Walter Lacarbonara

Achille Paolone

Fabriziovestroni

Fabrizio Vestroni

Nonlinear modal properties of nonshallow cables

Keywords: Nonshallow cables, nonlinear normal mode, direct method of multiple scales, reduced-order models

A nonlinear mechanical model of nonshallow linearly elastic suspended cables is employed to investigate the nonlinear modal characteristics of the free planar motions. An asymptotic analysis of the equations of motion is carried out directly on the partialdifferential equations overcoming the drawbacks of a discretization process. The direct asymptotic treatment delivers the approximation of the individual nonlinear normal modes.

General properties about the nonlinearity of the in-plane modes of different type -geometric, elasto-static and elasto-dynamic -are unfolded. The spatial corrections to the considered linear mode shape caused by the quadratic geometric forces are investigated for modes belonging to the three mentioned classes. Moreover, the convergence of Galerkin reduced-order models is discussed and the influence of passive modes is highlighted.

Introduction

The linear and nonlinear dynamics of suspended elastic cables have received considerable attention due to their use in several applications in the fields of communications, electricity, mooring systems, transportation, and crane-operation systems. The linear vibration theory of suspended cables is attributed to the work of Irvine and Caughey [START_REF] Irvine | The linear theory of free vibrations of a suspended cable[END_REF], Irvine [START_REF] Irvine | Cable Structures[END_REF], and Triantafyllou and co-workers [START_REF] Burgess | The elastic frequencies of cables[END_REF][START_REF] Triantafyllou | The dynamics of taut inclined cables[END_REF]. The modal properties of shallow cables have been shown to depend on one elasto-geometric parameter, the so-called Irvine's parameter.

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Free nonlinear vibrations were studied in [START_REF] Luongo | Planar non-linear free vibrations of an elastic cable[END_REF][START_REF] Cheng | Closed-form vibration analysis of sagged cable/mass suspensions[END_REF] to mention only a few works. Several studies dealt either theoretically or experimentally with harmonically forced oscillations both for the nonresonant and the resonant cases, the latter including a plethora of modal interactions. In particular, it has been demonstrated that the responses of shallow suspended cables near the first crossover exhibit complex behavior due to the presence of multiple internal resonances involving in-plane and out-of-plane modes. Examples include the coexistence of different types of periodic motions and the occurrence of quasiperiodic and chaotic oscillations [START_REF] Lee | Three-dimensional oscillations of suspended cables involving simultaneous internal resonances[END_REF]- [START_REF] Rega | Nonlinear vibrations of suspended cables -Part I: Modeling and analysis. Part II: Deterministic phenomena[END_REF].

The great majority of the works, especially those addressing nonlinear vibrations, deals with shallow cables described by approximate mechanical models based on the static condensation of the longitudinal dynamics. In particular, a characterization of the nonlinear normal modes of shallow cables has been addressed in [START_REF] Lacarbonara | Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems[END_REF][START_REF] Arafat | Non-linear responses of suspended cables to primary resonance excitations[END_REF]. An extensive and updated review of the state of the art on shallow cables can be found in [START_REF] Rega | Nonlinear vibrations of suspended cables -Part I: Modeling and analysis. Part II: Deterministic phenomena[END_REF].

On the contrary, a few studies have addressed linear and nonlinear dynamic behaviors of nonshallow cables whereas nonshallow configurations may occur in a number of engineering applications such as in cables used for cable railways, transmission lines, mooring lines or tag-lines. Hence, there is a practical and theoretical interest in investigating nonshallow cable configurations and the leading dynamics around them.

In [START_REF] Lacarbonara | Shallow versus nonshallow cables: linear and nonlinear vibration performance[END_REF][START_REF] Lacarbonara | Elastodynamics of nonshallow suspended cables: Linear modal properties[END_REF], a nonlinear mechanical model of nonshallow cables, describing the fully coupled longitudinal and transverse dynamics, was presented. Therein, results of the investigations into the spectral properties of linear free vibrations around the catenary configurations were reported. Differently from shallow cables, whose linear dynamics depend solely on Irvine's parameter, it is shown that the linear vibration properties of nonshallow cables depend on two parameters separately regulating the cable elastic and geometric stiffnesses. Among other properties, it was also determined where, in parameter space, the three classes of modes appear, namely, geometric, elasto-static and elasto-dynamic modes which were already partly mentioned in [START_REF] Burgess | The elastic frequencies of cables[END_REF]. It was demonstrated that the elastic modes belong to a complete sequence of symmetric and skew-symmetric stretching modes starting

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from the lowest elasto-static stretching mode with nearly constant elongation. These modes are manifested in the neighborhood of the various crossovers -the well-known (elasto-static) lowest crossovers and the highlighted higher-order (elasto-dynamic) crossovers. Away from these crossovers, the modes are geometric modes, in the sense that they are prevalently governed by the geometric stiffness, and exhibit leading transverse displacements.

In this paper, the primary focus is on the effects of the geometric nonlinearity on the modal properties; in particular, the objective is to characterize the nonlinear properties of the individual in-plane modes with a clear effort towards unfolding general properties.

Further, the a priori knowledge of the nonlinear modal properties is the basis for the prediction of the features of unimodal forced responses and interaction phenomena (the type of bifurcations, the possible routes to chaotic solutions,...).

The nonlinear partial-differential equations of motion and boundary conditions are recast in first-order form to make the employed asymptotic scheme suitable for higher-order approximations. Then, the asymptotic analysis of individual nonlinear normal modes is presented. The main results on the nonlinear characteristics of the modal motions are summarized. They mostly relate to the so-called effective nonlinearity coefficient of the considered mode which regulates the bending of the backbone and to the shape functions dictating the spatial corrections to the considered linear mode shape at second order. General conclusions are drawn about the nonlinear laws of the modes, depending on whether they are geometric, elasto-static or elasto-dynamic.

Equations of motion

We denote (O, i, j, k) the orthonormal basis of a fixed inertial reference frame with origin in O (Fig. 1), let N * 0 and N * describe the static axial force due to gravity in the initial configuration C 0 and the incremental dynamic force arising in the change from the initial to the current configuration C. Imposing the balance of linear and angular momentum yields

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the equation of free undamped motions as

∂ [N * 0 (a -â0 )] ∂x * + ∂ (N * a) ∂x * = m 0 sec θ 0 ∂ 2 u * ∂t * 2 (1)
where x * indicates the horizontal coordinate along the fixed i direction. In Eq. ( 1), m 0 is the mass per unit cable length in its initial configuration C 0 lying in the (i, j)-plane -here expressed as p * 0 (x * ) = x * i + y * (x * )j; u * is the displacement vector, henceforth conveniently decomposed as u * (x * , t * ) = û * + w * k with û * being the in-plane displacement (i.e., lying in the (i, j)-plane); â0 (x * ) is the unit tangential vector in C 0 and θ 0 (x * ) is the angle between â0 and i given by θ 0 (x * ) = arctan(dy * /dx * ); a is the unit vector tangent to the current configuration of the cable axis. Henceforth, the same notation will be employed throughout the manuscript, namely, lowercase bold italic letters indicate vectors in E 3 , hatted lowercase bold italic letters denote vectors lying in the (i, j)-plane, and lowercase bold italic letters denote the corresponding algebraic vectors.

The cable axial dilatation associated with the deformation from

C 0 to C is ν = dp * ds * = cos θ 0 dp * dx * = cos θ 0 (1 + u ) 2 + (tan θ 0 + v ) 2 + w 2 (2)
where |.| represents the magnitude of the vectorial argument and the prime indicates differentiation with respect to the nondimensional coordinate x = x * , s * is the arclength along the cable axis in C 0 . Further, the displacement components are nondimensionalized as u := u * , v := v * , w := w * . The unit vector in the current tangential direction is expressed as

a := p |p | ≡ cos θ 0 (1 + u )i + (tan θ 0 + v )j + w k ν (3)
Due to the relatively high axial stiffness of typical engineering cables, the initial configuration of the cable, represented by the catenary, and the axial load are, respectively,

y(x) = 1 γ cosh γ 2 -cosh γ 1 2 -x , N 0 (x) = cosh γ 1 2 -x ( 4
)
where y := y * , γ := mg H * 0 is solution of the geometric compatibility condition sinh γ 2 = γ 2 η 0 with η 0 := L 0 , L 0 is the initial total length of the cable, N 0 (x) :=

N * 0 (x * ) H * 0
, and H * 0 is the

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horizontal projection of N * 0 . The extensibility of the cable under its own weight is neglected as it is typically done for engineering cables [START_REF] Irvine | Cable Structures[END_REF] since the elastic elongation is indeed very small, hence, uninfluential on the static equilibrium. On the contrary, for cables with a low elastic stiffness relative to the geometric stiffness, the cable elastic deformation mode should be accounted for. This can be shown rigorously for an elastic cable subject to its own weight. The compatibility condition delivering the horizontal component of the equilibrium axial force is

η 0 k + 2 γ sinh -1 η 0 γ 2 = 1 (5)
where

k := E A 0 H * 0
, E is Young's modulus of elasticity and A 0 is the area of the undeformed cable cross section. When k is sufficiently large, the equation reduces to that governing the catenary, obtained neglecting the cable elasticity.

A linear constitutive elastic law relating the incremental axial load to the axial strain is adopted in the form N * = EA 0 (ν -1) where ν -1 =: e is the axial elongation.

(sec θ 0 )ü - cos θ 0 ν N 0 (u -ν + 1) + N (1 + u ) = 0 (sec θ 0 )v - cos θ 0 ν N 0 (v -tan θ 0 (ν -1)) + N (v + tan θ 0 ) = 0 (sec θ 0 ) ẅ -(N 0 + N ) cos θ 0 ν w = 0 (6)
The equations of motion are supplemented with the boundary conditions u(0, t) = 0 and u(1, t) = 0, for a cable suspended from two supports at the same level (Fig. 1).

