Q2. What is the common explanation for the instability of a cylinder array?
For rotated square arrays, even for low values of m8 The authorpD2 , the instability is of the two-degreeof-freedom variety; moreover, this type of array is prone to be subject to a recently discovered static instability, i.e. fluid-elastic divergence [7].
Q3. What is the analytic model used for the analysis?
the analysis is carried out by considering a single flexible cylinder, with all the other cylinders in the array rigid and immobile; the analytic model used is that of Price and Paidoussis [4].
Q4. What is the mechanism responsible for the instability of a cylinder array?
The first mechanism gives rise to single-degree-of-freedom flutter; hence array stability may be analyzed, approximately, by considering all cylinders of the array except one to be rigid, and considering motions of just that one cylinder [6; 4, 5].
Q5. What is the output of the equation used to display bifurcations?
The parameter varied is U, and the output utilized to display bifurcations is the approximate dimensionless displacement at the middle point of the cylinder, q(O· 5, t), obtained through equation ( 5).
Q6. What is the purpose of the Poincare map P?
It is fortunate that to fulfill the purpose of determining stability and bifurcations, an implicit form of the Poincare map P is sufficient.
Q7. What is the advantage of using the Poincare map technique for system 13?
The advantage of employing this technique for system (13) is that thereby one is effectively dealing with a two-dimensional discrete map rather than a three-dimensional discontinuous flow, and the stability information of a continuous periodic solution of (13) may be obtained by computing eigenvalues of P at the corresponding fixed point.
Q8. What is the importance of non-linearity in heat exchangers?
This non-linearity is considered to perhaps be more important for heat exchanger arrays than the fluid-force non-linearities, since impacting occurs even at relatively small amplitude (the gap to the baffle plate being generally small), when fluid non-linearities are still relatively unimportant.
Q9. What is the critical flow velocity for the onset of instability?
To find the critical flow velocity for the onset of instability, all system parameters are fixed as defined in the previous section, and the dimensionless flow velocity U is gradually increased.
Q10. How are the dynamics of the system studied?
The dynamics of the system are studied through simulation, obtaining bifurcation and phase flow diagrams and also Lyapunov exponents.
Q11. What is the important of the critical flow velocities?
Of all the critical flow velocities, the first one, U HI = 1· 785, is the most important, since thereafter an oscillatory state prevails for the double-span system of Figure 1.
Q12. What is the importance of the increase in response frequencies with flow?
The increase in response frequencies with flow is of particular importance, since that would result in increased wear of the tubes in heat exchangers.
Q13. How can the authors determine the stability of periodic solutions?
In order to determine the stability of a periodic solution emanating from (to, Yo), the Jacobian matrix of the Poincare map P at (to, Yo) must be computed:a(t, y) DP(to, Yo) = .a( to, Yo) (18)Because of the discontinuity in y, the computation of the derivative (18) must be divided into four parts:(19)All the individual derivative terms can easily be determined using the impact rule and the functions f and g defined in equations ( 16):au1, YI) = act3, y3) [1 o ]. a( to, Yo) a(tz, Yz) 0 -ra(tz,yz) - ::; :~ - :~;:~ act~, yJ) ag atz ag --+-ag at2 ag --+- atz at! at! atz ay! ay!(20a)(20b)and the derivative iJ(t4 ,y4)ja(t3 ,y3) can be obtained from equation (20b) by shifting the numbers from 1 to 3 and 2 to 4.
Q14. What is the simplest possible model for the impact oscillator?
The mathematical model for the impact oscillator described above is a second order piecewise linear differential equation for which there exist explicit solutions for the time intervals between any two consecutive impacts.
Q15. What is the effect of the trilinear model on the response to the impact?
the results obtained with this model suggest that the chaoticlooking response obtained with the trilinear model of the impact constraint beyond, but close to, the Hopf bifurcation point may in fact be quasi-periodic.
Q16. What are the reasons for the impacting on the baffle plates?
For the reasons given in the previous paragraph, the only non-linearities considered here are those associated with impacting on the baffle plates.
Q17. What is the phase flow diagram for the system with a trilinear spring model?
Phase flow and (c, d) power spectral density diagrams for the system with trilinear modelling of constraint stiffness: (a, c) for U== 1·9, just beyond the Hopf bifurcation; (b, d) for U= 3·0 showing strongly chaotic motion.