Detection of cracks in rotor based on the 2X and 3X super-harmonic frequency components and the crack-unbalance interactions

Transcription

1 Detection of cracs in rotor based on the X and 3X super-harmonic frequency components and the crac-unbalance interactions Jean-Jacques Sinou To cite this version: Jean-Jacques Sinou. Detection of cracs in rotor based on the X and 3X super-harmonic frequency components and the crac-unbalance interactions. Communications in Nonlinear Science and Numerical Simulation, Elsevier, 8, 3, pp <0.0/j.cnsns >. <hal > HAL Id: hal Submitted on 8 Feb 03 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Detection of cracs in rotor based on the and 3 super-harmonic frequency components and the crac-unbalance interactions Abstract Jean-Jacques Sinou Laboratoire de Tribologie et Dynamique des Systèmes UMR-CNRS 553 Ecole Centrale de Lyon, 3 avenue Guy de Collongue 93 Ecully Cedex, France jean-jacques.sinou@ec-lyon.fr The purpose of this paper is to investigate the use of the and 3 super-harmonic components for detecting the presence of a single transverse breathing crac in a non-linear rotor system. This procedure is based on the detection of the super-harmonic components of the non-linear dynamical behaviour at the associated sub-critical resonant peas. The non-linear behaviour of the rotor system with a breathing crac is briefly analysed numerically: it will be illustrated that the effects of the crac size and location induce the variation of non-linear responses and the emerging of new resonance - antiresonance peas of the craced rotor at second, third and fourth harmonic component. Then, the influence of the crac-unbalance interactions and more particularly the relative orientation between the front crac and the unbalance are also undertaen with considerations of various crac depths, and unbalance magnitudes. It is demonstrated that for a given crac depth, the unbalance does not only affect the vibration amplitude of the amplitudes, but also the and 3 sub-critical resonant peas. Finally, it is illustrated that the emerging of super-harmonic components provides useful information on the presence of crac and may be used on an on-line crac monitoring rotor system. Using this methodology, the detection of small levels of damage may be easily undertaen. Keywords: craced detection, rotor system, non-linear vibration, super-harmonic components. Introduction Detection of damage in rotor systems is an important concern to engineering communities. The importance of early detection of cracs has led to continuous efforts due to the fact that unpredictable occurrence of damage may cause catastrophic failure. It is very difficult but also highly desirable to pursue effective engineering solutions to detect and locate the damage situation in rotating systems at the earliest possible stage. Reviews on the dynamical behaviour of rotors with transverse crac were published by Wauer [], Gasch [] and Dimarogonas [3]. During the past several decades, significant amount of research has been conducted in the area of crac detection in systems using only theoretical modelling method [ ], combined both theoretical and experimental methods [ ] or only experimental method [5]. The main idea of these approaches is that a change in a rotor system due to damage crac will manifest itself as changes in the rotor dynamic behaviour: first of all, the presence of a transverse crac induces a slight decrease of the natural frequencies [, 0, ]. Secondly, resonances appear when the rotational speeds of the shaft reach and 3 of the critical speeds of the rotor system. Therefore, with the increase of the crac depth, the and

3 3 sub-critical resonant peas increase [,, 7]. Finally, some researchers [8] indicated that the shaft executes two and three loops per shaft revolution at the and 3 sub-critical speeds, respectively. In most of the studies for crac detection in rotor systems, researchers used changes in natural frequencies and evolution of the non-linear behaviour of the system at the super-harmonic components as the diagnostic tools. In this paper it will be shown that an appropriate use of the super-harmonic components may be useful for crac detection in rotor systems. So the present study attempts to propose a complete analysis of the crac-unbalance interactions on the super-harmonic components at the and 3 sub-critical resonant peas. Numerical example will be conducted on variety of damage location, crac size and unbalance parameters (magnitude and relative orientation with the front crac) to verify the suitability of the use of the super-harmonic components in order to detect the presence of a transverse crac in rotor. One of the advantages of the proposed approach is that the emerging of super-harmonic components may be easily undertaen for the detection of a crac in rotating shafts, especially in the early stage of the damage where the ability to discriminate changes of modal parameters caused by damage from those caused by other environmental condition changes is very difficult. The paper is set up as follows: firstly, the description of the non-linear rotor system and the modeling of the breathing crac are investigated. Then, the non-linear periodic response of the craced rotor is undertaen by approximating the non-linear dynamic by truncated Fourier series with m harmonics. Moreover, the state vectors of the complete craced rotor will be partitioned into subvectors relating to the Fourier components which are associated with the degrees of freedom at the crac location, and the Fourier components which are associated with the others degrees of freedom. Then, the emerging of the and 3 super-harmonic components for detecting the presence of a crac is investigated. Numerical examples including various crac parameters (location and depth) and unbalance parameters (magnitude and orientation with the crac) are considered in order to validate the detection of a crac based on the resonance peas at the or 3 sub-critical resonances and the determination of the associated super-harmonic frequency components. The model of the craced rotor In this study, the rotor is composed of a shaft with one disc at the mid-span, as illustrated in Figure. All the values of the physical parameters are given in Table.. Shaft elements The shaft is discretized into 0 Timosheno beam finite elements with four degrees of freedoms at each node (two lateral displacements and two rotations). At each node of the Timosheno beam finite elements, we have [9, 0] (M e T +Me R )Ẍe +(ηk e B ωge )Ẋ e +(K e B +ηωke C )Xe = F e () where ω is the rotational speed. M e T and Me R, and Ge are the translational, rotary mass and gyroscopic matrices of the shaft element, respectively. K e B and Ke C are the stiffness and circulatory matrices due to shaft internal damping. η defines the coefficient of damping that is associated to the modal damping for the first mode of the system at rest ( ω = 0 ). F e includes the gravitational forces and unbalance forces.

