lumped mass at 27 m height Lumped mass at 20 m height mX = 228 kg mXI = 924 kg Distributed mass per unit height Gravitational acceleration

E = 205 GPa ρ = 7850 kg m−3

m0 = 880 kg

¯ = 40 kg m−1 m g = 9.806650 m/s2

Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM 12

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

The heights and geometric properties of the sections of each segment are as follows: For the first segment, from the base where x = 0, up tox = L1 = 7.60m,D1 = 153.07 cm,e1 = 0.87 cm,d1 (x) = D(x) − 2e1 , 2 π (D (x )4 − d (x )4 ). For the second segment, from x = 7.60 m up tox = L = 33.44m,e = A1 (x ) = π4 (D2 − d1 (x ) ), and I1 (x ) = 64 1 2 2 2 2 π (D (x )4 − d (x )4 ). For the third segment, from 0.84 cm,d2 (x) = D(x) − 2e2 ,A2 (x ) = π4 (D(x ) − d2 (x ) ), and I2 (x ) = 64 2

2 2 x = 33.40 m up tox = L = 60.80m,D3 = 94 cm,e3 = 0.64 cm, d3 (x) = D(x) − 2e3 , A3 (x ) = π4 (D(x ) − d3 (x ) ), and I3 (x ) = 4 4 π 64 (D (x ) − d3 (x ) ). For the D9 −D0 D(x ) = L x + D0 , where

whole structure, the external diameter varies linearly with the height, following the expression D9 is the diameter of the superior extremity and D0 is the diameter of the inferior extremity,

with x varying from 0 to L. The coordinates and thickness of the sections of each segment are the follows: on the base, wherex = 0, one has: D0 = 82 cm, t1 = 0.76 cm. On the first segment, from the base of the pole up to L1 = 5.90m,t1 = 0.76cm. On the second segment, between L1 and L2 = 6.10m, t2 = 1.52cm. On the third segment, between L2 and L3 = 11.90m, t3 = 0.76cm. On the fourth segment, between L3 and L4 = 12.10m, t4 = 1, 52cm. On the fifth segment, between L4 and L5 = 17.90m, t5 = 0.60cm. On the sixth segment, between, betweenL5 and L6 = 18.10m,t6 = 1.20cm. On the seventh segment, between L6 and L7 = 23.90m, t7 = 0.60cm. On the eighth segment, betweenL7 and L8 = 24.10m, t8 = 0.60cm. On the ninth segment, between L8 and L = 30.00m, one hast9 = 0.60cm. The geometric proprieties, as well the internal diameter, area, and moment of inertia, were obtained 2 2 π (D (x )4 − d (x )4 ), where i characterizes with these general expressions: di (x) = D(x) − 2ti , Ai (x ) = π4 (D(x ) − di (x ) ), Ii (x ) = 64 i the analyzed segment. 4.3.1.2. Calculation of generalized mass. The generalized mass is calculated as

 mi =

Li

¯ , where being i = 1, 2 . . . 9 and i = I, II . . . IX. mι (x )φ (x ) dx, mι (x ) = Ai (x )ρ + m 2

Li−1

(27)

The other lumped masses are

m10 = mX φ (x10 )2 , with x10 = 27.00 m; m11 = mXI φ (x11 )2 , with x11 = 20.00m.

(28)

The generalized distributed mass is given by

m=

11 

mi ,

(29)

i=1

and the total generalized mass is given by

M = m0 + m.

(30)

The total generalized mass of the structure is 2131.60 kg. 4.3.1.3. Calculation of generalized geometric stiffness. The normal lumped efforts are given by

F0 = m0 g, F11 = mXI g, F10 = mX g and in each segment by

 Fi =

Li Li−1

mI (x )gdx,

(31)

where g is the gravitational acceleration. The generalized axial force F is given by

F=

11 

Fi .

