Najib Kacem
email: najib.kacem@insa-lyon.fr

Sebastien Hentz

Sébastien Baguet

Régis Dufour

Forced large amplitude periodic vibrations of non-linear Mathieu resonators for microgyroscope applications

Keywords: Microgyroscope, periodic vibrations, nonlinear dynamics, Mathieu resonator, bifurcation topology, scale factor

This paper describes a comprehensive nonlinear multiphysics model based on the Euler-Bernoulli beam equation that remains valid up to large displacements in the case of electrostatically actuated Mathieu resonators. This purely analytical model takes into account the fringing field effects and is used to track the periodic motions of the sensing parts in resonant microgyroscopes. Several parametric analyses are presented in order to investigate the effect of the proof mass frequency on the bifurcation topology. The model shows that the optimal sensitivity is reached for resonant microgyroscopes designed with sensing frequency four times faster than the actuation one.

Introduction

Microelectromechanical gyroscopes have attracted lots of attention due to their small size, batch fabrication, Integrated Circuit (IC) compatibility, low cost, and acceptable moderate performance for most applications. They can be used either as a low-cost miniature companion with micromachined accelerometers to provide heading information for inertial navigation purposes or in other areas, including automotive applications for ride stabilization and rollover detection; some consumer electronic applications, such as video-camera stabilization, virtual reality, and inertial mouse for computers; robotics applications; and a wide range of military applications [START_REF] Yazdi | Micromachined inertial sensors[END_REF][START_REF] Geen | Single-chip surface micromachined integrated gyroscope with 50 deg;/h allan deviation, Solid-State Circuits[END_REF].

Micromachined gyroscopes typically rely on the coupling of an excited vibration mode into a secondary mode due to the Coriolis acceleration based on the combination of vibration of a proofmass and an orthogonal angular-rate input. As shown in Fig. 1, when the gyroscope is subjected to an angular rotation, a sinusoidal Coriolis force is induced in the direction orthogonal to the drive-mode oscillation at the driving frequency. To achieve high sensitivity in conventional micro rate gyroscopes based on harmonic oscillators, the drive and the sense resonant frequencies are typically designed and tuned to match, and the device is controlled to operate at or near the peak of the response curve (where amplitude is defined by the Q-factor) [START_REF] Clark | Micromachined vibratory rate gyroscopes[END_REF]. However, current micro fabrication processes produce asymmetries causing frequency mismatching between modes, translating to drastic loss of sensitivity [START_REF] Yazdi | Micromachined inertial sensors[END_REF]. Although solutions to overcome frequency mismatching have been pursued [START_REF] Park | Adaptive control for the conventional mode of operation of mems gyroscopes[END_REF][START_REF] Shkel | Dynamics and control of micromachined gyroscopes[END_REF], many of them involve adding complexity to the system by including additional controllers, additional degrees of freedom [START_REF] Acar | Nonresonant micromachined gyroscopes with structural mode-decoupling[END_REF] or utilizing multiple drive mode oscillators [START_REF] Acar | An approach for increasing drive-mode bandwidth of mems vibratory gyroscopes[END_REF].

Due to the complexity of the proposed control schemes [START_REF] Park | Adaptive control for the conventional mode of operation of mems gyroscopes[END_REF][START_REF] Shkel | Dynamics and control of micromachined gyroscopes[END_REF], alternate approaches were considered. One of them is the use of resonant sensing [START_REF] Seshia | An integrated microelectromechanical resonant output gyroscope[END_REF] of the Coriolis force instead of displacement sensing employed in most conventional microgyroscopes. Profiting from the high sensitivity of the resonant detection, the matching of the drive and the sense frequencies is not mandatory to achieve a high resolution. Consequently, the number of states that have to be simultaneously controlled and the number of variables that require identification are much smaller and the dynamics is simplified from a minimally two-dimensional system to a series of coupled one-dimensional mass-spring-damper systems. Nevertheless, resonant sensing suffers from the soon occurrence of nonlinearities [START_REF] Kacem | Nonlinear dynamics of nanomechanical beam resonators: improving the performance of nems-based sensors[END_REF][START_REF] Kacem | Dynamic range enhancement of nonlinear nanomechanical resonant cantilevers for highly sensitive nems gas/mass sensor applications[END_REF][START_REF] Kacem | Nonlinear phenomena in nanomechanical resonators: mechanical behaviors and physical limitations[END_REF][START_REF] Kacem | Computational and quasi-analytical models for non-linear vibrations of resonant mems and nems sensors[END_REF] which reduces the gyroscope performances. Moreover and unlike resonant accelerometers, the sensing parts are Mathieu resonators since their stiffnesses are modulated periodically by a time-varying Coriolis force at the proof mass frequency.

In this paper, a typical resonant gyroscope is described and equations of motion are derived for both actuation and sensing elements. Then, the nonlinear dynamics of electrostatically driven Mathieu resonators is investigated for resonant gyroscope applications using a complete model which includes the mechanical nonlinearities, nonlinear electrostatic terms up to the fifth order as 2 Produit par HAL -18 Sep 2014 well as the Coriolis parametric excitation. The model is based on the Galerkin expansion method coupled with a perturbation technique; the resonant and periodic case under primary excitation has been considered. The main idea is to provide practical rules for microgyroscopes designers based on the proof mass frequency effect analysis in order to enhance the scale factor of the resonant sensor.

The Resonant Gyroscope

The resonant gyroscope, as its name implies, utilizes resonant sensing as a basis for Coriolis force detection. In its simplest form, the device consists of three resonating elements, a proof mass vibrating in the tens of kiloHertz and two resonating sense elements, with a designed resonant frequency, generally, an order of magnitude higher than that of the proof mass in order to avoid possible parametric instabilities in detection. Although the principle is general enough to accommodate devices that sense rotation rate about in-plane axes with minor modifications (such as driving the proof mass perpendicular to the substrate), the focus of this paper concerns gyroscopes that sense rotation rate about a single axis orthogonal to the plane of the device substrate. In addition, gyroscope topologies for a dual-mass to cancel common-mode acceleration signals and gyroscope suspensions to reduce quadrature error carry over for a resonant gyroscope. A schematic of the z-axis resonant gyroscope is shown in Fig. 2. The device consists of a proof mass suspended by flexures attached to lever mechanisms [START_REF] Roessig | Integrated MEMS Tuning Fork Oscillators for Sensor Applications[END_REF] for Coriolis force amplification. The proof mass is driven along the y axis using embedded lateral comb drive actuators. If an external rotation is applied to the chip about the z-axis, the Coriolis forces acting on the proof mass is transmitted to the lever mechanisms that amplify these forces prior to its being communicated axially onto two resonators placed on each side of the proof mass for a differential output. The two resonators vibrate out of phase and parallel to the direction of motion of the proof mass. The periodic compression and tension of the resonators by the Coriolis force at the proof mass drive frequency modulates the resonant frequency of these force sensors. Each force sensor comprises mechanical resonator shown in Fig. 3, electrostatically actuated and embedded in the feedback loop of an oscillator circuit. Thus, by demodulating the oscillation frequency, the rotation rate applied to the device can be estimated. The dynamics of the device (Figs. 2 and3) can be described by a series of coupled differential equations. The proof mass dynamics can be described for most part by a classical spring-massdamper equation. The dynamics of the resonator subjected to an axial time-varying Coriolis force is described by a nonlinear Mathieu partial differential equation. The respective equations can be written:

