Harris'''' Shock and Vibration Handbook Part 3 ppt

Introducing the second of these into Eq. (4.65) and employing the condition that x 0 is also a solution, ¨ δ+κ 2 (1 + 3µ 2 x 0 2 )δ=0 (4.66) Now an expression for x 0 must be obtained; assuming a one-term approximation of the form x 0 = A cos ωt, Eq. (4.66) becomes + (λ+γcos ϕ)δ=0 (4.67) where κ 2 (1 + 3 ⁄2µ 2 A 2 ) = 4ω 2 λ (4.68) and 3 ⁄2κ 2 µ 2 A 2 = 4ω 2 γ 2ωt =ϕ Equation (4.67) is known as Mathieu’s equation. Mathieu’s equation has appeared in this analysis as a variational equation char- acterizing small deviations from the given periodic motion whose stability is to be investigated; thus, the stability of the solutions of Mathieu’s equation must be stud- ied.A given periodic motion is stable if all solutions of the variational equation asso- ciated with it tend toward zero for all positive time and unstable if there is at least d 2 δ ᎏ dϕ 2 NONLINEAR VIBRATION 4.41 FIGURE 4.36 Generalized phase-plane solution of Bessel’s equation. 8434_Harris_04_b.qxd 09/20/2001 11:30 AM Page 4.41 one solution which does not tend toward zero. The stability characteristics of Eq. (4.67) often are represented in a chart as shown in Fig. 4.37. From the response diagram of Duffing’s equation, the out-of-phase motion hav- ing the larger amplitude appears to be unstable. This portion of the response dia- gram (Fig. 4.16C) corresponds to unstable motion in the Mathieu stability chart (Fig. 4.37), and the locus of vertical tangents of the response curves (considering un- damped vibration for simplicity) corresponds exactly to the boundaries between sta- ble and unstable regions in the stability chart. Thus, the region of interest in the response diagram is described by the free vibration ω 2 =κ 2 (1 + 3 ⁄4µ 2 A 2 ) (4.69) and the locus of vertical tangents 3 ⁄2κ 2 µ 2 A 2 +=0 (4.70) The corresponding curves in the stability chart are taken as those for small posi- tive values of γ and λ which have the approximate equations γ= 1 ⁄2 − 2λ (4.71) γ=− 1 ⁄2 + 2λ (4.72) Now, if Eq. (4.69) is introduced into Eqs. (4.68), the resulting equations expanded by the binomial theorem (assuming µ 2 small), and Eq. (4.72) introduced, the result is an identity.Therefore, the free vibration-response curve maps onto the curve of pos- itive slope in the stability chart.The locus of vertical tangents to the response curves maps into the curve of negative slope in the stability chart; this may be seen from the identity obtained by introducing the equations obtained above by binomial expan- sion into Eq. (4.71) and then employing Eq. (4.70). In any given case, it can be determined whether a motion is stable or unstable on the basis of the values of γ and λ, according to the location of the point in the stabil- ity chart. p ᎏ A 4.42 CHAPTER FOUR FIGURE 4.37 Stability chart for Mathieu’s equation [Eq. (4.67)]. 8434_Harris_04_b.qxd 09/20/2001 11:30 AM Page 4.42 The question of stability of response also can be resolved by means of a “stability criterion” developed from the Kryloff-Bogoliuboff procedures. The differential equation of motion is considered in the form ¨x +κ 2 x + f(x,˙x) = p cos ωt Proceeding in the manner of the Kryloff-Bogoliuboff procedure described earlier, ˙ A = f(x,˙x) sin (κt +θ) − cos ωt sin (κt +θ) ˙ θ= f(x,˙x) cos (κt +θ) − cos ωt cos (κt +θ) Expanding the last terms of these equations, the result contains motions of fre- quency κ, κ+ω, and κ−ω. The motion over a long interval of time is of interest, and the motions of frequencies κ+ωand κ−ωmay be averaged out; this is accomplished by integrating over the period 2π/ω: ˙ A = S(A) − sin(Φ−ωt) ˙ θ= − cos(Φ−ωt) where S(A) = ͵ 2π 0 f(A cos Φ, −Aκ sin Φ) sin Φ dΦ C(A) = ͵ 2π 0 f(A cos Φ, −Aκ sin Φ) cos Φ dΦ The steady-state solution may be determined by employing the conditions A = A 0 , ψ=Φ−ωt =ψ 0 : = S 2 (A 0 ) + [C(A 0 ) + A 0 (κ−ω)] 2 tan ψ 0 = This steady-state solution