Viscous damping on flexural mechanical resonators

Accepted Manuscript Title: Viscous damping on flexural mechanical resonators Author: id="aut0005" orcid="0000-0002-0300-1987" biographyid="vt0005"> Guillaume Aoust Rapha¨el Levy B´eatrice Bourgeteau Olivier Le Traon PII: DOI: Reference:

S0924-4247(15)00181-8 http://dx.doi.org/doi:10.1016/j.sna.2015.04.004 SNA 9147

To appear in:

Sensors and Actuators A

Received date: Revised date: Accepted date:

12-12-2014 13-2-2015 1-4-2015

Please cite this article as: Guillaume Aoust, Rapha¨el Levy, B´eatrice Bourgeteau, Olivier Le Traon, Viscous damping on flexural mechanical resonators, Sensors & Actuators: A. Physical (2015), http://dx.doi.org/10.1016/j.sna.2015.04.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Viscous damping on flexural mechanical resonators

ONERA–The French Aerospace Lab, F-91123 Palaiseau cedex, France

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Guillaume Aousta , Rapha¨el Levya,∗, B´eatrice Bourgeteaua , Olivier Le Traona

Abstract

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A new analytical formula predicting the quality factor of a wide range of flexural mechanical resonators operating in a viscous fluid medium is presented. The formula is derived from a fluid analogy used in the single beam Euler-Bernoulli bending theory, and an extension to tuning forks is proposed. Comparisons with experimental data extracted from the literature are presented. New experiments have also been carried out on different kinds of resonators to investigate the pressure dependency of their quality factors in air. In both cases, very good agreement is obtained for resonator sizes ranging from micrometric to macroscopic sizes.

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Keywords: Viscous damping, resonators, fluid mechanics, Q-value, Viscosity 1. Introduction

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Flexural mechanical resonators operated in vacuum have been extensively used in the past century for various applications, taking the most of their sharp resonance characteristics and their relative small size. These resonators have been widely used for time and frequency applications(1), especially since the invention of the electronic watch based on high quality factor quartz tuning forks(2). They have also been used as sensing elements for miniature vibrating inertial sensors(3), mainly since the 80’s. These micro electromechanical systems (MEMS) known as Coriolis Vibrating Gyrometers (CVG)(4; 5; 6) ∗

Corresponding author Email address: [email protected] (Rapha¨el Levy)

Preprint submitted to Sensors and Actuators A: Physical

April 7, 2015

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and Vibrating Beam Accelerometers (VBA)(4; 7) have been designed for accelerations and rotations ultra-sensitive measurements using a capacitive or piezoelectric method of detection. Accelerations are deduced from a shift of their resonance frequency which is continuously tracked. Rotations are deduced from the measurement of the Coriolis force applied to such resonators primarily and constantly excited. Clamped-Clamped beams have been used for example as pressure sensors(8). The pressure to be measured stresses a membrane bounded to one of the beam’s anchor, which in turn shifts the resonance frequency of the beam. Flexural mechanical cantilevers have been used as probes in the atomic force microscopy domain(9). In tapping mode of operation, their sharp resonance is the key for accurate tip-to-sample distance measurement. Several applications of flexural MEMS cantilevers as magnetic field sensors have been reported(10; 11). These MEMS are covered by a ferromagnetic material which interacts with an external magnetic field, hence modifying their resonance frequency. More recently, the use of nano electromechanical systems (NEMS) for weigh applications also appeared, and already achieved mass measurements below the zeptogram scale under vacuum(12; 13). Some applications however require a complete immersion of the resonant structure inside a fluid medium. It is for example used for highly sensitive chemical sensing platforms (microbalances). Masses below the attogram have already been detected in air at atmospheric pressures(12). Contact force measurements using tuning forks also have to be performed in situ. We can for example cite photo acoustic force detections (14), resonant optothermoacoustic detection (15) or static pressure measurements. Temperature measurements within a 0.001 ◦ C precision can also be achieved through the dependence of the resonance frequency to the temperature(16). In any case, the surrounding fluid strongly modifies the characteristics of the resonance, and hence can dramatically reduce the sensors performance compared to vacuum. An effective tool for modeling such reduced characteristics is hence crucial for any optimization or prediction purpose. In the following, we will restrain to the common case of resonators made of beams with a circular or rectangular cross section. We also focus on viscous damping, which is most of the time the dominant damping source(19).

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Macroscale beams

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Macroscale tuning forks

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Microscale cantilevers

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Microscale tuning forks

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Figure 1: Aspect comparison of flexural mechanical resonators used to carry out experimental comparisons detailed in section 4, illustrating various geometrical sizes among existing resonators. HTF is a large in-house quartz tuning fork operated on its out-ofplane first flexural mode, whereas TF 1 and TF 2 are commercially available quartz tuning forks operated in their in-plane first flexural symmetric mode. TF 3 is a diamond tuning fork reported in the literature(17). CC 1 and CC 2 refer to in-house clamped-clamped beams, and CF refer to typical cantilevers reported in the literature(18). Exact dimensions are listed in table 2.

