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AERODYNAMIC FORCES ON A SQUARE PRISM IN CONTROLLED
MOTION: A PHENOMENOLOGICAL DISCUSSION
Luigi Carassale+1, Lorenzo Banfi+2, Andrea Freda+3
+
Dept. of Civil, Chemical and Environmental Engineering, University of Genova, Italy
The generation of aerodynamic forces of oscillating bodies is the result of several physical
phenomena interacting each other. The classification of Vortex-Induced Excitation (VIE) and
Motion-Induced Excitation (MIE), together with their modeling, is only partially satisfactory
mostly due to the mentioned interactions. In this paper, we describe the apparently simple case
of a square prism oscillating within a smooth flow demonstrating the occurrence of VIE-MIE
interactions and multiple lock-in conditions. The discussion is largely based on the distinction
between synchronous and asynchronous force components and their separated analysis.
Keywords: Vortex Induced Excitation, Motion Induced Excitation, lock-in, Square
Prism
1. INTRODUCTION
The aerodynamic forces acting on an oscillating bluff body are produced by a combination of different
phenomena interacting with each other. Conventionally, it is common to distinguish between Vortex-Induced
Excitation (VIE) and Motion-Induced Excitation (MIE), but this separation is largely artificial and is mostly
invoked for modelling simplicity. A possible way to investigate this problem involves wind-tunnel experiments
on bodies oscillating in controlled motion and the processing of the measured forces by means of deterministic
or stochastic tools. Most of the contributions on this subject are related to the case of circular cylinders in water
and focus on relatively low Reynolds number and high reduced frequency1-3).
In this work we consider a square prism oscillating in cross-flow direction with a harmonic controlled
motion. The Reynolds number is relatively high, while the RF ranges from a very-low value up to a value close
to the Strouhal number. The data processing starts with a spectral analysis of the lift force used to recognize
several non-standard lock-in conditions4, 5). Then, the synchronous components of the lift force are estimated
and represented according to a Hammerstein model6) and compared to the classical aeroelastic derivatives7) and
the quasi-steady formulation8). Finally, we discuss, from a qualitative point of view, the spectral structure of the
aerodynamic forces in different lock-in conditions, including the case of non-integer lock-in, which, to our
knowledge, was never documented before.
2. EXPERIMENTAL SETUP
The experimental setup is constituted by a rigid prism with square cross section and sharp edges
constrained to oscillate in cross-flow direction (vertical). The motion is harmonic with frequency and amplitude
variable in a relatively large range. The forced motion is produced by a crankshaft system connected to an
electric motor driven through a closed-loop controller. A heavy flywheel is used to reduce possible highfrequency fluctuations of the angular velocity. The prism is mounted on linear-motion bearings and connected
to the flywheel by two long rods. The non-linearity of the crank-shaft mechanism generates a small secondorder harmonics of the motion, whose amplitude is below 0.2% of the first-harmonics amplitude for the range
of the parameters considered.
(1) Definitions and scaling
Let y(t) be the vertical position of the prism, which can be conveniently expressed through the complex
notation:
+2
+3
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+1
+1
+2
+3
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y  t   Y e i t
(1)
where Y and  are, respectively, the amplitude and the rotational frequency of the imposed motion and t is the
time. The lift force (per unit span length) acting on the prism is calculated from the pressure field as:
L  t    ny  s  p  s, t  ds
(2)
C
where p(s,t) is the pressure in the point with curvilinear abscissa s of the prism cross section, considered as
positive if directed outwards (suction) and ny is the projection along the motion direction of the unit vector
normal to the cross-section border. The pressure field is measured through an instrumented ring of 20 pressure
taps located at the mid-span of the model and connected to a pressure scanner mounted on board. Accordingly,
the integral in Eq. (2) is calculated for each time sample using the rectangle method.
The physical quantities involved in the experiment are scaled in traditional fashion, i.e.:

y t  by
t  bY ei t ;

