ABSTRACT The aim of this work is to model the vibrational behaviour of thin plates joined to a stiff orthogonal side plate using the technique of &apos;roll swaging&apos;. Swage joints are typically found in plate-type fuel assemblies for nuclear reactors. Since they are potentially liable to flow-induced vibrations, it is crucial to be able to predict their dynamic characteristics. It is shown that the contact between the plates resulting from the swage can be modelled assuming a perfect clamp of all the degrees of freedom but the rotational around the axis parallel to the swage. A modal analysis was performed on different specimens and the values of the first natural frequencies are used to find the equivalent torsional spring stiffness, by matching these frequencies with the results obtained from a finite element model (FEM)

David Wassink

Mauro Caresta

CiteSeerX

 Proceedings of 20
th
 International Congress on Acoustics, ICA 2010 
23-27 August 2010, Sydney, Australia 
 
ICA 2010 1 
Vibrations of roll swage jointed plates 
Mauro Caresta (1), David Wassink (2) 
(1) School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, NSW 2052, Australia              
(2) Australian Nuclear Science and Technology Organisation (ANSTO), Lucas Heights, NSW 2234, Australia 
 
PACS: 43.40.At; 43.40.Dx 
ABSTRACT 
The aim of this work is to model the vibrational behaviour of thin plates joined to a stiff orthogonal side plate using 
the technique of ‘roll swaging’. Swage joints are typically found in plate-type fuel assemblies for nuclear reactors. 
Since they are potentially liable to flow-induced vibrations, it is crucial to be able to predict their dynamic character-
istics. It is shown that the contact between the plates resulting from the swage can be modelled assuming a perfect 
clamp of all the degrees of freedom but the rotational around the axis parallel to the swage. A modal analysis was 
performed on different specimens and the values of the first natural frequencies are used to find the equivalent tor-
sional spring stiffness, by matching these frequencies with the results obtained from a finite element model (FEM).
INTRODUCTION 
Plate-type fuel assemblies are used in several research reac-
tors where a high neutron flux is desired. A plate-type as-
sembly consists of several thin plates containing a uranium 
mixture, clad with aluminium and mounted in a box-type 
assembly. They are potentially affected by structural insta-
bilities due to the interaction with the coolant flow [1-3]. 
Miller [4] used the wide beam theory to investigate the static 
instability of the plates. Other researchers modelled the plates 
using the thin plate theory assuming simply-supported 
boundary conditions [5] or fully-clamped edges [6]. Kim and 
Davis [7] improved the previous works assuming plates as 
laterally fixed but elastically restrained in rotation. The aim 
of this work is to give a theoretical and experimental justifi-
cation of the model used in Ref. [7]. In a typical fuel assem-
bly the plates are inserted into slots machined into the side 
walls of the fuel box. The clamping of the plates to the box is 
generally assured by a swage between adjacent plates. The 
swage is obtained by forcing a swage cutting wheel into the 
aluminium ridge between the slots. The shape of the cutting 
edge results in plastic deformation of the area surrounding 
the swage, and presses the material onto the plate, creating 
the clamp as shown in Figure 1.  
Figure 1. Schematic diagram of a typical swage joint. 
 
The nature of the clamping is crucial to predict the vibra-
tional behaviour of a fuel assembly. A perfect clamp com-
pletely constrains all six degrees of freedom at the edges of 
the fuel plates. In this work it is suggested that the swaging 
process results in a clamp that fixes all the degrees of free-
dom but the rotational around the axis parallel to the swage. 
For small rotations, as assumed by the linear vibrations the-
ory, the effect of the swage joint is shown to be a torsional 
spring whose stiffness is related to the quality of the swage. 
A good swage leads to a very high stiffness that approaches 
the ideal case of a perfect clamp, while a poor swage is likely 
to result in a lower stiffness value tending to the case of a 
simple support. The model used in this work is built accord-
ing to the data of some specimens that were made at the Aus-
tralian Nuclear Science and Technology Organisation 
(ANSTO) in order to show that the changing of some pa-
rameters in the swaging process results in a shift of the natu-
ral frequencies. The aim of this work is to present a theoreti-
cal explanation of the abovementioned experimental evi-
dence. The specimens, shown in Figure 2, were tested by 
clamping the bottom of the support in a vice and performing 
an impact hammer test to detect the natural frequencies by 
inspection of the Frequency Response Function (FRF). A 
laser vibrometer was used to measure the response close to 
the corner of the plate to maximise the visibility of all the 
modes. 
 
plate 
swage support plate 
slot 
23-27 August 2010, Sydney, Australia Proceedings of 20th International Congress on Acoustics, ICA 2010 
2 ICA 2010 
 
 
Figure 2.  A specimen used for the modal analysis tests. 
 
