Chapter 2Variable Elasticity Effects in RotatingMachineryAbstract The effects of variable elasticity in rotating machinery occur with alarge variety of mechanical, electrical, etc., systems, in the present case, geo-metrical and/or mechanical problems. Parameters affecting elastic behavior do notremain constant, but vary as functions of time. Systems with variable elasticity aregoverned by differential equations with periodic coefﬁcients of the Mathieu-Hilltype and exhibit important stability problems. In this chapter, analytical tools forthe treatment of this kind of equations are given, including the classical Floquettheory, a matrix method of solution, solution by transition into an equivalentintegral equation and the BWK procedure. The present analysis is useful for thesolution of actual rotor problems, as, for example, in case of a transversely crackedrotor subjected to reciprocating axial forces. Axial forces can be used to controllarge-amplitude ﬂexural vibrations. Flexural vibration problems can be encoun-tered under similar formulation.2.1 IntroductionVariable elasticity effects occur with systems in which the parameters affectingelastic behavior do not remain constant, but vary as functions of time. Equations ofmotion pertaining to such systems remain linear, but they possess time-dependentcoefﬁcients. Similar phenomena appear in all ﬁelds of physics and are generallyassociated with wave propagation in periodic media, a problem encountered asearly as 1887 by Lord Rayleigh, and subsequently by other prominent physicists. It was recognized that such phenomena are described by means of Hill andMathieu differential equations. A detailed and comprehensive review on the workon waves in periodic structures was given by Brillouin .A. D. Dimarogonas et al., Analytical Methods in Rotor Dynamics, 25Mechanisms and Machine Science 9, DOI: 10.1007/978-94-007-5905-3_2,Ó Springer Science+Business Media Dordrecht 2013
26 2 Variable Elasticity Effects in Rotating Machinery Omitting a large number of non-mechanical phenomena mentioned in the abovereferences, it is interesting to concentrate attention on the following vibratingmechanical systems [3–10]:1. A rotating shaft with non-circular cross-section, i.e. non-uniform ﬂexibility.2. A mass suspended from a taut string with time-varying tension.3. A pendulum with time-varying length.4. An inverted pendulum attached to a vertically vibrating hinge.5. The side-rod system of electric locomotives, exhibiting torsional vibrations.6. The rotating parts of small motors, which are subject to the time-varying action of electromagnetic ﬁelds.7. A rotating ﬂywheel carrying radially moving masses.8. A rotating ﬂywheel eccentrically connected to reciprocating masses. The list is by no means complete, but the examples are characteristic of theproblems encountered. If one of these structures is constrained to move at constant circular frequency,the respective parameters are periodic functions and under proper conditions largevibration amplitudes may develop. The reason is that, within each cycle, positivework is introduced into the system, which results in a gradually increasingamplitude, i.e. instability. This effect can be caused, for example, by a smallaccidental force producing an initial velocity and being enhanced, under certainconditions, by the action of gravity or static forces . In Fig. 2.1, the response ofa simple oscillator with a different spring constant in tension and in compression ispresented. The oscillator is assumed to perform harmonic oscillations at constantcircular frequency x. By using the analysis given in Sect. 2.2, the total energy stored in the systemduring the ﬁrst and second half-cycles can be evaluated. This may be positive, zeroor negative. The second and third cases lead to stable situations; however, the ﬁrstone leads to increasing vibration amplitudes, i.e. to instability.Fig. 2.1 Response of asimple oscillator with adifferent spring constant intension and in compression