(32)

i=0

Thus, the geometric stiffness is obtained by the expressions

 Kg9 =



L8

 Kg8 =

L8 L7

 Kg7 =

L

L7 L6

2 



d F0 + F11 + mIX (x )(L − x )g φ (x ) dx





(a )

,

2

d F0 + F11 + F9 + mV III (x )(L8 − x )g φ (x ) dx





(b )

,

2

d F0 + F11 + F9 + F10 + F8 + mV II (x )(L7 − x )g φ (x ) dx

,

(c )

(33)

Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

13

Table 11 Normal forces on the structure. L

Analytical procedure

FEM

(m)

(kN)

(kN)

Absolute

(%)

30.00 24.10 23.90 18.10 17.90 12.10 11.90 6.10 5.90 0.00

8.6240 0 0 17.820587 18.228409 34.636570 35.079190 42.936723 43.518196 53.486859 54.112409 64.908508

8.6240 0 0 17.820587 18.228409 34.636570 35.079190 42.936724 43.518196 53.486860 54.112410 64.908509

0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0

0.0 0 0 0 0 0 –0.0 0 0 0 01 –0.0 0 0 0 01 –0.0 0 0 0 01 –0.0 0 0 0 01 –0.0 0 0 0 01 –0.0 0 0 0 01 –0.0 0 0 0 01 –0.0 0 0 0 01 –0.0 0 0 0 01

which are repeated for other segments

 Kgi =

Li Li−1



F0 +

10  i+1



Difference

2 

d Fi+1 + mι (x )(Li − x )g φ (x ) dx

.

(34)

Therefore, the total generalized geometric stiffness is

Kg =

9 

Kgi .

(35)

i=1

The generalized geometric stiffness of the structure is 1.139550 kN/ m. 4.3.1.4. Calculation of generalized elastic stiffness. Similar to geometric stiffness, the elastic geometric is

 K0i =

Li Li−1



E Ii (x )

2

d2 φ (x ) d x2

dx.

(36)

The generalized elastic stiffness is given by the sum of the above components

K0 =

9 

K0i .

(37)

i=1

The generalized conventional stiffness is 25.481173 kN/m. 4.3.1.5. Calculation of frequency. The frequency of the first mode calculated numerically using Eq. (7) in hertz, is 0.537826 Hz. If the geometric stiffness is not considered, by linear analysis, the frequency is 0.550271 Hz. In order to employ Eq. (8) directly, it is necessary to adopt a criterion for weighting the geometric properties of the structure. One such approach is performed by thickness weighted as

t0 L1 + t1 ( L2 − L1 ) + t0 ( L3 − L2 ) + t1 ( L4 − L3 ) + t2 ( L5 − L4 ) + t3 ( L6 − L5 ) + t2 ( L7 − L6 ) + t3 ( L8 − L7 ) + t2 ( L − L8 ) t= = 0.69 cm, L Dt + Db π 2 D= ,d = D − 2t, A = D − d2 . 2 4

(38)

with t0 = 0.76 cm, t1 = 1.52 cm, t2 = 0.62 cm, and t3 = 1.2 cm. The frequencies calculated using the weighting parameters is 0.408091 Hz. 4.4. Computational modeling The structure presented in this work was modeled using frame elements with transverse sections, constant and variable, depending on the case. The forces described in Table 9 with the corresponding masses were applied to the model. The connecting regions were treated as variable section frame elements of 0.2 m length and thickness corresponding to the sum of the thickness of the sections immediately above and below the joint zone. The frequencies obtained by FEM were: without geometric nonlinearity (Linear), 0.543837 Hz; and with geometric nonlinearity (GNL), 0.531972 Hz. The normal forces on each segment of the structure are compared to assess the quality of the analytical results of this work, as summarized in Table 11. As can be seen, the results show there are no differences between the analytical and computational solutions. Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

ARTICLE IN PRESS

JID: APM 14

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

(a) Top structure

(b) Accelerometer positioned

(c) Acquisition Data System

Fig. 4. Instrumentation of the structure.

(a) Temporal acceleration series

(b) Signal in the frequency domain

Fig. 5. Signal from experimental activity.