In [START_REF] Lacarbonara | Shallow versus nonshallow cables: linear and nonlinear vibration performance[END_REF][START_REF] Lacarbonara | Elastodynamics of nonshallow suspended cables: Linear modal properties[END_REF] it was shown that, differently from shallow cables as far as the linear dynamics are concerned, nonshallow cables are governed by two independent parameters, relating to the geometric and elastic stiffness, namely, γ and k or λ := γ k ηe with η e := 1 0 cos 3 θ 0 dx. In [START_REF] Lacarbonara | Elastodynamics of nonshallow suspended cables: Linear modal properties[END_REF] it was shown that typical values of k for engineering cables are within the range [10 3 , 10 4 ].

A c c e p t e d m a n u s c r i p t 3 Asymptotic analysis of individual nonlinear normal modes

An asymptotic approach is well-suited to characterize the nonlinear properties of the individual nonlinear normal modes of cables. This type of nonlinear analysis can not be pursued via standard numerical solution approaches such as those based on weak formulations (finite elements, weighted residuals) unless a good initial estimate of the nonlinear modes is available. When the individual nonlinear modal manifolds of a nonlinear unforced and undamped system are not known a priori, the numerical responses would be affected by many modes (in principle, all the modes captured by the numerical discretization). The ensuing numerical responses would not be helpful for disclosing the featured nonlinear behaviors of individual normal modes. Furthermore, the employed direct asymptotic expansion is the most accurate local solution approach since it does not discard any spatial information of the motion but it embodies all of it into the solutions of a few boundary-value problems at second order.

The cable motions are assumed to occur in the neighborhood of the initial configuration C 0 and are such that the cable tension never vanishes. The cable responses are consequently described seeking the solutions of the following third-order Mac Laurin series expansion of the equation of motion:

sec θ 0 ü -[N 0 a 1 + ke 1 â0 ] -[N 0 a 2 + ke 2 â0 + 2ke 1 a 1 ] -[N 0 a 3 + ke 3 â0 + 3ke 2 a 1 + 3ke 1 a 2 ] = 0 (7)
where e j and a j are the jth-order terms of e and a, respectively. Letting u := (u, v, w)

and v := ( u, v, ẇ) represent the algebraic vectors associated to the Euclidean displacement and velocity vectors, the ensuing equations can be cast as:

u -v = 0 I v + Lu = N 2 (u, u) + N 3 (u, u, u) (8)
along with the boundary conditions previously described.

In [START_REF] Nayfeh | Multimode interactions in suspended cables[END_REF], I = (sec θ 0 )(x)I (I is the identity tensor) and L denote the inertial and linear elasto-geometric operators, respectively. Namely, L := diag[ L, L] due to the uncoupling

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of the equations of motion where L and L denote the in-plane and out-of-plane stiffness operators, respectively (reported in [START_REF] Lacarbonara | Elastodynamics of nonshallow suspended cables: Linear modal properties[END_REF]). On the other hand, the quadratic and cubic restoring forces, in operator form, are

N 2 (u, u) =[N 0 a 2 (u, u) + ke 2 (u, u)â 0 + 2ke 1 (u)a 1 (u)] N 3 (u, u, u) =[N 0 a 3 (u, u, u)] + [ke 3 (u, u, u)â 0 + 3ke 2 (u, u)a 1 (u) + 3ke 1 (u)a 2 (u, u)] (9)
Their componential forms are given in Appendix A. For computational reasons, it is worth

noting that N 2 is non-commutative, i.e., N 2 (u, w) = N 2 (w, u).
In the following, the analysis is first developed for individual planar modes, away from internal resonances. To this end, the ansatz on the form of the solutions of Eqs. ( 8) is

û(x, t) = 3 k=1 k ûk (x, t 0 , t 2 ) + • • • , v(x, t) = 3 k=1 k vk (x, t 0 , t 2 ) + • • • ( 10
)
where û := (u, v) , v := ( u, v) are the restrictions of the displacement u and velocity v to the (i, j)-plane, t 0 := t is the fast time scale, t 2 := 2 t is the stretched time scale, and is a small dimensionless number introduced to measure the order of magnitude of the deviations from the initial configuration C 0 . Then, the first derivative with respect to time is defined as

∂/∂t = D 0 + 2 D 2 + • • • where D k := ∂/∂t k .
The solution is considered independent of the slow time scale t 1 = t because no resonant terms arise at second order away from 2:1 internal resonances.

Substituting [START_REF] Rega | Nonlinear vibrations of suspended cables -Part I: Modeling and analysis. Part II: Deterministic phenomena[END_REF] into the system of first-order (in time) equations of motion and boundary conditions, using the independence of the time scales, and equating coefficients of like powers of yields

Order : D 0 û1 -v1 = 0 Î(D 0 v1 ) + Lû 1 = 0 (11)

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Order 2 :

D 0 û2 -v2 = 0 Î(D 0 v2 ) + Lû 2 = N 2 (û 1 , û1 ) (12)
Order 3 :

D 0 û3 -v3 = -D 2 û1 Î(D 0 v3 ) + Lu 3 = -D 2 v1 + N 2 (û 1 , û2 ) + N 2 (û 2 , û1 ) + N 3 (û 1 , û1 , û1 ) (13)
where N 2 and N 3 denote the restrictions of the quadratic and cubic forces to the (i, j)plane, and we further note that use of the non-commutativity of N 2 was made.

Because the considered mode is away from internal resonances with other modes, the solution at order can be assumed as

û1 = A m (t 2 )e iωmt 0 φ m (x) + cc, v1 = iω m A m (t 2 )e iωmt 0 φ m (x) + cc (14)
where ω m is the linear eigenfrequency of the mth in-plane mode; i := √ -1, and A m ∈ C is the complex-valued amplitude of the mode at leading order; cc denotes the complex conjugate of the preceding terms. Substituting ( 14) into the second-order problem, Eq. ( 12), yields

D 0 û2 -v2 = 0 Î(D 0 v2 ) + Lu 2 = A 2 m e 2iωmt 0 + A m Ām Q(x) + cc ( 15
)
where the bar indicates the complex conjugate and Q(x) := N 2 (φ m , φ m ). The particular solution of the second-order problem can be expressed as

û2 =A 2 m e 2iωmt 0 Φ (∞) (x) + A m Ām Ψ (∞) (x) + cc v2 =2iω m Φ (∞) (x)A 2 m e 2iωmt 0 + cc (16)
where the functions Φ (∞) and Ψ (∞) are solutions of the following boundary-value problems:

L(x)Φ (∞) (x) -4ω 2 m I(x)Φ (∞) (x) = Q(x), L(x)Ψ (∞) (x) = Q(x) (17)
where, for sake of clarity, the space-dependence of the inertial and stiffness operators has been made explicit. The boundary conditions are

Φ (∞) (0) = 0 and Φ (∞) (1) = 0, Ψ (∞) (0) = 0 and Ψ (∞) (1) = 0.

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Since the inertial and stiffness operators depend on the position coordinate x, closedform solutions of (17) do not exist in general. However, the solutions can be conveniently expressed in series form in terms of the eigenfunctions as follows:

Φ (∞) (x) = ∞ j=1 α j φ j (x), Ψ (∞) (x) = ∞ j=1 β j φ j (x) (18)
where

α j := 1 [(ω j ) 2 -4(ω m ) 2 ] 1 0 φ j (x) • Q(x) dx , β j := 1 (ω j ) 2 1 0 φ j (x) • Q(x) dx (19)
and the dot indicates the standard inner product. We observe that the coefficient in Eq.

(19) 1 diverges when ω j = 2ω m , hence the displacement would grow indefinitely large, due to the fact that a two-to-one internal resonance between the jth and mth mode may be activated and the expansion based on the individual mode breaks down. In this case, a two-mode expansion is necessary to properly account for the interaction as it is discussed further on. Substituting Eq. ( 16) into the third-order problem and imposing the solvability condition [18] at this order yields the following modulation equation:

i 4 Ȧm = Γ (∞) A 2 m Ām (20)
The effective nonlinearity coefficient Γ (∞) regulates the bending of the backbone of the cable oscillating in the mth mode, that is, the nonlinear frequency variation with the amplitude

ω (∞) m := ω m -Γ (∞) a 2 m ( 21
)
The coefficient can be regarded as a nonlinear modal constitutive parameter embodying the combined modal effects of the quadratic and cubic forces as it is given by

Γ (∞) :=Γ (∞) 2 + Γ 3 , Γ (∞) 2 := 1 8ω m 1 0 φ m (x) • F (∞) 2 (x) dx, Γ 3 := 1 8ω m 1 0 φ m (x) • F 3 (x) dx (22)

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The vectors F (∞) 2 (x) and F 3 (x) represent the resonant geometric forces at third order arising from the overall quadratic and cubic nonlinear forces, respectively, and are expressed as

F (∞) 2 (x) := N 2 (Φ (∞) , φ m ) + N 2 (φ m , Φ (∞) ) + 2 N 2 (Ψ (∞) , φ m ) + 2 N 2 (φ m , Ψ (∞) ) F 3 (x) :=3 N 3 (φ m , φ m , φ m ) (23) In Eq. (22), Γ (∞) 2
denotes the softening-type contribution of the quadratic forces which, in principle, depends -through the second-order functions Φ (∞) and Ψ (∞) -on all of the cable eigenfunctions as emphasized by the superscript ∞, whereas Γ 3 is the hardening-type contribution of the cubic forces depending only on the considered active mode ([18], [16]).

