Vibration Control of Fractionally-Damped Beam Subjected to a Moving Vehicle and Attached to Fractionally-Damped Multiabsorbers

Abstract

This paper presents the dynamic response of Bernoulli-Euler homogeneous isotropic fractionally-damped simply-supported beam. The beam is attached to multi single-degree-of-freedom (SDOF) fractionally-damped systems, and it is subjected to a vehicle moving with a constant velocity. The damping characteristics of the beam and SDOF systems are described in terms of fractional derivatives. Three coupled second-order fractional differential equations are produced and then they are solved by combining the Laplace transform with the decomposition method. The obtained numerical results show that the dynamic response decreases as (a) the number of absorbers attached to the beam increases and (b) the damping-ratios of used absorbers and beam increase. However, there are some critical values of fractional derivatives which are different from unity at which the beam has less dynamic response than that obtained for the full-order derivatives model. Furthermore, the obtained results show very good agreements with special case studies that were published in the literature.

1. Introduction

Many physical problems were adequately described by differential equations of fractional order. The list of these problems is long and the areas of applications are broad. Polymeric damping, fluid mechanics, and the theory of viscoelasticity [1–8] are some of these applications. The solution in the closed form to a class of SDOF fractional oscillators subjected to an impulse response is investigated [9]. Relationship between fractional oscillator processes and the corresponding fractional Brownian motion processes is well established in research. However, the increment process of fractional Brownian motion can be described by a random function with long-range dependence fractional Gaussian noise model [10]. A fractal time series is taken as the solution of differential equation of a fractional order, or a response of a fractional system, or a fractional filter driven with a white noise in the domain of stochastic processes [11, 12]. A general approach for approximating ideal filters that are based on fractional calculus from the point of view of systems of fractional order was introduced [13, 14]. A new direct operational inversion method is introduced for solving coupled linear systems of ordinary fractional differential equations, where the obtained solutions are expressed explicitly in terms of multivariate Mittag-Leffler functions [9, 15, 16]. The solutions of fractional Langevin equation of two different orders which are known as the fractional Ornstein-Uhlenbeck processes, based on Weyl and Riemann-Liouville fractional derivatives were obtained [9, 17]. Power-laws-type data may further, in earthquake and wind engineering applications, be governed by stochastically differential equations of fractional order [18, 19]. Analysis of signals generated by fractional-order process and applications of such model in a wide range of physical, mechanical, and biological systems are discussed in [20]. However, as noted from the literature, a clear physical interpretation of the fractional derivative has been elusive. Good examples are (1) transverse vibration of homogeneous beams whose damping behavior is described by a fractional derivative of predefined constant orders was an important subject of matter [11, 21–24]; (2) diffusion process where fractional differential equations have been employed to describe an anomalous diffusion regime including both subdiffusion and superdiffusion [25]; and (3) many linear viscoelastic damping materials exhibit a macroscopic constitutive behavior which has been the subject of many investigations involving fractional order derivative [26]. However, in the fractional-derivative model of viscoelastic materials over extended ranges of time and frequency, the deformation work corresponding to springs and losses corresponding to dashpots (described by fractional derivatives) have both energy types (stored and dissipated) at any point of the utilized materials [27, 28].

The dynamic response characteristics of beams resulting from the passage of different moving load systems, and with the existence of dampers and oscillators, have been studied within many applications. Some of the applications consist of the vibrations existing in bridges, aircraft carriers, and railroad tracks due to moving vehicles, and vibrations existing in pipeline systems due to fluid flows [29–31]. One of the main objectives of studying such problems is to minimize or to suppress the dynamic response characteristics or vibration levels of beams when subjected to different load systems. The simplest and most economical way to achieve this objective is by having different lumped appendages attached to the beam at different locations. In investigating the dynamic response of beams, if the travelling force is constant, then the time-dependent coefficients can be obtained in analytical forms [32] which make it easy to find a solution to any particular problem. However, in many cases, it is not just the response of the beam (which is a function of the spatial variable and time and depends on several parameters such as beam characteristics, the magnitude of the force, its velocity, number of appendages attached to the beam, and their locations) that the designer is interested in. However, it has long been observed that, as a structure is subjected to moving loads, the induced dynamic deflection and stress can be significantly higher than those observed in the static case [33]. In this regard, the majority of the literature has been devoted to the study of the so-called moving force [34], moving mass [35–38], moving oscillator problems [39–41], and multiple moving oscillators [42].

The vibration of a beam excited by a moving unmaintained oscillator is discussed in [43]. In this study, for certain values of system parameters and travelling speeds of the moving oscillator, separation of the oscillator from the beam can occur frequently and even more than once during the travel of the load [43]. The problem of an oscillator traversing on an elastically-supported continuum is studied, whereas the flexibility in the boundaries of the continuum is modeled by linear-transverse springs [33]. Green functions formulation to analyze the free vibration of a linear undamped beam subjected to grounded SDOF oscillators appendages at different discrete points is used [44], where researchers compared their results with the approximate solution obtained by Galarkin's method.