d 2 Y( t) d t2 + δ Q dY( t) d t + δ2 Y( t) = F e M cos δt (1)
EI ∂ 4 w( x, t) ∂ x4 + ρbh ∂ 2 w( x, t) ∂ t2 + c∂ w( x, t) ∂ t - Ebh 2l l 0 ∂ w( x, t) ∂ x 2 d x∂ 2 w( x, t) ∂ x2 -Ñ + A l Fc cos δt ∂ 2 w( x, t) ∂ x2 = 1 2 ε 0 bC n Vdc + Vac cos( Ωt ) 2 (g -w( x, t)) 2 (2)
Fc = 2Mθ dY( t) d t ( 3
)
where x is the position along the resonator length, E and I are the Young's modulus and moment of inertia of the cross section. Ñ is the applied tensile axial force due to the residual stress on the silicon, t is time, ρ is the material density, h is the microbeam thickness, g is the capacitor gap width, and ε 0 is the dielectric constant of the gap medium. The last term on the left-hand side of Eq. ( 2) represents an approximation of the electric force assuming a complete overlap of the area of the microbeam and the stationary electrode including the edge effects by the coefficient C n [START_REF] Nishiyama | Capacitance of a strip capacitor, Components, Hybrids, and Manufacturing Technology[END_REF]. Furthermore, Ỹ is the mass displacement along the vertical axis, M is the mass of the drive part of the resonant gyroscope, δ is its resonance frequency, Q the drive quality factor, F e is the electrostatic force induced by the lateral comb drive actuators, A l is the amplification coefficient of the Coriolis F c force due to the lever mechanism. The perturbation term A l Fc cos δt , which represents a modulation of the spring constant of the resonant sensor at the gyroscope drive frequency ( δ 2π ), is directly contributed by the amplified Coriolis force impinging axially on the resonator. Thus, the Coriolis force ( Fc ) modulates the spring constant of the resonator system. Clearly, Eq. ( 2) governing the micro-beam resonator dynamics is similar to the equation of motion of the sensing parts in the case of a resonant accelerometer [START_REF] Kacem | Nonlinear dynamics of nanomechanical beam resonators: improving the performance of nems-based sensors[END_REF]. The only difference is the time-varying term proportional to the Coriolis force that modulates the spring constant of the resonator. Obviously, for a very low actuation frequency δ with respect to the resonator natural frequency, the Coriolis force is quasi-static and then sensing dynamics for both resonant inertial sensors become equivalent.

Note that the two resonators placed on each side of the structure experience an equal and opposite axial force. The output of the device is the resonant frequency shift difference (∆ f = ∆ f 1 -∆ f 2 ) between the two resonators, measured at the gyroscope proof mass drive frequency.

Actuation part

The details about the dimensions (proof mass+spring) are described in Fig. 4. An equivalent mechanical model was used in order to compute analytically the stiffness K m of the spring-mass-damper system as follow:

K m = 8Ee M b s 3L s1 + L s2 (4)
where e M is the MEMS level thickness (the resonator thickness could be smaller than the proof mass, springs and lever mechanisms thickness which is the case of a M&NEMS gyroscope). The natural frequency of the actuation system is then:

δ = 8Eb s ρL 2 m 3L 3 s1 + L 3 s2 ( 5
)
The electrostatic force generated by the comb drive actuators is:

F e = ∂U ∂g c = n ǫ 0 e M Vm dc + Vm ac cos δt 2 g c (6) n = 2L m + g c -W c g c + W c (7) 5
where ⌊⌋ denotes the floor function, n is the number of interdigitated fingers, U is the energy associated with the applied electric potential (a drive voltage Vm ac and a polarization Vm dc ), W c is the width of a finger, g c is the gap between two fingers as shown in Fig. 4.

The quality factor of the mass-spring-damper system being very high (10 4 < Q m < 10 6 ), the static displacement is negligible with respect to the dynamic displacement and then the proof mass displacement at resonance can be formulated as follows:

Y M = Q m F e ρ δ2 L 2 m e M ( 8
)
As the proof mass is actuated using a lateral comb drive topology and is allowed to displace along the sense direction as well, then it becomes susceptible to a pull-in like phenomenon since the actuator topology looks more like a parallel-plate capacitor along the sense direction. As a result, motion along the sense direction due to non-idealities or asymmetries in the electromechanical structure places an upper limit on the actuation voltage and the displacement that can be allowed along the sense axis. As it depends on the resonator stiffness along the X axis, this issue is negligible for our designs of resonant gyroscope. If an external rotation rate θ( • /s) is applied to the chip about the z-axis, the Coriolis force acting on the proof mass is:

Fc = 2π 180 L 2 m e M ρθ δY M (9)

Lever mechanism

Since a high scale factor is required for a low noise sensor, a lever mechanism suitable for surface micromachined technology is used in order to amplify the Coriolis force acting on the proof mass. It is made of flexural pivots for leverage and to link to the input (F c ) and output (F ca ) forces (true pivots are unavailable in the fabrication processes). As long as the torsional stiffness of the flexures is not too high, the structure will effectively approximate a lever, magnifying the input force communicated to the resonators and increasing the scale factor of the sensor.

The model for determining the actual amount of magnification provided by the lever is shown in Fig. 5. The behavior of the system must be solved by simultaneously solving for the vertical deflection and rotation of the system which gives the following equations:

X = 1 k piv 1 - k r L a1 L a2 k Φ + k r L 2 a2 Fc (10) Φ = L a1 k Φ + k r L 2 a2 Fc (11) Fca = k r L a1 L a2 k Φ + k r L 2 a2 Fc ( 12
)
where X is the deflection of the structure, Φ is the rotation of the structure, Fca is the output force (the amplified Coriolis force), k piv is the vertical stiffness of the pivot beam, k r is the stiffness of the output structure (the resonator), k Φ is the sum of the link rotational stiffnesses k r,Φ and k piv,Φ , L a2 is the distance from the pivot to the output, L a1 is the distance from the pivot to the input and Fc is the input force (the Coriolis force acting on the proof mass). 6

Produit par HAL -18 Sep 2014 In the ideal case, the torsional stiffness of the flexures is zero and the vertical stiffness of the pivot is infinite, and above equations to:

X = 0 (13) Φ = L a1 k r L 2 a2 Fc (14) Fca = L a1 L a2 Fc ( 15
)
In order to obtain the maximum possible amplification, the position of the resonator with respect to the lever mechanism structure has been optimized by FE simulations.

Model

A sketch of a Mathieu resonator is presented in Fig. 3. In order to model and investigate the nonlinear dynamics of such electromechanical structure, the nonlinear partial differential equation ( 2) is considered with the following boundary conditions:

w(0, t) = w(l, t) = ∂ w ∂ x (0, t) = ∂ w ∂ x (l, t) = 0 ( 16
)
The Mathieu equation has been widely studied in the context of parametric resonance. Newman et al [START_REF] Newman | Dynamics of a nonlinear parametrically excited partial differential equation[END_REF] investigated the dynamics of a parametrically excited partial differential equation and particularly the dependence of the steady state behavior on parameter values and initial conditions. In [START_REF] Abraham | Approximate asymptotics for a nonlinear mathieu equation using harmonic balance based averaging[END_REF], a new technique derived from [START_REF] Chatterjee | Harmonic balance based averaging: Approximate realizations of an asymptotic technique[END_REF] based on an approximate realization of the method of averaging has been used to tackle weakly nonlinear Mathieu equations whose unperturbed dynamics is close to points corresponding to simple resonances between response and parametric forcing. 7

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Rand et al [START_REF] Rand | 2:2:1 resonance in the quasiperiodic mathieu equation[END_REF] constructed analytical expressions for the transition curves of the quasiperiodic Mathieu equation in the vicinity of the resonance 2 : 2 : 1 using a double-perturbation procedure.

In [START_REF] Zounes | Global behavior of a nonlinear quasiperiodic mathieu equation[END_REF], the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation has been investigated. Belhaq and fahsi [START_REF] Belhaq | 2:1 and 1:1 frequency-locking in fast excited van der polmathieuduffing oscillator[END_REF] showed that in the vicinity of the 2 : 1 and 1 : 1 resonances in a fast harmonically excited Van Der Pol-Mathieu-Duffing oscillator, fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift. In [START_REF] Michon | Duffing oscillator with parametric excitation: Analytical and experimental investigation on a belt-pulley system[END_REF], the motion of a sample automotive belt-pulley system subjected to tension fluctuations governed by a Mathieu-Duffing equation was theoretically and experimentally investigated. Note that the transition curves of a linear Mathieu equation (Fig. 6) can be approximately plotted using a perturbation technique. The number of tongues of instability corresponds to the truncation order of the asymptotic expansion. Figure 6 displays three instability tongues that correspond to a third order expansion of the perturbation technique. The curves are determined analytically using the Floquet theory for small Coriolis forces. Here, ∆ is the ratio between the proof mass frequency and the resonator frequency and F ca is the dimensionless input time-varying force axially applied to the resonator. These instability tongues emanate from the points 1 ∆ = n 2 on the 1 ∆ axis. Dufour and Berlioz [START_REF] Dufour | Parametric instability of a beam due to axial excitations and to boundary conditions[END_REF] showed that the dynamic stability of parametrically exited beams depends on the type of parametric excitation, the forcing frequencies and the boundary conditions and demonstrated that the existence of the instability zone is in relationship with the topology of the modal geometric stiffness matrices due to axial force and torque.