L t  qbL t 
(3)
where b is the cross-section size, t the non-dimensional time, Y the non-dimensional motion amplitude,  the
non-dimensional motion frequency, referred to as motion Reduced Frequency (RF), q is the kinetic pressure:
t t
U
;
b
Y
Y
;
b

b
;
U
q
1
U 2
2
(4)
in which U is the free-stream velocity.
(2) Experimental parameters
The experimental results that are discussed herein refers to a prism with size b = 50 mm and length l=
500 mm, oriented at the angles of incidence α = 0°, 6°, 9°, 12° (positive angles are nose up), oscillating with
non-dimensional amplitude Y = 10%, 20%, 30%. The motion frequency range in the interval  / 2 = 1.25 
14.4 Hz. The wind velocity is U = 5.5m/s, corresponding to the Reynolds number Re = 1.8104; the freestream
turbulence intensity is about 1%. The motion RF is in the interval  / 2 = 0.011  0.13, corresponding to the
reduced velocity range 7.7  88.6.
The pressure signals are sampled at 500 Hz and synchronized with the instantaneous position of the
prism through a tachometric signal obtained by a laser probe.
During the measurements, the motion frequency is changed according to a stepped sweep. The length of
the constant-frequency steps is 30 s.
3. SPECTRAL STRUCTURE OF THE LIFT FORCE
Considering an experimental condition in which the motion frequency  and amplitude Y are constant,
the lift force may be idealized as a sum of three contributions: the static component L(0), the synchronous
component L(s) and the asynchronous component L(a)
L t  
L0  L s t   La t 
(5)
The synchronous component is periodic with the same period of the motion and is assumed to be
deterministically related to it by a possibly non-linear transformation, while, the asynchronous component may
be idealized as a zero-mean stationary or cycle-stationary random process whose characteristics are determined
by the motion frequency and amplitude. From a different point of view, it can be observed that the synchronous
component has a discrete spectrum comprising harmonics whose frequency is an integer multiple of the motion
frequency, while, the asynchronous component usually has a continuous spectrum with the possible presence of
pure tones whose frequency is not an integer multiple of the motion frequency.
This section describes, from a qualitative point of view, the spectral structure of the lift force identifying
synchronous and asynchronous force components for the case of Y = 30% and  = 0°, 6°, 9°, 12°. The discussion
is based on the Power Spectral Density function (PSD) estimated for each value of the motion frequency  and
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represented as color maps in Figure 1. The frequency reported in the vertical axes is scaled in the same fashion of
the motion frequency, i.e.  = b/U, while the color scale represents the logarithm of the PSD amplitude. The
maps show several spectral details of the lift force that have been preliminary described in9). In particular, the most
evident features are: 1) the frequency band of the Karman Vortex Shedding (KVS), 2) the synchronous components
determined by the motion frequency and its super-harmonics (straight lines directed upwards), 3) asynchronous
components at the sum or difference of the KVS frequency and motion frequency.
For α = 0°, in the very low RF range, the Strouhal number St of the oscillating prism decreases from its
static value St* until  /2  ≃ 0.04; then St increases progressively as  increases, and its trace becomes parallel
to the motion frequency before disappearing at lock-in. The trace of KVS disappears at  /2  ≃ 0.09, while the
1st-order synchronous force component increases its intensity from  /2  ≃ 0.07.
For α = 6°, the trace of the KVS is interrupted at  /2  ≃ 0.065 and  /2  ≃ 0.045 where the secondary
and tertiary lock-in (i.e. synchronization of the wake with the second and third harmonics of the motion) take
place. In the low RF range, before the tertiary lock-in, St increases linearly with  . Between the tertiary and
secondary lock-in St is constant and the intensity of the VIE is lower than in the low RF range. The passage
through the secondary lock-in produces a reduction of St and a further reduction of VIE intensity. Beside the
trace of KVS, the line with non-dimensional frequency St   is clearly visible.
For α = 9°, most of the spectral features are common to the case α = 6°, but two relevant differences can
be noted. First, KVS is present, even if weak, up to  /2  ≃ 0.11 and deviates upwards just before disappearing
(likewise for α = 0°). Second, it is clearly visible a lock-in phase at  /2  ≃ 0.09, corresponding to St = 1.5  .
At the same RF, it is also visible the harmonics with frequency 0.5  . In this case, it is not possible to speak
about vortex-motion synchronization, since St and  are not integer multiples. We refer to this condition as
non-integer lock-in.
Figure 1. PSD of the lift force for Y/b = 30%. Colors are in logarithmic scale.
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For α = 12°, in the low RF range St decreases until the secondary lock-in, than it increases becoming
roughly parallel to the motion line before disappearing at  /2  ≃ 0.11. The tertiary lock-in is very weak, while
the lock-in with 1.5  is well visible like for α = 9°. Besides, it can be observed that just below  /2  = 0.12,
i.e. where KVS disappears, the 1st-order synchronous component loses most of its intensity within a quite small
RF range.
4. SYNCRONOUS COMPONENTS OF THE LIFT FORCES
The spectral analysis of the lift force showed the simultaneous presence of both synchronous and
asynchronous components. In this section we focus on the synchronous force components addressing their
estimation and modeling.
(1) Estimation
The synchronous component of the lift force is periodic by definition, thus it can be expanded into the
Fourier series:

