Dynamic stability of smart sandwich beams with electro-rheological core resting on elastic foundation

Abstract

This article focuses on dynamic stability of smart sandwich beams resting on Winkler elastic foundation subjected to harmonic axial loads. To increase the dynamic buckling load and the stability region of the beam, an electro-rheological layer is adhered as a core. The finite element method is employed to derive a three layer smart sandwich beam element. By inspecting dynamic response of the beam in different load amplitudes, critical dynamic loads are calculated. Parametric study is done to investigate effects of various parameters such as beam geometry, foundation stiffness, static load, applied voltage and properties of core layer on critical dynamic loads and stability regions of the beam. This study indicates that by applying electric field to the electro-rheological core, dynamic critical load and consequently, dynamic stability of the beam increase efficiently. Furthermore, the elastic foundation reduces the unstable region and increases the critical dynamic load of the smart beam. Proper use of these parameters makes the beam less sensitive to axial harmonic loading by relocating the instability region.

Keywords

Dynamic stability elastic foundation smart sandwich beam electro-rheological fluids finite element method

Introduction

Various structural members are frequently subjected to time depending periodic loads. These loads may cause large amplitude vibration, which prevents the normal operation of the structure due to dynamic instability. Therefore, investigating the effective parameters and the method of preventing or controlling this type of instability in such structural members have been an interesting research subject.

An efficient method of increasing the dynamic stability of a system is to control the high amplitude vibrations of the structure. There are different methods to control vibration of a system, such as using viscose dampers [1] or applying smart materials like piezoelectric [2], magno-rheological (MR) fluids [3] or electro-rheological (ER) fluids. The ER fluids are a class of smart fluids, which contain semi-conductive particles suspended in non-conductive liquids. By applying an electric field to the fluid, particles are polarized and as a result the characteristic of the ER fluids changes from a Newtonian flow in which particles move freely to a fluid in which its particles are aligned in chains in the direction of electric field. Several experimental studies have been done to determine mechanical properties of the ER fluid under the influence of the electric field. Investigations on ER fluid properties led to various mathematical models, which express dynamic behavior of ER fluids in the pre-yield and post-yield stages [4,5]. Bingham’s model is the most well-known model which has been proposed for describing the post-yield behavior of ER fluids [6]. Park et al. [7] investigated material characteristics of an ER fluid subjected to electric field and temperature. Wahed et al. [8] studied rheological characteristics of ER fluids in time-dependent loading. Another salient characteristic of ER fluids is their fast response to the electric filed. These two significant characteristics, namely reversible changes of mechanical properties and fast response to the electric field, have drawn researcher’s attention to apply these fluids in active-control of vibrations [9,10]. By expanding effectiveness of ER fluids in control and vibration problems, researchers have tried to use these fluids as a core in sandwich structures. This model, which is called constrained layer damping, has been used intensively by researchers before [11]. In this model, a visco-elastic layer is added to the vibrating system supported by a constraining layer to damp structural vibrations. Using an active ER fluid instead of previous passive visco-elastic materials, increases controllability of the structure. In the recent years, there has been a growing interest in using ER fluid as an active constrained layer damping. Yeh et al. [12,13] studied dynamic stability of sandwich beams and plates with ER core. They have used Kevin’s model to describe the visco-elastic behavior of the ER fluid. Narayana, and Ganesan [14] compared visco-elastic damping and ER fluid core damping in composite sandwich skew plates employing finite element method (FEM). Rezaeepazhan and Pahlavan [15] investigated the transient response of a sandwich beam with ER core. They have used Bingham’s model to describe the behavior of the ER fluid under the influence of the electric field.

Increasing the stiffness of the system, i.e. applying elastic foundation, enhances the dynamic stability of the structural members. Because of widespread application of elastic foundation in many engineering fields, such as rail road, civil and aerospace engineering, many researchers have focused on vibration, buckling and stability response of beams/plates on the elastic foundations. Different models have proposed for elastic foundations in which their differences refer to type and order of generated load in interface of the structure and the foundation. The Winkler foundation is one of the simplest models of the elastic foundation which defines a contact pressure proportional to the vertical displacement at each interface point [16]. Eisenberger and Clastornik [17] studied vibrations and buckling of a beam on a variable Winkler elastic foundation. Later, Eisenberger et al. [18] investigated stability of beams on the elastic foundations by employing the FEM. Engel [19] studied dynamic stability of a simply-supported beam on an elastic foundation which was subjected to a periodic axial load. Lee [20] investigated dynamic stability of a tapered cantilever beam on an elastic foundation subjected to a follower force.