Figure 6 shows the instability chart which correspond to the Strutt diagram for 1 ∆ . Inside the tongues, the resonator displacement grows exponentially in time. Outside the tongue, the displacement becomes the sum of terms each of which is the product of two periodic (sinusoidal) functions with generally incommensurate frequencies, that is, the displacement is a quasiperiodic function of time. Also, the resonator is very weakly damped (high quality factors) which makes these transition curves approximately valid for a damped linear Mathieu equation. Figure 6 shows unbounded solutions to Mathieu's equation which can result from resonances between the forcing frequency and the oscillator's unforced natural frequency. However, real physical systems do not exhibit unbounded behavior. The difference lies in the fact that the Mathieu equation is linear. The 8 effects of nonlinearity can be explained as follows: as the resonance causes the amplitude of the motion to increase, the relation between period and amplitude (which is a characteristic effect of nonlinearity) causes the resonance to detune, decreasing its tendency to produce large motions.

The equation that governs the Mathieu resonator is a nonlinear partial differential equation under parametric and external excitation and consequently, further complicated than those already studied in literature. In addition, the structure of the stability regions of the quasiperiodic Mathieu equation is much more complicated than for the Mathieu equation.

Belhaq et al. [START_REF]Asymptotic solutions for a damped non-linear quasi-periodic mathieu equation[END_REF] and Guennoun et al. [START_REF] Guennoun | Quasi-periodic solutions and stability for a weakly damped nonlinear quasi-periodic mathieu equation[END_REF] consider a homogeneous Mathieu equation with quasiperiodic linear coefficients and a constant nonlinear coefficient. The small parameter technique of multiple scales is applied twice to the system to obtain an approximate time-invariant system. In another study (see Belhaq and Houssni, [START_REF] Belhaq | Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations[END_REF] the system under investigation contains quadratic and cubic nonlinearities as well as parametric (linear terms) and external excitations of incommensurate frequencies. The small parameter techniques of generalized averaging and multiple-scale perturbation are employed to obtain a solution. Rand and his associates [START_REF] Zounes | Global behavior of a nonlinear quasiperiodic mathieu equation[END_REF][START_REF] Rand | 2:1:1 resonance in the quasi-periodic mathieu equation[END_REF][START_REF] Zounes | Transition curves for the quasi-periodic mathieu equation[END_REF] analyze a linear homogeneous quasiperiodic Mathieu equation via several methods such as numerical integration, Lyapunov exponents, regular perturbation, Lie transform perturbation and harmonic balance.

The purpose of this paper is to provide practical rules for MEMS designers in order to enhance the performances of resonant microgyroscopes. Since quasiperiodicity is an undesirable phenomenon which can lead to chaotic oscillations, we restrict our investigations to the periodic motions of the forced nonlinear Mathieu equation.

Following Rand and Morrison [START_REF] Rand | 2:1:1 resonance in the quasi-periodic mathieu equation[END_REF], in our case for an external excitation tuned around the resonator primary resonance and by analogy, the Mathieu equation is quasiperiodic when ∆ 2-m n for ∆ ∈ [0, 1], m ∈ Z and n ∈ N. Consequently, the quasiperiodic domains in the (∆, F ca ) plan are very limited for low frequencies actuation ( δ) with respect to the periodic motions. This ensures the generality of our parametric analysis of the gyroscope sensitivity out of quasiperiodicity. However, a first order averaging method is valid only for low Coriolis forces which ensure a negligible effect of the superharmonic resonances at each ∆ < 1. Otherwise, high order averaging is required. For simplicity, all these conditions are assumed to be satisfied.

Nondimensionalization

For convenience and equations simplicity, the following nondimensional variables are introduced:

w = w g , x = x l , t = t τ (17)
where τ = 2l 2 h 3ρ E . Substituting Eq. ( 17) into Eqs. ( 2) and ( 16), yields:

∂ 4 w ∂x 4 + ∂ 2 w ∂t 2 + c ∂w ∂t -α 2 [Vdc + Vac cos(Ωt)] 2 (1 -w) 2 =       N + F c cos δt + α 1 1 0 ∂w ∂x 2 dx       ∂ 2 w ∂x 2 (18) w(0, t) = w(1, t) = ∂w ∂x (0, t) = ∂w ∂x (1, t) = 0 (19)
The parameters appearing in Eq. ( 18) are:

c = cl 4 EIτ , N = Ñl 2 EI , F c = A l Fc l 2 EI , δ = δτ α 1 = 6 g h 2 , α 2 = 6C n ε 0 l 4 Eh 3 g 3 , Ω = Ωτ (20)

Reduced order model

In practice, the quality factors Q of Mathieu resonators are in the range of 10 3 -5.10 4 which makes the static displacement negligible with respect to the dynamic displacement of the microbeam. A reduced-order model is generated by modal decomposition transforming Eq. ( 2) into a finite-degree-of-freedom system consisting of nonlinear Mathieu ordinary differential equations in time. The undamped linear mode shapes of the straight microbeam are used as basis functions in the Galerkin procedure. To this end, the deflection is expressed as:

w(x, t) = n k=1 a k (t)φ k (x) ( 21
)
where a k (t) is the k th generalized coordinate and φ k (x) is the k th linear undamped mode shape of the straight microbeam, normalized such that 1 0 φ k φ j = δ k j where δ k j is the Kronecker symbol.

The linear undamped mode shapes φ k (x) are governed by:

d 4 φ k (x) dx 4 = λ 2 k φ k (x) ( 22
)
φ k (0) = φ ′ k (0) = φ ′ k (1) = φ ′′ k (1) (23)
Here, λ k is the k th natural frequency of the resonator. Since some high order nonlinearities proves to be relevant by enabling the capture of 5 possible amplitudes for a given frequency in the mixed behavior [START_REF] Kacem | Bifurcation topology tuning of a mixed behavior in nonlinear micromechanical resonators[END_REF], the electrostatic force in Eq. ( 18) is expanded in a fifth order Taylor series. Then, Eq. ( 21) is substituted into the resulting equation, Eq. ( 22) is used to eliminate d 4 φ k (x) dx 4 , and the outcome is integrated from x = 0 to 1. Thus, a system of coupled nonlinear Mathieu ordinary 10

Produit par HAL -18 Sep 2014

differential equations in time is obtained. The DC voltage, which is generally at least ten times higher than the AC voltage, makes the second harmonic negligible. Also, the first mode should be the dominant mode of the system. According to this assumption, the study can be restricted to the case n = 1. then, it gives:

ä1 + cȧ 1 + (500.564 -2α 2 ξ 0 + 12.3N) a 1 +α 2 ξ 0 0.831 + 4a 1 2 + 13.255a 1 4 +12.3F c a 1 cos δt -23.17α 2 ξ 0 a 1 5 + (151.354α 1 -7.403α 2 ξ 0 ) a 3 1 = 0 ( 24
)
where ξ 0 = V 2 dc + V ac cos Ωt. To analyse the equation of motion [START_REF]Asymptotic solutions for a damped non-linear quasi-periodic mathieu equation[END_REF], it proves convenient to invoke perturbation techniques [START_REF] Nayfeh | Introduction to Perturbation Techniques[END_REF] which work well with the assumptions of "small" excitation and damping, typically valid in MEMS resonators. Nevertheless, in order to avoid quasiperiodicity, we chose δ ω n ∈ Q ∩ [0, 1] and we assume that the Coriolis forces are weak enough to make the possible superharmonic resonances negligible with respect to the fundamental primary resonance. Since near-resonant behavior is the principal operating regime of the proposed system, a detuning parameter, σ is introduced, as given by:

Ω = ω n + εσ (25)
where ω n = 500.564 + 12.3N -2V 2 dc α 2 . Separating the resulting equations and averaging them over the period 2π Ω in the t-domain results in the system's averaged equations, in terms of amplitude and phase, which are given by:

Ȧ = ǫ c 2 A -ǫ AF c ω n sin[π∆] sin[π∆ -2β] (∆ -2) +ǫ 0.831 + A 2 + 1.657A 4 α 2 ξ 1 ω n sin β +ǫ AF c ω n sin[π∆] sin[π∆ + 2β] (2 + ∆) + O ǫ 2 (26) β = ǫσ -ǫ F c ω n cos[π∆ -2β] sin[π∆] (∆ -2) -ǫ F c ω n cos[π∆ + 2β] sin[π∆] (2 + ∆) + sin[2π∆] ∆ -ǫ α 2 ξ 1 ω n 0.831 A + 3A + 8.285A 3 cos β +ǫ α 2 A 2 V 2 dc ω n 2.776 + 7.241A 2 -ǫ 56.757A 2 α 1 ω n + O ǫ 2 (27)
where ξ 1 = V dc V ac and ∆ = δ ω n . The steady-state motions occur when Ȧ = β = 0, which corresponds to the singular points of Eqs. ( 26) and [START_REF] Rand | 2:1:1 resonance in the quasi-periodic mathieu equation[END_REF]. Thus, the frequency-response can be written in its parametric form for β ∈ [0, π] and

∆ ∈ Q ∩ [0, 1]. Ω = f 1 (β, ∆) (28) A = f 2 (β, ∆) (29)
This analytic expression (set of two equations) makes the model suitable for MEMS and NEMS designers as a fast and efficient tool for resonant gyroscope performances optimisation.

