自動(dòng)烹飪機(jī)送料機(jī)構(gòu)的結(jié)構(gòu)設(shè)計(jì)【說(shuō)明書+CAD+SOLIDWORKS】,說(shuō)明書+CAD+SOLIDWORKS,自動(dòng)烹飪機(jī)送料機(jī)構(gòu)的結(jié)構(gòu)設(shè)計(jì)【說(shuō)明書+CAD+SOLIDWORKS】,自動(dòng),烹飪,機(jī)送料,機(jī)構(gòu),結(jié)構(gòu)設(shè)計(jì),說(shuō)明書,仿單,cad,solidworks Engineering Fracture Mechanics Vol. 34, No. 3, pp. 721-728, 1989 Printed in Great Britain. 0013-7944/89 $3.00 + 0.00 0 1989 Perpmon Press pk. FATIGUE CRACK PROPAGATION IN RESONATING STRUCTURES A. J. DENTSORAS Lecturer, Machine Design Laboratory, University of Patras, Greece and A. D. DIMAROGONAS W. Palm Professor of Mechanical Design, Washington University, St. Louis, Missouri, U.S.A. Ahatraet-Fatigue crack propagation in structures excited at one of their resonances is investigated. The structure is discretixed and represented by a system of linear differential equations. Cracks are modelled as local flexibilities. Application on a 4-node rotor with a circumferential crack illustrates the dynamic crack arrest. The depth of the crack determines the local stiffness introduced by the crack, which in turn influences the dynamic response of the system under external excitation of constant amplitude at a resonant frequency. The propagation of the crack introduces additional flexibility which causes gradual shift from resonance. The dynamic response is reduced and, under certain circumstances, becomes less than a threshold value characteristic of the material considered. This phenomenon, known as dynamic crack arrest is drastically influenced by the loss coefficient of the material which is the main factor to determine the crack propagation rate at the initial stages of the crack growth when the loading of the cracked section becomes maximum. 1. INTRODUCTION UNDER the influence of the dynamic loads, a propagating crack might cause failure of a structural member. Consequently, the determination of the crack propagation rate is of major importance for the safe operation of machines. The equation of Paris and Erdoganl3 is used to determine the propagation rate: $ = B(AK)” (1) where B, m are material constants and AK is the stress intensity factor range. Equation (1) involves the stress range only and not the mean stress. It is also restricted to crack running ahead. To overcome these limitations, Sih4-6 proposed the extension of the strain energy density factor concept to predict the growth of fatigue cracks where S, n are material constants and AS, is the strain energy density factor range. Crack nucleation and propagation is natural to be expected in vibrating structures, in par- ticular when local or general resonance occurs. Chondros and Dimarogonas7-91 and Dimarogonas and Massouros lo have shown that the appearance of a crack influences considerably the vibration characteristics of such structures. Because of this variation, the stress field in the vicinity of the crack will change. Dentsoras and Dimarogonas 1 l-l 31 have shown for simple structures that, under such conditions, it is possible for the crack propagation rate to become lower than a threshold value characteristic of the material considered. This process, known as dynamic crack arrest, was shown that is strongly affected by material damping being the dominate damping mechanism of the system considered. Here this principle is applied for a vibrating cracked structure modelled as a multi- degree of freedom system and it was applied for a 4-stage cracked rotor under torsional vibrations. 721 722 A. J. DENTSORAS and A. D. DIMAROGONAS 2. ANALYSIS Consider a lumped mass vibrating system Ml, K with n-degrees of freedom, where M is the diagonal mass matrix and Kj the stiffness matrix. If: x = x,(r)x,(t) . . .x,(t) (3) is the response vector and: Kl = k3vGl Ml = wwul (4) where KU, MU are the dimensionless mass and stiffness matrices respectively and m, k. are constants, Zj2 + 0 i-eigenvalue. By introducing: x = MIS, (7) G-9 (9) eq. (5) can be modified as follows: mi” + ml” = PI (10) where A = diag(R,) is the eigenvalue matrix. Equation (10) corresponds to uncoupled vibration. If the structural damping of the system is the dominant damping mechanism and the damping matrix C can be normalized then 141 d,lCl d,l= moJllo diag( s (&,)-d(u/R,). (45) a,/& In Fig. 4, the results of the numerical integration are shown for various values of loss coefficient and for mild steel with the following characteristics: S, = 4.3 x lONlm* m = 2.51 B = 2.832 x 10-26m(2 +“)*/cycle Nm Kth = 3.788 x 106N/m3*. .l . 