Micky Rakotondrabe

Ioan Alexandru Ivan

member, IEEE Sofiane Khadraoui

Philippe Lutz

member, IEEE Nicolas Chaillet

member, IEEE Ioan Alexandru

Simultaneous Displacement/Force Self-Sensing in Piezoelectric Actuators and Applications to Robust Control

Keywords: Self-sensing, piezoelectric actuator, displacement and force estimation, robust H∞ control

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I. INTRODUCTION

M Icropositioning (or precise positioning) tasks, the basis of microassembly and micromanipulation applications, require a very high accuracy which is in the order of micrometer or even submicrometer. To reach such severe performances, the used microrobots, micromanipulators and actuated microsystems are often based on active and smart materials instead of hinges and joint systems which generate imprecision due to the mechanical clearances. Among the used active materials, piezoelectric materials are the most recognized. They have been successfully used to develop stepper microrobots [START_REF] Bergander | Mobile cm3-microrobots with tools for nanoscale imaging and micromanipulation[END_REF] [START_REF] Rakotondrabe | Development, modelling and control of a micro/nano positioning 2DoF stick-slip device[END_REF], Atomic Force Microscopes (AFM) [START_REF] Bining | Atomic Force Microscope[END_REF] and continuous microactuators such as piezocantilevers and microgrippers [4] [5]. This recognition is mainly thanks to the high resolution (at the nanometer level), the high bandwidth (more than the kHz) and the relatively high force density that they offer. For all that, piezoelectric materials (in particular piezoceramics) exhibit hysteresis and creep nonlinearities. These nonlinearities become non-negligible when the applied electrical field -and therefore the range of deformation-are large. Hysteresis and creep limit the accuracy of the developed actuators. In order to enhance the performances, several works have been devoted to the modeling and to the control of piezoelectric actuators within these ten last years. These works include closed-loop control techniques [START_REF] Leang | Hysteresis, creep, and vibration compensation[END_REF]- [START_REF] Lee | Fast Robust Nanopositioning -A Linear Matrix Inequalities Based Optimal Control Approach[END_REF], open-loop control techniques [START_REF] Fleming | A grounded load charge amplifier for reducing hysteresis in piezoelectric tube scanners[END_REF]- [START_REF] Rakotondrabe | Complete open loop control of hysteretic, creeped and oscillating piezoelectric cantilevers[END_REF] and the combination of both [START_REF] Devasia | A survey of control issues in nanopositioning[END_REF].

In feedback techniques, basic control schemes like PIDstructure and more advanced control laws (H ∞ , passivity, interval, nonlinear control, etc.) have been successfully used. The main advantage of feedback is the robustness: 1) the specifications are satisfied in almost all the cases, and 2) the effects of model uncertainties and of the external disturbances can be rejected efficiently. However, feedback requires sensors. This constitutes the main limitation of feedback techniques at the micro/nano world and related applications. On the one hand, sensors that have the required performances (accuracy, bandwidth and low noises) are bulky and expensive: for instance triangular optical sensors, interferometers, etc. On the other hand, sensors that are packageable and cheap are fragile, difficult to handle and noisy: for instance strain gages. Further, feedforward techniques have been successfully introduced to bypass this limitation as they do not need sensors. They have the particularity of yielding really packaged controlled systems. Nevertheless, feedforward techniques fail with uncertain models and in presence of external disturbances. To resume, finding the compromise between satisfying the desired performances and satisfying the available space is difficult.

Promising approaches avoiding external sensors in meso and micro systems are the so-called self-sensing methods. They allow both actuation and sensing capabilites, and are especially (but not necessarily) used for piezoelectric materials based actuators and systems. Given the reversibility of the piezoelectric effect, self-sensing does or should not influence the actuation capabilities in terms of range or dynamics. It could provide simultaneous quantitative information of end-effector displacement, manipulation force, close contact detection or even temperature evaluation. If well designed, self-sensing approaches could combine the advantages from feedback (robustness) and from feedforward (packageability) for piezoelectric actuators.

The first use of "self-sensing" term goes back to 1992 when J.J. Dosch et al [START_REF] Dosch | A Self-Sensing Piezoelectric Actuator for Collocated Control[END_REF] successfully damped the vibration of a piezoelectric beam without using the aid of external sensors. In their works, the voltage drop provided from a capacitive bridge was processed in an analog circuit, amplified and returned back to the piezoelectric element. In that time, several independent applications began to emerge for beam vibration control or stack piezo devices for micropositioning. Several years later T. Taigami et al [START_REF] Takigami | Application of selfsensing actuator to control of a soft-handling gripper[END_REF] applied the method for force selfsensing and for the control of a large size bimorph actuator by using a half-bridge circuit, a voltage follower and a computerbased data acquisition system. The authors experimentally verified that the stiffness of the grasping object does not affect the measurement; however electronic schematic limited the applied voltage range and nonlinearities (hysteresis and creep) were not compensated. Since, self-sensing is widely used for vibration control of flexible structures [START_REF] Cui | Compounding Control of Piezoceramic Micro-Motion Worktable Based on Integrator[END_REF] [START_REF] Putra | Self-sensing actuation with adaptive control in applications with switching trajectory[END_REF]. All these existing self-sensing works usually offered shortterm measurements: measurement is not available for no more than 5 seconds in general. Moreover, very few works refer to the use of self-sensing method for the control of positioning. Consequently, these existing works could not be applied to measure signals (displacement and force) and to control systems in micropositioning tasks such as micromanipulation or microassembly. Indeed, such applications require the measurement of displacement/force and the combination of dynamic and static sensing/measurement with duration from several tens of seconds to several minutes. This duration corresponds to the transport of the manipulated micro-objects where the displacement has to be controlled precisely and the manipulated force to be known. Newly, in our previous works [START_REF] Ivan | Quasi-static displacement self-sensing method for cantilevered piezoelectric actuators[END_REF], a new scheme of self-sensing has been proposed for precise measurement of the displacement in its steady-state for more than 600s. Further we have extended the proposed scheme to include the measurement of steady-state force signal [START_REF] Ivan | Current integration force and displacement self-sensing method for cantilevered piezoelectric actuators[END_REF]. But as the proposed scheme was for quasi-static selfsensing, it could not be used for control application. This paper aims to develop a scheme for static and dynamic selfsensing in order to allow the control of piezoelectric actuators used in micropositioning tasks. The proposed approach provides a measurement and estimation of the displacement in both steady-state and dynamics enabling the positioning control for long term period. Furthermore, the scheme also allows the measurement and estimation of the steady-state force at the same time which is very useful for the user to overlook the manipulation force applied to the object. Finally, to demonstrate these advantages, we employ the proposed selfsensing to a feedback control application. Experimental tests were carried to validate the self-sensing techniques and to validate its applicability to control applications. They clearly demonstrate the efficiency of the proposed self-sensing.

The paper is organized as follows. In section-II we remind the quasi-static self-sensing as in [START_REF] Ivan | Quasi-static displacement self-sensing method for cantilevered piezoelectric actuators[END_REF] and [START_REF] Ivan | Current integration force and displacement self-sensing method for cantilevered piezoelectric actuators[END_REF]. In section-III, we present the proposed self-sensing allowing the steady-state and dynamic displacement measurement and the steady-state force measurement at the same time. Finally, section-IV is devoted to a control application of micropositioning tasks.

II. REMIND OF THE QUASI-STATIC DISPLACEMENT AND FORCE SELF-SENSING

A. The framework of micromanipulation and microassembly

Within the framework of micromanipulation and microassembly applications, microgrippers based on piezoelec-tric actuators with cantilever structure are used. In general two actuators are employed to constitute a microgripper. Each actuator has a rectangular section and is usually unimorph. Bimorph actuators are rarely used because of the difficulty to fabricate them and to setup the electrical connection. A unimorph piezoelectric actuator is composed of two layers glued themselves: a piezoelectric layer (often based on lead zirconate titanate (PZT) material) and a passive layer (often based on copper or nickel materials) (Fig. 1-a). When a voltage U is applied to the piezoelectric layer, it contracts or expands following the converse principle of piezoelectric effect. As a result, the whole cantilever bends (Fig. 1-b). The microgripper is able to position very precisely (i.e. to pick, transport and place very precisely) a small object if each actuator is conveniently controlled [START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to automated pick-and-place tasks[END_REF] (Fig. 1-c). An essential feature of self-sensing applied to microgripper's actuators is the ability to simultaneous measure and estimate the displacement at the tip and the force applied to the object (Fig. 1-d). This is what this paper claims to propose.

B. Quasi-static displacement self-sensing

Self-sensing, i.e. the use of an actuator as also a sensor, is possible thanks to the reversibility of piezoelectric effect. Its principle scheme is pictured in Fig. 2-a [START_REF] Ivan | Quasi-static displacement self-sensing method for cantilevered piezoelectric actuators[END_REF]. When a voltage U is applied to the actuator, a bending y is obtained (actuation). During this bending, an electrical charge Q appears on the electrodes of the actuator. This charge can be amplified by a convenient electrical circuit and transformed into an exploitable voltage U o . From the available signals U and U o , an observer calculated from the electrical circuit and from the model of the piezoelectric actuator provides an estimate ŷs of the displacement y at its steady-state. Both the electrical circuit and the observer constitute the sensing/measurement part of the self-sensing.

Fig. 2-b pictures the proposed electrical circuit. In order to explain its principle, the equivalent electrical circuit of the piezoelectric actuator has been introduced. This equivalent circuit is composed of three parallel components: the internal generator which supplies charge Q s due to internal stress of the material when deformed, the internal piezoelectric capacitor C p and the leakage resistor R f p . The total charge Q that appears on the electrodes of the actuator is therefore composed of the internal charge Q s (piezoelectric effect) and the charge

Q U = C p U (dielectric effect): Q = Q s + C p U .
Experiments shown however that Q s is very weak relative to Q U . This will drastically decrease the accuracy of a selfsensing measurement as the sensitivity to the internal stress, and therefore to the deformation, is very weak. To reduce this unwanted property, we proposed to put in parallel with the piezoelectric actuator a reference capacitor C r such that C r = C p . By inverting the voltage applied to C r relative to the voltage U applied to the actuator, the charge due to the dielectric effect (Q U = C p U ) can be compensated. Consequently, the final charge that will be amplified by the op-amp is:

Q -C r U = Q s + C p U -C r U = Q s .
However, the piezoelectric capacitor varies following the applied voltage U [START_REF] Ishikiriyama | Improvement of self-sensing piezoelectric actuator control using permittivity change detection[END_REF], i.e. C p = C p (U ). Hence the dielectric effect is not completely compensated by the reference capacitor C r .