Analytical results and device specifications

All the numerical simulations were carried out with the following set of parameters:

• Proof mass: L m = 100 µm, e M = 2 µm, l c = 6 µm, g c = 2 µm, d c = 4 µm and W c = 1µm.

• Resonator: l = 100 µm, b = 2 µm, h = 5 µm, g = 300 nm, Q = 1000.
The estimated quality factor of the system mass-spring-damper is around 10 5 . The actuation voltages Vm ac and Vm dc are adjusted with respect to a constant 2.5µm proof mass oscillation at resonance. The ratio between the proof mass and the resonator frequencies (∆) as well as V ac and V dc were used for parametric studies.

Proof mass frequency effect

Figure 7 shows four nonlinear hardening frequency responses at a proof mass frequency ten times smaller than the resonator frequency (∆ = 0.1) and for several angular rates 0 -900 • /s. The DC polarization of the resonator is low enough (V dc = 1V) to keep the global nonlinear stiffness dominated by the mechanical nonlinearities and hence, the predicted hardening behavior.

In this configuration, the Coriolis force that modulates the resonator stiffness at the proof mass frequency represents a slow dynamic with respect to the resonator dynamic ten times faster. Therefore, the frequency effect is negligible and only the resulting stress is considered which implies a positive frequency shift proportional to the external rotation rate θ. Then, for ∆ = 0.1 and for several angular rates going from 0 • /s up to 900 • /s, the DC voltage applied to the resonator is increased from 1V up to 9V while decreasing the AC voltage from 0.2V down to 30mV (see Fig. 8). This increases significantly the negative nonlinear stiffness due to the electrostatic force that dominates in this case the global stiffness of the resonator and hence, the predicted softening behavior. For low actuation frequency with respect to the sensing frequency, the microgyroscope behaves as a resonant accelerometer. Indeed, the Coriolis force is seen by the resonator as a quasi-static force. Consequently, the close-form solutions of the critical amplitudes and the mixed behavior initiation amplitude established in [START_REF] Kacem | Nonlinear dynamics of nanomechanical beam resonators: improving the performance of nems-based sensors[END_REF] can be used here in order to improve the performances of the resonant gyroscope. Particularly, the compensation of the nonlinearities is possible when the mechanical and electrostatic critical amplitudes are equilibrated. This results on an optimal DC voltage [START_REF] Kacem | Dynamic range enhancement of nonlinear nanomechanical resonant cantilevers for highly sensitive nems gas/mass sensor applications[END_REF] for which the obtained frequency resonance peak is linear beyond the critical amplitude. 12 In a second step, the frequency of the proof mass is increased up to a quarter the resonator frequency. In the same way, for this configuration, Figures 9 and10 show respectively several nonlinear hardening and softening forced frequency curves for several rotation rates. Thus, the compensation for the nonlinearities is possible for specific DC voltage. However, the apparition of additional bifurcation points 3 and 4 is notable for external rotation rates (θ) beyond 600 • /s. The maximum is close to the bifurcation point 4 and no more situated at β = π 2 . This strange behavior can be explained by an important contribution of the superharmonic resonance of order quarter the resonator primary resonance for θ beyond 600 • /s. Therefore, even out of quasiperiodicity, the averaging method is not valid except at a higher order defined by the transition curve at the corresponding superharmonic resonance. Indeed, at this level the full scale here is limited by the proof mass frequency (for a valid first order averaging).

The classical specifications of current MEMS gyroscopes include a dynamic range at best around 100 • /s [START_REF] Kacem | Nonlinear dynamics of M&NEMS resonant sensors: design strategies for performance enhancement[END_REF] which ensures the validity of the first averaging. The curves in Fig. display the variation of the Mathieu resonator displacement at its middle point and at resonance when the phase β = π 2 for several values of angular rates θ. Uncommonly, it appears that the symmetry can be broken between negative and positive Coriolis stress effect when the resonator dynamic becomes highly nonlinear for large angular rates and frequencies. This corresponds to a strong nonlinear parametric spring softening effect for which the resonator displacement averaged over the period 2π

ω n is amplified (parametric perturbation) and then a high instability such as the pull-in [START_REF] Nayfeh | Dynamic pull-in phenomenon in mems resonators[END_REF] could be suddenly reached. Since the sensitivity of the resonant gyroscope could be extremely reduced for a large dynamic range, working at ∆ < 0.25 is indispensable for high grade resonant gyroscopes.

Resonant gyroscope scale factor

The scale factor of the sensor, S F that relates the output frequency shift difference (δ f ) between the two resonators to the external input rotation rate (θ) is given by: Note that the scale factor corresponds to the derivate of Eq. ( 28) at the phase β = π 2 with respect to the rotation rate θ for a dynamic range (DR) that ensures stable motions out of quasiperiodicity and negligible superharmonic resonance effects. The expression can be also written in terms of a ratio of some drive and sense parameters multiplied by a lever gain and a constant dependent on the mode shape of the resonating element. The scale factor is only dependent on material and geometrical parameters and the displacement of the gyroscope proof mass at resonance (Y M ). Consequently and unlike classical gyroscopes, the goal of the control scheme is now simplified to the requirement of maintaining constant amplitude motion for the gyroscope proof mass and the resonator-sensing elements at their respective resonant frequencies. Figure 12 shows the variation of the resonant gyroscope scale factor with respect to the frequency ratio between the actuation and the sensing parts (∆) for a resonator driven in the linear regime. Obviously, to use this curve, one should exclude the non-rational values of ∆ (quasiperiodic regime). Remarkably, the maximum of sensitivity is located at ∆ = 0.25 which corresponds to a superharmonic resonance in the nonlinear regime as shown in Figs. 9 and 10. Moreover the scale factor is approaching zero for three cases:

S F = d f dθ Π 2 , θ ∈ DR (30) 14
• ∆ = 0 which is obvious since the Coriolis force is proportional to the proof mass frequency.

• ∆ = 0.5. It corresponds to a simultaneous superharmonic and primary resonances in the nonlinear regime. In this configuration, the performances of the Mathieu resonator are drastically reduced in the linear regime and furthermore, in the nonlinear regime, the secular terms coming from the secondary resonance should be taken into account for analytical investigation in a large dynamic range.

• ∆ = 1. It corresponds to a simultaneous parametric and primary resonance in the nonlinear regime. We arrive to the same conclusions as the case ∆ = 0.5. When the resonator is driven beyond its critical amplitude (in the nonlinear regime), unlike the linear case where the scale factor is constant for a given ∆ and θ inside the dynamic range, the resonant gyroscope sensitivity is highly dependent on the external rotation rates for a frequency ratio δ > 0.1 as shown in Fig. 13. Curiously, the scale factor is infinitely reduced and the gyroscope becomes completely unstable if we assume a negligible secondary resonance effect which is valid up to ∆ = 0.25 as shown in Fig. 12. Furthermore, for an admissible scale factor nonlinearity when the Mathieu resonator is driven in the nonlinear regime, the resonant gyroscope must be designed with ∆ < 0.05.

Conclusions

In this paper, the nonlinear dynamics of Mathieu resonator for resonant gyroscope applications has been modeled using the Galerkin discretization coupled with a perturbation technique and under few assumptions that lead to steady-state periodic motions. The relatively simple dynamic model of the nonlinear Mathieu resonator utilized here, is able to predict the measured resonator response for various parameter settings qualitatively and in many cases even quantitatively. Characteristic nonlinear dynamic steady-state behavior is very well predicted by the model. Therefore, it represents a good first step in the modeling process and a suitable starting point for understanding and predicting the dynamic behavior of resonant MEMS gyroscopes. The resulting benefits include nonlinear dynamics control, improved scale factor stability over micromechanical gyroscopes utilizing open-loop displacement sensing, large dynamic range and high resolution.

