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A model for impact dynamics and its application to frequency analysis of tapping-mode atomic force microscopes

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A model for impact dynamics and its application to frequency analysis of tapping-mode atomic force microscopes
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  A Model for Impact Dynamics and its Application to Frequency Analysisof Tapping-Mode Atomic Force Microscopes D. Materassi, M. Basso, R. GenesioDipartimento di Sistemi e Informatica,Universit`a di Firenzevia di S. Marta 3, I-50139 Firenze (Italy)  Abstract —The problem of two-body impact dynamics is con-sidered providing a general class of models based on hysteresisfunctions. The structure of the model and its flexibility allowsfor a direct application of harmonic balance techniques for theanalysis of periodic impacts when the forces involved are eitherrepulsive, repulsive-attractive and dissipative. An applicationto the oscillation analysis of a tapping-mode Atomic ForceMicroscope (AFM) provides useful analytical results which givea qualitative explanation of a number of known experimentalphenomena. I. INTRODUCTIONPhysical systems showing impact phenomena are frequentin many research fields [1]. Main applications are, obviously,in mechanics where macroscopic bodies are considered. Inthis case, an impulsive approximation for interaction forceswith a pure repulsive nature can be correctly assumed.Moreover, energy losses are often considered by introducingthe coefficient of restitution [2] [3]. On the other hand, insome novel applications as, for instance, in nanotechnologies,there are impacts between microscopic particles that cannot be considered impulsive, since they involve complicatedinteractions that can be repulsive or attractive depending onthe distance of the bodies.The aim of this work is to propose a simple and flexibleway to describe impacts using hysteresis functions to modelthe related interaction forces. The hysteresis model can beviewed as a generalization of the impulsive case and allows,in some way, for the use of potential functions even if thesystem is dissipative. It also presents advantages when theinteraction forces are either repulsive and attractive, or whenthe duration of the impact must be modeled too. In addition,considering systems with periodic impacts, the hysteresismodel allows for the use of powerful analysis techniques,such as harmonic balance [4], which can not be used forimpulsive forces.To show how the above impact model can be successfullyemployed in complicated analysis problems, we consideran application to a tapping-mode Atomic Force Microscope(AFM) [5]. In this operating mode the cantilever of the AFMis sinusoidally forced by a piezo positioned under its supportinducing a periodic oscillation that is naturally influenced bythe interaction forces between the tip of the cantilever andthe sample to analyze. The sample profile can be inferred bymoving the cantilever along the surface of the sample andobserving the characteristics of the periodic motion. Sincethere is a low sample-cantilever interaction, this operatingmode is indicated to analyze soft samples where there is ahigh possibility of causing lacerations and scratches.In [6] and [7] models that accurately — yet numerically —describe the device behavior are proposed. Analytical resultscan be found in [8], where a simple impulsive impact modelis developed. However, since the model employed neglectsattractive forces, it is not able to reproduce some importantphenomena of the AFM. In this work, we provide a completefrequency analysis of a tapping-mode AFM exploiting theproposed hysteresis model and taking also into accountattractive forces in the sample-cantilever interaction.The paper is organized as follows. In Section II webriefly describe the general problem of modeling an impactand propose a new model which makes use of hysteresisfunctions. In Section III we exploit such model to describethe AFM tapping-mode dynamics and in Section IV weanalyze it with harmonic balance techniques. In Section V themodel is extended to account also for attractive phenomena.In Section VI main