Consequently the exploitable charge becomes:

Q -C r U = Q s + C p (U )U -C r U = Q s + ∆ Q (U ) (1)
where ∆ Q (U ) is residual charge that depends on the applied voltage U . If ∆ Q (U ) = 0, the charge due to the dielectric effect was fully compensated and the self-sensing measurement technique would be highly sensitive to the deformation of the piezoelectric material. Although ∆ Q (U ) = 0, it is still much less than Q s making the sensitivity of the technique still high. In Fig. 2-b, the capacitor C is used to amplify the charge while the resistor R disc and the relay k disc allow resetting the output U o in order to avoid any unexpected saturation issues. Finally, the op-amp is considered to have a very high input impedance. In the sequel, the voltage U is also called input voltage, control voltage or driving voltage. It corresponds to the input control signal of the actuator. Noticing that Q = αy, where α is the displacement-charge coefficient, the output and exploitable voltage U o is expressed as follows [START_REF] Ivan | Quasi-static displacement self-sensing method for cantilevered piezoelectric actuators[END_REF]:

U o = - 1 C   αy -C r U + 1 R f p t 0 U dt + Q DA (t, U )   (2)
where Q DA (t, U ) is the dielectric absorption of the equivalent capacitor C p in the piezoelectric actuator [START_REF] Kundert | Modeling dielectric absorption in capacitors[END_REF]. Dielectric absorption is the property of a dielectric, like piezoelectric materials, which prevents a capacitor from totally discharging, even when short-circuited for a short period of time. The dielectric absorption can be approximated by a linear model: The observer that gives an estimate ŷs of y at its steady-state is therefore described by (Eq. 4) [START_REF] Ivan | Quasi-static displacement self-sensing method for cantilevered piezoelectric actuators[END_REF]:

Q DA (s) = k DA (1+τ DA s) U (s) ⇔ Q DA (s) = Q tf DA (s)U (s) (3
ŷs = 1 α   (C r U -CU o ) -Q DA (t, U ) - 1 R f p t 0 U dt   (4)
The different parameters are identified following an experimental procedure or computed from the physical and geometrical properties of the piezoelectric actuator and from the parameters of the electrical circuit (see [START_REF] Ivan | Quasi-static displacement self-sensing method for cantilevered piezoelectric actuators[END_REF]).

C. Quasi-static displacement and force self-sensing

The previous self-sensing only provides the (steady-state) displacement information. This is only utilizable when the actuator has a free displacement and thus when no object is manipulated. When the actuator is in contact with an object and consequently when a manipulation force exists, the observer defined by (Eq. 4) of the previous self-sensing is not anymore valuable. In that case, the force and the displacement need to be modelled at the same time. By using an appropriate observer from this model, an observer that estimates the real displacement y and the real force F can again be computed. Then, the displacement and force selfsensing is derived. Fig. 3-a shows the new principle scheme of the quasi-static displacement and force self-sensing [START_REF] Ivan | Current integration force and displacement self-sensing method for cantilevered piezoelectric actuators[END_REF].

We remind that the electrical scheme is still similar than that of the quasi-static displacement self-sensing (see Fig. 2-b). The observer that gives an estimate ŷs of the displacement and an estimate Fs of the force at their steady-state is [START_REF] Ivan | Current integration force and displacement self-sensing method for cantilevered piezoelectric actuators[END_REF]:

                     Fs = 1 β   (C r U -CU o ) -F cr (t) -F hys (t) - 1 R f p t 0 U dt   ŷfree s = 1 α   (C r U -CU o ) -Q DA (t, U ) - 1 R f p t 0 U dt   ŷs = ŷfree s -s p Fs (5)
where β is the force sensitivity coefficient. It relates the electrical charge appearing on the actuator's surface with the applied external force. s p is the piezoelectric compliance that relates the displacement with the applied external force. Signal ŷfree s corresponds to the estimate steady-state displacement when there is no external force. It corresponds to the estimate displacement presented in (Eq. 4). Finally, signals F cr (t) and F hys (t) corresponds to the effects of the creep and the hysteresis nonlinearities of the actuator. While the creep is often described by a linear operator, the hysteresis is described by a nonlinear one such as the Bouc-Wen, the Prandtl-Ishlinskii or the Preisach operators. Their compact notations are:

F cr (s) = F tf cr (s)U (s) ; F hys (s) = Γ (U ) (6)
with F tf cr (s) a transfer function that captures the creep and Γ (U ) the nonlinear operator that captures the hysteresis.

III. COMPLETE DISPLACEMENT SELF-SENSING WITH STEADY-STATE FORCE ESTIMATION

The displacement and force self-sensing previously presented is static. It can estimate these signals at their steadystate regime or when working at low frequency, i.e. if F (t) and y(t) are constant we have:

Fs (t) = F (t) f or t → ∞ ŷs (t) = y(t) f or t → ∞ (7)
In the Laplace domain, (Eq. 7) is equivalent to:

Fs (s) = F (s) f or s → 0 ŷs (s) = y(s) f or s → 0 (8)
The main advantage of the quasi-static self-sensing is its suitability for micropositioning tasks in micromanipulation and microassembly applications. During these tasks, the object is maintained by the microgripper for several hundreds of seconds with constant force and/or displacement. However, "static measurement" is not proper for feedback control. Dynamic model as well as "dynamic measurement" are required. In this section, we propose a dynamic self-sensing, called complete self-sensing, able to provide steady-state and "dynamic" signals at the same time. More precisely, the proposed selfsensing will be able to provide the steady-state information of the manipulation force and both the steady-state and the "dynamic" information of the displacement which makes the latter signal usable for feedback control.

A. Principle scheme Fig. 4 gives the principle scheme of the proposed complete self-sensing. It is composed of:

• an electrical circuit that amplifies and transforms the charge Q into an exploitable voltage U o . This electrical circuit is similar to that presented in Fig. 2-b. • a static displacement and force observer that furnishes the steady-state estimates ŷs and Fs . The static observer is described by (Eq. 5).

• and a dynamics observer that furnishes complete information (stead-state and dynamics) ŷ on the displacement. Both the static observer and the dynamics observer constitute the complete observer. The synthesis of the dynamics observer is presented in the next parts.

B. Conditions for the complete observer

In order to have a steady-state and dynamics estimation of the displacement, i.e. a full measurement at low and high frequency, the following condition must be respected:

ŷ(s) = y(s); ∀s ∈ C (9)
In the temporal domain, (Eq. 9) is equivalent to:

ŷ(t) = y(t); ∀t ∈ R + (10)
Combining the condition of complete displacement selfsensing in (Eq. 9) and (Eq. 10) with the condition of quasistatic force self-sensing from (Eq. 7) and (Eq. 8), we can write the conditions of complete displacement and force self-sensing as follows:

Fs (s) = F (s) f or s → 0 ŷ(s) = y(s); ∀s ∈ R + ⇔ Fs (t) = F (t) f or t → ∞ ŷ(t) = y(t); ∀t ∈ R + (11)

C. Dynamics observer synthesis

The model of the piezoelectric actuator that relates the complete information of displacement y(s) with the applied input control U (s) and with the applied steady-state external force F s (s) is (see for eg. [START_REF] Rakotondrabe | Quadrilateral modeling and robust control of a nonlinear piezoelectric cantilever[END_REF]):

y(s) = d p D(s)U (s) -s p F s (s) (12)
where d p is the piezoelectric coefficient and D(s) is the dynamics such that D(s = 0) = 1 (normalization of the dynamics). In this equation, y = d p D(s)U corresponds to the free displacement, i.e. displacement without external force applied. The parameters d p and D(s) can be identified experimentally from the piezoelectric actuator by using a step response method, a harmonic method or a pseudo-random binary sequence methods.

On the other hand, if we consider again the third equation in (Eq. 5):

ŷs (s) = ŷfree s (s) -s p Fs (s) (13)
we observe that a dynamics we denote ∆(s) can be introduced in the estimate steady-state model (Eq. 13) in order to have a similar structure than (Eq. 12), and then to have a more complete estimate model:

ŷ(s) = ∆(s)ŷ f ree s (s) -s p Fs (s) (14)
Notice that ∆(s) will certainly be in the dynamics observer.

