Elsevier

Computers & Structures

Volume 83, Issues 25–26, September 2005, Pages 2162-2174
Computers & Structures

Critical load, dynamic characteristics and parametric instability of electrorheological material-based adaptive beams

https://doi.org/10.1016/j.compstruc.2005.02.028Get rights and content

Abstract

This study derives formulae for the critical load, natural frequency, and loss factor of a simply supported adaptive beam with embedded electrorheological fluid. The parametric instability and dynamic response of the beam subjected to a periodic axial force are investigated. The theoretical model is developed from DiTaranto sandwich beam theory and extended to a transverse vibration model. Galerkin’s method is applied to simplify the governing equation of motion to the form of Mathieu equation. The incremental harmonic balance (IHB) method is employed to determine the parametric instability of the electrorheological material-based adaptive beams. The fourth-order Runge–Kutta method is used to analyze the dynamic response of the adaptive beams. The effects of electric field, core thickness ratio, and length of beam on the critical load, natural frequencies, loss factor, and parametric instability are investigated. The influence of the static load parameter factor on the parametric instability is also addressed.

Introduction

Electrorheological (ER) fluids are suspensions of fine dielectric particles in non-conducting viscous fluids. The material properties of such fluid vary with the strength of the electric field applied to them [1]. Winslow was the first discovered this phenomenon in 1947. It was found that when an electric field is applied to a structure with an embedded electrorheological fluid, the damping and stiffness of the structure may change reversibly in milliseconds. Therefore, electrorheological fluids are regarded as smart materials [2] with a wide range of applications in numerous industries and technologies [3].

Electrorheological materials are viscoelastic in the pre-yield regime, and plastic in the post-yield regime on the shear strain versus shear stress diagram. These regimes are defined as those in which the material is strained below and above a critical yield strain, respectively. Several researchers have measured the pre-yield viscoelastic characteristics of electrorheological fluids. Weiss et al. [4] measured the shear modulus and loss factor of electrorheological fluids at various frequencies and strains under various applied fields. Don [5] measured the storage modulus and loss modulus of electrorheological materials, and proposed that an electrorheological fluid can be modeled as a complex linear viscoelastic solid, whose dynamic characteristics can be modeled as complex shear modulus which is function of the applied electric field and in general increases with the applied electric field.

Ross et al. [6] developed a method for modeling of laminated beams and plates with viscoelastic material sandwiched in between elastic layers. They derived a fourth-order differential equation for a simply supported beam and introduced the complex flexural modulus of composite structure. Later, DiTaranto [7] extended the Ross et al.’s model to lateral vibration problems of laminated beams of finite lengths with viscoelastic cores under various boundary conditions. They derived a sixth-order linear homogeneous differential equation of motion. Later, Mead and Markus [8], [9] modified DiTaranto’s approach to obtain a sixth-order equation for transverse motion of the forced vibration of a three-layered damped sandwich beam with arbitrary boundary conditions. The differential equations that govern the vibration of sandwich beams are well known and were derived by Ross, Ungarand and Kerwin, DiTaranto and Mead and Markus.

Sato et al. [10] extended the theory used in the study of DiTaranto to investigate the effectiveness of damping of the lateral vibration of sandwich beam that is symmetrically comprised of a viscoelastic core layer and thin elastic outer layers. Rao [11] used the Mead and Markus’ model to derive a complete set of equations of motion and boundary conditions that governed the vibration of sandwich beams.

Recently, electrorheological materials that exhibit typical properties of a viscoelastic material are regarded as smart material. Many studies have applied the sandwich beam model to the electrorheological beam. Yalcintas et al. [12], Don and Coulter [13], and Yalcintas and Coulter [14] extended Ross et al.’s model to vibration problems of laminated adaptive beams that incorporate electrorheological material as a controllable damping layer. Yalcintas and Coulter [12], [15], [16] applied the Mead and Markus model to the semi-active control of vibration in electrorheological material-based adaptive beams.