4 . Rigid disc The rotor system has one disc at the mid-span that is modelled as a rigid dis and may be written as ( M d T +M d R)Ẍd ωg d Ẋ d = F d () where M d T and Md R are the translational mass and rotary mass matrices respectively. Gd is the gyroscopic matrix, and F d corresponds to the unbalance and gravitational forces. Figure : Finite-element model of the rotor and the craced-beam section Notation Description Value R radius of the rotor shaft 0.005m L length of the rotor shaft 0.5m R D outer radius of the dis 0.05m h D thicness of the dis 0.05m E Young s modulus of elasticity. 0 N.m G shear modulus N.m ρ density 7800g.m 3 ν Poisson ratio 0.3 η coefficient of damping 0 5 m e mass unbalance 0.00g d e eccentricity of the mass unbalance 0.0m φ relative orientation between the crac and the unbalance 0degree K s stiffness of supports 0 N.m ω,ω first double frequency of the uncraced rotor (at rest) 37rad.s ω 3,ω second double frequency of the uncraced rotor (at rest) 898rad.s Table : Value of the physical parameters 3

5 .3 Modelling of the breathing crac.3. The crac model If a transverse crac appears in a rotor system, additional flexibilities are generated at the location of the crac due to strain energy concentration in the vicinity of the crac tip under load. There are a number of approaches for modelling cracs in shafts; we refer the interested reader to [] and [3] for comprehensive literature survey of various crac modelling techniques. In this paper, the model proposed by Mayes and Davies [3, ] is used in order to locally represent the stiffness properties of the crac cross section. This model considers the reduction of the second moment of area I of the element at the location of the crac that ay be defined by R ( ν ) F (µ) I = I 0 l + R ( ν ) (3) F (µ) l where I 0, R, l, and ν are the second moment of area, the shaft radius, the length of the section and the Poisson s ratio, respectively. µ is the non-dimensional crac depth and is given by µ = h R where h defines the crac depth of the shaft. F (µ) defines the non-linear compliance function varied with the non-dimensional crac depth µ that may be obtained from a series of experiments with chordal cracs [3, ]. So, the stiffness matrix K crac of the crac cross section is given by (to the principal axes of the crac front) K crac = E l 3 I X 0 0 li X I X 0 0 li X I Y li Y 0 0 I Y li Y 0 l I Y 0 0 li Y l I Y 0 l I X li X 0 0 l I X I X 0 0 li X I Y li Y 0 Sym. l I Y 0 l I X The moments of inertia about the parallel centroidal axes, I X and I Y, are given by [0] () I X = ĨX (5) I Y = ĨY A X () where X and A define the uncraced area of the cross-section and the distance from the axis X to the centroid of the cross section X = 3A R3 γ 3 (7) ( A = R ( µ)γ + α ) (8) where α defines the crac angle and is given by α = cos ( µ). Then, the asymmetric area moments of inertia ĨX and ĨY about the X and Y-axes are defined as Ĩ X = A Y da = R ( ( ( µ) µ+µ ) γ + α ) (9)

6 Ĩ Y = A where γ is equal to X da = πr +R µ µ for convenience. ( 3 ( µ)γ3 + ( ( µ) µ+µ ) ) γ +sin (γ) (0).3. The breathing mechanism When a craced rotor rotates slowly under the load of its own weight, the crac will open and close once per revolution. This periodic opening and closing of the crac is called breathing phenomenon [8]. Due to this mechanism, the stiffness matrix of the shaft at the crac position is non-linear and periodical time varying during the rotation of the rotor system. As previously demonstrated by Gasch [, ], the opening and closing of the crac during its rotation is mainly due to the shaft self-weight. So, assuming that the static deflection is much greater that the dynamic response of the craced rotor, the breathing of the crac may be expressed by a cosine function f(t) f (t) = cosωt () where ω defines the rotational speed of the rotor. During the shaft s rotation, the crac opens and closes: the associated breathing action of the crac is illustrated in Figure. When the crac is fully closed the rotor may be treated as uncraced, due to the fact that the crac has no effect on the dynamic behaviour of the rotor (i.e. f(t) = 0). If f(t) =, the crac is fully open. As previously explained, this opening and closing of the crac (described in Equation ) assumes that the gravity determinates the breathing of the crac due to weight dominance (i.e. the static deflection is much greater than the rotor vibration). Figure : Breathing crac due to the rotation of the shaft (white = portion of opened crac, blac=portion of closed crac) 5