results are summarized.II. IMPACT MODELINGLet  P  1  and  P  2  be two material points with masses  m 1  and m 2 , respectively, moving along the  x  axis, with positions x 1 ( t )  and  x 2 ( t )  ( x 1  ≤  x 2 ). We consider  P  1  and  P  2  subjectto external forces,  F  1 ( t )  and  F  2 ( t ) , and to a mutual internalforce. Let  H  1  be the intensity of the internal force on  m 1 and  H  2  the intensity on  m 2 , with  H  1  =  − H  2 . We assume H  1  and  H  2 , in a general case, depending only on time  t , therelative position  x 2 − x 1  and the relative velocity  ˙ x 2 −  ˙ x 1 .The following dynamical model describes the system  m 1 ¨ x 1  =  F  1 ( t ) + H  1 ( t,x 2 − x 1 ,  ˙ x 2 −  ˙ x 1 ) m 2 ¨ x 2  =  F  2 ( t ) + H  2 ( t,x 2 − x 1 ,  ˙ x 2 −  ˙ x 1 ) .  (1)This approach is general and, by a theoretical point of view,completely solves the problem. However, the integration of (1) could be difficult or not possible for the analytical formof   F  1 ,  F  2 ,  H  1  and  H  2 . In particular, the the functions  H  i are critical since they define the “coupling” of the dynamicalsystem.We intend to limit the study of an impact consideringthe case where  P  1  and  P  2 , interacting, get closer at thebeginning, and then further. Therefore, we assume that thesystem dynamics can be split in two different phases: an“approaching phase” and a “departing phase”. Defining  d  := x 2 − x 1 , we consider a time interval  [ t s ,t f  ]  where  d ( t s ) = d ( t f  ) , and  ˙ d ( t s )  ≤  0  ≤  ˙ d ( t f  ) . The impulse-momentumchange law allows for the determination of the trajectoryof the center of mass, while it is needed another informationabout the “mode of impact” to know the trajectories of thetwo masses. For isolated systems ( F  1 ( t ) =  F  2 ( t ) = 0 ), if  ˙ d ( t s )  = 0 , we can define the coefficient of restitution of the Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003  FrE04-3 0-7803-7924-1/03/$17.00 ©2003 IEEE 6218  impact [2] [3] e  :=  − ˙ d ( t f  )˙ d ( t s )=  −  ˙ x 2 ( t f  ) −  ˙ x 1 ( t f  )˙ x 2 ( t s ) −  ˙ x 1 ( t s ) .  (2)It is important to note that  e  is non negative and that it de-pends in general on the masses of the material points, on theirinitial velocities and on the functions  H  i ( · ) . The coefficientof restitution can be seen as a parameter summarizing all theimpact dynamics: from  e ,  ˙ x 1 ( t f  )  and  ˙ x 2 ( t f  )  can be obtained.Further, it is straightforward to prove that  e  is related to theloss of kinetic energy through the equation ∆ E  cin  = (1 − e 2 ) m 1 m 2 2( m 1  + m 2 ) [˙ x 1 ( t i ) −  ˙ x 2 ( t i )] 2 .  (3)In general, the analytical evaluation of   e  from  m i ,  ˙ x i ( t s ) and  H  i  is not possible, therefore  e  is approximated bymeans of simplifying assumptions. Usually,  e  is consideredindependent on the initial velocities of the two masses, be-coming a constant that completely characterizes the impact.Better approximations are clearly possible. For a non-isolatedsystem it is not possible to define a coefficient of restitutionunless it is considered dependent also on the expressions of the external forces. Such a definition changes the nature of  e  since we are oriented to consider a parameter representingonly the internal forces during the impact and not the externalones. In addition, this solution is quite difficult to deal with.In impacts between macroscopic bodies, usually, the inter-nal forces provide a finite impulse in a quite “infinitesimal”interval, which allows to assume the system approximatelyisolated. In this case the use of the coefficient of restitutionsimplifies the problem providing useful results. If the systemcan not be considered isolated (not even approximatively)during the interaction, a solution is to introduce a simplifiedmodel for the forces  H  i . In order to account for energylosses we have to consider a force depending not only onthe distance between the two masses, since the related forcefield is conservative. Models with linear dependence bothon the distance and relative velocities are typically usedassuming energy losses for viscous friction [9]:  H  1 ( d,  ˙ d ) = − H  2 ( d,  ˙ d )  ≃  Kd  +  c  ˙ d . If we consider