Designoptimizationoftheramstructureoffrictionstirweldingrobot

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Design optimization of the ram structure of friction stir welding robot
Article  in  Mechanics of Advanced Materials and Structures · January 2019
DOI: 10.1080/15376494.2018.1471758
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ORIGINAL ARTICLE
Design optimization of the ram structure of friction stir welding robot
Haitao Luoa , Jia Fua, Peng Wangb, Jinguo Liua, and Weijia Zhoua
aDepartment of Space Automation Technologies & Systems, State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy
of Sciences (CAS), Shenyang, P.R. China; bInstitute of Modern Transmission and Digital Technology, Northeastern University, Shenyang, China
ABSTRACT
This paper proposes a method for designing the ram structure of friction stir welding (FSW) robots
using finite element analysis. Under the given working condition, force analysis for the ram struc-
ture is performed. By analyzing the boundary and load cases on the ram structure, we optimize
the topology and size. Through ram-structure optimization considering the static and dynamic
characteristics, we can achieve lightweight design of the ram structure and effectively improve the
FSW robot’s welding precision. The optimization flow and method can be applied to heavy-load
robots and high-stiffness structure.
ARTICLE HISTORY
Received 17 April 2018
Accepted 22 April 2018
KEYWORDS
Designing; ram structure;
friction stir welding; robots;
dynamics characteristics;
topology; size; optimization
1. Introduction
Friction stir welding (FSW) is a kind of solid connection
method, which has many advantages compared with the
traditional fusion welding due to their attractive joint mech-
anical properties and environmental friendliness [1]. In add-
ition, neither welding wire nor welding pretreatment is
required [2]. Therefore, FSW has widely been adopted in
various industries worldwide.
To meet the welding requirements in aeronautics and
astronautics, FSW robots are usually huge and complex. If
their components or parts are too heavy, the welding preci-
sion of FSW robots will seriously degrade because the
weight has a great impact on the static and dynamic charac-
teristics [3]. Therefore, the design optimization of the major
parts of FSW robots is of great practical significance for the
research and development of FSW robots.
In order to effectively reduce the weight of FSW robots,
researchers have adopted a number of methods to enhance
its structure performance in accordance with the actual
working condition by removing redundant material and
then adding rib-plates. Based on these measures, the final
design still cannot resist structural deformation and dynamic
flutter. To date, the development of advanced design meth-
ods for the simultaneous optimization of topology and size
is in strong demand. However, in order to put structure
optimization technology into practice, we not only need to
establish a reliable optimization model, but also an effective
and efficient optimization algorithm. There are a variety of
existing methods for structure optimization [4, 5]. The opti-
mal criterion method is one of the early methods for struc-
tural design optimization. In the 1970s, people took the
Kuhn–Tucker condition as the criteria for structure
optimization, which is very generic and mathematically
sound [6]. Based on the early work of Prager and Taylor,
Venkayya et al. proposed a rule-based method for dealing
with large-scale optimization problems [7]. In 1977, Fleury
and Sander showed that the optimal criterion method could
also serve as the dual problem of mathematical program-
ming method, and one significant contribution of their
work is to combine two methods of structural optimization
[8]. The optimal criterion method has been extended to a
more general system optimization approach, thereby putting
forward the so-called composite beam adjustment to ensure
the feasibility of the solution [9, 10]. For structural opti-
mization, Schmit formulatedthe section optimization of
component as a mathematical programming problem and
proposed to deal with a variety of structural optimization of
load cases using mathematical programming methods [11].
It has shown that it was feasible to solve structural optimiza-
tion problems with finite element method. And then, math-
ematical programming methods were further developed to
integrate the advantage of optimal criterion methods by tak-
ing into account the mechanical properties [12], such as
explicit approximation, variable link, effective constraint
selection, inverse variable introduction, and dual solution
technology, which led to great improvement of the compu-
tation efficiency. In recent years, intelligent optimization
algorithms have also become popular in the ram structure
optimization [13–20]. In addition, the constructed method
based on sampling points is a new kind of modeling
method. For the large-scale optimization problems of engin-
eering structural, entity test is not only expensive but also
computationally very intensive. With the development of
high speed computers and finite element analysis, numerical
simulation technologies have attracted more and more
CONTACT Haitao Luo luohaitao@sia.cn State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences (CAS),
Shenyang 110016, P.R. China.
Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/umcm.
� 2019 Taylor & Francis Group, LLC
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES
2018, VOL. 0, NO. 0, 1–11
https://doi.org/10.1080/15376494.2018.1471758
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attention, as they can significantly reduce the computation
time. Such methods mainly include the response surface
(RS) method [21], radial basis function method [22], aug-
mented radial basis function method [23], and Kriging
method [24].
In this paper, we focus on the optimal design of the ram
structure of an FSW robot with the help of finite element
modeling. Force analysis under the given working condition
is also provided. In order to determine the final optimal
structure of the ram, we establish a finite element model
and optimize the topology and size based on the variable
density method and the external penalty function method
assisted by an approximate RS method. Our optimization
results show that the ram mass decreases greatly while the
end deformation just increases a little. Meanwhile, the nat-
ural frequency of the ram has been considerably increased.
The success of the proposed optimization method indicates
that the proposed ideas are promising, providing a new
means for the lightweight design of the relative structures.
The optimization design on the ram structure of FSW
robots is carried out in two stages, including topological
optimization and size optimization described as follows.
1. In the initial structure design, the ram can be taken as a
cuboid. The topology optimization can help designers to
quickly find the ram’s optimal material distribution under
the given objective function and constraint conditions.
Here, the objective function is ram’s minimum flexibility or
maximum stiffness and the constraint is volume percentage
or mass fraction. However, topology optimization only is
not sufficient, as the ram structure is just a framework. An
appropriate lattice-rib needs to be filled in its internal struc-
ture to further enhance its mechanical properties.
Meanwhile, topology optimization method also takes the
installation, debugging and process factors into account
during ram’s structure design. Therefore, topology opti-
mization is only the initial stage of ram structure design.
2. Size optimization focuses on different lattice-rib styles and
its specific geometry sizes located in the internal structure
of the ram. The optimization goal is to minimize the ratio
of total mass and its first order natural frequency. In other
words, the lighter the weight and the higher the natural
frequency of the ram structure, the better. The constraint
is the percentage of the natural frequency before and after
the optimization. By means of size optimization, the stiff-
ness of the ram will be further enhanced. Although its
structure mass increases slightly, the benefit is significant
compared with the static deformation of the ram before
and after optimization. That is, size optimization is the
more detailed stage of ram structure optimization.
2. Load cases about the ram structure of FSW robot
2.1. Brief introduction of the FSW robot
In recent years, aerospace industry has posed high demands for
advanced welding technologies for high-strength and low weight
aluminum alloys, which are widely used in launching vehicles,
missiles, and fight jets. Such aluminum alloys cannot be welded
with the traditional fusion welding techniques; otherwise the
required performance cannot be achieved, such as aluminum
alloy material AL-7075 and AL-2024. To enhance welding per-
formance, minimize the weight and increase fatigue strength,
FSW robots have been developed and became popular.
Most existing FSW devices were transformed from numeric-
ally controlled machine tools with a single function, which lack
flexibility and are not suited for complex work piece welding [25].
In the aerospace field, most workpieces are of large-scale and have
a thin-wall surface, where the weld seam is always represented as a
complex curve in 3D space. Furthermore, a very welding precision
is required [26, 27]. Another challenge in using these welding
devices is the complicated path planning and programming [28].
The new FSW robots with seven joint axis co-motions have
been developed to address the above difficulties. As Figure 1
shows, the FSW robot that we designed consists of X � Y � Z
axis, A� B axis, a stir-welding head and a rotary table. The main
body of the robot includes X� Y� Z axis, A� B axis, and a stir-
welding head. The X� Y� Z axis contains a transmission system
composed of a ball screw and a linear guide, a gravity compensa-
tion system consisting of balance weights and a gravity compensa-
tion system. The A � B axis includes worm and gear driving
systems and stir-welding head components. Rotary table (T axis)
has a rotary DOF. The workpiece to be welded is fastened by a
special fixture on the rotary table. The rotary table has a scale
mark, precise positing and locking function, which can meet the
demands of different working conditions. There are seven DOFs
in the whole system, where, the X� Y� Z axis has three transla-
tional DOFs, A � B axis has two rotational DOFs, and the stir-
welding head has translational and rotational DOF. The rotational
DOF of the stir-welding head is not relevant for the systemmodel.