Although a considerable amount of works has been devoted to the dynamic response of smart beams with active constrained layer damping and the dynamic of beams on an elastic foundation, little attention has been paid to the dynamic response of beams with ER core resting on an elastic foundation. It would be thus of interest to study the effect of elastic foundation on dynamic buckling of smart beams with an active ER layer.

In present work, a three-layered sandwich beam resting on the Winkler elastic foundation subjected to an axial parametric excitation is considered. This type of load leads to dynamic instability which in addition to the load amplitude, the frequency of the load plays a crucial rule in the instability of the structure. To improve the stability of the structure, an ER layer is added to the beam. Based on Bingham’s model for the ER layer, a finite element model is developed. A smart sandwich beam element is derived, and the corresponding governing equations are obtained. Applying FEM has produced the element stiffness, damping and mass matrices and a controllable damping force vector due to imposing electric field. The direct integration method is used to obtain the transient response of the beam. Dynamic buckling loads in different frequencies are determined by analyzing responses of the beam to assorted load amplitudes.

Description of the problem

A three-layered sandwich beam with ER core of length L, width b, and thickness H = h₁ + h₂ + h₃ is considered in this study. As shown in Figure 1, the beam is resting on a Winkler elastic foundation of modulus K_w. The ER layer is constrained by two isotropic elastic faces. The thickness of each layer is denoted by h_i and i = 1, 2, or 3.

As mention before, electric field changes the behavior of the ER fluid from a Newtonian fluid to a fluid in which polarized particles are aligned in chains. Consequently, the fluid resists shear stresses more firmly. The shear stress of the ER fluid in post-yield stages is expressed by Bingham’s relation as follows [7].

τ = η \overset{\cdot}{γ} + τ_{y} (E)

(1)

In this relation, τ is shear stress, $\overset{\cdot}{γ}$ is shear rate, η viscosity and $τ_{y} (E)$ is shear yield stress of ER fluid. $τ_{y} (E)$ in equation (1) is a function of electric field and increases exponentially with electric field. Hence equation (1) could be rewritten as

τ = η \overset{\cdot}{γ} + α E^{β}

(2)

α and β are inherent properties of ER fluid, which are usually determined experimentally [7].

Applying different voltages to the elastic layers of the beam generates an electric field in the area between these layers, which leads to change in the rheological properties of the core layer. The mathematical model of this structure is similar to assumptions used for the transient response of the sandwich beam with ER core [15]. In summary, rotary inertia, shear deformation, and transverse direct strains in both core and face-plates of the elastic layers are negligible. No slipping occurs between the neighboring layers. Moreover, the material line in the core layer remains straight after deformation, and no normal stress can be produced in the core layer [15].

Finite element modeling of the sandwich beam with ER core

To model a three-layered sandwich beam, the element shown in Figure 2 is considered. This element consists of two nodes with four degree of freedom per node. u₁ and u₃ represent axial displacement of elastic layers, w shows transverse displacement and θ represents the angular rotation of the nodes.

Displacement vector ${U} = {u_{1} (x, t) u_{3} (x, t) w (x, t)}^{T}$ is defined as

{U} = [N (x)] {q (t)}

(3)

q(t) is the vector of nodal degrees of freedom of the element, and N(x) are common beam shape functions at nodes i and j.

