On damping vibrations of three-layered beam containing magnetorheological elastomer

Abstract

In this article, we present the results of investigations of viscoelastic properties of magnetorheological elastomer containing carbonyl iron particles. Frequencies of natural vibrations of three-layered beam, supporting constructions of which are made from aluminum, and the inner layer—from magnetorheological elastomer—are calculated, and the dependence of vibrations on induction of the applied magnetic field is obtained. Nonstationary vibrations of the beam at pulse impact of magnetic field are found.

Keywords

damping magnetorheological elastomer layered structure vibration

Introduction

Magnetorheological elastomers (MREs) are one of the smart materials, elastic properties of which change depending on the value of the applied magnetic field. They are composed of magnetic particles in a deformed polymer matrix. The possibility to control viscoplastic and viscoelastic properties of MRE in a wide range allows one to use them in vibroprotecting devices.

A laminated thin-wall structure with electrorheological or magnetorheological medium as an interlayer smart material is one of the methods of vibroprotection constructions. Calculation of such structures is a complex task for mechanical engineers who deal with development of new methods of active and semiactive damping of construction vibrations. Although many works are dedicated to vibroprotecting devices of different constructions with the use of electrorheological and magnetorheological media (Bilyk et al., 2009; Coulter and Duclos, 1989; Korobko et al., 2009; Park et al., 1998; Reizina et al., 2009; Shaw, 2000; Yalcintas and Dai, 1999), most research have been done for the case when the smart material is an electrorheological and magnetorheological fluid. However, elastomers have an advantage in comparison with fluids: they are stagnant materials and keep their geometrical shape at low field levels. There are many articles studying the properties of MRE in the static regime (Chen et al., 2007; Zhou, 2003; Zhou and Wang, 2006), but only few research on the dynamic calculation of the adaptive sandwich beams and shells containing MRE appear in the literature (see, e.g. Mikhasev et al., 2011). This is explained by the fact that the response of composite constructions containing MREs significantly depends on the ratio of timescale of the controlling signal providing time reaction of MRE and dynamic characteristics of the controlled construction (Choi and Wereley, 2001; Mikhasev et al., 2010).

In this article, the principle of damping vibrations on the example of three-layered beam in which bearing layers are made of aluminum and the inner layer is made of the MRE is discussed. First, the MRE was made and its viscoelastic characteristics in the magnetic field were experimentally determined. Second, the equations of motion of the beam at free and forced vibrations in the magnetic field have been solved at fixed viscoelastic characteristics of prepared MRE. Then, the reaction of the beam to the nonstationary signal of magnetic field resulting in an abrupt change of MRE characteristics has been investigated.

Experimental

As a matrix for MRE, a natural inorganic polymer (bentonite clay with the size of laminar particles in the range 1–10 µm) in synthetic oil was used, and as a filler particles of carbonyl iron (particle size about 20 µm) was used. The matrix of MRE was prepared by a thorough rubbing of polymer in the oil with the addition of surfactant. In the prepared matrix, carbonyl iron particles were introduced (about 30 wt.%).

The dependence of components of complex modulus of shear G′ and G″ on the induction of the magnetic field B is determined by means of a Physica MCR 301 rheometer from Anton Paar using the measuring cell of the plate—plate type with diameter 20 mm and with the gap between plates 1 mm. G′ and G″ are determined in the regime amplitude sweep (deformation amplitude was changed in the range of 10⁻⁴–0.02) at fixed deformation frequency of 10 Hz.

Beam vibrations at fixed magnetic field intensity

Dependencies of components of complex shear modulus on the induction of magnetic field (at the frequency of external impact 10 Hz) are presented in Figure 1.

Presented curves indicate the nonlinear dependence of shear modulus on the induction of the magnetic field of high intensity. Only at B < 200 mT, our results correlate with that supposition of Yalcintas and Dai (2004) about linear dependence of complex shear modulus of magnetorheological material on induction of the magnetic field.