The body, upright pillar and slip saddle are casted of cast iron.
However, the main load bearing components, such as the ram, is
made up of alloy steel because of its configuration. The mass of
the main body of the robot is about 71 tons, with an envelope size
of 1.8m� 1.8m� 1.6m.
During the welding process, the ram of FSW robot is
always in cantilever and subject to an enormous force and
Figure 1. An illustration of a friction stir welding robot.
2 H. LUO ET AL.
torque as well as considerable amount of variation in FSW
processing. The static and dynamic performances of the ram
will have a strong impact on the geometric accuracy of the
weld seam. Therefore, optimization design of the ram struc-
ture is of great importance for improving the welding per-
formance of the FSW robot.
2.2. Modeling of the ram structure
The ram structure of FSW robots can be analyzed using the 3D
beam element theory. On that basis, we are able to get various
performance indicators for its static and dynamic characteris-
tics. Each node of the 3D beam element has six degrees of free-
dom, where thesix directions of the nodal displacements
correspond to the six nodal forces, as shown in Figure 2.
Under the given configuration, a finite element model of
the simplified elastic structure can be created by two nodes
of space beam element. For each space beam element, each
node has six DOFs. Therefore, the nodal displacement vec-
tor of the beam element can be expressed as
de ¼ dTi d
T
j
h iT
(1)
In Eq. (1),
di ¼ uiviwihxihyihzi
� �T
dj ¼ ujvjwjhxjhyjhzj
� �T
(
(2)
where ln, �n, wn(n ¼ i, j) denotes the linear displacement of
node i, j relative to XYZ axis in the local coordinate system,
respectively, hxn, hyn, hzn(n ¼ i, j) denotes the angular dis-
placement of the cross section located on node i, j relative
to XYZ axis, respectively, hxn is the twist angle of the cross
section around the X axis, and hyn, hzn is the turn angle of
cross section in XZ and XY coordinate planes, respectively.
The corresponding nodal force vector can be described as
follows.
Fe ¼ FTi F
T
j
h iT
(3)
In Eq. (3),
Fi ¼ NiQyiQziTxiMyiMzi
� �T
Fj ¼ NjQyjQzjTxjMyjMzj
� �T
(
(4)
where Nn(n ¼ i, j) is the axial force of the linear displace-
ment relative to node i, j, Qxn or Qyn(n ¼ i, j) is the shear
force in the XZ or XY coordinate system, respectively,
Txn(n ¼ i, j) is the torque of the angular displacement rela-
tive to node i, j, and Myn or Mzn(n ¼ i, j) is the bending
moment in the XZ or XY coordinate system, respectively.
We can then convert the nodal vector into nodal axial dis-
placement, nodal deflection and torsion angle vector as follows.
uf g ¼ uiuj½ �T
vf g ¼ tihzitjhzj
� �T
wf g ¼ wihyiwjhyj
� �T
hf g ¼ hxihxj
� �T
8>>>>>>>:
(5)
The displacement mode of the axial displacement u can be
expressed by a linear function, while the deflection � and x can
be expressed by a cubic polynomial. The displacement mode of
the torsion angle h is also expressed by a linear function.
According to the displacement vector of an element node, we
can get the displacement mode of 3D beam element as follows.
ff g ¼ uvwhx½ �T ¼ Nde (6)
In the above equation, N denotes the displacement shape
functional matrix of the 3D beam element.
Based on the theories of mechanics of materials, the
strain and displacement components of each point on the
flexible body comply with the conditions of geometric equa-
tion. Consequently, the linear strain and shear strain equa-
tions of the 3D beam element are as follows.
ex ¼ du=dx
ey ¼ �ydt2=dy2
ez ¼ �zdw2=dz2
c ¼ qdhx=dx
8>>>: (7)
Rewriting the above in the form of matrix, we obtain:
ef g ¼ Bde (8)
Since the stress and strain components have a linear rela-
tionship and meet the generalized Hooker’s law, the element
stress expressed by the nodal displacement is shown as follows.
rf g ¼ De ¼ DBde (9)
In the above equation, D denotes the elastic matrix:
D ¼
E 0 0 0
0 E 0 0
0 0 E 0
0 0 0 G
2
664
3
775 (10)
According to the virtual displacement principle of an elastic
body, all the work of external force on the elastic body along
the virtual displacement is equal to its virtual strain energy.
dTF ¼
ð ð ð
V
deTrdxdydz (11)
Then, we obtain the formula of the 3D beam element
between equivalent nodal force and equivalent nodal dis-
placement.
