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Experimental and numerical study of forces during oblique impact
Journal
of Sound
and Vibration
EXPERIMENTAL
(1988)
125(3),
AND
403-412
NUMERICAL
DURING
OBLIQUE
A. D. LEWISAND Department of Mechanical Engineering,
STUDY
OF FORCES
IMPACT
R. J. ROGERS
University of New Brunswick, Fredericton, New Brunswick, Canada, E3B 5A3
(Received 4 June 1987, and in revised form 2 November 1987)
The forces transmitted during the oblique impact of a sphere with a flat surface have been studied by using experimental and numerical techniques. Trends in the friction coefficient as a function of time, impact velocity and impact angle were observed. During contact the friction coefficient was observed to rise to a “plateau” value and then decline to zero as contact is lost. The plateau value was found to be independent of impact velocity and a bilinear function of impact angle. A model is proposed to predict the instantaneous tangential force given the normal force. Such a model will be useful for the simulation of heat exchanger tubes obliquely impacting against support plates.
1. INTRODUCTION
In any thermal power plant there may be up to several kilometres of heat exchanger tubes. Each tube is supported by baffle plates which have support holes larger than the tube outside diameter in order to allow for assembly and thermal expansion. Flow-induced vibration causes the tubes to impact and rub against the support plates. As a result, the tubes wear and must be repaired at great expense and loss of production. Several researchers [l-4] have developed finite element computer simulation techniques to predict the motion and contact forces of multi-span heat exchanger tubes. Models of tube wear are based on a work rate given by the product of the normal contact force and the sliding distance [3-51. From repeated oblique impact studies, Ko [6] has shown that the peak shear force varies with the impact angle and that wear rate is closely correlated with the shear force. The first of these findings implies that the friction coefficient (defined as the ratio of the shear and normal forces) depends on the impact angle. In the simulations done to date, however, a constant value for the friction coefficient has been assumed. If the friction coefficient could be predicted more realistically, then more accurate numerical results could be obtained. An accurate friction model would also facilitate the development of a wear rate model based on shear force. The bending vibration of a heat exchanger tube is such that it rarely makes contact across the whole width of a support plate, but rather has a small contact area near the edge of the plate. The time history of the shape of the contact area depends on the angular orientation (i.e., slope) of the tube and its angular velocity in both the x, y-plane and the x, z-plane (where the x axis passes through the centres of consecutive support plate holes). To simplify the problem and eliminate the effect of the alignment of the contacting surfaces, a sphere impacting the centre of a small plate was used. The plate was supported by a piezoelectric force transducer which had separate crystal wafers for the normal and shear directions. 403 0022-460X/88/180403+
10 $03.00/O
0 1988 Academic
Press Limited
404
A. D. LEWIS AND
R. J. ROGERS
Work of qualitative relevance to this paper is the study of elastic compliance during oblique impact by Maw, Barber and Fawcett [7], who have defined conditions of stick and slip that may exist on the contact face. The work is useful as the range of impact angles is divided into three regions based on two dimensionless parameters. The first region corresponds to small impact angles (measured away from normal). In this case the faces stick as soon as impact begins but when the maximum compression is passed, the outer areas begin to slip until the entire face is in “gross slip”. The second region corresponds to intermediate impact angles. In this region, the impact begins and ends with a condition of gross slip. In the middle of the impact, the impact face sticks first only in the centre and then over an expanding annular region until the entire area is sticking. The process then reverses until the impact face is in gross slip again. The third region represents large impact angles and a condition of gross slip exists over the entire impact face for the duration of the impact. In the work presented here, normal and shear force time histories during oblique elastic impact were measured for a range of impact angles and impact velocities. The instantaneous ratio of the shear and normal forces was plotted and a model for the friction coefficient was developed. The new model was introduced into the VIBIC [l] code and tested by simulating the experimental apparatus. Both the experimental and numerical results are presented in this paper and a comparison between the old and new friction models is given. 2. EXPERIMENTAL
RESULTS
Figure 1 shows a photograph of the apparatus constructed to produce the impact of a steel ball with a steel plate. The impact forces were measured in the normal and shear directions by using a biaxial piezoelectric force transducer. The ball was suspended from a rod of length 1.8 m which provided the ball with relatively low impact velocities (up to 50 mm/s). The rod was prevented from having rigid body twisting motion by a double knife-edge support assembly.