will now be perturbed and the stability of the ensuing motion investigated. Let A(t) = A 0 +ξ(t)[ξ<<A 0 ] ψ(t) =ψ 0 +η(t)[η<<ψ 0 ] By Taylor’s series expansion: ˙ ξ=ξS′(A 0 ) −ηcos ψ 0 ˙η= [(κ−ω) + C′(A 0 )] +ηsin ψ 0 where primes indicate differentiation with respect to A. These two differential equa- tions are satisfied by the solutions ξ=Ae zt η=Be zt p ᎏ 2κA 0 ξ ᎏ A 0 p ᎏ 2κ S(A 0 ) ᎏᎏᎏ C(A 0 ) + A 0 (κ−ω) p 2 ᎏ 4κ 2 1 ᎏ 2πκ 1 ᎏ 2πκ p ᎏ 2κA C(A) ᎏ A p ᎏ 2κ p ᎏ Aκ 1 ᎏ κ p ᎏ κ 1 ᎏ κ NONLINEAR VIBRATION 4.43 8434_Harris_04_b.qxd 09/20/2001 11:30 AM Page 4.43 where A and B are arbitrary constants and z = Ά [S(A 0 ) + A 0 S′(A 0 )] ± Ί [S(A 0 ) + A 0 S′(A 0 )] 2 − 4A 0 ¯p · and ¯p = p/2κ. For stability, the real parts of z must be negative; hence, the following criteria can be established: [S(A 0 ) + A 0 S′(A 0 )] < 0, > 0, ensures stability [S(A 0 ) + A 0 S′(A 0 )] < 0, < 0, ensures instability [S(A 0 ) + A 0 S′(A 0 )] > 0, ѥ 0, ensures instability [S(A 0 ) + A 0 S′(A 0 )] = 0, > 0, ensures stability These criteria can be interpreted in terms of response curves by reference to Fig. 4.14. For systems of this type, [S(A 0 ) + A 0 S′(A 0 )] < 0; when d¯p/dA 0 > 0, ¯p increases as A 0 also increases. This does not hold for the middle branch of the response curves, thus confirming the earlier results. SYSTEMS OF MORE THAN A SINGLE DEGREE-OF-FREEDOM Interest in systems of more than one degree-of-freedom arises from the problem of the dynamic vibration absorber. The earliest studies of nonlinear two degree-of- freedom systems were those of vibration absorbers having nonlinear elements. The analysis of multiple degree-of-freedom systems can be carried out by various of the methods described earlier in this chapter and are generally completely analo- gous to those given here for the single degree-of-freedom system, with analogous results. REFERENCES 1. Thompson, J. M. T., and H. B. Stewart:“Nonlinear Dynamics and Chaos,” pp. 310–320, John Wiley & Sons, Inc., New York, 1987. 2. Ehrich, F. F.: “Stator Whirl with Rotors in Bearing Clearance,” J. of Engineering for Indus- try, 89(B)(3):381–390, 1967. 3. Ehrich,F.F.:“Rotordynamic Response in Nonlinear Anisotropic Mounting Systems,” Proc. of the 4th Intl. Conf. on Rotor Dynamics, IFTOMM, 1–6, Chicago, September 7–9, 1994. 4. Ehrich, F. F.: “Nonlinear Phenomena in Dynamic Response of Rotors in Anisotropic Mounting Systems,” J. of Vibration and Acoustics, 117(B):117–161, 1995. 5. Choi, Y. S., and S. T. Noah: “Forced Periodic Vibration of Unsymmetric Piecewise-Linear Systems,” J. of Sound and Vibration, 121(3):117–126, 1988. d¯p ᎏ dA 0 d¯p ᎏ dA 0 d¯p ᎏ dA 0 d¯p ᎏ dA 0 d¯p ᎏ dA 0 1 ᎏ 2A 0 4.44 CHAPTER FOUR 8434_Harris_04_b.qxd 09/20/2001 11:30 AM Page 4.44 6. Ehrich, F. F.:“Observations of Subcritical Superharmonic and Chaotic Response in Rotor- dynamics,” J. of Vibration and Acoustics, 114(1):93–100, 1992. 7. Nayfeh,A.H., B. Balachandran, M.A. Colbert, and M. A. Nayfeh:“An Experimental Inves- tigation of Complicated Responses of a Two-Degree-of-Freedom Structure,”ASME Paper No. 90-WA/APM-24, 1990. 8. Ehrich, F.F.: “Spontaneous Sidebanding in High Speed Rotordynamics,” J. of Vibration and Acoustics, 114(4):498–505, 1992. 9. Ehrich, F. F., and M. Berthillier: “Spontaneous Sidebanding at Subharmonic Peaks of Rotordynamic Nonlinear Response,” Proceedings of ASME DETC ’97, Paper No. VIB- 4041:1–7, 1997. 10. Ehrich, F. F.: “Subharmonic Vibration of Rotors in Bearing Clearance,” ASME Paper No. 66-MD-1, 1966. 11. Bently, D. E.: “Forced Subrotative Speed Dynamic Action of Rotating Machinery,”ASME Paper No. 74-Pet-16, 1974. 12. Childs, D. W.: “Fractional Frequency Rotor Motion Due to Nonsymmetric Clearance Effects,” J. of Eng. for Power, 533–541, July 1982. 13. Muszynska,A.: “Partial Lateral Rotor to Stator Rubs,” IMechE Paper No. C281/84, 1984. 