{q} = {\begin{matrix} u_{1 i} & u_{3 i} & w_{i} & θ_{i} & u_{1 j} & u_{3 j} & w_{j} & θ_{j} \end{matrix}}^{T}

(4)

[N] = [\begin{matrix} N_{u_{1} i} & 0 & 0 & 0 & N_{u_{1} j} & 0 & 0 & 0 \\ 0 & N_{u_{3} i} & 0 & 0 & 0 & N_{u_{3} j} & 0 & 0 \\ 0 & 0 & N_{wi} & N_{θ \begin{matrix} i \end{matrix}} & 0 & 0 & N_{wj} & N_{θ \begin{matrix} j \end{matrix}} \end{matrix}]

(5)

In order to use the variational principle for dynamic analysis, Hamilton’s principle, the Lagrange (the functional L) is written as

L = T - V + W

(6)

where T, V and W are the kinetic energy, potential energy and work of external forces, respectively. Two elastic faces are modeled as Euler–Bernoulli beams. The energy lost by ER layer ( E_er ), strain energy of elastic layers ( V_s ), and kinetic energy of the element ( T_e ) would be obtained in the similar way which presented in our previous work [15].

However, the potential energy produced by the Winkler elastic foundation, is obtained as

V_{F} = \frac{1}{2} \int_{L} {q}^{T} [D_{F}]^{T} K_{W} [D_{F}] {q} d x

(7)

D_{F} = [\begin{matrix} 0 & 0 & 1 \end{matrix}] \begin{matrix} [N] \end{matrix}

(8)

Moreover, the work done by external axial periodic force P(t) would be achieved as follows.

W = \frac{1}{2} \int_{l} P (t) (\frac{\partial w}{\partial x})^{2} d x = \frac{1}{2} \int_{l} {q}^{T} [N]^{T} [Q]^{T} P (t) [Q] [N] {q} d x

(9)

Where,

[Q] = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \partial / \partial x \end{matrix}]

(10)

Applying Hamilton’s principle to the total energy of the beam element yields the finite element equation of motion for the smart sandwich beam with ER core as below

[M] {\overset{··}{q}} + [C] {\overset{\cdot}{q}} + ([K_{e}] + [K_{F}] - \begin{matrix} P (t) [K_{G}] \end{matrix}) {q} = {F_{ER}}

(11)

In equation (11) [M], [C] and [K_e] are the element mass matrix, damping matrix and stiffness matrix similar to transient analysis [15]. [K_F] and [K_G] are the foundation and geometric stiffness matrices of the beam element. {F_ER} is a damping force vector which has been generated by the second term of Bingham’s relation and represents the effect of electric field. These matrices are defined as

[M] = \int_{v} [D_{2}]^{T} J_{2} [D_{2}] d v + \sum_{i = 1}^{3} \int_{v} [R_{i}]^{T} ρ_{i} [R_{i}] d v

(12)

[C] = \int_{v} [D_{2}]^{T} η_{p} [D_{2}] d v

(13)

[K_{e}] = \int_{v} [D_{1}]^{T} E_{1} [D_{1}] d v + \int_{v} [D_{3}]^{T} E_{3} [D_{3}] d v

(14)

[K_{F}] = \int_{L} [D_{F}]^{T} K_{w} [D_{F}] d x

(15)

[K_{G}] = \int_{l} [N]^{T} [Q]^{T} P (t) [Q] [N] d x

(16)

{F_{ER}} = \int_{v} [D_{2}]^{T} τ_{E} (E) sgn (\overset{\cdot}{γ}) d v

(17)

Matrices [D_i] and [R_i] (i = 1,2,3) are presented in Appendix. The function sgn( $\overset{\cdot}{γ}$ ) in equation (17) presents the sign of the shear rate $\overset{\cdot}{γ}$ .

Determination of critical dynamic load

In the present study, critical dynamic load of the beam in specific load frequency is calculated based on vibrating behavior of the system. To achieve this, the transient response of the beam to an initial excitation must be obtained. Rezaeepazhan and Pahlavan [15] obtained transient response of a cantilever sandwich beam with ER core by taking advantage of direct integration method. This method is known as an effective general algorithm for solving dynamic problems. Substitution of the equivalent central difference expression for velocity and acceleration [15] in the derived finite element equation of motion for the smart sandwich beam resting on elastic foundation (equation (11)), leads to a recursive relation for the vector {q}.