Re ¼ Kede (12)
In Eq. (12), Ke means the element stiffness matrix of the
3D beam element.
Figure 2. Finite element model of 3D beam element.
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 3
Ke ¼
ð ð
BTDBdt (13)
Assume that A is the cross section area of the 3D space
beam element. In the XZ and XY coordinate planes, Iy and
Iz are the section moment-of-inertia. Jk is the element tor-
sion moment-of-inertia. E and G denote the material elastic
modulus and shear modulus, respectively. According to the
classical beam theory, the space beam element stiffness
matrix relative to the local coordinate system can be
expressed as in Eq. (14)
k ¼
EA
l
0
12EIz
l3
0 0
12EIy
l3
0 0 0
GJk
l
Sym
0 0 � 6ELy
l2
0
4EIy
l
0
6EIz
l2
0 0 0
4EIz
l
�EA
l
0 0 0 0 0
EA
l
0 � 12EIz
l3
0 0 0 � 6ELz
l2
0
12EIz
l3
0 0 � 12EIy
l3
0
6ELy
l2
0 0 0
12EIy
l3
0 0 0 �GJk
l
0 0 0 0 0
GJk
l
0 0 � 6ELy
l2
0
2EIy
l
0 0 0
6ELy
l2
0
4EIy
l
0
6ELz
l2
0 0 0
2EIy
l
0 � 6ELz
l2
0 0 0
4EIz
l
2
66666666666666666666666666666666666666666664
3
77777777777777777777777777777777777777777775
(14)
By combining the element stiffness matrix, we get the
global stiffness matrix K of the 3D beam element. And then
we obtain the relationship between the external load F and
the nodal displacement X. In the next, we need to obtain
the inherent structure characteristics, which is the basis of
the subsequent dynamics analysis. The un-damped free
vibration equation of the mechanical structure is as follows.
M€a tð Þ þ Ka tð Þ ¼ 0 (15)
In the above equation, M means the global mass matrix,
which also contains the element mass matrix. The element
mass matrix expression is as follows.
Me ¼
ð ð
qNTNdv (16)
By converting the above equation into solving the gener-
alized eigenvalue and eigenvector problem, we have
Ku�x2Mu ¼ 0 (17)
In this way, we can obtain the natural frequency x1,
x2,… ,xn and natural model of vibration u1, u2,… ,un.
2.3. Force analysis of the ram structure
FSW robots have five kinds of typical working conditions
under the actual welding process, such as melon-flap longi-
tude welding, melon-top ring welding, melon-bottom ring
welding, cylinder ring welding and cylinder longitude weld-
ing. But under each typical working condition, the ram’s
load is different. Comparing these five typical working
conditions, we can see that melon-flap longitude welding is
the worst working case, which will be taken as an example
to perform force analysis, as shown in Figure 3.
Under the melon-flap longitude welding condition, the
loads mainly include weight G, the plug-in resistance Fn of
the FSW head, feeding resistance Ft, transverse fluctuating
force Fn(t), rotating torque Tn, and the pulling force F of
steel wire rope. Moreover, e means the distance between
shaft axis of the FSW head and inner side of ram, l1 means
the distance between the ram centroid and fixed surface, l2
the distance between the ram centroid and the front-end
hole axis, l3 the distance between steel wire rope and fixed
surface, n the angular between the FSW head plug-in resist-
ance and the vertical direction, and a and b the length and
width of the ram, respectively. According to the above ana-
lysis, the six-dimensional forces and toques of the ram can
be given as follows.
1. X axial force:
Nx ¼ Fn sin nþ Ft cos n (18)
2. Y axial shear force:
Qy ¼ Fn cos n�Ft sin nþ 2Ft�G (19)
3. Z axial shear force:
Qz ¼ Fa tð Þ (20)
4. X axial toque:
Tx ¼ �Tn sin nð Þ� eþ a=2ð Þ Fn cos nð Þ þ Ft sin nð Þ½ �
(21)
5. Y axial bending moment:
My ¼ Tn cos nð Þ� eþ a=2ð Þ Fn sin nð Þ þ Ft cos nð Þ½ � (22)
6. Z axial bending moment:
Mz ¼ Fn cos nð Þ � Ft sin nð Þ½ � l1 þ l2ð Þ�Gl1 þ 2Fll3 (23)
Through the above force analysis, we can see that the
ram is a biaxial bending part mainly subjected to bending
moment. In order to comprehensively evaluate its mechan-
ical property against terminal load, the principal model
needs to be created and its total structure stiffness needs to
be acquired. Under the melon-valve welding configuration,
the elastic deformation of the ram is as shown in Figure 4.