In order to provide some design rules, the variation of the gyroscope sensitivity with respect to the ratio between the proof mass and the resonator frequencies was investigated. The analytical parametric analysis showed that in the resonator linear or slightly nonlinear regimes, a frequency ratio ∆ = 0.25 provides the greatest scale factor. However, once the resonator dynamics becomes strongly nonlinear (large oscillations beyond the critical amplitude), the sensitivity is significantly reduced if the resonator frequency is not at least an order of magnitude higher than that of the proof mass (∆ < 0.1). This drastic limit could be avoided by the compensation of the nonlinearities [START_REF] Kacem | Nonlinear dynamics of nanomechanical beam resonators: improving the performance of nems-based sensors[END_REF][START_REF] Kacem | Bifurcation topology tuning of a mixed behavior in nonlinear micromechanical resonators[END_REF] out of quasiperiodicity for ∆ = 0.25 which gives the maximum of sensitivity.

Figure 1 :

 1 Figure 1: Lumped parameter model of a vibratory gyroscope.

Figure 2 :

 2 Figure 2: Schema of a simple mass resonant gyroscope.

Figure 3 :

 3 Figure 3: Sketch of a Mathieu resonator.

Figure 4 :

 4 Figure 4: Proof mass and spring designs for a resonant gyroscope.

Figure 5 :

 5 Figure 5: Model used to predict leverage force magnification.

Figure 6 :

 6 Figure 6: Transition curves in a linear autonomous Mathieu equation. S denotes stable quasiperiodic domains and U denotes the unstable domains.

Figure 7 :

 7 Figure 7: Predicted forced frequency responses displaying a hardening behavior for ∆ = 0.1. W max is the displacement of the beam normalized by the gap g at its middle point l 2 , {1, 2} are the bifurcation points. The frequency shift is due to the variation of the external angular rate θ.

Figure 8 :

 8 Figure 8: Predicted forced frequency responses displaying a softening behavior for ∆ = 0.1. W max is the displacement of the beam normalized by the gap g at its middle point l 2 , {1, 2} are the bifurcation points. The frequency shift is due to the variation of the external angular rate θ.

11

 11

Figure 9 :

 9 Figure 9: Predicted forced frequency responses displaying a hardening behavior for ∆ = 0.25. W max is the displacement of the beam normalized by the gap g at its middle point l 2 , {1, 2, 3, 4} are the bifurcation points. The frequency shift is due to the variation of the external angular rate θ.

Figure 10 :

 10 Figure 10: Predicted forced frequency responses displaying a softening behavior for ∆ = 0.25. W max is the displacement of the beam normalized by the gap g at its middle point l 2 , {1, 2, 3, 4} are the bifurcation points. The frequency shift is due to the variation of the external angular rate θ.

Figure 11 :

 11 Figure 11: Variation of the Mathieu resonator displacement at its middle point and at resonance when the phase β = π2

Figure 12 :

 12 Figure 12: Scale factor variation with respect to the proof mass frequency in the resonator linear regime and inside the dynamic range of the resonant gyroscope. S R and PR are superharmonic and parametric resonances.

Figure 13 :

 13 Figure 13: Scale factor variation curves with respect to the proof mass frequency for a nonlinear resonator and several rotation rates.

Produit par HAL -18 Sep 2014

acknowledgments

The authors gratefully acknowledge financial support from the CEA LETI and I@L Carnot institutes (NEMS Project) and from the ANR (M&NEMS Project).


Dr Fiona M Burns
email: fburns@gum.ucl.ac.uk

Alison R Evans

Catherine H Mercer

Violetta Parutis

Christopher J Gerry

Richard Mole

Rebecca S French

John Imrie

Graham J Hart

Sexual and HIV risk behaviour in Central & Eastern European migrants in London

Keywords: Sexual behaviour, migration, UK, Europe, knowledge & attitudes

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers,

Introduction

Over the past decade the United Kingdom (UK) has experienced large migratory fluxes from Central and Eastern Europe (CEE). In particular the accession on 1 May 2004 of Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Slovakia and Slovenia (the 'A8') into the European Union (EU), and of Bulgaria and Romania (the 'A2') on 1 January 2007, were followed by large-scale, though often circular migration from these countries (1). Circular migration is a form of migration that allows some degree of legal mobility back and forth between two countries.

The uptake of safer sex measures and patterns of health service use in these migrant populations are unknown. Also the demographic profile of CEE migrants (young and frequently single)(1); the often high background prevalence of sexually transmitted infections (STIs) and HIV in their countries of origin [START_REF] Uuskula | Trends in the epidemiology of bacterial sexually transmitted infections in eastern Europe, 1995-2005[END_REF][START_REF] Hamers | HIV in central and eastern Europe[END_REF][START_REF]Global Report: UNAIDS report on the global AIDS epidemic 2010[END_REF]; and their lack of experience of the British health system, may place these new migrant communities at higher risk of sexual ill health and reproductive morbidity compared to the general population As research on the sexual behaviour of the general populations of Central and Eastern

Europeans is sparse and no information exists on the sexual behaviour of CEE migrants within the UK, our aim was to establish an understanding of the sexual lifestyles and health service needs of these communities. This study is part of the SALLEE project (sexual attitudes and lifestyles of London's Eastern Europeans); papers examining the sexual risk of CEE migrant men who have sex with men [START_REF] Evans | Central and East European migrant men who have sex with men: an exploration of sexual risk in the UK[END_REF] and CEE migrant attendance at genitourinary medicine clinics and STI diagnoses [START_REF] Evans | Factors associated with GUM clinic attendance and STI diagnosis among Central and East European migrants in London[END_REF], also to arise from this project have already been published.

Methods

A detailed description of the methodology has been previously published [START_REF] Evans | The sexual attitudes and lifestyles of London's Eastern Europeans (SALLEE Project): design and methods[END_REF]. A brief summary is provided below.

Participants and procedure

Eligible respondents were literate men and women aged over 17 who self-identified as migrants from one of the ten CEE countries. The community sample was recruited in London and the web survey was advertised on websites for CEE nationals in the UK as described in reference 7. Web survey respondents who gave their home post-code outside London were excluded from the study.

Fieldwork took place over a nine-month period (July 2008 -March 2009). The nine fieldworkers involved in the recruitment of respondents for the community sample were native speakers of six of the languages of the CEE countries.

Study instruments

The survey instrument was a self-completed questionnaire designed using SNAP 9 survey software (8) that was fielded using hand-held computers for the community sample and a web survey for the Internet sample. The community and internet survey questions were identical. The questionnaire was anonymous. The questionnaire was piloted to examine its feasibility and acceptability and to explore understanding of the question items and underlying constructs: the use of the hand-held computers, question routing, and technical usability were also tested during piloting.

The questionnaire was translated into 11 languages (the ten official languages of the CEE countries plus Russian) and bilingual native speakers of the 11 languages checked the translation accuracy.

Sampling

There is no adequate sampling frame of this new migrant population from which to draw a probability sample of CEE nationals in London. The study therefore relied on convenience sampling in order to generate a cost-effective sample that would be sufficiently robust for detailed analysis. This study adopted two sampling strategies (community and Internet) in order to ensure representation of key elements of the population, and minimise bias. A detailed social mapping exercise was conducted prior to recruitment of the community sample [START_REF] Evans | The sexual attitudes and lifestyles of London's Eastern Europeans (SALLEE Project): design and methods[END_REF]. This provides some confidence that the selected boroughs and locations capture a broad cross-section of CEE migrants in London. Due to the sample size it was not possible to analyse by individual countries. Rather, because of the differing dates of accession (2004 vs. 2007), we grouped respondents into two regions (the A8 and A2). Accession into Europe was dependent on countries fulfilling certain political and economic conditions, both of which potentially influence the sexual attitudes and behaviours of populations; in addition there were tighter controls on A2 (compared to A8) migration to the UK. Also within the socio-political sciences, 'Eastern Europe' is often sub-divided into Central Europe (the A8 states) and South-Eastern Europe (the A2). Historically Central European political culture has been characterised as more rational, contractual and individualistic, while South-Eastern Europe is more essentialist, collectivistic and arbitrary. Over the centuries this has resulted in different ways of thinking about individual freedoms, the boundaries between public and private and the role of the state -all of which influences attitudes and behaviours in society [START_REF] Berend | The historical evolution of Eastern Europe as a region[END_REF][START_REF] Stokes | Eastern Europe's Defining Fault Lines[END_REF][START_REF] Szucs | Three Historical Regions of Europe[END_REF].