Figure
1. Photograph
of experimental
apparatus.
FORCES
DURING
OBLIQUE
IMPACT
405
The ball was released from various points on a semi-circular track to allow for various impact angles. The impact velocity was changed by releasing the ball at different distances from the impact surface. The velocity of the ball was measured with a strobe light and a Polaroid camera. Enough photographs were taken to enable the determination of the impact velocity as a function of the distance of the ball’s release from the impact point. The force transducer signals were amplified and then stored on a digital oscilloscope. The sampling rate of the scope was 5 microseconds. This relatively fast rate allowed for accurate depiction of the force traces since their duration was typically 250-400 microseconds. The force data was then transferred onto a computer disk from which it could be retrieved for processing. Each impact actually produced multiple impacts, but only the first was used for analysis. This enabled data to be stored for a range of 18 impact angles (up to 85 degrees measured away from normal) and four velocities (up to 50 mm/s). Typical normal and shear force traces are shown in Figures 2(a) and 2(b), respectively. It can be seen that the shear force contains a small amount of sinusoidal noise between the impacts. The noise is thought to be the result of a shear resonance of the force transducer which was excited by the short duration impact. The level of noise was reduced by making the ball holder heavier which resulted in longer impact durations. It is unlikely that the noise was due to torsional oscillation of the rod since the same distinctive noise frequency could be excited by tapping the side of the plate. The small amount of noise was completely removed during post-processing. Since the shear force signal had to remain undisturbed
60 c
-2.51
Figure 2. Experimental (a) normal; (b) shear.
0
I 500
force time history
I I I 1000 1500 2000 Time (micro-seconds)
for impact
I 2500
angle of 45 degrees
I 3000
and impact
velocity
of 45.6 mm/s:
406
A. D. LEWIS
AND
R. J. ROGERS
as much as possible, a filtering process based on an FFT algorithm [8] was developed. The forward transform of the time signal’was computed, from which the spectrum could be plotted. The noise produced a distinct peak which was removed from the spectrum by using the interactive capabilities of the Hewlett-Packard 9816 computer. This modified spectrum was then inverse transformed to produce the filtered time signal. For each experiment, a plot of the friction coefficient as a function of time was obtained. A typical plot is shown in Figure 3. It can be noticed that the friction coefficient starts and ends at zero and rises to an approximate “plateau” in the middle. The mean plateau value was visually estimated for each case. Figure 4 shows 72 plateau values plotted as a function of impact angle for various approach velocities. The peak normal and shear forces are similarly plotted in Figures 5(a) and S(b), respectively. It is noticed that the shear force peaks at an impact angle of about 40 degrees.
0 0
Figure 3. Experimental of 33,6 mm/s.
friction
50
coefficient
100 150 Time (micro-seconds)
time history
for impact
200
250
angle of 45 degrees
and impact
velocity
velocity.
Velocity
0
Figure 4. Variation of friction coefficient (mm/s): 0, 9.3; +, 20.5; *, 33.6; W, 48.6.
plateau
value
with impact
angle
and impact
FORCES 100
I
,
I
DURING ,
I
,
OBLIQUE
I
,
I
,
407
IMPACT
,
,
I
,
1
,
I
(a)
so 8Ob
..m..
1 +++++ +*y i$** + .
70
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.
+++
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IO -
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. . . .
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*
1 10
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IIIIIIIIIII 20 30 40
of peak experimental
50
angle
.
+ + * :
OO:h _
.
.
+
t
++++
Impact
Figure 5. Variation Key as Figure 4.
.
??
d
000
??
* +
0
.
??
+++
Oooo
.
.
II
B
. *
60
??
70
Q 80
90
(degrees)
force with impact
angle and impact
velocity:
(a) normal;
(b) shear.
3. FRICTION MODEL From the experimental results it was possible to construct a model of the friction coefficient. Two factors had to be modelled to obtain an accurate representation of the experimental results.