If we find a dynamic observer ∆(s), the self-sensing will be able to track any transient part of the output displacement that is due to the electrical excitation U . However, the transient part of the displacement as response to any applied force F will not be estimated. In this case, only the steady-state value will be tracked. Finally, when both electrical excitation U and mechanical excitation F are applied to the piezoelectric actuator, we have the following results: 1) the self-sensing will still track and estimate the transient part and the steady-state value of the output displacement due to the voltage U , 2) the self-sensing will track and estimate the steady-state only of the displacement due to the force F . The elastic parts in (Eq. 12) and (Eq. 13) being similar, our study can be restricted to the displacement without external force, i.e.:

y(s) = d p D(s)U (s) ŷ(s) = ∆(s)ŷ f ree s (s) (15)
which is equivalent to:

       y(s) U (s) = d p D(s) ŷ(s) ŷfree s (s) = ∆(s) (16)
In order compute the dynamics observer and its gains, we need to write the transfer ŷ(s) U (s) between the estimate complete information ŷ(s) and the input control U (s) such that we can compare it with the transfer y(s) U (s) in (Eq. [START_REF] Dubra | Preisach classical and nonlinear modeling of hysteresis in piezoceramic deformable mirrors[END_REF]. For that, remind that:

ŷ(s) U (s) = ŷfree s (s) U (s) • ŷ(s) ŷfree s (s) (17)
where ŷ(s) ŷfree s (s) = ∆(s) according to (Eq. 16). We now need to compute the transfer ŷfree s (s) U (s) . To this end, we apply the Laplace transform to (Eq. 4) and use (Eq. 3), we derive:

ŷfree s (s) = 1 α C r - 1 1 R f p s -Q tf DA (s) U -CU o (18)
We know that the relation between the input voltage U and the exploitable voltage U o can be approximated by a linear transfer function H(s) such that [START_REF] Rakotondrabe | Dynamic displacement self-sensing and robust control of cantilevered piezoelectric actuators dedicated to microassembly tasks[END_REF]:

U o (s) = H(s)U (s) (19)
Replacing Q tf DA (s) by its model in (Eq. 3) and introducing (Eq. [START_REF] Mokaberi | Compensation of scanner creep and hysteresis for AFM nanomanipulation[END_REF]), (Eq. 18) becomes:

ŷfree s (s) U (s) = Cr α -1 α R f p s - k DA α(1+τ DA s) -1 α H(s); ∀s ∈ C (20)
Based on (Eq. 20), the transfer (Eq. 17) becomes:

ŷ(s) U (s) = C r α - 1 α R f p s - k DA α (1 + τ DA s) - 1 α H ŷ(s) ŷfree s (s) (21)
Let us now consider the transfer ŷ(s) ŷfree s (s) = ∆(s) which is in fact the basis of the dynamics observer. Finding this transfer function is a deconvolution problem because its input is a steady-state information while the output is a complete information. Remind that complete information means that it includes the steady-state and the transient parts information. Such problem comes back to the estimation of a transient part after yet reaching a steady-state and is therefore equivalent to a dynamics inversion. The inversion of dynamics have attracted the attention in different applications. In fact, inverting dynamics is delicate in the sense that some conditions should be satisfied [START_REF] Devasia | Should Model-based Inverse Inputs be used as Feedforward under Plant Uncertainty?[END_REF]. For instance, when using a direct inversion of models, conditions like bi-causality, bi-stability, minimum phase of the models, etc. should be satisfied. Several approaches have therefore been proposed to minimize the number of these conditions which finally limited the usable models. In [START_REF] Fleming | Measurement and control for high-speed sub-atomic positioning in scanning probe microscopes[END_REF], Fleming and Leang propose the use of an optimal equivalent inverse model as an approximate feedforward dynamics controller (compensator). In our previous works, an inverse multiplicative structure was proposed as inverse model for nonlinearity compensator in piezoelectric actuators [START_REF] Rakotondrabe | Bouc-Wen modeling and inverse multiplicative structure to compensate hysteresis nonlinearity in piezoelectric actuators[END_REF] [17] and for estimating signal in a compliant platform [START_REF] Rakotondrabe | Presentation, Force Estimation and Control of an Instrumented platform dedicated to Automated Micromanipulation Tasks[END_REF]. Inverse multiplicative structure does not require specific conditions, contrary to direct inversion of models. In addition, no computation is required to derive the inverse model. Indeed, as soon as the (direct) model is identified or given, the inverse model is yielded without calculation. This is because the inverse model uses a structure, called inverse multiplicative structure, that contains the initial (direct) model. For these reasons, we propose in this paper to impose the structure of the transfer ŷ(s) ŷfree s (s) = ∆(s) (the dynamics observer) as inverse multiplicative structure. Let Fig. 5 be the dynamics observer with the proposed structure, where G 1 (s), G 2 (s) and G 3 (s) are the observer gains to be synthesized. As said before, these gains are directly based on a direct model and no additional computation is required to determine them.

1 ( ) G s 2 ( ) G s 3 ( ) G s ˆ( ) free s y s ˆ( ) y s ( ) s ∆ 1 + -
Fig. 5: Dynamics observer with inverse multiplicative structure.

From Fig. 5, we have:

ŷ(s) ŷfree s (s) = G 1 (s) G 2 (s) + G 3 (s) (22)
Using (Eq. 21) and (Eq. 22), we yield the relation between complete estimate information ŷ(s) and input control U :

ŷ(s) U (s) = C r α - 1 α R f p s - k DA α (1 + τ DA s) - 1 α H G 1 G 2 + G 3 (23)
Using the first equation in (Eq. [START_REF] Dubra | Preisach classical and nonlinear modeling of hysteresis in piezoceramic deformable mirrors[END_REF]) and (Eq. 23), we have:

G 1 (s) G 2 (s) + G 3 (s) = d p D(s) Cr α -1 α R f p s - k DA α(1+τ DA s) -1 α H(s) (24)
We can therefore define the observer gains as follows:

           G 1 (s) = d p D(s) G 2 (s) = - 1 α R f p s - k DA α (1 + τ DA s) - 1 α H(s) G 3 (s) = C r α (25)
As we can remark, there is no need of dynamics inversion in the observer gains. The only gain to be inverted is G 3 (see Fig. 5) and this is a non-null constant (G 3 = Cr α = 0) and is always invertible. Also, there is no additional computation or algorithm (such as least-square, etc.) required for the gains. As soon as the electrical and piezoelectric models are known and identified, the observer is directly obtained.

D. Governing equations and block schemes of the complete observer

Gathering the static force and displacement observer in (Eq. 5), the dynamics observer in (Eq. 22) and the observer gains in (Eq. 25), the governing equations of the complete observer are:

                                                   Fs (t) = 1 β   (C r U (t) -CU o (t)) -F cr (t) -F hys (t) - 1 R f p t 0 U dt   ŷfree s (t) = 1 α   (C r U (t) -CU o (t)) -Q DA (t, U ) - 1 R f p t 0 U dt   ŷ(s) = G 1 (s) G 2 (s) + G 3 (s) ŷfree s (s) -s p Fs (s) (26)
with (according to (Eq. 3), (Eq. 6) and (Eq. 25)):

                               F cr (s) = F tf cr (s)U (s) F hys (s) = Γ (U ) Q DA (s) = k DA (1 + τ DA s) U (s) = Q tf DA (s)U (s) G 1 (s) = d p D(s) G 2 (s) = - 1 α R f p s - k DA α (1 + τ DA s) - 1 α H(s) G 3 (s) = C r α (27)
Fig. 6 presents the block diagram of the complete observer defined by (Eq. 26).

In the proposed observer (Eq. 26), we choose the hysteresis operator Γ(U ) to be a Prandtl-Ishlinskii operator. The main advantage of this hysteresis operator are: i) its high accuracy, and ii) the fact that it is more suitable for real-time implementation.. The Prandtl-Ishlinskii hysteresis operator is based on the superposition of several elementary hysteresis called backlash or play-operator defined by:

y(t) = max {u(t) -r, min {u(t) + r, y(t -T s )}} y(0) = y 0 ( 28
)
where r is the threshold. A more complex hysteresis based on n h number of backlashes is therefore defined as:

     y(t) = n h i=1 w i • max {U (t) -r i , min {U (t) + r i , y i (t -T s )}} y(0) = y 0 (29
) where w i , r i and y i are the weighting and the threshold parameters and the output of the i th backlash, respectively. The creep operator F tf cr (s) in the observer is a transfer function (linear operator):

F tf cr (s) = m k=0 b k s k n l=0 a l s l (30)
where b k and a l are the parameters to be identified and m and n (m ≤ n) are the degrees of the polynomials. The operator's order is equal to n.

E. Experimental results and setup

The complete self-sensing was implemented and tested. The setup used, presented in Fig. 7, is composed of:

• a unimorph piezoelectric cantilever with dimensions of 15mm × 2mm × 0.3mm, where the thickness of the piezoelectric layer (lead zirconate titanate -PZT-5H) and the thickness of the passive layer (Nickel) are 0.2mm and 0.1mm respectively, • a dSPACE-board (DS1103) and a computer material for the data acquisition, observer implementation and control signal. MATLAB-SIMULINK is the software used for that.

Based on the dynamics of the actuator, the sampling time is set equal to T s = 50µs. • a displacement optical sensor (LC-2420 from Keyence,

[33]) to measure the deflection (displacement) at the tip of the actuator. This sensor is set to have a resolution of some tens of nanometers and an accuracy of 100nm, • a force sensor (FT-S270 from femtotools [34]) to measure the force applied by the actuator at its tip. The sensor has a range of ±2mN , a sensitivity of 1 µN mV and a resolution of 0.4µN .

• a custom-made electronic circuit based on Fig. 2-b. The used op-amp (OPA111) features very low bias currents (1pA) and a low noise, • and a high voltage (HV) amplifier with differential output (KH7602M from Krohn-Hite) used to amplify the input voltage U from the dSPACE-/computer. This amplifier can provide up to ±200V at its output.

It is noticed that the displacement and the force sensors are used here to capture the real displacement y and the real force F in order to compare them with the estimate ŷ and F for the validation of the proposed approach. They are also used to identify some model parameters such as d p D(s).

The tip of the piezoelectric cantilevered actuator is positioned between the optical sensor head and the probe of the force sensor. The force sensor is mounted on a XYZ mechanical table which will bring it on close vicinity to the actuator's tip. Force may be applied either externally by adjusting the sensor position (e.g. pushing it against the cantilevered actuator), or through the cantilever subject to an applied voltage, which will solely enter in contact with the sensor, thus simulating a micromanipulation procedure. This second method permits to have a very high dynamics force, simulating a step micromanipulation force when applying a step voltage to the actuator entering in contact with an object. This paper presents the estimation of the final value of this step force, additionally to the bending of the actuator.

To identify the hysteresis Γ(U ), we choose a number of backlashes n h = 15. Then the parameters have been identified following the procedure proposed in [START_REF] Rakotondrabe | Complete open loop control of hysteretic, creeped and oscillating piezoelectric cantilevers[END_REF]. The creep operator was identified based also on the procedure in [START_REF] Rakotondrabe | Complete open loop control of hysteretic, creeped and oscillating piezoelectric cantilevers[END_REF].