2.1 Introduction 27Fig. 2.2 A beam on circularsupports of ﬁnite radiusexhibiting periodic lengthvariation during ﬂexuralvibration Consider now a rotating shaft, or the more general case of a beam executingﬂexural vibrations. The following cases of variable elasticity can be distinguished.2.1.1 Variable Length lA beam resting on circular supports with ﬁnite radius undergoes periodic changesof length during ﬂexural vibrations. This case is obviously not related to rotatingmachinery; however, it is an interesting example of periodic changes in elasticity(Fig. 2.2).2.1.2 Variable Stiffness EJThe ﬂexural stiffness EJ may vary periodically if either elastic modulus E of thestructure material or the second area moment J of the respective cross-sections orboth vary accordingly. Rotor-to-stator rub often occurs in rotating machinery.Methods for detecting the rub-impact effect were developed based on the variablestiffness issue for the rotor system [11, 12]. Therefore, one can distinguish thefollowing particular cases: Variable elastic modulus E: Many materials, especially polymers and theircomposites, exhibit different elastic modulus in tension Es and in compression Ec[12–15]. A beam performing longitudinal vibration would exhibit a variablestiffness of ripple form, as shown in Fig. 2.1. With ﬂexural vibration, in thepresence of an axial load, a respectively oscillating neutral axis would result,giving rise to periodically varying ﬂexural stiffness. Variable second area moment J: The second area moment J depends on theparticular form of the respective cross-sections. With circular cross-sections,J remains constant in all directions and the ﬂexural stiffness is not affected.However, with non-circular cross-sections, for example rectangular, if the beamrotates while the direction of the load or the couple vector remains constant,a periodically varying J results. With constant rotating speed, this variation issinusoidal. Non-circular cross-sections are produced by keyways, etc., specialformations of rotors, and also by longitudinal or transverse cracks. The latter mayhave a considerable effect and, alternatively, their presence can be detected by thedynamic behavior of the vibrating element. A quasi-sinusoidal experimental curve
28 2 Variable Elasticity Effects in Rotating MachineryFig. 2.3 Variation of ﬂexuralstiffness of a transverselycracked rotorrepresents the periodic variation of the ﬂexural stiffness  (Fig. 2.3). Thesimplest example of a system with periodically varying stiffness is a straightrotating shaft, whose cross-section has different principle moments of inertia.A classical example of systems with periodically varying stiffness was the drivesystem of an electric locomotive incorporating a coupling link for force trans-mission, applied during the ﬁrst years of electric locomotion.2.1.3 Variable Mass or Moment of InertiaSuch examples have already been mentioned. The case of a reciprocating engine isa very common one, by which the inertia of the piston mechanism varies peri-odically [17–19]. A heavy ﬂywheel is necessary in order to maintain uniformrotating speed, a device important for all systems exhibiting variable elasticity. In this chapter the various problems of such systems are discussed in detail,along with the respective mathematical techniques used for their treatment.2.2 The Problem of StabilityConsider a simple oscillator, i.e. a mass m suspended from a massless spring,whose constant, however, assumes the value k(1 ? d) for tension and the valuek(1 - d) for compression. Moreover, assume that at t = 0, its deﬂection is x = 0and its velocity x ¼ v0 . The free vibration of the system is governed by the _equations
2.2 The Problem of Stability 29Fig. 2.4 Variation ofdisplacement and velocitywith time of a simpleoscillator with a differentspring constant in tensionand compression ) € þ x2 ð1 þ dÞx ¼ 0 x 0 for x [ 0 ðtensionÞ ð2:1Þ € þ x2 ð1 À dÞx ¼ 0 x 0 for x0 ðcompressionÞwhere x2 ¼ k=m: During the ﬁrst half-cycle, the natural frequency of the system is 0 x1 ¼ x0 ð1 þ dÞ1=2 ð2:2aÞwhile during the second half-cycle x2 ¼ x0 ð1 À dÞ1=2 : ð2:2bÞ The period T of a complete cycle is p p 2p T ¼ T1 þ T2 ¼ þ ¼ ð2:3Þ x1 x2 xwhere x is the circular frequency of the free vibration, which, however, is notuniform over the two half-cycles. From Eq. (2.3) we obtain 1 1 1 ¼ þ ð2:4aÞ x x1 x2or rﬃﬃﬃﬃ À Á1=2 k 1 À d2 x ¼ 0:5 ð2:4bÞ m À1 þ d2 Á1=2 þ sÀ1 À d2 Á1=2 In Fig. 2.4, the forms of the displacement function xðtÞ and the velocityfunction xðtÞ are presented for d = 0.324. Amplitudes and velocities remain _constant, as the system is conservative. However, the oscillation is clearly notsimple harmonic, and can be analyzed as a Fourier series:
