IMECE IMPACT WORK-RATE AND WEAR OF A LOOSELY SUPPORTED BEAM SUBJECT TO HARMONIC EXCITATION. T Non-dimensional wear process time

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1 Proceedings of 5th FSI, AE & FIV+N Symposium 00 ASME International Mechanical Engineering Congress & Exposition November 7-, 00, New Orleans, Louisiana, USA IMECE IMPACT WORK-RATE AND WEAR OF A LOOSELY SUPPORTED BEAM SUBJECT TO HARMONIC EXCITATION Jakob Knudsen Λ Materials Science Malmö University SE Malmö, Sweden tsjakn@ts.mah.se Ali R. Massih Quantum Technologies AB Uppsala Science Park Uppsala, SE 7583 and Malmö University SWEDEN alma@quantumtech.se ABSTRACT Impact work-rate of a weakly damped beam with elastic two-sided amplitude constraints subject to harmonic excitation is calculated. Impact work-rate is the rate of energy dissipation to the impacting surfaces. The beam is clamped at one end and constrained by unilateral contact sites near the other end. This system was an object of a vibro-impact experiment which was analyzed in our earlier paper (Knudsen and Massih 000). Detailed nonlinear dynamic behavior of this system is evaluated in our companion paper (Knudsen and Massih 00b). Computations show that the work-rate for asymmetric orbits is significantly higher than for symmetric orbits at or near the same frequency. For the vibro-impacting beam, under conditions that exhibit a stable attractor, calculation of work-rate allows us to predict the lifetime of the contacting beam due to fretting-wear damage by extending the stable branch and using the local gap between contacting surfaces as a control parameter. That is, upon computation of the impact work-rate, the fretting-wear process time is calculated through back-substitution of the work-rate and gap-width in a given wear law. NOMENCLATURE T Wear process time T Non-dimensional wear process time Λ Address all correspondence to this author. V Σ ρ ω Ω ω 0 τ A A C c k c m D E F f a f c I k K k c k rot k tra L M P Material volume loss due to wear Poincaré section Beam density Non-dimensional forcing frequency Forcing frequency Fundamental eigenfrequency Non-dimensional time Forcing amplitude Cross section area Consistent damping matrix Stiffness proportional damping Mass proportional damping Flexural rigidity (= EI) Young s modulus Applied force Non-dimensional applied force vector Non-dimensional contact force vector Moment of inertia Wear parameter Consistent stiffness matrix Contact spring stiffness Torsion spring stiffness Lateral spring stiffness Beam length Consistent mass matrix Return map