The second-order functions regulate the spatial corrections to the linear mode shape

φ m in the displacement field û(∞) (x, t) =a m cos(ω (∞) m t + ψ m )φ m (x) + a 2 m 2 cos(2(ω (∞) m t + ψ m ))Φ (∞) (x) + Ψ (∞) (x) (24)
where ψ m is a constant. It is clear that the cable oscillates with frequency ω (∞) m around the displaced configuration given by p = p0 + 1/2 a 2 m Ψ (∞) (x). As it was pointed out in [START_REF] Nayfeh | Nonlinear normal modes of a continuous system with quadratic nonlinearities[END_REF], the calculated individual nonlinear normal modes are not synchronous and, hence, do not possess all the features of the original definition given by Rosenberg [19].

Instead of the direct perturbation approach, we can employ a full-basis Galerkin discretization, û(x, t) = ∞ j=1 q j (t)φ j (x), and, subsequently, the method of multiple scales may be applied to the resulting infinite-dimensional set of ODE's. It turns out that the cubic part of the effective nonlinearity coefficient is the same as in Eq. ( 22) whereas the contribution from the quadratic forces is expressed as

[16] Γ (∞) 2 = ∞ j=1 S j , S j = 1 8ω m (Λ mmj + Λ mjm ) 2Λ jmm ω 2 j + Λ jmm ω 2 j -4ω 2 m , Λ jkh := 1 0 φ j • N 2 (φ k , φ h ) dx (25)
In agreement with our previous observation, the coefficient in Eq. (25) diverges when ω j = 2ω m due to a 2:1 internal resonance between the jth and mth mode. To account for

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the 2:1 interaction, the generating solution must include the two interacting modes and a third-order expansion has to be pursued. The expansion was obtained for a general system with quadratic and cubic nonlinearities in [START_REF] Lacarbonara | Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems[END_REF]. Here, we summarize only the modulation equations governing the slow variations of the amplitudes and phases of the two interacting modes; that is,

i 4 Ȧm = 1 8ω m (Λ mmn + Λ mnm ) A n Ām e i δ t + Γ∞ A 2 m Ām + 1 8ω m Γ∞ mn A m A n Ān i 4 Ȧn = 1 8ω n Λ nmm A 2 m e -i δ t + ∆∞ A 2 n Ān + 1 8ω n Γ∞ mn A n A m Ām (26)
where A m and A n are the complex-valued amplitudes of the interacting modes at first order, δ is a small parameter expressing the detuning of the internal resonance, ω n = 2ω m + δ.

The softening part of the nonlinearity coefficient of the mth mode, Eq. ( 25), is modified as follows in the presence of the 2:1 resonance: the summation does not include the nth term (i.e., the term corresponding to the high-frequency mode) which, on the contrary, is

9/(4ω 2 n )Λ nmm (Λ mmn + Λ mnm ) /(8ω m ).
Here and henceforth, the modified coefficient will be denoted Γ(∞) . The other coefficients appearing in Eq. ( 26) are given in Appendix B.

On the other hand, when a truncation to the lowest M modes is performed, a reducedorder model yields an effective nonlinearity coefficient given by

Γ M = M j=1 S j + Γ 3 (27)
and, in turn, the nonlinear frequency is

ω M m := ω m -Γ M a 2 m .

Limits of validity of the no-compression cable model

The adopted cable model does not account for the fact that, when the tensile force vanishes, the cable can not resist compression. Several studies have addressed the effect of cable loosening on nonlinear vibrations of shallow cables [20]. Here, we are interested in calculating the amplitude range of validity of the no-compression cable model. To this end, the elongation, to within second order, is expressed as

e m (x, t) = a m cos ϕ m (t)Υ 1 (x) + a 2 m [cos 2ϕ m (t)Υ 2 (x) + Υ 3 (x)] (28)

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where ϕ m (t) := ω

(∞) m t + ψ m Υ 1 (x) = cos θ 2 0 φ 1(m) + tan θ 0 φ 2(m) Υ 2 (x) = 1 2 cos θ 2 0 Φ 1 + tan θ 0 Φ 2 + 1 2 cos θ 2 0 φ 1(m) -tan θ 0 φ 2(m) 2 Υ 3 (x) = 1 2 cos θ 2 0 Ψ 1 + tan θ 0 Ψ 2 + 1 2 cos θ 2 0 φ 1(m) -tan θ 0 φ 2(m) 2 (29)
Here, φ j(m) (j = 1, 2) indicates the jth component of the mth mode, Φ j and Ψ j are the jth components of the functions Φ (∞) and Ψ (∞) , respectively. Then, the total nondimensional tension is

N (x, t; a m ) = N 0 (x) + k e m (x, t; a m ) ( 30
)
The tension is assumed to attain the minimum when the modal elongation attains its maximum (i.e., the elastic cable shortening e m = ν m -1 is maximum) at x m (e.g., x m = 1 2 or 1 4 for the first symmetric and skew-symmetric modes, respectively). Hence, N (x m , t; a) =: Ň (t; a m ) becomes a function of time t and the amplitude a m at the natural frequency.

To determine the minimum of Ň (t; a m ), the extremal values of ěm := e m (x m , t; a m ) are sought. They are obtained when (i) sin ϕ m (t) = 0 or (ii) cos ϕ m (t) = -Υ1 /(4a m Υ2 ) where Υj := Υ j (x m ). In the first case,

ěm = ±a m Υ1 + a 2 m Υ2 + Υ3 (31)
and the amplitude that makes Ň (a) vanish is

a m = 1/2 ± Υ1 ± ( Υ1 ) 2 -4/k Ň0 ( Υ2 + Υ3 ) ( Υ2 + Υ3 ) -1 (32) provided that Υ 2 1 ≥ 4/k| Ň0 ( Υ2 + Υ3 )|. In case (ii), ěm = a 2 m ( Υ3 -Υ2 ) Υ2 - Υ 2 1 8 ( 33
)
and the amplitude where Ň(a) vanishes is

a m = Υ 2 1 8 Υ2 -Ň0 Υ3 -Υ2 -1 (34)
provided that the argument of the square root be greater than zero. Consequently, the admissible upper bound of the amplitude for maintaining positive tension in the cable is the minimum of the positive values expressed by Eqs. (32) and (34).

A c c e p t e d m a n u s c r i p t 4 Nonlinear modal motions

The nonlinear properties of the individual modes are investigated with specific efforts towards unfolding general properties. Besides regulating the system free nonlinear oscillations, these nonlinear modal properties allow also to predict the features of the unimodal forced responses or interactions. The nonlinear laws of the modes, as discussed in the preceding section, are dictated by the effective nonlinearity coefficient Γ (∞) . Its investigation is conducted under different static regimes and for different cable parameters generalizing previous results on shallow cables. Further, the convergence of the effective nonlinearity coefficient is studied outlining the reliability of low-dimensional Galerkin-reduced models.

Thereafter, the nonlinear modifications of the modal motions are discussed.

The nonlinearity of the individual normal modes

The effective nonlinearity coefficient of low-order modes belonging to the three classes of modes are considered; namely, geometric, elasto-static and elasto-dynamic modes. In the (γ, λ)-plane (Fig. 2b), we consider three regions corresponding to three principal static regimes [START_REF] Lacarbonara | Elastodynamics of nonshallow suspended cables: Linear modal properties[END_REF]: γ ∈ [0, 0.5] is the region of shallow profiles (the dark grey region in Fig. 2b) whereas γ > 1 is identified with the region of nonshallow profiles (the lightly shaded region in Fig. 2b). The interposed region, γ ∈ [0.5, 1], is a transition region between the two static regimes where the cable is considered as being neither shallow nor nonshallow. These regions ensue from appreciating how the sag-to-span ratio d varies with γ (Fig. 2a). Further, in In Fig. 4, the nonlinearity of the second symmetric mode (m = 3) of the transition cables is investigated. This mode undergoes a crossover with the second skew-symmetric mode for λ slightly below 4π where the mode becomes elasto-static. For low values of λ the mode is slightly hardening, thereafter around the crossover it becomes softening.

There are divergences for 2:1 internal resonances between the considered mode and the seventh or ninth modes, respectively, occurring in a very narrow parameter range. Further increasing λ, the mode (above the crossover, m = 4) becomes hardening again with a strong divergence in the region around the 2:1 internal resonance with the ninth mode. The coefficient Γ(∞) accounting for the interaction is always negative indicating a hardening behavior. In the interaction region, there occurs a crossover between the frequencies of the ninth and the tenth mode indicating that the ninth mode is elasto-static. The nonlinearity

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of the considered mode (m = 3) decreases significantly due to the softening contribution delivered by the coupled high-frequency elasto-static mode.

Then, the nonshallow regime, γ = 1.5, is investigated in the lowest symmetric modes;

namely, the third symmetric mode, m = 5 (Fig. 5), the fourth symmetric mode, m = 7 (Fig. 6a), and the ninth symmetric mode, m = 17 (Fig. 6b). The third symmetric mode (Fig. 5) is hardening in the whole range except for the region where it becomes elasto-static around λ = 5.7π while undergoing a crossover with the third skew-symmetric mode. In Fig. 6a, the nonlinearity of the fourth symmetric mode predicted with Γ (∞) follows the same pattern as that of the second symmetric mode of the transition cable; it is hardening, then around the crossover it becomes softening, then hardening again and diverges due a 2:1 resonance with the ninth symmetric mode (n = 17). In this case, Γ(∞) indicates that, in the interaction region, the mode is hardening then softening with a change of curvature occurring where the interacting mode undergoes a crossover. Away from the resonance region, the mode regains its hardening signature. As shown in Fig. 6b, the high-frequency interacting mode is hardening except for the region where it undergoes the crossover thus attaining its peak elasto-static strain energy; hence, its softening contribution to the low-frequency coupled mode renders this mode softening while engaged in the interaction.