L s   t    Lr ei rt
(6)
r 1
where Lr are the Fourier coefficients depending on the motion frequency and amplitude. In principle, the Fourier
coefficients can be calculated from the time history of L(s) by the orthogonal projection:
Lr 
T
1 s
L  t  e i rt dt
T 0
(7)
where T  2 / . In practice, Eq. (7) cannot be directly used for the estimation of Lr, as L(s) is not accessible,
being additively combined with L(a). On the other hand, since L(a) is random and zero-mean, Eq. (7) can be
rewritten in the form:
1 T

Lr  E   L t  e i rt dt 
T 0

(8)
where E[] represents the expectation operator, through which L(a) is cancelled due to the zero-mean assumption.
Practically, the expectation is estimated, invoking ergodicity, by averaging the projection of the lift force on the
harmonics of the motion for all the motion periods available in the data, i.e.:
Lr
kT
1 NT
 L t  e i rt dt
nT T k 1  k 1T
(9)
where nT is the number of complete motion periods.
(2) Modeling
The synchronous lift force is related to the body motion by a non-linear transformation depending on
the motion amplitude and frequency. The dependency on the frequency, in particular, reflects the presence of
memory in the transformation, which is the basic feature of unsteady forces. A relatively wide class of nonlinear systems with memory can be represented by a Hammerstein model, which is the parallel combination of
a set of linear systems with memory preceded by non-linear memoryless systems. Accordingly, we assume that
the synchronous non-dimensional lift force L( s )  L( s ) /qb is related to the non-dimensional displacement y by
the model:

L s  t 


  g  t    y    d
r 1 
r
r
(10)
where gr are kernel functions to be identified. Substituting Eq. (3) into Eq. (10) yields:
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
L s  t 



  g t  Y
r 1 
r

 Y r ei rt
r 1


r 1
r
ei r  d

 g   e
r
 i r 
d
(11)

Y G  r e
r
r
i r t
in which Gr are the Fourier transforms of the kernels gr and are, therefore, the Frequency Response Functions
(FRF) of the dynamical systems mapping the integer powers of the motion into the force. The comparison
between the last of Eqs. (11) and Eq. (6) provides the FRFs of the Hammerstein model as:
G
r r
1
Lr , Y
qbY r


(12)
in which the Fourier coefficients depend on the motion frequency and amplitude and are estimated through Eq.
(9) for the range of parameters explored during the test.
(3) Identity with flutter derivative notation
In bridge aerodynamics, self-excited forces are traditionally represented through the flutter derivatives.
Accordingly, for a pure-plunge motion, the lift force is given as:
H* y H* y

