# Joints Parameters Identification in Numerical Modeling of Structural Dynamics

Authors: Yuedong Yang; Jiqing Chen; Fengchong Lan; Fei Xiong; Zicong Zeng
Journal: Shock and Vibration
 (2018)

---

## Abstract

The dynamics of assembled structures are significantly dependent on joints. Joints parameters, owing to be difficultly measured, are always ignored by pure rigid in numerical modeling and it will result in unreliable or even error descriptions. 1~3 Hz deviations may cause resonance especially in engineering optimization issues; hence it is necessary to identify joints parameters for structural dynamics investigation. In present work, multiobjective optimization algorithms are used to identify joints parameters by approaching actual test results in each series of structure to decrease unreliable or error for numerical modeling. Taking automotive dynamics with seam-welding and spot-welding as examples, the relationship of joints parameters perturbation and structural dynamics is derived to give the selecting reason of parameters’ identification. The actual dynamics of an SUV’s frame and a thin-walled part in BIW (body in white) are utilized to validate the methodology. Results demonstrate that the validated model has enough accuracy to reflect the dynamics of the actual structure. The methodology provides reliable guarantee for dynamic analysis and the design of structure.


---

## Body

## 1. Introduction

All assembled structures are joined to each other through different types of joints, such as welds, rivets, and bolts [1]. Mechanical joints have significant influences on modal parameters, and the dynamic behavior of the assembled structure is highly dependent on joints [2]. In order to have an accurate understanding of the dynamics of an assembled structure, joints properties need to be accurately identified [1]. Structural joints as local characteristics are always ignored as pure rigid in simulations [3], which will result in unreliable predictions and even error. Finite element (FE) predictions are often called into question when they are in conflict with test results [4]. Inaccuracies in FE model and errors in results predicted by them can arise due to the use of incorrect modeling of boundary conditions, incorrect modeling of joints (degree of flexibility varies), and difficulties in modeling of damping. Hence, accurate mathematical models are required to describe the vibration characteristics of structures, which subsequently can be used for design purposes to limit the negative effects of vibrations [5].Bograd et al. [6] reviewed different methods for modeling the dynamic behavior of mechanical joints in assembled structures. In practice, the validation of finite element dynamic models was mainly through comparing experimental modal analysis results with analytical predictions [7]. Joint parameters could be experimentally determined by the frequency response functions measured, respectively, with and without the joints [8–14]. However, only use of natural frequencies in a model updating process is probably advantageous for a couple of reasons: (
1
) natural frequencies are usually more sensitive to the variations of model parameters, and (
2
) they are normally measured with higher accuracy in a modal test [15]. An example containing the modeling of joints with uncertain parameters was shown in [16]. Li et al. [17, 18] used fuzzy theory and a genetic algorithm to identify joint parameters for modular robots. Friswell et al. [19] proposed a layer of special interface elements having material properties that may be adjusted to improve the prediction of the joint structures. Several parameters’ selection methods were proposed in [20–22]; the high sensitive parameter was selected as the updating parameter, but a real uncertain or error parameter still needs to be further studied [7].The rest of this article is organized as follows. Section2 interprets the relation of joints parameters perturbation theory and dynamics, which reveals joints parameters selecting reason, illustrates some example descriptions for joints parameters, and introduces multiobjective optimization algorithms in identifying joints parameters. Sections 3 and 4 take seam-welding in an SUV frame and spot-welding in a thin-walled part of BIW as examples, respectively, to conduct structural actual dynamic tests and make some discussions. And in Section 5 summaries are drawn.


## 2. Theory

### 2.1. Some Example Descriptions for Joints Parameters

The first given description is seam-welding parameters in numerical modeling of an automotive frame. The uncertainty is embodied in parameters of thin interface elements. Young’s modulusE, Poisson’s ratio μ, density ρ, and thickness t of thin interface elements are regarded as uncertainty, which represent local rigidity K
i
j. The uncertainty damping C
i
j of thin interface elements represents local damping. A segment schematic diagram of seam-welding is illustrated in Figure 1(a). The whole seam-welding structure damping is set to G, which is also unknown parameters in numerical modeling.Figure 1
Some example descriptions for joints parameters.
(a)
Local seam-welding parameters description
(b)
Local spot-welding parameters descriptionAnother example description is spot-welding parameters as uncertainty in numerical modeling of thin-walled BIW parts. The uncertainty parameters are diameterD, Young’s modulus E, Poisson’s ratio μ, density ρ, structural whole damping G, and local damping C in connection positions. These parameters, as design variables, are determined by multiobjective optimization algorithm, which are determined by experience or simulating key parts of similar models in conventional modeling. Figure 1(b) shows a schematic diagram of spot-welding with joints parameters uncertainty.