Searching the relevant literature shows only few papers dealing with the vibration of beams with constant values for the fractional damping behavior [23, 24, 45]. However, in this study, the fractional damping behaviors of the beam and absorbers are taken with arbitrary orders. Therefore, this study is important for the following key reasons: the damping behavior of the dynamic response of the beam can be analyzed over a continuous wide range by means of fractional-derivative parameters. Moreover, the fractional derivative damping behavior may better converge to realistic experimental models than the well-known first order damping model. Hence, the fractional derivative model may precisely prescribe a nonlinear damping behavior than other existing damping models.

In this paper, a homogeneous isotropic Euler beam with different appendages attached at different locations subjected to a uniformly moving vehicle is investigated. The solution to the handled problem is based on decomposing the generalized transverse displacements of the beam, attached oscillators, and the moving vehicle into infinite series components. Then, the generalized displacements are recursively solved by in the Laplace transform domain. The solutions in this work mainly focus on the following characteristics: the maximum response of the beam for a given velocity of the vehicle, the maximum response over all possible velocities and the velocity at which it occurs, the optimal beam and absorbers fractional-damping characteristics, and the effect of the number of attached absorbers.

2. Formulation of the Problem

Transverse vibration of Bernoulli-Euler homogeneous isotropic fractionally-damped simply-supported beam is investigated. The beam is assumed to be subjected to a vehicle moving from left-to-right with a constant velocity (𝑣). The governing equation of the beam under investigation, whose damping characteristics are described by a fractional derivative of order 𝛽, can be written as𝜕𝐸𝐼⁴𝑤(𝑥,𝑡)𝜕𝑥⁴𝜕+𝐶^𝛽𝑤(𝑥,𝑡)𝜕𝑡^𝛽𝜕+𝜇²𝑤(𝑥,𝑡)𝜕𝑡²=−𝑚_𝑣̈𝛿()𝐻𝐿𝑞−𝑔𝑥−𝑣𝑡𝑣−−𝑡_𝑁𝑖=1𝑀_𝑖𝑧_𝑖̈(𝑡)𝛿𝑥−𝑑_𝑖()()(),𝑥∈0,𝐿,𝑡>0,BC's∶𝑤0,𝑡=0,𝑤𝐿,𝑡=0,𝑤_𝑥𝑥()𝑤0,𝑡=0,_𝑥𝑥()()()𝐿,𝑡=0,IC's∶𝑤𝑥,0=0,𝑑𝑤𝑥,0𝑑𝑡=0, (1a)(1b)(1c)where 𝐻(⋅) refers to the unit step Heaviside function and 𝛿(⋅) refers to the Kronecker delta function. The equations corresponding to the 𝑁-SDOF absorbers that are appendaged to the beam are given as𝑀_𝑖𝑧_𝑖̈(𝑡)=𝐾_𝑖𝑤𝑑_𝑖,𝑡−𝑧_𝑖(𝑡+𝐶_𝑖)𝑑^𝛾𝑤𝑑_𝑖,𝑡−𝑧_𝑖(𝑡)𝑑𝑡^𝛾𝑧,𝑡>0,IC's∶_𝑖(0)𝑑𝑧=0,^𝑖(0)𝑑𝑡=0. (2a)(2b)The equations corresponding to the moving vehicle can be expressed as𝑚_𝑣𝑞̈(𝑡)−𝑔=𝑘_𝑣𝑤(𝑣𝑡,𝑡)−𝑞(𝑡)+𝑐_𝑣𝑑𝑤(𝑣𝑡,𝑡)−𝑞(𝑡)𝑣𝑑𝑡,0<𝑡<𝐿,(0)(0)IC's∶𝑞=0,𝑑𝑞𝑑𝑡=0, (3a)(3b)where the beam is assumed to be initially at rest and {𝐿,𝐸,𝐼,𝜇,𝐶} are the length, modulus of elasticity, moment of inertia of cross-sectional area (𝐴), mass per unit length, and the coefficient of external damping of the beam, respectively. Furthermore, these values are assumed to be constants. The 𝑁-rigid-body SDOF fractionally-damped absorbers of order 𝛾, masses 𝑀_𝑖, stiffnesses 𝐾_𝑖, and damping constants 𝐶_𝑖 are attached to the beam, as shown in Figure 1 (for 𝑖=1,2,…,𝑁). These elements are located at distances 𝑑_𝑖 measured from the left-end of the beam. However, the parameters {𝑚_𝑣,𝑘_𝑣,𝑐_𝑣} refer to mass, stiffness, and damping constants of the vehicle, respectively. Furthermore, the set {𝑤(𝑥,𝑡),𝑧_𝑖(𝑡),𝑞(𝑡)} represents the beam transverse deflection at location (𝑥) and time (𝑡), the transverse dynamic displacements of the masses 𝑀_𝑖, and vehicle's transverse displacement, respectively.

Figure 1 

A Simply-supported beam attached to 𝑁-absorbers and subjected to a moving vehicle with a constant speed.