“‘“I
Figure 6. Numerical 33.6 mm/s.
friction
coefficient
time history
for impact
angle of 45 degrees
and impact
velocity
of
408
A. D. LEWIS AND
R. J. ROGERS
The first factor was the plateau level of the friction coefficient during impact as a function of impact angle and velocity. In Figure 4 the plateau value of the friction coefficient is plotted as a function of impact angle for various impact velocities. The data points seem to follow a bilinear type of curve. By using least squares curve fitting, the equation was found to be o-00448 e
p=
for for
1
o-179
o< e<400 e>400
I ’
(1)
where p and 8 are the friction coefficient and the impact angle (measured away from the normal), respectively. The second part of the model was the reproduction of the friction coefficient time history. As was observed in Figure 3, the friction coefficient starts and ends at zero. The problem was one of producing a means of coding this effect on the computer. It was decided to make the friction coefficient a linear function of the velocity of the projectile for the increasing part of the time trace until the friction coefficient reached its plateau value. For the declining part of the friction coefficient trace, a linear function of the deflection of the impact surfaces was used. Figure 6 shows a numerical representation of the friction coefficient corresponding to the same test conditions as in Figure 3.
Aluminum knifeuniversal joint
Aluminum
Figure
7. Finite element
edge
tube
model of apparatus.
FORCES
DURING
OBLIQUE
4. NUMERICAL
409
IMPACT
RESULTS
Both to test the friction model and to provide some comparison with the experimental results, the experimental apparatus was modelled numerically. To accomplish this the three dimensional beam vibration program, VIBIC [I], was used. This program simulates the motion of a beam by calculating the modal frequencies given the material composition and geometry of the beam. The program also allows the application of a driving force to the beam. For this work the beam is allowed to move freely in its first (pendulum) mode to generate motion. For the beam used in this work, the first mode period was measured to be l-96 seconds. The primary feature of VIBIC that made.it useful for the purpose intended is its ability to simulate impact of the beam with clearance supports. The program required that the beam be divided into beam finite elements along its length. The model chosen to represent the experimental apparatus has 17 elements and is shown in Figure 7. Two elements represent the hinge and the fastening of the hinge to the aluminum tube. The tube itself is divided into 10 elements. At the bottom end of the tube there are two elements comprising the fastening of the ball holder to the tube, one being steel and the other being aluminum. The ball holder is represented by two elements. The remaining element is a “massless” element connecting the concentrated inertia of the ball to the ball holder. The concentrated inertia is given the mass and rotational inertia values of the 25.4 mm diameter ball used in the experiment. 200,
II
I
I
I
I
I
I
,I000
I
(al
900
100 IO
E x ; E P
20
30
40
Contact
0 5oooox
,
,
,
,
50
60
stiffness,
,
I
70
8090
k (kN/m)
,
I
,
,
I
,
1000
(b)x\
; I: 5
1
-\
5000
$d
0
‘; 2
x\X
-800
500-
z u”
50-
-
600
-
400
Ax.x ‘X
5-
‘X
‘X
0.5-
‘X ‘X 200
0.05
-
‘X\
Y
0.005 0
0.2
0.4
0.6 Deflectlon
0.8 exponent
I.0
I.2
1.4
0
n
Figure 8. Sensitivity of peak normal force and impact duration for normal impact 45.6 mm/s: (a) for n = 0.5 and various contact stiffness k; (b) for k = 34 500 N/m exponents. X, Normal force; 0, contact duration.
and impact velocity of and various deflection
410
A.
D.
LEWIS
AND
R. J. ROGERS
When the experimental results were to be simulated, there turned out to be several program parameters that had to be tuned. These centered mainly around the normal force developed upon impact. This force is represented by P= kan,
(2)
where compression of the surfaces is represented by (Y.The stiffness k is the contact stiffness and thus is given by the reciprocal of the sum of the reciprocals of the stiffnesses for the ball and the transducer mounting. This number was calculated by using standard stiffness formulas but the results appeared to be sensitive to changes in this value. The exponent in the equation was also subject to some tuning. It is generally accepted that for two spheres n is 3/2 and for two parallel cylinders n is 1. The sensitivity of the peak normal force and the contact time to k and n for normal impact is shown in Figures 8(a) and 8(b), respectively. The best agreement with the experimental results over the range of data was found with values of the contact stiffness of 34 500 N/m and the exponent as n =0*5. Initially, the torsional degrees of freedom were included in the model, however, their effect was negligible and so they were removed from the model for the final runs. It should also be mentioned that zero damping was assumed. Plots of the numerically generated peak normal and shear forces are shown in Figures 9 and 10 for the old and new friction models, respectively. The results are based on 16 modes in each plane and a time step of 11.4 microseconds. The new model produces results that provide a fairly close match with the experimental results. Both shear force 100
I
,
1
,
I
,
I
,
I
,
I
,
I
,
I
,
I
(a)
90
c
60. . .