impulsive forcesmodeled by Dirac distributions we can, for example, findthe model with constant coefficient of restitution  e  andinstantaneous interaction time.The model proposed in this work defines a particular formfor the interaction forces and, at the same time, allows one togeneralize the case of constant coefficient of restitution, notonly for an instantaneous impact time. In addition, it presentssome advantages in the study of impacting systems withperiodic behaviors. The main idea is to consider two differentpositional forces during the impact, the first one when themasses are approaching, the second one when the massesare departing. We account for energy losses by assuming aforce that depends only on the sign of the relative velocityof the impacting bodies, that is H  ( d,  ˙ d ) :=  H  a ( d ) ˙ d >  0 H  b ( d ) ˙ d <  0 H  a ( d )+ H  b ( d )2 ˙ d  = 0 . (4)For integrable  H  a  and  H  b  we can introduce the potentialfunctions U  a ( d ) = −   dd ( t s ) H  a ( x )d x  ˙ d >  0 U  b ( d ) = −   dd ( t s ) H  b ( x )d x  ˙ d <  0 . (5) For  H  a ( d )  ≥  H  b ( d ) ,  ∀  d < d ( t s ) , then we have also that U  a ( d )  ≥  U  b ( d ) ,  ∀  d < d ( t s ) . Considering the instant  ¯ t  whenthe relative distance  d ( t )  is minimal, we can state that thepotential interaction energy of the system is  U  a ( t )  if   t <  ¯ t ,while it is  U  b ( t )  if   t >  ¯ t . At any instant the interaction forceis conservative except for  t  = ¯ t  when the relative velocity iszero and we have an energy loss equal to U  a [ d (¯ t )] − U  b [ d (¯ t )] .  (6)The energy lost in the impact can be also graphically inter-preted as the area between the curves  H  a ( d )  and  H  b ( d ) . Theimpact model with constant coefficient of restitution can beobtained equating (3) to (6) In this way, it is only establisheda relation between  U  a ( · )  and  U  b ( · ) , without fixing the form of the two potential functions. Such a “degree of freedom” canbe exploited to model, for example, the duration of the impactor other important characteristics of the physical system.III. A HYSTERESIS MODEL FOR THE AFMConsider the model of the tapping-mode AFM in [8] m ¨ x + c ˙ x + Kx  =  F  ( t ) + H  ( x,  ˙ x )  ,  (7)where  x  is the position of the cantilever tip,  m  is the tip mass, K   the elastic constant of the cantilever,  F  ( t )  the sinusoidalexcitation force of the tip,  c  the constant of viscous frictionof the medium in which the cantilever is dipped, and  H  the tip-sample interaction force. The srcin of the  x  axiscorresponds to the tip equilibrium in absence of any forcedue to the sample. Normalizing (7) we can write ¨ x + 2 ξω n  ˙ x + ω 2 n x  =  u ( t ) − h ( x,  ˙ x )  (8)where  ω n  =   K/m ,  2 ξω n  =  c/m ,  u ( t ) =  F  ( t ) /m  e h ( t ) =  − H  ( t ) /m . The function  h ( x,  ˙ x )  can be viewed asa static nonlinear feedback of a linear dynamic subsystemwith transfer function L ( s ) = 1 s 2 + 2 ξω n s + ω 2 n .  (9)The above class of models is well-known in control literatureand usually referred to as Lur’e systems [4] [10]. The freeoscillation of the cantilever (in absence of the sample) iscompletely defined by the linear block, while the interactionforces are characterized by the feedback block.It is well-known that the cantilever dynamics alwaysevolves into a stable periodic motion [8], [6]. In [8] it is alsoproved that any periodic solution of system (8), with forcingsignal of frequency  ω , has a frequency equal to  ω/n , with n  ∈ N . In the following we will assume  n  = 1 . Usually  ω  ischosen equal to the natural frequency  ω n . The “separation” isthe parameter defining the distance between the sample andthe equilibrium position of the tip in absence of the sample.It is important the analysis of the separation-amplitude curveobtained by plotting the oscillation amplitude as a functionof the separation. 