\begin{matrix} {q}_{i + 1} & = [\frac{1}{(Δ t)^{2}} [M] + \frac{1}{2 Δ t} [C]]^{- 1} ({F_{ER}}_{i} - [[K_{e}] + [K_{F}] - P (t) [K_{G}] - \frac{2}{(Δ t)^{2}} [M]] {q}_{i} \\ \begin{matrix}  \end{matrix} - [\frac{1}{(Δ t)^{2}} [M] - \frac{1}{2 Δ t} [C]] {q}_{i + 1}) \end{matrix}

(18)

where,

Δ t

is the time step which is considered equal for all steps. By employing equation (18) in each step, values of vector {q} are calculated in terms of values of this vector in previous steps. In the first step {q}-₁ is required, which could be obtained by the following relation

{q}_{- 1} = {q}_{0} - {\overset{\cdot}{q}}_{0} Δ t + \frac{1}{2} {\overset{··}{q}}_{0} (Δ t)^{2}

(19)

where (·) represents the time derivatives. By employing equation (18), the maximum deflection of the beam for different dynamic load amplitudes is obtained. If the values of maximum deflections are plotted versus load amplitudes, a diagram similar to Figure 3 is achieved.

As observed in this figure the maximum transverse displacement increases by increasing the dynamic load amplitude. However, the slope of the maximum transverse displacement curve is not constant. For small amplitude of dynamic load, maximum displacement increase gradually. However, for specific values of the dynamic load amplitude, the maximum displacement increases sharply by a small increment in dynamic load amplitude. This condition displays a critical state. The critical dynamic load is calculated by finding the junction of the asymptotic of the displacement curve and the load axis [21].

Validation

In order to validate the accuracy of the derived element and the solution procedure, results are compared with the results presented by other researchers. Considering, lack of investigation on the dynamic buckling problem of a sandwich beam with ER core resting on an elastic foundation, different parts of solution procedure are validated separately.

The finite element model presented for three-layered sandwich beam with ER core, has been validated similar to Rezaeepazhan and Pahlavan [15]. In order to evaluate method of finding critical load and finite element modeling done for Winkler elastic foundation, the results of present FEM are compared with results of Abbas and Thomas [22]. They have studied dynamic stability of different beams on the elastic foundation and obtained corresponding stability regions. In this case, a clamped-free beam on Winkler elastic foundation is considered, which is subjected to a harmonic axial load as below

P = P_{0} + P_{1} \begin{matrix} cos (θ \begin{matrix} t \end{matrix}) \end{matrix}

(20)

In which, P₀ static component of axial load and P₁ is the amplitude of dynamic or variable component of axial load and θ is the frequency of dynamic load. To determine the stability region of the beam, different load frequencies are considered and the critical dynamic load for each frequency is calculated.

Figure 4 compares two stability regions of the beam obtained by present study and those presented by Abbas and Thomas [22]. As displayed in this figure, proper convergence has been achieved and indicates that the present method of finding critical load and finite element modeling done for Winkler elastic foundation are accurate. In Figure 4, parameters P_e, P_e* and γ are defined as below [22].

P_{e} = 9.844 \sqrt{\frac{EI}{ρ {AL}^{4}}}

(21)

P_{e}^{*} = 9.874 \frac{EI}{L^{2}}

(22)

γ = \frac{K_{w} L^{4}}{π^{4} EI}

(23)

Numerical results

In order to investigate the dynamic stability of a smart sandwich beam with ER core resting on an elastic foundation, a clamped-free beam with properties presented in Table 1, is considered. The ER core consists of Arabic gum particles as the dispersed phase suspending in the transformer oil as the base fluid. According to Choi’s study [5], coefficients of the Bingham’s model for this fluid are equal to α = 36.81 and β = 1.52. The density and viscosity of fluid are 1180 kgs/m³ and 10 CSt, respectively.

Table 1.

Geometry and material properties of the sandwich beam

Length of the beam (L)	600 mm
Width of the beam (b)	30 mm
Thickness of constraining layer (h₁)	1 mm
Thickness of base layer (h₃)	3 mm
Thickness of ER layer (h₂)	4 mm
Young’s modulus of elastic layers (E₁, E₃)	210 GPa
Density of elastic layers (ρ₁, ρ₁)	7800 kg/m³

It is assumed that, the beam is resting on a Winkler elastic foundation with the modulus K_w = 10 kN/m·m and the ER layer is subjected to an electric field with intensity of 1 kV/mm. These properties are considered as the base properties of the beam while in each case study, all these properties except for the investigating parameter, are constant.

