Journal of Applied Mathematics and Physics, 2013, 1, 31-36
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.16007
Open Access JAMP
Vibration Analysis of Steel Strip in Continuous
Hot-Dip Galvanizing Process
Peng Li, Han Chen*
School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan, China
Received September 2013
ABSTRACT
In hot-dip galvanizing lines, undesirable vibration of the moving steel strip occurs due to the impingement of the
high-speed turbulence jet, which leads to non-uniformity of zinc thickness, as well as splash of the liquid zinc. In this
paper, the turbulent jet flow field is firstly numerically obtained using the CFD method. Then, the influence of the tur-
bulent jet flow on the steel strip is simplified as a harmonic force at the impingement line, and t he response of the steel
strip is obtained by means of finite element analysis for different strip lengths, thickness and pretension forces. Influ-
ences of impingement distance, air knife slot gap and jet pressure , on vibration of the steel strip are also analyzed. The
results will provide theoretical bas is for the reduction of steel strip vibration in continuous hot-dip galvanizing process.
Keywords: Vibration; Zinc Ripp le; Turbulence; LES (Large Eddy Simulation); Harmonic Force
1. Introduction
In the hot-dip galvanized process, the high-speed gas jet
impinges on the moving steel strip coated with liquid
zinc, and leads to the formation of a runback flow of the
liquid zinc down to the bath, to control the coating
thickness. In continuous hot-dip galvanization lines, vi-
bration of the moving steel strip has been observed [1],
which results in reduced coating quality, such as zinc
ripple. More seriously, splash of the liquid zinc could
occur when the steel strip vibrates strongly enough, and
coating quality will be even worse.
Focused on vibration and zinc ripples on the steel strip,
some scholars have carried out so me research work. Lin
et al. [2] analyzed the natural frequency and response of
the steel strip under different pretension forces, impulse
loads, and harmonic forces. Zhou et al. [3] proposed a
magnetic levitation technology using PID to control vi-
bration of the steel strip. Chen et al. [4] studied spindle
coupling clamp and roll grinding machines on chatter
marks by vibration measurement and mode analysis for
both mill and roll grinding machines. Li et al. [5] th eo-
retically studied the impact of strip thickness, rolling
speed, friction coefficient of the roll gap, strip tension
and rolling lubrication on self-excited and vertical vibra-
tion during cold rolling. Mi et al. [6] investigated the
variation characteristics of the mill and confirmed the
vibration order by analyzing and comparing the vibration
signal during the rolling cycle. Hardwicka [7] pr oposed a
simple, instrumented method for obtaining an objective
measurement of the actual depth of any marks in the roll
surface, and easily determined whether the mill or the
roll shop was responsible for chatter marks on the strip.
Li et al. [8] studied the online control theory in the gal-
vanization process in order to control the steel strip vi-
bration. Hong et al. [9] investigated active vibration co n-
trol of a tensioned, elastic, axially moving string. Albeit
the above work on the vibration of steel strips in the con-
tinuous hot-dip galvanization process, little work has
been done on the mechanism of this vibration, as well as
its characteristics and influenci ng parameters.
In this paper, the turbulent jet flow fiel d is firstly numeri-
cally obtained using the CFD method. Then, the influ-
ence of the turbulent jet flow on the steel strip is simpli-
fied as a harmonic force at the impingement line, and the
response of the steel strip under this harmonic line force
is obtained by means of finite element analysis. Mech an-
ism of the vibration of steel strips in the galvanization pro-
cess is revealed and its characteristics are analyzed. Effects
of impingement distance, air knife slot gap and jet pres-
sure on vibration of the steel strip are also investigated.
2. Modeling the Jet Flow
2.1. LES of the Turbulent Jet Flow
The large eddy simulation (LES) method is adopted to
numerically solve for the 3D unsteady turbulent jet flow
*Corresponding a uthor.

P. LI, H. CHEN
Open Access JAMP
32
field. The gas jet calculation domain and the boundary
conditions are shown in Figure 1. X is th e direction per-
pendicular to the steel strip surface, y is the steel strip
motion direction, and z is the steel strip width direction.