70.* * 60 t
??
. .
??
**
??
* +
+
. ??
+
.. *
. .
*
??
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. i
I
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0
. ??
+++ 300
_
+ 0
0
O
+
??
*
.
+
??
0
+*
??
-
“0,‘; 0
(b)
I
1
I
IO
I
20
I,, 30 Impact
Figure 9. Variation of peak numerical (a) normal; (b) shear. Key as Figure 4.
I 40 angle
I
,
50
1
I 60
I
. 9c
(degrees)
force with impact angle and impact velocity
for old friction model:
FORCES 100
I
,
I
DURING
,
I
,
I
OBLIQUE
,
,
,
,
IMPACT
,
I
,
411
,
,
,
(a)
so z
,,i
g
70,
z 8 2 z
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+ +
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00 d
20-
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00
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0
IO-
+
r
o+0
0
1
201
0
IO
20
30
40
50
60
70
80
SO
Impact angle (degrees) Figure 10. Variation of peak numerical (a) normal; (b) shear. Key as Figure 4.
force with impact
angle and impact
velocity
for new friction
model:
results peak at impact angles of about 40 degrees. The new model provides a better match with the experimental results especially for low impact angles. The shear force for the old model drops to zero at low impact angles as it should, but not as fast as seen experimentally. This drop is controlled by specifying a tangential cut-off velocity below which the shear force is tapered to zero. It may be argued that the cut-off velocity could be chosen so that the old model predicts the same results as the new model. However, the prediction of this cut-off velocity boils down to modelling the effects seen in the new model. The work of Maw, Barber and Fawcett [7] provides some qualitative support for the results observed experimentally and with the new friction model. Of their contact conditions of stick and slip, the condition of stick should produce a larger shear force if all other factors are the same. Thus the impact angles that produce stick conditions for the longest time should give the largest shear forces. This corresponds to the intermediate impact angles and indeed the measured shear forces are the largest in the area of 40 degrees.
5. CONCLUSIONS
AND FUTURE
WORK
Detailed time history measurements of the normal and shear forces during oblique impact of a steel ball on a steel plate have shown that the coefficient of friction rises to a plateau value and then declines to zero as contact is lost. The plateau value has been found to be a linear function of impact angle up to values of about 40 degrees measured
412
A.D.LEWISAND
R.J.ROGERS
away from the normal direction. At larger angles the friction plateau is constant. The coefficient of friction has been found to be independent of approach velocity for values between 10 and 50 mm/s. A computer model of the friction coefficient has beem introduced into a program which simulates heat exchanger tube vibration. Simulations of the experimental apparatus give good agreement with the experimental results providing the force-deflection relationship for the normal case is tuned. Further work in this area would be to try non-linear functions to model the rising and declining of the instantaneous friction coefficient. Experiments where the contact occurs in a liquid medium would provide results more directly relevant to liquid heat exchanger vibration modelling. A new apparatus based on the techniques used in the present apparatus could be built for this purpose. Further work is also required to develop a theoretical basis for the contact stiffness and deflection exponent corresponding to the results found here.
