Elsevier

Applied Mathematical Modelling

Volume 80, April 2020, Pages 911-928
Applied Mathematical Modelling

A novel analysis method for damping characteristic of a type of double-beam systems with viscoelastic layer

https://doi.org/10.1016/j.apm.2019.11.008Get rights and content

Highlights

  • A new damping characteristic analysis method for complicated structures is proposed.

  • Explicit expression of the damping ratio of a class of viscoelastic double-beam is derived.

  • The model can consider the effect of multiple factors, which is helpful to design and optimize this type of structures.

  • The analysis results are introduced to the design of a kind of double-sheathing cable systems in engineering.

Abstract

The double-beam system with a viscoelastic layer is a classical mechanical model for many beam-type composite structures. However, few studies have been able to optimize the structure from the perspective of structural damping characteristics. To fully understand the damping characteristics of the viscoelastic double-beam system, an analysis method based on dynamic stiffness method and Wittrick-Williams algorithm is presented in this paper. Through numerical case studies, five typical parameters of the viscoelastic double-beam system are discussed to investigate their influence on the damping characteristic of the system. Finally, the conclusions are used to parametric analysis for a kind of double-sheathing cable systems. Results show that the damping coefficient of the connection layer have a significant effect on the damping characteristic of the double-sheathing cable system compared with other design parameters. The proposed methods and conclusions obtained in this paper are helpful to design and optimize the structural parameters of engineering structures, thus having certain application and promotional value.

Introduction

Damping is one of the main properties of materials defined as loss of energy and can be used to suppress the vibration of structures [1]. It has an important influence on the dynamic behavior of engineering structures such as beams, plates [2] and shells [3].

Structural damping is different from additional damping. It only uses the viscoelastic properties of the structural material itself to dissipate the vibration energy and does not require additional damping devices [4], [5], [6], [7], [8], [9]. However, in general, the damping properties of materials are inconsistent with their strength characteristics. The damping properties of high-strength materials are usually weak, while the damping properties of low-strength materials are high. Therefore, it is difficult to find a kind of classical material to meet the dual requirements of structural strength and vibration control simultaneously. The damped composite structure with a additional viscoelastic core can well balance the contradiction between the strength and damping of the material [10]. A type of double-beam structure connected by a viscoelastic layer is a typical damped composite structure [11]. The two beams are used to bearing bending moments and shear forces, while the viscoelastic connection layer is used for vibration suppression of excessive amplitudes in the vertical or lateral direction without extra damping components. Its excellent structural performance is of concern in engineering, while its dynamic problems have become one of the research focus [12].

In fact, double-beam structures have been extensively studied and widely used in the past few decades. Many engineering structures like sandwich beams [13], [14], [15], continuous dynamic vibration absorbers [16,17], composite layered foundations [18], floating-slab tracks [19], embedded fluid-conveying single-walled nanotubes [20,21] can be analyzed through the double-beam system. However, most studies have many limitations such as the damping effect of the connection layer is usually ignored during the dynamic analysis, the boundary conditions of the two beams in the same side must be identical [22]. Some studies considered the damping effect of connection layer based on some assumptions and simplifications on the model. For example, with the assumption of the simply supported boundary condition, Chen and Sheu [23,24] studied the free vibration and dynamic response of a type of double-beam system with a viscoelastic core, but the two beams must be identical; Abu-Hilal [25] gave a closed-form solution of the dynamic response of a double-beam system under moving dead loads by simplifying the damping of the connection layer to the viscous damping of the two beams. Under the assumption that the two beams are identical and simply supported, Oniszczuk et al. [26] considered the viscoelastic properties of the connection layer and gave an analytical solution for the transverse vibration of a viscoelastic double-string system; Wu and Gao [27] gave the analytical solution of the dynamic response of a viscous double-beam system under a moving harmonic load; Han and Dan [28] obtained the damping ratio of a viscoelastic double-beam system considering the damping of the two beams and connection layer, but ignored the effect of axial loads. Obviously, there is still a gap between these studies and real engineering structures, in addition, existing researches mainly focused on the effect of the damping on dynamic characteristic of the system, not considered the influence of the design parameters of the two beams on the system from the aspect of damping ratio.