14. Ehrich, F. F.: “High Order Subharmonic Response of High Speed Rotors in Bearing Clear- ance,” J. of Vibration,Acoustics, Stress and Reliability in Design, 110(9):9–16, 1988. 15. Masri, S. F.: “Theory of the Dynamic Vibration Neutralizer with Motion Limiting Stops,” J. of Applied Mechanics, 39:563–569, 1972. 16. Shaw, S. W., and P. J. Holmes: “A Periodically Forced Piecewise Linear Oscillator,” J. of Sound and Vibration, 90(1):129–155, 1983. 17. Shaw, S. W.: “Forced Vibrations of a Beam with One-Sided Amplitude Constraint: Theory and Experiment,” J. of Sound and Vibration, 99(2):199–212, 1985. 18. Shaw, S. W.: “The Dynamics of a Harmonically Excited System Having Rigid Amplitude Constraints,” J. of Applied Mechanics, 52:459–464, 1985. 19. Choi,Y. S., and S. T. Noah:“Nonlinear Steady-State Response of a Rotor-Support System,” J. of Vibration,Acoustics, Stress and Reliability in Design, 255–261, July 1987. 20. Moon, F. C.:“Chaotic Vibrations,” John Wiley & Sons, Inc., New York, 1987. 21. Sharif-Bakhtiar, M., and S. W. Shaw: “The Dynamic Response of a Centrifugal Pendulum Vibration Absorber with Motion Limiting Stops,” J. of Sound and Vibration, 126(2):221– 235, 1988. 22. Ehrich, F. F.: “Some Observations of Chaotic Vibration Phenomena in High Speed Rotor- dynamics,” J. of Vibration and Acoustics, 113(1):50–57, 1991. 23. Duffing, G.: “Erzwungene Schwingungen bei veranderlicher Eigenfrequenz,” F. Vieweg u Sohn, Brunswick, 1918. 24. Rauscher, M.: J. of Applied Mechanics, 5:169, 1938. 25. Kryloff, N., and N. Bogoliuboff:“Introduction to Nonlinear Mechanics,” Princeton Univer- sity Press, Princeton, N.J., 1943. NONLINEAR VIBRATION 4.45 8434_Harris_04_b.qxd 09/20/2001 11:30 AM Page 4.45 CHAPTER 5 SELF-EXCITED VIBRATION F. F. Ehrich INTRODUCTION Self-excited systems begin to vibrate of their own accord spontaneously,the amplitude increasing until some nonlinear effect limits any further increase. The energy supply- ing these vibrations is obtained from a uniform source of power associated with the system which, due to some mechanism inherent in the system, gives rise to oscillating forces. The nature of self-excited vibration compared to forced vibration is: 1 In self-excited vibration the alternating force that sustains the motion is created or controlled by the motion itself; when the motion stops, the alternating force dis- appears. In a forced vibration the sustaining alternating force exists independent of the motion and persists when the vibratory motion is stopped. The occurrence of self-excited vibration in a physical system is intimately associ- ated with the stability of equilibrium positions of the system. If the system is dis- turbed from a position of equilibrium, forces generally appear which cause the system to move either toward the equilibrium position or away from it. In the latter case the equilibrium position is said to be unstable; then the system may either oscil- late with increasing amplitude or monotonically recede from the equilibrium posi- tion until nonlinear or limiting restraints appear. The equilibrium position is said to be stable if the disturbed system approaches the equilibrium position either in a damped oscillatory fashion or asymptotically. The forces which appear as the system is displaced from its equilibrium position may depend on the displacement or the velocity,or both. If displacement-dependent forces appear and cause the system to move away from the equilibrium position, the system is said to be statically unstable. For example, an inverted pendulum is stati- cally unstable. Velocity-dependent forces which cause the system to recede from a statically stable equilibrium position lead to dynamic instability. Self-excited vibrations are characterized by the presence of a mechanism whereby a system will vibrate at its own natural or critical frequency, essentially independent of the frequency of any external stimulus. In mathematical terms, the motion is de- scribed by the unstable homogeneous