Kar and Sujata [17] considered the stability of a fixed-free tapered symmetric sandwich beam, whose core material is viscoelastic with a complex modulus, under a pulsating axial force. In their study, the method of multiple scales was used to obtain the boundaries between stable and unstable regions. Ray and Kar [18] studied the parametric instability of a three-layered symmetric sandwich beam under various boundary conditions. They derived equations of motion using Hamilton’s principle; and used Galerkin’s method to establish a set of coupled Hill equations with complex coefficients. By using the same method, Kar and Ray [18] investigated the dynamic stability of a pre-twisted three-layered symmetric sandwich cantilevered beam subjected to a periodic axial load at the free end.

Lau et al. [19] proposed an incremental harmonic balance (IHB) method to determine the parametric instability of viscous damped column or beam system that exhibits nonlinear vibration. Pierre and Dowell [20] extended the IHB method to investigate the dynamic instability of viscous damped plates. Yuan and Lau [21] applied the IHB method to elucidate the effect of an in-plane load on nonlinear panel flutter. The incremental harmonic balance method (IHB) is simple and can be conveniently used in numerical analysis, it has been successfully applied to determine the dynamic instability of a structural system with viscous damping.

This study develops formulae for the critical load, natural frequencies, and loss factor of a simply supported sandwich beam that contains electrorheological fluid. It also investigates the parametric instability and dynamic response of the beam subjected to periodic axial force. The effects of the parameters including the electrical field, the core thickness ratio, the length of beam and the static load parameter factor, on the critical load, the natural frequency, the loss factor and the parametric instability are considered.

In this study the theoretical model is developed from the DiTaranto’s theory and is extended to transverse vibration model. As an example, we have derived a sixth-order equation for the transverse motion of a three-layered sandwich beam. The dependence of the complex shear modulus of the electrorheological fluid to applied electric field is considered. The governing equation of motion obtained by applying the generalized Galerkin’s method is simplified to the complex Mathieu equation with coefficients that depend on the applied electric field. These coefficients can be used to obtain the natural frequencies and structural loss factor of the sandwich beam. It can then be used to derive the critical load for the simply supported sandwich beam, by aforementioned simplified procedure. The complex Mathieu equation can be expressed as a general Mathieu equation, whose coefficients are not of complex form. The incremental harmonic balance (IHB) method is used to determine the parametric instability of the electrorheological material-based adaptive beams; and the fourth-order Runge–Kutta method is used to determine the dynamic response of such beams.

Section snippets

Adaptive beam with electrorheological core

Consider a simply supported electrorheological material-based adaptive beam under a periodic axial load Pf(t) = P0 + P1 cos ωt at x = L, as indicated in Fig. 1; where, ω is the frequency of the applied load; and P0 and P1 are the amplitudes of the static load and dynamic load, respectively, and y(x, t) is the transverse displacement of the beam.

Fig. 2 depicts the conformation of the adaptive beam. The length and width of are L and b, respectively. The two elastic faces have thickness 2H1 and 2H3, with

Incremental harmonic balance method

In this study the incremental harmonic balance (IHB) method proposed by Lau et al. [19] is implemented to determine the boundary of parametric instability of a simply supported electrorheological material-based adaptive beam. For simplicity, the general Mathieu equation of the system, Eq. (19), is rewritten in compact form.M(Ωn,μn,fn,τ,ω)=0

The first step of the IHB method to obtain the periodic solutions is adding small increments to initial guess of the solution of Eq. (20), Ω0, μ0, f0.

Numerical studies and discussion

The numerical study here comprises the evaluations of critical load, natural frequency, loss factor, parametric instability, and dynamic response of a simply supported symmetric sandwich beam structure with an electrorheological core and upper and lower elastic faces made of aluminum, as shown in Fig. 1. Geometrical data are indicated in Fig. 2: L = 381 mm, b = 25.4 mm, H1 = H3 = 0.36765 mm and H2 = 0.36765 mm. The electrorheological material used in this study is same as those described by Yalcintas and

Conclusions

In this study, the modified DiTaranto model was implemented to derive the governing equation of transverse motion for the simply supported adaptive beam with an electrorheological core subjected to a periodic axial force. The generalized Galerkin’s method is applied to simplify the governing equation to the form of Mathieu equation of which coefficients are functions the applied electric field. The formulae for the critical load, the natural frequency, and the structural loss factor are derived

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