7 . Equation of motion of the craced rotor After assembling the different shaft elements, the rigid disc and the discrete bearing stiffness that are located at the two ends of the shaft, the equation of motion of the complete craced rotor system in a fixed co-ordinate system can be written as MẌ+DẊ+(K f (t)k c )X = Q+W () where overdots indicate differentiation with respect to time. The mass matrix M includes mass matrices of the shaft and rigid disc. The matrix D considers the shaft internal damping, the damping of the supports and the gyroscopic moments. The matrix K includes the stiffness matrices of the shaft and supports, and the circulatory matrix due to shaft internal damping. K c is the stiffness matrix due to the crac. The terms of this matrix are equal to zero except at the crac location degree-of-freedom where the 8 8matrix K crac is present. Q and W are the vector of gravity and imbalance forces due to the dis and the shaft, respectively. As previously indicated, the above equations of the craced rotor have a time-dependent coefficient due to the fact that the crac breathes when the system rotates. The amount of open part of the crac constantly varies with the rotation of the shaft, thereby changing the stiffness of the craced rotor. The global stiffness matrix of the rotor consists of a constant part K and a time dependent part f (t)k c. 3 Non-linear analysis 3. Non-linear responses of the craced rotor Due to the time-dependent coefficient of Equation, the system of the crac rotor may be rewritten in a non-linear form as MẌ+DẊ+KX = Q+W+f NL (X,ω,t) (3) with f NL (X,ω,t) = ( cosωt)k cx () In the following the term f NL will be treated as a non-linear term due to its dependence on X that maes Equation 3 non-linear. A frequency-domain method such as the harmonic balance methods with continuation schemes that are well-nown numerical tools, may be applied in order to study non-linear dynamics vibrations in rotating systems [, 3]. This approach may be used as an alternative to timedomain methods when periodic solutions of the on-linear system exist, and so is a very efficient way of approximating the vibration of a craced rotor. We refer the interested reader to [ 5] for a survey of some recent developments and alternative approaches. The general idea of the harmonic balance method is to represent the periodic solution of the non-linear system by its frequency content. So, the non-linear dynamical responses of the craced rotor system are represented as truncated Fourier series with m harmonics: X(t) = B 0 + m (B cos(ωt)+a sin(ωt)) (5) = where ω defines the fundamental frequency. B 0, A and B (with =,,m) define the unnown coefficients of the finite Fourier series. The number of harmonic coefficients m is selected on the basis

8 of the number of significant harmonics expected in the non-linear dynamical response. Moreover, the non-linear force f NL, the gravity force Q and the global unbalance force W are represented as truncated Fourier series. First of all, the non-linear force due to the presence of the crac f NL is approximated by finite Fourier series of order m f NL (X,ω,t) = C f m 0 + ) (C f cos(ωt)+sf sin(ωt) = Then, it may be observed that the unbalance force components without considering the crac (for the shaft and the dis) in the horizontal and vertical directions (Y-direction and X-direction as indicated in Figure ) are given as m e d e cos(ωt+φ) and m e d e sin(ωt+φ), respectively. m e and d e are the mass unbalance and the eccentricity for each element of the rotor system. φ defines the initial angular position with respect to the Z-axis. So, the gravity force Q and the global unbalance force W are exactly defined by constant components and first-order periodic components in the frequency domain, respectively. We have () Q(X,ω,t) = C Q 0 (7) W(X,ω,t) = C W cos(ωt)+sw sin(ωt) (8) Substituting these last fourth expressions 5,, 7 and 8 into the rotor equation of motion 3 and balancing the harmonic terms yields a set of (m+) n equations where n is the number of degreeof-freedom for the complete craced rotor system. The constant terms B 0 that are given by the first n th relations are given by KB 0 = C Q 0 +Cf 0 (9) Then, the first harmonic components A and B are determined by resolving the following equations [ ][ ] [ K ω M ωd A S W ωd K ω = +S f ] M B C W (0) +Cf Finally, the m (n ) remaining equations that define the th Fourier coefficients A and B for m are given by [ K (ω) ][ ] [ ] M ωd A S f ωd K (ω) = M B C f () The non-linear expression f NL (X,ω,t) is a function of the non-linear responses X(t) and the associated Fourier coefficients B 0, A and B (with m). So, the Fourier coefficients C f 0, Sf and Cf (with m) may be determined from B 0, A and B (with m) by using the following iteration process, called the Alternate Frequency/Time domain approach (AFT method []) [B 0 A B A m B m ] T X(t) f NL (X,ω,t) [C 0 S C S m C m ] T () Then, the (m+) n non-linear equations of motion 9, 0 and can be solved by using a solver such as the Newton-Raphson method [7]. Moreover, a continuation scheme in conjunction with the harmonic balance method and based on the path following continuation and Lagrangian polynomial extrapolation [0, 8], is used to give a first approximation of the Fourier coefficients B 0,A andb (with m) of the craced rotor system when the rotational speed ω increases. 7