2. Viscous damping state of the art

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How does viscous damping exerted by a fluid affect the bending of a beam? To the best of our knowledge, Newell(20) is the first to give an analytical expression for a beam immersed in an unbounded viscous fluid. He chose to apply the very well-known Stokes’ expression for the hydrodynamic force per unit area applied by any fluid on a structure. This approach is extremely simple since neither the thickness of the vibrating structure nor the exact geometry is taken into account. The presence of a nearby static structure is

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also excluded. Later on, Tang (21) proposed a model to estimate the quality factor of laterally driven resonant structures close to a static structure. Even if the oscillating plates considered are not flexural mechanical resonators strictly speaking, they are still free-free beams oscillating in a fluid medium and hence perfectly suit the problem. Couette flow underneath the plate is found to be the dominant dissipative process when the distance between the oscillating plate and the surface is sufficiently small. Observing large overestimations of the quality factor predicted by the previous model, air drag on the top surface is introduced (22) and is assumed to behaves as a Stokes flow. An empirical formula based on the superposition principle is presented, mainly because the front air resistance description and edge effects were only accessible through experiment. All of these first attempts were nevertheless not sufficient to correctly describe fluid damping on the constantly increasing resonators’ diversity. The rigorous fluid mechanics problem constitutes a formidable challenge and neither simple analytical formula nor universal model has yet appeared in the literature. For the past twenty years, two very different kind of approaches have been applied to improve viscous damping models. The first approach tends to solve the full problem analytically under limited approximations, by inserting the fluid forces into the Euler-Bernoulli bending theory. It started with Sader(23) and has been constantly refined since then (24; 25; 26; 27). Although a general numerical method is proposed, no simple expression can be extracted for resonators with an arbitrary cross section. The case of infinitely thin rectangular cantilevers on any flexural mode number however has a complex analytical solution and as well as a rather straightforward numerically approximated solution. The rigorous model developed is also able to take into account a static surface nearby or the compressibility of the fluid, although the two latter effects can only be apprehended through advanced numerical simulations. The Euler-Bernoulli bending formalism has also been used for laterally driven resonators by Cox & al(28), and also deals with infinitely thin cantilevers immersed in an unbounded fluid. The crucial point is the use of Stokes’ fluid damping force per unit area. The analytical work assumes that the edge and pressure effects acting on the thickness are negligible compared to the shear damping force. The second approach tries to establish analogies in order to get an approximate but simple analytical formula. The pioneering work of Hosaka(29) established an analogy between a flexural mechanical cantilever and a string

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of spheres. Indeed, the exact solution of an oscillating sphere immerged in an unbounded fluid exists and is very simple(30). Hosaka used this well-known result to express a new simple expression of the quality factor. However, the solution only applies to the first flexural mode of a very thin cantilever. The discrepancies between theory and experiment can also be sometimes substancial, especially for macroscopic size resonators. Using the same analogy, Vignola(17) established a refined formula by adjusting the effective surface used by Hosaka. In the past 10 years, increments applied on both approaches have mostly tried to generalize the solutions using numerical simulations. Lee extended this way the work of Hosaka for micro cantilevers(31), leading to a more reliable result although limited in its validity to some particular dimensions and experimental conditions. Cox also extended his own results for laterally driven micro resonators with numerical simulations(32) and derived a more accurate expression for the quality factor of any beam with a rectangular cross section. The method used in this paper lies between the two previously described approaches. We include a new analogy in the rigorous formalism first introduced by Sader in order to obtain a simple analytical expression. The formalism can be applied to any flexural mode number, for any rectangular cross section and for any boundary conditions imposed on the beam. As we will see in section 3 and 4, the obtained formula is fully analytical and gives very good results for various kinds of flexural resonators over a wide range of dimensions. 3. Theoretical Model

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Starting with a single beam theoretical description in subsection 3.1, we will introduce an analogy allowing us to split the total fluid forces into several independant components. Each component is then linked to a known solution of a simpler problem. The quality factor of the resonating beam is then deduced and displayed in equation (10). An extension to the tuning fork theoretical description is also proposed in subsection 3.2. 3.1. The single beam 3.1.1. The theoretical model We first consider an oscillating mechanical beam within a homogeneous fluid unbounded in space. Each end of the beam is clamped, pinned, sliding 5

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or free. The present theoretical model hence covers general beam boundary conditions , although beams used as part of flexural mechanical resonators are often clamped-clamped or clamped-free. A schematic depiction of the mechanical cantilever considered in our model is given in figure 2. The extended case of the tuning fork will be discussed in section 3.2. Following a similar approach to Sader’s(23), we consider the following approximations and assumptions:

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Figure 2: Schematic of a beam oscillating in a fluid medium, supposed unbounded in the x > 0 half space.

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• The beam’s cross section is rectangular and uniform over its entire length L. • The length of the beam L greatly exceeds its transverse dimensions e and l. • We consider only flexural modes of vibration in the y-direction, whose 6

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amplitudes are supposed far smaller than e and l. Plane sections hence remain plane and normal to the axis of the beam.