4.5. Experimental field investigation The investigation of the natural frequency of the structure under ambient excitement was carried out using piezoresistive accelerometers, manufactured by Bruel and Kjaer [45], with DC response and sensitivity of 1021 mV/g, provided with integrated cable, and suitable for measuring accelerations in the range of ±2 g. This device was fixed to the superior extremity of the pole, as seen in Fig. 4(a) and (b). The ADS-20 0 0 automatic data-acquisition system AqDados [46] was used with AI-2161 conversing plates, an AC-2122VA controlling plate (LYNX Informatics), and 16-bit resolution. Fig. 4(c) shows the setup of the data-acquisition system that was connected to a portable computer for the recording of the signals. The devices were taken to the top of the pole, where they were placed on the surface of the working platform, and protected from bad weather. The electric energy system of the station was used as a source to feed power to the electronic equipment. The signal acquisition was carried out at a rate of 50 Hz, and its elapsed time was 40 h 33 min 22 s. The acceleration time series can be seen in in Fig. 5(a). It can be seen that the structure was under sufficient excitement by wind, because rain and strong winds occurred during this period. The fundamental frequency of the structure was obtained from the time series of the signal acquisition by fast Fourier transform (FFT) in the program AqDAnalysis 7.02 [47]. The obtained result was 0.53 Hz. In Fig. 5(b), the analysis results of the signal in the frequency domain are presented. 5. Conclusions In this study, the Rayleigh method was applied to calculate the fundamental frequency of a mobile phone mast system, considering the geometric stiffness function of concentrated forces and the self-weight of the structure. For comparison, an FEM-based computer simulation was performed using a formulation developed in this study. The difficulty associated with calculating the geometric stiffness of a real structure is due to the change in the properties of the structure along its height. Therefore, the entire analysis method must be resolved within the limits set for each interval and must include the selfweight of the structural elements. Considering the historical importance of the Rayleigh method in the field of mechanical vibrations, this work presents an opportunity to evaluate its application to a real structure, along with a comparison of the analytical results with the results obtained using modern computing methods. Based on the results obtained in this study, the following conclusions can be drawn: 5.1. Case 1 (1) The formulation proposed in this paper for calculating the geometric stiffness is reviewed by the FEM, by comparing the results of the axial forces on each segment of the structure (Table 4). No major differences are found between the results. Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

JID: APM

ARTICLE IN PRESS A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

[m3Gsc;October 21, 2016;20:45] 15

(2) The vibration frequency of the fundamental mode, calculated using the proposed method under GNL, is 0.569646 Hz, and that obtained by the FEM is 0.489339 Hz; that is, the result obtained using the proposed method is 16.41% greater than that obtained by the FEM. It is important to note that the principal characteristic of the Rayleigh method is the assumption that the mathematical function chosen to represent the first mode of vibration is a trigonometric function, whereas in FEM it is a polynomial function. That function is considered to be valid in the entire domain of the structure, requiring that the integrals of the method are resolved into the limits established by the geometry of the structure. (3) The geometric stiffness is 97.47% lesser than the conventional stiffness, and the frequency computed without the geometric stiffness segment is 0.577001 Hz. This implies that the reduction in the stiffness reduces the frequency by only 1.27%. The same analysis by the FEM gives a linear frequency of 0.499057 Hz, which represents a reduction of 1.95%. From Eq. (9), the frequencies calculated using the weighting parameters is 0.403193 Hz (nonlinear). This implies that a correction factor of 1.4 should be applied to the closed-form equation to obtain the correct values of the nonlinear frequency. 5.2. Case 2 (4) This study employed the same analytical procedure of the case 1 to calculate the initial undamped frequency of a 30-m-high mast under geometric nonlinearity, for which a simplified closed-form equation was proposed. When calculated analytically, under geometric nonlinearity, the vibration frequency of the natural fundamental mode was only 1.1% lesser than that obtained by the FEM using a compatible formulation (0.543837 Hz – linear and 0.531972 Hz – nonlinear). (5) When computed without the geometric stiffness component, this frequency was 0.550271 Hz. Further, it was observed that the geometric stiffness was 95.53% smaller than the conventional stiffness. This implies that the reduction in the stiffness reduces the frequency by only 2.26%. From Eq. (8), the frequencies calculated using the weighting parameters was 0.408091 Hz (nonlinear). This represents the need to apply a correction factor of 1.32 to obtain the correct values. 5.3. Conclusion Based on the results obtained for these cases, a factor of correction equal to 1.36 must be applied to Eq. (8) in order to find the correct value of the frequency, considering the geometric stiffness, which attributes a geometric nonlinearity characteristic to the system. The results obtained for analytical and computational solution were evaluated by comparing the normal forces at a key point defined by the geometry of the structure, allowing the evaluation of the quality of the solutions in relation to each other. The calculation of the normal forces is the most important operation in a geometric nonlinear consideration, because it constitutes the central factor that changes the geometric stiffness parcel of the structural stiffness. In addition to both mathematical studies an experimental field investigation was performed for the second case, in order to settle possible differences relating to the results obtained between the analytical solution and FEM. That exploration revealed a very good agreement of results of the frequency between methods. Directions for future research include application of the proposed analytical method to other structures with different geometry and load conditions. The results can be used to format a set of values such that an average correction factor can be finally applied to the closed-form equation in order to obtain the frequency of the first natural mode of vibration. This equation and factor can be included in a code of standardizations that would enable professionals in relevant fields to obtain appropriate results without the use of computers, thereby resolving practical engineering problems in a relatively quick and simple manner. One of these fields that can be cited is related to wind action, where the first natural frequency is the most important parameter for calculation of the dynamic forces. Acknowledgment The authors thank the National Council for Scientific and Technological Development (CNPq) of Brazil for their financial support through a research and productivity grant. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