The remaining parameters in (Eq. 26) are identified following the procedures in [START_REF] Ivan | Quasi-static displacement self-sensing method for cantilevered piezoelectric actuators[END_REF] [START_REF] Ivan | Current integration force and displacement self-sensing method for cantilevered piezoelectric actuators[END_REF]. The electrical elements are: C = 47nF and C r = 8.2nF . Finally for the given actuator, we identified and calculated k DA = -0.028µm/V , τ DA = 60s, α = 273mV /µm, β = 1.03nC/mN and R f p = 0.435T Ω. The piezoelectric compliance s p has been identified by using the above mentionned external force sensor and displacement optical sensor. The force sensor is used to push the tip of the piezoelectric cantilever until a constant force F is obtained and the resulting steady-state displacement is measured by the optical sensor. Different values of force ranging between 0mN and 2mN were used. The results are given in Fig. 8 where we can notice the linearity of the force-displacement relation. From experimental data shown in the figure, a linear regression allows us to identify an average of s p = 1.2858µm/mN with a standard deviation of 0.08867µm/mN .

The complete observer with the identified parameters has been implemented. The first experiment consists in comparing the quasi-static displacement self-sensing presented in section. II-B with the proposed complete displacement selfsensing in free condition F = 0. In such a way, we can observe the efficiency of the complete self-sensing to track high frequency or high rate evolution of the displacement. The experiment consists in applying a step input voltage U = 20V to the actuator and to report the resulting real displacement y thanks to the optical sensor. At the same time, the estimate displacement from the quasi-static displacement self-sensing and from the proposed complete displacement self-sensing are also reported. Then, parts of the displacement step response are observed. The observation is performed within only 200ms because of the low settling time of the piezoelectric cantilever. Fig. 9 present the results. As we can see, the quasi-static self-sensing does not estimate the transient part of the real displacement (Fig. 9-a) and only estimates the steady-state. In contrary , the complete self-sensing captures both transient part and the steady-state value (Fig. 9-b).

The next experiment consists in applying a series of force at the tip of the piezoelectric actuator thanks to the probe of the force sensor and thanks to a manual positioning table used to move the sensor towards the actuator. The reaction force of the actuator which is F is the force to be observed/estimated by the self-sensing. At the same time, the displacement y (deflection of the actuator) generated by the force is also estimated by the self-sensing. The aim of this experiment is to validate and demonstrate the efficiency of the proposed self-sensing to estimate both displacement and force. As the force is estimated in a quasti-static way, the measurement here is performed during several tens of seconds. Fig. 10 picture the results. From these figures, we remark that the estimate F derived by the sefl-sensing well captures the real force F measured by the sensor (Fig. 10-a). Concerning the displacement, the estimate ŷ captures enough the real displacement y despite the slight inaccuracy of ∼ = 200nm for 3.1µm of range. This inaccuracy is acceptable for the considered applications since it is very negligible (much less than the micron). Comparing the results in Fig. 9 and in Fig. 10, we notice that the accuracy of the proposed self-sensing is much more better when the force F is zero. The inaccuracy obtained when F is non-zero can be reduced by using a more precise model of the piezoelectric compliance s p . In this paper, we have proposed a linear model (see Fig. 8) but a high degree polynomial could be used in order to reduce this average error (standard deviation) and consequently to reduce the error of estimation of the force. It is worth to notice that the force and the displacement at the steady-state, pictured in Fig. 10-a and b respectively, are not constant. In fact this is due to the creep phenomenon that typifies piezoelectric ceramics based actuators. Such a drift is observed at low frequencies or when a constant input voltage is applied. Our experiments clearly show that the self-sensing can efficiently track the appearance of this creep. To summarize, the characterization demonstrates that the proposed self-sensing well estimates both the transient part (high frequency) and the steady-state (low frequency) of the displacement of the piezoelectric actuator. In the steadystate, we see that the error of displacement estimation is less than 1%. Concerning the force estimation, which is in the steady-state, the accuracy is slightly worse than that of the displacement: 6% of error over the period of observation (100s) and for an amplitude of (3mN ). Despite this error, this is compatible with microassembly or micromanipulation tasks where the displacement is feedback controlled and the force is only displayed.

IV. APPLICATION TO A ROBUST FEEDBACK CONTROL OF THE DISPLACEMENT

In this part, we apply the proposed complete self-sensing method in a feedback control: the estimate displacement from the self-sensing will be used as feedback signal, the estimate force being displayed. We use the the standard H ∞ technique to calculate the controller. Such technique has already demonstrated its efficiency to improve piezoelectric actuators performances in previous works [START_REF] Rakotondrabe | Quadrilateral modeling and robust control of a nonlinear piezoelectric cantilever[END_REF]. But, relative to these previous works, the measurement for feedback in this paper is from self-sensing technique, not from an external sensor. A. The system to be controlled Fig. 11 presents the system to be controlled. In this, the real displacement y, which is unavailable for measurement, is estimated by the previous self-sensing. This estimate ŷ is used for feedback. In parallel to the control of the displacement, the force F applied by the actuator to the environment (manipulated object) is estimated by the same self-sensing (cf. previous section). Therefore, during the displacement control, the manipulation force can be displayed in real time. It is worth to notice that the optical sensor is still used to measure the real displacement y and to compare the latter with the estimate during the control, and therefore for the validation.

B. Sepcifications for the closed-loop

To design the controller, we propose the following specifications. They are in accordance with the requirements in micropositioning tasks during micromanipulation and microassembly.

Tracking performances -The overshoot, which was initially 64% (see the step response of the system in Fig. 9), should be cancelled. The statical error should be negligible. Finally the settling time should be less than 100ms.

Limitation of the input voltage U -In order to avoid the destruction of the piezoelectric actuator by eventual overvoltage, the input control U should be limited. We choose a maximal ratio of Umax ymax = 100 20 V µm = 5 V µm where y max corresponds to the maximal range of use.

C. H ∞ controller synthesis

One of the main advantages of the H ∞ approach is the explicit consideration of the specifications. These specifications are transcribed into weighting functions during the synthesis. Consider in Fig. 12-a the closed-loop scheme for the controller design, where the weighting W 1 (s) and W 2 (s) are used to include the specified tracking performances and the specified voltage limitation respectively. In the figure:

• y r is the reference displacement, • ε = y r -y is the error, • e 1 and e 2 are the weighted outputs used for the synthesis, • G(s) is the system to be controlled, with G(s) = d p D(s),

• and K(s) is the controller to be synthesized. The standard scheme corresponding to Fig. 12-a is pictured in Fig. 12-b. This is used for the standard H ∞ synthesis. In this, P (s) is called augmented system since it contains the system G(s) augmented by the weightings W 1 (s) and W 2 (s). The exogenous input is the reference y r and the output to be controlled are the weighted signals e 1 and e 2 . The standard H ∞ problem consists in finding an optimal value γ > 0 and a controller K(s) stabilizing the closed-loop in Fig. 12-b (the standard scheme) and guaranteeing the following inequality [START_REF] Balas | µ-analysis and synthesis toolbox[END_REF]:

F l (P (s), K(s)) ∞ < γ (33)
where F l (P (s), K(s)) is the lower linear fractionar transformation (LFT) between P (s) and K(s) and is defined by:

e 1 e 2 = F l (P (s), K(s)) • y r (34)
From Fig. 12-a, we have:

e 1 = W 1 Sy r ; e 2 = W 2 KSy r (35)
where S = (1 + KG) -1 is the sensitivity function. Using (Eq. 34) and (Eq. 35), we have the LFT of the standard scheme:

F l (P (s), K(s)) = W 1 S W 2 KS (36)
Using the condition of the H ∞ standard in (Ineq. 33) and the LFT in (Eq. 36), we infer the more detailed conditions:

W 1 S ∞ < γ ; W 2 KS ∞ < γ (37)
which is also respected if we have

|W 1 | < γ |W1| ; |KS| < γ |W2| (38)

D. Computation of the controller

The weighting functions were chosen according to the specifications in Section. IV-B. We choose the following set:

W 1 (s) = 0.03s+1 s+0.001 ; W 2 = 2×10 -5 s+1 6.25×10 -5 s+5 (39)
After computation, we obtain a controller of order 5:

         K(s) =
163 s + 8 × 10 4 (s + 3876) s 2 + 54s + 1.36 × 10 7 s + 5 × 10 4 (s + 4137) (s + 0.001) s 2 + 519s + 1.4 × 10 7 γ optimal = 0.2 (40) γ being less than 1 indicates that the controller will ensure the performances described in the specifications.

( ) K s ( ) K s 1 ( ) W s 1 ( ) W s 2 ( ) W s 2 ( ) W s ( ) G s ( ) P s

E. Experimental results

The controller K(s) in (Ineq. 40) has been implemented in the MATLAB-SIMULINK software. The first experiment consists in applying a series of steps for the reference input y r to the closed-loop system. No force sensor (or object) is in contact with the actuator in this first experiment such that F = 0. The aim is to just validate the displacement tracking performances. Both the estimate displacement ŷ that was used for the feedback and the real displacement y that is measured thanks to the optical sensor are captured. Fig. 13 pictures the results. As we can see, the estimate displacement ŷ well coincides with the real displacement y. On the other hand, they well tracks the input reference y r . The specifications are particularly respected: there is no overshoot, the statical error tends towards zero and the settling time is less than 100ms.