Finally, it is of interest to investigate the lowest elasto-dynamic mode exhibited by the nonshallow cable, with γ = 1.5. The considered mode is the seventeenth and its effective nonlinearity coefficient variation is shown in Fig. 7. This mode is hardening except for the region around its crossover with the eighteenth mode where it appears to be softening. This region corresponds to that where the mode becomes elasto-dynamic possessing a longitudinal motion much larger than the transverse component as shown in the modal displacements for λ = 6.81 π (right below the elasto-dynamic crossover). On the other hand, the mode is predicted as hardening with the one-mode discretization where it is indeed softening. However, where the coefficient becomes nearly vanishing, the one-mode discretization suggests the need of more passive modes in the discretization.

A c c e p t e d m a n u s c r i p t 4.2 Spatial characteristics of the nonlinear modal motions

In this section, the quadratic spatial corrections to the active linear mode shape are discussed with the aim of quantifying the extent of its influence and the spectral content of these nonlinear corrections.

The displacement field at the instant of time when the velocity field vanishes takes the form (24) with cos ϕ m (t) = cos 2ϕ m (t) ≡ 1. In Fig. 8, the lowest (first skew-symmetric) mode of a nonshallow cable, when γ = 1.5 and λ = 7 π, is considered. The u and v displacement components of the first-and second-order approximations are shown in parts (a) and (b), respectively, with the amplitude corresponding to the maximum admissible value (such that the corresponding motion maintains non-negative tension in the cable). It can be observed that the quadratic nonlinearities introduce appreciable spatial distortion into the shape of the leading motion making the horizontal symmetric and the vertical skew-symmetric displacement components neither symmetric nor skew-symmetric. To gain more insight into the characteristics of the motion, the modal coefficient α k in Φ (∞) and the function itself, and the coefficient β k and Ψ (∞) are shown in parts (c) and (d), respectively. The considered skew-symmetric mode generates symmetric quadratic forces, hence, contributions from the even (symmetric) modes only are expected. However, besides the second, fourth and sixth mode shapes, there is a meaningful contribution of the seventh mode shape and minor contributions of the ninth and eleventh modes. The leading contributing modes, namely the sixth and seventh modes, are on the ramping parts of the ω nλ curves (see Fig. 2d),

where, they turn out to be both symmetric elasto-static modes thus breaking down the natural sequence of symmetric and skew-symmetric modes as illustrated in [START_REF] Lacarbonara | Elastodynamics of nonshallow suspended cables: Linear modal properties[END_REF]. In Fig. 9, we show how these signatures reflect themselves onto the convergence of the effective nonlinearity coefficient. As expected, the main contribution to the softening part of the effective nonlinearity coefficient comes from the two elasto-static modes, the sixth and seventh modes. The symmetric quadratic forces generate stretching of the cable axis which is almost entirely captured by the lowest symmetric elasto-static modes.

In Fig. 10, again the displacement field of the lowest mode (skew-symmetric) is shown In this case, although with the same sag-to-span ratio, the cable possesses a higher axial elastic stiffness. Besides the second mode, all of the even modes up to, and including the twelfth mode, contribute although a remarkable contribution comes from the twelfth and thirteenth modes. The reason for this participation is the same as in the previous case;

that is, the twelfth and thirteenth modes are the lowest two elasto-static symmetric modes.

Similarly, these modes determine the convergence of the effective nonlinearity coefficient.

Concluding remarks

A mechanical model describing finite motions of nonshallow cables around their initial cate- Moreover, the nonlinear shape modifications of the modal motions have been studied

with a particular attention on the modal spatial content. It was found that quadratic modal forces involve modes with stretching, hence those modes that belong to the family of elastostatic modes whose frequency locus unfolds along the successive lowest crossovers. From the convergence analysis, the contribution of these modes was shown to be important; therefore, they must be included in the nonlinear description by reduced-order models, although they are far from the considered individual modes.

The issues pointed out about the suitability of reduced-order models of nonshallow cables are felt to be general and applicable also to those systems possessing stiffnesses/strain energies of different type and different orders of magnitude such as arches, membranes and shells.

A c c e p t e d m a n u s c r i p t A c c e p t e d m a n u s c r i p t A The nonlinear force and strain expansions

The quadratic and cubic geometric force operators are

N 2 (u, u) • i = cos θ 0 (k -N 0 ) e 2 1 -e 2 -e 1 u N 2 (u, u) • j = cos θ 0 (k -N 0 ) tan θ 0 e 2 1 -e 2 -e 1 v N 2 (u, u) • k = cos θ 0 (N 0 -k) w e 1 (35) N 3 (u, u, u) • i = cos θ 0 (N 0 -k) e 3 1 -2e 1 e 2 + e 3 -e 2 1 u + e 2 u N 3 (u, u, u) • j = cos θ 0 (N 0 -k) tan θ 0 e 3 1 -2e 1 e 2 + e 3 -e 2 1 v + e 2 v N 3 (u, u, u) • k = cos θ 0 (k -N 0 ) e 2 1 -e 2 w ( 36
)
The first-, second-, and third-order strains are

e 1 = cos 2 θ 0 (u + tan θ 0 v ) e 2 = 1 2 cos 4 θ 0 (u tan θ 0 -v ) 2 + w 2 cos 2 θ 0 e 3 = - 1 2 cos 6 θ 0 (u + tan θ 0 v ) (u tan θ 0 -v ) 2 + w 2 cos 2 θ 0 (37)

B 2:1 interaction coefficients

The coefficients in Eq. ( 26) are [START_REF] Lacarbonara | Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems[END_REF] ∆

(∞) = 1 8ω n ∞ j=1 (Λ nnj + Λ njn ) 2Λ jnn ω 2 j + Λ jnn ω 2 j -4ω 2 n + 3 8ω n Θ nnnn (38) Γ(∞) mn = ∞ j=1,j =m [(Λ mjn + Λ mnj ) (Λ jmn + Λ jnm ) × 1 ω 2 j -9ω 2 m + 1 ω 2 j -ω 2 m + 2Λ jnn ω 2 j (Λ mmj + Λ mjm ) + (Λ mmn + Λ mnm ) 2 8ω 2 m + 4 ω 2 m (Λ mnn Λ mmm ) + 2 (Θ mnnm + Θ mnmn + Θ mmnn )

0 m 0 2 ,

 02 Introducing a suitable nondimensional time t := ω c t * with ω c := H * and the nondimensional axial force N := N * H * 0 = k (ν-1), the ensuing nondimensional equations of motion, in componential form, are then expressed as

Fig. 2b , 2 and k 3 = 5 •A c c e p t e d m a n u s c r i p t

 2b235 Fig. 2b, the region of admissible elastic stiffness k in the (γ, λ)-plane is shown. The isostiffness curves are drawn according to the definition of Irvine's parameter. The two lateral thick curves denote the boundaries of the admissible region and correspond to k 1 = 5 • 10 2 and k 3 = 5 • 10 4 , respectively. In Figs. 2c and 2d, the frequencies of the transition cables with γ = 0.75 and those of nonshallow cables with γ = 1.5 are shown, respectively. The shaded area denotes regions of non physically admissible cable parameters. The nonlinear characteristics of the modal properties are discussed considering variations of the effective nonlinearity coefficient with Irvine's parameter λ and fixed values

A c c e p t e d m a n u s c r i p t

  along with the coefficients α k (and Φ(∞) ) and β k (and Ψ(∞) ) when γ = 1.5 and λ = 12.6 π.

A c c e p t e d m a n u s c r i p t

  nary configurations has been employed to investigate the nonlinear vibration characteristics of individual in-plane modes. An asymptotic treatment based on the method of multiple scales has been applied directly to the partial-differential equations of motion and boundary conditions, overcoming the drawbacks of a discretization process.The nonlinear characteristics of the modes of different type have been studied, namely, those relating to geometric modes (with prevalent transverse motion and negligible stretching), elasto-static modes (with prevalent transverse motion and appreciable stretching), elasto-dynamic modes (with prevalent longitudinal motion and stretching).The general results about the investigated nonshallow cables, in line with known results on shallow cables, indicate that the geometric modes are hardening. Conversely, in the neighborhood of the localized regions where the frequencies undergo crossovers, the modes suffer a transition into elasto-static or elasto-dynamic modes; this transition, occurring within the linear eigenvalue structure, further makes the nonlinear mode of the softeningtype. The physical phenomenon inherent in the change of the nonlinearity may be explained accounting for the fact that the relevant mode, around the crossovers, exhibits a shape with an appreciable transverse displacement inducing stretching which is quite sensitive to the upward or downward displacement directions. Indeed, a significant drift is caused by the quadratic geometric forces towards the upper configurations where the tension in the cable can vanish thus leading to unsymmetrical softening behavior.

A c c e p t e d m a n u s c r i p t where Θ ijkh = 1 0A c c e p t e d m a n u s c r i p t 10 AFigure 1 :

 1101 Figure

Figure 2 : 1 = 2 . 5 • 10 2 and k 2 = 5 • 10 4 .

 2125254 Figure 2: a) Variation of γ with the sag-to-span ratio d and (b) region of admissible physical nondimensional parameters in the γ-λ plane. The curves indicate iso-stiffness curves: k 1 = 2.5 • 10 2 and k 2 = 5 • 10 4 . Variation of the frequencies of the in-plane modes with λ π when (c) γ = 0.75 and (d) γ = 1.5. A (S) denotes skew-symmetric (symmetric) modes.

4 Figure 4 :Figure 5 :

 445 Figure 4: Variation of Γ (∞) , Γ(∞) and Γ (m) with λ/π of the third mode (m = 3) when γ = 0.75.

A c c e p t e d m a n u s c r i p t 2

 2

Figure 6 :

 6 Figure

A c c e p t e d m a n u s c r i p t 4 Figure 7 :Figure 9 :Figure 10 : 1 ,

 479101 Figure 7: Variation of Γ (∞) and Γ (m) with λ/π of the lowest elasto-dynamic mode (m = 17) of a nonshallow cable (γ = 1.5).

Acknowledgment

This work was partially supported under a FY 2005-2006 PRIN Grant from the Italian Ministry of Education, University and Scientific Research.