2 S t U u v x x i X i Cross sectional area of contact site Time Impact work-rate Non-dimensional impact work-rate Non-dimensional velocity vector Non-dimensional displacement vector Non-dimensional gap size Initial gap size k c Αcos Ω t k c INTRODUCTION Oscillating mechanical systems, excited by flow-induced vibration, confined within barriers are frequently encountered in many industrial equipment such as steam generator tubes, heat exchanger tubes (Chen 99) and fuel rods in nuclear power plants (Pettigrew, Taylor, Fisher, Yetisir, and Smith 988). These oscillators exhibit highly nonlinear behavior due to impacting. The response includes, periodic oscillations, chaotic vibration and random vibration. Commonly, impacting is associated with increase wear of the components of the oscillator (Frick, Sobek, and Reavis 984), which is related to the dynamical behavior of the system. Hence, study of the vibro-impact dynamics is important for understanding and analyzing wear of components that are under such motions. On a macroscopic scale removal of material from the surface due to impacts may be modeled by Archard s wear relationship (Archard 953). To relate wear volume to non-linear quantities, such as the contact forces and relative sliding motion, Frick, Sobek, and Reavis (984) introduced the concept of workrate. Recently, the notion of the shear work-rate was proposed by Pettigrew, Yetisir, Fisher, Smith, and Taylor (999). It is defined as the integral of the shear or sliding force times the sliding distance per unit time. Hence, the shear work-rate is identical to the mechanical energy dissipated at contact points. Recently, Monte Carlo simulations have been adopted to cope with the complex dynamic response as well as uncertainties in material parameters involved in vibro-impacting systems with wear. Delaune, de Langre, and Phalippou (999) studied how uncertainties of parameters in wear tests influenced work-rate calculations and Charpentier and Payen (000) applied a probabilistic method to compute wear work-rate and lifetime of steam generator tubes. In our previous work we have analyzed vibro-impact dynamics in connection with an experiment concerning wear workrate of a beam oscillator confined within elastic barriers subject to harmonic loads (Knudsen and Massih 000) and stochastic loads (Knudsen, Massih, and Gupta 000). Recently, (Knudsen and Massih 00a & Knudsen 00), we have evaluated details of dynamic stability of weakly damped beam oscillators with elastic supports using the Poincaré map method to determine stable periodic solutions and the work-rate associated to these solutions. Figure. k tra X X k rot NON-LINEAR BEAM OSCILLATOR In the present work we evaluate the work-rate for stable periodic solutions, exhibiting asymmetric wear of the contact sites, as the gap increases through wear. This allows us to predict the lifetime of the studied system, i.e. the time to reach a specified wear depth of the contact sites. NON-LINEAR OSCILLATOR A long slender beam is considered. The beam is supported at one end by a pair of stiff linear springs, namely a torsion spring with stiffnesses k rot and a lateral spring with stiffness k tra, suppressing end rotation and end deflection, respectively. The motion at the other end is constrained by two contact sites, which are modeled by linear springs with stiffness k c. The contact sites are initially placed with a symmetric gap of X = X = X 0 as shown in Figure. The beam structure is modeled with finite elements (FE) and it s time variation is discretised using Newmark s time integration method (Johansson 997). Dissipation is introduced in the system through Rayleigh type damping (Petyt 990). The beam is excited with a harmonic force, F(t) =A cos(ωt), positioned at the same axial position as the contact sites. DYNAMICS The dynamics of the FE system, described in the previous section, can be written as a system of first order non-dimensional differential equations

3 x; τ = v v; τ = M [f a + f c Cv Kx] τ; τ = where M, C and K are the consistent mass, damping and stiffness matrices, respectively, x = X=X 0 and v are the non-dimensional displacement and velocity vectors, respectively, τ = ω 0 t is the non-dimensional time, ω 0 is the fundamental eigenfrequency of the system, t is the time, f a is the applied force and f c is the contact force. Derivatives with respect to non-dimensional time are denoted by ( ); τ. Consequently we define the state vector for the system by (x;v;τ). The vector field defined in () has dimension n dof +, where n dof is the degrees of freedom (DOF) of the system. For a harmonic load with forcing frequency Ω, the field is periodic in τ with period π=ω, where ω = Ω=ω 0. Since we apply a node-to-node contact algorithm, let subscript c denote nodal quantities at the place of structural contact. Next we define a Poincaré section at the right hand contact site of Figure by () Σ = fx;v;τ)jx c =+x ;v c > 0g () where x = X =X 0 is the non-dimensional gap connected to the right hand contact site. Note that, the non-dimensional gap connected to the left hand contact site is x = X =X 0. The corresponding return map P is defined by are described in Knudsen and Massih (00a) and Knudsen and Massih (00b) and therefore will not be repeated here. WEAR MODEL Wear of material can be modeled with Archard s wear law (Archard 953), which states that loss of material is related to contact force times the sliding distance through some material parameters. In a one-dimensional system this makes no sense, since no sliding can take place. For the purpose of this paper we assume that the material loss or damage can be related to the impact energy and that all damage occurs at the contact sites, according to V = k hui T (5) where V is the volume loss due to wear, k is a material parameter, hui is the time averaged impact work-rate and T is the wear process time. The impact work-rate parameter, is thus essentially a measure of available power to produce damage at the supports. Following Knudsen and Massih (000) we define the incremental impact work-rate to be du = pds, where p and s denote contact force acting on the support and its displacement, respectively. Returning to the global variables, defined in the foregoing section, we write the work-rate increment connected to contact site i as P = Σ! Σ or (ˆτ; ˆx r ; ˆv) =P(τ;x r ;v) (3) where x r is the reduced displacement vector, i.e. we exclude the contact node x c =+x (+ initially), and use the hat symbol ˆ to denote the quantities at the subsequent structural contact. Harmonic motion of the non-linear oscillator described by equations (), i.e. a fixed point of the return map (3), satisfies the following condition τ πn + ω ; xr ; v = P m ( τ; x r ; v) (4) where n denotes the sub-harmonic of the m iterated map and the bar over the variable indicates a periodic point. The local stability of a periodic solution is given by the eigenvalues of the Jacobian matrix evaluated at the corresponding fixed point of the map (3). Fixed points of equation (4) are found by a predictor-corrector method using the Newton- Raphson scheme to correct the predicted value (Nayfeh and Balachandran 995). The details of the method of computation du i = k c (jxj X i )d (jxj X i ) = k c (jxj X i )jẋjdt where Ẋ = dx=dt. Equation (6) can be rewritten in non-dimensional form as (6) du i =[jxj x i ]jvjdτ (7) where du = du=(k c X 0 ) and v = Ẋ=(X 0 ω 0 ). Note that we account for impacts occurring at x =+x and x by taking the absolute value of x. The time averaged work-rate is consequently hui i = τ m τ 0 Z τm τ 0 [jxj x i ]jvjdτ (8) where τ 0 is the time from which the iteration is initiated and τ m is the time after m iterations of the map P. We also mark that the change of sign in v is accounted for in the integrand.