L  t  qb  1*  *24 
U U U b 
(13)
where U* 1 / is the reduced velocity, while H1* and H4* are the flutter derivatives, which are estimated
experimentally and depends on the RF  . Substituting Eq. (1) into Eq. (13) and taking into account the scaling
(4), the lift force remains expressed as:
L  t  
qY 2 i H1*  H 4*  ei t
(14)
The comparison with Eqs. (6) and (12) suggests that the 1st-order (r = 1) Hammerstein FRF is related to
the flutter derivatives H1* and H4* as:
G1    2 i H1*    H 4*   
(15)
(4) Identity with quasi-steady modeling
When the motion frequency is very low compared to the characteristic frequency of the wake, i.e. if
 St , the motion-excited forces may be modeled invoking the quasi-steady assumption. Accordingly, the lift
force is given by the expression:
y
(16)
LQS  t  
qb  CD  CL 
U
where CD and CL are, respectively, the static drag coefficient and the slope of the static lift coefficient estimated
for the steady angle of incidence of the body.
Besides, the effect of the added mass may be modelled as proportional to the force necessary to
accelerate a volume of fluid equal to the body volume, i.e.:
L AM  t   b2 Ay
(17)
where A is the added-mass coefficient. Substituting Eq. (1) into Eqs. (16)-(17) and taking into account the
scaling (4), the lift force becomes:
L  t
 qY  i   CD  CL   22 A ei t
(18)
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The comparison with Eqs. (6) and (18) suggests that, in the quasi-steady limit, the 1st-order Hammerstein
FRF is related to the static aerodynamic coefficients and to the added-mass coefficient as:
G1    i   CD  CL   2 2 A
(19)
(5) Stability analysis
The 1st-order synchronous forces have a fundamental role in stability analysis of aeroelastic systems.
When the body is supported by visco-elastic devices, indeed, the imaginary part of G1 can be interpreted as a
frequency-variant aerodynamic damping, while its real part may be considered as an aerodynamic stiffness. For
pure-plunge motion, the effect of the aerodynamic stiffness is usually negligible10), while a negative value of
the aerodynamic damping can easily lead to galloping instability. In this case, the stability analysis is carried
out by studying the sign of the galloping coefficient ag = -(CD+CL) or of the flutter derivative H1*.
Figure 2 shows the imaginary and real part of the 1st-order Hammerstein FRF G1. As proposed by11)the
imaginary part is divided by  , while the real part is divided by  2 , so that the former can be interpreted as a
frequency-variant galloping coefficient and the latter as a frequency-variant added mass coefficient. In the plots
of the imaginary part, the galloping coefficient is reported by a black dashed line.
For the case =0°, Im(G1 )/ is almost constant up to the primary lock-in with a value that is up to 70%
greater (more unstable) than the quasi-steady prediction. Also Re(G1 )/2 is rather constant until the primary
lock-in, with the exception of the very-low frequency range for which the estimation may be inaccurate due to
the very low acceleration level.
The case =6° is characterized by a galloping coefficient that is very close to zero due to the equal
opposite contribution of CD and CL. The function Im(G1 )/ attains this value for very low RF, while it becomes
positive (unstable) with values similar to the case =0°, even for relatively low RF. Crossing the secondary
lock-in, Im(G1 )/ has a jump downwards, becoming negative for Y =30%. Re(G1 )/2 is similar to the case
=0°, with the exception of the low-RF range (which may be not significant), a discontinuity at the secondary
lock-in and a smoother variation at the primary lock-in.
The case =9° is qualitatively similar to =6°. The function Im(G1 )/ seems to tend to ag for   0 ,
however, the quasi-steady limit has not been reached in the investigated conditions.
The case =12° is very close to the critical angle of incidence for which the flow reattaches on the lateral
face exposed to the flow12). The flow reattachment produces the change of sign of CL, making the galloping
coefficient to pass from positive (unstable) to negative (stable). In the figure, both the values of ag estimated
before and after the critical angle of incidence are reported. The function Im(G1 )/ tends to the post-critical
quasi-steady limit for   0 , even if for the steady prism at =12° the flow is still fully separated. Besides, it
remains negative up to the primary lock-in. The added mass coefficient is smaller than for the other angles
considered.
5. AERODYNAMIC FORCES AT LOCK-IN
In this section we describe the aerodynamic behavior of the prism at lock-in. The discussion is based on
the time histories of the forces acting on the lateral faces defined as:
Ft t  
1
 p  s,t  ds;
qb top
face
Fb t   
1
 p  s,t  ds
qb bottom
(20)
face
Ft and Fb being, respectively, the forces per unit span acting on the top and bottom faces of the prism. For =0°,
the sum of Ft and Fb provides the (non-dimensional) lift force; for the other considered angles of incidence it is
not rigorously true, but for small  the contribution of the windward and leeward faces is negligible. The
advantage of considering the two lateral faces separately is twofold. First, for =0°, this gives the chance to see
the effects of harmonic components that in the lift force are cancelled due to symmetry. Second, at incidence,
the pressure field acting on the two lateral faces has different characteristics that should be investigated
separately.
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(a)
(b)
(c)
(d)
Figure 2. Imaginary and real part of the 1 -order Hammerstein FRF for α = 0° (a), 6° (b), 9° (c), 12° (d). The
black lines represent the galloping coefficient ag = -(CD+CL)
st
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(a)
(b)
(c)
(d)
(e)
Figure 3. Time histories of the forces on the side faces of the prism at lock-in: α = 0°, primary lock-in (a);
α=9°, primary (b), secondary (c) and tertiary (d) lock-in; α=9°, lock-in at 2St/ = 1.5 (e).