SWAGING PROCESS SIMULATION 
To understand the nature of the contact between the plate and 
the support, a simple two dimensional model is created to 
simulate the swaging process. It is solved using Nastran Im-
plicit Non-Linear (Solver 600) [8]. The swage wheel is mod-
elled as a rigid wedge and is moved toward the support. The 
material is Aluminium with density ρ = 2700 Kgm-3, Youngs 
modulus E = 69 MPa, Poisson ratio ν = 0.33 and perfect plas-
tic behaviour with Yield stress of 280 MPa. A more precise 
stress-strain curve should be used in order to improve the 
results. Figure 3 shows the deformation from the simulation. 
It can be seen that the deformed material presses the top of 
the plate to create the swage clamp. The different colours are 
related to the Von Mises stress. 
 
 
Figure 3. Swaging process simulation with Nastran. 
 
DYNAMIC BEHAVIOUR OF THE SWAGE 
Inspecting the results from the swaging simulation, a scheme 
is reproduced in Figure 4 to study the kinematics of the joint. 
It is reasonable to assume that after the swage is completed 
the plate can not translate along the y and z directions and it 
can not rotate around the y axis. In theory a translation in x 
direction and a rotation around z are also possible but it 
would not affect the dynamics of the plate in bending vibra-
tions. Only the in plane motion will be altered, but this hap-
pens at much higher frequency and is therefore considered 
constrained in this work. A small rotation θ in x direction is 
possible around the contact line passing by point C0 allowing 
a compression along the contact line L, as shown in Figure 4.  
 
Figure 4. Small rotation of the plate in the swage joint 
 
Considering a small clockwise rotation θ  around the point 
C0, the component of the displacement normal to the contact 
line, which coordinate is ξi∈[0,L], results in the compression 
of the material and then in a reaction force per unit rotation 
given by 
 
d (d ) cos
i i
f E A rε δ= ,   d (d )A w ξ=                                     (1) 
 
where ε  is the strain, A and w are the area and width of the 
contact respectively. Other parameters arise from the geome-
try of the swage and are given by 
2 2
0 0
2 cos
i i i
r r rξ ξ γ= + − ,
2
π
γ ε= + , 1tan ( / )
s s
h dε −=       (2) 
2 2 2
1 0cos
2
N i
i
N
r r
r l
ξ
δ −
 − +
=   
 
                 (3) 
 
 
 
C0 
θ 
L 
z 
y 
x 
23-27 August 2010, Sydney, Australia Proceedings of 20th International Congress on Acoustics, ICA 2010 
ICA 2010 3 
 
 
Figure 5. Kinematics of the swage joint. 
All the forces normal to the contact line are equivalent to a 
force of magnitude   
 
0
L
s i
F df
ξ
ξ
=
=
= ∫                    (4) 
 
and with a lever arm b respect to point C0 given by b = 
Hbsinε + 2/3L.  Hb is the height of the plate from point C0 to 
point C-0 shown in Figure 6. The global effect is then a mo-
ment to point 
0
C  given by Ms = Fsθb that can be rewritten as 
 
s s
M K θ= , 
s s
K F b=                  (5) 
 
s
K  is the equivalent torsional spring of the swage joint. The 
exact value has to be found by matching experimental and 
FEM results for the first natural frequency. 
 
 
 
 
 
 
Figure 6. Force resultant from a small otation of the plate. 
 
A similar model can be used to calculate the reaction moment 
when the plate is rotating anticlockwise. In this case the plate 
is likely to rotate around the point C-0. The moment is likely 
to have a different value leading to a non-linear spring char-
acteristic, with different stiffness for the positive and nega-
tive rotations. 
 
 
 
FEM MODEL OF THE JOINED PLATES 
 
An FEM model is built according to the dimension of the 
specimens. The plate and the support are meshed using 
QUADR plate elements. The support is clamped at the base 
to simulate the clamping of the real model to a vice. The 
plate is connected to the support using bush elements with 
adjustable torsional spring around the x axis. The first five 
mode shapes are reported in Figure 7 for a perfect clamp 
situation.  
 
  
  
   
 
Figure 7.  The first five mode shapes for a perfect clamped 
plate. 
 
 
1st 
2nd 
3rd 
4th  
5th  
C0 
Fs 
b 
df(L) 
df(0) C-0 
Hb 
rN    
C0 C0 
θ  
cos
i i
rθ δ  
i
rθ  
s
d  
hs 
i
ξ  
ri 
 
r0 
βi 
i
δ  
ri 
23-27 August 2010, Sydney, Australia Proceedings of 20th International Congress on Acoustics, ICA 2010 
4 ICA 2010 
In the case where a simple support was used, the first mode 
degenerated into a rigid body rotation (0 Hz) around the 
swage axis. The other mode shapes were practically the same 
except for a slightly bigger rotation at the connection with the 
support. Different values for the torsional spring were used to 
simulate the conditions between a perfect clamp (Ks → ∞) 
and a simple support (Ks = 0). The first five natural frequen-
cies are were normalised with respect to the perfect clamp 
case and are plotted versus the spring stiffness in Figure 8. 
Convergence to the perfect clamp case is achieved for a value 
of around Ks = 200 Nm/rad. Observing the slope of the 
curves it can be seen that the sensitivity first increases, reach-
ing a maximum at 10 Nm/rad for the first natural frequency, 
and at 100 Nm/rad for the fifth natural frequency.  The slope 
decreases again approaching the ideal clamped case. The 
lower order modes are more affected by the spring stiffness. 
 
