Abstract

1. On the concept of negative dampingWorkshop on “Mathematical challenges and modelling of hydro elasticity”International Centre for Mathematical Sciences, Edinburgh, June 2010Ove T. Gudmestad and Axel A. BonnaudUniversity of Stavanger, Norway

2. Abstract • Damping is limiting the motions of an oscillator, a dynamic system. • Different formulations for damping are suggested in the literature • In case the forcing function of a dynamic system contains terms proportional to the velocity of motion of the oscillator, the effects will contribute to damping the oscillations. • Should the total damping under certain conditions become negative, the oscillations will grow until the damping again has become positive. • Investigations into negative damping effects and discussions where negative damping might appear in practical applications are given

3. 1) Basic equation • We consider a one-degree-of-freedom forced oscillator with viscous damping described by the equation of motion • Structural damping is associated with the constant c0, and critical damping is obtained for the case =1 • Should the value of the damping c0 take on a negative value, the oscillations will have increasingly large amplitudes, see Figure 1.

4. Effect of negative damping • Figure 1 Dynamic response for different negative values of damping in the equation of motion

5. Variable damping term that could be negative • Increasing amplitudes resulting from equation (1) would be non-physical, representing energy generation in the system. • In case of certain formalism for the external loading, the oscillations could grow to large values. • Let us, for example investigate the case when the external forcing term fe (t) of the dynamic system contains terms proportional to the velocity of motion of the oscillator, the effects of these will contribute to dampen the oscillations.

6. Variable damping term that could be negative • The following one-degree-of-freedom oscillator described by the equation of motion is then to be considered: • (2) Here: • g0(t) is the “remaining” external, time-varying excitation load, reduced by the load proportional to the velocity term, dy(t)/dt • c (t)  c0 + c(t)is the damping, consisting of a constant term c0(that, for example, would represent the structural damping)and a time-varying term c(t).

7. An example, c0=0 • Let us, now for simplicity consider a linear oscillator of the form: (3) • Let a(t) be a periodic or almost periodic function with −1 ≤ a (t) ≤ 1 (the min value of a(t) = −1 and max a(t) = 1). This choice will clearly fulfil our other aims. For simplicity we may choose a (t) to be periodic, for example: (4) • The damping is slowly changing compared with a timescale of t of order 1. • In order to mimic a drag force loading term, Morison et al. (1950) we will choose: b(t) = sin(kt)|sin(kt)| (5)

8. Typical solution y(t) of equation (3) as function of time with a(t) and b(t) as in equations (4) and (5) We have thus demonstrated that an apparent negative damping give rise to “burst type” of displacements (an oscillatory instability). We will look into different examples of apparent negative damping terms.

9. 2) Galloping • Case 1: Galloping of a one-degree of freedom system in steady flow with magnitude U, see e.g. Blevins (1994) • When a non-circular cross-section structure experiences fluid forces that changes unfavourably with the orientation in the flow it may cause the structure to start vibrating. • If this force tends to increase the vibration, then the structure will become unstable. • This phenomenon is known as galloping. Examples of galloping phenomenon can be seen on the vibration of ice-coated power line cables or on twin pipelines, e.g. piggyback pipelines. • This condition often met at higher reduced velocity; such that: Ured =

10. The model of galloping for a one-degree of freedom system.

11. Lift force in case of galloping • FD is directed along the relative velocity vector while the lift force FL is similarly acting in a direction α relative to the vertical with positive direction upwards: • By assuming α to be small so that and by expanding in Mac Laurin series, the lift force and equation of motion are

12. Equivalent damping • The total damping ratio (including structural and flow damping effects) can now be written: • For positive, i.e. destabilizing, the total damping may become negative and structural failure could therefore be expected

13. 3) Example: Piggyback pipeline geometry • MSc students Hang and Lubis did in 2009 study the response of a piggyback pipeline configuration (Figure 3) in a constant flow. • The flow is from left to right. • Figure 3 “Piggyback” pipeline configuration with two pipes of diameters D + 1/2 D. • Galloping-like instabilities were observed for a “piggyback” configuration with pipes of diameters D + ½ D where the flow is from left to right, see figure 3.

14. They found that for this geometry the large response starts at Ured = 4 and increases with increasing Ured. Reference is made to Figure 4. • Further tests should be carried out to verify the onset of galloping for these low values of Ured. • It is possible that vortex induced vibrations onset at Ured = 4 and that galloping takes over at a higher value of Ured.

15. Piggyback geometry and results Amplitude of the response of a “piggyback” pipeline geometry. D + ½ D plotted against Ured, Hang (2009).

16. 4) Galloping of cables • As for the special pipeline configuration, galloping on ice-coating cables occurs when certain amount of ice is developed on the cables, see Figure 5. • It can be caused by glaze ice and rime ice or wet snow on the conductor Figure 5 Ice Accretion on cables; (a) rough condition, (b) round profile due to continuous rotation (Havard, D.G and Lilien, J.L, 2007)

17. 5) Motion of a slender offshore structure taking relative velocity motion term into account • Case 2: Motion of a slender offshore structure taking relative velocity motion term into account: • The general second order ordinary differential equation for the horizontal response y (t) of a one degree of freedom slender offshore structure when subjected to constant nonlinear drag loading (that is; the loading generated by the velocity U of a current) which, according to experiments is given by the term per unit length of the structure; see for example Sarpkaya and Issacsson (1981):

18. Inclusion of added mass and relative motion • We should include the added mass term • We could also include current in the loading term of the equation of motion. • We should also include the effect of the motion of the structure itself on the forcing term, that is, we should consider relative acceleration and velocity terms in (2) as was suggested by Gudmestad and Connor (1983):

19. Effects of the relative velocity term • We could consider the nonlinear relative velocity drag forcing term: • Thus, the damping term becomes of the form c(t)  c0 + c(t) consisting of one linear term c0 and one time-varying term c(t) as was discussed previously. • We have postulated that this could represent “burst type of response”: • Gudmestad, O. T.; T. M Jonassen, C.-T. Stansberg and A. N. Papusha, (2009). Nonlinear one degree of freedom dynamic systems with “burst displacement characteristics” and “burst type response”. Proceedings of “Fluid Structures Interaction”, 2009, WIT Press, Crete, May.

20. 6) Study of damping function • we have recently studied the effect of the relative importance of the terms contributing to the damping • We take with • We, furthermore, define Amd = c1-c0 , see Figure 7 • Figure 7 The damping function studied

21. Influence of the damping amplitude (here denoted L=0.5)on the horizontal displace- ment for Td/TL = 2.0. Larger displacements observed

22. 7) Splash zone lifting analysis Snapshot of the time series showing the structure’s vertical velocity at the “suction can” top is larger than water particle velocity at the same location indicating role of drag as damping

23. 8) Ice engineering • Theoretical work has been conducted to increase the understanding of the dynamic response of offshore structures. • Määttänen (1978) proposed that steady-state vibration of a narrow vertical offshore structure is a self-excited process where the non-linear forces due to ice crushing provide an apparent negative damping effect to the structure. • Kärnä et al (2005), furthermore, presented a model of dynamic ice forces on vertical offshore structure and their model makes use of the concept of negative damping

24. 9) Conclusions and recommendations • The concept of negative damping could explain large motions of vibrating structures • Further analysis of the concept should be conducted with reference to onset of ”burst type of displacements” • Nonlinear analysis using nonlinear equations as Van der Pool oscillator and the Duffing equation should be revisited