Lord Rayleigh, Theory of Sound, Dover Publications, New York, 1877 (two volumes)re-issued 1945. S. Timoshenko, Theory of Elastic Stability, McGraw-Hill Book Company, Inc., New York and London, 1936. J. Ratzersdorfer, Buckling. Prismatic Bars, Vol. 2, Institute for Technological Research (IPT), São Paulo, 1954 (in Portuguese). M.L. Gambhir, Stability Analysis and Design of Structures, Springer, Berlin and Heidelberg, 2004 ISBN: 978-3-642-05866-0 (Print), 978-3-662-09996-4 (Online), doi:10.1007/978- 3- 662- 09996- 4. R.M.L.R.F. Brasil, M.A. Silva, RC large displacements: optimization applied to experimental results, Comput. Struct. 84 (2006) 1164–1171, doi:10.1016/j. compstruc.2006.01.027. C.W. Bert, Application of a version of the Rayleigh technique to problems of bars, beams, columns, membranes, and plates, J. Sound Vib. 119 (2) (1987) 317–326, doi:10.1016/0022- 460X(87)90457- 3. A. Nikkhoo, A. Farazandeh, M.E. Hassanabadi, On the computation of moving mass/beam interaction utilizing a semi-analytical method, J. Braz. Soc. Mech. Sci. Eng. (2014) 1–11, doi:10.1007/s40430- 014- 0277- 1. C.H. Nguyen, A.F.G. Solari, F. Tubino, Aeroelastic instability and wind-excited response of complex lighting poles and antenna masts, Eng. Struct. 85 (15) (2015) 264–276, doi:10.1016/j.engstruct.2014.12.015. R.B. Bhat, Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method, J. Sound Vib. 102 (4) (1985) 493–499, doi:10.1016/S0022-460X(85)80109-7.

Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020

JID: APM 16

ARTICLE IN PRESS

[m3Gsc;October 21, 2016;20:45]

A.d.M. Wahrhaftig, R.M.L.R.F. Brasil / Applied Mathematical Modelling 000 (2016) 1–16