Assume that the ram is a constant-section cantilever, and
point A is a fixed end. Three concentrated forces are loaded
Figure 3. Force analysis of the ram structure.
4 H. LUO ET AL.
from point B to point D. According to the basic theory of
material mechanics, the shear force and bending moment in
the fixed end section can be described by:
FRA ¼ 2Fl�Gþ Qy
MA ¼ 2Fll3�Gl1 þ Qy l1 þ l2ð Þ
�
(24)
As three concentrated forces are applied to different loca-
tions of the ram, the bending moment equation of the ram
beam section can be obtained using the piecewise calculation
method. The bending-moment equations of AB, BC, and CD
segments can be written as follows.
M x1ð Þ ¼ 2Fl � Gþ QyðÞx1� 2Fll3 � Gl1 þ Qy l1 þ l2ð Þ� �
;
0 � x1 � l3ð Þ
M x2ð Þ ¼ Qy l1 þ l2 � x2ð Þ�G l1 � x2ð Þ; l3 � x2 � l1ð Þ
M x3ð Þ ¼ Qy l1 þ l2 � x3ð Þ; l1 � x3 � l1 þ l2ð Þ
8>>>>>:
(25)
Since the moment equations are different, the corre-
sponding deflection line equations are also different. In add-
ition, the elastic deformation belongs to a small
deformation, so its deflection is far less than its span.
Therefore, the approximate differential equation of deflec-
tion line and turn angle equation can be described as fol-
lows.
y ¼ Ð Ð M
Ely
dx
� �
dx þ Cxþ D
h ¼ Ð M
Ely
dxþ C
8>>>>>:
(26)
In the fixed end A, the deflection displacement and turn
angle are equal to zero. Therefore, the deflection equations
of AB, BC, and CD segments can be shown as follows.
yAB ¼ 2Fl þ Uð Þx31�3 2Fll3 þ Vð Þx21
6EIy
yBC ¼ �Ux32 þ 3Vx22 þ C2x2 þ D2
6EIy
yCD ¼ Qyx33�3Qy l1 þ l2ð Þx23 þ C3x3 þ D3
6EIy
8>>>>>>>>>>>>>>>:
(27)
To clearly formulate the problem, we introduce two con-
stants, U and V, where U ¼ Qy � G and V ¼ Qy(l1 þ l2) � Gl1.
By virtue of deflection equation’s continuity condition, the four
integral constants in Eq. (26) can be determined. The expres-
sions corresponding to C2, D2, C3, and D4 can be expressed as
follows.
C2 ¼ U � Flð Þl23�2Vl3
D2 ¼ Fl � 2U þ 3Vð Þl33
3
C3 ¼ 1
2
Qy � Uð Þl21 þ U � Flð Þl23 þ Vl1�2Vl3 þ Qyl1l2
D3 ¼ 2U � Qyð Þl31 þ 6V � 4U þ 2Flð Þl33�3Vl21�3Qyl21l2
8>>>>>>>>>>>:
(28)
Based on the deflection line equation, the turn angular
equations of AB, BC, and CD segment can be obtained. In
Figure 4, the maximum displacement and turn angle in the
endpoint D are expressed as follows, respectively.
jyjmax ¼ yD ¼ 1
6
Qyl
3
1�
1
2
Qy l1 þ l2ð Þl21 þ C3l1 þ D3
jhjmax ¼ y0D ¼ 1
2
Qyl
2
1�Qy l1 þ l2ð Þl1 þ C3
8>: (29)
Based on Eq. (29), we may limit the maximum deflection
and turn angle, and predefine them according to the actual
demand. In this way, the stiffness conditions of the ram can
be defined as follows.
jyjmax � y½ �
jhjmax � h½ �
(
(30)
where [y] and [h] are allowable deflection and angle.