9 3. Partition and condensation on the craced element The state vectors A and B (for m) are partitioned into subvectors relating to the Fourier components A c and Bc which are associated with the degrees of freedom at the crac location, and the Fourier components A u and Bu which are associated with the others degrees of freedom. [ ] A c [ ] U c U = B U u = c A u = Ψ A (3) B B u The subscript represents th harmonic components, the superscript u represents uncraced, and the superscript c represents craced. Hence, for the present case (i.e. the rotor system has only one crac), the vectors A c and Bc have the size of 8, and the vectors A u and Bu have the size of 3. Then, the vectors Uc have the size of, and the vector U u have the size of 7. Considering Equation 3, Equation 0 and which is associated with the th harmonic components can be partitioned as Θ U = F () with [ Θ cc Θ = Θ uc ] Θ cu Θ uu [ = Ψ T K (ω) M ωd ωd K (ω) M ] Ψ (5) Each of the matricesθ have the size of88 88;Θ cc,θcu,θuc andθuu are, 7,7, and 7 7 matrices, respectively. The expressions of F which is associated with the first harmonic components is given by F = [ F c F u ] = S W,c +S f,c C W,c +C f,c S W,u C W,u [ S = W ΨT +S f ] C W +Cf and the expressions of F (for m) which are associated with the th harmonic components can be rewritten as [ ] [ ] S f,c [ ] F c F = F c F u = = C f,c S f 0 0 = ΨT C f (7) 0 The vectors F have the size of 88. The vectors S f,c, Cf,c, SW,c and C W,c have the size of 8, and the vectors S W,u and C W,u have the size of 3. The vectors F (for m) represent the excitation due to the presence of the crac. So the vector F u is a is a zero vector, as indicated in Equation 7. Moreover, it may be remained that the vector F corresponds not only to the excitation due to the presence of the crac, but also to the contribution of the unbalance force: the terms of this vector are zero except at the crac and unbalance locations. So F u may only contain an unbalance contribution, as indicated in Equation. () 8

10 So, considering Equations 3,, 5 and, Equations 0 that define the first harmonic components A and B can be partitioned as [ ][ ] [ ] Θ cc Θ cu U c F c Θ uc Θ uu = (8) Then, considering 3,, 5 and 7, them (n ) Equations that define the th Fourier coefficients A and B for m may be partitioned as [ Θ cc Θ uc U u ][ ] Θ cu U c Θ uu U u = F u [ F c 0 Finally, considering Equations 8 and 9, the vectors U c and Uc that correspond to the Fourier components of the crac element may be determined by solving ( ) ( ) U c = Θ cc Θcu Θuu Θ uc F c Θcu Θuu F u U c = ( Θ cc Θ cu Θ uu Θ uc ] (9) (30) ) F c (3) Then, the Fourier components vectors U u and Uu of the uncraced elements are given by U u = Θ uu F u Θ uu U u = Θuu Numerical simulations ( ( ) Θ cc Θ cu Θ uu Θ uc ( F c Θ cu Θ uu F) ) u (3) Θ uc. Effects of the crac size and location ( Θ cc Θcu Θuu ) F Θ uc c (33) In this section, the main effects of the crac size and location on the non-linear behaviour of the craced rotor system are briefly summarized. Firstly, Figures 3 illustrate the effects of crac depth on the vertical and horizontal responses corresponding to the first harmonic component (see Figure 3(a)), the super-harmonic frequency components (see Figure 3(b)), the super-harmonic frequency components (see Figure 3(c)), and the superharmonic frequency components (see Figure 3(d)) at the node position of the shaft 0.5m. Due to the presence of the crac, the second harmonic components increase when the rotational speed reaches and of the critical speeds. The third harmonic components (respectively, fourth harmonic components) increase near the rotational speeds at 3, and of the critical speeds (respectively near the rotational speeds at, 3, and of the critical speeds). A decrease in the critical speeds of the rotor system due to the reduction in system stiffness resulting from the presence of the crac is also observed. Moreover, it is clear that the vibration amplitudes of the second, third and fourth harmonic components depend on the craced depth: with the increase of the crac depth, these harmonic components increase. Considering the first harmonic component, the vibration amplitudes of the crac rotor system do not greatly change with respect to the crac size. However, it may be remind that, for a given crac depth, the first harmonic component of the crac rotor system is associated with the rotor imbalance and the relative position between the crac direction and the imbalance [7]. Then, the effects of crac position on the harmonic components of the nonlinear response of the rotor are 9