30 2 Variable Elasticity Effects in Rotating Machinery X 1 x ð t Þ ¼ a0 þ ðak sin kxt þ bk cos kxtÞ ð2:5Þ k¼1 Considering the odd (sine) functions, we obtain Z 2 T ak ¼ xðtÞ sin kxt dt ð2:6Þ T 0where ( ) x1 sin x1 t ð0 t T1 Þ xðtÞ ¼ ð2:7Þ x2 sin x2 t ðT1 t T Þ As evident from Eq. (2.5), the system possesses in fact an inﬁnite number ofeigen frequencies x, 2x, 3x …, multiples of the circular frequency x of a fullcycle, as deﬁned in Eq. (2.4a). Now, if the system is forced to oscillate at uniform circular frequency x,stability problems may arise. The respective analysis is due to van der Pol andStrutt [3, 4] and is based on the solution of Eq. (2.1) under the proper boundaryconditions. Namely, Eq. (2.1) possess the following general solutions: ) x1 ¼ A1 sin x1 t þ A2 cos x1 t ð2:8Þ x2 ¼ B1 sin x2 t þ B2 cos x2 t Now, considering that at the end of the ﬁrst half-cycle (t = p/x), displacementsand velocities for both solutions [Eq. (2.8)] must be equal and, moreover, that atthe end of the ﬁrst complete cycle (t = 2p/x) displacements and velocities must bek times as large as at t = 0, four homogeneous linear equations result, containingthe integration constants as unknowns. The condition for these equations to givesolutions other than zero for the latter leads to the quadratic k2 À 2qk þ 1 ¼ 0 ð2:9Þgiving for k the values pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k¼qÆ q2 À 1 ð2:10Þwhere px1 px2 x2 þ x2 px1 px2 q ¼ cos cos À 1 2 sin sin ð2:11aÞ x x 2x1 x2 x xor h 1 x0 i h x0 i 1 q ¼ cos pð1 þ dÞ2 cos pð1 À dÞ1=2 À À Á1 x x 2 1 À d2 2 h i h i ð2:11bÞ 1 x0 1 x0 Â sin pð1 þ dÞ2 sin pð1 À dÞ2 x x
2.2 The Problem of Stability 31Fig. 2.5 Regions of stabilityand instability for a simpleoscillator with a differentspring constant in tension andin compression. (Accordingto van der Pol and Strutt[3, 4]) Now, the system is stable for q = ±1, in which case k = q, and for |q| 1,when k is complex, but its modulus equals one. Alternatively, instability occurswith |q| [ 1. In this case, if q is positive, displacements and velocities at the end ofeach consecutive cycle of the spring ﬂuctuation will increase, retaining the samesign. This means that the system vibrates at the same frequency X of springﬂuctuation or at a multiple of it: 2X, 3X..... If q is negative, displacements andvelocities will also increase at the end of each consecutive cycle of the springﬂuctuation, but will have alternating sign. This means that the system vibrates atfrequencies X/2, X/3….. As q is a function of d and the ratio x0/x, according to Eq. (2.11b) the aboveconditions of stability can be plotted in the classical diagram of Fig. 2.5. The shadedregions in Fig. 2.5 correspond to stable situations (-1 q 1), blank regions tounstable ones jq j [ 1 and full and dotted lines to q = ?l and q = -1 respec-tively. Details of this representation can be found in the relevant literature [2–4]. Inthe presence of damping, the motion is again not simple harmonic, even for the freevibration, and a solution for x must be sought in the form of Fourier series .2.3 The Mathieu-Hill EquationThe general form of an equation of motion, pertaining to a system with variableelasticity, is d2 x þ f ðtÞx ¼ 0 ð2:12Þ dt2where f(t) is a single-valued periodic function of fundamental period T, which canbe represented by a general Fourier series of the form
32 2 Variable Elasticity Effects in Rotating Machinery X 1 f ð t Þ ¼ A0 þ ðAk cos kxt þ Bk sin kxtÞ ð2:13Þ k¼1where x = 2p/S. Equation (2.12), if the function f(t) has the general form (2.13),is known as Hill’s equation. If the Fourier series (2.13) can degenerate in thesimple form f ðtÞ ¼ A0 þ A1 cos xt ð2:14Þthen Eq. (2.12) is known as Mathieu’s equation. This equation can be solved byusing the tabulated Mathieu functions . An important property of a system expressed by the Mathieu equation is thatinstabilities are exhibited for certain ranges of the parameters A0 and A1, i.e. if thesystem is displaced from the equilibrium, x = 0, a motion of increasing amplitudefollows. The classical method for the solution of Hill’s equation, available in mosttextbooks , is as follows: Following the form (2.13) of the function f(t), onecan look for a similar expansion for the