A c c e p t e d m a n u s c r i p t

[16] Lacarbonara, W., Direct treatment and discretizations of nonlinear spatially continuous systems, Journal of Sound and Vibration 221 (1999) 849-866.

[17] Antman, S.S., Nonlinear problems of elasticity (2005) Springer-Verlag, New York.

[18] Nayfeh, A. H., Lacarbonara 2 ) when γ = 1.5 and λ = 7 π.


R Boisgard

J P Aimé

G Couturier

Dynamic operation modes of AFM: Non-linear behavior and theoretical analysis of the stability of the AFM oscillator

The first part of this paper gives a theoretical study of the mechanics of contact of an AFM tip on viscous materials. Analytical expressions are derived showing the non-linear behaviors specifically related to the use of dynamic operation modes of AFM on viscous materials. A detailed analysis of the dissipated energy as a function of the tip indentation is presented. The second part is dedicated to a theoretical analysis investigating the domain of stability of the oscillator and the influence of the machine. The theoretical approach includes the electronic feedback loop used with the Frequency Modulation mode. Because the interaction between the tip and the sample produces a dynamical non-linear behavior, an unstable branch occurs that can change the stability of the oscillator. In particular, a sudden jump of the oscillating tip can be produced. In spite of the complexity of the problem, the analytical approach ends with two simple equations. The two equations provide an unambiguous way of discriminating between the contributions from the machine and the tip sample interaction.

A c c e p t e d m a n u s c r i p t I General Introduction

The dynamical modes of Atomic Force Microscopes (AFM) are commonly used to investigate a large variety of molecular systems and samples. The concept of dynamical modes is based on the oscillation of a tip cantilever system at, or near, the resonance frequency and on changes in the oscillating behavior as the tip is moved towards the surface [START_REF] Gleyzes | Bistable behavior of a vibrating tip near a solid surface[END_REF][START_REF] Albrecht | Frequency modulation detection using high-Q cantilevers for enhanced force microscope sensitivity[END_REF][START_REF] Zhong | Elings Fractured polymer/silica fiber surface studied by tapping mode atomic force microscopy[END_REF][START_REF] Garcia | Dynamic atomic force microscopy methods[END_REF][START_REF] Stocker | Surface Reconstruction of the Lamellar Morphology in a Symmetric Poly(styrene-block-butadiene-block-methyl methacrylate) Triblock Copolymer: A Tapping Mode Scanning Force Microscope Study Macromolecules[END_REF].

Dynamical modes were first conceived in order to minimize the contact between the tip apex and the surface. This way, images can be recorded with only slight intermittent contact over a cycle of oscillation or even with no contact at all. As a result, shear forces are significantly reduced. The emergence of dynamical modes was a breakthrough because of their ability to probe minute forces. AFM dynamical modes have opened avenues for routine imaging of soft materials [START_REF] Stocker | Surface Reconstruction of the Lamellar Morphology in a Symmetric Poly(styrene-block-butadiene-block-methyl methacrylate) Triblock Copolymer: A Tapping Mode Scanning Force Microscope Study Macromolecules[END_REF][START_REF] Leclère | Microdomain Morphology Analysis of Block Copolymers by Atomic Force Microscopy with Phase Detection Imaging[END_REF][START_REF] Magonov | Tapping-mode atomic force microscopy study of the near-surface composition of a styrene-butadiene-styrene triblock copolymer film[END_REF][START_REF] Leclère | Direct Observation of Microdomain Morphology in "All-Acrylic" Thermoplastic Elastomers Synthesized via Living Radical Polymerization[END_REF][START_REF] Konrad | Volume Imaging of an Ultrathin SBS Triblock Copolymer Film[END_REF][START_REF] Kopp-Marsaudon | Quantitative measurement of the mechanical contribution to tapping mode atomic force microscopy images of soft materials[END_REF][START_REF] Knoll | Tapping Mode Atomic Force Microscopy on Polymers: Where Is the True Sample Surface?[END_REF][START_REF] Dubourg | Dynamic Force Microscopic Study of a triblock copolymer with AFM non contact resonant mode[END_REF][START_REF] Reiter | Direct Visualization of Random Crystallization and Melting in Arrays of Nanometer-Size Polymer Crystals[END_REF][START_REF] Nony | DNA properties investigated by dynamic force microscopy[END_REF][START_REF] Dubourg | Apparent hardening of soft samples through Q factor change in AFM[END_REF][START_REF] Dubourg | Probing the relationship between the scales of space and time in an entangled polymer network with an oscillating nanotip[END_REF] and for achieving atomic resolution in ultra-high vacuum conditions [START_REF] Giessibl | Atomic resolution of the silicon (111)-(7x7) surface by atomic force microscopy[END_REF][START_REF] Sugawara | Defect motion on an InP (110) surface observed with Non Contact atomic force microscopy[END_REF][START_REF] Kitamura | Observation of 7× 7 Reconstructed Structure on the Silicon (111) Surface using Ultrahigh Vacuum Noncontact Atomic Force Microscopy[END_REF][20][START_REF] Lantz | Low Temperature Scanning Force Microscopy of the Si(111)-(7 x 7) Surface[END_REF][START_REF] Schwarz | Dynamic-mode scanning force microscopy study of n-InAs(110)-(1 x 1) at low temperatures[END_REF].

AFM dynamic modes are also able to produce dissipation maps with contrast at the atomic scale. However, measurements are usually recorded with cantilevers of resonance frequency between 150-300 kHz. Therefore, the time during which the oscillating tip interacts with the surface-about a microsecond-is much larger than any characteristic relaxation times at the atomic scale. In order to explain a drastic slowing down of the dynamics at such small scales, numerous approaches have been considered; for instance influence of a stochastic resonance process, influence of a displacive soft mode of a second order phase transition, or mechanical hysteresis. None of these analyses are able to explain the atomic contrast of a dissipation image. Therefore, it has been recognized as being of major interest to study the stability of the dynamic modes when the tip interacts with a surface.

After a brief description of the two dynamic operation modes of the AFM, we present an analysis of the non-linear behavior of the damping signal when the tip oscillates during a fraction of the period inside a viscous material. Then, a theoretical analysis, including the electronic feedback loop that maintains the oscillation amplitude constant, is presented. In this section, the main purpose is to investigate the domain of stability of the oscillator and the influence of the parameters of the machine.

A c c e p t e d m a n u s c r i p t II-AM and FM modes: simple theoretical modelling.

The two dynamic operating modes of AFM are briefly described below:

-The amplitude modulation (AM) mode [START_REF] Gleyzes | Bistable behavior of a vibrating tip near a solid surface[END_REF][START_REF] Zhong | Elings Fractured polymer/silica fiber surface studied by tapping mode atomic force microscopy[END_REF][START_REF] Garcia | Dynamic atomic force microscopy methods[END_REF], also called "Tapping" mode, is a dynamical mode in which the tip-cantilever system oscillates at a fixed excitation frequency and with a fixed amplitude excitation. The AM mode is able to produce different types of images and is the most commonly used. The first success was its ability to record images on polymers without inducing the severe damage associated with the contact mode [START_REF] Zhong | Elings Fractured polymer/silica fiber surface studied by tapping mode atomic force microscopy[END_REF]. As a result, a significant literature is dedicated to studies on very soft samples like biological samples and polymers [START_REF] Stocker | Surface Reconstruction of the Lamellar Morphology in a Symmetric Poly(styrene-block-butadiene-block-methyl methacrylate) Triblock Copolymer: A Tapping Mode Scanning Force Microscope Study Macromolecules[END_REF][START_REF] Leclère | Microdomain Morphology Analysis of Block Copolymers by Atomic Force Microscopy with Phase Detection Imaging[END_REF][START_REF] Magonov | Tapping-mode atomic force microscopy study of the near-surface composition of a styrene-butadiene-styrene triblock copolymer film[END_REF][START_REF] Leclère | Direct Observation of Microdomain Morphology in "All-Acrylic" Thermoplastic Elastomers Synthesized via Living Radical Polymerization[END_REF][START_REF] Konrad | Volume Imaging of an Ultrathin SBS Triblock Copolymer Film[END_REF][START_REF] Kopp-Marsaudon | Quantitative measurement of the mechanical contribution to tapping mode atomic force microscopy images of soft materials[END_REF][START_REF] Knoll | Tapping Mode Atomic Force Microscopy on Polymers: Where Is the True Sample Surface?[END_REF][START_REF] Dubourg | Dynamic Force Microscopic Study of a triblock copolymer with AFM non contact resonant mode[END_REF][START_REF] Reiter | Direct Visualization of Random Crystallization and Melting in Arrays of Nanometer-Size Polymer Crystals[END_REF][START_REF] Nony | DNA properties investigated by dynamic force microscopy[END_REF][START_REF] Dubourg | Apparent hardening of soft samples through Q factor change in AFM[END_REF][START_REF] Dubourg | Probing the relationship between the scales of space and time in an entangled polymer network with an oscillating nanotip[END_REF]. However, while the AM mode provides a convenient and easy way to image soft materials, quantitative analysis of the experimental data remains difficult. The non-linear behavior of the AFM oscillator in proximity of the surface makes the analysis of the variations of the oscillation amplitude and of the phase at a fixed driven frequency and fixed driven excitation truly difficult [START_REF] Giessibl | Atomic resolution of the silicon (111)-(7x7) surface by atomic force microscopy[END_REF][START_REF] Sugawara | Defect motion on an InP (110) surface observed with Non Contact atomic force microscopy[END_REF][START_REF] Kitamura | Observation of 7× 7 Reconstructed Structure on the Silicon (111) Surface using Ultrahigh Vacuum Noncontact Atomic Force Microscopy[END_REF][20][START_REF] Lantz | Low Temperature Scanning Force Microscopy of the Si(111)-(7 x 7) Surface[END_REF][START_REF] Schwarz | Dynamic-mode scanning force microscopy study of n-InAs(110)-(1 x 1) at low temperatures[END_REF][START_REF] Anczykowski | Cantilever dynamics in quasinoncontact force microscopy: Spectroscopic aspects[END_REF][START_REF] Boisgard | Hysteresis generated by attractive interaction: oscillating behavior of a vibrating tip-microlever[END_REF][START_REF] Cleveland | Energy dissipation in tapping-mode atomic force microscopy[END_REF][START_REF] Wang | Analytical descriptions of the tapping-mode atomic force microscopy response[END_REF][START_REF] Nony | Non-linear dynamical properties of an oscillating tipcantilever system in the tapping mode[END_REF][START_REF] Dürig | Conservative and dissipative interactions in dynamic force microscopy[END_REF][START_REF] Garcia | Attractive and repulsive tip-sample interaction regimes in tapping-mode atomic force microscopy[END_REF][START_REF] Wang | The role of damping in phase imaging in tapping mode atomic force microscopy[END_REF][START_REF] Aimé | Non-linear dynamic behavior of an oscillating tip-microlever system and the contrast at the atomic scale[END_REF][START_REF] San Paulo | Tip-surface forces, amplitude, and energy dissipation in amplitude-modulation (tapping mode) force microscopyPhys[END_REF][START_REF] Dürig | of Physics Interaction sensing in dynamic force microscopy[END_REF][START_REF] Stark | From Images to Interactions: High-Resolution Phase Imaging in Tapping-Mode Atomic Force Microscopy[END_REF][START_REF] Hölscher | Quantitative analysis of dynamic-forcespectroscopy data on graphite (0001) in the contact and noncontact regimes[END_REF][START_REF] Giessibl | Forces and frequency shifts in atomic-resolution dynamic-force microscopy[END_REF]. In particular, changes in the oscillation amplitude can be the result of both conservative and dissipative interactions. A non-intuitive result is that a decrease in the oscillation amplitude can either be due to an attractive or a repulsive interaction [START_REF] Boisgard | Hysteresis generated by attractive interaction: oscillating behavior of a vibrating tip-microlever[END_REF].