4 Gap Evolution In Knudsen and Massih (00a) we found that the impacting beam oscillator, presented in section Non-Linear Oscillator, has stable periodic solutions, for which the symmetrically placed contact sites experience asymmetric work-rate. We regard the contact sites as one-dimensional and assume that damage only occurs at contact sites, hence the volume loss is proportional to the gap increment. Using equation (5) we write the gap increment for the individual contact sites as X i οhui i T for i = ;;:::;N (9) where X i denotes the gap increment at contact site i and N is the number of contact sites. Using the gap connected to contact site as a control parameter in our computations, the gap evolution at all other contact sites is found through X i = hui i hui X (0) The time to reach this wear depth (gap) is found by back substitution into equation (5), viz T = X i k 0 hui i () where k 0 = k=s and S is the cross sectional area of the contact site. Equations (0) and () expressed in non-dimensional form become x i = hui i hui x and T = x i hui i () Note that the non-dimensional wear process time T comprises the material parameter k 0. COMPUTATIONS AND RESULTS In this section we present results of our computations for the considered non-linear oscillator. We have utilized the Poincaré map method to find fixed periodic points of the system. Using the gap x, at the right hand contact site in Figure (initially at x =+), as a control parameter, we then employ the sequential continuation method (Nayfeh and Balachandran 995) to compute the evolution of the wear depth. All computations have been Table. PROPERTIES OF BEAM OSCILLATOR Beam density ρ kg/m Cross section area A m 7: Beam length L m 0.64 Mass proportional damping c m s Stiffness proportional damping c k s : Flexural rigidity D Nm 4.90 Support lateral spring stiffness k tra N/m : Support torsion spring stiffness k rot Nm/rad 5:35 0 Contact spring stiffness k c N/m 3750 Force amplitude A N 6 Initial gap X 0 m :5 0 4 made using the structural properties and conditions listed in Table. These properties concern a portion of a nuclear fuel rod vibrating in air (Knudsen and Massih 000). In an earlier analysis (Knudsen and Massih 00a) the dynamic response in the frequency interval ω [3;] was studied and branches of stable period-one solutions were identified. Furthermore, the impact work-rate was evaluated along these branches. It was found that a stable periodic solution could lead to asymmetric wear of the symmetrically placed contact sites. In the present work we have attempted to analyze the time evolution of such points through wear of the contact sites. Figures a and b show the impact work-rate along two different branches of stable period-one solutions, exhibiting different work-rates at the initially symmetrically placed contact sites. These figures show the work-rate evaluated at right hand contact site (at x =+x ) and at the left hand contact site (at x = x ). From the stable branches shown in Figure we have selected two frequencies for further evaluation, namely ω = 5:60 and ω = 9:900. At these frequencies the evolution of gap and workrate are computed as the gap increases due to wear, according to Archard s law (Archard 953). Figures 3a and 3b show the gap ratio, i.e. gap at site (x ) normalized with the gap at site (x ) plotted against the gap at site ; whereas Figures 4a and 4b show the work-rate for each contact versus the gap at site. In Figure 3a we see that initially x increases faster than x due to the difference in their work-rates (Figure 4a). However, this difference quickly diminishes as x increases, then at x ß : the trend suddenly reverses. This is visible in Figure 3a as the gap ratio starts to increase. At x ß :34 we see an abrupt change in the work-rate (Figure 4a) and the gap ratio increases at a slower rate, see Figure 3a. Note that, as the wear process continues the system tends towards a symmetric gap. At the higher