unknown periodic function x(t): X 1 x ð t Þ ¼ a0 þ ðak cos kxt þ bk sin kxtÞ ð2:15Þ k¼1 Equation (2.12) then leads to an inﬁnite system of simultaneous linear homo-geneous equations for the unknown coefﬁcients ak and bk. In order to obtain a non-trivial solution, the corresponding inﬁnite determinant must be zero. However, ifthe series of the coefﬁcients Ak and Bk in Eq. (2.13) are absolutely converging, thelatter can be easily evaluated or, alternatively, higher terms can be neglected, andthe problem can be solved, by long computations in any case. The method is notvery practical; however, it appears to be the only one operating with functions f(t),for which no analytical expression is available, as occurs with experimentallydetermined forms. Apart from Hill’s method of inﬁnite determinants, further methods have beendeveloped for the treatment of coupled Mathieu equations , based on theexpansion of the converging inﬁnite determinant, which also provides a criterionfor stability of the solution. Moreover, a stability criterion for the general Hillequation can be developed in a similar manner .2.4 The Classical Floquet TheoryEquations of the Mathieu-Hill type can be solved by means of the Floquet theory[2, 22–25]. This theory basically deals with equations of the form y ¼ AðtÞy _ ð2:16Þ
2.4 The Classical Floquet Theory 33where y = (y1, y2,…, yn) and A(t) = [aij(t)] is a continuous n 9 n matrix deﬁnedin the range -? t ?. A(t) satisﬁes the relation A ðt þ T Þ ¼ A ðt Þwhere the period T is a non-zero quantity. The complete Floquet theory, leading tothe general form of solution of Eq. (2.16) can be found in the literature [24, 25]. Abasic property of the solution, however, is described by the Floquet theoremstating that For the system expressed by the matrix Eq. (2.16), there exists a non-zeroconstant k, real or complex, and at least one non-trivial solution y(t) having theproperty that yðt þ T Þ ¼ kyðtÞ The various values k1, k2,…, km of the parameter k are the distinct character-istic roots of a constant n 9 n matrix C deﬁned by the relation Uðt þ T Þ ¼ UðtÞcwhere U(t) = [uij(t)] is a fundamental system of solutions of Eq. (2.16). Theabove values are called characteristic factors or multipliers of Eq. (2.16).The following corollary is valid: The system (2.16) has periodic solution of period T, if and only if there is atleast one characteristic factor equal to one . The quantities r1, r2,…, rm deﬁned by the relations À Á kj ¼ exp rj T j ¼ 1; 2; . . .; mare called the characteristic exponents of the system (2.16) and, according to thefollowing theorem There are at least m solutions of the system (2.16) having the form xi ðtÞ ¼ pi ðtÞ expðri tÞ i ¼ 1; 2; . . .; mwhere the functions pi(t) are periodic with period T. Now, consider Hill’s equation: € þ f ðtÞx ¼ 0 x ð2:12Þwhere f(t) is real–valued, continuous and periodic with period T, assumed tradi-tionally as equal to p. Let a fundamental system of solutions be x1(t) and x2(t), and x1 ð0Þ ¼ 1 x1 ð0Þ ¼ 0 x2 ð0Þ ¼ 0 x2 ð0Þ ¼ 1 _ _whose Wronskian is equal to one. That is " # x1 ðt Þ x 2 ðt Þ UðtÞ ¼ x1 ðt Þ x 2 ðt Þ _ _
34 2 Variable Elasticity Effects in Rotating Machineryand ! 0 1 AðtÞ ¼ Àf ðtÞ 0when the matrix C is equal to " # x1 ðpÞ x2 ðpÞ C¼ x1 ðpÞ x2 ðpÞ _ _whose characteristic roots k1, k2 are roots of the equation ðkI À CÞ ¼ k2 À ½x1 ðpÞ þ xðpÞk þ 1 ¼ 0 _where k1 k2 ¼ 1 and r1 þ r2 0. Hence a number r can be deﬁned satisfying theconditions expðirpÞ ¼ k1and expðÀirpÞ ¼ k2 The following cases are distinguished:1. k1 = k2, leading to two linearly independent solutions: u1 ðtÞ ¼ expðirtÞf1 ðtÞ u2 ðtÞ ¼ expðÀirtÞf2 ðtÞ where f1(t), f2(t) are periodic with period p.2. k1 = k2, leading to one non-trivial periodic solution with period p (with k1-k2 = 1) or 2p (with k1 = k2 = -1). In order to apply the above results, information about the matrix C is required,which can be derived from the conditions imposed on A(t), guaranteeing theexistence of stable, unstable or oscillatory solutions. Finally, for a non-homogeneous system x ¼ AðtÞx þ BðtÞ _ ð2:17Þwhere A(t) and B(t) are periodic matrices with period T, the following theorem isvalid: For the system (2.17), a period solution with period S exists for every B(t), ifand only if the corresponding homogeneous system has no non-trivial solution ofperiod T.