11 illustrated in Figures. It may be remind that the crac location clearly affect the decrease in the critical speeds of the craced rotor and the vibration amplitudes in the sub-critical resonances []. Finally, it may be observed that antiresonances for the, 3 and super-harmonic frequency components of the craced rotor system appears due to the presence of the crac. The emerging and location of new antiresonances and the shift in the antiresonances depend on the crac size and location. In conclusion, the variation of non-linear responses and the emerging of new resonance - antiresonance peas of the craced rotor at second, third and fourth harmonic components may provide useful information on the presence of a crac and may be used on an on-line crac monitoring rotor system. Vertical displacement (m) Horizontal displacement (m) Vertical displacement (m) Horizontal displacement (m) Rotating speed (rad.s ) Rotating speed (rad.s ) (a) harmonic components Rotating speed (rad.s ) Rotating speed (rad.s ) (c) 3 super-harmonic components Vertical displacement (m) Horizontal displacement (m) Vertical displacement (m) Horizontal displacement (m) Rotating speed (rad.s ) Rotating speed (rad.s ) (b) super-harmonic components Rotating speed (rad.s ) Rotating speed (rad.s ) (d) super-harmonic components Figure 3: Effects of the crac size at the node position of the shaft 0.5m with a crac situated at L crac = 0.75m and the unbalance located at0.m of the left end ( µ =,...µ = 0.75,. µ = 0.5, µ = 0.5) 0

12 Vertical displacement (m) Horizontal displacement (m) Horizontal displacement (m) Vertical displacement (m) Rotating speed (rad.s ) Rotating speed (rad.s ) (a) harmonic components Rotating speed (rad.s ) Rotating speed (rad.s ) (c) 3 super-harmonic components Vertical displacement (m) Horizontal displacement (m) Vertical displacement (m) Horizontal displacement (m) Rotating speed (rad.s ) Rotating speed (rad.s ) (b) super-harmonic components Rotating speed (rad.s ) Rotating speed (rad.s ) (d) super-harmonic components Figure : Effects of the crac location at the node position of the shaft 0.5m with a non-dimensional crac depth µ = and the unbalance located at 0.m of the left end (position of the crac L crac = 0.075m, L crac = 0.5m,. L crac = 0.75m,...L crac = 0.5m)

13 . Damping and unbalance effects First of all, it is well nown that increasing the rotor unbalance increases the amplitudes of the craced rotor [7]: this fact is indicated in Equations 8 and 0. However, it may be noted that the other n amplitudes (with n ) may also be affected by the rotor imbalance. Figure 5(a) illustrates the evolution of the vibration amplitudes in the sub-critical resonances with the variation of the unbalance of the craced rotor. With the increase of the rotor unbalance, the amplitudes at the sub-critical resonances increase, due to the interaction of the crac breathing mechanism, gravity and rotor unbalance (as indicated in Equations 9 and ). Effectively, it may be remained that the amplitudes of the non-linear terms due to the presence of the crac depend on the amplitudes of the rotor s vertical and horizontal displacements (as indicated in Equation ) and so the rotor unbalance. Moreover, the vibration amplitudes in the sub-critical resonances depend on the damping of the craced rotor system, as illustrated in Figure 5(b). With any decrease of damping, the amplitudes increase drastically and the presence of the crac may be clearly detected. However, if the damping of the rotor system is relatively high, the resonant amplitudes in the sub-critical resonances will disappear due to the fact that the and 3 super-harmonic frequency components are suppressed. These informations can be used as indexes for the detection of cracs in the rotor system: if the damping remains constant, increasing the unbalance of the rotor system may change and increase the n amplitudes (with n ) when the rotor reaches the n sub-critical resonances. However, it is clear that the vibration amplitudes in the n sub-critical resonances (with n ) depend not only on the rotor damping, unbalance, position and depth of the crac, but also on the combinations of the unbalance and the crac parameters. So, the effects of crac-unbalance interaction are analysed in the following section of this paper. Vertical displacement (m) x Vertical displacement (m) x x 0 3 Unbalance (g.m) (a) Unbalance effect x Damping factor (b) Damping effect 5 5 Figure 5: Influence of damping and mass unbalance on the vertical vibration amplitudes around subcritical resonances (with µ =, L crac = 0.5m) and the unbalance located at 0.5m