Statistical analysis

Standard

The sample size of 2000 people provides 80% power to detect as significant the association with an explanatory factor where the difference in prevalence is around 6%.

A sample of this size also allows for adequately powered subgroup analysis. Analysis was performed using Intercooled STATA 8.0 (12) and SPSS12.0 [START_REF]SPSS [computer program[END_REF].

The study was approved in 2008 by the Camden & Islington Community Research Ethics Committee, UK (07/H0722/110).

Role of the funding source:

The sponsor of the study had no role in study design; the collection, analysis or interpretation of the data; the writing of the report; or the decision to submit the paper for publication.

Results: Sample characteristics:

This CEE population sample (n=2648) was derived by merging the community sample (n=2276) with the Internet sample (n=372). The mean age of respondents was 29.0, 51.4% were married or cohabiting, and almost half (48.3%) of respondents were male (Supplementary Table 1). The majority of respondents were Polish (n=1082, 40.9%), Romanian (n=492, 18.6%), or Lithuanian (n=449, 17.0%). The Internet sample was more likely to be born in an A8 country (97.8% vs. 74.2%, p<0.001), to be educated to degree level (43.9% vs. 29.4%, p<0.001) and in paid employment (84.1% vs. 72.8%, p<0.001), than the community sample. Most respondents had migrated post accession, 79.3% arriving after May 2004. Those who arrived in the UK post accession had been in the UK for a mean of around 1.5 years (median 17-19 months). Three quarters (74.6%) of respondents had returned to their home country at least once in the past year and 15.0% had returned four or more times.

Sexual behaviours:

Partners or sexual partners were defined as people who have had sex together -whether just once, or a few times, or as regular partners, or as married partners. This definition was made explicit within the questionnaire. Table 1 shows the distribution of reported numbers of partners (in the past year and past five years) by age and gender. There was substantial heterogeneity in numbers of heterosexual partners reported in the past five years; 70.3% of men and 48.8% of women reported more than one partner, whereas 29.1% of men and 6.9% of women reported more than five. Men consistently reported higher numbers of heterosexual partners than women over all time periods. Same sex partnerships were reported by 3.4% of men and 4.0% of women.

Nearly half (44.9%) of men and 29.1% of women had formed a new heterosexual partnership over the past year. The mean number of new partnerships in the past year varied from 5.3 among previously married men aged 25-34 to 0.1 among married women aged over 34. The mean number of new partners declined with increasing age for men but not for women. Younger respondents were less likely to be in a married or cohabiting relationship. Across all age groups and both genders new partner acquisition was highest among the single or previously married. Over half (55.6%) of all male respondents were single or previously married; these men formed 74.0% of all new heterosexual partnerships by men in the past year.

Among those respondents reporting sex in the past year a quarter of men and 7.6% of women reported relationships that were concurrent (overlapped or were simultaneous).

Just over half (51.5%) of men and most (80.2%) women reported only one partner. The majority of new heterosexual partnerships in the past year were consummated in the UK.

Approximately three quarters of most recent sexual partnerships were with nationals from the home country of the respondent (75.6% for men and 71.1% for women). A UK national was the most recent sexual partner for 10.4% of men and 14.5% of women.

Sexually transmitted infections and HIV:

An ever previously diagnosed STI (excluding HIV) was reported by 11.1% of respondents. Most infections had not been diagnosed in the UK (Table 2). A third of respondents had ever had an HIV test, and a third of these had their last HIV test in the UK. Just over 1% of respondents reported they were HIV positive (18 men and 11 women), with 31.0% being diagnosed in the UK. The majority of respondents who reported they were HIV positive were Polish (n=23, 79%). Reported risk factors for HIV were: previously injecting drugs (n=2), sex between men (n=1), six (21%) had ever paid for sex but none reported having been paid for sex, and eight (28%) reported a previous STI diagnosis.

Amongst the 546 (21%) respondents who reported recreational drug use in the past year, marijuana was most widely used (87.8%), followed by ecstasy (31.0%), cocaine (23.4%), speed (18.6%), and crystal-meth (5.4%).

Risk behaviours by region and sex:

Table 3 presents the prevalence of sexual behaviours, attitudes and drug use by sex and region of origin. Many risk behaviours varied by region and sex. Compared to their A2 counterparts, A8 respondents were more likely to report heterosexual oral or anal sex and recreational drug use in the past year. A8 men were more likely to have ever injected drugs (6.6% vs. 1.2%, p<0.001), whilst A8 women were more likely to report concurrent relationships in the past year, a same sex partner ever, and more heterosexual partners in the past year than those from the A2. Having paid for sex was widely reported across both regions but was more likely among men from the A2 countries (44.1% vs. 28.5%, p<0.001), as was concurrency in the past year (32.2% vs. 23.4%, p=0.019). A2 men were also more likely to have had an HIV test than A8 men (37.5% vs. 31.0%, p=0.032), and to have used condoms consistently in the past four weeks, especially among those reporting two or more partners in the past year (55.3% vs. 34.8%, p=0.002). A2 men and women were significantly less likely to be registered with a general practitioner (GP) although this association was lost when adjusted for time in the UK.

To examine high-risk sexual behaviour we calculated the odds ratio (OR), 95%CI and the aOR of factors associated with one or more new heterosexual partners in the past year (Supplementary Table 2). Male respondents reporting other risk behaviours in the same time period such as recreational drug use (aOR 1.37, 95%CI 1.01-1.87)), drinking alcohol on average three or more days a week (aOR 1.62, 95%CI 1.14-2.28) and anal sex (aOR 1.89, 95%CI 1.35-2.64) had increased odds of new heterosexual partners, as were those reporting a previous STI diagnosis (aOR 1.69, 95%CI 0.99-2.86) or oral sex (aOR 3.61 95%CI 1.64-3.31); whilst married or cohabiting men and men registered with a GP were less likely (aOR 0.35, 95%CI 0.25-0.47, and aOR 0.69, 95%CI 0.51-0.93 respectively). CEE women were more likely to have had new sexual partner(s) in the past year if they reported a same sex partner ever (aOR 2.31, 95%CI 1.26-4.26), anal or oral sex in the past year (aOR 1.48, 95%CI 1.01-2.15 and aOR 1.76, 95%CI 1.23-2.52 respectively), drinking alcohol more frequently (aOR 1.68, 95%CI 1.02-2.80), and a previous STI diagnosis (aOR 1.75, 05%CI 1.13-2.73); they were less likely if they were married or cohabiting (aOR 0.28, 95%CI 0.20-0.38), from the A8 compared to the A2 (aOR 0.60, 95%CI 0.37-0.97), and over the age of 24 (aOR 0.71, 95%CI 0.51-0.98).

No association between time in the UK and most of the risk behaviours was found (data not shown). The notable exception was reporting of ever having had sex with someone of the same sex, which was more likely with increasing time in the UK (p=0.034).

Comparison with British national data:

Among respondents who reported heterosexual sex ever, the CEE sample was younger (27.7 vs. 31.8 yrs, p<0.001) than the Natsal sample, more likely to be single (40.2% vs.

35.0%, p<0.001), and less likely to have a degree (31.8% vs. 35.8%, p<0.001).

Significant behavioural differences between samples remained after adjusting for sociodemographic variables in multivariate analyses (Table 4). Male CEE respondents were more likely to report two or more partners in the past year (aOR 2.1, 95%CI 1.3-2.1) and in the past five years (aOR 1.7, 95%CI 1.6-2.6), but were less likely to report having had an STI (aOR 0.7; 95%CI 0.5-1.0) than Natsal respondents. CEE men were more than three times as likely to have paid for sex with a woman (aOR 3.2; 95%CI 2.5-4.0) and twice as likely to have injected non-prescribed drugs (, aOR 2.2; 95%CI 1.3-3.9). CEE women were three times more likely to have injected non-prescribed drugs than Natsal women (aOR 3.0, 95% CI 1.1-8.1) and were less likely to report having had an STI (aOR 0.7, 95%CI 0.6-1.0). The proportion of female respondents reporting two or more partners in the past five years and in the past year were similar across surveys.

Consistent condom use in the past four weeks was higher in the CEE sample (aOR 1.3, 95%CI 1.0-1.8).