Let us consider a three-layered beam of the length L (Figure 2), external layers of which are not sensitive to the magnetic field, and the inner layer represents a viscoelastic MRE. As the governing equations describing the composite beam motion, the system of differential equations obtained by Yalcintas and Dai (2004) will be used

ρ \frac{\partial^{2} w^{*}}{\partial {t^{*}}^{2}} + 2 EI \frac{\partial^{4} w^{*}}{\partial x^{4}} - \frac{G^{*} b {(h_{1} + h_{2})}^{2}}{h_{2}} (\frac{\partial^{2} w^{*}}{\partial x^{2}} - \frac{\partial ϕ^{*}}{\partial x}) = f

J \frac{\partial^{2} ϕ^{*}}{\partial {t^{*}}^{2}} - \frac{b h_{1} E {(h_{1} + h_{2})}^{2}}{2} \frac{\partial^{2} ϕ^{*}}{\partial x^{2}} - \frac{b G^{*} {(h_{1} + h_{2})}^{2}}{h_{2}} (\frac{\partial w^{*}}{\partial x} - ϕ^{*}) = 0

(1)

where w* is the normal deflection of the beam, ϕ* is the cross-sectional rotation, x is a coordinate at the beam medium line, f is the external force per unit length, t* is time, b is beam width, h₁ = h₃ is thickness of the elastic bearing layers, h₂ is thickness of the MRE layer, E is the Young modulus of the surface layers, G* is the complex shear modulus of MRE depending on the induction B of the external magnetic field, ρ is density of the beam per unit length, I is the geometric inertia moment of the cross section, and J is the inertia mass moment per unit length. Here, magnitudes $ρ, I, and J$ are introduced as

ρ = 2 ρ_{1} + ρ_{2}

I = \frac{2}{3} b {(h_{1} + \frac{h_{2}}{2})}^{3}

J = \frac{2}{3} \frac{ρ_{1}}{h_{1}} [{(h_{1} + \frac{h_{2}}{2})}^{3} - \frac{h_{2}^{3}}{8}] + \frac{ρ_{2} h_{2}^{2}}{12}

(2)

where $ρ_{1} and ρ_{2}$ are densities of the surface and MRE layers, respectively, per unit length.

Let us consider the simply supported boundary conditions

w^{*} = \partial^{2} w^{*} / \partial x^{2} = \partial ϕ^{*} / \partial x = 0 at x = 0, L

(3)

Then, the natural forms of vibrations are given by functions

w_{n}^{*} (x, t^{*}) = \sin λ_{n} x e^{i ω_{n} t^{*}}, ϕ_{n}^{*} (x, t^{*}) = C_{n} \cos λ_{n} x e^{i ω_{n} t^{*}}, n = 1, 2, \dots, \infty

(4)

λ_{n} = π n / L L, C_{n} = \frac{2 λ_{n} G^{*}}{λ_{n}^{2} h_{1} h_{2} E + 2 G^{*}}

(5)

where

ω_{n} = λ_{n}^{2} \sqrt{\frac{E}{ρ} [2 I + \frac{G^{*} b {(h_{1} + h_{2})}^{2}}{E h_{2} λ_{n}^{2}} (1 + \frac{2 G^{*}}{λ_{n}^{2} h_{1} h_{2} E + 2 G^{*}})]}

(6)

is the sought complex frequency of natural vibrations, depending on complex shear modulus G*. Moreover, we should note here that the similar formula for the frequency $ω_{n}$ , deduced by Yalcintas and Dai (2004), is erroneous.

The calculation was carried out for the beam with parameters (h₁ = h₂ = 0.7 mm, L = 390 mm, and b =25 mm), bearing layers of which are prepared from aluminum, and the inner layer is made from MRE with shear modulus presented in Figure 1. The analysis of the results indicates that there is a weak dependence of the natural frequency on induction of magnetic field for all modes. In particular, by changing the induction from 0 to 400 mT, the lowest natural frequency grows almost linearly from 278 Hz up to 480 Hz, and then nonlinearly decays by the following increase of the magnetic field intensity. With the increase of the mode number, this dependence weakens. The dependence of the imaginary part of the vibration frequency on the intensity of the magnetic field is shown in Figure 3. As it is seen already at B > 350 mT, the growth of the intensity of the magnetic field results in the drop in the speed of vibrations dampening, which indicates the inefficiency of the further rise of induction. Therefore, for the considered MRE, the most optimal from the point of view of dampening low-frequency vibrations is the value of induction of B = 350 mT.

Let the beam undergo an impact from an external force of intensity $f = ρ F_{0} (x) e^{i Ω t^{*}}$ , where Ω is the frequency of forced vibrations. Solution of equation (1) with boundary condition (3) is

w^{*} (x, t^{*}) = \sum_{n = 1}^{\infty} [A_{n} e^{i ω_{n} t^{*}} + \frac{N_{n}}{ω_{n}^{2} - Ω^{2}} e^{i Ω t^{*}}] \sin (λ_{n} x)

(7)

where $N_{n} = \int_{0}^{L} F_{0} (x) \sin λ_{n} x d x$ denotes the generalized forces, and A_n denotes the arbitrary complex numbers that are found from the initial conditions of the problem.