Pressure boundary conditions are applied on the inlet of
the air knife, and no slip boundary conditions are speci-
fied on the surfaces of strip and air knife. Boundary con-
ditions on the top and the bottom of the computational
domain are pressure outlet conditions. The distance of
the air knife to the strip is L = 10 mm, air knife slot gap d
= 1.3 mm, strip velocity Vp = 2.5 m/s, air knife inlet
pressure P0 = 70 kPa.
Figure 2 shows the velocity vector s of the ga s flow on
the x-y plane in the middle of the strip width. Entrain-
ment of the air around the air knife into the jet can be
observed and alternating pairs of vortices with opposed
rotation directions appear above and below the gas jet
centre plane. These vortices move towards the steel strip,
then upward or downward along the strip surface, until
being dissipated far away from the impingement point. In
this manner, the non-uniform and unsteady flow filed is
formed near the jet-strip impinging region.
Under the influence of the alternating vortex motions,
the flow vector field will also fluctuate periodically.
Figure 3 shows the evolution of velocity vectors in one
period. An observation point is placed on the center of
the strip surface to observe the change of pressure with
time (Figure 4). Average and standard deviation of the
pressure, Pm and P’, are also calculated.
2.2. Harmonic Line Force by the Jet
The influence of the turbulent jet flow on the steel strip is
simplified as a harmonic line force, whose period can be
obtained from Figure 4. The magnitude of the harmonic
force is obtained as follows.
Figure 1. Computational domain and boundary conditions
for the jet flow.
Figure 2. Velocity vectors on the x-y plane.
Figure 3. Evolution of the velocity vectors in one period.
Figure 4. Oscillation of static pressure at the observation
point.
The pressure distribution (Figure 5) on the strip sur-
face in the x-y plane can be described by a Gaussian dis-
tribution [10],
2
0.693
max
pp e
ξ −
= ×
(1)
where Pmax is the maximum pressure at the stagnation

P. LI, H. CHEN
Open Access JAMP
33
Figure 5. Ty pical impingement pressure pr ofile on the strip
surface on the x-y plan.
point; ξ = y/b, b is the distance between the jet axis and
the location of Pmax/2. Since the characteristic length of
this Gaussian distribution 10 mm is much smaller than
the strip length (typically 17.5 m), the jet impact on the
steel strip can be simplified as a line force.
The jet impact force of unit length, F, is calculated as
the integral of pressure P along the y direction as
2
0.693
y
Fp edy
ξ −
= ×
∫
(2)
The jet impact force, F’, is then th e integral of F along
the direction of strip width.
3. Finite Element Analysis of Steel Strip
Vibration
3.1. Model Parameters
Density of the galvanized steel strip is 7800 kg/m, its
elasticity modulu s is E = 2 .06 *1011 pa, Poisson’s ratio ν
= 0.3, the thickness ranges from 0.5 to 2.0 mm, the width
is 1.5 m, the length ranges from 15 to 25 m, and the dis-
tance of air knife to the bottom of strip is 1.5 m. Both
ends of the strip in the y direction are fixed. The numeri-
cal model of the steel strip is shown in Figure 6.
3.2. Vibration Analysis
3.2.1. Effect of Pretension Force
The natural frequency and the maximum displacement of
the steel strip vibration are studied, when the strip has a
length of 17.5 m, thicknes s of 1.5 mm, and the pretension
force takes the values of 0, 10 kN, 20 k N, 30 kN, 40 kN,
and 50 kN.
Figure 7 shows the vibration modal of the strip with
the pretension force being 40 kN. The 1st, 3rd, and 6th
order vibration modes are mainly vertical bending de-
formation, while 2nd, 4th, and 5th order vibration modes
Figure 6. Model of the steel strip.
Figure 7. Vibration mod e under the 40 kN prestress.

P. LI, H. CHEN
Open Access JAMP
34
are mainly tensional deformation.