2.5 Matrix Solution of Hill’s Equation 352.5 Matrix Solution of Hill’s EquationConsider Eq. (2.12) again and two linearly independent solutions x1(t) and x2(t) inthe interval 0 B t B T. The general solution for displacement and velocity is givenby the equations ) x ð t Þ ¼ A 1 x 1 ð t Þ þ A2 x 2 ð t Þ ð2:18aÞ x ð t Þ ¼ A 1 x 1 ð t Þ þ A2 x 2 ð t Þ _ _or in matrix form # # # x ðt Þ x1 ðtÞ x2 ðt Þ A1 ¼ ð2:18bÞ x ðt Þ _ x1 ðtÞ _ x2 ðt Þ _ A2 The Wronskian of the two solutions x1(t) and x2(t), given by the followingequation x ðtÞ x ðtÞ 1 2 W ¼ ¼ x1 ðtÞ_ 2 ðtÞ À x1 ðtÞx2 ðtÞ x _ ð2:19Þ x1 ðtÞ x2 ðtÞ _ _is constant in the interval 0 t S  and, since x1(t) and x2(t) are linearlyindependent, we have also W = 0; hence the 2 9 2 matrix in Eq. (2.18b) is non-singular and has the following inverse: !À1 ! x1 ðtÞ x2 ðtÞ 1 x2 ðtÞ Àx2 ðtÞ _ ¼ ð2:20Þ x1 ðtÞ x2 ðtÞ _ _ W À_ 1 ðtÞ x1 ðtÞ x If the initial conditions are given: # # x ðt Þ x0 ¼ ð2:21Þ x ðt Þ _ _ x0 t¼0 From Eq. (2.18b) we derive # # # x0 x 1 ð 0Þ x 2 ð 0Þ A1 ¼ ð2:22Þ _ x0 x 1 ð 0Þ _ x 2 ð 0Þ _ A2 or # #À1 # A1 x 1 ð 0Þ x2 ð0Þ x0 ¼ ð2:23Þ A2 x 1 ð 0Þ _ x2 ð0Þ _ _ x0which determines the column of arbitrary constants, and, consequently, the ﬁnalsolution assumes the form
36 2 Variable Elasticity Effects in Rotating Machinery ! ! ! ! x ðt Þ 1 x1 ðtÞ x2 ðtÞ x 2 ð 0Þ _ Àx2 ð0Þ x0 ¼ ð2:24Þ x ðt Þ _ W x1 ðtÞ x2 ðtÞ À_ 1 ð0Þ _ _ x x 1 ð 0Þ _ x0 At the end of the ﬁrst period, i.e. at t = T, Eq. (2.24) becomes ! ! x ðt Þ x ¼M 0 ð2:25Þ xðtÞ t¼T _ _ x0where # # 1 x1 ðT Þ x2 ðT Þ x2 ð0Þ À x2 ð0Þ _ M¼ ð2:26Þ w x1 ðT Þ x2 ðTtÞ _ _ À_ 1 ð0Þ x1 ð0Þ x It can be easily shown that, at the end of the nth period of f(t), i.e. at t = nT, wehave # # x ðt Þ n x0 ¼M ð2:27Þ x ðt Þ _ _ x0 t¼nTwhich provides the complete solution of Hill’s equation in terms of the initialconditions, and two linearly independent solutions of Hill’s equation in the fun-damental interval 0 B t B T. The nth power of the matrix M can be computed bymeans of Sylvester’s theorem [26, 27]. Solutions have been given for variousforms of the function f(t), such as rectangular ripple, sum of step functions,exponential function, sawtooth variation, etc. .2.6 Solution by Transition into an Equivalent Integral EquationEquation (2.12), with the function f(t) developable in a Maclaurin series, can besolved by means of a transition to an equivalent Volterra integral equation of thesecond type. Again, by using the notation of Eq. (2.21), we obtain the followingform : Zt xðtÞ ¼ x0 t þ x À ðt À sÞf ðsÞxðsÞds ð2:28Þ 0 The classical method of solution is by means of successive approxi–mations.Problems of a practical nature may appear with the determination of the iteratedkernels, depending on the function f(t). Particular examples, such as for f(t) = tncan be found in the literature .