6219  d H Fig. 1. Qualitative behavior of microscopic interaction forces. In Fig. 1 a qualitative plot of the interaction forces as afunction of the tip-sample distance is shown (dotted curve).We can single out three zones: at large distances there isa negligible attractive force, then, approaching, there is aweak attractive force and finally at short distances the forceis highly repulsive. The behavior is highly nonlinear and canbe experimentally obtained by static measures. We intendat first to model only the repulsive phenomenon, neglectingany possible attraction. We consider this case because, asobserved in [8], the duration of the impact is short withrespect to the period of the cantilever oscillation and theattractive region is narrow with respect to the amplitude of such an oscillation. Let us consider a sample positioned onthe negative zone of the  x  axis and assume that the interactionforces acting on the tip are negligible if the tip position x ≥− l . For  x < − l  we assume a linear force with slope  K  (solid curve in Fig. 1), therefore   h ( x ) = 0  ∀  x ≥− lh ( x ) =  K  ( x + l )  ∀  x < − l.  (10)A model with a linear repulsive force can not have a generalvalidity since the actual repulsive force grows more thanlinearly [5]. Linear growth is an acceptable approximationif the tip “grazes” the sample with a low impact velocity.However, this is a case of interest since in tapping-mode thetip-sample interaction is quite limited. In addition, the force(10) is conservative. If we want to introduce dissipation,as seen before, we can modify it into a hysteresis law.More precisely, we consider two different repulsive forcesaccording to the fact that the tip is approaching the sampleor is departing from it   h ( x,  ˙ x ) = 0  ∀  x ≥− lh ( x,  ˙ x ) =  K  a ( x + l )  ∀  x < − l ∧  ˙ x <  0 h ( x,  ˙ x ) =  K  b ( x + l )  ∀  x < − l ∧  ˙ x >  0 . (11)In Fig. 2 a qualitative plot is shown. We want to exploitthe suggested model to derive an analytical form for theseparation-amplitude curve. The parameter  l  of the model canbe assumed, apart from an additive constant, to correspondto the separation.IV. HARMONIC BALANCE ANALYSISConsidering physical values for the mass and the elasticconstant of the cantilever, we note that the linear part of the h x K (x+l) K (x+l) a b Fig. 2. Linear repulsive hysteresis model for the interaction force. Lur’e system given by  L ( s )  in (9) has a marked filteringeffect beyond the resonance peak due to a very small damp-ing. This suggests to exploit the harmonic balance methodto analyze the system periodic behavior [4]. Limiting to thefirst order harmonic, we can assume  x ( t ) ≃ Re[ A  +  B e iωt ] and approximate the corresponding output of the nonlinearhysteresis block   h  x ( t ) ,  ˙ x ( t )  ≃ Re[ N  0 A + N  1 B e iωt ] , where  N  0  =  N  0 ( A,B,ω ) := 1 A 1 T     T  0 h  x ( t ) ,  ˙ x ( t )  d tN  1  =  N  1 ( A,B,ω ) := 1 B 2 T     T  0 h  x ( t ) ,  ˙ x ( t )  e − iωt d t (12) are the constant and harmonic gains of the nonlinear block usually denoted as the describing functions of the nonlinear-ity. In particular, for the function (11) we obtain N  0 ( A,B ) =  0  q >  1 ( K a + K b )2 πBA  q   acos( q  ) −   1 − q  2   | q  |≤ 1 K a + K b 2 BA  q q < − 1 (13) N  1 ( A,B ) =  0  q >  1 − ( K a + K b )2 π  q    1 − q  2 − acos( q  )  ++ K a − K b 2 π  (1 − q  ) 2 i  | q  |≤ 1 K a + K b 2  + 2 K a − K b π  q i q < − 1 (14) where q   :=  A + lB .  (15)Note that  N  0  and  N  1  are not dependent on the frequency  ω and that  N  1  can be formally expressed as a function of   q  only. In the sequel we will indicate it as  N  1 ( q  ) . The variable q   can represent the “penetration” of the tip in the sample. Infact, assuming as exact the first harmonic approximation, wehave that for  q >  1  the tip does not get in contact with thesample, for  q   = 1  the tip grazes the sample and for  q <  1 we have an effective interaction. The case for  q <  − 1  ismeaningless in this model since it would indicate that thetip completely oscillates “into” the sample; so we will notanalyze it deeply.The periodic solutions of system (8) with  u ( t ) =Re[Γe i ( ωt + φ ) ]  can be computed through the classical describ-ing function method which srcinates the following equationto be solved in  A ,  B ,  φ A + B e iωt = − L (0) N  0 A + L ( iω )[ − N  1 B  + Γe iφ ]e iωt (16) 6220  or, equivalently,   [1 + L (0) N  0 ( A,B )] A  = 0[1 + L ( iωt ) N  1 ( A,B )] B  =  L ( iωt )Γe iφ .  (17) Such system yields  l  =  q  | L ( iω ) | Γ  q   ≥ 1  q  − L (0)( K  a  + K  b )2 π   1 − q  2 − q   acos(q)  ··  Γ | L ( iω ) − 1 + N  1 ( q  ) | | q  | <  1  1 + L (0)( K  a  + K  b )2   q  Γ | L ( iω ) − 1 + N  1 ( q  ) |  q   ≤− 1 A  =  0  q   ≥  1Γ L (0)( K  a  + K  b )2 π [   1 − q  2 − q   acos( q  )] | L ( iω ) − 1 + N  1 ( q  ) | | q  | <  1 − Γ L (0)( K  a  + K  b )2 q  | L ( iω ) − 1 + N  1 ( q  ) |  q   ≤− 1 B  = Γ | L ( iω ) − 1 + N  1 ( q  ) | φ  = atan  Im  L ( iωt ) − 1 + N  1 ( q  )  Re[ L ( iωt ) − 1 + N  1 ( q  )]  . (18) The variable  q  , for its definition, depends on  A ,  B  and  l ,therefore equations (18) are only implicit relations, “formallymasked” as explicit ones. The functions  A ( q  ) ,  B ( q  )  and l ( q  )  have some regularity properties, such as continuity,differentiability and monotony, which are analyzed in [11].System (18) can not be solved in closed form since it involvestranscendental equations. However, it is possible to obtainits solution through a conceptually easy method. Since thevariable  l  is a known parameter of the model, it is possible, inprinciple, by the first of (18), to determine the correspondingvalues of   q   and then  A ,  B  and  φ  by exploiting the remainingequations. In other words, we have transformed the problemof solving the whole system (17) into the easier problem of solving a single real equation in the unknown  q  .Experimentally the separation-amplitude curve can be ob-tained slowly moving the sample towards the cantilever andmeasuring both the amplitude of the first harmonic and theseparation. Although it is not possible to derive an explicitanalytical form for  B  =  B ( l ) , we can give a parametricform for it. By using the “ q  -explicit” equations in (18) wecan consider the parametric curve  l  =  l ( q  ) B  =  B ( q  )  ∀  q   ∈ R .  (19)If   l ( q  )  is monotone, to any value of   l  corresponds no morethan one value of   q   and, consequently, no more than onevalue of   B . The curve (19) is, in this case, a function  B ( l ) and can be sufficient to study some characteristics of theseparation-amplitude curve. For example, the slope of thecurve at any point can be evaluated exploiting the fact that dBdl  =  dBdq  dldq  − 1 . Moreover, if   l ( q  )  is monotone, we canconsider  lim q →−∞ l ( q  ) . This quantity, if it exists and is finite,represents the minimum distance between the equilibrium po-sition of the cantilever and the sample to have an oscillation.The tip-sample interaction is present only for  q <  1 . Thecontinuity of the functions  l ( q  ) ,  A ( q  ) ,  B ( q  ) ,  φ ( q  )  impliesthat all possible parametric curves that can be obtained, suchas  ( B,l ) ,  ( A,l )  and  ( φ,l ) , are continuous too. In addition,the monotony of   l ( q  ) , numerically verified in a wide rangeof physical values, implies monotony for the separation-amplitude curve. This simple model predicts that, reducingthe distance between the sample and the cantilever, the am-plitude of the tip oscillation is always decreasing too and theperiodic motion changes with continuity its characteristics.This two properties are not fully confirmed by experimentaldata as outlined in [6]. In fact, the separation-amplitude curveoften has a more complex behavior; it can present a non-monotone form, or even jumps. These phenomena, as wewill analyze further, can be explained either qualitatively andquantitatively by introducing attractive forces in the model.In spite of these limits, the present repulsive model is ableto predict with some accuracy the behavior of the system.Moreover, usual repulsive forces have a steep slope. Thissuggests to analyze the particular case where  K  a  and  K  b are large, which, in the limit, is equivalent to model theimpact as instantaneous, with coefficient of restitution equalto  e  =   K  b K  a . We easily note that for  q   ∈ [1 , + ∞ )  B ( q  ) =Γ | L ( iω ) |  is continuous and monotone for any value of   K  a and  K  b . By increasing  K  a  and  K  b , with constant  e , wefind a sequence of continuous functions  B K  a ( q  )  and  l K  a ( q  ) whose plot converges toward the vertical segment at  q   = 1 .Practically, we obtain that for a small left interval of   q   = 1 , B  and  l  assume a wide range of values, as shown for  l ( q  ) in Fig. 3. These properties are proved in detail in [11].Following the last observation we assume, for large  K  a ,that all the solutions of the equation  l  =  l ( q  ) , with  l < Γ | L ( iω ) |  having physical meaning, correspond to  q   ≃  1 .Therefore, we have l  =  qB − A ≃ B − A  ∀  l <  Γ | L ( iω ) | .  (20)For large values of   K  a , an upper bound of   A ( q  )  can becomputed 0 ≤ A ( q  ) ≤ Γ L (0) =:  A max .  (21) If   A max  is negligible with respect to  l  we obtain l ≃ B  ∀  l <  Γ | L ( iω ) |  (if Γ L (0) ≪ l ) .  (22)In [8] the same result ( l  ≃  B ) is achieved for an impulsiveimpact model assuming small  ξ   and  ω  =  ω n  and usingtime-domain methods. In that situation we have  A max  =  Γ ω 2 n and  Γ | L ( iω n ) |  =  Γ ξω 2 n . In fact, in such a case the condition A max  ≪ l  holds for a wide range of   l .We can also evaluate  sin( φ )  using the identity sin(atan( XY    )) =  Y   | X + iY   | , that is sin( φ ) = Im  L ( iωt ) − 1 + N  1 ( q  )  | L ( iωt ) − 1 + N  1 ( q  ) |  .  (23) Substituting in the first of (18) we find, in the case  | q  | <  1 ,that l ( q  ) =  q   +  K  a  + K  b 2 π  q   acos(q) −   1 − q  2  ··  Γ sin( φ )Im[ L ( iωt ) − 1 + N  1 ( q  )] . (24) 6221  0.20.30.40.50.60.70.80.911.1−5051015202530 q    l   (  n  m   ) K a Fig. 3. Parameter  l  as a function of   q  for different  K  a . −5051015202530051015202530 l (nm)    B   (  n  m   ) K a Fig. 4. Separation curves with increasing force slope. For  q   ≃ 1  and  ω  =  ω n  we have l ( q  )  ≃  Γ2 ξω 2 n sin( φ ) .  (25) Again the latter relation has been obtained in [8]. Howeverboth formulas (24) and (25) do not give good agreement withexperiments. As we will explain later and observed in [7] and[6], this is due to having neglected the attractive effect of theinteraction force.V. EXTENSION TO A REPULSIVE-ATTRACTIVEFORCELet us modify the nonlinear function used in the previoussection with the aim to consider also the attractive region.In order to derive useful analytical results we propose thenonlinearity   h ( x,  ˙ x ) = 0  ∀  x ≥− lh ( x,  ˙ x ) =  F  a  + K  a ( x + l )  ∀  x < − l ∧  ˙ x <  0 h ( x,  ˙ x ) =  F  b  + K  b ( x + l )  ∀  x < − l ∧  ˙ x >  0 (26)reported in Fig. 5, where  F  a  and  F  b  are two additiveconstants describing in some way the attractive forces andsatisfying the relation  0  ≤  F  a  ≤  F  b  to account for energylosses. The new function is the sum of two simple non-linearities: the preceding linear hysteresis plus a constanthysteresis. The describing function is the sum of the two h x −l F F a b K (x+l) K (x+l) a b Fig. 5. Qualitative behavior of the hysteresis with repulsive attractive effect. separate describing functions  N  0  =  12 πA  { ( F  a  + F  b )acos( q  )+ − ( K  a  + K  b )[   1  − q  2 − q  acos( q  )] B  N  1  =  −  1 BF  a + F  b π   1  − q  2 −  K a + K b 2 π  [ q    1  − q  2 −  acos( q  )]++ i  F  b − F  a πB  (1  − q  )  −  K b − K a 2 π  (1  − q  ) 2  . (27) If we define, for convenience, Σ F   :=  F  a  + F  b  Σ K  :=  K  a  + K  b ∆ F   :=  F  b  − F  a  ∆ K  :=  K  b  − K  a ¯Φ R ( q  ) :=   1  − q  2 π ¯Ψ R ( q  ) := acos( q  )  − q    1  − q  2 2 π ¯Φ I  ( q  ) := 1  − q π ¯Ψ I  ( q  ) := (1  − q  ) 2 2 π ¯ χ ( q  ) := acos( q  )2 π ¯Ω( q  ) :=   1  − q  2 − q  acos( q  )2 π (28) we can write N  0  =Σ F   ¯ χ ( q  )  −  Σ K  ¯Ω( q  ) BAN  1  =  − ¯Φ R ( q  ) B  + Σ K  ¯Ψ R ( q  ) + i  ∆ F   ¯Ψ R B  −  ∆ K  ¯Ψ I  ( q  )  . (29) Finally, by the substitution Φ :=  − Σ F   ¯Φ R  + i ∆ F   ¯Φ I   Ψ := Σ K  ¯Ψ R  − i ∆ K  ¯Ψ I   (30) we can obtain a formally handy expression for the describingfunctions N  0  = 1 A [ χ ( q  ) + Ω( q  ) B ] N  1  =Φ( q  ) B  + Ψ( q  ) . (31)We want to perform an analysis similar to the previoussection. By imposing the harmonic balance we can obtaina system analogous to (18). Note that now  N  1  can not bewritten in a form dependent on  q   only. However, it is possibleto write again  B  as a function of   q  , since B  = Γ | L ( iω ) − 1 + N  1 |  (32) implies  L ( iω ) − 1 B  + BN  1  2 =  | L ( iω ) − 1 B  + Φ + Ψ B | 2 = Γ 2 .  (33) 6222
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