REFERENCES 1. R. J. ROGERS and R. J. PICK 1976 NuclearEngineering and Design 36, 81-90. On the dynamic spatial response of a heat exchanger tube with intermittent baffle contacts. 2. R. G. SAUVE and W. W. TEPER 1987 Journal of Pressure Vessel Technology 109,70-79. Impact simulation of process equipment tubes and support plates-a numerical algorithm. and B. VILLARD 1986 Flow-Induced Vibration-1986, 147-160. 3. F. AXISA, J. ANTUNES Overview of numerical methods for predicting flow-induced vibration and wear of heat-exchanger tubes. (ASME Pressure Vessels and Piping Conference and Exhibition, Chicago.) 4. T. M. FRICK, T. E. SOBEK and J. R. REAVIS 1984 Symposium on Flow-Induced Vibrations 3, Vibration of Heat Exchangers, 149-161. Overview on the development and implementation of methodologies to compute vibration and wear of steam generator tubes. (ASME Winter Annual Meeting, New Orleans.) 5. P. J. HOFMANN, T. SCHETTLER and D. A. STEININGER 1986 ASME 86-PVP-2. Pressurized water reactor steam generator tube fretting and fatigue wear considerations. (ASME Pressure Vessels and Piping Conference, Chicago.) 6. P. L. KO 1985 Wear 106,261-281. The significance of shear and normal force components on tube wear due to fretting and periodic impacting. 1976 Wear 38, 101-114. The oblique impact of 7. N. MAW, J. R. BARBER and J. N. FAWCE~ elastic spheres. 8. 1979 Programs for Digital Signal Processing, 1.2.7-1.2.18. See Chapter 1: Discrete Fourier Transform Programs New York: IEEE Press.
of Sound
and Vibration
EXPERIMENTAL
(1988)
125(3),
AND
403-412
NUMERICAL
DURING
OBLIQUE
A. D. LEWISAND Department of Mechanical Engineering,
STUDY
OF FORCES
IMPACT
R. J. ROGERS
University of New Brunswick, Fredericton, New Brunswick, Canada, E3B 5A3
(Received 4 June 1987, and in revised form 2 November 1987)
The forces transmitted during the oblique impact of a sphere with a flat surface have been studied by using experimental and numerical techniques. Trends in the friction coefficient as a function of time, impact velocity and impact angle were observed. During contact the friction coefficient was observed to rise to a “plateau” value and then decline to zero as contact is lost. The plateau value was found to be independent of impact velocity and a bilinear function of impact angle. A model is proposed to predict the instantaneous tangential force given the normal force. Such a model will be useful for the simulation of heat exchanger tubes obliquely impacting against support plates.
1. INTRODUCTION
In any thermal power plant there may be up to several kilometres of heat exchanger tubes. Each tube is supported by baffle plates which have support holes larger than the tube outside diameter in order to allow for assembly and thermal expansion. Flow-induced vibration causes the tubes to impact and rub against the support plates. As a result, the tubes wear and must be repaired at great expense and loss of production. Several researchers [l-4] have developed finite element computer simulation techniques to predict the motion and contact forces of multi-span heat exchanger tubes. Models of tube wear are based on a work rate given by the product of the normal contact force and the sliding distance [3-51. From repeated oblique impact studies, Ko [6] has shown that the peak shear force varies with the impact angle and that wear rate is closely correlated with the shear force. The first of these findings implies that the friction coefficient (defined as the ratio of the shear and normal forces) depends on the impact angle. In the simulations done to date, however, a constant value for the friction coefficient has been assumed. If the friction coefficient could be predicted more realistically, then more accurate numerical results could be obtained. An accurate friction model would also facilitate the development of a wear rate model based on shear force. The bending vibration of a heat exchanger tube is such that it rarely makes contact across the whole width of a support plate, but rather has a small contact area near the edge of the plate. The time history of the shape of the contact area depends on the angular orientation (i.e., slope) of the tube and its angular velocity in both the x, y-plane and the x, z-plane (where the x axis passes through the centres of consecutive support plate holes). To simplify the problem and eliminate the effect of the alignment of the contacting surfaces, a sphere impacting the centre of a small plate was used. The plate was supported by a piezoelectric force transducer which had separate crystal wafers for the normal and shear directions. 403 0022-460X/88/180403+
10 $03.00/O
0 1988 Academic
Press Limited
404
A. D. LEWIS AND
R. J. ROGERS
Work of qualitative relevance to this paper is the study of elastic compliance during oblique impact by Maw, Barber and Fawcett [7], who have defined conditions of stick and slip that may exist on the contact face. The work is useful as the range of impact angles is divided into three regions based on two dimensionless parameters. The first region corresponds to small impact angles (measured away from normal). In this case the faces stick as soon as impact begins but when the maximum compression is passed, the outer areas begin to slip until the entire face is in “gross slip”. The second region corresponds to intermediate impact angles. In this region, the impact begins and ends with a condition of gross slip. In the middle of the impact, the impact face sticks first only in the centre and then over an expanding annular region until the entire area is sticking. The process then reverses until the impact face is in gross slip again. The third region represents large impact angles and a condition of gross slip exists over the entire impact face for the duration of the impact. In the work presented here, normal and shear force time histories during oblique elastic impact were measured for a range of impact angles and impact velocities. The instantaneous ratio of the shear and normal forces was plotted and a model for the friction coefficient was developed. The new model was introduced into the VIBIC [l] code and tested by simulating the experimental apparatus. Both the experimental and numerical results are presented in this paper and a comparison between the old and new friction models is given. 2. EXPERIMENTAL
RESULTS
Figure 1 shows a photograph of the apparatus constructed to produce the impact of a steel ball with a steel plate. The impact forces were measured in the normal and shear directions by using a biaxial piezoelectric force transducer. The ball was suspended from a rod of length 1.8 m which provided the ball with relatively low impact velocities (up to 50 mm/s). The rod was prevented from having rigid body twisting motion by a double knife-edge support assembly.