2.7 The Bwk Procedure 372.7 The Bwk ProcedureThe BWK1 procedure, discovered while dealing with problems of wave mechanics[23, 29, 37] is suitable for the solution of Eq. (2.12) in the following manner: Let afunction be considered, such that Zt À1=2 u¼g exp ðÀiSÞ S ¼ Gdx ð2:29Þ awhere G(t) is any given function of t. The following equation can then be derived: _ 2 # 3 G € 1G 2 €þu G À u þ ¼0 ð2:30Þ 4 G 2Gwhich, when compared with Eq. (2.12), leads to the condition _ 2 € 3 G 1G f ¼ G2 À þ ð2:31Þ 4 G 2G If Eq. (2.31) can be solved to a certain approximation, then the function u inEq. (2.29) represents an approximate solution of Eq. (2.12). Moreover, in case that_ €G and G can be omitted in Eq. (2.31), an acceptable approximation can beobtained by taking G ¼ G0 ¼ ½f ðtÞ1=2 ð2:32Þwhich is a zero–order approximation to the BWK procedure. As a particular case,with G ¼ A=ða þ tÞ2 ð2:33Þ _ €terms in G; G cancel out in Eq. (2.30), and a rigorous solution of Eq. (2.12)results . An accurate approximation to G can be determined by means of a series G ¼ G0 þ G1 þ G2 þ . . . ð2:34Þunder the assumption that G0 =G2 e _ G=G3 e2 € e2 ( 11 Named after three authors: L. Brillouin (J. Phys., 7, 1926, 353); G. Wentzel (Z. Phys., 38,1926, 518); and E.C. Kemble (The Fundamental Principles of Quantum Mechanics, McGraw-Hill, New York, 1937).
38 2 Variable Elasticity Effects in Rotating Machinerywhen the successive terms would be of the order e2, e4….. By substitution inEq. (2.30) we obtain 9 G0 ¼ f 1=2 2 = 3 G _0 €0 1G 2G0 G1 ¼ 4 G0 À 2 G0 ð2:35Þ 2 _0 G _1 G G_1 € 0 G1 1G € G1 ; 2G0 G1 þ G2 ¼ 3 G0 1 2 _ À G0 À 2 G0 G À G0 G € 0 0and the expansion of G would yield a ﬁrst solution from Eq. (2.29). A secondindependent solution t can be found either with f [ 0, G real and positive andS real, or with f 0, G and S purely imaginary and i1=2 u real. The ﬁrst casecorresponds to propagating waves, the second to attenuated waves without prop-agation. A detailed analysis can be found in the literature [23, 37].2.8 Vibrations of Different-Modulus MediaAs an example, consider the longitudinal vibrations of a bar with uniform cross-section made of a material with a different modulus in tension and in compression. The one-dimensional equation of motion is o2 u o2 u ¼a 2 ð2:36Þ ot2 oxwhere a = (E/q)1/2, E is Young’s modulus and q is density. Moreover 1 ET 2 ou a ¼ aT ¼ for r [ 0 or [ 0 ðtensionÞ q ox 1 EC 2 ou a ¼ aC ¼ for r 0 or 0 ðcompressionÞ q ox Assuming that the bar is initially tensioned by a force producing displacementu0 at x = l and released at t = 0, the initial and boundary conditions become: 9 t¼0 ou ot ¼0 u ¼ u0 x = l x¼0 u¼0 ð2:37Þ ou ; x¼l ox ¼ 0 By applying separation of variables, the general solution of Eq. (2.36) can begiven the following form: X x uðx; tÞ ¼ xn ð xÞTn ðtÞ ð2:38Þ n¼1