3. Ram-structure topology of optimization
3.1. Topology optimization model
According to the specified welding condition, the ram’s load
case is analyzed. For example, under the melon-flap longi-
tude working condition, the ram (Z axis) is in cantilever
configuration. Because of the self-weight and overload from
the FSW head end, the static and dynamic characteristics of
the ram will degrade sharply, which severely affect the weld-
ing precision.
In order to reduce the weight of the ram and maximize
the structural stiffness, we discretize the structure into finite
elements and determine which position of the material can
be removed. Finally, we can get the optimal topology of
the ram.
Topology optimization has three main advantages
[29, 30]:
1. it is possible to use the material configuration to opti-
mize the structure performance and composition;
2. it is able to determine the element choices according to
the best force transmission path of materials;
3. it can automatically form hole-like internal structure.
Due to the above reasons, topology optimization in
engineering application is of great practical significance.
Methods for the topology optimization mainly include the
homogenization method, the variable density method, the
variable thickness method, evolutionary structural optimiza-
tion method, level set method, independent continuous
mapping method, isoperimetric method, and bubble method.
Among them, the variable density method has gained much
Figure 4. Bending deformation sketch of ram.
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 5
interest due to its adjustable punishment factor, high com-
putation efficiency and effective and global search ability.
The variable density method introduces a kind of imaginary
variable density material, which is set as variable. An index
function of cell density can be used to simulate properties of
materials.
E xð Þ ¼ E0q xð Þp (31)
In Eq. (31), E(x) denotes Young’s modulus, Eo the initial
Young’s modulus, q the relative density of element, and P
the penalty factor. P is also referred to as a driving factor,
which drives the element relative density to 0 or 1. P is
related to Poisson ratio as shown Figure 5.
The topology optimization of the ram structure using the
solid isotropic material with penalization (SIMP) can be for-
mulated as follows, where structure flexibility is set to be the
objective to be minimized:
Find : q ¼ q1; q2; :::; qnð ÞT 2 Rn
Min : C qð Þ ¼ FTX ¼ XTKX
S:T : F ¼ KX
V qð Þ ¼
Xn
i¼1
qiti � fV0 ¼ Vmax
0>>>>>>>>>>>>>>>>:
(34)
where partial derivatives @(KX)/@qi, @V/@qi, and @C/@qi are
sensitivity of displacement, volume, and objective function,
respectively.
Ck
i
� �n
qki ; qmin>>: (35)
In the above equation, the Ck
i ¼ pðqiÞðp�1ÞuTi K0ui=k1vi ¼ 1
is the design criterion. n is damping factor, which is used to
ensure the stability and convergence of numerical calculation.
The main steps for iterative optimization of the ram top-
ology using the SIMP method can be summarized as
given below.
1. Define the boundary conditions such as design domain,
constraints and loads. The element relative density in
design domain can change with the iteration process.
2. Discretize the structure into finite element meshes, and
compute the element stiffness matrix before optimization.
3. Initialize the element design variables and specify the
initial relative density of elements in the design domain.
4. Calculate the material characteristic parameters and the
stiffness matrix of each discrete element and calculate
the node displacement.
5. Calculate the compliance and sensitivity value of the
overall structure, and solve the Lagrange multiplier.
6. Update the design variables using optimization crite-
ria method.
7. Check if the stop criterion is met. If not, go to step (4).
3.2. Topology optimization results
When the penalty factor takes the intermediate value five, the
topology optimization result of the ram structure of the FSW
robot is shown in Figure 6(a). From the figure, it can be found
Figure 5. The influence of the penalty factor on the relationship between elas-
tic modulus and relative density.
6 H. LUO ET AL.
that the density range distributes in accordance with the layer
way. Inthe figure, the red area denotes the region where the
material is kept, and the blue area indicates the region where
the material is removed, while the other colors represent tran-
sition areas. We can see that under the melon-valve configur-
ation, the final optimized topology of the ram structure is
similar to an arch bridge. Figure 6(b) shows the convergence
profile of the objective function, which indicates that the
objective to be minimized decreases very rapidly in the begin-
ning and converges after approximately 10 iterations.