In the second experiment, we apply arbitrary levels of reference input, but still with F = 0. The objective is to evaluate the input voltage U and the ability of the complete self-sensing to work in long term measurement (over a period of 200s) of displacement signal and with high voltage. Fig. 14a pictures the reference displacement y ref , the measured displacement y from the external optical sensor and the estimate displacement ŷ from the self-sensing. Results shown in these figures demonstrate the efficiency of the control techniques using the self-sensing, where it can be verified that the output of the piezoelectric actuator well tracks the reference input y ref . Fig. 14-b shows the high input voltage U (up to 100V ) applied by the controller K(s) to the actuator. Fig. 14-c shows a zoom on the displacement response at time between 168s and 187s where the reference input displacement is y r = 0µm. From this latter figure, we observe that the signal obtained from the self-sensing technique is less noisy than that provided by the optical sensor. Although the reference y r is null and the output y (and its estimate ŷ) is null, we remark from Fig. 14c and Fig. 14-b that the corresponding driving voltage U is non-null (about -10V ). This is due to the fact that this driving voltage is compensating for the nonlinearities (hysteresis and creep) of the piezoelectric actuator which demonstrates the efficiency of the designed controller. The last experiment consists in performing a tracking (tracking of y r ) when the actuator is in contact with the force sensor (F = 0). During the experiment, the reference displacement y r is modified slightly in order to avoid brusque evolution that may destroy the force sensor. Fig. 15 picture the results. In Fig. 15-a where the displacement ŷ is shown, the values are shifted of 50µm. In Fig. 15-b where the real force F is shown, the values are shifted of -0.3mN . The noises arrowed on the curve are due to a slight sliding between force sensor's probe and the actuator. Finally in Fig. 15-c where the estimate force F is shown, the values are shifted of about 1mN . As we can see in the two latter figures, the estimate force well estimates the real force, in particular when the contact is reached. When the contact is off, there is a slight drift of the estimate force (Fig. 15-c). This drift causes a shift of the estimate force relative to the real force during contact. This drift, which can be positive or negative, is due to the sensitivity of the self-sensing to the environmental conditions (temperature variation,...) but also in part due to the model of the compliance s p . This drift can be minimized by utilizing a controlled environment for the experimental setup. Another approach consists in employing an environment-dependent self-sensing technique. In this, the parameters of the selfsensing are dependent on the evolution of the temperature. Finally, a more precise model of the compliance s p could also increase the accuracy of the force estimation.

V. CONCLUSION

This paper has presented the development of a self-sensing technique, called complete self-sensing, that is able to estimate the force and the displacement in a piezoelectric actuators dedicated to micropositioning tasks. While the force is estimated at its steady-state (static), the displacement is estimated at both its transient part (dynamics) and its steady-state. The advantage is that it is possible to develop a feedback displacement control and at the same time to display the force applied by the actuator. Another advantage relative to existing self-sensing techniques is that the proposed approach can capture the two signals for long term (several hundred of seconds). The experiments carried on a feedback control have demonstrated these advantages.

The accuracy of self-sensing techniques depends on the accuracy of the identified parameters. If they are badly identified, the self-sensing will provide inaccurate estimate of the displacement or force. These parameters may also change over the time, for instance due to thermal variation.

Fig. 1 :

 1 Fig. 1: (a) and (b), principle of a unimorph piezoelectric cantilevered actuator. (b) a microgripper with two piezoelectric cantilevered actuators. (c) simultaneous estimation of the displacement and of the manipulation force by using self-sensing.

  )where k DA is the static gain, τ DA is the constant time and s is the Laplace variable. Expression Q tf DA (s) = k DA (1+τ DA s) is the transfer function of the dielectric absorption.

Fig. 2 :

 2 Fig. 2: (a) Principle scheme of the quasi-static displacement self-sensing. (b) the electrical circuit [25].

Fig. 3 :

 3 Fig. 3: Principle scheme of the quasi-static displacement and force self-sensing.

  Fig. 4: Principle scheme of the complete displacement and force self-sensing.

Fig. 6 :

 6 Fig. 6: Block diagram of the complete observer.

Fig. 7 :Fig. 8 :

 78 Fig. 7: Setup used for the experimental tests. The identification of d p and D(s) is performed by applying a step voltage U to the actuator without force at the tip and then by capturing the output y thanks to the optical sensor. After applying an ARMAX (Auto Regressive model with eXternal input) method to the captured data with MATLAB [38], we obtain:           

Fig. 9 :

 9 Fig. 9: Experimental results. (a): displacement estimation (with force null) using the quasi-static self-sensing in [25] [26]. (b) results using the proposed complete displacement self-sensing.

Fig. 10 :

 10 Fig. 10: Experimental results. (a): force F measured by an external force sensor and estimate force F from the selfsensing. (b): displacement y measured by an external optical sensor and estimate displacement ŷ from the self-sensing. To solve the problem (Ineq. 38) (or (Ineq. 37)), we use the Glover-Doyle algorithm which is based on the Riccati equations [36] [37].

Fig. 11 :

 11 Fig. 11: The system to be controlled.

Fig. 12 :

 12 Fig. 12: (a): the closed-loop scheme augmented with the weightings. (b): the corresponding standard scheme.

Fig. 13 :

 13 Fig. 13: Step response of the closed-loop system with F = 0.

Fig. 14 :

 14 Fig. 14: Long term experiment with F = 0 and with high input voltage U . (a): the reference displacement y ref and the output displacement measured by the optical sensor (y) and estimated (ŷ). (b): the driving voltage U . (c): a zoom on the displacement when the reference input y r = 0µm.

Fig. 15 :

 15 Fig. 15: Experimental result on tracking of a reference displacement y r with F = 0 (the actuator is in contact with an object). (a): the estimate displacement ŷ. (b): the measured (real) force F . (c): the estimate force from the self-sensing measurement technique.


Hussein Hussein
email: hussein.hussein@femto-st.fr

Patrice Moal

Gilles Bourbon
email: gilles.bourbon@femto-st.fr

Yassine Haddab
email: yassine.haddab@femto-st.fr

Philippe Lutz
email: philippe.lutz@femto-st.fr

Patrice Le Moal
email: patrice.lemoal@femto-st.fr

Modeling and Stress Analysis of a Pre-Shaped Curved Beam : Influence of High Modes of Buckling

Keywords: Pre-shaped curved beam, buckling, bistable mechanism, stress state, MEMS

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

In microsystems, numerous applications can benefit from bistability, such as switches, valves, relays, positioners [START_REF] Chen | Synthesis of compliant multistable mechanisms through use of a single bistable mechanism[END_REF], braille displays [START_REF] Niu | Bistable electroactive polymer for refreshable braille display with improved actuation stability[END_REF], reconfigurable robotic devices [START_REF] Hafez | Optimized binary modular reconfigurable robotic devices[END_REF] and digital microrobotics. [START_REF] Chen | Digital microrobotics based on bistable modules: Design of compliant bistable structures[END_REF][START_REF] Chen | Characterization and control of a monolithically fabricated bistable module for microrobotic applications[END_REF][START_REF] Chalvet | Digital microrobotics based on bistable modules: Design of a non-redundant digital micropositioning robot[END_REF][START_REF] Chalvet | A microfabricated planar digital microrobot for precise positioning based on bistable modules[END_REF]. Bistable structures are generally simple and allow assigning a binary state to every reachable position.

The stability of a bistable system can be ensured either by active or passive maintain. In active maintain [START_REF] Fukuta | Design, fabrication, and control of mems-based actuator arrays for air-flow distributed micromanipulation[END_REF][START_REF] Abadie | Modeling of a new sma micro-actuator for active endoscopy applications[END_REF][START_REF] Driesen | Nickel-plated thermal switch with electrostatic latch[END_REF], the mechanism needs some external energy to keep its place, while 1550055-1 it is not the case in passive maintain where the mechanism is inherently stable at each position. Generally, passive maintain has many advantages such as low energy consumption due to short actuation time and self-stability which allows improving the repeatability with an open loop control. Many passive maintain examples can be found in literature, we can mention magnetic maintain [START_REF] Zhang | Preparing of a high speed bistable electromagnetic rf mems switch[END_REF][START_REF] Wu | Design and implementation of a bistable microcantilever actuator for magnetostatic latching relay[END_REF][START_REF] Barth | A bistable magnetically enhanced shape memory microactuator with high blocking forces[END_REF], bistable electro active polymers [START_REF] Yu | Bistable electroactive polymers (BSEP): Large-strain actuation of rigid polymers[END_REF], and compliant mechanisms [START_REF] Oberhammer | Mechanically tri-stable, true single-pole-double-throw (spdt) switches[END_REF]Chen et al., 2009a[START_REF] Yu | Bistable electroactive polymers (BSEP): Large-strain actuation of rigid polymers[END_REF][START_REF] Huang | A mems bistable device with push-on-push-off capability[END_REF].

Further, compliant mechanisms exhibit many advantages such as increased precision and reliability, no friction, reduced wear and low manufacturing costs.

Curved beam can be considered as a bistable system that combines the advantages of passive maintain and compliant mechanisms.

In this paper, we develop the analytical solution of the snapping forces and stresses with considering high modes of buckling. The impact of high modes of buckling on the snapping forces, the bistable behavior and the stress state is discussed based on a mathematical approach.

Bistable curved beam designs are still paying the price for the lack of physical intuition by resorting to the FEM modeling, where there is a difficulty to consider all complex and nonlinear effects. Euler elastica theory was the starting point to study the beam buckling for large deformation. Based on Lagrangian approach, [START_REF] Vangbo | An analytical analysis of a compressed bistable buckled beam[END_REF] carried out one of the first studies on pre-compressed curved beams that takes compressibility into account for small deformations. The obtained expressions consider high modes of buckling. Buckling behavior of a suspended microbeam as a result of electrothermal expansion is investigated by [START_REF] Chiao | Self-buckling of micromachined beams under resistive heating[END_REF]. Emam and Nayfeh examine in their studies [START_REF] Emam | On the nonlinear dynamics of a buckled beam subjected to a primary-resonance excitation[END_REF][START_REF] Nayfeh | Exact solution and stability of postbuckling configurations of beams[END_REF] the vibration and dynamics of postbuckling configurations of a beam. [START_REF] Park | Pre-shaped buckled-beam actuators: Theory and experiments[END_REF] demonstrate that bistability of buckled beams depends not only on a ratio of its initial rise to its thickness, but also on its residual stress. [START_REF] Cazottes | Actionnement des systeèmes bistables: modélisation et études expérimentales[END_REF] in his thesis has studied the bistability of a pre-compressed curved beam when it is actuated either by force or by moment. Elastica models for static and dynamic analysis with solutions and experiments are given by [START_REF] Camescasse | Actionnements statique et dynamique dun mecanisme bistable: aspects modélisation, conception et expérimental[END_REF] in his thesis. Chen et al. [START_REF] Chen | Exact snapping loads of a buckled beam under a midpoint force[END_REF][START_REF] Chen | Static snapping load of a hinged extensible elastica[END_REF], carried out one of the last studies on extensible elastica theory of a curved beam.