2.8 Vibrations of Different-Modulus Media 39 where ) Xn ð xÞ ¼ An sinðxn x=aÞ þ Bn cos ðxn x=aÞ ð2:39Þ Tn ðtÞ ¼ Cn sin xn t þ Dn cos xn twhere the constants An, Bn, Cn, Dn, xn assume different values for tension and forcompression (index c or T, respectively) and can be determined by the initial andboundary conditions. By virtue of the ﬁrst boundary and the ﬁrst initial condition[Eqs. (2.37), (2.38) assumes the form X x ðxTn xÞ u¼ An sin cos xTn t ð2:40Þ n¼1 aTand the second initial condition gives xTn ¼ ð2n À 1ÞpaT =2l ð2:41Þ Moreover, we have x X x ð2n À 1Þ u0 ¼ An sin px l n¼1 2lleading to 8u0 ðÀ1ÞnÀ1 An ¼ ð2:42Þ p2 ð2n À 1Þ2which gives Eq. (2.38) the following form: 8u0 X ðÀ1ÞnÀ1 x 2n À 1 ð2n À 1Þ u¼ 2 2 sin px cos paT t ð2:43Þ p n¼1 ð2n À 1Þ 2l 2lvalid for the interval 0 B t B l/aT, during which the bar is in the tensile state, i.e.during the ﬁrst half-cycle. For the second half-cycle, when the bar enters thecompressive state, the initial conditions are 1 ou t¼ u¼0 ¼ Àu0 aT =l aT otas Eq. (2.43) yields, with the same boundary conditions. From Eq. (2.36), witha = ac, we have X1 xcn x l u¼ Bn sin sin xcn t À ð2:44Þ n¼1 ac aT
40 2 Variable Elasticity Effects in Rotating Machinery where 2n À 1 xcn ¼ pac ð2:45Þ 2l 8u0 aT Bn ¼ À ð2:46Þ p2 ac ð2n À 1Þ2leading eventually to the equation 8u0 X 1 1 2n À 1 2n À 1 l 2l u¼ sin px sin paT t À À ð2:47Þ p2 n¼1 ð2n À 1Þ2 2l 2l aT acvalid within the interval l l 2l t þ aT a T acduring which the bar is in the compressive state. The period of vibration for the barparticles is equal to l l T ¼ 2l þ aT ac or qﬃﬃﬃﬃ Ec pﬃﬃﬃ 1 þ ET T ¼ 2l q qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2:48Þ Ec 1 þ ET The present analysis is useful for the solution of actual rotor problems, as, forexample, in the case of a transversely cracked rotor subjected to reciprocating axialforces. The problem of ﬂexural vibrations can be encountered in a similar manner.Axial forces are very important in this case. The latter, even if not appliedexternally, develop during large deﬂections, and the problem assumes a rathercomplex form. An axial force can be used to control large-amplitude ﬂexuralvibrations [30, 31]. Other problems of instability including crack breathing,damping or parametric instabilities have to be treated with non-linear formulation[32–37].References 1. Elachi, C.: Waves in active and passive periodic structures: a review. Proc. IEEE 64, 1666–1698 (1976) 2. Brillouin, L.: Wave Propagation in Periodic Structures. Dover Publications, New York (1953)
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42 2 Variable Elasticity Effects in Rotating Machinery33. Bykhovsky, I.I.: Fundamentals of Vibration Engineering. Mir Publishers, Moscow (1972)34. Changhe, L., Bernasconi, O., Xenophontidis, N.: A generalized approach to the dynamics of cracked shafts. J. Vib. Acous. Str. Reliab. Des. 111, 257–263 (1989)35. Han, Q.K., Chu, F.L.: Parametric instability of two disk-rotor with two inertia assymetries. Int. J. Struct. Stab. Dyn. 12(2), 251–284 (2012)36. Kanwal, R.P.: Linear Integral Equations. Academic Press, New York (1971)37. Khachatryan, A.A.: Longitudinal vibrations of prismatic bars made of different-modulus materials. Mekhanika Tverdogo Tela 2(5), 140–145 (1967)