[10] S. Chakraverty, L. Behera, Free vibration of non-uniform nanobeams using Rayleigh–Ritz method, Physica E 67 (2015) 38–46, doi:10.1016/j.physe.2014. 10.039. [11] K. El Bikri, R. Benamar, M.M. Bennouna, Geometrically non-linear free vibrations of clamped–clamped beams with an edge crack, Comput. Struct. 84 (7) (2006) 485–502, doi:10.1016/j.compstruc.2005.09.030. [12] L.E. Monterrubio, S. Ilanko, Proof of convergence for a set of admissible functions for the Rayleigh–Ritz analysis of beams and plates and shells of rectangular planform, Comput. Struct. 147 (2015) 236–243, doi:10.1016/j.compstruc.2014.09.008. [13] Y.K. Cheung, D. Zhou, Vibration of tapered Mindlin plates in terms of static Timoshenko beam functions, J. Sound Vib. 260 (4) (2003) 693–709, doi:10.1016/S0022-460X(02)01008-8. [14] M. Chiba, T. Sugimoto, Vibration characteristics of a cantilever plate with attached spring–mass system, J. Sound Vib. 260 (2) (2003) 237–263, doi:10. 1016/S0 022-460X(02)0 0921-5. [15] X.X. Hu, T. Sakiyama, H. Matsuda, C. Morita, Fundamental vibration of rotating cantilever blades with pre-twist, J. Sound Vib. 271 (1-2) (2004) 47–66, doi:10.1016/S0 022-460X(03)0 0262-1. [16] P.A.A. Laura, U. Masiáb, D.R. Avalos, Small amplitude, transverse vibrations of circular plates elastically restrained against rotation with an eccentric circular perforation with a free edge, J. Sound Vib. 292 (3-5) (2006) 1004–1010, doi:10.1016/j.jsv.2005.09.024. [17] S. Kandasamy, A.V. Singh, Free vibration analysis of skewed open circular cylindrical shells, J. Sound Vib. 290 (3-5) (20 06) 110 0–1118, doi:10.1016/j.jsv. 2005.05.010. [18] S. Wang, Unified Timoshenko beam B-spline Rayleigh-Ritz method for vibration and buckling analysis of thick and thin beams and plates, Int. J. Numer. Methods Eng. 40 (1997) 473–491, doi:10.1002/(SICI)1097-0207(19970215)40:3473::AID- NME753.0.CO;2- U. [19] D. Zhou, Y.K. Cheung, Vibrations of tapered Timoshenko beams in terms of static Timoshenko beam functions, J. Appl. Mech. 68 (4) (2001) 596–602, doi:10.1115/1.1357164. [20] R.W. Clough, J. Penzien, Dynamic of Structures, Second Edition, McGraw-Hill International Editions, Taiwan, 1993. [21] A.M. Wahrhaftig, Analysis of the first modal shape using case studies, In Proceedings of the 6th International Conference on Computational Methods, 1st–4th August 2016, Berkeley, ScienTech Publisher, Paper ID 1378 (ISSN 2374-3948, online). [22] A.M. Wahrhaftig, R.M.L.R.F. Brasil, J.M. Balthazar, The first frequency of cantilever bars with geometric effect: a mathematical and experimental evaluation, J Braz. Soc. Mech. Sci. Eng. 35 (4) (2013) 457–467, doi:10.1007/s40430- 013- 0043- 9. [23] F.V. Filho, Matrix Analysis of Structures (Static Stability, Dynamics), in: Almeida Neves (Ed.), Technological Institute of Aeronautics, Rio de Janeiro, 1975. [24] R. Levy, C.-S. Chen, C-W. Lin, Y-B. Yang, Geometric stiffness of membranes using symbolic algebra, Eng. Struct. 26 (2004) 759–767, doi:10.1016/j. engstruct.2003.12.011. [25] W.R. Spillers, Geometric stiffness matrix for space frames, Comput. Struct. 36 (1) (1990) 29–37, doi:10.1016/0045-7949(90)90171-W. [26] E.L. Wilson, K.J. Bathe, Numerical Methods in Finite Element Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976. [27] Y.-H. Lin, M.W. Trethewey, Finite element analysis of elastic beams subjected to moving dynamic loads, J. Sound Vib. 136 (2) (1990) 323–342, doi:10. 1016/0022- 460X(90)90860- 3. [28] S. Ormarsson, O. Dahlblom, Finite element modeling of moisture related and visco-elastic deformations in inhomogeneous timber beams, Eng. Struct. 49 (2013) 182–189, doi:10.1016/j.engstruct.2012.10.019. [29] P.W. Likins, Geometric stiffness characteristics of a rotating elastic appendage, Int. J. Solids Struct. 