14 .3 Effects of the crac-unbalance interaction on the super-harmonic frequency components Figures 7 illustrate the second super-harmonic frequency components of the middle of the rotor for various crac-unbalance orientations and unbalance in the vertical and horizontal directions. It clearly appears that the relative orientation angle between the unbalance of the craced rotor and the crac and their interaction drastically affect the evolutions of the second super-harmonic frequency component at the sub-critical resonances. Then, the evolutions of the sub-critical resonances pea with respect to the unbalance-crac angle change due to the magnitude of the unbalance in both the vertical and horizontal directions (see for example Figures 7(b), (d) and (f)). Therefore, it may be observed that the interaction between the crac and the unbalance may mas the presence of the crac: effectively, the second super-harmonic frequency component and the resonant amplitudes in the sub-critical resonances may disappear (see for example Figure 7(b) when the angle of the unbalance is at 70 degrees). Figures indicate the evolutions of the third super-harmonic components of the middle of the rotor when the rotor reaches the 3 sub-critical resonances. As previously seen for the second super-harmonic frequency component, depending on the relative angle between unbalance and crac vectors, the third super-harmonic frequency component can increase or even decrease in vertical and horizontal amplitudes. With the decrease of the rotor unbalance, the magnitudes of the super-harmonic frequency components decrease in the vertical and horizontal directions. If the crac effect is predominant, the magnitude of the sub-critical resonances peas does not greatly change, as illustrated in Figures 7(e-f) and (e-f). If the crac unbalance is more important than the crac, it is well nown that the the magnitude of the sub-critical resonances peas is constant as shown in Figure 7(a) for the horizontal direction. However, it may be noted that the associated magnitude of the sub-critical resonances pea slightly changes in the vertical direction: effectively, the highest changes in the stiffness of the crac cross section (see Equation ) occur in the vertical direction due to the orientation of the crac and the shaft self-height. This is why the sensibility of the magnitudes of and 3 sub-critical resonances with respect to the unbalance angle and the unbalance-crac interactions are different in the vertical and horizontal directions. Moreover, the influence of the crac on the non-linear dynamic of the rotor system increases when the unbalance magnitude decreases. In this case, two resonant peas appear in the horizontal direction due the coupling between the two bending directions, as indicated in Figure 7(e). With the decrease of the unbalance, the ratio between the first resonance pea (at 03rad/s) and the second resonance pea (at 05rad/s) decreases (see Figures 7(a), (c) and (e)). The same phenomenon is observed at the 3 subcritical resonances, as indicated in Figures (a), (c) and (e). It may be noted that the crac-imbalance magnitudes and relative angle are not nown a priori, maing crac detection very difficult. All these information can be used to identify the crac-unbalance interaction and the predominance of the crac or unbalance on the dynamic of the rotor system. In conclusion, for a given crac depth and position, the magnitudes of the second and third superharmonic frequency components (at the and 3 sub-critical resonances, respectively) are associated with the rotor unbalance and the position of the unbalance relative to the front crac direction. With the decrease and increase of the rotor unbalance, the magnitudes of the second and third super-harmonic components in both the vertical and horizontal directions may drastically change due to the interaction of gravity, the rotor unbalance and the crac breathing action. All these phenomena may also be observed for first order harmonic frequency components of the first critical speed, as illustrated in Figure 8(a), but also for the super-harmonic frequency components of the second critical speed of the craced rotor, as indicated in Figure 8(b): the rotor unbalance and the 3

15 position of the unbalance relative to the crac direction greatly influence the maximum of the resonance pea at the critical speed and the second super-harmonic frequency components of the second critical speed.. Effects of the crac-unbalance orientation and crac depth In this section the influence of the crac depth with the interaction of the crac-unbalance orientation is investigated. For the sae of clarity, we focus the study at the 3 sub-critical resonances. Figures 9 show the third super-harmonic frequency components of the degree-of-freedom situated at the middle of the rotor, for various crac-unbalance orientations and three non-dimensional crac depth. These figures may be compared with the Figures (c) and (d) of the previous section (with a non-dimensional crac depth that is equal toµ =, corresponding to the loss of half the shaft s area). Due to the crac depth and the crac-unbalance interaction, the magnitudes of the third super-harmonic frequency components at 3 sub-critical resonances of the first critical speed change: with the decrease of the non-dimensional crac depth, the influence of the crac is less predominant in the horizontal direction. Moreover, the value of the associated resonance pea decreases with the increase of the crac due to the reduction of the second moment of area at the location of the crac. For a deep crac (µ = in Figure (c)), two resonance peas appear due to the breathing crac and the associated coupling between the horizontal and vertical direction. When the crac depth decreases, the first resonance pea disappear (as shown for µ = 0.75 in Figure 9(a)). Then, the crac-unbalance interaction is more predominant in the vertical direction: when the crac depth decrease, the ratio between the minimum and maximum of the third super-harmonic frequency components (as a function of the orientation between the crac and the unbalance) decrease or increase. This reflects the fact that for a deep crac, the crac effect is predominant, whereas the unbalance effect is more important when the crac depth is small. All these results illustrate that the detection of a crac can be difficult due to the interaction of the effects of the crac and the unbalance. However, the influence of the orientation between the crac and the unbalance appear to be clearly identified if the evolutions of the n super-harmonic frequency components at the n sub-critical resonances are investigated. It may be observed that the classical non-linear responses of the craced rotor at the or 3 sub-critical resonances may be very complex, as indicated in Figures 0. Effectively, the evolution of the complete non-linear magnitudes as a function of the relative orientation angle between unbalance and the crac and their interaction drastically affect the system response, maing crac detection very difficult. The magnitudes of the or 3 sub-critical resonances correspond to the combination of all the harmonic components. So these evolutions of resonance peas do not permit a vibration characterization of the craced rotor system due to the influence and interaction between all the super-harmonic frequency components. 5 Conclusion The evolution of the super-harmonic components of and 3 revolution in the sub-critical speed region can be used as an index to detect a crac in the rotor. However, due to crac-unbalance interaction the evolutions of the super-harmonic frequency components and the associated resonance peas may be very complex. It was demonstrated that for a given crac depth, the unbalance does not only affect the vibration amplitude of the amplitudes, but also the and 3 sub-critical resonant peas. With the increase of the unbalance magnitude, the n sub-critical resonant peas increase obviously due to the non-linear behaviour of the breathing crac and the interaction between the crac, gravity and unbalance