Concerning the case of a pre-shaped curved beam, [START_REF] Qiu | A curved-beam bistable mechanism[END_REF] have calculated analytically the snapping forces by neglecting high modes of buckling and they provide an approximation of the solution with high modes for high values of the initial height-to-thickness ratio. Their analytical results are approved by FEM simulations and experiments.

This paper recalls in a first stage the solution of the snapping forces of a preshaped curved beam with clamped-clamped boundary conditions. The solution is obtained in [START_REF] Qiu | A curved-beam bistable mechanism[END_REF] using an approximation of neglecting high modes of buckling.

1550055-2

In Sec. 3, we develop analytically the solution of the snapping forces with considering high modes of buckling. The analytical expressions obtained with high modes implicitly include the effects of all modes of buckling on the curved beam behavior. Existence and magnitude of bistability are discussed with respect to the snapping forces evolution.

In Sec. 4, evolution of stresses inside the curved beam is calculated analytically with and without the approximation of high modes. Calculating the stress state is important for the design, particularly for miniaturization and optimization purposes under elastic or failure limits.

Finally, the analytical results with and without high modes of buckling are compared with FEM simulations. This comparison shows the interest to consider high modes of buckling in the modeling of the curved beam behavior.

Buckling of a Beam

Buckled beam model

Buckling a beam is defined as a sudden deformation which occurs when the excess of compression energy stored in the beam is converted into bending energy. In other words, transverse deflection occurs when the compressive force P exceeds a critical value P 0 , the beam enters in the first buckling mode (Fig. 1).

The postbuckling configuration of the beam can be considered as a compliant mechanism that shows bistability. The buckled beam shows stability of its position at two possible configurations which are symmetrically distributed in the two buckling sides (Fig. 2).

1550055-3

In most cases, and in this paper, the action can be a lateral force applied in the middle of the beam. Otherwise, the action can be also a force applied in different points [START_REF] Camescasse | Actionnements statique et dynamique dun mecanisme bistable: aspects modélisation, conception et expérimental[END_REF], an electromagnetic field [START_REF] Park | Pre-shaped buckled-beam actuators: Theory and experiments[END_REF], an electrostatic field [START_REF] Krylov | Dynamic stability of electrostatically actuated initially curved shallow micro beams[END_REF][START_REF] Tajaddodianfar | Dynamics of bistable initially curved shallow microbeams: Effects of the electrostatic fringing fields[END_REF] or moments applied in determined locations on the beam [START_REF] Stoimenov | Manufacturing of ionic polymermetal composites (IPMCs) that can actuate into complex curves[END_REF][START_REF] Cazottes | Actionnement des systeèmes bistables: modélisation et études expérimentales[END_REF].

There are two possible approaches to deal with the post-buckling problem, based on static and dynamic models. In dynamic modeling, there are two types of modes, buckling modes that depend on axial stress and resonance modes that depend on the system frequency. In contrast, since it does not consider the time, static modeling exhibits only buckling modes.

Generally, the resonance frequency of a clamped-clamped pre-shaped curved beam is higher while reducing the beam dimensions. MEMS devices generally range in size from 20 µm to few millimeters. An interest to dynamic modeling can be referred to studies in [START_REF] Emam | On the nonlinear dynamics of a buckled beam subjected to a primary-resonance excitation[END_REF], [START_REF] Nayfeh | Exact solution and stability of postbuckling configurations of beams[END_REF], [START_REF] Cazottes | Actionnement des systeèmes bistables: modélisation et études expérimentales[END_REF], [START_REF] Camescasse | Actionnements statique et dynamique dun mecanisme bistable: aspects modélisation, conception et expérimental[END_REF], [START_REF] Tajaddodianfar | Dynamics of bistable initially curved shallow microbeams: Effects of the electrostatic fringing fields[END_REF]. In this study, we investigate the static modeling.

Curved beams can be divided on the basis of the fabrication process to three kinds [START_REF] Zaidi | Méthode d'apport d'énergie sélective et sans contact: applications aux micro-actionneurs bistables[END_REF]:

• Pre-compressed curved beams where the beam is fabricated rectilinear then it is compressed [START_REF] Vangbo | An analytical analysis of a compressed bistable buckled beam[END_REF][START_REF] Cazottes | Actionnement des systeèmes bistables: modélisation et études expérimentales[END_REF][START_REF] Camescasse | Actionnements statique et dynamique dun mecanisme bistable: aspects modélisation, conception et expérimental[END_REF]. • Pre-stressed curved beams where during fabrication, the beam is buckled due to residual stress which is added by heating or oxidation [START_REF] Pane | Investigation on bistability and fabrication of bistable prestressed curved beam[END_REF]]. • Pre-shaped curved beams where the beam is directly fabricated at the first buckling shape mode without residual stresses [START_REF] Qiu | A centrally-clamped parallel-beam bistable mems mechanism[END_REF].

The behavior of the pre-compressed curved beam is symmetrical between the two sides of buckling, in terms of the snapping force, beam shape and stresses evolution. However, in the case of microfabrication, the monolithic constraint makes this solution difficult to use. The pre-stressed curved beam makes the integration into a monolithic device easier, but the residual stress is difficult to control by fabrication [START_REF] Zaidi | Méthode d'apport d'énergie sélective et sans contact: applications aux micro-actionneurs bistables[END_REF]. However, for the pre-shaped curved beam, despite the fact that symmetry is lost and that bistability exists under some conditions, fabrication and integration are simpler. Some solutions are presented in the literature to improve the symmetry property of pre-shaped curved beams [START_REF] Jensen | Design optimization of a fully-compliant bistable micro-mechanism[END_REF][START_REF] Qiu | A curved-beam bistable mechanism[END_REF].

In this paper, we investigate the modeling of a pre-shaped curved beam fabricated directly at the first buckling shape mode. The model of the curved beam presented in Fig. 3 has the following characteristics: axial force P , thickness t, depth b, span l, deflection d, beam shape w(x) and applied lateral force in the middle of span f . Moreover, h is the height of the fabricated curved beam at x = l 2 .

1550055-4 Fig. 3. Clamped-clamped curved bistable beam at initial position and after deflection.

Buckling equation

Based on small deformation hypothesis, [START_REF] Timoshenko | Theory of Elastic Stability[END_REF] gives the buckling equation of a curved beam:

d 2 dx 2 EI d 2 w dx 2 + P d 2 w dx 2 = 0, ( 1
)
where E is the Young's modulus and I = z 2 dydz is the quadratic moment. Considering a uniform beam in terms of material properties and sections:

EI d 4 w dx 4 + P d 2 w dx 2 = 0. (2)
Introducing the clamped-clamped boundary conditions implies the modal nature of the solution. The solution is an infinite sum of buckling modes:

w(x) = ∞ j=1 a j w j (x), (3)
where a j is the jth constant mode which reflects the contribution of each buckling mode in the total solution, and w j (x) is the jth buckling shape mode:

w j = 1 -cos N j x l , N j = (j + 1)π, j = 1, 3, 5, . . . , w j = 1 -2 x l -cos N j x l + 2 sin Nj x l Nj , N j = 2.86π, 4.92π, 6.94π, 8.95π, . . ., j = 2, 4, 6, . . . , (4)
where N j is the jth mode of the normalized axial force N and N = P l 2 EI .

Snapping forces without high modes of buckling

The approximation of neglecting high modes of buckling simplifies the calculation [START_REF] Qiu | A curved-beam bistable mechanism[END_REF]. Normalization is taken in order to simplify the solution:

X = x l ; W j (X) = w j (x); W (X) = w(x) h = ∞ j=1 A j W j (X); F = f l 3 EIh ; ∆= d h .
(5)

1550055-5 The variational calculation in [START_REF] Qiu | A curved-beam bistable mechanism[END_REF] leads to three kinds of solutions. The first kind is when the curved beam is in the compressible phase:

       F = F 1 , N < N 2 mode 2 is not constrained, N 3 mode 2 is constrained, A j = 0; j = 1, 5, 9, 13, . . . . (6)
The second kind is when N reaches N 2 without mechanical constraints:

   F = F 2 , N = N 2 , A j = 0; j = 1, 2, 5, 9, 13, . . . . (7)
The third kind is when mode 2 is constrained and

N reaches N 3 :    F = F 3 , N = N 3 , A j = 0; j = 1, 3, 5, 9, 13, . . . . (8)
Mode 2 is constrained by connecting two curved beams in the middle as shown in Fig. 4. Otherwise, for a simple curved beam without constraints, the stability margin for the second stable position is too low. In this paper, we are interested only to the first and third kinds of solutions. The results for the second kind are similar to those of the third kind in calculation.

The terms A j , for j = 1, 5, 9, 13, . . . have the following values:

A 1 = - 1 2 N 2 1 N 2 -N 2 1 + 4F N 2 1 (N 2 -N 2 1 ) , ( 9
)
A j = 4F N 2 j (N 2 -N 2 j ) for j = 5, 9, 13, . . . . ( 10
)
Two characteristics equations that describe the curved beam behavior are derived from the variational calculation:

∆ = 1 -2 j=1,5,9,... A j , ( 11
)
N 2 12Q 2 = N 2 1 16 - ∞ j=1 A 2 j N 2 j 4 , ( 12
)
where Q is the height-to-thickness ratio Q = h/t.

1550055-6

The following equations summarize the first and third kinds of solution the snapping forces that are calculated in [START_REF] Qiu | A curved-beam bistable mechanism[END_REF] without high modes of buckling. For the first kind:

             F 1 = 3π 4 Q 2 2 ∆ ∆ 2 -3∆ + 2 + 4 3Q 2 , N 2 = 3π 2 Q 2 -∆ 2 + 2∆ , W (X) = A 1 W 1 (X); A 1 = 1 -∆ 2 . ( 13
)
For the third kind:

                   F 3 = 6π 4 4 3 -∆ , N = N 3 , W (X) = A 1 W 1 (X) + A 3 W 3 (X), A 1 = 1 -∆ 2 ; A 2 3 = - 1 16 ∆ 2 -2∆ + 16 3Q 2 . ( 14
)

Snapping Forces with High Modes of Buckling

In this section, we develop the solution of the snapping forces taking into account the high modes of buckling. The stability equilibrium is then analyzed in the two sides of buckling based on the snapping forces solution.