solution to the homogeneous equations of motion. In contradistinction, in the case of “forced,” or “resonant,” vibrations, the fre- quency of the oscillation is dependent on (equal to, or a whole number ratio of) the frequency of a forcing function external to the vibrating system (e.g., shaft rotational 5.1 8434_Harris_05_b.qxd 09/20/2001 11:28 AM Page 5.1 speed in the case of rotating shafts). In mathematical terms, the forced vibration is the particular solution to the nonhomogeneous equations of motion. Self-excited vibrations pervade all areas of design and operations of physical sys- tems where motion or time-variant parameters are involved—aeromechanical sys- tems (flutter, aircraft flight dynamics), aerodynamics (separation, stall, musical wind instruments, diffuser and inlet chugging), aerothermodynamics (flame instability, combustor screech), mechanical systems (machine-tool chatter), and feedback net- works (pneumatic, hydraulic, and electromechanical servomechanisms). ROTATING MACHINERY One of the more important manifestations of self-excited vibrations, and the one that is the principal concern in this chapter, is that of rotating machinery, specifically, the self-excitation of lateral, or flexural, vibration of rotating shafts (as distinct from torsional, or longitudinal, vibration). In addition to the description of a large number of such phenomena in standard vibrations textbooks (most typically and prominently, Ref. 1), the field has been sub- ject to several generalized surveys. 2–4 The mechanisms of self-excitation which have been identified can be categorized as follows: Whirling or Whipping Hysteretic whirl Fluid trapped in the rotor Dry friction whip Fluid bearing whip Seal and blade-tip-clearance effect in turbomachinery Propeller and turbomachinery whirl Parametric Instability Asymmetric shafting Pulsating torque Pulsating longitudinal loading Stick-Slip Rubs and Chatter Instabilities in Forced Vibrations Bistable vibration Unstable imbalance In each instance, the physical mechanism is described and aspects of its preven- tion or its diagnosis and correction are given. Some exposition of its mathematical analytic modeling is also included. WHIRLING OR WHIPPING ANALYTIC MODELING In the most important subcategory of instabilities (generally termed whirling or whipping), the unifying generality is the generation of a tangential force, normal to 5.2 CHAPTER FIVE 8434_Harris_05_b.qxd 09/20/2001 11:28 AM Page 5.2 an arbitrary radial deflection of a rotating shaft, whose magnitude is proportional to (or varies monotonically with) that deflection. At some “onset” rotational speed, such a force system will overcome the stabilizing external damping forces which are generally present and induce a whirling motion of ever-increasing amplitude, limited only by nonlinearities which ultimately limit deflections. A close mathematical analogy to this class of phenomena is the concept of “nega- tive damping” in linear systems with constant coefficients, subject to plane vibration. A simple mathematical representation of a self-excited vibration may be found in the concept of negative damping. Consider the differential equation for a damped, free vibration: m¨x + c˙x + kx = 0 (5.1) This is generally solved by assuming a solution of the form x = Ce st Substitution of this solution into Eq. (5.1) yields the characteristic (algebraic) equation s 2 + s +=0 (5.2) If c < 2͙m ෆ k ෆ , the roots are complex: s 1,2 =− ±iq where q = Ί − ΂΃ 2 The solution takes the form x = e −ct/2m (A cos qt + B sin qt) (5.3) This represents a decaying oscillation because the exponential factor is negative, as illustrated in Fig. 5.1A. If c < 0, the exponential factor has a positive exponent and the vibration appears as shown in Fig. 5.1B. The system, initially at rest, begins to oscillate spontaneously with ever-increasing amplitude. Then, in any physical sys- tem, some