First, effects of various parameters on critical dynamic load are studied. In all these cases, the frequency of harmonic dynamic load is considered equal to two times of the first natural frequency of the beam [23]. In Figures 5 to 10 the non-dimensional dynamic buckling load λ is plotted in terms of different parameters. λ is defined as

λ = \frac{P_{cd} L^{2}}{{bh}^{3} E}

(24)

In this relation, P_cd is the critical dynamic load and h = h₁ + h₃.

Figure 5 displays the effects of electric field on critical dynamic load. In Figure 5 it is clearly observed that, the non-dimensional dynamic buckling load (λ) increases by increasing the applied electric field. As it was expected, increasing the intensity of the electric field causes more damping force {F_ER} and as a result, the beam would be capable of withstanding a higher dynamic load. Figure 6 illustrates the effect of stiffness of elastic foundation on the critical dynamic load. It is obvious that elastic foundation makes the system stiffer. Therefore, beam could resist higher dynamic loads. As seen in Figure 6, this effect is more sensible in the presence of electric field.

In Figure 7, the effects of thickness of ER layer on the critical dynamic load are presented. By increasing the thickness of ER layer, equivalent mass of the system is increased and vibration is hardly damped consequently. On the other hand, increasing the thickness of ER layer produces more damping force and therefore, damping ability of the system is improved. As shown in Figure 7, by increasing thickness of ER layer, the non-dimensional dynamic buckling load is increased. It seems for this case, the effect of damping forces is stronger than equivalent mass of the system.

Effect of the thickness ratio of elastic layers on critical dynamic load is also investigated. In this case, the thickness ratio of the base layer to constraining layer is changing while the thickness of core layer and equivalent mass of the system is remained constant. To achieve this, total thickness of elastic layers is kept constant while the thickness ratio is changed. Figure 8 illustrates the effect of this ratio on non-dimensional dynamic buckling load of the beam. According to Figure 8, the dynamic buckling load (λ) increases by increasing the thickness ratio of the elastic layers. It means that, by choosing a thicker base layer than constraining layer, dynamic load carrying capacity of the beam is improved.

Furthermore, effect of change in the width of the applied electric field is considered. In this case, the width of core layer is constant while the width of the electric field (electrode) is changed such that only a strip of ER core is affected by electric field. b_e represents the width of the electrode or width of that part of the core, which is subjected to an electric field. Figure 9 indicates that, with a constant equivalent mass, by increasing the width of the electrode, the dynamic load-carrying capacity of the beam increases.

Another parameter which is studied in this part is the effect of prestressed or static component of the applied load (P₀). By increasing P₀, maximum amplitude of the load which is imposed to the beam is increased. Therefore, beam could accept a smaller dynamic load. As illustrated in Figure 10, compressive prestressed load, positive P₀, decreases the critical dynamic load while, tensile prestressed, negative P₀, increases the critical dynamic load of the beam.

In the next part, the first instability region of the beam is determined, and the effects of various parameters are investigated. In Figures 11 to 16, r and μ are the non-dimensional parameters which are defined as follows [23]

r = \frac{θ}{2 ω}

(25)

μ = \frac{P_{cd}}{2 (P_{cr} - P_{0})}

(26)

P_cr is the static critical buckling load and ω is expressed as

ω = ω_{n} \sqrt{1 - \frac{P_{0}}{P_{cr}}}

(27)

In which, ω_n is the natural frequency of the beam. In the absence of P₀, the first instability region is occurred around r = 1 [23]. This happens when the frequency of the dynamic load reaches 2ω₁. The instability boundary has a minimum point at r = 1 and this point is considered as a special critical point for the stability analysis of the beam. As mentioned before, to determine the stability region, the critical dynamic load should be extracted for each frequency of dynamic load.