5 1 5 10 50 (iT,J, / IO3 Fig. 3. Variation of dimensionless stress intensity factor vs crack depth. Loss coefficient is the parameter. Crack propagation in resonating structures 121 a * 6 t : e * 3 . 2 II - 2 0 I 0 i : I 4 , I .4. * ,I, /I I / / 1 1 i&f: I / / R0 / / / 0 5 /- / : 4 0 0 / 0 r/ 0 0 : ,/ : 0 / / Arrest line . 1- 1 5 10 50 100 500 N/IO Fig. 4. Number of fatigue cycles vs crack depth for mild steel. Loss coefficient is the parameter. The arrest line shown is the locus of all points where transition to low rate crack propagation takes place on every curve. 4. CONCLUSIONS The Paris and Erdogan model for the fatigue crack propagation is used to study the behaviour of a cracked resonant system. A general relation is established relating crack depth with number of fatigue cycles applicable generally. Crack propagation at or near resonance appears very often in machines and structures and is one of the usual causes of machine failure. In situations where the introduction of the crack can alter the dynamic characteristics of the system, such dynamic analysis is used to determine the mission profile for the loading of the cracked section, for a certain constant external excitation. This information is used as feedback to the fatigue crack propagation model. Initially, crack propagates at or very closely to resonance with very high rates. Since the crack depth changes, there is a considerable reduction of the system natural frequency which causes a “shift away” from resonance and, as a consequence, a drop of the crack propagation rate. This situation, under certain circumstances, which strongly depend on the damping factor, can lead to dynamic crack arrest shown in Fig. 4. Here, it seems that crack arrest almost always occurs. This is true up to a certain point since if the crack propagates to a certain extent, which can happen before the arrest point, the structural member might fail by unstable crack growth or some other mechanism which could destroy the structural integrity of the system at hand. Finally, it can be easily concluded that the magnitude of the excitation and the geometry of the system can only alter the results quantitatively. Crack arrest or not depends primarily on the damping of the corresponding node. REFERENCES I P. C. Paris, The fracture mechanics approach to fatigue-an interdisciplinary approach. Proc. 10th Materials Research Conf., Syracuse University Press, New York (1964). 2 P. C. Parts and F. A. Erdogan, Critical analysis of crack propagation laws. J. Bas. Engng 85, 528 (1963). 3 F. A. Erdogan, Crack propagation theories, in Fracture (Edited by H. Liebowitx), Vol. 2. Academic Press, New York (1968). S G. C. Sih, Some basic problems in fracture mechanics and new concepts. Engng Fracture Med. 5, 365-377 (1973). S G. C. Sih, Strain energy density factor applied to engineering problems. Inr. J. Fracture 10, 305-321 (1974). 6 G. C. Sih and B. M. Barthelemy, Mixed mode fatigue crack growth predictions. Engng Fracture Mech. 13.439-451 (1980). A T. G. Chondros and A. D. Dimarogonas, Identification of cracks in circular plates welded at the contour. 1979 ASME Design Engineering Technical Conf. St. Louis, U.S.A., Paper No. 79-DET-106 (September 1979). 728 A. J. DENTSORAS and A. D. DIMAROGONAS 8 T. G. Chondros and A. D. Dimarogonas, Identification of cracks in welded joints of complex structures. J. Sound Vibration 69, 531 (1980). 9 A. D. Dimarogonas, Vibration Engineering. West, St. Paul (1976). IO A. D. Dimarogonas and G. Masouros, Torsional vibration of a shaft with a circumferential crack. Engng Fracture Mech. 15, 439444 (1981). I I A. J. Dentsoras and A. D. Dimarogonas, Resonance controlled fatigue crack propagation. Engng Fracrure Mech. 17, (1983). l2 A. J. Dentsoras and A. D. Dimarogonas, Resonance controlled fatigue crack propagation in a beam under longitudinal vibration. Inr. J. Fracture 23, 15-22 (1983). 13 A. J. Dentsoras and A. D. Dimarogonas, Resonance controlled fatigue crack propagation in cylindrical beams under combined loading. ASME Winter Annual Meeting, Boston (1983). 14 S. Timoshenko, Vibration Problems in Engineering. Wiley, New York (1984). 15 A. D. Dimarogonas and S. Paipetis, Rotor Dynamics. Applied Science, London (1983). 16 G. C. Sih and B. McDonald, Fracture mechanics applied to engineering problems, strain energy density fracture criterion. Engng Fracrure Mech. 6, 493-507 (1974). 17 A. P. Parker, The Mechanics of Fracture and Fatigue-An Introduction. E.F.N. Spon. Ltd., London-New York (1981). (Received 17 October 1988)