10
−4
10
−2
10
0
10
2
0
0.2
0.4
0.6
0.8
1
N
o
rm
al
is
ed
 n
at
. 
fr
eq
u
en
cy
Spring stiffness [Nm/rad]
 
 
f
1
f
2
f
3
f
4
f
5
 
Figure 8. Variation of the natural frequencies with the spring 
stiffness (FEM results). 
 
Figure 9 shows the same data as in the previous figure but 
arranged with respect to the natural frequency order. A char-
acteristic stepwise variation between the two extreme situa-
tions of perfect clamp and simple support is observed. The 
presented arrangement of the results is useful to compare the 
frequencies with the experimental results.  
1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Natural frequency order
N
o
rm
al
is
ed
 n
at
. 
fr
eq
u
en
cy
 
 
increasing K (0−inf)
 
 
Figure 9. Normalised first five natural frequencies (FEM 
results). 
Figure 10 shows the first natural frequencies of some speci-
mens obtained by setting the roll swaging wheel at different 
heights with respect to the plates. It can be seen that the trend 
of the curves is similar to the FEM results presented in Fig-
ure 9. 
 
1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.2
0.4
0.6
0.8
1
Natural frequency order
N
o
rm
al
is
ed
 n
at
. 
fr
eq
u
en
cy
 
 
K=0 case (FEM)
 
Figure 10. Normalised first five natural frequencies (experi-
mental results). 
 
Figure 11 shows the results updating the value of the tor-
sional spring in the FE model to match the first natural fre-
quency of two experimental results. The maximum error for 
the other frequencies is around 3%. A better estimation of the 
spring stiffness would need an update considering the global 
effect on more resonances in a desiderate frequency range.  
1 1.5 2 2.5 3 3.5 4 4.5 5
0.4
0.5
0.6
0.7
0.8
0.9
1
Natural frequency order
N
o
rm
al
is
ed
 n
at
. 
fr
eq
u
en
cy
 
 
Exp result
K=0.65 Nm/rad (FEM)
K=4.8 Nm/rad (FEM)
Exp result
 
Figure 11. Matching of the natural frequencies by FEM up-
dating. 
 
 
CONCLUSIONS 
The work presented shows a first step to model the dynamic 
behaviour of swage joints and open a variety of issues that 
need to be studied in more depth. In particular a more precise 
model of the swaging process using the non-linear capabili-
ties of FE modelling is required. A more sophisticated FEM 
updating process able to match the results of more natural 
frequencies will give an improved value for the equivalent 
torsional spring. In order to validate the results it is also nec-
essary to set up a consistent method of swaging the speci-
mens with the aim of finding the sensitivity of the natural 
frequencies to parameters such as height and depth (applied 
force) of the swage and the cutting profile of the, swage 
wheel. The results can be then used to validate the swaging 
simulations from FEM. From a reliable FE model it may be 
possible to figure out the equivalent torsional spring stiffness 
without requiring an updating of the linear model with the 
experimental results. A non linear behaviour is also evident 
from the Frequency Response Functions of some specimens, 
by the leaning forward of some resonance peaks. The reason 
could be the asymmetry of the torsional moment as discussed 
before or a hardening effect as given by cubic stiffness as in 
Duffing oscillator [9]. The more plausible explanation is 
another matter worthy of further exploration. 
23-27 August 2010, Sydney, Australia Proceedings of 20th International Congress on Acoustics, ICA 2010 
ICA 2010 5 
REFERENCES 
 
1 R. D. Blevins, "Flow-induced vibration in nuclear reactors: 
a review", Progress in Nuclear Energy, 4, 25-49 (1979) 
2 M. P. Paidoussis, in Practical Experiences with Flow-
Induced Vibrations, Vol. 829-832 (Ed: E. E. Naudascher 
and D. Rockwell), Springer-Verlag, Berlin 1980. 
3 M. Ho, G. Hong and A. N. F. Mack, Experimental 
investigation of flow-induced vibration in a parallel plate 
reactor fuel assembly, 15th Australasian Fluid Mechanics 
Conference, Sydney, Australia,13-17 December 2004 
4 D. R. Miller, "Critical flow velocities for collapse of reactor 
parallel-plate fuel assemblies", Journal of Engineering for 
Power-Transactions of the ASME, 82, 83-95 (1960) 
5 S. J. Pavone and H. A. Scarton, "Laminar flow-induced 
deflections of stacked plates", Nuclear Engineering and 
Design, 74, 79-89 (1982) 
6 D. C. Davis and H. A. Scarton, "Flow-induced plastic 
collapse os stacked fuel plates", Nuclear Engineering and 
Design, 85, 193-200 (1985) 
7 G. Kim and D. C. Davis, "Hydrodynamic instabilities in 
flat-plate-type fuel assemblies", Nuclear Engineering and 
Design, 158, 1-17 (1995) 
8 MSC Nastran Implicit Nonlinear (SOL 600) User’s Guide, 
2007. 
9 S. S. Rao, Mechanical Vibrations, Prentice Hall, 4th 
Edition, Singapore, 2005. 
 
 


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