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• The material composing the beam is supposed isotropic and homogeneous. Its young modulus E and density ρb are constant values.

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∂4 ˜ (x, ω) − F˜hydro (x, ω) = F˜ext (x, ω). (2) EI 4 − ω 2 ρb el W ∂x √  The Knudsen number Kn = kB T / 2πs2 P Lc , defined by the ratio between the mean free path of the molecules inside the surrounding fluid and the characteristic geometrical dimension of the problem, is usually introduced to know which formulation of fluid dynamics should be used. In the latter formula, T is the absolute temperature, kB the Boltzmann contant, P the total presssure, Lc the representative length scale and s the gas particles typical diameter. Four main regimes exist(33): the continuum regime (Kn < 0.01), the slip regime (0.1 < Kn < 0.01), the transitional regime (0.1 < Kn < 10) and the free molecular regime (10 < Kn ). We now detail the approximations and assumptions made about the surrounding fluid: #

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(1)

where I = e3 l/12 is the moment of inertia of the beam. Ftot (x, t) is the total force per unit length (along the x axis) applied on the contour of the beam section. This latter force gathers in particular the hydrodynamic force Fhydro (x, t) due to the action of the surrounding fluid. It also contains a possible excitation force Fdrive (x, t) and an equivalent force due to the Brownian motion of molecules inside the resonator FBM (x, t). In the following, we will gather as Fext (x, t) = Ftot (x, t) − Fhydro (x, t) any force that is not due to the fluid action. Taking the Fourier transform of equation (1), we obtain for any angular frequency: "

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∂2 ∂4 EI 4 + ρb el 2 w (x, t) = Ftot (x, t), ∂x ∂t

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Using the beam theory of Euler-Bernoulli, the general equation governing the deflection w(x, t) of the beam is given by:

• We suppose that the fluid is in the continuum regime and unbounded in space.

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• The velocity field at any section along the beam is well approximated by that of an infinitely long beam oscillating in the y-direction with the same amplitude. This approximation is guaranteed by the beam approximation (L  l, e) already made to apply the Euler-Bernoulli beam theory. In our model, the use of Navier-Stokes equations implies that our final formula is expected to fail when Kn > 0.01. For high Knudsen numbers, especially if Kn > 10, a different physical description has to be considered, taking into account the molecular nature of the surrounding fluid(34). Since the amplitude of vibration has already been assumed small compared to the beam’s dimensions, all nonlinear convective inertial effects in the fluid can be neglected. We can hence use the linearized Navier-Stokes equation which yield to an expression of F˜hydro (x, ω) as a linear function of ˜ (x, ω). the displacement W It is often convenient to introduce the Reynolds number Re = ρf ωl2 /4µ (the factor of fourth has been added in order to be consistent with previous works(32; 23)), and the dimensionless hydrodynamic function Γhydro (ω) defined as follows:

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• The fluid is supposed incompressible with a homogeneous density ρf and dynamic viscosity µ. This approximation stands as long as the largest dimension of the beam (e or l) is much smaller than the wavelength of the cantilever’s vibration (λ = c/ (2πωres )), valid in most practical cases. In terms of damping sources, this approximation implies that we neglect the acoustic radiation damping(19).

π ˜ (x, ω). F˜hydro (x, ω) = ρf ω 2 l2 Γhydro (ω)W 4

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The challenge is now to find the right expression for Γhydro (ω).

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3.1.2. The analogy In our model, the fluid is supposed laminar and the actual streamlines follow a pattern labeled Situation A in figure 3. The actual pattern can be seen as the juxtaposition of three consecutive domains. Domains (1) and (3) can be joined: we obtain a pattern very close to the one of an infinitely thin beam along the y-direction, labeled as situation B in figure 3. The remaining pattern on domain (2) can be identified as the pattern resulting

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Figure 3: Streamlines of the incompressible fluid around a section of the oscillating beam displayed in figure 2. Situation A refers to the case of an arbitrary beam’s rectangular cross section, whereas situation B (resp. situation C) refers to the case of an infinitly thin beam in the y direction (resp. in the z direction).

from an infinitely thin vibrating beam in the z-direction, labeled in figure 3 as situation C. According to the superposition principle allowed by the use of the linearized fluid problem detailed in subsection 3.1.1, the total hydrodynamic force Fhydro (x, t) can be considered quite close to the sum of two contributions: • A front and back force Ff b (x, t) equal to the total hydrodynamic force applied on the beam’s section of situation B. • An up and down lateral force Fud (x, t) close to the total hydrodynamic force applied on the beam’s section of situation C.