Equations that are similar to (1a), (1b), (1c), (2a), (2b), (3a), and (3b) are presented in [42], but with the conventional first order derivative damping behaviors. The fractional derivatives of order 𝛽 or 𝛾, presented in (1a) and (2a), may be defined in many ways [47]. Among these are the two most frequently encountered, the Riemann-Liouville and Caputo fractional derivatives of order 𝛽 which are defined by the following convolution integrals, respectively:𝜕^𝛽𝑤(𝑥,𝑡)𝜕𝑡^𝛽=1Γ𝜕𝑘−𝛽𝜕𝑡^𝑘𝑘𝑡₀𝑤(𝑥,𝑢)(𝑡−𝑢)^{𝛽+1−𝑘}𝜕𝑑𝑢,_𝛽𝑤(𝑥,𝑡)𝜕𝑡^𝛽=1Γ𝑘−𝛽^𝑡₀1(𝑡−𝑢)^{𝛽+1−𝑘}𝜕^𝑘𝑤(𝑥,𝑢)𝜕𝑢^𝑘𝑑𝑢, (4a)(4b)where 𝑘 is a positive integer such that (𝑘−1≤𝛽<𝑘).

In modal form, the transverse deflection of the simply-supported beam can be written as [31, 32]: 𝑤(𝑥,𝑡)=^∞_𝑛=1𝑌_𝑛(𝑡)𝑋_𝑛(𝑥), (5) where 𝑌_𝑛(𝑡) is the generalized displacement or the modal response of the beam and 𝑋_𝑛(𝑥) are the normal modes of the undamped free vibration of the simply-supported beam written as [46] 𝑋_𝑛(𝑥)=sin𝜅_𝑛𝑥, (6) where (𝜅_𝑛=𝑛𝜋/𝐿) is the frequency parameter associated with the simply-supported beam. Equation (6) should satisfy the following fourth-order partial differential equation: 𝜕𝐸𝐼⁴𝑋_𝑛(𝑥)𝜕𝑥⁴𝑌_𝑛(𝑡)𝜕+𝜇²𝑌_𝑛(𝑡)𝜕𝑡²𝑋_𝑛(𝑥)=0. (7) Substituting (5) into (1a), (2a), and (3a) leads to the following coupled-system of fractional differential equations:𝐸𝐼^∞_𝑛=1𝑌_𝑛(𝑡)𝑑⁴𝑋_𝑛(𝑥)𝑑𝑥⁴+𝐶^∞_𝑛=1𝑑^𝛽𝑌_𝑛(𝑡)𝑑𝑡^𝛽𝑋_𝑛(𝑥)+𝜇^∞_𝑛=1𝑑²𝑌_𝑛(𝑡)𝑑𝑡²𝑋_𝑛(𝑥)=−𝑚_𝑣̈𝛿()𝐻𝐿𝑞−𝑔𝑥−𝑣𝑡𝑣−−𝑡_𝑁𝑖=1𝑀_𝑖𝑧_𝑖̈(𝑡)𝛿𝑥−𝑑_𝑖,𝑀_𝑖𝑧_𝑖̈(𝑡)=−𝐾_𝑖𝑧_𝑖(𝑡)+𝐶_𝑖𝑑^𝛾𝑧_𝑖(𝑡)𝑑𝑡^𝛾+^∞_𝑟=1𝐾_𝑖𝑌_𝑟(𝑡)𝑋_𝑟𝑑_𝑖+𝐶_𝑖𝑑^𝛾𝑌_𝑟(𝑡)𝑑𝑡^𝛾𝑋_𝑟𝑑_𝑖,𝑚_𝑣𝑞̈(𝑡)=𝑚_𝑣𝑔−𝑘_𝑣𝑞(𝑡)+𝑐_𝑣𝑑𝑞(𝑡)+𝑑𝑡_∞𝑝=1𝑘_𝑣𝑌_𝑝(𝑡)𝑋_𝑝(𝑣𝑡)+𝑐_𝑣𝑑𝑌_𝑝(𝑡)𝑋𝑑𝑡^𝑝(𝑣𝑡). (8a)(8b)(8c)Multiplying (8a) by 𝑋_𝑚(𝑥) and integrating over the domain (0≤𝑥≤𝐿) yields ^∞_𝑛=1^𝐿₀𝐸𝐼𝑌_𝑛(𝑡)𝑑⁴𝑋_𝑛(𝑥)𝑑𝑥⁴𝑋_𝑚(𝑥)+𝑑𝑥_{∞𝑛=1𝐿0}𝑑^𝛽𝑌_𝑛(𝑡)𝑑𝑡^𝛽𝐶𝑋_𝑛(𝑥)𝑋_𝑚(𝑥)+𝑑𝑥_{∞𝑛=1𝐿0}𝑑²𝑌_𝑛(𝑡)𝑑𝑡²𝜇𝑋_𝑛(𝑥)𝑋_𝑚(𝑥)𝑑𝑥=−^𝐿₀𝑚_𝑣𝑞̈−𝑔𝑋_𝑚(𝑥)𝛿()𝐻𝐿𝑥−𝑣𝑡𝑣−−𝑡𝑑𝑥_{𝑁𝑖=1𝐿0}𝑀_𝑖𝑧_𝑖̈(𝑡)𝑋_𝑚(𝑥)𝛿𝑥−𝑑_𝑖𝑑𝑥. (9)