10 (2) (1974) 161–167, doi:10.1016/0020-7683(74) 90015-8. [30] P.W. Likins, Finite element appendage equations for hybrid coordinate dynamic analysis, Int. J. Solids Struct. 8 (5) (1972) 709–731, doi:10.1016/ 0 020-7683(72)90 038-8. [31] R. Carrion, E. Mesquita, J.L. Ansoni, Dynamic response of a frame-foundation-soil system: a coupled BEM–FEM procedure and a GPU implementation, Journal of the Brazilian Society of Mechanical Sciences and Engineering (2014) 1–9, doi:10.1007/s40430- 014- 0230- 3. [32] H. El-Absy, A.A. Shabana, Geometric stiffness and stability of rigid body modes, J. Sound Vib. 207 (4) (1997) 465–496, doi:10.1006/jsvi.1997.1051. [33] S.R. Idelsohn, A. Cardona, A load-dependent basis for reduced nonlinear structural dynamics, Comput. Struct. 20 (1–3) (1985) 203–210, doi:10.1016/ 0 045-7949(85)90 069-0. [34] E.L. Wilson, A. Habibullah, Static and dynamics analysis of multi-story buildings, including p-delta effects, Earthq. Spectra 3 (2) (1987) 289–298, doi:10.1193/1.1585429. [35] T. Lou, S.M.R. Lopes, A.V. Lopes, FE analysis of short- and long-term behavior of simply supported slender prestressed concrete columns under eccentric end axial loads causing uniaxial bending, Eng. Struct. 85 (2015) 52–62, doi:10.1016/j.engstruct.2014.12.023. [36] R. Avilés, A. Hernández, E. Amezua, O. Altuzarra, Kinematic analysis of linkages based in finite elements and the geometric stiffness matrix, Mech. Mach. Theory 43 (2008) 964–983, doi:10.1016/j.mechmachtheory.2007.07.007. [37] G. Li, Y. Zhang, H.-N. Li, Nonlinear seismic analysis of reinforced concrete frames using the force analogy method, Earthq. Eng. Struct. Dyn. 43 (14) (2014) 2115–2134, doi:10.1002/eqe.2439. [38] E. Kalkan, D.F. Laefer, Seismic based strengthening of steel and RC telecommunication poles based on fem analysis, Eng. Struct. 26 (2004) 2101–2111, doi:10.1016/j.engstruct.20 04.07.0 06. [39] L. Jingbo, W. Zhenyu, Dynamic properties and seismic response of RC structures characterized by abrupt change in horizontal connection stiffness, in: Proceedings of the International Conference on Advances in Building Technology, Hong Kong, China, 1, December 2002, pp. 4–6, doi:10.1016/ B978-0 0804410 0-9/50 037-1. [40] S. Yuana, K. Yea, C. Xiaoa, F.W. Williams, D. Kennedy, Exact dynamic stiffness method for non-uniform Timoshenko beam vibrations and Bernoulli– Euler column buckling, J. Sound Vib. 303 (2007) 526–537, doi:10.1016/j.jsv.2007.01.036. [41] I. Kovacic, M.J. Brennan, B. Lineton, Effect of a static force on the dynamic behaviour of a harmonically excited quasi-zero stiffness system, J. Sound Vib. 325 (2009) 870–883, doi:10.1016/j.jsv.2009.03.036. [42] A.M. Wahrhaftig, R.M.L.R.F. Brasil, Initial undamped resonant frequency of slender structures considering nonlinear geometric effects: the case of a 60.8 m-high mobile phone mast, Soc. Mech. Sci. Eng. (2016) 1–11, doi:10.1007/s40430- 016- 0547- 1. [43] M.A. Silva, R.M.L.R.F. Brasil, Nonlinear dynamic analysis based on experimental data of RC telecommunication towers subjected to wind loading, Math. Probl. Eng. 2006 (2006) 1–10 Article ID 46815, doi:10.1155/MPE/2006/46815. [44] M.A. Silva, J.S. Arora, R.M.L.R.F. Brasil, Dynamic analysis of pre-cast RC telecommunication towers using a simplified model, Des. Anal. Mater. Eng. Struct. Adv. Struct. Mater. 32 (2013) 97–116, doi:10.1007/978- 3- 642- 32295- 2_8. [45] Brüel & Kjaer, Accelerometers & Conditionings, Product Catalogue, Brüel & Kjaer, Nærum, Denmark, 2015. [46] AqDados 7.02, Program of Signals Acquisition, User Guide, Lynx Electronic Technology, Ltd., São Paulo, 2013. [47] AqDAnalysis 7.02, Program of Signals Analysis, User Guide, Lynx Electronic Technology, Ltd., São Paulo, 2013.

Please cite this article as: A.d.M. Wahrhaftig, R.M.L.R.F. Brasil, Vibration analysis of mobile phone mast system by Rayleigh method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.10.020