0 Phase angle in the assumed shape of the non-locked model response 0ss Phase angle of the steady-state non-locked model response. 1.1.1) ceases to be valid and the actual vortex frequency is very close to the natural frequency of the system [29].
SCOPE OF THIS INVESTIGATION
CHAPTER II
A BRIEF HISTORICAL REVIEW
They range from studies with fixed cylinders in steady and sheared flow to studies with prototype-sized cables in quasi-steady flow. This topic still arouses a lot of interest not only in academia, but also in practice, where engineers encounter problems caused by eddy vibrations [13].
EXPERIMENTAL INVESTIGATIONS
It should be noted that this relatively good match was only achieved when the cylinder was vibrated under lock-in conditions, i.e. when the swirl frequency wv is very close to the frequency w at which the cylinder is forced to vibrate (ie wv = w). However, outside the lock-in region, the vortices dissipate at Strouhal frequency ws, while the cylinder is forced at frequency w (ie, wv = ws* w).
ANALYTICAL MODELING
While Hall [29] used several datasets obtained from air-only experiments to determine model constants with the following values. Coefficient of inertia Cmh and coefficient of drag Cdh versus reduced speed obtained via the Wake Oscillator Nodel using a set of constants (2.3.13).
0 FENG EXP 1 .TAL DATA -MODEL
INTRODUCTION
CHAPTER III
STABILITY ANALYSIS OF STEADY-STATE RESPONSE
Multiple solutions are expected due to the nonlinear nature of the current problem [9], so determining the stability of each solution becomes essential. The method of slowly varying parameters, when applied to the equation of motion of a single degree of freedom oscillator, yields a pair of first-order ordinary differential equations given in terms of exactly the same variables used in section 3.2, that is, the amplitude and frequency of the system. Otherwise the solution will be unstable. 3.3.26) will have a non-trivial solution only if the determinant of the coefficient matrix is zero.
However, due to the complexity of each aij term (i,j = 1,2), no sufficient condition for instability can be derived in terms of the basic variables involved in Eq. A parametric study has shown that for most of the parameter ranges considered here1, the real part of at least one of the roots is positive, whenever a. According to the current approximation, the endurance limits are independent of the structural damping ratio.
The system of equations Eq. 3.3.55) will have a non-trivial solution, only if the determinant of the coefficient matrix is zero. Once again, due to the complexity of the quantities involved, the above condition cannot be translated in terms of the basic variables present in Eq.
INTRODUCTION
A PURELY EMPIRICAL APPROACH
4.2.1) and (4.2.2), Qn is evaluated as a function of Q for each of the four values of B and at each experimental data point for Cmh and Cdh given in Fig. But a close inspection of the actual experimental data points [61] shows that there appears to be a shift between the Cmh and Cdh coefficients obtained in each experimental set. It is clear from the above that appropriate data interpolation will be required to produce reasonable model response curves.
Ideally, the interpolation expressions should be chosen to reflect the nature of the vortex-induced vibration phenomenon. But if the nature of the fluid-structure interaction were known, there would be no need for an approximate model. In this procedure, an expression for the curves in the Vr direction is chosen to preserve the features considered most important, but no attempt is made to best fit the actual data points.
A similar examination of the behavior of the Cml(Vr,B) curves presented in Figure 2.2.3 shows that. Vro , zero crossing of the curves. assume as the asymptotic value to which Cmh would tend for relatively large values of Vr•.
NCJRMRLIZED RMPL I TUDE, B
The constants in the expression above were determined by a least-squares fit of the experimental data. In section 4.2.1 it is mentioned that careful examination of the actual experimental data shows that there is a shift between the Cmh and Cdh coefficients obtained for a same Vr• But only based on the analysis of these coefficients, a one does not say which coefficient is actually shifted with respect to which. A left or right shift in the analytical interpolation expression for Cmh is accomplished by the value of the parameter Vro• Thus, if Vro = 5.15.
It is still noted that the fitting approach used herein, which is modular, allows independent changes in each of the parameters. Furthermore, the frequency is normalized by Wn, so that the response curves for the lock-in case (i.e. w = wn) can lie along the horizontal w/wn 1 and for the non-lock-in case (i.e. w = ws ), on the diagonal. The results from one case to another are very similar and it can be noted that the model exhibits many of the features associated with the vortex-induced vibration phenomenon.
The locking response presents large vibration amplitudes at w = wn, where the vortex shedding frequency is known to be locked to the structural frequency of vibration. Relatively smaller vibration amplitudes are given by the non-key model, with a vibration frequency equal to the Strauhal frequency.
Hodel versus experimental amplitude response .. that the response corresponding to smaller amplitudes is always. However, the same stability analysis applied to the nonblocking response shows that the solutions are stable everywhere. A jump in solvation is expected when the transition from locked to unlocked solution occurs with increasing flow rates.