Table 1 shows the first three order vibration modes of
the steel strip with different prestress. With an increase
of the pretension force, the natural frequency f increases,
and the maximal displacement d decreases. Therefore,
for steel strips with smaller preten sion forces, strip vibra-
tion is more severe under the impact of the gas jet.
3.2.2. Effect of Strip Length
The natural frequency and the maximum displacement of
the steel strip are studied, when th e pretension force is 40
kN, strip thickness is 1.5 mm, and the strip length takes
the values of 15 m, 17.5 m, 20 m, 22.5 m, and 25 m.
Table 2 shows the first three order vibration modes of
steel strip with different strip lengths. With an increase in
the strip length, the natural frequency and the maximal
displacement both decrease. Therefore, for shorter steel
strips, strip vibration is more severe under the impact of
the gas jet.
3.2.3. Effect of Strip Thickness
The natural frequency and the maximum displacement of
the steel strip are studied, when th e pretens ion force is 40
kN, the strip length is 17.5 m, and the strip thickness takes
the values of 0.5 mm, 1.0 mm, 1.5 mm, and 2.0 mm.
Table 3 shows the first three order vibration modes of
the steel strip with different strip thickness. With an in-
crease in strip thickness, the natural frequency and the
maximal displacement both decrease. Therefore, for
thinner steel strips, strip vibration is more severe under
the impact of the gas jet.
3.3. Harmonic Response Ana lysis
The effects of imp ing emen t d ist ance , air knife slot gap, and
jet pressure on the harmonic response of the steel strip
are investigated with a pretension force of 40 kN in the
Table 1. First three order vibration modes with different
pretension forces.
Pretension Force 1st order 2nd order 3rd order
0 f (Hz) 0.026 0.072 0.141
d (m) 0.09109 0.08704 0.08786
10 kN f (Hz) 0.689 0.713 1.382
d (m) 0.08130 0.14051 0.08173
20 kN f (Hz) 0.972 0.989 1.945
d (m) 0.08116 0.14030 0.08159
30 kN f (Hz) 1.189 1.203 2.378
d (m) 0.08111 0.14020 0.08153
40 kN f (Hz) 1.372 1.384 2.744
d (m) 0.08107 0.14014 0.08149
Table 2. First three order vibration modes with different
strip lengths.
Strip Length 1st order 2nd order 3rd order
15 m f (Hz) 1.602 1.616 3.204
d (m) 0.08766 0.15143 0.08827
17.5 m f (Hz) 1.372 1.384 2.744
d (m) 0.08107 0.14014 0.08149
20 m f (Hz) 1.199 1.210 2.399
d (m) 0.07578 0.13105 0.07608
22.5 m f (Hz) 1.066 1.075 2.131
d (m) 0.07140 0.12353 0.07163
25 m f (Hz) 0.95859 0.96706 1.91700
d (m) 0.06771 0.11716 0.06788
Table 3. First three order vibration modes with different
strip thic kness.
Strip Thickness 1st order 2nd order 3rd order
0.5 mm f (Hz) 2.365 2.366 2.669
d (m) 0.14012 0.24220 0.27952
1.0 mm f (Hz) 1.676 1.68 2.989
d (m) 0.09916 0.17143 0.19784
1.5 mm f (Hz) 1.372 1.384 2.744
d (m) 0.08107 0.14014 0.08149
2.0 mm f (Hz) 1.192 1.216 2.384
d (m) 0.07032 0.12154 0.07068
Table 4. Effects of impingement distance on steel strip
vibration.
S(mm) 8 9 10 11 12
T (10−5s) 6.43 6.65 6.69 6.95 7.08
Pm (kPa) 26.64 25.32 22.79 21.57 20.03
P’(kPa) 8.13 9.72 11.03 12.22 12.71
f’ = 1/T(Hz) 15552.1 15037.6 14947.7 14388.5 14124.3
F’(N) 189.92 227.06 257.66 285.45 296.91
Dmax (μm) 1.21 1.50 1.51 1.56 1.76
Table 5. Effects of jet pressure on the steel strip vibration.