Discussion

The accession of ten CEE countries to the EU has resulted in one of the largest migratory influxes in peacetime British history. This project provides estimates of sexual behaviour patterns in these new communities. As would be expected we found wide variability in sexual lifestyles by gender, age, relationship status, and region of origin.

Whilst reports of prior STIs are lower than in the general British population, CEE migrants, especially male migrants, report high rates of behaviours associated with increased risk of HIV and STI transmission. The benefits of more consistent condom use may be offset by higher rates of partner acquisition, paying for sex and injecting drug use.

Like other communities, heterosexual CEE migrants demonstrate assortative sexual mixing (whereby their most recent partner was from their home country) (14;15). Nearly three-quarters of last sexual partnerships in the general CEE population sample were with a national from the home country of the CEE respondent. Risk of HIV and other STIs may increase as CEE communities become more integrated with the British population. Currently the reported high-risk behaviours appear offset by low prevalence of infections within the CEE migrant community. With increasing time in the UK it is possible that CEE migrants will have sexual partners from more diverse backgrounds, potentially increasing the likelihood of exposure to infections [START_REF] Evans | Central and East European migrant men who have sex with men: an exploration of sexual risk in the UK[END_REF]. STI screening opportunities differ in the UK compared to many CEE countries [START_REF] Riedner | Recent declines in reported syphilis rates in eastern Europe and central Asia: are the epidemics over?[END_REF], which may impact on STI reports, especially on infections that are often asymptomatic.

However over 30% of respondents reported ever having an HIV test, higher than the 13% reporting an HIV test ever in the last Natsal survey [START_REF] Mcgarrigle | Investigating the relationship between HIV testing and risk behaviour in Britain: Analysis of the National Survey of Sexual Attitudes and Lifestyles 2000[END_REF].

The reported HIV prevalence of 1.1% is substantially higher than the estimated prevalence of 0.09% in the general British population [START_REF]Sexually transmitted infections in black African and black Caribbean communities in the UK: 2008 report[END_REF]. This burden of infection is not, however, reflected in national HIV surveillance data. Although the numbers remain relatively small, there was a tenfold increase between 2000 and 2007 in the total number (8 to 84), and proportion (0.3% to 2.3%), of all new HIV diagnoses that are in people from the A8 [START_REF] Delpech | The impact in the UK of the Central and Eastern European HIV epidemics[END_REF]. Eastern Europe does have the highest rate of HIV across Europe but Poland has one of the lowest [START_REF] Eurohiv | HIV/AIDS Surveillance in Europe. Mid-year report[END_REF]. In Eastern Europe although mode of acquisition is often unreported [START_REF]HIV/AIDS Surveillance in Europe 2008[END_REF], the epidemic is believed to be largely driven by injecting drug use (a behaviour reported by ~ 4% of our respondents), although heterosexual transmission is also on the increase (20;21).

Only 31% of the reported HIV was diagnosed in the UK. Potentially migrants may be aware of their HIV infection but not accessing services in the UK, and hence not impacting on national surveillance data as yet. This seems unlikely but is not impossible, especially as our qualitative data suggests people continue to access health care in home countries and use the Internet to obtain medications (Burns, unpublished data). In Central & Eastern Europe there is a high degree of stigma and discrimination attached to being HIV positive (22;23;23;24); given this data relies on self-reports a bias towards underreporting of HIV seropositivity would be expected. Conversely HIV positive people may be more interested in sexual health matters which may impact on participation.

Limitations

The dilemma over how to interpret the HIV findings highlights one of the major limitations of our survey, the absence of biological samples. The high-risk behaviours reported suggest that blood-borne virus screening in this population would be informative and should be considered in future studies. A further limitation is that the data is based on self-reports. Central and Eastern Europeans are comprised of heterogeneous populations that differ sociologically in ways that may impact on sexual attitudes and lifestyles, for example religiosity and social liberalism. Numbers precluded analysis by specific country of origin, Polish, Lithuanian and Romanian respondents accounted for 76% of all data. A separate analysis was undertaken to ensure Polish 27); it may be that this also holds true in other populations. Voluntary migrants are individuals who take a risk to travel to, and work in, environments that they hope will be beneficial to them. This risk-taking may permeate into the choices they make in other areas of their lives [START_REF] Duckett | International organisation for migration[END_REF].

Public health implications:

Migrants are often viewed as a health threat, yet evidence shows that the process of migration can present a health threat to migrants themselves. The 'healthy migrant' effect (whereby because of the health and human capital required for migration, immigrants are on average healthier than the population they originate from, and often also the population in their host country [START_REF] Marmot | Lessons from the study of immigrant mortality[END_REF]) is likely to be evident in these new communities. Research, however, suggests that migrant health deteriorates more rapidly than the general population with time [START_REF] Ronellenfitsch | Deteriorating health satisfaction among immigrants from Eastern Europe to Germany[END_REF]. The mobile nature of migrant communities, language barriers and confusion over entitlement to services often means members are unlikely to benefit fully from public health programmes or access to health services.

Men in particular were less likely to be registered with a GP yet reported significant risk behaviours for blood-borne viruses. Whilst this study focused on sexual and reproductive health there are, of course, many other aspects to health likely to be relevant to CEE migrants, and subsequently for their host nations. These include high rates of smoking, alcohol consumption, and cardiovascular disease [START_REF] Yusuf | Global Burden of Cardiovascular Diseases[END_REF][START_REF] Kern | Cardiovascular risk factors in Croatia: struggling to provide the evidence for developing policy recommendations[END_REF][START_REF] Mckee | Understanding the toll of premature death among men in eastern Europe[END_REF].

A feature of CEE migration to the UK is that many people migrate for relatively short but recurring periods (1). Similarly our respondents returned home frequently. In a borderless EU management of chronic health conditions and surveillance of communicable diseases will increasingly involve trans-national collaborations. Systems to monitor and facilitate trans-national health-care and disease surveillance for migrant communities are needed. Improving health outcomes for migrants are likely to benefit both the receiving (or host) and home countries [START_REF]Migrant Health: Background note to the 'ECDC Report on migration and infectious diseases in the EU[END_REF]. This study helps to illuminate our understanding of the sexual lifestyles, sexual and reproductive health risks, and health service needs of these migrant communities. Results from this study will help inform service planning and identify where STI and HIV interventions should be targeted. It also provides baseline data to help evaluate the effectiveness of interventions and provides a useful adjunct to interpreting data derived from other community and clinic based surveys. ¶ A2 refers to Bulgaria and Romania § Comparing A8 with A2 † Among those who had vaginal or anal sex in the past year # May also include men who have paid for sex elsewhere in addition to UK or home country Ψ Restricted to those reporting sex in past 4 weeks, refers to anal or vaginal sex.

  statistical tests, e.g. Chi 2 test and Student's t-test, were used to examine associations between variables. Multivariate analysis of factors associated with one or more new heterosexual partner in previous 12 months used a backwards-stepwise model; all variables with a p-value <0.1 for the crude association were retained for incorporation in the multivariate model. To evaluate reported behaviours with those of the British population we conducted. a comparison with individual-level data from the National Survey of Sexual Attitudes and Lifestyles (Natsal) 2000, the most recent national probability sample survey at the time of writing, conducted between May 1999 and February 2001. Logistic regression modelling was used to obtain adjusted odds ratios (aOR) to control for any variation in the age, gender, marital status, and education between the two data sources.

Table 1 : Distribution of numbers of partners over past 5 years and past year by gender and age-group

 1

		Men (age-group [years])			Women (age-group [years])
	18-24	25-34	35 +	All	18-24	25-34	35+	All

1 Among those who've had sex in past year

2

Only includes men reporting anal intercourse with a homosexual partner

Table 2 : Self-reported HIV and sexually transmitted infections 1 and place of diagnosis.