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Figure 3 shows the time histories of Ft and Fb along with the trajectory of the motion y(t). The vertical
lines of the grid represent the time instants corresponding to the zero-level up-crossing of the displacement. To
simplify the interpretation, the time histories of the forces are low-pass filtered at the RF /2 = 0.5. All the
plots refers to the motion amplitude Y = 30%.
The primary lock-in for =0° (Fig. 3a) is reported mostly as a reference case. The forces on the lateral
faces are almost harmonic with a phase delay about /2 with respect to the motion. Beside these expected
characteristics, a number of relevant features can be observed. First, the forces acting on the top and bottom side
of the prism are rather different in terms of amplitude of fluctuation, while are practically equal as far as their
mean value is concerned. It should be emphasized that the experimental setup is nominally symmetric and that
the static lift coefficient is practically zero. Second, the fluctuation of Fb is not symmetric, having the suction
peaks (downwards) generally sharper than the overpressure peaks. This behavior is mostly determined by the
second harmonics of the motion. On the other hand, the fluctuation of Ft is symmetric, but with peaks that are
sharper than for a harmonic wave. This feature is mostly related to the presence of the 3rd harmonics of the
motion.
In the case =9° at primary lock-in (Fig. 3b) the forces Ft and Fb are significantly different both in terms
of amplitude and waveform. On the bottom face, which is the one exposed to the wind, a series of intense suction
peaks appear at the motion frequency with a phase delay about /2 (like for  = 0°). These peaks are followed
by shorter peaks appearing during the downward motion of the prism (when for  = 0° we see the overpressure
peak). These secondary peaks are probably produced by the interference between the separated shear layers and
the lateral face of the prism. From a spectral point of view, they reflects the presence of an intense 2nd-harmonics
of the motion. On the top side, the force is quite irregular. The main suction peaks generally appear with the
same timing as for =0°, but the 3rd harmonics of the motion is very evident and often is dominant.
For  = 9° at the secondary lock-in (Fig. 3c), both the forces Ft and Fb have two suction peaks per motion
period suggesting the shedding of vortexes at the frequency St = 2St 2 . Those vortexes come in pairs so
that two strong vortexes appearing about phase zero are followed by two weaker vortexes delayed about half of
the motion period. The different amplitude of those pairs of vortexes is due to an additive harmonic component
at the motion frequency. This is quite evident on the bottom face where the separated shear layers stay closer to
the body and, therefore, are more influenced by the body motion.
For  = 9° at the tertiary lock-in (Fig. 3d), the behavior is similar to the secondary lock-in, with the
obvious difference that the forces are dominated by the harmonics at the frequency 2St 3 . A pair of intense
vortexes are shed near phase zero, then two pairs of weaker vortexes follow. In addition, the variation of the
peak amplitude is more evident on the bottom face where the separated shear layers are more influenced by the
motion.
Figure 3e shows the non-integer lock-in observed for 2St  1.5 . The forces are quite irregular,
however it can be noted that the main suction peaks on the bottom face appear once per motion cycle. On the
top face the peaks appear more erratically, however it happens relatively often that three suction peaks occur
evenly distributed along two consecutive motion cycles (e.g. between the non-dimensional times 106 and 130
or between 154 and 177).
6. CLOSING REMARKS
The discussion of the experimental results concerning a square prism in controlled motion suggests some
observations whose relevance goes beyond the treated case study.
The distinction between synchronous and asynchronous forces seems to be more rational than the
traditional classification of vortex-induced and motion-induced actions, as it includes force components
generated by the non-linear interaction between VIE and MIE.
Synchronous forces can be represented by a Hammerstein model, which, at the first order, reproduces
the classical concept of flutter derivative and can be interpreted, in the limit for   0 , consistently with the
quasi-steady formulation. The interpretation of higher-order Hammerstein FRFs, which have not been reports
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herein due to space limitations, is still to be investigated, and may be related to higher-order terms of the quasisteady expansion or to other weakly non-linear models for the motion-excited forces13-15).
The study of the synchronous forces reviled the existence of numerous lock-in regions. Beside the wellknown primary lock-in (for 2St 
 ), a secondary and tertiary lock-in conditions are documented for the prism
with incidence. In these conditions, the forces acting on the lateral faces of the prism are weaker than at the
primary lock-in (30 to 50% less), but may be relevant in practical applications.
It has been discovered a non-integer lock-in condition related to the intermittent occurrence of three
vortexes in two consecutive motion cycles. This phenomenon may be regarded as a quantization effect produced
by the motion on the vortex shedding. Accordingly, the number of vortexes that are released during an integer
number (typically one) of consecutive motion cycles must be an integer.
The analysis of the symmetric configuration (=0°) showed the existence of some RF ranges in which
the unavoidable lack of symmetry of the experimental setup produces a substantial asymmetric pressure field.
The reason of this behavior is not clear yet and should be carefully investigated.
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