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The overall solution still satisfies the infinitely distant boundary condition that airflow and pressure are null. The superposition argument is mathe9

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The expression for Γf b (ω) Obtaining a reliable expression for Γf b (ω) is a very difficult task. It has been extensively studied, mostly by Sader(23). An analytical solution exists in terms of Meijer G-functions but is rather cumbersome and has no direct expression(25). We will prefer in this article the exact analytical result obtained for the circular section beam:  √  4iK1 −i iRe  √ . (5) Γf b (ω) = 1 + √ iRe K0 −i iRe K0 and K1 are modified Bessel functions of the second kind. The latest funcR +∞ −x cosh t tions can also Rbe expressed in their integral forms as K0 (x) = 0 e dt +∞ −x cosh t and K1 (x) = 0 cosh(t)e dt. Expression (5) has been numerically refined by Sader(23), in order to give a more accurate result for infinitely thin rectangular sections. However, the two solutions remain quite close(23), that’s why we use the analytical expression (5) in our comparison in section 4.

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Γhydro (ω) = Γf b (ω) + Γud (ω).

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matically not rigorous here, since the domains are of course not independent. However, a good approximation is expected because the general behavior of the fluid is conserved quite accurately. This analogy also takes into account some edge effects included in situation B but neglects the edge effect originally present in domain (2). The superposition principle is then obviously retrieved in the expression of the corresponding hydrodynamic function:

The expression for Γud (ω) The up and down lateral force is very well approximated by the expression first derived by Stokes. His model assumes an infinitely thin plane oscillating at a fixed angular frequency ω. Under the no slip condition, the linearized incompressible Navier-Stokes equation leads to: √ 2 2e (6) Γud (ω) = √ (1 + i) . πl Re For further details on how to obtain expression 6, the reader should report to Cox’s work(32). 10

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∂4 πρf l EI 4 − ω 2 ρb el 1 + Γhydro (ω) ∂x 4ρb e

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In equation (8), Γrhydro (ω) is the real part of Γhydro . Γihydro (ω) will be refered as its imaginary part. The eigen angular frequencies ωn are defined by the implicit equation:

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The resolution of this kind of equation has been known for long. First, the eigenmodes φn (x) labeled by integer n are extracted by solving the following undamped homogeneous equation:

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3.1.3. Resolution Back to equation (2), we obtain in the angular frequency domain:

αn4 12ωn2 πl = ρb + ρf Γrhydro (ωn ) . 4 2 L Ee 4e !

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Depending on the beam mechanical boundary conditions, the αn numbers are known and detailed in table 1 for some particular boundary conditions.

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Table 1: Numbers αn depending on the beam’s mechanical boundary conditions, allowing to find the beam’s resonance frequencies.

Boundary Conditions

Implicit equation for αn 1 − cosh (αn ) cos (αn ) = 0 1 + cosh (αn ) cos (αn ) = 0 sin (αn ) = 0 tanh (αn ) + tan (αn ) = 0 tanh (αn ) − tan (αn ) = 0

Clamped-Clamped Clamped-Free Pinned-Pinned Clamped-Sliding Clamped-Pinned

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The variations of Γrhydro (ω) with respect to ω are sufficiently slow compared to ω 2 to consider that Γrhydro (ω) is almost equal to Γrhydro (ωn ) around

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The previous formalism implies an infinite value of the quality factor in the absence of fluid. In practice, this value is finite due to internal dampings ( thermo elastic diffusion, mechanical dissipation of electrodes, etc.(35)). The expression of the quality factor could be further simplified since we have [4ρb e/ (πρf l)]  Γrhydro for most practical cases, particularly when the resonator operates within a slightly viscous fluid such as air. Under this approximation, the well-known additive rule for high-Q resonators is directly P deduced from equation (10): Q1tot = i Q1i . 3.2. The tuning fork Tuning forks are also very common and widely used flexural mechanical resonators. Their quality factor can be apprehended once the case of cantilevers has been treated. A tuning fork operated on its in-plane eigenmodes is indeed composed of two beams which vibration is symmetric. The symmetry imposes the same fluid boundary conditions as that of one of the two prongs vibrating at a fixed distance D = g/2 from an infinite static plane, with g the size of the gap between the two prongs. Similarly, a tuning fork operated on its out-of-plane eigenmodes corresponds to the vibration of two beams which vibration is anti-symmetric. The problem hence reduces to the vibration of one of the two prongs, at a fixed distance D = g/2 from an infinite static plane imposing a zero-speed condition to the fluid. A sketch of those two possible situations is displayed in figure 4. The literature extensively reports the study of a cantilever vibrating near a static planar surface(36; 37). If the beam is too close from the static surface, a squeeze-film damping can appear when the vibration direction is directed towards the static plane (see a) in figure 4). If the vibration direction is parallel to the static plane, a couette damping can appear instead (see b) in figure 4). As we still search for descriptions leading to simple analytical expressions, we can push our analogy detailed in subsection 3.1.2 a step further to take into account the two latter additional damping sources. The total hydrodynamic force can now be seen as the sum of four independent contributions exerced

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Γihydro (ωn )

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the nth resonance peak. As a direct consequence, we can write the quality factor as an explicit analytical expression:

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Out-of-plane TF mode

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𝒈 Fluid Plane of Symmetry

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Fluid Plane of Anti-Symmetry

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Figure 4: Sketch of the streamlines around a tuning fork section when it vibrates on a) an in-plane symmetric eigenmode and b) an out-of-plane anti-symmetric eigenmode. The four edges of the oscillating beam are labelled 1,2,3 and 4.

on each edge of a beam’s section , labelled in figure 4. In the case of a beam vibrating far from any static surfaces, Γf b /2 is naturally chosen as the corresponding hydrodynamic function for the front or back edges (with respect to the beam displacement direction), and similarly Γud /2 is chosen for the up or down edges. When one of the edges faces a static planar surface, another hydrodynamic function Γsqueeze (ω) or Γcouette (ω) has to be taken into account to obtain the complete load on this particular edge. As an example, we consider situation a) in figure 4. The hydrodynamic function for edge number 3 has to take into account Γsqueeze , for example by addition: Γ3 = Γf b /2 + Γsqueeze . This simple expression has the correct asymptotic values, even if we expect it to be quantitatively less accurate. However, it allows to take into account a wider range of situations. We will see that this approach is sufficient to explain some experimental results displayed in section 4. The other edges are supposed to be unaffected by the 13

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3.2.1. The squeeze force The expression for Γsqueeze (ω) can be approached by considering the Reynolds equations. Under the assumptions and approximations detailed by Bao(38), the squeeze force per unit length is given by F˜squeeze (x, ω) = ˜ (x, ω)/d, with Pa the surrounding pressure and σ = −Pa l [fe (σ) − ifd (σ)] W 2 2 12µωl / (Pa d ) a dimensionless number. The functions fe and fd have been introduced by Langlois(39):  q sinh √ σ +sin √ σ ( ) ( )    fe (σ) = 1 − σ2 cosh √ σ2 +cos √ 2σ ,   ( 2) ( 2)  (11)  q sinh √ σ −sin √ σ   ( ) ( )  2  √2 √2 .  fd (σ) = σ cosh( σ )+cos( σ ) 2 2

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presence of the static wall: Γ1 = Γf b /2, Γ2 = Γud /2 and Γ4 = Γud /2. For situation b) in figure 4, we would propose to use Γ3 = Γcouette , and leave the contributions on the other edges unchanged. Again, we expect discrepancies since the couette flow affects at least the front and back damping on edges 2 and 4. Nevertheless, the physics of the situations should be reasonably well captured. Experiments on the latter proposition are not carried on in the present work.

We hence extract the function Γsqueeze (ω):

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Γsqueeze (ω) = −

4Pa [fe (σ) − ifd (σ)] . πdlρf ω 2

(12)

The main challenge is to estimate the correct superposition of the squeeze and front force on the edge facing the static wall. In the comparison presented in section 4, we simply added the two hydrodynamic functions, even though a better total hydrodynamic function could be proposed. It however ensures the right assymptotic limits on variable d, even if discrepancies are expected when the two contributions have the same magnitude. We will see that we correctly reproduce the observed quality factors on commercially available quartz tuning forks subject to squeeze film damping. 3.2.2. The Couette force Similarly to the squeeze film damping, we could also consider the case of a static planar surface nearby the beam, but parallel to the vibration direction. It is indeed known that a viscous drag can occur if the resonator and the 14

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surface are too close(22). The fluid behavior is then referred as a Couette flow. The analogy applied on that particular surface of the beam would hence result in replacing the contribution Γud (ω)/2 by Γcouette (ω), extracted from the resolution of an infinitely extended plate oscillating parallel to a static wall nearby(22). The replacement makes sense only if the Couette flow damping becomes significant compared to the lateral damping.

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4. Experimental validation

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3.3. Summary We used a physical analogy based on the observation of the fluid behavior to estimate the viscous damping forces applied on each edge of a beam immersed in a viscous fluid. We obtained a fully analytical formula predicting its quality factor (eq.10). We point out that the latter formula only implicitly depends on the length L of the beam through the resonant frequency. We also notice that the theoretical damping does not only depend on the ratio between the section dimensions, which means that the absolute dimensions of the resonator also matters.

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We first show in subsection 4.1 that our formula is consistent with some experimental results randomly picked in the literature, although the geometries and sizes are very different from one experiment to the other. New experiments with our own resonators are then reported in subsection 4.2. 4.1. Literature comparisons Zhang(18) published a set of experimental results concerning micrometric cantilevers of fixed length L = 600µm with various width ranging from l = 20µm to l = 50µm. These typical microscale cantilevers have been represented in figure 1 as ”CF”, and their thickness e = 5µm is the same for every cantilever. Each cantilever is driven on several eigenmodes of flexure, allowing an investigation over viscous damping’s frequency dependency. The relation between the damping coefficient used by the authors and the quality factor is Cdamping = πωρf l2 Γihydro (ω) /4. A typical vacuum quality factor Qstruct = 10000 has been chosen, however this value does not affect the final results significantly. Our result shows very good agreement with the trend of the reported experimental values, and viscous damping is stronger at lower frequencies. Discrepancies tend to appear only for the highest reported mode. The mode 15