nonlinear effect enters and Eq. (5.1) fails to represent the system realisti- cally. Equation (5.4) defines a nonlinear system with negative damping at small amplitudes but with large positive damping at larger amplitudes, thereby limiting the amplitude to finite values: m¨x + (−c + ax 2 )˙x + kx = 0 (5.4) Thus, the fundamental criterion of stability in linear systems is that the roots of the characteristic equation have negative real parts, thereby producing decaying amplitudes. In the case of a whirling or whipping shaft, the equations of motion (for an ideal- ized shaft with a single lumped mass m) are more appropriately written in polar coordinates for the radial force balance, −mω 2 r + m¨r + c˙r + kr = 0 (5.5) and for the tangential force balance, 2mω˙r + cωr − F n = 0 (5.6) where we presume a constant rate of whirl ω. c ᎏ 2m k ᎏ m c ᎏ 2m k ᎏ m c ᎏ m SELF-EXCITED VIBRATION 5.3 8434_Harris_05_b.qxd 09/20/2001 11:28 AM Page 5.3 In general, the whirling is predicated on the existence of some physical phe- nomenon which will induce a force F n that is normal to the radial deflection r and is in the direction of the whirling motion—i.e., in opposition to the damp- ing force, which tends to inhibit the whirling motion. Very often, this normal force can be characterized or approxi- mated as being proportional to the radial deflection: F n = f n r (5.7) The solution then takes the form r = r 0 e at (5.8) For the system to be stable, the coeffi- cient of the exponent a = (5.9) must be negative, giving the require- ment for stable operation as f n <ωc (5.10) As a rotating machine increases its rotational speed, the left-hand side of this inequality (which is generally also a function of shaft rotation speed) may exceed the right-hand side, indicative of the onset of instability.At this onset condition, a = 0+ (5.11) so that whirl speed at onset is found to be ω= ΂΃ 1/2 (5.12) That is, the whirling speed at onset of instability is the shaft’s natural or critical fre- quency, irrespective of the shaft’s rotational speed (rpm). The direction of whirl may be in the same rotational direction as the shaft rotation (forward whirl) or opposite to the direction of shaft rotation (backward whirl), depending on the direction of the destabilizing force F n . When the system is unstable, the solution for the trajectory of the shaft’s mass is, from Eq. (5.8), an exponential spiral as in Fig. 5.2.Any planar component of this two- dimensional trajectory takes the same form as the unstable planar vibration shown in Fig. 5.1B. GENERAL DESCRIPTION The most important examples of whirling and whipping instabilities are Hysteretic whirl Fluid trapped in the rotor k ᎏ m f n − cω ᎏ 2mω 5.4 CHAPTER FIVE FIGURE 5.1 (A) Illustration showing a decay- ing vibration (stable) corresponding to negative real parts of the complex roots. (B) Increasing vibration corresponding to positive real parts of the complex roots (unstable). 8434_Harris_05_b.qxd 09/20/2001 11:28 AM Page 5.4 [...]... Paper 73- DET- 131 , September 19 73 29 Ref 1, pp 31 7 32 1 30 Ehrich, F F.: Trans ASME, (A) 90(4) :36 9 (1968) 31 Taylor, E S., and K A Browne: J Aeronaut Sci., 6(2): 43 49 (1 938 ) 32 Houbolt, J C., and W H Reed: “Propeller Nacelle Whirl Flutter,” I.A.S Paper 61 34 , January 1961 33 Trent, R., and W R Lull: “Design for Control of Dynamic Behavior of Rotating Machinery,” ASME Paper 72-DE -39 , May 1972 34 Ehrich,... Phenomenon in Turbomachinery Rotors,” ASME Paper 73- DET-97, September 19 73 35 Floquet, G.: Ann l’école normale supérieure, 12:47 (18 83) 36 McLachlan, N W.: “Theory and Applications of Mathieu Functions,” p 40, Oxford University Press, New York, 1947 37 Ref 1, pp 33 6 33 9 38 Brosens, P J., and H S Crandall: J Appl Mech., 83( 4):567 (1961) 39 Messal, E E., and R J Bronthon: “Subharmonic Rotor Instability... Wiley & Sons, Inc., New York, 1 939 53 Dimarogonas, A D., and G N Sander: Wear, 14 (3) :1 53 (1969) 54 Dimarogonas, A D.: “Newkirk Effect: Thermally Induced Dynamic Instability of HighSpeed Rotors,” ASME Paper 73- GT-26, April 19 73 55 Ehrich, F and D Childs: Mech Eng., 106(5):66 (1984) 8 434 _Harris_06_b.qxd 09/20/2001 11:26 AM Page 6.1 CHAPTER 6 DYNAMIC VIBRATION ABSORBERS AND AUXILIARY MASS DAMPERS F Everett... Forward 8 434 _Harris_05_b.qxd 09/20/2001 11:28 AM Page 5. 