The dynamic response of the beam represented by (1a) is analyzed after projecting the 4th order partial derivative into a complete orthogonal basis. For the present problem, the eigenfunctions of the linear operator representing the simply-supported beam without appendages given in (6) can be used. These eigenfunctions should satisfy the following orthogonality conditions [32, 42]:^𝐿₀𝜇𝑋_𝑛(𝑥)𝑋_𝑚(𝑥)𝑑𝑥=𝜇𝐿2𝛿_𝑛𝑚,_𝐿0𝐸𝐼𝑋_𝑛^′′(𝑥)^′′𝑋_𝑚(𝑥)𝑑𝑥=𝐸𝐼𝑛⁴𝜋⁴2𝐿³𝛿_𝑛𝑚,for𝑛,𝑚=1,2,3,…. (10a)(10b)By considering these orthogonality conditions, the differential equation of the 𝑛th mode of the generalized beam-displacement, (9), can be rewritten as ^∞_𝑛=1𝑌_𝑛(𝑡)𝐸𝐼𝑛⁴𝜋⁴2𝐿³𝛿_𝑛𝑚+^∞_𝑛=1𝑑^𝛽𝑌_𝑛(𝑡)𝑑𝑡^𝛽𝐶𝐿2𝛿_𝑛𝑚+^∞_𝑛=1𝑑²𝑌_𝑛(𝑡)𝑑𝑡²𝜇𝐿2𝛿_𝑛𝑚=−𝑚_𝑣𝑞̈−𝑔𝑋_𝑚()𝐻𝐿𝑣𝑡𝑣−𝑡−^𝑁_𝑖=1𝑀_𝑖𝑧_𝑖̈(𝑡)𝑋_𝑚𝑑_𝑖. (11) Substituting (6) into (8a), (8b), (8c), and (11) and noting that (𝑚=𝑛) lead to the following coupled system of fractional differential equations:𝑑²𝑌_𝑛(𝑡)𝑑𝑡²+2𝜔_𝑛𝜁_𝑛𝑑^𝛽𝑌_𝑛(𝑡)𝑑𝑡^𝛽+𝜔²_𝑛𝑌_𝑛(𝑡)=−𝜂₁𝑚_𝑣𝑞̈−𝑔sinΩ_𝑛𝑡𝐻𝐿𝑣+−𝑡_𝑁𝑖=1𝑀_𝑖𝑧_𝑖̈(𝑡)Ωsin^𝑛𝑑_𝑖𝑣,𝑧_𝑖̈(𝑡)=−𝜔²_𝑖𝑧_𝑖(𝑡)+2𝜔_𝑖𝜁̂_𝑖𝑑^𝛾𝑧_𝑖(𝑡)𝑑𝑡^𝛾+^∞_𝑟=1𝜔²_𝑖𝑌_𝑟(𝑡)Ωsin^𝑟𝑑_𝑖𝑣+𝑑^𝛾𝑌_𝑟(𝑡)𝑑𝑡^𝛾2𝜔_𝑖𝜁̂_𝑖Ωsin^𝑟𝑑_𝑖𝑣,𝑞̈(𝑡)=𝑔−𝜔²_𝑣𝑞(𝑡)+2𝜔_𝑣𝜁_𝑣𝑑𝑞(𝑡)+𝑑𝑡_∞𝑝=1𝜔²_𝑣𝑌_𝑝(𝑡)sinΩ_𝑝𝑡+2𝜔_𝑣𝜁_𝑣𝑑𝑌_𝑝(𝑡)Ω𝑑𝑡sin_𝑝𝑡, (12a)(12b)(12c)where 𝜔_𝑛=𝜅²_𝑛𝐸𝐼𝜇,𝜁_𝑛=𝐶2𝜇𝜔_𝑛,𝜁̂_𝑖=𝐶_𝑖2𝑀_𝑖𝜔_𝑖,𝜁_𝑣=𝑐_𝑣2𝑚_𝑣𝜔_𝑣,𝜔_𝑣=𝑘_𝑣𝑚_𝑣,𝜔_𝑖=𝐾_𝑖𝑀_𝑖,Ω_𝑛=𝑛𝜋𝑣𝐿 (13) are the undamped natural circular frequency (𝜔_𝑛), beam damping ratio (𝜁_𝑛), 𝑖th oscillator damping ratio (𝜁̂_𝑖), vehicle damping ratio (𝜁_𝑣), the natural frequency of the vehicle (𝜔_𝑣), the natural frequency of the 𝑖th oscillator (𝜔_𝑖), and the load frequency (Ω_𝑛), respectively, where (𝜂₁=2/𝜇𝐿).