-The second mode uses the frequency modulation (FM) technique, also called non-contact resonant mode [START_REF] Albrecht | Frequency modulation detection using high-Q cantilevers for enhanced force microscope sensitivity[END_REF]. The FM mode was first used under ultra-high-vacuum experimental conditions to image semiconductor surfaces. With the FM mode, a negative resonant frequency shift, at constant oscillation amplitude, is used as the error signal to control the distance between the tip and the surface. In doing so, the interaction between the tip and the surface remains attractive and the tip either does not touch the surface at all or touches it only slightly. The key experimental achievement was to show the capability to record images with a contrast at the atomic scale [START_REF] Giessibl | Atomic resolution of the silicon (111)-(7x7) surface by atomic force microscopy[END_REF][START_REF] Sugawara | Defect motion on an InP (110) surface observed with Non Contact atomic force microscopy[END_REF]. The experimental data are the resonance frequency variation and the damping signal extracted from the energy per cycle needed to keep the oscillation amplitude constant. Contrary to AM, when the oscillation amplitude is kept constant, the physical origin of the change of the oscillating properties at the resonance frequency is straightforwardly attributed. The frequency shift is only related to conservative interaction, and the additional damping signal gives the dissipative interaction [START_REF] Lantz | Low Temperature Scanning Force Microscopy of the Si(111)-(7 x 7) Surface[END_REF][START_REF] Hölscher | Quantitative analysis of dynamic-forcespectroscopy data on graphite (0001) in the contact and noncontact regimes[END_REF].

However, the high sensitivity of the FM mode requires samples of great quality with a nearly atomic flat surface and without significant pollution.

A c c e p t e d m a n u s c r i p t 1-Conservative interaction.

A shift of the resonant frequency of a harmonic oscillator corresponds either to a change in the mass of the oscillator or to a change in an effective force gradient. For experiments with an AFM, where a mass transfer is not expected, an effective spring constant must be considered. A general expression describing the resonant frequency shift is [START_REF] Dürig | Conservative and dissipative interactions in dynamic force microscopy[END_REF][START_REF] Dürig | of Physics Interaction sensing in dynamic force microscopy[END_REF][START_REF] Giessibl | Forces and frequency shifts in atomic-resolution dynamic-force microscopy[END_REF]:

int 0 1 eff c k k ν ν     ∆ = + -     1 (1)
Here, 0 ν is the resonant frequency when the tip does not interact with the surface, k is the effective spring constant describing changes of the interaction between the tip and the surface over an oscillation period, and k is the cantilever spring constant. The high sensitivity of commercially available electronics allows one to record low values of effective spring constant. For example, with a cantilever spring constant k int eff c 10 c = Nm -1 , an effective spring constant less than can be measured with frequency demodulators [START_REF] Dietzel | Analysis of mechanical properties of single wall carbon nanotubes fixed at a tip apex by AFM[END_REF].

1 - 3 int 10 - = Nm k eff
The first goal is to deduce the laws of interaction between the tip and the surface.

There are numerous works attempting to relate the frequency shifts to the attractive tipsample interaction. The analysis of the resonance frequency shift was first derived using the Hamilton-Jacobi formalism. Assume a power law dependence for the attractive interaction:

( ) ( ) ( ) ( ) int n C F D x t D x t - =- - ( 2
)
where D is the distance between the vertical location of the tip at rest and the surface and

) cos( ) ( 0 t A t x ω =
. The frequency shift is given by [START_REF] Giessibl | Forces and frequency shifts in atomic-resolution dynamic-force microscopy[END_REF]:

( ) ( ) 0 0 1 3/ 2 1/ 2 0 1 , , , , 2 n C k A C n I n kA ν ν π - ∆ ∆ - ∆ ; (3)

A c c e p t e d m a n u s c r i p t

where n is the power law dependence of the tip surface interaction (see equation 2, [START_REF] Giessibl | Forces and frequency shifts in atomic-resolution dynamic-force microscopy[END_REF]), and where ( )

∫ ∞ ∞ - + = dy y n I n 2 1 1 1 ) (
has tabulated values (see for example reference [START_REF] Garcia | Dynamic atomic force microscopy methods[END_REF]). Here, ∆ is the closest distance between the tip and the surface, A 0 is the resonance amplitude, and C is a constant giving the strength of the attractive interaction. A separate development based on the principle of least action gives similar results. This latter method includes the influence of the quality factor, and is thus not restricted to interpretation of the resonant frequency shifts [START_REF] Boisgard | Hysteresis generated by attractive interaction: oscillating behavior of a vibrating tip-microlever[END_REF][START_REF] Aimé | Non-linear dynamic behavior of an oscillating tip-microlever system and the contrast at the atomic scale[END_REF].

2-Dissipative interaction:

In addition to conservative forces of the tip sample interaction, one must also consider dissipative forces that lead to additional energy losses during each oscillation period.

Dissipation manifests itself as a hysteresis of the force versus displacement curve. Dissipation mechanisms may involve electrical losses, time delays, viscoelastic material effects [START_REF] Dubourg | Probing the relationship between the scales of space and time in an entangled polymer network with an oscillating nanotip[END_REF][START_REF] Dürig | Conservative and dissipative interactions in dynamic force microscopy[END_REF][START_REF] Dürig | of Physics Interaction sensing in dynamic force microscopy[END_REF][START_REF] Boisgard | Surface mechanical instabilities and dissipation under the action of the oscillating tip[END_REF], or mechanical instabilities due to adhesion [START_REF] Dürig | of Physics Interaction sensing in dynamic force microscopy[END_REF].

Energy loss due to viscous effect exhibits an oscillation amplitude dependence and, when soft materials are involved, an indentation depth dependence [START_REF] Dubourg | Probing the relationship between the scales of space and time in an entangled polymer network with an oscillating nanotip[END_REF]. For mechanical instability with adhesion hysteresis, the dissipated energy is a constant [START_REF] Dürig | of Physics Interaction sensing in dynamic force microscopy[END_REF][START_REF] Dietzel | Mechanical properties of a carbon nanotube fixed at a tip apex: A frequency-modulated atomic force microscopy study[END_REF]. The total damping coefficient is [START_REF] Cleveland | Energy dissipation in tapping-mode atomic force microscopy[END_REF]:

0 i tot n t γ γ γ = + (4)
where

Q m 0 0 ω γ =
is the damping coefficient in the absence of interactions between the tip and the surface. Typically in vacuum,

c diss k A E Q π 0 1 9 - = ; J. In air, Q 500 = and the dissipated energy is 2 0 c diss k A Q π = 1 7 10 E - ≈
J. The measured damping signal, in Volts, is given by , the additional damping coefficient int γ can be converted into the dissipated energy. Thus in ambient conditions, a damping increase of about 10 J can be measured, while in vacuum at room temperature it can be as low as .

0 int 0 1 a a D D γ γ   = +     (5)
19 -

B k T

III Non-linear contact mechanics: an example with a viscoelastic material.

Only a few analyses have been dedicated to the study of the interaction when the tip makes significant indentation in the surface. This case is by far the most difficult to solve, and only phenomenological approaches, including contact mechanics, are used [START_REF] Garcia | Dynamic atomic force microscopy methods[END_REF][START_REF] Dubourg | Probing the relationship between the scales of space and time in an entangled polymer network with an oscillating nanotip[END_REF][START_REF] Wang | The role of damping in phase imaging in tapping mode atomic force microscopy[END_REF][START_REF] San Paulo | Tip-surface forces, amplitude, and energy dissipation in amplitude-modulation (tapping mode) force microscopyPhys[END_REF][START_REF] Boisgard | Surface mechanical instabilities and dissipation under the action of the oscillating tip[END_REF].