P0(kPa) 20 30 40 50 60
T (10−5s) 8.92 7.25 6.65 6.04 4.52
Pm (kPa) 10.88 17.34 22.79 27.98 31.73
P’(kPa) 7.75 9.89 11.03 11.59 12.28
f’ = 1/T(Hz) 11210.8 13869.6 15037.6 16556.3 22123.9
F’(N) 181.04 231.03 257.66 270.74 286.86
Dmax (μm) 1.91 1.59 1.51 1.31 0.774

P. LI, H. CHEN
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35
Table 6. Effects of slot gap on the steel strip vibration.
H(mm) 0.8 0.9 1.0 1.1 1.2
T (10−5s) 6.02 6.19 6.25 6.44 6.96
Pm (kPa) 16.75 20.03 22.79 24.59 26.37
P’(kPa) 5.49 7.84 11.03 12.91 15.73
f’ = 1/T(Hz) 16611.3 16155.1 15037.6 14619.9 14367.8
F’(N) 128.25 183.14 257.66 301.58 399.79
Dmax (μm) 0.614 0.927 1.51 1.87 2.56
steel strip.
For different impingement distances, air knife slot gaps,
and jet pressures, the mean (in time) pressure Pm and its
standard deviation P’ on the surface of steel strip are listed
in Tables 4-6, as well as the pressure fluctuation period
T. With the aid of the Equations (1) and (2), the amplitude
of the harmonic force F’ can be obtained. Listed in Ta-
bles 4-6 is also maximum displacement of the strip steel.
The effects of impingement distance are shown in Ta-
ble 4. With the increase of the impingement distance S,
the jet fluctuation frequency f’ = 1/T increases, the mean
pressure decreases, standard deviation of pressure in-
creases, the amplitude of the harmonic force increases,
and the maximum displacement of strip vibration in-
creases. Therefore, the steel strip vibrates more seriously
when the air knife is farther away (within the range of 8 -
12 mm) from the steel strip.
The effects of jet pressure are shown in Table 5. With
the increase of the jet pressure P0, jet fluctuation fre-
quency increases, average pressure increases, the stan-
dard deviation of pressure increases, the amplitude of the
harmonic force increases, and the maximum displace-
ment of strip vibration decreases. Although the average
pressure and the standard deviation of pressure increase
with the increasing of the jet pressure, normalized pres-
sure fluctuation P’/Pm decreases. Therefore, the steel strip
vibrates more seriously when the jet pressure is lowered
(within the range of 20 - 60 kPa).
The effects of the air knife slot gap are shown in Table
6. With the increase of the slot gap H, jet fluctuation
frequency decreases, average pressure increases, standard
deviation of pressure increases, the amplitude of the
harmonic force increases, and the maximum displace-
ment of strip vibration increases. Therefore, the steel
strip vibrates more seriously when the air knife slot gap
is opened wider.
4. Conclusions
Vibration of the steel strip in the continuous hot-dip gal-
vanization process can be attributed to the turbulent
fluctuation of the jet flow field. In this paper, the turbu-
lent jet flow field is first numerically obtained using the
CFD method and the influence of the turbulent jet flow
on the steel strip is simplified as a harmonic line force.
Then, vibration modes of the steel strip with different
pretension forces, strip lengths and thicknesses are ana-
lyzed using the finite element method. Our results indi-
cate that both frequency and magnitude of steel strips
vibration increases with the decrease of the pretension
force, the strip length and thickness.
Effects of impingement distances, air knife slot gaps,
and jet pressure, on vibration of the steel strip are also
analyzed. It is demonstrated that the steel strip vibrates
more seriously when the distance of the air knife to the
steel strip increases, the jet pressure decreases, and the
air knife slot gap in creases.
Our results will help gain insights into the formation
mechanism of strip vibration in continuous hot-dip gal-
vanizing processes, and lay the oretical foundation for the
further reduction of zinc ripples and increase of galvani-
zation quality.
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