16 x 0 7 x 0 5 Horizontal amplitude (m) Vertical amplitude (m) (a) Horizontal amplitudes - m e d e = 0 g.m (b) Vertical amplitudes - m e d e = 0 g.m 08 Horizontal amplitude (m) x (c) Horizontal amplitudes - m e d e = 0 5 g.m Vertical amplitude (m) x (d) Vertical amplitudes - m e d e = 0 5 g.m 08 Horizontal amplitude (m) x (e) Horizontal amplitudes - m e d e = 0 7 g.m Vertical amplitude (m) x (f) Vertical amplitudes - m e d e = 0 7 g.m 08 Figure : Evolution of the 3 super-harmonic frequency components on the 3 sub-critical resonances (at the middle of the shaft0.5m) with respect to the crac-unbalance orientation (with µ =,L crac = 0.5m and the unbalance located at0.5m) 5

17 Horizontal amplitude (m) x (a) Horizontal amplitudes - m e d e = 0 g.m Vertical amplitude (m) x (b) Vertical amplitudes - m e d e = 0 g.m Horizontal amplitude (m) x 0 8 Vertical amplitude (m) x (c) Horizontal amplitudes - m e d e = 0 5 g.m (d) Vertical amplitudes - m e d e = 0 5 g.m 0 Horizontal amplitude (m) x (e) Horizontal amplitudes - m e d e = 0 7 g.m Vertical amplitude (m) x (f) Vertical amplitudes - m e d e = 0 7 g.m 0 Figure 7: Evolution of the super-harmonic frequency components on the sub-critical resonances (at the middle of the shaft0.5m) with respect to the crac-unbalance orientation (with µ =,L crac = 0.5m and the unbalance located at0.5m)

18 x 0 5 x 0 5 Horizontal amplitude (m) 5 3 Vertical amplitude (m) (a) µ =, L crac = 0.5m, m e d e = 0 7 g.m (b) µ =, L crac = 0.5m, m e d e = 0 5 g.m Figure 8: Evolution of the amplitudes with respect to the crac-unbalance orientation (a) at the middle of the shaft 0.5m for the first harmonic frequency components in the first critical speed with the unbalance located at 0.5m, (b) at the one third of the shaft 0.5m for the super-harmonic frequency components in the sub-critical resonances of the second critical speed with the unbalance located at 0.5m of the rotor. Even if the crac-unbalance orientation and the unbalance magnitude are unnown, both and 3 super-harmonic frequency components can be used to detect the presence of crac in rotor. The suitability of this approach was verified for various numerical example on a variety of damage location, crac size, unbalance magnitude, and crac size orientation. It appears that the detection of the resonances peas at the or 3 sub-critical resonances and the determination of the associated super-harmonic frequency components may be useful and acceptable to the industrial community. References [] Wauer, J., 990. Dynamics of craced rotors: Literature survey. Applied Mechanics Review, 3, pp [] Gasch, R., 993. A survey of the dynamic behaviour of a simple rotating shaft with a transverse crac. Journal of Sound and Vibration, 0(), pp [3] Dimarogonas, A., 99. Vibration of craced structures: a state of the art review. Engineering Fracture Mechanics, 55, p [] Sehar, A.,. Crac identification in a rotor system:a model-based approach. Journal of Sound and Vibration, 70, p [5] Gounaris, G. D., and Papadopoulos, C. A.,. Crac identification in rotating shafts by coupled response measurements. Engineering Fracture Mechanics, 9, p