 2

	Sexually Transmitted Infection	%, 95% CI, (n)	Proportion diagnosed in the UK
		N=2296	%, 95% CI
	Chlamydia	2.7, 2.1-.3.4 (63)	44.4, 31.9-57.5
	Genital herpes	1.8, 1.3-2.4 (42)	28.6, 15.7-44.6
	Genital warts	1.1, 0.7-1.6 (25)	32.0, 14.9-53.5
	Gonorrhoea	2.0, 1.4-2.6 (45)	15.6, 6.5-29.5
	Non-specific urethritis	0.5, 0.2-0.8 (11)	18.2, 2.3-51.8
	Pelvic inflammatory disease	2.1, 1.4-3.1 (25)	8.0, 1.0-26.0
	(women only)
	Syphilis	0.6, 0.3-1.0 (13)	15.4, 1.9-45.4
	Trichomonas vaginalis	1.2, 0.8-1.7 (28)	21.4, 8.3-41.0
	HIV	1.1, 0.8-1.8 (29)	31.0, 15.3-50.8

Table 3 : Prevalence of sexual behaviours and drug use: Men and women by region of origin

 3 A8 refers to Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Slovakia and Slovenia.

			Men			Women
		A8 * n=939	A2 ¶ n=329 p-value §	A8 n=1097	A2 n=264	p-value
	Heterosexual practices
	Vaginal intercourse in past month	61.2%	53.4%	0.057	70.1%	76.0%	0.374
	Oral-genital contact in the past year	69.0%	60.6%	0.008	61.6%	44.6%	<0.001
	Anal sex in the past year	30.6%	23.5%	0.018	22.9%	13.3%	0.001
	Condom used on all occasions, past
	4 weeks Ψ and:	n=224	n=76		n=135	n=18
	One partner past year	31.6%	29.6%	0.913	24.9%	30.1%	0.197
	Two or more partners past year	34.8%	55.3%	0.002	34.1%	33.3%	0.950
	All	33.6%	42.7%	0.050	27.0%	31.5%	0.389
	Risk perception
	More at risk of HIV in UK than	n=767	n=271		n=884	n=225
	home country	57.0%	52.0%	0.066	52.8%	47.1%	0.271
	Sexual partnerships
	Number of heterosexual partners
	past year: mean (SD)	2.36 (3.7) 3.29 (8.5) 0.064 1.42 (3.8) 1.07 (1.0)	0.010
	Ever had homosexual partners	5.0%	2.8%	0.091	6.1%	1.9%	0.007
	Concurrent partnerships, past year †	23.4%	32.2%	0.019	8.9%	3.3%	0.010
	Ever paid for sex	28.5%	44.1% <0.001	NA	NA	NA
	Where paid for sex (n=382)			0.911			NA
	UK #	31.7%	31.6%		NA	NA
	Home country #	35.0%	38.2%		NA	NA
	Both in UK and home country #	17.1%	15.4%		NA	NA
	Elsewhere	16.3%	14.7%		NA	NA
	Drug use
	Ever injected non-prescribed drugs	6.6%	1.2%	<0.001	2.5%	0.0%	NA
	Injected non-prescribed drugs past
	year	2.4%	0.0%	NA	1.3%	0.0%	NA
	Recreational drug use in past year
	(excluding intravenous drugs)	37.3%	14.2% <0.001	13.3%	5.0%	<0.001
	Sexual health
	Ever had HIV test	31.0%	37.5%	0.032	32.4%	34.9%	0.450
	Ever diagnosed with a STI	n=798	n=277		n=988	n=236
	(excludes candidiasis)	8.6%	11.6%	0.153	12.8%	11.0%	0.455
	Registered with a GP	45.1%	35.1%	0.002	67.5%	58.6%	0.006

*

Table 4 : Multivariate analysis of risk behaviours comparing SALLEE respondents to Natsal Ψ respondents.

 4 National Survey of Sexual Attitudes and Lifestyles 2000 † Restricted to London respondents aged 18-44 who reported ever having had sex (age range related to Natsal upper limit and SALLEE lower limit). * Unweighted, weighted denominator ¶ Adjusted for age, relationship status, education: comparing SALLEE to Natsal respondents # Condoms used on all occasions of vaginal and or anal sex in the past 4 weeks

		SALLEE †	Natsal †
	Behaviour	(n=2323)	(n=2883,	Crude OR	Adjusted OR	p value
		%	1526)*	(95% CI)	(95% CI) ¶
			%
	Men	n=1102	n=1216, 773*
	2+ partners (past 5 years)	77.3	58.9	2.4 (1.9, 2.9) 2.1 (1.6, 2.6) <0.001
	2+ partners (past year)	45.0	29.4	2.0 (1.6, 2.4) 1.7 (1.3, 2.1) <0.001
	Ever paid for sex with a woman	35.5	15.8	2.9 (2.3, 3.7) 3.2 (2.5, 4.0) <0.001
	Ever injected drugs	5.9	2.4	2.6 (1.5, 4.5) 2.2 (1.3, 3.9)	0.005
	Consistent condom use (past 4 weeks) #	36.6	29.2	1.4 (1.1, 1.8) 1.2 (1.0, 1.6)	0.099
	Ever diagnosed with STI	8.9	13.6	0.6 (0.5, 0.8) 0.7 (0.5, 1.0)	0.045
	Women	n=1221	n=1667, 753*
	2+ partners (past 5 years)	52.9	40.9	1.6 (1.3, 2.0) 1.2 (1.0, 1.5)	0.058
	2+ partners (past year)	18.7	15.6	1.2 (1.0, 1.6) 1.0 (0.8, 1.3)	0.915
	Ever injected drugs	2.2	0.7	3.3 (1.3, 8.8) 3.0 (1.1, 8.1)	0.035
	Consistent condom use (past 4 weeks) #	28.2	21.7	1.4 (1.1, 1.8) 1.3 (1.0, 1.8)	0.034
	Ever diagnosed with STI	12.8	18.1	0.7 (0.5, 0.9) 0.7 (0.6, 1.0)	0.038

Ψ

Data based on self-reports of ever-diagnosed infections rather than biological samples.

Acknowledgements:

We are grateful for the expert guidance provided by our expert and community advisory boards, for the hard work and commitment of the fieldworkers who collected the data. We would also like to thank the staff and management at the Mortimer Market Centre, the Archway Sexual Health Clinic and the Margaret Pyke Centre; the websites that fielded the questionnaire; the many commercial, social and educational premises that facilitated recruitment for our fieldworkers; and everyone who took part in the study. This study was supported by the MRC Sexual Health and HIV Research Strategy Committee, the views expressed are those of the authors and not necessarily those of the MRC or the Health Departments.

Funding: This research was supported by the MRC Sexual Health and HIV Research Strategy Committee (Grant: G0601703).

responses did not substantially differ from responses from the other 'A8' nationalities combined; they did not (data not shown). The limitations relating to convenience samples have previously been published [START_REF] Evans | The sexual attitudes and lifestyles of London's Eastern Europeans (SALLEE Project): design and methods[END_REF]. Natsal 2000 data was collected nine years prior to this data. It is possible that the frequency of reported behaviours would have changed in the British population over this time. However, there is no more recent survey with which to compare our data until Natsal 2010 data collection is completed. It is also possible that the sampling method may have created participation bias. Internet surveys of men who have sex with men show higher risk behaviours than the general population (ref), however these surveys use social networking sites often used to find sexual partners. None of the websites used to recruit for this study were social networking sites.

The economic recession has seen the rate of influx decline, however large numbers of CEE nationals continue to migrate into the UK (1), and many migrants have now made the UK their permanent home. The UK has a duty of care to ensure appropriate risk reduction strategies are in place and that these new communities are aware of and able to access these services. Paying for sex was reported by over a third of all CEE men. Further research is needed on the cultural factors associated with commercial sex, what type of sex is occurring and where, and the risks involved.

No association between time in the UK and many of the risk behaviours was found. The notable exception was reporting of ever having had sex with someone of the same sex, which was more likely with increasing time in the UK (p=0.034). High-risk behaviours, regardless of time in the UK, support the concept of the migrant as a 'risk-taker'. In Africa migration has been identified as a critical factor in high-risk sexual behaviour independent of marital and cohabitation status, social milieu, or awareness of HIV (25-

Key messages:

The accession of ten Central and Eastern European (CEE) countries to the EU has resulted in the largest migratory influx in peacetime British history.

• This is the first study to provide estimates of sexual behaviour patterns in these new communities.

• High rates of partner acquisition, paying for sex and injecting drug use were reported; behaviours associated with increased risk of HIV and STI transmission.

• Reports of prior STIs were lower in CEE migrants to London than in the general British population however 1.1% reported being HIV positive.

• Heterosexual CEE migrants demonstrate assortative sexual mixing.

Contributors: FB, CM, CG, RM, RF, JI, GH participated in the design of this study. AE, VP and FB were responsible for study coordination and data collection. FB, AE and CM conducted the statistical analysis. FB was the lead writer of this paper. All authors have seen and approved the final version of this manuscript.

Conflicts of Interest: No conflicts of interest.

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