In addition, at the nanometer scale, surface forces become dominant and contact mechanics must be described as the result of an interplay between the attractive and the repulsive interaction. The problem can be partly solved with the introduction of a length as a contact distance between the tip and the sample. However, because there is a non-linear relationship between the elastic repulsive force and the indentation depth, a simple analytical expression cannot be obtained [START_REF] Wang | Analytical descriptions of the tapping-mode atomic force microscopy response[END_REF][START_REF] Nony | Non-linear dynamical properties of an oscillating tipcantilever system in the tapping mode[END_REF].

When viscoelastic materials are investigated, we face the difficult task to describe the contact between the tip and the viscous sample. For AFM dynamic modes, the difficulty is increased as the nanotip oscillates and has an intermittent contact with the sample. However, using crude assumptions, we can borrow approaches from common rheological experiments.

In this section we give a detailed description of a theoretical analysis that should help to interpret indentation experiments with an AFM tip oscillating in a viscoelastic material. In order to simplify the theoretical analysis, we consider a shear experiment, in which the viscoleastic material is sheared with a flat oscillating plate.

For the AFM experiment the plate is the tip inside the polymer. To write the constitutive equation, we describe the viscoelastic material with a simple Maxwell mechanical element [START_REF] Dubourg | Probing the relationship between the scales of space and time in an entangled polymer network with an oscillating nanotip[END_REF][START_REF] Ferry | Viscoelastic Properties of Polymers 3rd[END_REF]. The mechanical element is made of a spring and a dashpot in series. The constitutive equation is

( ) ( ) ( ) s t t t σ τ σ ηγ + =- & & ( 6
)
where σ is the stress in the mechanical element, γ& the shear rate, s τ is a relaxation time, and η is the zero frequency viscosity.

When intermittent contacts occur between the tip and the sample, the tip induces a sample displacement assuming that the material is incompressible. However, contrary to usual shear experiments, the imposed displacement is not a simple sinusoidal perturbation. Because the tip moves inside the sample only for a fraction of time of the oscillation period, the time dependence of the indentation depth ( )

t δ is: ( ) ( ) cos for 2 res t A t D t τ δ ω = - ≤ (7)
( )> cos ω D cos 2 res A D ωτ   =     or 1 1 2 cos res D T A A τ ω π - 2∆   =     ; f n t o r the maximum indentation. D A -= A ∆ =
The indentation depth writes ( ) ( )

0 cos n n t δ δ ∞ = = ∑
ω with the Fourier coefficient δ n given in the appendix A.

The expression of the Fourier coefficients δ n is cumbersome to use, and a few simplifications can be made. Noting that the coefficients vanish rapidly with increasing n, we follow a method identical to the one given in reference [START_REF] Boisgard | Surface mechanical instabilities and dissipation under the action of the oscillating tip[END_REF].

A maximum harmonic number 0 ( ) ( ) ( )

' ' t t G t t t σ -∞ = - - ∫ & ' dt γ (8)

A c c e p t e d m a n u s c r i p t

where ( )

exp s s t η G t τ τ   = -   
 is the material relaxation modulus.

For sake of simplicity, we consider a simple shear strain. An estimation of the shear rate γ& is obtained assuming a no slip condition at the surface of the tip and a polymer at rest at a distance of the tip of the order of the diameter of contact ρ between the tip and the material.

Hence, we set γ ρ ∆ :

and γ ρ ∆ & &:
. The residence time is of the order of 1 µs, giving a shear rate s 6 10 γ&:

-1 . Note that in spite of a large shear rate, the volume of material involved is only a few nm 3 , and the dissipated energy is always very low.

Asymptotic behaviors correspond to long and fast relaxation processes with respect to the oscillation period. Starting from equation A10 (appendix A), the two asymptotic forms of int

where the power law dependence corresponds to relaxation processes with a time constant much smaller than the residence time

3/ 2 A - res τ .
Similarly, for 1 , give a simple way to interpret the viscoelastic properties of the sample and possibly the type of molecular motion induced by the tip [START_REF] Dubourg | Probing the relationship between the scales of space and time in an entangled polymer network with an oscillating nanotip[END_REF][START_REF] Boisgard | Surface mechanical instabilities and dissipation under the action of the oscillating tip[END_REF]. In particular the formulas were successfully used to analyze the experimental data obtained on a polymer melt [START_REF] Dubourg | Probing the relationship between the scales of space and time in an entangled polymer network with an oscillating nanotip[END_REF].

s τ ω ? , ( ) ( ) 2 '

IV-Modeling the Oscillating behavior and the machine for the FM mode: Bandwidth and Coarse Graining approach

As discussed in the previous sections, FM mode provides unambiguous information on the conservative and dissipative interaction. However, the non-linear behavior of the oscillator in proximity to the surface and its intricate play with the electronic feedback loop must be investigated to ensure an accurate analysis of the physical origin of changes in the AFM oscillator properties. In the presence of non-linearity, an unstable behavior of the AFM oscillator may occur that can induce frequency shifts and damping variations. For the specific case of the FM-mode, we must consider the feedback loop, with its own characteristic time, required to keep constant the oscillation amplitude. To do so, an additional differential equation is considered. The later equation takes into account the effective time dependence of the forcing. Until now, this type of study has been mostly achieved through numerical computations [START_REF] Couturier | Aimé Damping and instability in non-contact AFM: contribution of the Instrument[END_REF]42].

In this part, we outline an analytical approach aiming to extract the influence of the electronic feedback loop that compensates change of the oscillating properties and improves the oscillator stability. The aim is to compute the evolution of the damping signal, in order to discriminate between contributions coming from the tip sample interaction and those from the machine.

Because each experimental data point represents a time average over several hundreds oscillation periods, the variation of any parameters occur on a time scale much larger than the

A c c e p t e d m a n u s c r i p t

time period of the oscillation. Therefore, it is worth to extract information upon the time evolution at a time scale larger than that of the oscillation period. To do so, we use a method of coarse graining that has been shown to be efficient to investigate the stability of the AFM oscillator when the tip interacts with a surface [43].

A Lagrangian able to describe the FM-mode including the electronic loop keeping the oscillation amplitude constant is (a more precise description of the calculus is given in Appendix B):

µ [ ] 2 2 0 1 1 2 2 a c d m k x D x C L x x x D x ξ γ ω       = - - + - - x (12)
where µ

d a D x x ξ   
 is a forcing term, which is given by the electronic loop that keeps the oscillation amplitude constant [START_REF] Couturier | Aimé Damping and instability in non-contact AFM: contribution of the Instrument[END_REF]. The terms with underscore will not be varied when the variationnal procedure will be applied. Here, µ d ξ is a phase shifter operator and D is the damping signal. A direct consequence of the time dependence of the forcing term is that the oscillation amplitude A becomes a time dependent parameter. Therefore, we look for a general solution of the form:

a ( ) ( ) ( )cos ) ( x t A t t =

A c c e p t e d m a n u s c r i p t

A consequence of the fact that the average process on time scaleτ obeys inequality 14 is that the variation of any quantity characterizing the oscillator over a time interval that is small compared to the characteristic time t ∆ τ , must also be small.

The principle of least action states that the integral of the Lagrangian between the two instants and must be optimal along the physical path.

S L

= ∫ dt (16) where is now a continuous function of the slowly varying variables , Once the effective Lagrangian is expressed in terms of the corresponding variables, we obtain the long time behavior by applying the Euler-Lagrange equations.

Let us begin with the kinetic term:

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 cos sin 2 sin cos 2 1 1 2 2 1 2 x A t A t A A t A A A α α α α α α α α = + - A c c e p t e d m a n u s c r i p t 2 2 2 1 cos ( ) 2 2 x A t α = = A (18)
The average interaction potential is ( )

int 1/ 2 2 2 1 1 cos( ( )) CG CG t t t t CG CG C C C V d t d t D x D A t D A τ τ τ τ α + + = = - - - ∫ ∫ ; (19)
Using equation 13, the damping term is:

( ) ( 1 cos cos cos sin cos sin 2 xx A A A AA AA ) α α α α α α α α α α   = - = -+ - 
only the average difference is not negligible.

The main difference with the previous approaches [START_REF] Boisgard | Hysteresis generated by attractive interaction: oscillating behavior of a vibrating tip-microlever[END_REF][START_REF] Nony | Non-linear dynamical properties of an oscillating tipcantilever system in the tapping mode[END_REF], is that in addition to the dependence on phase φ and oscillation amplitude A, L now explicitly depends on the time derivatives of e α and .