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number indirectly appears through the resonance frequency in our analytical formula 10. In the work of Vignola(17), a microscale diamond tuning fork has been fabricated and its quality factor has been measured in ambient air. The length of each prong is L = 5.8µm, its width is e = 0.8µm and its thickness is l = 0.2µm. This tuning fork has been represented in figure 1 as TF 3. Our model predicts Qair = 776, which is consistent with the experimental value reported of Qexp = 813. Even if the size of the resonator is very small, the Knudsen number Kn ∼ 0.05 remains small enough to consider the regime as close to continuum. The gap between the two prongs is also sufficient to consider that the influence of the squeeze damping is negligible. The criterion Γisqueeze (ω)/Γif b (ω) ∼ 4 10−11  1 is indeed largely satisfied. As a third and last comparison with the literature, we use the work of Lee& al.(31) who obtained experimental measurements of micrometric size cantilevers quality factors. They fabricated a silicon cantilever with dimensions set to L = 240µm, e = 1.15µm and l = 30µm. For those particular dimensions, we obtained Qair = 31, which is very close to Lee’s measurement

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Figure 5: Quality factor comparisons in ambient air between Zhang experimental data (dots series) and our analytical formula (line series). The experimental study on various modes of flexure allows to compare the dependency of the formula with respect to the frequency.

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4.2. Pressure dependency in air Relatively few experimental studies have been published about air damping on macroscopic size mechanical resonators in the continuum regime, compared to the micro/nano resonators literature(34; 40; 41; 42). As their surface to volume ratio decreases, the hydrodynamic force applied on the surfaces becomes weaker compared to the inertial forces directly proportional to the resonator’s volume. However, for macroscopic size resonators such as commercially available quartz tuning forks, the residual damping still limits some applications such as quartz enhanced photoacoustic spectroscopy(14).

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Qexp = 29. Since the thickness of the cantilever e is very thin, our formula naturally reduces to Sader’s. The lateral damping becomes negligible compared to the front and back damping. Based on the comparison with the three previous papers, our model seems consistent with microscopic size cantilevers and tuning forks, with typical length of 1µm to 100µm. However, some formulas given in the literature are very far from reality when it comes to macroscopic size resonators. For example, Newell’s formula (20) largely overestimates the quality factor of commercially available quartz tuning forks vibrating in ambient air at atmospheric pressure, by at least one order of magnitude (see figure 9). As it will be shown with our own experiments, our model also applies to macroscopic size resonators.

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4.2.1. Experimental setup We investigated the quality factor’s pressure dependency of a large cantilever, clamped-clamped beams and tuning forks immersed in air. In order to cover various kinds of dimensions and different boundary conditions, we selected the following resonators also displayed in figure 1: • Two commercially available quartz tuning forks TF 1 and TF 2, with a few hundred micrometers typical length. Their relatively small gap between the two prongs is expected to behave like the presence of a static wall perpendicular to the vibration plane as explained in subsection 3.2. • An in-house tuning fork HTF, which looks like a large quartz tuning fork with a millimeter typical length operated on its first out-of-plane mode. This resonator should emphasize the importance of the absolute size on the quality factor. 17

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Their characteristics are summarized in table 2, and a sketch of their mode shape is available in figures 6,7 and 8 respectively.

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• Two in-house clamped-clamped resonators CC 1 and CC 2, which display a typical length of the order of a few tens of micrometers. In contrast with TF 1 and TF 2, these resonators operate with the complete absence of any static wall nearby.

Table 2: Characteristics of the differents resonators we investigated. Type ”cc” (resp. ”cf”) refers to ”Clamped-Clamped” (resp. ”Clamped-Free”) boundary conditions. The quality factor Qstruct has been measured experimentaly under vacuum.

e (mm)

l (mm)

TF 1

3.75

0.6

0.34

TF 2

3.12

0.4

HTF

16

CC 1

CC 2

Qstruct

fres (Hz)

T ype

77000

32762

cf

0.20

120000

32765

cf

0.31

d

0.33

2

2

240000

5672

cf

4.1

0.108

0.054

/

25000

35950

cc

2.26

0.06

0.03

/

12000

61780

cc

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1.8

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g (mm)

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L (mm)

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Resonator

Each of those resonators has been inserted under a sealed vacuum bell. A pressure sensor Pfeiffer TPR280 has been used to monitor the pressure. The vacuum bell is directly linked to a vacuum pump through a sluice gate, allowing to set any pressure in the range 1 Pa-105 Pa. The vacuum quality factor Qstruct is considered to be the measured quality factor when the pressure is below 0.1 Pa. The corresponding theoretical value would derive from the competition of multiple internal dampings as explained in subsection 3.1.3. 18