The optimization result can provide very instructive insight
into the design of ram structures. Note that in order to meet
these external constraint conditions and stiffness strength
design requirements, the optimized structure needs to be
“repaired” by filling up some of the regions where the materials
are removed in the topology optimization, such as add rib-plate
and beam column. The final design of the ram structure is pre-
sented in Figure 7(a). The deformation of the ram along the Z-
axis is as shown in Figure 7(b) under the melon-valve configur-
ation with the external load acting on the end of the ram and
the fixed constraint applied to the other end. Compared with
original displacement, the maximum displacement of the opti-
mized ram has significantly increased by nearly 40%.
4. Size optimization of ram structure
4.1. Ram size optimization model
In order to enhance the stiffness and vibration resistance of
the ram structure and to improve its deformation resistance
and external interference, we conduct a further step for
refining the design obtained by topology optimization.
Considering the processing and installation demand, we
configured the rib-plate type and frame size to achieve the
optimal performance of the ram in accordance with several
existing design schemes. Different styles of rib-plate cell are
shown in Figure 8.
Because the volume of the ram is huge, the computation
time will become prohibitively long using the real simulation
model. Hence, we adopt an approximation model, also
known as surrogates, for estimating the objective value.
Using approximation models for design optimization can be
traced back to when the work reported in 1960s, Schmit pro-
posed the approximate model concept in structural optimiza-
tion design, which accelerated the optimization process and
promoted the application of optimization algorithm in engin-
eering [11]. Many approximate models have been employed as
surrogates, such as the RS methods, Kriging method, and radial
basis function neural network models. In this work, we use the
Kriging method as the approximate model. The size optimiza-
tion process using the Kriging method is given in Figure 9.
For constructing Kriging approximation model, training
samples can be obtained by the Latin hypercube design of
experimental (DOE) method. Latin hypercube DOE is based on
Latin square method plus a uniformity criterion, when its value
reached maximum. So this method is also called Uniform Latin
square DOE. Latin hypercube DOE can generate uniform sam-
ple points with in the design space, which can greatly reduce
needed sample size for approximating nonlinear problem.
In addition to the samples obtained by the Latin hyper-
cube DOE, the accuracy of the Kriging model was further
checked using some random sample points. Once the accur-
acy reaches a predefined threshold, the Kriging model will
no longer be updated and the optimization will be based on
the Kriging only. In this way, the computation for optimiza-
tion can be significantly reduced.
Many optimization algorithms, such as sequential quad-
ratic programming, adaptive simulated annealing and the
gradient based method can be employed for size optimiza-
tion assisted by the Kriging model. In this paper, the exter-
nal penalty function method was adopted. When Kriging
model’s precision meets the predefined accuracy, the finite
element model is no longer need in optimization.
Size optimization of the ram structure can be formulated
as follows.
Figure 6. Topology configuration of the ram structure with (a) material density distribution, P¼ 5, and topology configuration according to (b) convergence curve.
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 7
Find : X ¼ x1x2x3x4½ �T ¼ dtl1l2½ �T
Min : F Xð Þ ¼ W=f 01
S:T : g dtl1l2ð Þ ¼ f1�f 01 � 0
X 2 R
(36)
In the above equation, W denotes the mass of the ram; d
is the bore diameter of the rib-plate cell; t is the thickness of
the rib-plate cell; f1/f 01 is the natural frequency of the ram
before and after optimization; and l1 and l2 are the length
and width of the rib-plate cell.
The relationship between the sectional dimensions and
the output response of the ram is can be described by a
Kriging model as follows.
F xð Þ ¼ ~F xð Þ þ e (37)
where F(x) is the function describing the actual response
value given the design, which is unknown, ~FðxÞ the approxi-
mate response value, estimated using the RS model, and e is
a random error between the approximation and the actual
Figure 7. The results of (a) configuration after optimization, and the comparison between the topology (b) path diagram and original path diagram.
Figure 8. Six kinds of different style rib-plates, (a) “W” style, (b) “*” hole style, (c) round hole style, (d) cross hole style, (e) “*” style, and (f) matts style.
8 H. LUO ET AL.
value, usually assumed to be a Gaussian noise zero mean
and standard deviation of r.
Based on the complexity of the actual function F(x), the
order of the polynomial model to approximate the response
can be chosen to be from 1 to 4. Note that the minimum num-
ber of training samples for constructing the approximate
model is dependent on the model order and the number of
decision variables. For size optimization of the space beam
structure, we can choose a second-order polynomial to meet
the accuracy requirement. Then the initialization requires a
minimum of 1/2(Nþ 1)(Nþ 2) for sample points. In this case,
the approximate model can be described as follows.