Snapping forces

Combining Eqs. ( 9) and ( 10) with ( 11) and ( 12), the two infinite series become:

j=1,5,9,...

A j = - 1 2 N 2 1 N 2 -N 2 1 + 4F • Sum 1 , ( 15
) ∞ j=1 A 2 j N 2 j = 1 4 N 2 1 (N 4 1 -16F ) (N 2 -N 2 1 ) 2 + 16F 2 • Sum 2 , ( 16
)
where Sum 1 and Sum 2 have the following shapes:

Sum 1 = ∞ j=1,5,9,... 1 N 2 j (N 2 -N 2 j ) , ( 17
) Sum 2 = ∞ j=1,5,9,... 1 N 2 j N 2 -N 2 j 2 . ( 18
)
Imposing j = 4k + 1, Sum 1 in ( 17) can be decomposed in two infinite sums:

1 4π 2 N 2 ∞ k=0 1 (2k + 1) 2 - ∞ k=0 1 (2k + 1) 2 -( N 2π ) 2 . ( 19
)
1550055-7

The first Sum in ( 19) is equal to:

∞ k=0 1 (2k + 1) 2 = π 2 8 . ( 20
)
The second Sum can be concluded from the following equation [START_REF] Remmert | Modes of convergence in function theory[END_REF]:

π tan π 2 x = ∞ k=0 4x (2k + 1) 2 -x 2 . ( 21
)
Then, Sum 1 is equal to:

Sum 1 = 1 8N 3 N 4 -tan N 4 . ( 22
)
Introducing ( 22) in ( 11), a new equation is derived:

F = N 3 N 4 -tan N 4 N 2 N 2 -4π 2 -∆ . ( 23
)
The sum in ( 18) can be obtained by deriving the sum in ( 17) with respect to N :

∂(Sum 1 ) ∂N = -2N • Sum 2 . ( 24
)
Then, Sum 2 is equivalent to:

Sum 2 = 3 64N 4 1 - tan( N 4 ) N 4 + tan 2 ( N 4 ) 3 . ( 25
)
Introducing ( 25) in ( 12), the following equation is obtained for the first kind of solution:

3 16N 4 1 + tan 2 N 4 3 - tan N 4 N 4 F 2 1 - 4π 2 (N 2 -4π 2 ) 2 F 1 + N 2 12Q 2 - π 2 N 2 N 2 -8π 2 4 (N 2 -4π 2 ) 2 = 0. (26)
Equations ( 23) and ( 26) are the characteristic equations which allow defining the relations between F , N and ∆ for the first kind of solution. In this case, the problem can be solved by numerical method. The idea is to change N in (26) from 0 to the point where there are no real solutions. Two values of F are obtained for each value of N . Introducing these values in (23), the relations N -∆ and F -∆ are obtained.

Figure 5 shows evolution of N in function of ∆ for different values of Q. Shapes of the curved beam during snapping between two sides of buckling are shown in Figure 5 with first, second and third kinds of solution. Noting that the normalized axial force N is equivalent to zero at (∆ = 0, F = 0) and at (∆ = 20/π 2 , F = 3840/π 2 ). The normalized displacement at the end of deflection is close to 2, but not exactly as it is for the pre-compressed beam.

1550055-8 Furthermore, Fig. 5 illustrates the values of Q providing the transitions between the first, second and third kinds of solution. In the first kind of solution, the maximum value of N that can be reached during snapping increases with increasing the value of Q. N exceeds N 1 only for:

Q ≥ 64π 2 117 -7π 2 ≈ 1.16. ( 27
)
Noting that at N = N 1 , the normalized force has a unique value F = 2π 4 . On the other side, the third kind of solution is simpler. Making N constant at N 3 = 4π, the third kind simplify the previous equations. Evolution of the force can be directly concluded from (23):

F 3 = 64π 2 4 3 -∆ . ( 28
)
Equation ( 28) exhibits a perfect linear interaction between force and displacement in the third kind of solutions. Figure 6 shows evolution of F in function of ∆ for different values of Q.

On the other side, the mode constant A 3 which appears in the third kind of solution is obtained by recalculating (26) without canceling A 3 , then introducing (23) in the new equation:

A 2 3 = - 3 4π 2 ∆ 2 + 14 9π 2 ∆ + 1 18 - 20 27π 2 - 1 3Q 2 . ( 29
)
Noting that the sign of A 3 changes with the direction of deflection.

1550055-9 In light of the equation of A 3 , the third mode cannot appear unless Q respects the following condition:

Q ≥ 162π 2 27π 2 + 32 ≈ 2.314. (30)
Further, ∆ top and ∆ bot (Fig. 6), which are the exact positions where the third mode appears, can be concluded from (29):

∆ top , ∆ bot = 28 27 ± 2π 3 1 6 + 16 81π 2 - 1 Q 2 . ( 31
)
The exact values of F top and F bot (Fig. 6) can be concluded simply by introducing the values of ∆ top and ∆ bot , respectively in (28).

Bistability conditions

Physically, the curves in Fig. 6 present the amount of the lateral force produced by the beam in the center point after displacement. In this context, the bistability that we look for is provided by the negative portions of F produced by the beam that will push to the other buckling side.

As we can conclude from Fig. 6, the value of the snapping force is not symmetric between the two sides of buckling. This comes from the bending energy which starts its evolution from the initial shape per fabrication.

The shift-up of the curves in Fig. 6 affects the bistability behavior. Mechanical conditions must be considered in order to involve bistability, while the greater margin of stability remains in the first side of buckling.

The stable positions are the points where F is equivalent to zero and the beam tends to return to its position after a small displacement. Thus, canceling F in (26) for the first kind of solution, three values of N are obtained. Putting these values 1550055-10 in ( 23), three values of ∆ are obtained:

∆ = 0, 3 2 ± 1 4 - 4 3Q 2 . ( 32
)
We conclude from (32) and Fig. 6 that the beam exhibits two stable positions, only for Q > 16 3 ≈ 2.31. The second ∆ is unstable position because every variation of its state will create a tendency to move away.

Moreover, due to mode 3, when Q ≥ √ 6 ≈ 2.45, there are three new values of ∆ where F is equal to zero:

∆ = 0, 4 3 , 3 2 + 1 4 - 4 3Q 2 . ( 33
)
The first and last ∆ values in (33) are the two stable points. In the other side, the ∆ positions which cancel F for the second kind of solution are:

∆ = 0, 1.96, 3 2 + 1 4 - 4 3Q 2 . ( 34
)
A curved beam where the unsymmetrical modes of buckling are not constrained (including mode 2) will never show bistability unless Q > 5.65. The bistability in this case is very limited and no important force is obtained in the second side of buckling.

Table 1 summarizes the conditions on Q in order to reach N 1 , N 2 , N 3 and the bistability feature. The normalized axial force N inside the curved beam reaches N 1 during deflection between the two sides of buckling only for Q > 1.16 and reaches N 2 for Q > 1.65 when mode 2 is not constrained and reaches N 3 for Q > 2.31 when mode 2 is constrained. The bistability exists only for Q > 5.65 when mode 2 is not constrained and for Q > 2.31 when mode 2 is constrained.

Stress State

In this section, axial and bending stresses are calculated with and without the approximation of high modes. This allows obtaining the evolution of the maximal total stress inside the curved beam during deflection.

Stresses without high modes

Stresses inside the beam are decomposed into axial stress and bending stress. Axial stress is constant along the beam and has a maximum when the deflection is around the middle while bending stress changes along the beam sections and increases as far as the deflection is closer to the other side. The axial stress p is equivalent to:

p = Et 2 12l 2 N 2 . ( 35
)
The bending stress T starts evolution from the initial shape:

T = Ez d 2 w dx 2 - d 2 w dx 2 , ( 36
)
where w(x) is the initial shape of the beam and z is the distance in the cross-section to the neutral line.

For the first kind of solution, axial stress is simply concluded from ( 13) and ( 35):

p = π 2 Eth l 2 Q 4 (-∆ 2 + 2∆). ( 37
)
Also, bending stress is obtained using ( 13) and ( 36):

T = 2π 2 Ezh l 2 ∆ cos 2π x l . ( 38
)
The same for the third kind of solution, axial and bending stresses are calculated from ( 14), ( 35) and ( 36):

p = π 2 Eth l 2 4 3Q , ( 39
)
T = 2π 2 Ezh l 2 ∆ cos 2π x l -2 -∆ 2 + 2∆ - 16 3Q 2 cos 4π x l . ( 40
)
The total stress inside the beam is simply the sum of the axial and bending stresses. However, analyzing ( 38) and ( 40), the extremum of the bending stress is noticed at the midpoint x = l 2 and at boundaries x = [0, l] when the first kind of solution is present and only at the middle when mode 3 is present. Also, the stress is maximized when z is at the limits z = | t 2 |. Then, putting these values in the stress equations, the absolute value of the maximal total stress during deflection can be written as follows, for the first kind of solution:

σ ∆ max = π 2 Eth l 2 - Q 4 ∆ 2 + 1 + Q 2 ∆ . ( 41
)
And for the third kind of solution:

σ ∆ max = π 2 Eth l 2 ∆ + 2 -∆ 2 + 2∆ - 16 3Q 2 + 4 3Q . ( 42
)
Analyzing the last two equations, we notice the presence of three different forms of evolution curves for the maximal stress during deflection, as shown in Fig. 7. The first form of stress is when only the first kind of solution exists. The second form is when the third kind of solution appears during deflection. The third form is when the maximal stress point is higher in the third kind of solution domain.

1550055-12

The first form exists for Q < q 16 3 ≈ 2.31, while the third one appears for Q > 2

q 31 3 -4 √ 5 ≈ 2.36.
The second form of stress exists between the last two values of Q. Noting that there is a small difference between the last two values of Q, which means that the second form of stress is a rare case.