Despite the good agreement between the predicted and experimentally measured maximum vibration amplitudes, the overall amplitude response curve for the model is consistently shifted to the left relative to the experimental data. A correction in Cdh coefficients (rather than in Cmh coefficients) has not been attempted, but it is expected that such a correction would shift the model predictions to the right relative to the model predictions presented herein. In principle, there seems to be no reason to assume that the force coefficients Cmh and Cdh would reach the same values for the same experiment performed in different liquid media (e.g. air, water).
In case 1, the assumed mass parameter value is within the range that Feng used in his experiments and the behavior of the model is essentially similar to that described previously. The lock-in bandwidth and amplitude are, as expected, larger because the assumed damping ratio is an order of magnitude smaller than the values used by Feng, but the overall shape of the amplitude response does not resemble the previous model response curves depicted in Fig.
One can also note that the branching of unstable solutions has completely unfolded from that of stable solutions. However, it is believed that the current model will perform best in predicting response for structures in water. Yet it seems impossible to infer or infer any dependence between the maximum amplitude and the 3.2.16.
It should be emphasized that in the previous equation, the structural fraction of critical damping s must be measured in vacuum. 4.3.11) will clearly only approximately predict the maximum amplitude of vibration, a parametric study in n and s is undertaken in the next section, with the aim of determining how good an approximation it is. So that the solution of Eq. 4.3.11) can be directly compared with other published results [25], the variable reduced damping s is defined as follows. The predicted maximum amplitude response obtained as a function of reduced damping is shown in Fig.
Based on the presented approximations, Eq. 4.3.13) will always give the correct value for the amplitude at ws/wn = 1, although it may or may not coincide with that of the maximum amplitude. To evaluate the prediction capabilities of the Lock-in Model and to evaluate approximate solutions for stability limits (Eq. 3.3.34)) and for maximum amplitudes (Eq.
REDUCED DRMP I NGP '(
Approximate results for both the maximum amplitudes and the stability limits agree practically with those obtained by exact analysis. In each of the four figures, the corresponding n is kept constant while the fraction of critical damping ~ is varied so that the ratio ~/n assumes the values given in the first column of the table for predicting the approximate maximum amplitudes. thoroughly tested for a wide range of values of n. How these results compare with those obtained from the exact analysis is shown in Table 4.3.6.
NORMALIZED VELOCITY,
In terms of stability limits, the region is bounded by Eq. 3.3.34) generally coincides with the actual unstable region. the results deteriorate as n and ~ increase, yet all unstable solutions remain within the estimated stability limit given by Eq. From the above it is now clear that large amplitudes of vibration and frequency entrainment can occur over a wide range of normalized speeds. In general, one can conclude that, for a small damping and a large mass ratio (e.g. a light cylinder in water), the amplitude response curve will be narrow at Bmaxf /2 (6w1 = 0.15), while being associated with a relatively larger frequency. entrainment area (6w2 >> 0.60).
As the damping increases, one bandwidth tends to the other, then much smaller regions of frequency insertion can be expected. Note that the amplitude and frequency responses for structures in water, that is, for values of
CHAPTER V
9] Feng, C.C., "The measurement of vortex-induced effects in flow past stationary and oscillating circular and D-section cylindeis'', M.A.Sc. 11] Ferguson, N., "The measurement of wake and surface effects in the subcritical Flow Past a Circular Cylinder at Rest and in Vortex-Excited Oscillations", M.A.Sc. 48] Parkinson, G.V., "Mathematical Models of Flow-Induced Vibrations of Bluff Bodies", IUTAM-IAHR Symposium, Karlsruhe 1972, Technical Session B, Technical Session.
52] Ramberg, S.E., Griffin, O.M., and Skop, R.A., "Some resonant vibration properties of marine cables with application to the prediction of vortex-induced structural vibrations", ASME Winter Meeting, Ocean Eng. 54] Ramberg, S.E., and Griffin, O.M., "The Effects of Vortex Coherence, Spacing, and Circulation on the Flow-Induced Forces on Vibrating Cables and Bluff Structures", Naval Research Laboratory, Report 7945, January 1976. 55] Ramberg, SE, "The Influence of Yaw Angle on the Vortex Wake of Stationary and Vibrating Cylinders," Naval Research Laboratory, Memorandum Report 3822, August 1978.
60] Sarpkaya, T., "An Analytical and Experimental Study of the In-plane and Transverse Oscillations of a Circular Cylinder in Uniform Flow", Report No. 63] Sarpkaya, T., and Shoaff, R.L., "A Discrete Vortex Analysis of Flow Around Stationary and Oscillating Transverse Circular Cylinders", Report No.
