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On the concept of negative damping
Workshop on “Mathematical challenges and modelling of
hydro elasticity”
International Centre for Mathematical Sciences,
Edinburgh, June 2010

Ove T. Gudmestad and Axel A. Bonnaud
University of Stavanger, Norway

1
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
c

2m 0

1) Basic equation

• We consider a one-degree-of-freedom forced oscillator
with viscous damping described by the equation of
motion
d 2 y( t )      dy( t )
m              c0          k 2 y( t )  f e ( t )
dt 2           dt
• Structural damping is associated with the constant c0,
and critical damping is obtained for
c
the case             =1
2m 0

• Should the value of the damping c0 take on a negative
value, the oscillations will have increasingly large
amplitudes, see Figure 1.
c

2m 0

Effect of negative damping

c

2m 0

• Figure 1 Dynamic response for different negative values
of damping in the equation of motion
Variable damping term that could be negative

• Increasing amplitudes resulting from equation (1) would
be non-physical, representing energy generation in the
system.

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.
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:

d 2 y( t )          dy( t )
•      m      2
 c( t )          ky ( t )  g 0 ( t )   (2)
dt                 dt

Here:
• g0(t) is the “remaining” external, time-varying
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).
An example, c0=0

• Let us, now for simplicity consider a linear oscillator of
the form:
d 2 y( t )         dy(t ) 2
2
 a (t)         y( t )  b ( t )   (3)
dt                dt
• 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:
t
a(t )  sin( )        (4)
10
• The damping is slowly changing compared with a
timescale of t of order 1.

et al. (1950) we will choose:
b(t) = sin(kt)|sin(kt)|             (5)
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.
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;
U
such that: Ured =         20
fn D
The model of galloping for a one-degree of
freedom system.
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:
FL  1  DU rel CL
2
2

FD  1  DU rel CD
2
2

• By assuming α to be small so that y / U   and by
expanding in Mac Laurin series, the lift force and
equation of motion are


           CL           y

FL   DU CL  0  
1       2
 CD       
2

                    0 
D


       DU C y        
               DU 2
y  2 y  y                    y  y y  
2
CL   0

      4m y      0 
                2m
Equivalent damping

• The total damping ratio (including structural and flow
damping effects) can now be written:
 DU C y
T   y 
4m y      0

• For positive, C y  i.e. destabilizing, the total
damping may become negative and structural failure
could therefore be expected
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.
• 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.
Piggyback geometry and results

Amplitude of the response of a “piggyback” pipeline
geometry. D + ½ D plotted against Ured, Hang (2009).
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)
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
current) which, according to experiments is given
by the term per unit length of the structure; see
for example Sarpkaya and Issacsson (1981):

d 2 y( t )    dy( t )             1
m            c          ky ( t )  C'd DU U
dt 2         dt                 2
Inclusion of added mass and relative motion
• We should include the added mass term                                      d 2 u (t )
C m D     2

4    dt 2
• 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):

d 2y (t ) dy(t )
m 2 c               ky(t ) 
dt            dt
 2  d 2 u (t ) d 2 y(t )   2 d 2 u (t )
  C m  1 D  2  2    D
1          du(t ) dy(t ) du(t ) dy(t )
C d D{              }      
2           dt      dt dt        dt                 4  dt
              dt      4     dt 2
Effects of the relative velocity term

• We could consider the nonlinear relative velocity drag
forcing term:
1         du(t ) dy(t ) du(t ) dy( t )
C d D{             }      
2          dt     dt     dt     dt

• 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.
6) Study of damping function

• we have recently studied the effect of the relative
importance of the terms contributing to the damping
•
• We take c  c0  c1 sin(d  d ) with c0  c1
• We, furthermore, define Amd = c1-c0 , 
am p  Am d / c 0
see Figure 7

• Figure 7 The damping function studied
Influence of the damping amplitude (here denoted
L=0.5)on the horizontal displace- ment for Td/TL = 2.0.
Larger displacements observed
7) Splash zone lifting analysis

Structure's vertical velocity at the top of suction can and relative velocity between water
partcle and structure Vs Simulation time
2
1.5
1
Velocity (m/s)

0.5
0
-0.5
-1
-1.5
-2
-2.5
400                      450                                       500                             550                                       600
Time (s)

relative velocity between structure (at suction can top) and water particle   structure velocity (at suction can top)

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
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
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

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