Figure 9. Size optimization process using approximate models.
Figure 10. (a) The configuration results of the ram after size optimization, and the size optimization (b) path diagram vs. topology and original path diagram.
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 9
~F xð Þ ¼ b0 þ bxxþ b2xþ :::þ bNxN
þ bNþ1x
2
1 þ bNþ2x
2
2 þ :::þ b2Nþ2x
2
2N þ
X
i6¼j
bijxixj
(38)
4.2. Size optimization results
In order to reduce the ram weight and enhance its stiffness,
based on topology optimized results, we optimize its internal
structure elements which together form the whole ram frame.
We find the lattice gabion is the best pattern element which
makes up the internal structure of the ram frame. The optimal
thickness size of each rib-plate from lattice gabion is shown in
Figure 10(a). Finally, we conduct the simulation analysis under
the specified condition and got the path diagram as shown in
Figure 10(b). Comparing these results, we can see that the stiff-
ness performance of the ram has improved and the mass distri-
bution is more reasonable after the size optimization.
In order to reflect the extent of the variations of each
basic performance index more intuitively, the mass, displace-
ment and natural frequency of the ram are listed in Table 1.
Fromthetable, we find that the effect of topology optimiza-
tion has considerably reduced the ram’s weight, which also
sheds light on the general design direction. Meanwhile, size
optimization has greatly improved the structural stiffness
with only slight increase in the mass of the design. Through
topology and size optimization of the ram structure, the end
displacement of the ram in the welding process has signifi-
cantly decreased while ensuring the natural frequency of the
FSW robots to be in the feasible range.
Based on topology and size optimization, a ram structure
has been manufactured, as shown in Figure 11(a). The entire
friction stir robot is shown in Figure 11(b).
5. Conclusion
This paper introducesa type of environment-friendly tech-
nology termed as FSW robots, which can be used for high-
strength lightweight aluminum alloy welding highly
demanded in the aerospace industry. This kind of welding
facilities can significantly enhance the transportation cap-
acity of the space shuttle rockets, and improve the welding
quality of complex curved surface structures. Thereafter, in
order to improve the geometric accuracy and mechanical
properties of the seams, design and development of heavy-
load and high-stiffness FSW robots, especially those are able
to be used for welding of large thin-wall aluminum alloy
structures will be indispensable. Considering the influence of
the gravity and the endload on the ram due to its cantilever
configuration, we have established a detailed mathematical
model for force analysis, based on which topology and size
optimization using effective and computationally efficient
optimization algorithms have been conducted. By perform-
ing the topology and size optimization, we are able to
obtained optimized material distribution, structure style and
key size of ram structure, resulting in much better mechan-
ical properties.
Through the dynamic characteristic analysis and struc-
tural optimization design on the ram structure of FSW
robots, we have also gained useful insight into vibration
resistance, high stiffness and structure lightweight design,
leading to improve welding precision of FSW robots.
Table 1. Comparison between various performances indicators of the ram structure before and after optimization.
Indicator Primary structure After topology optimization After size optimization Contrast between them
Mass (t) 30.15 11.25 14.36 52.4% (reduce)
Displacement (mm) 0.67 0.98 0.76 13.4% (increase)
Natural frequency (Hz) 86 114 133 54.6% (enhance)
Figure 11. (a) The manufactured ram structure and (b) the friction stir robot.
10 H. LUO ET AL.
Acknowledgments
This work was supported by the National Natural Science Foundation
of China under grant number 51505470; Youth Innovation Promotion
Association, CAS [grant no. 2018237].
Declaration of interest
The authors declared no potential conflicts of interest with respect to
the research, authorship, and/or publication of this article.
ORCID
Haitao Luo http://orcid.org/0000-0001-8865-6466
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	Abstract
	Introduction
	Load cases about the ram structure of FSW robot
	Brief introduction of the FSW robot
	Modeling of the ram structure
	Force analysis of the ram structure
	Ram-structure topology of optimization
	Topology optimization model
	Topology optimization results
	Size optimization of ram structure
	Ram size optimization model
	Size optimization results
	Conclusion
	Acknowledgments
	Declaration of interest
	References