To find the solution of the system in (12a), (12b), and (12c), associated with the boundary and initial conditions given in (1b), (1c), (2b), and (3b), the dependent variables 𝑌_𝑛(𝑡), 𝑧_𝑖(𝑡), and 𝑞(𝑡) can be decomposed into the following infinite series:𝑌_𝑛(𝑡)=^∞_𝑘=0𝑌^𝑘_𝑛(𝑡)=𝑌⁰_𝑛(𝑡)+𝑌¹_𝑛(𝑡)+𝑌²_𝑛(𝑡)𝑧+⋯,_𝑖(𝑡)=^∞_𝑘=0𝑧^𝑘_𝑖(𝑡)=𝑧⁰_𝑖(𝑡)+𝑧¹_𝑖(𝑡)+𝑧²_𝑖(𝑡)𝑞(𝑡)=+⋯,_∞𝑘=0𝑞^𝑘(𝑡)=𝑞⁰(𝑡)+𝑞¹(𝑡)+𝑞²(𝑡)+⋯, (14a)(14b)(14c)where 𝑌^𝑘_𝑛(𝑡), 𝑧^𝑘_𝑖(𝑡), and 𝑞^𝑘(𝑡) for (𝑘=0,1,2,…) are the components of 𝑌_𝑛(𝑡), 𝑧_𝑖(𝑡), and 𝑞(𝑡), respectively. The components of 𝑌_𝑛(𝑡), 𝑧_𝑖(𝑡), and 𝑞(𝑡) can be determined recursively, by substituting (14a), (14b), and (14c) into (12a), (12b), and (12c), from the following equations:𝑑²𝑌⁰_𝑛(𝑡)𝑑𝑡²+𝜔²_𝑛𝑌⁰_𝑛(𝑡)=𝜂₁𝑚_𝑣𝑔sinΩ_𝑛𝑡𝐻𝐿𝑣,𝑑−𝑡₂𝑧⁰_𝑖(𝑡)𝑑𝑡²=^∞_𝑟=1Ωsin^𝑟𝑑_𝑖𝑣𝜔²_𝑖𝑌⁰_𝑟(𝑡)+𝑑^𝛾𝑌⁰_𝑟(𝑡)𝑑𝑡^𝛾2𝜔_𝑖𝜁̂_𝑖,𝑑²𝑞⁰(𝑡)𝑑𝑡²=𝑔+^∞_𝑝=1sinΩ_𝑝𝑡𝜔²_𝑣𝑌⁰_𝑝(𝑡)+2𝜔_𝑣𝜁_𝑣𝑑𝑌⁰_𝑝(𝑡),𝑑𝑑𝑡₂𝑌_𝑛^𝑘+1(𝑡)𝑑𝑡²+𝜔²_𝑛𝑌_𝑛^𝑘+1(𝑡)=−2𝜔_𝑛𝜁_𝑛𝑑^𝛽𝑌^𝑘_𝑛(𝑡)𝑑𝑡^𝛽−𝜂₁𝑚_𝑣𝑑²𝑞^𝑘(𝑡)𝑑𝑡²sinΩ_𝑛𝑡𝐻𝐿𝑣+−𝑡_𝑁𝑖=1𝑀_𝑖𝑑²𝑧^𝑘_𝑖(𝑡)𝑑𝑡²Ωsin^𝑛𝑑_𝑖𝑣,𝑑²𝑧_𝑖^𝑘+1(𝑡)𝑑𝑡²=−𝜔²_𝑖𝑧^𝑘_𝑛(𝑡)+2𝜔_𝑖𝜁̂_𝑖𝑑^𝛾𝑧^𝑘_𝑛(𝑡)𝑑𝑡^𝛾+^∞_𝑟=1Ωsin^𝑟𝑑_𝑖𝑣𝜔²_𝑖𝑌_𝑟^𝑘+1(𝑡)+𝑑^𝛾𝑌_𝑟^𝑘+1(𝑡)𝑑𝑡^𝛾2𝜔_𝑖𝜁̂_𝑖,𝑑²𝑞^𝑘+1(𝑡)𝑑𝑡²=−𝜔²_𝑣𝑞^𝑘(𝑡)+2𝜔_𝑣𝜁_𝑣𝑑𝑞^𝑘(𝑡)+𝑑𝑡_∞𝑝=1sinΩ_𝑝𝑡𝜔²_𝑣𝑌_𝑝^𝑘+1(𝑡)+2𝜔_𝑣𝜁_𝑣𝑑𝑌_𝑝^𝑘+1(𝑡),𝑑𝑡 (15a)(15b)(15c)(15d)(15e)(15f)for 𝑘≥0. Now, using the Laplace transform for the Caputo fractional derivative which is defined in (4b) as [47] ℓ𝑑^𝛽𝑌^𝑘_𝑛(𝑡)𝑑𝑡^𝛽=𝑠^𝛽𝑦^𝑘_𝑛(𝑠)−^𝑚−1_𝑗=0𝑠^{𝛽−𝑗−1}𝑦^𝑘_𝑛^𝑗(0),for𝑚−1<𝛽≤𝑚, (16) the Laplace transforms of (14a), (14b), (14c), (15a), (15b), (15c), (15d), (15e), and (15f), with the related homogeneous initial conditions, can be rewritten as𝑦_𝑛(𝑠)=^∞_𝑘=0𝑦^𝑘_𝑛(𝑠),𝑍_𝑖(𝑠)=^∞_𝑘=0𝑍^𝑘_𝑖(𝑠),𝑄(𝑠)=^∞_𝑘=0𝑄^𝑘(𝑠), (17a)(17b)(17c)where𝑦⁰_𝑛(𝑠)=𝜂₁𝑚_𝑣𝑔𝑠²+𝜔²_𝑛ℓsinΩ_𝑛𝑡𝐻𝐿𝑣,𝑍−𝑡_0𝑖(𝑠)=^∞_𝑟=1Ωsin^𝑟𝑑_𝑖𝑣𝜔²_𝑖+2𝜔_𝑖𝜁̂_𝑖𝑠^𝛾𝑠²𝑦⁰_𝑟(𝑠),𝑄⁰(𝑠)=𝑔𝑠³+^∞_𝑝=1𝜔²_𝑣ℓsinΩ_𝑝𝑡𝑌⁰_𝑝(𝑡)𝑠²+2𝜔_𝑣𝜁_𝑣ℓsinΩ_𝑝𝑡𝑑𝑌⁰_𝑝(𝑡)/𝑑𝑡𝑠²,𝑦_𝑛^𝑘+1(𝑠)=−2𝜔_𝑛𝜁_𝑛𝑠^𝛽𝑠²+𝜔²_𝑛−𝜂₁𝑚_𝑣ℓ𝑑²𝑞^𝑘(𝑡)/𝑑𝑡²sinΩ_𝑛𝑡𝐻(𝐿/𝑣−𝑡)𝑠²+𝜔²_𝑛+^𝑁_𝑖=1𝑀_𝑖𝑠²𝑍^𝑘_𝑖(𝑠)sinΩ_𝑛𝑑_𝑖/𝑣𝑠²+𝜔²_𝑛,𝑍_𝑖^𝑘+1(𝑠)=−𝜔²_𝑖𝑍^𝑘_𝑖(𝑠)𝑠²+2𝜔_𝑖𝜁̂_𝑖𝑠^𝛾𝑍^𝑘_𝑖(𝑠)𝑠²+^∞_𝑟=1Ωsin^𝑟𝑑_𝑖𝑣𝜔²_𝑖𝑦_𝑟^𝑘+1(𝑠)𝑠²+2𝜔_𝑖𝜁̂_𝑖𝑠^𝛾𝑦_𝑟^𝑘+1(𝑠)𝑠²,𝑄^𝑘+1(𝑠)𝜔=−^2𝑣𝑄^𝑘(𝑠)𝑠²+2𝑠𝑄^𝑘(𝑠)𝜔_𝑣𝜁_𝑣𝑠²+^∞_𝑝=1𝜔²_𝑣ℓsinΩ_𝑝𝑡𝑌_𝑝^𝑘+1(𝑡)𝑠²+2𝜔_𝑣𝜁_𝑣ℓsinΩ_𝑝𝑡𝑑𝑌_𝑝^𝑘+1(𝑡)/𝑑𝑡𝑠², (18a)(18b)(18c)(18d)(18e)(18f)for 𝑘≥0 and (𝑖=1,2,…,𝑁), where 𝑁 is the number of the SDOF fractionally-damped absorbers attached to the beam. In the above equations ℓ(⋅)designates the Laplace transform of (⋅) where (ℓ(𝑧_𝑖(𝑡))=𝑍_𝑖(𝑠), ℓ(𝑌_𝑛(𝑡))=𝑦_𝑛(𝑠) and (ℓ(𝑞(𝑡))=𝑄(𝑠). Nevertheless, the explicit forms of (18a), (18b), (18c), (18d), and (18e) for 𝑘≥0 are rather long and complex. However, when the damping effect is fractional, these equations contain the well-known hepergeometric function defined as [48] hypergeom𝑛₁,𝑑₁,𝑑₂=,𝑧_∞𝑘=0𝑧^𝑘Γ𝑛₁+𝑘/Γ𝑛₁𝑘!∏²_𝑗=1Γ𝑑_𝑗+𝑘/Γ𝑑_𝑗. (19)