5 [-] x / ω [ ] (a) 5:604 < ω < 5:63 (a) 30 5 x / 0.99 [-] ω [ ] (b) 9:704 < ω < 0:74 Figure. IMPACT WORK-RATE hui AT CONTACT SITES & ALONG STABLE BRANCHES OF PERIOD-ONE SOLUTIONS IN THE FRE- QUENCY INTERVAL (a) 5:604 < ω < 5:63, (b) 9:704 < ω < 0:74. frequency the situation is different. The initially symmetric gap becomes gradually more asymmetric (Figure 3b), which is connected to an increasing difference in work-rate between the two contact sites (Figure 4b). Lifetime Estimate The lifetime or the time to reach a certain wear depth at the contact sites is estimated by utilizing equation (). Figures 5a (b) Figure 3. GAP RATIO x =x BETWEEN CONTACT SITES & FOR FORCING FREQUENCY (a) ω = 5:60, (b) ω = 9:900. and 5b show the gap size connected to the contact sites for ω = 5:60 and ω = 9:900, respectively. For ω = 5:60, the growth of the two gaps (x and x ) are virtually identical (cf. Figure 5a). The evolution of the gap follows the evolution of the work-rate (Figure 4a). For» x i» :34, i = ; the work-rate decreases as the gap increases, see Figure 4a. This behavior is also reflected in Figure 5a, where the gap growth rate (slope of curves) decreases in this interval. For gaps x i > :34, i = ;, the work-rate increases with gap size, which in turn leads to a gradual increase of gap growth rate, see Figure 5a.

6 x x gap [-] wear process time [-] (a) (a) x x gap [-] wear process time [-] (b) Figure 4. IMPACT WORK-RATE hui AT INDIVIDUAL CONTACT SITES & FOR FORCING FREQUENCY (a) ω = 5:60, (b) ω = 9:900. (b) Figure 5. GAP SIZE x AT INDIVIDUAL CONTACT SITES & FOR FORCING FREQUENCY (a) ω = 5:60, (b) ω = 9:900. For ω = 9:900 the gap growth rate increases as the wear process continues, see Figure 5b. We also see that the system becomes increasingly asymmetric with time, since the gap at contact site grows faster than at site, i.e. hui > hui. DISCUSSION In the foregoing section a sharp change in work-rate, as the gap increases through x = :34 for the case ω = 5:60 (Figure 4a), was calculated. This change is connected to a qualitative change in the dynamics of the system. In Knudsen and Mas- sih (00a) it was noted that a more wrinkled phase orbit displayed higher work-rate compared to a smooth orbit. The shape of the phase orbits for different gap sizes, x = :8 (solid line) and x = :34 (broken line) for ω = 5:60 are shown in Figure 6. As the gap increases we have two competing effects, i) the workrate decreases where the phase orbit becomes smoother and ii) the work-rate increases as the velocity of the impacting body increases with the growing gap. For ω = 9:900 we were able to follow the gap evolution through wear up to a gap size of x» 4:0, cf. Figures b, 3b and 4b. For x > 4:0, impacts no longer occur at the contact sites