The maximal stress point in the first two forms is at ∆ = 2 when Q ≤ 2 and at ∆ = 1 + 2/Q when Q ≥ 2. Thus, for Q ≤ 2, the maximal stress σ max is equivalent to:

σ max = 2π 2 Eth l 2 . ( 43
)
When Q is between [2; 2.36], σ max is equivalent to:

σ max = π 2 Eth l 2 1 + 1 Q + Q 4 . ( 44
)
In the third form, the maximal stress point is at:

∆ = 1 + 1 √ 5 1 - 16 3Q 2 . ( 45
)
Introducing this value in (42), for Q > 2.36, σ max is equivalent to:

σ max = π 2 Eth l 2 1 + 4 3Q + √ 5 1 - 16 3Q 2 . ( 46
)
1550055-13

H. Hussein et al.

Noting that the maximum of σ max in the last form is for Q = 16/ √ 3. However, based on the previous equations, σ max is ranging between: [2.01; 3.31[ Q ∈ [2.36; ∞[. (47) These values of σ max in (47) are calculated when ∆ is ranging between 0 and 2. Although, the end of deflection can be considered at the second stable position that corresponds to a transversal displacement ∆ lower than 2. In this case, the new range of σ max is as follows:

σ max = π 2 Eth l 2 •        2 Q < 2, [2; 2.01] Q ∈ [2; 2.36[,
σ max = π 2 Eth l 2 • [1.93; 1.97] Q ∈ [2.31; 2.34], [1.97; 3.31[ Q ∈ [2.34; ∞[. (48)

Stresses with high modes

In the other side, the problem with high modes is complex and hard to handle without approximations. Difficulty lies in the fact that the maximal stress point is difficult to determine analytically. Axial stress (35) remains the same, while bending stress changes with the consideration of high modes. Bending stress in this case is concluded by introducing (4) into (3) and (3) into (36):

T = π 2 Ezh l 2   2 cos 2πX - j A j (j + 1) 2 cos (j + 1)πX   . ( 49
)
The index j in the previous equation refers to j = 1, 5, 9, . . . for the first kind of solution, and to j = 1, 3, 5, 9, . . . for the third kind of solution.

Drawing equations with changing ∆, N, and Q variables along the beam shows that the midpoint at x = l 2 is a local maximum point, and in some cases, the global maximum will not remain at the midpoint but rather a point beside it. However, there are no big difference between stress values at the local and global maximum points. An approximation to suppose the midpoint as the global maximum point is taken. The infinite sum in ( 49) is calculated by referring to the second sum in (19). The maximal total stress for the first kind of solution is then concluded:

σ N,∆ max = π 2 Eth l 2 N 2 N 2 -4π 2 + N 2 12π 2 Q + N 2 N 2 N 2 -4π 2 -∆ tan N 4 4π 2 N 4 -tan N 4 . ( 50
)
1550055-14

The maximum of the stress in the first kind of solution is at the final point. Noting that the final point where N is equivalent to zero is at ∆ = 20/π 2 . Introducing these values in (50), the maximal stress becomes:

σ max = π 2 Eth l 2 × 240 π 4 . ( 51
)
Idem for the third kind of solution, the maximal total stress can be obtained by setting the value of N and taking into account the constant A 3 :

σ ∆ max = π 2 Eth l 2 4 3 + 4 3Q + 4 9 -243 π 2 ∆ 2 + 504 π 2 ∆ + 18 - 240 π 2 - 108 Q 2 . ( 52
)
The maximum of the third solution of stress is remarked at ∆ = 28/27. Introducing this ∆ value in (52), we obtain the total maximum stress for the third solution of stress:

σ max = π 2 Eth l 2 4 3 + 4 3Q + 4 27 192 π 2 + 162 - 972 Q 2 . ( 53
)
We should note here that the global maximum point is at x = l/2 when the deflection ∆ is at the two specific positions taken in ( 51) and ( 53). In this context, ( 51) and ( 53) are exact.

Based on the above, the maximal stress with considering high modes of buckling is ranging between: [2.46, 3.41[ Q ∈ [2.42, ∞[. (54) Noting that the maximum for the maximal stress in the third kind of solution is for

σ max = π 2 Eth l 2 • 2.46 Q < 2.42,
Q = 156 2+ 64 27π 2
.

Simulations and Discussion

In this section, results with and without the approximation of neglecting high modes of buckling, are compared with finite elements simulations using ANSYS.

Simulations are made on a mechanism of two curved beams connected in the middle in order to prevent unsymmetrical buckling modes from occurring. In theory, the snapping force f will be doubled with the number of beams while the deflection d and stresses inside the beam remain the same.

A comparison between the snapping force theory with and without high modes of buckling and simulation of a silicon curved beam with 5 mm length, 20 µm thickness, 10 mm depth, 80 µm height and a Young's modulus of 169 GPa is presented in Fig. 8.

Curves in Fig. 8 show a good agreement between the present theory and FEM simulation. The curve without high modes of buckling was compared previously by [START_REF] Qiu | A curved-beam bistable mechanism[END_REF] with FEM simulations and experiments. The presented modeling with high modes shows to be more similar to the FEM simulation in terms of the snapping forces. It allows obtaining the expressions of the different values of Q and the main snapping points (∆ top , F top , ∆ bot , F bot ) that include the effects of all modes of buckling. In the example given in Fig. 8, the relative difference between the negligence and the consideration of high modes of buckling for the values of the snapping points (∆ top , F top , ∆ bot , F bot ) is (38.53%, 2.93%, 2.32%, 1.39%), respectively.

1550055-15

The quite large error on the ∆ top parameter could be a problem for the design of a bistable system. Reminding that the bistable system, as we assume in this paper, is composed of a passive bistable mechanism such a pre-shaped curved beam and actuators for switching between both stable positions. The snapping points in terms of displacements and forces respectively define the stroke and the force that the actuators have to provide. Thus, the accurate knowledge of the relations between the snapping points and the pre-shaped curved beam dimensions is very important in order to achieve the best integration of the complete bistable system.

Noting that the curves in Fig. 8, with and without high modes, meet at the two stable positions.

In the other side, Fig. 9 shows a comparison of the evolution of maximal stresses, between the theory with and without high modes of buckling and simulation of a curved beam with the same dimensions and properties.

The importance of high modes of buckling is more obvious in the calculation of stresses. As shown in Fig. 9, neglecting high modes will make a significant difference between the calculated stress and the simulation. Differences appear in the shape of the stress curves and in the highest stress point position.

1550055-16 Figure 10 shows the evolution of the maximal bending stress in the two cases, with and without high modes of buckling.

Comparing the results in Figs. 9 and 10, we conclude that the bending stress has the main contribution in the total stress value. The bending and total stress curves have almost the same shape. The contribution of high modes of buckling is more important in the calculation of the bending stresses.

In the first kind of solutions, the axial stress evolves after deflection from the two sides of buckling to the middle (Fig. 5), while the maximal bending stress is higher when the curved beam is far away from its initial position (Fig. 10).

1550055-17

In the third kind of solutions, the axial stress is constant along the beam (Fig. 5), while the bending stress has a maximum around the middle of deflection (Fig. 10).

Calculating internal stresses in the curved beam is important for design purposes, particularly for miniaturization and for determining the design limits under elastic or failure limits.

The accurate determination of the relation between the beam dimensions and the stress state allows identifying the limits of miniaturization and avoiding the fracture of a miniature curved beam.

Conclusion

We investigated the effect of high modes of buckling in the modeling of a pre-shaped curved beam. In a first stage, the presented modeling develops further the snapping forces solution and bistability conditions in order to include high modes of buckling. In a second stage, we developed the analytical solution of the stress inside the beam during deflection between the two sides of buckling.

The buckling with or without mechanical conditions on antisymmetric modes, the force characteristics, bistability conditions and internal stresses have been described in this paper based on mathematical approach in order to provide a clear physical understanding of the nonlinear bistable behavior of a curved beam. Finally, the calculus was compared with and without high modes and with FEM simulations.

The results show that considering high modes of buckling allows improving the accuracy of the snapping force curves, while the importance is more obvious for the calculation of the stress state.

The presented work in this paper constitutes a basis for the design and the optimization of a pre-shaped curved beam bistable mechanism. Future works aim to define the capabilities of miniaturization of such bistable mechanism as a function of the expected mechanical performances and the elastic or failure limits.

Fig. 1 .

 1 Fig. 1. Buckling of a beam before and after a critical axial compression.

Fig. 2 .

 2 Fig. 2. Transition between the two stable positions of a buckled beam as a result of a lateral force applied in the middle of the beam.

Fig. 4 .

 4 Fig. 4. Transition between the two stable positions of two curved beams connected in the middle, mode 3 appears during transition.

Fig. 5 .

 5 Fig. 5. Evolution of the normalized axial force N in the first kind of solution in function of Q ratio. N is constant in the second and third kinds of solution. The shape of the curved beam in the first, second and third case.

Fig. 6 .

 6 Fig. 6. Evolution of the normalized snapping force F of the pre-shaped curved beam for different Q values when mode 2 is constrained.

Fig. 7 .

 7 Fig. 7. Evolution of the maximal stress in the beam during deflection for different values of Q.

Fig. 8 .

 8 Fig. 8. Comparison of the snapping-force behavior during deflection between theory and FEM simulation.

Fig. 9 .

 9 Fig. 9. Comparison between the maximal stress value during deflection between theory and FEM simulation.

Fig. 10 .

 10 Fig. 10. Comparison between the bending maximal stress value during deflection with and without high modes of buckling.

Table 1 .

 1 Conditions on Q in order to reach modes 1-3 and the bistability feature for the curved beam.

		Mode 2	Mode 2
		unconstrained	constrained
	Mode 1	Q ≥ 1.16	Q ≥ 1.16
	Mode 2	Q ≥ 1.65	does not appear
	Mode 3	does not appear	Q ≥ 2.31
	Bistability	Q ≥ 5.65	Q ≥ 2.31
		1550055-11

Acknowledgment

This work has been supported by the Labex-ACTION Project (Contract "ANR-11-LABX-01-01").