Function (7) allows us to predict the reaction of the three-layered beam on the applied periodical external load with arbitrary distribution in coordinate X Due to the fact that the speed of vibration dampening depends on the complex shear modulus G*, formula (7) could be used as for determining the optimal regime of changing the intensity of the magnetic field with the aim of the most effective damping of vibrations (Yalcintas and Dai, 2004).

It should be noted that formulas (6) and (7) are obtained at the condition that the shear modulus G* is a constant value, that is, it does not depend on time. At the same time, the response of magnetorheological medium to the external magnetic pulse depends significantly on the ratio of the timescale of the controlling signal and the reaction of the very medium. So, at application of the signal, time of reaction of MRE is about 10⁻³–10⁻² s. At smooth change of induction of the magnetic field, time of reaction can vary substantially. In any case, the obtained relations (6) and (7) do not reveal the reaction of the beam in the time interval comparable with the time of MRE reaction and could be considered only at setting a certain stationary regime for viscoelastic characteristics of MRE. Therefore, relations (6) and (7) could be used for solving the problem of vibration damping only on low frequencies, when the frequency of natural vibrations and also the frequency of exciting vibrations are not comparable with the speed of MRE reaction to the magnetic field. Abrupt impact of the magnetic field is a kind of “parametric blow” for a mechanical system and can excite additional high-frequency modes that require further investigations.

Influence of the magnetic field signal on high-frequency vibrations

For MRE, the real and imaginary parts of G* at zero induction B of the magnetic field are equal G′ = 4.5 kPa and G″ = 17 kPa. Maximal values of G′ = 3000 kPa and G″ = 820 kPa are reached at induction B = 500 mT and B = 200 mT, respectively. At pulsed change of the induction of the magnetic field, the complex shear modulus G* = G*(v_rt*) is the time function, where v_r = 1/t_r is the speed of MRE reaction on the impact of the magnetic field (≈10²–10³ s⁻¹), and t_r = 10⁻³ s is assumed here as the characteristic time. Let us introduce the dimensionless values

w^{*} = Lw, x = Ls, t^{*} = t_{r} t, f (t) = 2 \frac{G^{*}}{ε^{\frac{1}{2}} E} \frac{l^{2}}{h^{2}}

ε = \frac{t_{r}}{T_{v}}, T_{v} = \frac{1}{3} \sqrt{\frac{2 ρ}{E} \frac{h}{b}} \frac{l^{2}}{h^{2}}

(8)

where T_v is the period of low-frequency vibrations of the beam at $h_{1} = h_{2} = h$ . Furthermore, it is assumed also that ε is a small parameter characterizing the speed of reaction of MRE on the magnetic field signal relative to the lowest frequency of bending vibrations.

Using the multiple scale method, the solution of equation (1) will be found in the form

w = W (t) \sin π ns, ϕ = Φ (t) \cos π ns

n = ε^{- 1 / 2} p, p ~ 1

W = W_{0} + ε^{1 / 2} W_{1} + ε W_{2} + \dots, Φ = Φ_{0} + ε^{\frac{1}{2}} Φ_{1} + ε Φ_{2} + \dots

(9)

where n is an integer of the order ε^−1/2, and $W_{j} and Φ_{j}$ are functions of independent arguments $τ_{0} = ε^{- 1 / 2} t$ , $τ_{1} = t$ , and $τ_{2} = ε^{1 / 2} t, \dots$ . Substituting equations (8) and (9) in (1) results in the sequence of differential equations with respect to unknown functions W_k, Φ_k. Considering each of these sequences and deleting secular solutions at $τ \to \infty$ , we find the sought $W_{j} and Φ_{j}$ . In particular, the consideration being limited by first two approximations gives

w \approx \sin (\frac{π ps}{ε^{1 / 2}}) {A_{01} \exp {i [\frac{{(π p)}^{2} t}{ε^{1 / 2}} + \frac{2}{9} \int_{0}^{t} f (τ) d τ]} + A_{02} \exp {- i [\frac{{(π p)}^{2} t}{ε^{1 / 2}} + \frac{2}{9} \int_{0}^{t} f (τ) d τ]}}

(10)

where unknown A₀₁ and B₀₁ are found from initial conditions.