19 x 0 8 x 0 7 Horizontal amplitude (m) Vertical amplitude (m) (a) Horizontal amplitudes - µ = 0.75 (b) Vertical amplitudes - µ = x 0 8 x 0 8 Horizontal amplitude (m) 3 Vertical amplitude (m) (c) Horizontal amplitudes - µ = 0.5 (d) Vertical amplitudes - µ = Horizontal amplitude (m) x Vertical amplitude (m) x (e) Horizontal amplitudes - µ = 0.5 (f) Vertical amplitudes - µ = Figure 9: Evolution of the3 super-harmonic frequency components in the 3 sub-critical resonances (at the middle of the shaft 0.5m) with respect to the crac-unbalance orientation and the non-dimensional crac depth (with L crac = 0.5m, m e d e = 0 5 g.m and the unbalance located at 0.5m) 8

20 Horizontal displacement (m) x (a) Horizontal amplitudes - m e d e = 0 5 g.m Horizontal displacement (m) x (b) Horizontal amplitudes - m e d e = 0 7 g.m Vertical displacement (m) x Horizontal displacement (m) x (c) Vertical amplitudes - m e d e = 0 g.m (d) Horizontal amplitudes - m e d e = 0 7 g.m Figure 0: Evolutions of the non-linear responses (a-b) sub-critical resonances, (c-d) 3 sub-critical resonances (with L crac = 0.5m and µ = ) 9

21 [] Chen, C., Dai, L., and Fu, Y.,. Nonlinear response and dynamic stability of a craced rotor. Communications in Nonlinear Science and Numerical Simulation, In press, p. 5. [7] Prabhaar, S., Sehar, A., and Mohanty, A.,. Transient lateral analysis of a slant-craced rotor passing through its flexural critical speed. Mechanism and Machine Theory 37 () 7 00, 37, p [8] Friswell, M., and Penny, J.,. Crac modelling for structural health monitoring. International Journal of Structural Health Monitoring, (), p [9] Pugno, N., Surace, C., and Ruotolo, R., 0. Evaluation of the non-linear dynamic response to harmonic excitation of a beam with several breathing cracs. Journal of Sound and Vibration, 35 (5), p [0] Sinou, J.-J., and Lees, A. W., 5. Influence of cracs in rotating shafts. Journal of Sound and Vibration, 85(-5), pp [] Sinou, J.-J., and Lees, A. W., 7. A non-linear study of a craced rotor. European Journal of Mechanics A/Solids, (), pp [] Mayes, I., and Davies, W., 97. The vibrational behaviour of a rotatingsystem containinga transverse crac. IMechE Conference on Vibrations in Rotating Machinery, C/8/7, pp. 53. [3] Mayes, I. W., and Davies, W. G. R., 98. Analysis of the response of a multi-rotor-bearing system containing a transverse crac in a rotor. Transactions of the ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 0, pp [] P. Pennacchi, N. Bachschmid, A. V.,. A model-based identification method of transverse cracs in rotating shafts suitable for industrial machines. Mechanical Systems and Signal Processing, In press, p. 3. [5] Adewusi, S., and Al-Bedoor, B.,. Experimental study on the vibration of an overhung rotor with a propagating transverse crac. Shoc and Vibration, 9, p [] Gasch, R., 97. Dynamic behaviour of a simple rotor with a cross-sectional crac. IMechE Conference on Vibrations in Rotating Machinery, C/78/7, p [7] Zhu, C., Robb, D., and Ewins, D., 3. The dynamics of a craced rotor with an active magnetic bearing. Journal of Sound and Vibration, 5, p [8] Henry, T., and Oah, B., 97. Vibration in craced shafts. IMechE Conference on Vibrations in Rotating Machinery, C//7, p [9] Nelson, H., and Nataraj, C., 98. The dynamics of a rotor system with a craced shaft. Journal of Vibration, Acoustics, Stress, and Reliability in Design, 08, p [0] Lalanne, M., and Ferraris, G., 990. Rotordynamics Prediction in Engineering, ed. John Wilet and Sons. [] Davies, W. G. R., and Mayes, I. W., 98. The vibrational behaviour of a multi-shaft, multibearing system in the presence of a propagating transverse crac. Transactions of the ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 0, pp

22 [] Nayfeh, A., and Balachandran, B., 995. Applied nonlinear dynamics : analytical, computational and experimental methods. John Wiley & Sons. [3] Nayfeh, A., and Moo, D., 995. Nonlinear oscillations. John Wiley & Sons. [] He, J.,. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 0(0), pp. 99. [5] Sinou, J.-J., Thouverez, F., and Jézéquel, L.,. Methods to reduce non-linear mechanical systems for instability computation. Archives of Computational Methods in Engineering: State of the Art Reviews, (3), pp [] Cameron, T. M., and Griffin, J. H., 989. An alternating frequency time domain method for calculating the steady state response of nonlinear dynamic systems. ASME Journal of Applied Mechanics, 5, pp [7] Flannery, B. P., Teuolsy, S. A., and Vetterling, W., 99. Numerical Recipes in Fortran, ed. Cambridge University Press. [8] Cardona, A., Lerusse, A., and Geradin, M., 998. Fast fourier nonlinear vibration analysis. Computational Mechanics,, pp. 8.