Figure 11 shows the influence of the electric field on the instability region. When an electric field is applied to the ER core, it produces a stronger damping force and as a result dynamic stability of the system is improved. As observed in Figure 11, applying electric field moves the boundary of the instability region upward. This effect is more sensible near the extreme point of the instability boundary namely at r = 1. This behavior is desirable since it shifts the most unstable status of the beam. However, applying the electric field doesn’t have any effect on the dynamic load which their frequency is far from 2ω_n.

Elastic foundation changes the instability boundary in a different way. Figure 12 displays the effect of the stiffness of the elastic foundation (K_w) on the stability region of the beam. As mentioned before, by increasing stiffness of elastic foundation, stiffness of the system increases considerably and makes the beam more stable. The results presented in Figure 12 show that, elastic foundation reduces the unstable region of the beam and also moves minimum point of instability boundary upward.

Figure 13 presents the effect of thickness of ER layer on the stability region of the beam. Increasing the thickness of ER layer causes the reduction of the unstable region of the beam. However, as shown in Figure 13, changing this parameter is not as efficient as changing the applied electric field. This fact was expected because as it was mentioned before, by increasing the thickness of ER layer, two paradox events are occurred simultaneously. Effect of the width of the electric field on the first stability region of the beam is also investigated in Figure 14. In this case, the equivalent mass of the system is kept constant and only the width of the electrodes is changed. As seen in Figure 14, increase of the width b_e moves the unstable region upward. That means the stability of the beam is improved.

According to Figure 8, by increasing the thickness ratio of elastic layers, the stability of the beam is improved. However, this expectation is not complied in Figure 15. For different thickness ratios, approximately identical stability boundaries are achieved. Such a behavior is caused by using non-dimensional parameters μ and r for presenting the instability boundaries. By increasing thickness ratio of elastic layers, the stiffness of system increases. Consequently, critical buckling load (P_cr) and critical amplitude of dynamic load are enhanced in such a way that the instability boundaries presented by equation (26) merge each other. In fact, this parameter improves the dynamic load-carrying capacity of the beam, but this effect is not visible in the stability diagram which is plotted by using these forms of non-dimensional parameters.

Influence of the static component of applied axial load is investigated next. To find the values of the non-dimensional parameter r, the first natural frequency of the beam should be replaced in equation (27). If the first natural frequency of unloaded beam is employed, instability regions present in Figure 16(a) are obtained. According to this figure, instability region is moved along r axis as the static component of the applied load increases. This behavior is occurred since by applying a static axial load, the natural frequency of the beam is changed. Consequently, minimum point of the instability region appears in a point different than r = 1. This effect could be useful when a dynamic load is applied to the structure with frequency around 2ω₁. For this case, structure reaches its unstable state easily. Since, applying a static axial load to the structure changes minimum point of the instability region, it could be an efficient way to put structure in more stable status.

Figure 16(b) shows another evaluation for parameter r. By considering the static component of the applied load as an initial static load, the beam is a prestressed beam; its natural frequencies are a function of this initial static load. If these natural frequencies are substituted in equation (27), all the minimum points of stability boundaries would be laid in a same point and as observed in Figure 16(b), by increasing the static load, instability region of the beam also increases.

Conclusion

In this article, dynamic stability of smart sandwich beams with ER core resting on Winkler elastic foundation was studied. A smart sandwich beam element was derived based on the Bingham’s model. Numerical investigation was performed for critical dynamic loads and stability regions of the beam. Presented numerical results showed that, change of the stiffness of elastic foundation leads to remarkable changes in the critical dynamic load and the stability region of the beam. Applying the electric field produces more damping effects by ER layer and consequently, increases the critical dynamic load of the beam. The presence of an electric field also extends the stability region of the beam. Increasing thickness of ER core and width of the applied electric field had a similar effect on the increase of critical dynamic loads. Moreover, these two parameters slightly reduce the unstable region of the beam. The relative change in thickness of the elastic layers also affected the critical dynamic loads of the beam. Results indicated that by increasing thickness ratio of elastic layers, the critical dynamic loads increase. By changing the static component of the applied axial load, the minimum point of instability boundary could be changed from unfavorable frequency to proper frequency.

As a recommendation for future research, extending the present work to more complex smart structures, i.e. frames, plates and shells, and study the performance of smart elastic foundations can be considered.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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