23 SELF-EXCITED VIBRATION 5. 23 IDENTIFICATION OF SELF-EXCITED VIBRATION Even with the best of design practice and application of the most effective methods of avoidance, the conditions and mechanisms of self-excited vibrations in rotating machinery are so subtle and pervasive that incidents continue to occur, and the major task for the vibrations... Eng., 43: 99–1 03 (1959) 47 Ref 1, pp 36 5 36 8 48 Van der Pol, B.: Phil Mag., 2:978 (1926) 49 Ehrich, F F.: “Bi-Stable Vibration of Rotors in Bearing Clearance,” ASME Paper 65WA/MD-1, November 1965 8 434 _Harris_05_b.qxd 09/20/2001 11:28 AM Page 5.25 SELF-EXCITED VIBRATION 5.25 50 Ref 1, pp 245–246 51 Newkirk, B L.: “Shaft Rubbing,” Mech Eng., 48: 830 (1926) 52 Kroon, R P., and W A Williams: “Spiral Vibration. .. 1961 25 Black, H F., and D N Jenssen: “Effects of High Pressure Seal Rings on Pump Rotor Vibrations,” ASME Paper 71-WA/FE -38 , December 1971 26 Alford, J S.: J Eng Power, 87(4) :33 3, October 1965 27 Ehrich, F F et al.: “Unsteady Flow and Whirl-Inducing Forces in Axial Flow Compressors Part II—Analysis,” ASME Paper 2000-GT-0566, May 2000 28 Black, H F.: “Calculation of Forced Whirling and Stability of Centrifugal... is diagnosis and correction Figure 5.3B suggests the forms for display of experimental data to perceive the patterns characteristic of whirling or whipping, so as to distinguish it from forced vibration, Fig 5.3A Table 5.2 summarizes particular quantitative measurements that can be made to distinguish between the various types of whirling and whipping, and other types of self-excited vibrations The... C., and E E Haft: “Stability of an Unsymmetrical Rotating Cantilever Shaft Carrying an Unsymmetrical Rotor,” ASME Paper 71-Vibr-57, September 1971 41 Eshleman, R L., and R A Eubanks: “Effects of Axial Torque on Rotor Response: An Experimental Investigation,” ASME Paper 70-WA/DE-14, December 1970 42 Wehrli,V C.:“Uber Kritische Drehzahlen unter Pulsierender Torsion,” Ing.Arch., 33 : 73 84 (19 63) 43 Ref 36 ,... clearances would minimize the influence of tip clearance on the unit’s performance and, hence, minimize the contribution to destabilizing forces 8 434 _Harris_05_b.qxd 09/20/2001 11:28 AM Page 5.15 SELF-EXCITED VIBRATION 5.15 PROPELLER AND TURBOMACHINERY WHIRL Propeller whirl has been identified both analytically31 and experimentally .32 In this instance of shaft whirling, a small angular deflection of the shaft... of the apparatus INSTABILITIES IN FORCED VIBRATIONS In a middle ground between the generic categories of force vibrations and selfexcited vibrations is the category of instabilities in force vibrations These instabilities are characterized by forced vibration at a frequency equal to rotor rotation (generally induced by unbalance), but with the amplitude of that vibration being unsteady or unstable Such . A 0 (κ−ω) p 2 ᎏ 4κ 2 1 ᎏ 2πκ 1 ᎏ 2πκ p ᎏ 2κA C(A) ᎏ A p ᎏ 2κ p ᎏ Aκ 1 ᎏ κ p ᎏ κ 1 ᎏ κ NONLINEAR VIBRATION 4. 43 8 434 _Harris_04_b.qxd 09/20/2001 11 :30 AM Page 4. 43 where A and B are arbitrary constants and z = Ά [S(A 0 ) + A 0 S′(A 0 )] ± Ί [S(A 0 ) + A 0 S′(A 0 )] 2 − 4A 0 ¯p · and ¯p. J. of Vibration and Acoustics, 117(B):117–161, 1995. 5. Choi, Y. S., and S. T. Noah: “Forced Periodic Vibration of Unsymmetric Piecewise-Linear Systems,” J. of Sound and Vibration, 121 (3) :117–126,. Mechanics, 39 :5 63 569, 1972. 16. Shaw, S. W., and P. J. Holmes: “A Periodically Forced Piecewise Linear Oscillator,” J. of Sound and Vibration, 90(1):129–155, 19 83. 17. Shaw, S. W.: “Forced Vibrations