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4.2.2. Results and comments The commercial quartz tuning forks Results concerning the two commercial quartz tuning forks are displayed in figure 6. Owing to the continuum approximation we made to obtain formula (10), discrepancies are expected when the Knudsen number Kn becomes greater than 0.01. The analytical trend in this regime is in very good agreement with the measured values. The predicted value is accurate within 20% for the two quartz tuning forks. The two resonators seem also to be sensitive to squeeze film damping occurring between the two prongs. The presence of such damping indeed explains the theoretical inflection around pressure P=100 Pa. Experimentaly, the intrinsic quality factor of TF 2 is better than the one of TF 1, even if it is worst when immersed in air at ambient pressure. An inflection seems to occur between the slip regime (0.1 < Kn < 0.01) and the transitional regime (0.1 < Kn < 10). This characteristic could be surprising at first because it does not appear in similar studies on cantilevers(42; 40; 41). We believe that this could be attributed to additional squeeze film damping. The inflection is much more important for TF 2 than for TF 1, which would imply a stronger squeeze film damping for TF 2 than for TF 1. This is consistent with previous works reporting a l/g dependency for this particular damping(36; 37). This is also consistent with our analytical formula, for which the inclusion of squeeze film damping (see subsection 3.2) is responsible for the analytical inflection reported in figure 6. Unfortunately, we cannot discriminate for certain the exact impact of discrepancies with the NavierStokes equation and squeeze film damping in this situation. Indeed, the inflection occurs in a regime (0.01 < Kn < 10) in which our formula (10) validity can be questioned. The presence of the squeeze film damping could also explain the 20%

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The pressure appears in formula (10) through the fluid density ρf . According to the ideal gas law, it decreases linearly with the pressure as ρf = ρat P/Pat , with ρat = 1, 2kg.m−3 the ambient air density at 20 ◦ C and Pat = 101 325P a the normal atmospheric pressure. As explained in subsection 3.2.1, we took into account the squeeze film damping due to the proximity of the two prongs when we modelled the response of tuning forks TF 1 and TF 2. Since HTF is vibrating on an out-of-plane mode of vibration, no squeeze film damping is considered. The eventual use of a Couette damping will be discussed in further details.

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Figure 6: Pressure dependency of the quality factor for the two commercially available quartz tuning forks TF 1 and TF 2. The dashed vertical lines delimit the fluid regimes according to the knudsen number Kn introduced in section 3.

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difference between the predicted and the measured values. It could show that the inclusion of this damping as an independent additive damping, as we proposed in 3.2.1, gives a rather accurate idea of the final damping. Specific experiments on squeeze film damping could be interesting to improve our model. Physically, it means that the squeeze film damping affects the fluid streamlines on the front face of the resonating beam, which no longer looks like our analogy in figure (3). The large in-house tuning fork Concerning the HTF, we could not perform any measurements below a pressure of P=10 Pa because the resonance bandwidth is too narrow for our experimental setup. Our frequency generator cannot be more precise than the millihertz, while the uncertainty on the bandwidth measurement has to be known with better precision when the quality factor is over 200000. The values could however be determined by an alternative method, for instance 20

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Figure 7: Pressure dependency of the quality factor for the HTF resonator, vibrating on its out-of-plane first mode of vibration. The dashed vertical lines delimit the fluid regimes according to the knudsen number Kn introduced in section 3.

by measuring the characteristic decay time. Unlike the previous quartz tuning forks, the HTF can be driven on an out-of-plane mode of vibration. The two prongs oscillations become parallel to one another and no squeeze film damping is likely to appear. The proximity of the prongs could modify the nature of the flow between the two parallel prongs, especially since the resonant frequency of theqstructure is low. If we 2µ use the expression of the penetration depth(22) δ = ωρ , it becomes larger f than the gap for pressures under 30P a, which remains below the threshold for the continuum regime. We hence do not expect Couette damping above that value, that is why it is relevant to consider each beam of the HTF as an independent vibrating beam within an unbounded fluid. The experimental results displayed in figure 7 do not evidence any transition different flow 21

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regimes, neither squeeze film damping nor Couette flow damping. We measure a 8% maximum relative error between theory and experiment over the entire continuum regime.

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The clamped-clamped beams

Figure 8: Pressure dependency of the quality factor for the two clamped-clamped beams CC 1 and CC 2. The dashed vertical lines delimit the fluid regimes according to the knudsen number Kn introduced in section 3.

495

As a third and final experiment, we selected resonators with a characteristic section size of the order of a few tens micrometers. The results are displayed in figure 8. We clearly notice the change from the continuum regime to the free molecular regime, resulting in large but expected discrepancies with the theoretical model. The analytical formula accurately predicts the observed values with a maximum 8% relative error over the entire continuum and slip regimes. 22

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0 .1 > K

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0 .1 < K

5

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1 0

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O u C o H o N e

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Q u a lity F a c to r

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4.2.3. Theoretical comparisons Finally, it is interesting to compare different existing theoretical models with our new formula. The particular case of a clamped-free beam oscillating in an unbounded fluid medium is hence considered, since most of these existing models have been derived to address this particular fluid problem. Three Q-factor formulas introduced in the state of the art (section 2) have been applied to the HTF case: the first and most simple introduced by Newell(20), the one derived from Hosaka’s analogy between a clamped-free beam and a string of spheres(29), and the numerically refined and most recent one reported by Cox(32). The results appear in figure 9.