Figure
1. Photograph
of experimental
apparatus.
FORCES
DURING
OBLIQUE
IMPACT
405
The ball was released from various points on a semi-circular track to allow for various impact angles. The impact velocity was changed by releasing the ball at different distances from the impact surface. The velocity of the ball was measured with a strobe light and a Polaroid camera. Enough photographs were taken to enable the determination of the impact velocity as a function of the distance of the ball’s release from the impact point. The force transducer signals were amplified and then stored on a digital oscilloscope. The sampling rate of the scope was 5 microseconds. This relatively fast rate allowed for accurate depiction of the force traces since their duration was typically 250-400 microseconds. The force data was then transferred onto a computer disk from which it could be retrieved for processing. Each impact actually produced multiple impacts, but only the first was used for analysis. This enabled data to be stored for a range of 18 impact angles (up to 85 degrees measured away from normal) and four velocities (up to 50 mm/s). Typical normal and shear force traces are shown in Figures 2(a) and 2(b), respectively. It can be seen that the shear force contains a small amount of sinusoidal noise between the impacts. The noise is thought to be the result of a shear resonance of the force transducer which was excited by the short duration impact. The level of noise was reduced by making the ball holder heavier which resulted in longer impact durations. It is unlikely that the noise was due to torsional oscillation of the rod since the same distinctive noise frequency could be excited by tapping the side of the plate. The small amount of noise was completely removed during post-processing. Since the shear force signal had to remain undisturbed
60 c
-2.51
Figure 2. Experimental (a) normal; (b) shear.
0
I 500
force time history
I I I 1000 1500 2000 Time (micro-seconds)
for impact
I 2500
angle of 45 degrees
I 3000
and impact
velocity
of 45.6 mm/s:
406
A. D. LEWIS
AND
R. J. ROGERS
as much as possible, a filtering process based on an FFT algorithm [8] was developed. The forward transform of the time signal’was computed, from which the spectrum could be plotted. The noise produced a distinct peak which was removed from the spectrum by using the interactive capabilities of the Hewlett-Packard 9816 computer. This modified spectrum was then inverse transformed to produce the filtered time signal. For each experiment, a plot of the friction coefficient as a function of time was obtained. A typical plot is shown in Figure 3. It can be noticed that the friction coefficient starts and ends at zero and rises to an approximate “plateau” in the middle. The mean plateau value was visually estimated for each case. Figure 4 shows 72 plateau values plotted as a function of impact angle for various approach velocities. The peak normal and shear forces are similarly plotted in Figures 5(a) and S(b), respectively. It is noticed that the shear force peaks at an impact angle of about 40 degrees.
0 0
Figure 3. Experimental of 33,6 mm/s.
friction
50
coefficient
100 150 Time (micro-seconds)
time history
for impact
200
250
angle of 45 degrees
and impact
velocity
velocity.
Velocity
0
Figure 4. Variation of friction coefficient (mm/s): 0, 9.3; +, 20.5; *, 33.6; W, 48.6.
plateau
value
with impact
angle
and impact
FORCES 100
I
,
I
DURING ,
I
,
OBLIQUE
I
,
I
,
407
IMPACT
,
,
I
,
1
,
I
(a)
so 8Ob
..m..