7 v [-] x =.8 [-] x =.40 [-] Figure 6. PHASE PORTRAITS FOR ω = 5:60 WITH GAPS x = :8 AND x = :40, RESPECTIVELY. for the steady-state motion of the oscillator. Hence, no further wear of the contact sites can take place. Lifetime predictions are usually made with the work-rate computed for the initial gap, hui 0 i, through τ = x i=hui 0 i (Charpentier and Payen 000), i.e. a linear relationship between wear depth and wear process time is assumed. This linear approach can lead to either an over- or an under-estimation of the lifetime by a factor of more than two, depending on the forcing frequency, compared to the incremental approach used to generate the curves presented in Figures 5a and 5b. Note that the workrate computed for the initial gap can be read from Figures 4a and 4b. For example, for the two cases studied here, to reach a wear depth twice the size of the initial gap, the linear approach leads to lifetime under-estimation by a factor.48 for ω = 5:60 and an over-estimation by a factor of. for the case with ω = 9:900, see Figures 5a and 5b. The difference in estimated lifetime between ω = 5:60 and ω = 9:900 is also noteworthy (compare Figures 5a and 5b). This difference is connected to the calculated differences in work-rate, see Figures 4a and 4b. The method described here can readily be extended to cover aperiodic response resulting from harmonic as well as stochastic loads, i.e. replacing the fixed point iteration with an averaging method. Furthermore, for industrial applications it will be necessary to consider other processes affecting the gap size, e.g. wear and creep down of the impacting body, wear scar geometry, thermal expansion etc. ACKNOWLEDGMENT The work was supported by the Swedish Foundation for Knowledge and Competence Development (KKS) under Award HÖG /0. References Archard, J. (953). Contact and rubbing of flat surfaces. Journal of Applied Physics 4, Charpentier, J. and T. Payen (000, June). Prediction of wear work rate and thickness loss in tube bundles under cross-flow by a probabilistic approach. In S. Ziada and T. Staubli (Eds.), Flow-Induced Vibration, pp A. A. Balkema. ISBN: Chen, S. S. (99). A review of dynamic tube-support interaction in heat exchanger tubes. In IMechE 99 C46/0, pp. 0. Delaune, X., E. de Langre, and C. Phalippou (999). A probabilistic approach to the dynamics of wear tests. In M. J. Pettigrew (Ed.), Flow-Induced Vibration 999, Volume 389 of PVP. ASME. Frick, T. M., E. Sobek, and J. R. Reavis (984, December). Overview on the development and implementation of methodologies to compute vibration and wear of steam generator tubes. In M. Paidoussis, J. Chenoweth, and M. Bernstein (Eds.), ASME Special Publication, Symposium on flow-induced vibrations in heat exchangers, New Orleans, Louisiana, pp Johansson, L. (997). Beam motion with unilateral contact constraints and wear of contact sites. Journal of Pressure Vessel Technology 9, Knudsen, J. (00). Vibro-Impact Dynamics of Fretting Wear. Licentiate thesis, Luleå University of Technology, Luleå, Sweden. ISSN: , ISRN: LTU - LIC - - 0/ SE. Knudsen, J. and A. R. Massih (000, May). Vibro-impact dynamics of a periodically forced beam. Journal of Pressure Vessel Technology, 0. Knudsen, J. and A. R. Massih (00a). Dynamic stabiblity of weakly damped oscillators with elastic impacts and wear. Journal of Sound and Vibration. In press. Knudsen, J. and A. R. Massih (00b, November 7 ). Nonlinear dynamics of a loosely supported beam subject to harmonic excitation. In Proceedings of the 5th FSI, AE & FIV+N Symposium, New Orleans, Louisiana, USA. American Society of Mechanical Engineers. Knudsen, J., A. R. Massih, and R. Gupta (000, June). Analysis of a loosely supported beam under random excitations. In S. Ziada and T. Staubli (Eds.), Flow-Induced Vibration, pp A. A. Balkema. ISBN: Nayfeh, A. and B. Balachandran (995). Applied nonlinear dynamics; Analytical, Computational and Experimental

8 Methods. John Wiley & Sons, Inc. Pettigrew, M. J., C. E. Taylor, N. J. Fisher, M. Yetisir, and B. A. W. Smith (988). Flow-induced vibration: recent findings and open questions. Nuclear Engineering and Design 85, Pettigrew, M. J., M. Yetisir, N. J. Fisher, B. A. W. Smith, and C. E. Taylor (999). Prediction of vibration and frettingwear damage: An energy approach. In M. J. Pettigrew (Ed.), Flow-Induced Vibration 999, Volume 389 of PVP, pp ASME. Petyt, M. (990). Introduction to Finite Element Vibration Analysis, Chapter 9. Cambridge, England: Cambridge University Press.

A Probabilistic Approach to the Dynamics of Wear Tests

A Probabilistic Approach to the Dynamics of Wear Tests X. Delaune Commissariat à l Energie Atomique, Département de Mécanique et de Technologie, 91191 Gif/Yvette, France E. de Langre LadHyX, Ecole Polytechnique, 91128 Palaiseau, France C. Phalippou Commissariat