The term-by-term inversion of (18a), (18b), (18c), (18d), and (18e) with (14a), (14b), and (14c) represents the corresponding solution of the handled problem. However, many problems can be treated as special cases of (18a), (18b), (18c), (18d), and (18e); the absorbers can be neglected by assuming that (𝜔_𝑖=0) and the moving vehicle can be treated as a moving load by assuming (𝜁_𝑣=1 and 𝑘_𝑣=1) or by replacing the Heaviside step function by unity, where (in this case) the gravity acceleration (𝑔) is further replaced by the intensity of the moving load.

3. Results and Discussion

In this Section, two examples are given to verify the validity of the formal technique used in this paper. Moreover, three more important case studies are introduced to show the effect of fractional-damping behaviors for both of the beam and the SDOF absorbers and the effect of the number of SDOF absorbers attached to the beam (𝑁). In all case studies, the dynamic response of the beam is mainly governed by the fundamental mode (𝑛=1) of the beam because this mode has the significant effect on the beam dynamic response.

3.1. Verification Problem I

This example verifies the validity of the technique used in this paper. In this example, an undamped beam with a moving oscillator is investigated [49]. The parameters adopted for the beam under investigation are given in Table 1. The displacement response for the beam versus the normalized time is shown in Figure 2; that is, when (𝑡=0) the load enters the beam from left-hand side (𝑥=0) and when (𝑡=𝐿/𝑣) the moving load system leaves the right-hand side of the beam (𝑥=𝐿). The curves of Figure 2 show the dynamic mid-span displacement response of the beam for the five cases of traveling velocities: 𝑣= 10 m/s, 𝑣= 27.778 m/s, 𝑣= 40 m/s, 𝑣= 60 m/s, and 𝑣= 80 m/s, respectively.