A

Since the damping is also a function varying slowly with time compared to the frequency domain where the phase shifter is effective, we assume:

a D µ [ ] µ [ ] a a d d D x D x ξ ξ ;
On the other hand the phase shifter µ d ξ acts on the tip motion ( )

x t following the criterion [START_REF] Couturier | Aimé Damping and instability in non-contact AFM: contribution of the Instrument[END_REF]:

µ [ ] [ ] 3 cos cos sin 2 d d x A A A π ξ ξ α α   = = - =-     α (21)
The average value becomes:

µ [ ] µ [ ] ( 1 sin cos sin 2 a a a a d d D x x D x x D AA D AA ) ξ ξ α α α α - - ; ; ; (22)

A c c e p t e d m a n u s c r i p t

The complete effective Lagrangian is given by: ( )

( ) [ ] ( ) 2 2 2 0 1/ 2 2 2 1 1 1 1 cos sin 4 4 2 2 a e c C L mA k A AA AA D D A α γ α α γα ω α = - + - -- -α - (23)
The first Euler-Lagrange equation

e L d dt A A e L ∂ ∂   =   ∂ ∂   & becomes: ( ) ( ) [ ] ( 2 0 3 / 2 2 2 2 0 c o s a c AC mA k A A A D D A ) s i n α γ α α γα ω α = - + - -- - -α (24)
As usual, writing that the solution is given by the parameters [START_REF] Boisgard | Hysteresis generated by attractive interaction: oscillating behavior of a vibrating tip-microlever[END_REF],α α

= , A A = ,α α = & & and A A = & & yields ( ) 2 3 / 2 2 2 2 0 c AC mA k A A D A α γ = - + - - & & ( 25
)
When the time variation of the oscillation amplitude is neglected, equation [START_REF] Cleveland | Energy dissipation in tapping-mode atomic force microscopy[END_REF] gives the known stationary result :

( )

2 3 0 0 3 / 2 2 2 2 0 0 1 2 a A D A ω ω ω κ ω ω   - - -   -   ; ; (26)
where, following [START_REF] Aimé | Non-linear dynamic behavior of an oscillating tip-microlever system and the contrast at the atomic scale[END_REF], α ω

∂   =   ∂ ∂   & becomes: 2 2 1 2 2 d A mA m mAA dt α α α   = +     ( ) [ ] ( 0 1 1 sin cos 2 2 e a L AA AA D ) γ α α γα ω α α α ∂ = -- - ∂ & & - (28)
+ - +    & 0  =  (29)
When the variations of the oscillator properties are restricted to a shift of the resonance frequency, the observed change of the damping parameter is given by:

2 0 0 a Q d D dt ω ω γ ω ω   +    ;  (30)
Therefore, when no extrinsic dissipation occurs, the FM mode is described with the two equations ( 25) and [START_REF] Wang | The role of damping in phase imaging in tapping mode atomic force microscopy[END_REF]. The first term means that the damping coefficient, scaling as 1 Q , is slightly modified because of the frequency shift. Typical variations are about 10 -4 , thus can be neglected. The second term rises when images are non-adiabatically recorded. The second term becomes noticeable when variations in the frequency occur within the time scale of the period, i.e. t Q ω ∆ : . Therefore, equation [START_REF] Wang | The role of damping in phase imaging in tapping mode atomic force microscopy[END_REF] tells us that the instrumental contribution to the damping coefficient should always be negligible when the FM mode is perfectly set at the resonance frequency. However, as shown recently [START_REF] Couturier | Aimé Damping and instability in non-contact AFM: contribution of the Instrument[END_REF]42] in a narrow domain of frequencies an unstable behavior occurs that can lead to a strong enhancement of the damping signal.

Such an instrumental contribution can be removed by varying the scan frequency.

Equation [START_REF] Cleveland | Energy dissipation in tapping-mode atomic force microscopy[END_REF] predicts an observed frequency shift given by: ( )

3 0 0 3/ 2 2 2 0 1 log 2 2 a A d A Q dt D A ω ω κ ω - = - + - (31)
Here again, for many experimental cases we may consider the contribution of changes in the oscillation amplitude as a function of time negligible regarding its the influence on the tip sample interaction.

Equations ( 30) and [START_REF] Aimé | Non-linear dynamic behavior of an oscillating tip-microlever system and the contrast at the atomic scale[END_REF] were found to be precise enough to describe the contribution of the machine on many of the experimental results obtained with a FM-AFM [START_REF] Couturier | Aimé Damping and instability in non-contact AFM: contribution of the Instrument[END_REF]. Therefore, in spite of the complexity of the machine, there are simple expressions that provide an unambiguous way to discriminate between contributions of the tip sample interaction and of the feedback electronic loop.

A c c e p t e d m a n u s c r i p t

where Ω is the surface of contact.

Using the approximate sample speed ( ) ( 1

n diss res s T n s E n T n τ η π ω ρ τ ω = Ω   ∆     + ∑ ; (A9)
Finally, the measured damping coefficient int γ is deduced from the relation

2 int diss T E A πγ ω = .
The great difficulty is to evaluate the geometrical parameter Ω and ρ in equation A9. Here, is the surface of contact between the tip and the material, Ω ρ is the contact diameter. These must both be expressed as a function of ∆ . The relation between the indentation depth and the geometrical parameter driving the shear displacement inside the polymer requires an accurate knowledge of the size and shape of the tip. Unfortunately, in AFM an accurate knowledge of the tip at the nanometer scale is often missing, thus, as an example, we only consider a simple tip shape. For a conical tip From equations B1 and B2 we write a lagrangian from which we can retrieve these equations by application of the least action principle.

A c c e p t e d m a n u s c r i p t

  where D a and D a0 are the damping signals measured when the tip is close to the surface and far from the surface respectively.

A c c e p t e d m a n u s c r i p t

  To calculate the stress and the dissipated energy we use the Boltzmann superposition principle relating the stress to the history of the strain[START_REF] Ferry | Viscoelastic Properties of Polymers 3rd[END_REF].

;

  relaxation times of the sample are: for all n (see equation A9 in the appendix) and thus, using an approximate expression for the sum,

τ

  Therefore when the relaxation time of the viscous material s τ obeys the inequality 1 s τ ω ? , the damping follows a ( ) 2 s τ ω -power law: thus relaxation processes with too long s τ do not A c c e p t e d m a n u s c r i p t contribute to dissipation. At fixed indentation depth ∆ the oscillation amplitude dependence corresponds to the asymptotic behavior for relaxation times much longer than that of the oscillation period T. Equations 10 and 11 give two power laws, one related to the indentation depth ∆ the second to the oscillation amplitude A. The power laws, A for short sample relaxation time 3/ 2 s , and A for long sample relaxation time 5/ 2 -

2 A & 2 ω

 22 that varies rapidly with time. Typically, the first time derivative isα ω ≈ & , while the time derivative of α & varies slowly with time. The expression of the tip velocity is: into account that A & and α ω = & & &are slow temporal variables, quadratic expressions of the form and & are neglected in the following.From an experimental point of view, the oscillating amplitude, or the frequency, are given by devices that have their own response time, i.e. the bandwidth. Let CG τ be this characteristic time, corresponding to the coarse-graining scale. The time CG τ must be greater than the period, typically a few milliseconds or several hundreds of periods. Therefore a precise temporal description can only be done on events whose experimental characteristic times τ are greater than CG τ . The following inequality then applies:

  and α &.

  this work, we have presented two non-linear behaviors observed with the dynamic modes of AFM. One deals with phenomena occurring when a viscoelastic material is investigated. The other case concerns the behavior of the oscillator when it becomes unstable because of the non-linearity induced by the tip sample interaction. The non-linearity can produce significant variations of the recorded damping coefficient that can be considered as artefacts and not as the result of additional dissipated energy. Therefore the effects of abrupt changes of the oscillating properties induced by an unstable behavior are investigated. A complete theoretical description, including the electronic feedback loop of the machine, is performed giving two simple equations. The two equations allow discriminating between effects from an unstable behaviour of the oscillator and a dissipation induced by the physical properties of the sample.

  Gauthier, M., Pérez R., Arai, T. et al Interplay between Nonlinearity, Scan Speed, Damping, and Electronics in Frequency Modulation Atomic-Force Microscopy. Phys. Rev. Let. 2002, 89, 146104-146107.[43] Nony, L., Boisgard, R., Aimé, J.P. Stability Criterions of an oscillating tip-cantilever system in dynamic force microscopy. Eur. Journ. of Phys. B 2001,24, 221-229.

  the cone half-angle. Then this simple geometrical consideration gives the expression of the damping coefficient int γ for a cone :

ξ

  When a mechanical oscillator is inserted within a feedback loop that maintains the oscillation amplitude constant, the general differential equation is: mass of the cantilever, k is the cantilever stiffness, m c γ the dissipation coefficient due to the hydrodynamic action to the surrounding fluid, F is the interacting force between the tip apex and the surface, and int ( )e f t is the, a priori unknown,forcing with a general time dependence that keeps the oscillation amplitude constant.Equation B1 aims to describe the oscillation of an AFM micro cantilever oscillating in a FM machine[42]. In this case the general expression of ( ) is a linear operator that imposes the phase to satisfy the Barkhausen condition and is the recorded damping signal. Let us note that we have introduce the parameter ( ) to give to the damping the dimensional signification of a dissipation coefficient. The diagram of figure B1 helps to understand the operation performed with the FM machine.

  Figure B1. Scheme of the FM-AFM electronic loop including the oscillating cantilever. The feedback loop keeps the oscillation amplitude constant

& &

& & &

& & & & & & & &

& & & & & (20)the result given by equation (20) is based on the assumption that α is quickly varying with time, as α is nearly the same as α , the difference α α varies slowly with time. Therefore,

& & &

& & & & &(27)

& &Here we would raise a difficulty associated with the use of the underscored variable notation.By so doing, it means that the variables are not varied during the minimization procedure that occurs in the application of the least action principle. It is a standard procedure to handle nonconservative interaction with the Lagrangian formalism.

; t [START_REF] Giessibl | Atomic resolution of the silicon (111)-(7x7) surface by atomic force microscopy[END_REF] where we have used the assumption that the quadratic term in A & is negligible.

For the potential term:

A c c e p t e d m a n u s c r i p t A c c e p t e d m a n u s c r i p t

Appendix A: Indentation of an oscillating tip in a Visco-elastic material.

The indentation depth can be written as 0 ( ) cos( )

with the Fourier coefficients :

( ) ( )

and the crude assumption :

with ρ the contact diameter. Thus, the corresponding stress generated by the tip motion:

Using the well know relations between the real and imaginary parts of complex viscosity and modulus [START_REF] Ferry | Viscoelastic Properties of Polymers 3rd[END_REF] ( ) ( )

The corresponding energy loss per period is given by 0 ( )

A c c e p t e d m a n u s c r i p t

To be complete, one has also to consider the differential equations describing the electronic feedback loop keeping the oscillation amplitude constant : a peak detector or diode rectifier followed by a proportional-integrator monitor (PI) [42]. Thus, together with equation B1 we obtain the set of equations :

The second and third equations describe the action of the diode rectifier that extracts the amplitude measurement. The fourth equation is the first order differential equation that controls the oscillation amplitude with integral and proportional gain according to a set point amplitude .