To make the analysis results more reliable and fully understand the influence law of the design parameters on damping characteristics of the viscoelastic double-beam system, it is necessary to firstly establish a refined double-beam model that can faithfully reflects the mechanical behavior of the actual structural, and then propose an exact dynamic analysis method to fully consider the effect of design parameters on damping characteristics of the system. For the continuum medium supporting such as foundation soil, Winkler [29] proposed a spring analogy method named Winkler model to simulate the foundation soil. It is the oldest and simplest structural model which involves distributed springs but it neglects the shear and damping capacity of the medium, Pasternak [30] proposed a two-parameter model which can consider the transverse shear deformation to make up the deficiency of the Winker model and it is widely used by many researchers [31], [32], [33]. When it comes to a long and thin pavement structure, the Euler–Bernoulli beam theory is suitable for the computation of flexural deformation. To this end, the Winkler type layer with distributed mass, stiffness and damping is employed in this paper to model the viscoelastic connection layer, the shear deformation of the Winkler layer is ignored here. Besides, two axially loaded Euler beams are employed to simulate the two beams of slender double-beam systems. It is worth noting that the distributed mass of the viscoelastic layer is also considered in this paper, this is because that for some engineering structures like the elastic bearing block track, the mass of the viscoelastic layer (bearing blocks) is usually not negligible. To investigate the damping characteristics of the viscoelastic double-beam system, the governing differential equation of the double-beam system is established in the Laplace domain first. Then, the dynamic analysis method (DMS) is employed to analyze the dynamic characteristic of the system.

The DSM has been widely studied and extended in recent decades [34], [35], [36]. The success of the method is due to the fact that the structure only needs to be specifically divided at geometric or material discontinuities, thus it only takes a few elements to calculate any order modes without discretizing the structure [37]. This property of DSM makes it have both high precision and high computational efficiency in any frequency range compared with the conventional structural dynamic analysis methods, such as the finite element method [38]. However, the DSM also has its inherent problems, namely the difficulties in solving the transcendental frequency equation derived from the ODE in frequency domain. To overcome these difficulties, the Wittrick–Williams (W-W) algorithm was put forward in 1970, which is used to obtain the roots of the frequency equation. The method can not only meet the requirement of arbitrary solution accuracy, but also perfectly solve the root-missing problem [39].

However, few studies have been able to optimize the structure from the perspective of structural damping characteristics [40,41]. In this paper, a novel analysis method based on the DSM and Wittrick-Williams algorithm is proposed, and the damping ratio of the viscoelastic double-beam system is obtained to fully understand the damping characteristics of the viscoelastic double-beam system. Numerical cases are used to investigate the influence law of design parameters on the damping characteristics of the viscoelastic double-beam system. The obtained results provide a theoretical basis for the optimal design and vibration control of such type of structures.

Section snippets

The mechanic model and governing differential equation of the viscoelastic double-beam system

The viscoelastic double-beam system investigated in this paper consists of two axially loaded parallel elastic beams with the same length l, the two beams are continuously connected by a Winkler-type viscoelastic interlayer. Both beams are assumed to be slender, and therefore the classical Euler-Bernoulli beam theory is applied during the derivation. As shown in Fig. 1, for a viscoelastic double-beam element, the upper beam (Beam1) is often used to bear the external load, while the lower beam

The verification of the damping ratio calculation method

Before applying the proposed method to the study of the damping characteristics of viscoelastic double beam system, it is necessary to verify the accuracy of the method. Therefore, a representative research about the double-beam system is elaborately selected here for validation [27]. In Reference [27], a simply supported double-beam system with a viscoelastic layer is studied analytically, where the two beams are identical and undamped. It can be seen that the boundary condition and the

Application

It can be seen from the above analysis that the damping characteristics of the system can be effectively improved by optimizing the parameters of the viscoelastic layer. Based on this, a double-layer sheathing cable system shown in Fig. 13 is investigated in this section. The outer sheathing and inner steel tendons are modeled by Beam 1 and Beam 2 respectively, because the cable sheath does not bear the axial force in the actual engineering, thus we can assume P1 = 0; the filling layer is

Conclusions

In this paper, the damping characteristic of a type of double-beam systems connected by a viscoelastic layer is investigated. To investigate the damping characteristic of the system, a method to analyze the damping characteristic of the viscoelastic double-beam system is proposed in this paper. Through numerical cases, five key parameters of the are discussed to investigate their influence on the damping characteristic of the system. Finally, the results are introduced to the damping

Acknowledgments

This work is supported by the National Nature Science Foundation of China (Grant no. 5187849); the National key R&D Program of China (2017YFF0205605); Shanghai Urban Construction Design Research Institute Project ‘Bridge Safe Operation Big Data Acquisition Technology and Structure Monitoring System Research’; and the Ministry of Transport Construction Science and Technology Project ‘Medium-Small Span Bridge Structure Network Level Safety Monitoring and Evaluation’.

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