Table 1 

Material and geometrical properties of beam, vehicle, and absorbers.

Figure 2 

The dynamic response of an elastic beam at mid-span for many load speeds.

In this example, it is shown that the dynamic displacement response increases as the traveling speed increase. This follows from the fact that the energy content of the elastic beam increases when the speed increases. Furthermore, Figure 2 shows that the amplitude of the displacement response for the force and mass moving systems increases as speed increases. The maximum displacement when 𝑣= 27.778 m/s is (𝑤_max= 2.17453 m/s) which satisfies the results obtained by Duhamel integral [32] and is almost the same as that presented in [49].

3.2. Verification Problem II

This verification problem presents a damped beam attached to a single absorber and subjected to a moving load of intensity 9.81 N; that is, the moving vehicle of unit mass approaches to the moving load (𝑘_𝑣= 0 N/m, 𝜁_𝑣= 1 Ns/m, 𝑚_𝑣= 1 kg). The data presented in [50, 51] are used for solving this problem, see Table 1. In this problem, the response of an initially stationary beam subjected to a moving load with a constant velocity is considered. The beam is assumed to be free or appendaged to one, two, or three absorbers (𝑁=0,1,2,3) and (𝑖=1,…,𝑁) at different locations of the beam, see Figures 3, 4, and 5.

Figure 3 

The maximum displacement of the beam at mid-span versus the velocity of the moving load for 𝜁=0.

Figure 4 

The maximum displacement of the beam at mid-span versus the velocity of the moving load for 𝜁=0.075.

Figure 5 

The maximum displacement of the beam at mid-span versus the velocity of the moving load for 𝜁=0.15.

In this problem, (𝑁=0) refers to a beam without any appendages; that is, all parametric properties of the oscillators are null. The case for (𝑁=1) was studied in [51] as a two-SDOF system, while in [50], it was analyzed using the Euler-Bernoulli theory. In [50], the Galerkin-Bubnov variation method is used to solve the beam problem with only one absorber attached at the mid-span of the beam. The order of the fractional derivative for both of the beam and the oscillator is assumed to be unity; that is, 𝛽=𝛾=1. Figures 3–5 show the maximum response of the beam at the mid-span versus the velocity of the moving load for three different values of the beam damping ratio; that is, 𝜁=0, 𝜁=0.075, and 𝜁=0.15, respectively. The upper two curves in Figure 3 fit precisely the published results in [50, 51]. The curve for (𝑁=0) in Figure 3 fits exactly the well-known exact solution which is obtained by the Duhamel integral [32]. Furthermore, we note from Figures 3–5 that, as both the number of absorbers which are attached to the beam and as the damping beam-ratio increases, the dynamic response decreases. This is clearly seen from the lowest curve in Figure 5 where 𝑁=3 and 𝜁=0.15.

3.3. Optimization of the Vehicle Dynamic Response

The moving load problem that was solved in Section 3.2 is solved here for the more general case of a moving vehicle (𝑘_𝑣= 700 N/m, 𝜁_𝑣=0.1, 𝑚_𝑣= 1 kg). Figures 6, 7, and 8 illustrate the maximum response of the beam at the mid-span versus the velocity of the moving vehicle for three different values of the damping ratio of the beam; that is, 𝜁=0, 𝜁=0.075, and 𝜁=0.15, respectively. Moreover, Figure 9 shows the maximum central beam displacement versus the velocity of the moving vehicle for different numbers of absorbers (𝑁=1,2). The curves in these figures show that the maximum central displacement of the beam decreases as the number of absorbers, beam damping ratio, and vehicle velocity increase. By comparing Figures 3–5 obtained for a moving load with Figures 6–8 obtained for a moving vehicle, it is clearly seen that the maximum displacement of the beam decreases for the moving vehicle case and this is due to the effect of the damping ratio of the vehicle. Moreover, the critical mean velocity of the moving vehicle at which the maximum displacement occurs is 𝑣= 25 m/s, see Figures 6–8.

Figure 6 

The maximum displacement of the beam at mid-span versus the velocity of the moving vehicle for 𝜁=0.

Figure 7 

The maximum displacement of the beam at mid-span versus the velocity of the moving vehicle for 𝜁=0.075.

Figure 8 

The maximum displacement of the beam at mid-span versus the velocity of the moving vehicle for 𝜁=0.15.

Figure 9 

The maximum displacement of the beam at mid-span versus the velocity of the moving vehicle with one and two (𝑁=1,2) absorbers attached.

3.4. Optimization of the Fractional Derivatives

In this Section, one example is analyzed to show the effect of the order of fractional derivatives for both of the beam and absorbers (0<𝛾<2, 0<𝛽<2). The 3D surface shown in Figure 10 (for the data given in Table 1) represents the variation of the maximum deflection of the beam at the mid-span (where the absorber is attached) with the beam fractional derivative (𝛽) and the fractional derivative of the absorber (𝛾). The obtained surface in Figure 10 leads to very interesting critical values of 𝛽 and 𝛾. From this surface, the optimal values obtained are 𝛽=1.245 and 𝛾=1.190, at which the minimum beam deflection occurs. This reduction is due to the effect of the damper added to the vehicle. The optimal values obtained from the surface shown in Figure 10 further show the physical meaning of fractional derivative, which is equivalent to increasing the damping characteristics of the beam.

