Flow-Induced Vibration Handbook for Nuclear and Process Equipment. Группа авторов

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for both resonance and non‐resonance data. The parameter L/h is defined in Ziada et al (1989).

Graph depicts proposed Damping Criteria (Resonance Parameter) for: a) In-Line (Square) Arrays; b) Staggered (Triangular) Arrays.

      Similarly, for in‐line arrays (square)

      (2‐43b)equation

      The parameter Gi is plotted against (XL)2 in Fig. 2-19a, where XL = L/D. For a satisfactory design, it must be shown that the heat exchanger tube bundle is in the non‐resonance parts of Fig. 2-19.

      Due to the complexity of a multi‐span steam generator tube, a computer code is used to predict vibration response accurately and to determine the susceptibility of a tube to periodic‐wake‐shedding resonance and fluidelastic instability. Such a computer code must be capable of calculating modeshapes and natural frequencies for a number of modes at least equal to the number of tube spans.

      Generally, the code will be in two parts: the first part calculates the free-vibration tube characteristics such as mode shapes and natural frequencies, while the second part is a forced vibration analysis that calculates the tube response to turbulence‐induced excitation and periodic wake shedding, and the fluidelastic instability ratios. A forced vibration analysis involves the application of the distributed flow velocities and densities that are determined using the methods outlined in Chapter 3.

      If the tube‐to‐support clearances are too large, the tube supports will not be effective and the assumption of hinged supports will not be valid. Tube‐to‐support diametral clearances of 0.38 mm (0.015 in.) or less are typically used in nuclear heat-exchanger design. Effective supports can become ineffective if a heat exchanger is subjected to significant corrosion or a chemical cleaning technique is used to clean a fouled unit. The effects of future chemical cleaning and corrosion should be considered in the vibration response analysis.

      The ultimate goal of vibration analyses is to ensure that neither fretting wear nor fatigue damage occurs in heat exchangers. Although a linear analysis does not predict fretting wear, fretting‐wear damage can be estimated, as discussed in Section 2.6.1.

      Within vibration prediction codes, the tube response y(x,t) at any point x along the tube and at any time t may be expressed as a normal mode expansion in terms of the generalized coordinates qi(t) as follows:

      where ϕi(x) is the mode shape of the ith mode and n is the number of modes to be used in the analysis. Using Lagrange’s equation and assuming that the damping is small and that it does not introduce coupling between modes, the equation of motion for the ith mode is as follows:

      where ζi is the damping ratio, ℓ is the tube length and ωi is the angular frequency of the ith natural mode. The natural modes are normalized so that

      (2‐46)equation

      where m is the mass per unit length.

      2.5.1 Fluidelastic Instability

      A fluidelastic instability ratio is determined by calculating an effective velocity based on the velocity and density profiles and the important mode shapes. For the ith mode, the critical velocity, Upc, at which instability occurs, may be expressed as:

      (2‐47)equation

      2.5.2 Random Turbulence Excitation

      Tube response to turbulence‐induced excitation forces must be calculated using random vibration theory, and the response must be summed over all significant modes of vibration. The mean square amplitude response, images, of a continuous uniform cylindrical tube to distributed random forces, g(x,t), may be expressed by:

      (2‐48)equation

      where Hr*and Hs are complex complementary frequency response functions for the rth and sth modes, R(x,x,f) is the spatial correlation density function and x and x′ are points along the length of the tube. More detail is found in Pettigrew et al (1978), Axisa et al (1990) and in Chapters 9 and 10.

      2.5.3 Periodic Wake Shedding

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