1 0

d e l d e l m o d e l m o d e l

4

1 0

-1

1 0

0

1 0

1

1 0

2

1 0

3

1 0

4

1 0

5

1 0

6

P re s s u re (P a )

Figure 9: Theoretical pressure dependency of the quality factor for the HTF resonator in air, predicted by different models. The dashed vertical line delimit the fluid continuous regimes (Kn < 0.1) introduced in section 3 for which the models are valid. The dotted black line corresponds to Newell’s analytical formula(20). The dashed-dotted black line represents Hosaka’s analytical formula(29). The dashed black line represents the numerically refined formula obtained by Cox(32) and the continuous dark yellow line represents our fully analytical formula (10).

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We used two known results from the literature, describing the behavior of infinitly thin structures vibrating in a viscous fluid, to build a general model for flexural mechanical resonators made of beams with an arbitrary rectangular cross section. We extracted a fully analytical formula expressing their quality factors and frequency shifts, and obtained a good agreement with experiments reported in the literature. We also investigated the pressure dependency of five different resonators in air, which also showed a good agreement with the analytical formula. The presence of squeeze film damping for some quartz tuning forks has also been reported, and could be more accurately predicted with further developements using our approach. This work could lead to geometry optimization towards a new class of high Qfactor resonators operated in fluid medium.

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5. Conclusion

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Newell’s formula does not depend on the surrounding fluid density, and hence is less and less accurate as the fluid density increases. For air at atmospheric pressure, the discrepancies are significant and the formula fails to even give a correct order of magnitude of the quality factor. Hosaka formula is better and the trend seems reasonable, although it clearly underestimates the fluid damping by a factor of 3. His formula remains useful to rapidly obtain the viscous damping’s dependency to the various mechanical and fluid parameters. Only Cox’s formula seems to be able to reproduce the good agreement we obtained in figure 7 for the HTF resonator. A more detailed comparison of our respective formulas on different resonator geometries confirmed the very good agreement. The two models obviously display the same asymptotic limits when l tends to zero, since they both use the result from Stokes’ second problem. However, the way the thickness e is included is radically different, since it is a numerical refinement for Cox whereas we used a second asymptotic behavior to remain fully analytical. Our own model remains more accurate when e tends to zero, since we use the analytical asymptotic result from Sader.

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[3] N. M. Barbour, Inertial Navigation Sensors, NATO 116 (2011) 1–28.

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[10] J. W. V. Honschoten, W. W. Koelmans, S. M. Konings, L. Abelmann, M. Elwenspoek, Nanotesla torque magnetometry using a microcantilever, in: Eurosensors XXII, 2008.

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[14] A. A. Kosterev, Y. A. Bakhirkin, R. F. Curl, F. K. Tittel, Quartzenhanced photoacoustic spectroscopy., Optics Letters 27 (21) (2002) 1902–4. URL http://www.ncbi.nlm.nih.gov/pubmed/18033396 [15] A. A. Kosterev, J. H. Doty, Resonant optothermoacoustic detection: technique for measuring weak optical absorption by gases and microobjects., Optics Letters 35 (21) (2010) 3571–3. URL http://www.ncbi.nlm.nih.gov/pubmed/21042353

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Biographies

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Guillaume Aoust received the M.Sc. degree from Ecole Polytechnique (Palaiseau, France) in 2011 in the field of lasers and optics, as well as the M.Sc. degree from the Ecole Nationale des Ponts et Chauss´ees (Champ-surMarne) in 2013 in the field of acoustics and mechanics. He is currently Ph.D. candidate at The French Aerospace Lab (ONERA, France) on photoacoustic spectroscopy and resonator design for force measurements. Rapha¨ el Levy obtained an Engineering degree from ENSICAEN in 2001, he obtained his PhD from University Paris-Sud in 2005. He is currently a researcher at The French Aerospace Lab (ONERA, France), and his current research interests include the development of MEMS resonator based sensors. B´ eatrice Bourgeteau obtained an Engineering degree from ENSAM in 2005, and she obtained her PhD from University Paris-Sud in 2008. She is currently a researcher at The French Aerospace Lab (ONERA, France). Olivier Le Traon obtained an Engineering degree from INSA Lyon (Institut National des Sciences Appliqu´ees) in 1986 and a master degree in Solid Mechanics and Modeling. He joined the Physics and Instrumentation Department of ONERA (the French Aerospace Lab) in 1991. His main research activities are Inertial Vibrating MEMS (VBA and CVG), and more particularly quartz inertial sensors. Since 2005, he is the head of the ”Sensors and Microtechnology Unit” in which are developed new generations of MEMS inertial measurement units, as well as thin film sensors for high temperature applications and new inertial instruments based on atomic interferometry.

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