Figure 10 

The maximum central deflection of the beam against the order of the beam's fractional derivative and the fractional derivative of the absorber (𝛽,𝛾).

3.5. Optimization of the Number of Fractionally-Damped Oscillators

In this section, the optimal values that were obtained in Sections 3.3 and 3.4 (𝛽=1.245, 𝛾=1.19, 𝑣= 25 m/s) are used to determine the dynamic response of the beam attached to many SDOF systems. The other parameters considered in this example are shown in Table 1. Figures 11, 12, 13, and 14 show the dynamic response of the beam for various values of beam damping ratio (𝜁=0, 𝜁=0.075, 𝜁=0.15), respectively. The effects of the beam damping ratio and the number of oscillators are clearly observed from the curves in Figures 11–14.

Figure 11 

The dynamic response (𝑤(𝐿/2,𝑡)) for the beam with 𝜁=0, subjected to a moving vehicle and attached to many absorbers.

Figure 12 

The dynamic response (𝑤(𝐿/2,𝑡)) for the beam with 𝜁=0.075, subjected to a moving vehicle and attached to many absorbers.

Figure 13 

The dynamic response (𝑤(𝐿/2,𝑡)) for the beam with 𝜁=0.15, subjected to a moving vehicle attached to many absorbers.

Figure 14 

The dynamic response (𝑤(𝐿/2,𝑡)) for the beam with 𝜁=0.15, attached to many absorbers for moving vehicle (MV) and moving load (ML).

The effect of the previously obtained optimal values is observed clearly when both Figures 5 and 8 are compared with Figure 13. The percent relative reduction in the maximum central deflection (when the beam subjected to a moving load is compared to that subjected to a moving vehicle) is computed for (𝑁=0, 𝑁=1, 𝑁=2, and 𝑁=3) as (14.5%, 13.85%, 12.74%, and 11.63%), respectively, see Figures 5 and 8, while the percent relative reduction in the maximum central deflection when Figure 8 (𝛽=1 and 𝛾=1) is compared with Figure 13 (𝛽=1.245 and 𝛾=1.19) of the beams subjected to a moving vehicle is computed for (𝑁=0, 𝑁=1, 𝑁=2, and 𝑁=3) as (27.58%, 41.27%, 48.99%, and 54.54%), respectively, see Figures 8 and 13. This reduction in the dynamic response of the beam is due to the effect of the damping of the beam, fractional damping, number of absorbers, and the vehicle-damping. However, the influence of the order of fractional derivatives, beam damping ratio, oscillators damping ratios, and the number of the attached oscillators on the beam dynamic response is clearly observed.

Figure 14 shows the dynamic response of the beam versus time for both of the moving load and vehicle for the beam damping ratio (𝜁=0.15) for which (𝑁=0, 𝑁=1, 𝑁=2, 𝑁=3). The fractional derivatives for both the beam and absorbers are taken as 𝛽=1.245 and 𝛾=1.19, respectively. The effect of these optimal values on the beam response is clearly observed. The reduction in the dynamic response of the beam is due to the effect of the damping of the beam, fractional damping, number of attached absorbers, and the vehicle-damping.

4. Conclusions and Future Work

In this paper, an analytical model is introduced to solve the transverse vibration of Bernoulli-Euler beam attached to 𝑁-SDOF absorbers with damping characteristics that are described in terms of fractional derivatives. The damping characteristics of the beam and absorbers are assumed to be of fractional-order derivatives. The method employed uses the Laplace transform with decomposing the displacement into infinite series to find the solution. However, the solution may be written in a closed form for some special cases; otherwise, it may be truncated and produced by using appropriate mathematical software such as Maple.

The numerical results obtained in this paper, illustrate that (1) the dynamic response decreases as the number of absorbers attached to the beam increases, (2) as both the damping-ratios of absorbers and beam increase, the dynamic response decreases, (3) the type of load further affects the dynamic response of the beam; hence, a moving vehicle reduces the response due to its damping property, and (4) there are some critical values of 𝛽 and 𝛾 at which the beam has less deflection than that which is obtained from the full-order derivative models (i.e., 𝛽=1 and 𝛾=1).

Finally, even though the fractional derivative model provides better damping models of fractionally-damped structures and materials, a limited number of papers have been presented in this field. This is probably due to the fact that the underlying mathematics of the fractional derivative models are sophisticated and not well-developed. Hence, developments in this field will lead to the development of the applications of fractional derivative models in several other engineering problems.

The approach of the fractional derivative damping, used in this paper, will allow researchers to choose suitable mathematical models that may better converge to realistic experimental models than the well-known conventional first-order damping models. Hence, the fractional derivative models may precisely represent a nonlinear damping behavior better than other existing damping models.

Future work will involve the following items for the application of the fractional calculus-based approach that is used in this paper within: (1) quantum and nano computing systems [52], (2) nanobeams, materials and medical applications [52–54], (3) fractal-based formulations to the microquasistatic thermoviscoelastic creep for rough surfaces in contact [55], and (4) other beam configurations for various types of nonlinearities, damping characteristics, and moving vehicles.