1 +++++ +*y i$** + .
70
.
??
60
??
*
??
.
??
.
+++
??
??
oooo
IO -
. . 5-
. . . .
?? ***c
*
+
*
1 10
+
IIIIIIIIIII 20 30 40
of peak experimental
50
angle
.
+ + * :
OO:h _
.
.
+
t
++++
Impact
Figure 5. Variation Key as Figure 4.
.
??
d
000
??
* +
0
.
??
+++
Oooo
.
.
II
B
. *
60
??
70
Q 80
90
(degrees)
force with impact
angle and impact
velocity:
(a) normal;
(b) shear.
3. FRICTION MODEL From the experimental results it was possible to construct a model of the friction coefficient. Two factors had to be modelled to obtain an accurate representation of the experimental results.
“‘“I
Figure 6. Numerical 33.6 mm/s.
friction
coefficient
time history
for impact
angle of 45 degrees
and impact
velocity
of
408
A. D. LEWIS AND
R. J. ROGERS
The first factor was the plateau level of the friction coefficient during impact as a function of impact angle and velocity. In Figure 4 the plateau value of the friction coefficient is plotted as a function of impact angle for various impact velocities. The data points seem to follow a bilinear type of curve. By using least squares curve fitting, the equation was found to be o-00448 e
p=
for for
1
o-179
o< e<400 e>400
I ’
(1)
where p and 8 are the friction coefficient and the impact angle (measured away from the normal), respectively. The second part of the model was the reproduction of the friction coefficient time history. As was observed in Figure 3, the friction coefficient starts and ends at zero. The problem was one of producing a means of coding this effect on the computer. It was decided to make the friction coefficient a linear function of the velocity of the projectile for the increasing part of the time trace until the friction coefficient reached its plateau value. For the declining part of the friction coefficient trace, a linear function of the deflection of the impact surfaces was used. Figure 6 shows a numerical representation of the friction coefficient corresponding to the same test conditions as in Figure 3.
Aluminum knifeuniversal joint
Aluminum
Figure
7. Finite element
edge
tube
model of apparatus.
FORCES
DURING
OBLIQUE
4. NUMERICAL
409
IMPACT
RESULTS
Both to test the friction model and to provide some comparison with the experimental results, the experimental apparatus was modelled numerically. To accomplish this the three dimensional beam vibration program, VIBIC [I], was used. This program simulates the motion of a beam by calculating the modal frequencies given the material composition and geometry of the beam. The program also allows the application of a driving force to the beam. For this work the beam is allowed to move freely in its first (pendulum) mode to generate motion. For the beam used in this work, the first mode period was measured to be l-96 seconds. The primary feature of VIBIC that made.it useful for the purpose intended is its ability to simulate impact of the beam with clearance supports. The program required that the beam be divided into beam finite elements along its length. The model chosen to represent the experimental apparatus has 17 elements and is shown in Figure 7. Two elements represent the hinge and the fastening of the hinge to the aluminum tube. The tube itself is divided into 10 elements. At the bottom end of the tube there are two elements comprising the fastening of the ball holder to the tube, one being steel and the other being aluminum. The ball holder is represented by two elements. The remaining element is a “massless” element connecting the concentrated inertia of the ball to the ball holder. The concentrated inertia is given the mass and rotational inertia values of the 25.4 mm diameter ball used in the experiment. 200,
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and impact velocity of and various deflection
410
A.
D.