Relation (10) describes nonstationary high-frequency bending vibrations with the frequency of the order ε^−1/2. Here, $2 / 9 \int_{0}^{1} f_{1} (τ) d τ$ and $2 / 9 \int_{0}^{t} f_{2} (τ) d τ$ are the nonstationary corrections to the frequency of bending vibrations and the damping ratio, respectively, where $f_{1} (t)$ and $f_{2} (t)$ are the real and imaginary parts of the functions $f (t)$ . Solution (10) could be interpreted as the high-frequency nonstationary reaction of the three-layered beam consisting of MRE on the quick signal of the magnetic field.

Conclusion

Viscoelastic properties of MRE containing carbonyl iron particles have been investigated. Shear modulus grows from a value about 4 kPa without magnetic field up to 3 MPa in magnetic fields with the induction of 300 mT and higher. Loss modulus is maximal (about 800 kPa) at induction of the magnetic field of 200–300 mT. Solution of motion equations for the three-layered beam, the bearing constructions of which are made from aluminum and the inner layer is made from MRE, has shown that first six natural vibration frequencies of the beam grow slowly with the growth of induction of the applied magnetic field up to 400 mT and then nonlinearly decrease with the further increase of the induction of the magnetic field. It is stated that for the beam under investigation from the point of view of vibration dampening, the magnetic field with induction of about 350 mT is optimal. The general solution of motion equations for the beam under forced bending vibrations has been found at stationary impact of the magnetic field. Also, the analytical solution of the normal bending of the beam at pulsed impact of the magnetic field is found.

Footnotes

Funding

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

References

Bilyk

Korobko

Reizina

. (2009) Vibration control of oscillating system based on governing viscoelastic characteristics of absorber by means of external electric signal. Solid State Phenomena 144: 220–225.

Chen

Gong

(2007) Microstructures and viscoelastic properties of anisotropic magnetorheological elastomers. Smart Materials and Structures 16(6): 2645–2650.

Choi

Wereley

(2001) Assessment of time response characteristics of electrorheological and magnetorheological dampers. In: Proceedings of SPIE, edited by Inman

D. J.

, Newport Beach, CA, 5–7 March 2001, vol. 4331, pp. 92–102.

Coulter

Duclos

(1989) Applications of electrorheological materials in vibration control. In: Proceedings of the 2nd international conference on ER Fluids, edited by Carlson

J. D.

Sprecher

A. F.

Conrad, Raleigh, NC, 7–9 August 1989, pp. 300–325

Korobko

Novikova

Bedzik

. (2009) Adaptive fluids for electrorheological dampers and their damping characteristics. In: Proceedings of the 8th international conference on “Vibroengineering 2009,” Klaipeda, Lithuania, 16–18 September 2009, Kaunas: Technologija, pp.27–30.

Mikhasev

Botogova

Korobko

(2011) Theory of thin adaptive laminated shells based on magnetorheological materials and its application in problems on vibration suppression. In: Altenbach

Eremeyev

(eds) Shell-Like Structures, Advanced Structures Materials, vol. 15. Berlin: Springer-Verlag, pp.727–750.

Mikhasev

Korobko

Novikova

(2010) On suppression of vibrations of three-layered beam containing magnetorheological composite. Mechanics of Machines, Mechanisms and Materials 4: 49–53(in Russian).

Park

Choi

Jung

(1998) Shape control of an electrorheological fluid based smart plates. In: Proceedings of SPIE, edited by Regelbrugge

M. E.

, San Diego, CA, 2 March 1998, vol. 3329, pp.824–835.

Reizina

Korobko

Bilyk

. (2009) Optimal control of transient processes in an oscillating system with an electrorheological shock-absorber. Journal of Physics: Conference Series 149: 012025.

10.

Shaw

(2000) Hybrid control of cantilevered ER sandwich beam for vibration suppression. Journal of Intelligent Material Systems and Structures 11(1): 26–31.

11.

Yalcintas

Dai

(1999) Magnetorheological and electrorheological materials in adaptive structures and their performance comparison. Smart Materials and Structures 8(5): 560–573.

12.

Yalcintas

Dai

(2004) Vibration suppression capabilities of magnetorheological materials based adaptive structures. Smart Materials and Structures 13(1): 1–11.

13.

Zhou

(2003) Shear properties of a magnetorheological elastomer. Smart Materials and Structures 12(1): 139–146.

14.

Zhou

Wang

(2006) Study on the adjustable rigidity of magnetorheological-elastomer-based sandwich beams. Smart Materials and Structures 15(1): 59–74.