LEWIS
AND
R. J. ROGERS
When the experimental results were to be simulated, there turned out to be several program parameters that had to be tuned. These centered mainly around the normal force developed upon impact. This force is represented by P= kan,
(2)
where compression of the surfaces is represented by (Y.The stiffness k is the contact stiffness and thus is given by the reciprocal of the sum of the reciprocals of the stiffnesses for the ball and the transducer mounting. This number was calculated by using standard stiffness formulas but the results appeared to be sensitive to changes in this value. The exponent in the equation was also subject to some tuning. It is generally accepted that for two spheres n is 3/2 and for two parallel cylinders n is 1. The sensitivity of the peak normal force and the contact time to k and n for normal impact is shown in Figures 8(a) and 8(b), respectively. The best agreement with the experimental results over the range of data was found with values of the contact stiffness of 34 500 N/m and the exponent as n =0*5. Initially, the torsional degrees of freedom were included in the model, however, their effect was negligible and so they were removed from the model for the final runs. It should also be mentioned that zero damping was assumed. Plots of the numerically generated peak normal and shear forces are shown in Figures 9 and 10 for the old and new friction models, respectively. The results are based on 16 modes in each plane and a time step of 11.4 microseconds. The new model produces results that provide a fairly close match with the experimental results. Both shear force 100
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force with impact
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results peak at impact angles of about 40 degrees. The new model provides a better match with the experimental results especially for low impact angles. The shear force for the old model drops to zero at low impact angles as it should, but not as fast as seen experimentally. This drop is controlled by specifying a tangential cut-off velocity below which the shear force is tapered to zero. It may be argued that the cut-off velocity could be chosen so that the old model predicts the same results as the new model. However, the prediction of this cut-off velocity boils down to modelling the effects seen in the new model. The work of Maw, Barber and Fawcett [7] provides some qualitative support for the results observed experimentally and with the new friction model. Of their contact conditions of stick and slip, the condition of stick should produce a larger shear force if all other factors are the same. Thus the impact angles that produce stick conditions for the longest time should give the largest shear forces. This corresponds to the intermediate impact angles and indeed the measured shear forces are the largest in the area of 40 degrees.
5. CONCLUSIONS
AND FUTURE
WORK
Detailed time history measurements of the normal and shear forces during oblique impact of a steel ball on a steel plate have shown that the coefficient of friction rises to a plateau value and then declines to zero as contact is lost. The plateau value has been found to be a linear function of impact angle up to values of about 40 degrees measured
412
A.D.LEWISAND
R.J.ROGERS
away from the normal direction. At larger angles the friction plateau is constant. The coefficient of friction has been found to be independent of approach velocity for values between 10 and 50 mm/s. A computer model of the friction coefficient has beem introduced into a program which simulates heat exchanger tube vibration. Simulations of the experimental apparatus give good agreement with the experimental results providing the force-deflection relationship for the normal case is tuned. Further work in this area would be to try non-linear functions to model the rising and declining of the instantaneous friction coefficient. Experiments where the contact occurs in a liquid medium would provide results more directly relevant to liquid heat exchanger vibration modelling. A new apparatus based on the techniques used in the present apparatus could be built for this purpose. Further work is also required to develop a theoretical basis for the contact stiffness and deflection exponent corresponding to the results found here.
REFERENCES 1. R. J. ROGERS and R. J. PICK 1976 NuclearEngineering and Design 36, 81-90. On the dynamic spatial response of a heat exchanger tube with intermittent baffle contacts. 2. R. G. SAUVE and W. W. TEPER 1987 Journal of Pressure Vessel Technology 109,70-79. Impact simulation of process equipment tubes and support plates-a numerical algorithm. and B. VILLARD 1986 Flow-Induced Vibration-1986, 147-160. 3. F. AXISA, J. ANTUNES Overview of numerical methods for predicting flow-induced vibration and wear of heat-exchanger tubes. (ASME Pressure Vessels and Piping Conference and Exhibition, Chicago.) 4. T. M. FRICK, T. E. SOBEK and J. R. REAVIS 1984 Symposium on Flow-Induced Vibrations 3, Vibration of Heat Exchangers, 149-161. Overview on the development and implementation of methodologies to compute vibration and wear of steam generator tubes. (ASME Winter Annual Meeting, New Orleans.) 5. P. J. HOFMANN, T. SCHETTLER and D. A. STEININGER 1986 ASME 86-PVP-2. Pressurized water reactor steam generator tube fretting and fatigue wear considerations. (ASME Pressure Vessels and Piping Conference, Chicago.) 6. P. L. KO 1985 Wear 106,261-281. The significance of shear and normal force components on tube wear due to fretting and periodic impacting. 1976 Wear 38, 101-114. The oblique impact of 7. N. MAW, J. R. BARBER and J. N. FAWCE~ elastic spheres. 8. 1979 Programs for Digital Signal Processing, 1.2.7-1.2.18. See Chapter 1: Discrete Fourier Transform Programs New York: IEEE Press.