### 2.2. Joints Parameters Perturbation Theory and Dynamics

In a free vibration system with multiple degrees of freedom, assume that some order circle frequency of system in numerical modeling isω
s and vibration mode is {
φ
s
}, which obtained system dynamic experiment corresponding ω
s as ω
t and its vibration mode as {
φ
t
}. If simulation dynamic characteristics are needed to express actual ones, the simulation value ω
s must be updated to ω
t and corresponding vibration mode is changed from {
φ
s
} to {
φ
t
}. Local corresponding rigidity, damping, and mass matrices are only needed to adjust Δ
[
K
], [
C
], and Δ
[
M
] to achieve that simulation behaviors describe actual ones in a local joints parameters identification problem. Actual stiffness matrix K
t and actual mass matrix M
t can be described as follows:(1)
K
t
=
K
s
+
Δ
K
,
(2)
M
t
=
M
s
+
Δ
M
,
(3)
C
t
=
0
s
+
C
,where K
s, M
s, and 0
s represent simulation rigidity matrix, mass matrix, and damping matrix in conventional rigid numerical modeling, respectively.The nature properties of matrices are used. Only local variable quantitiesΔ
[
K
], Δ
[
M
], and [
C
] are needed to be solved in (1)~(3), and the whole system coefficient matrices are not necessary to be solved. The identification simulation model which can express structural actual dynamic characteristics can be described as follows:(4)
K
s
+
Δ
K
+
j
ω
t
C
-
ω
t
2
M
s
+
Δ
M
φ
t
=
0
.In order to analyze easier, frequency characteristic functions can be defined as(5)
α
ω
t
=
K
s
-
ω
t
2
M
s
-
1
.Substituting (5) to (4),(6)
φ
t
=
α
ω
t
ω
t
2
Δ
M
-
Δ
K
-
j
ω
t
C
φ
t
.If the mass of element between system nodesi and j needs to be updated by δ
m
i
j, then, Δ
M
=
δ
m
i
j
v
m
i
j
v
m
i
j
T, where v
m
i
j
=
0,0
,
…
,
1
i
j
,
…
,
0
T. Similarly, if the joint stiffness of element between system nodes i and j needs to be updated by δ
k
i
j, then, Δ
K
=
δ
k
i
j
v
n
i
j
v
n
i
j
T, where v
n
i
j
=
0,0
,
…
,
1
i
j
,
…
,
0
T. If damping between system nodes i and j needs to be considered, C
=
C
i
j
v
l
i
j
v
l
i
j
T where v
l
i
j
=
0,0
,
…
,
1
i
j
,
…
,
0
T.Assume that the total numbers of identification mass, rigidity, and damping elements arem, n, and l in system, separately. Equation (6) can be transformed to the following:(7)
φ
t
=
α
ω
t
ω
t
2
∑
i
=
1
m
δ
m
i
j
v
m
i
j
v
m
i
j
T
-
∑
i
=
1
n
δ
k
i
j
v
n
i
j
v
n
i
j
T
-
j
ω
t
∑
i
=
1
l
C
i
j
v
k
i
j
v
k
i
j
T
φ
t
=
α
ω
t
ω
t
2
∑
i
=
1
m
δ
m
i
j
v
m
i
j
v
m
i
j
T
φ
t
-
α
ω
t
∑
i
=
1
n
δ
k
i
j
v
n
i
j
v
n
i
j
T
φ
t
-
α
ω
t
j
ω
t
∑
i
=
1
l
C
i
j
v
k
i
j
v
k
i
j
T
φ
t
.(
1
) For the situation that n joints rigidity are updated only, the parts of identification rigidity elements in vibration modes {
φ
t
} corresponding to certain order circle frequency ω
t can be described as X
12
∗
,
X
34
∗
,
…
,
X
i
j
∗
,
…
,
X
2
n
-
12
n
∗
T, total of n variables. The nodes matrix is converted to elements matrix. According to [23], the concrete forms of vibration mode transformation vectors {
φ
t
} and frequency characteristic transformation functions α
p
q
,
r
s
ω
t are as follows:(8)
X
12
∗
=
X
1
∗
-
X
2
∗
,
X
34
∗
=
X
3
∗
-
X
4
∗
,
…
,
X
i
j
∗
=
X
i
∗
-
X
j
∗
,
…
,
X
2
n
-
12
n
∗
=
X
2
n
-
1
∗
-
X
2
n
∗
α
p
q
,
r
s
ω
t
=
α
p
r
ω
t
+
α
q
s
ω
t
-
α
p
s
ω
t
-
α
q
r
ω
t
.Frequencies characteristic functionsα
ω
t can be concrete described by the form of elements:(9)
α
12,12
ω
t
α
12,34
ω
t
⋯
α
12
,
i
j
ω
t
⋯
α
12,2
n
-
12
n
ω
t
α
34,12
ω
t
α
34,34
ω
t
⋯
α
34
,
i
j
ω
t
⋯
α
34,2
n
-
12
n
ω
t
⋮
⋮
⋮
⋮
⋮
⋮
α
i
j
,
12
ω
t
α
i
j
,
34
ω
t
⋯
α
i
j
,
i
j
ω
t
⋯
α
i
j
,
2
n
-
12
n
ω
t
⋮
⋮
⋮
⋮
⋮
⋮
α
2
n
-
12
n
,
12
ω
t
α
2
n
-
12
n
,
34
ω
t
⋯
α
2
n
-
12
n
,
i
j
ω
t
⋯
α
2
n
-
12
n
,
2
n
-
12
n
ω
t
.Equation (7) can be described as follows through a series of changes:(10)
X
12
∗
,
X
34
∗
,
…
,
X
i
j
∗
,
…
,
X
2
n
-
12
n
∗
T
=
-
α
12,12
ω
t
α
12,34
ω
t
⋯
α
12
,
i
j
ω
t
⋯
α
12,2
n
-
12
n
ω
t
α
34,12
ω
t
α
34,34
ω
t
⋯
α
34
,
i
j
ω
t
⋯
α
34,2
n
-
12
n
ω
t
⋮
⋮
⋮
⋮
⋮
⋮
α
i
j
,
12
ω
t
α
i
j
,
34
ω
t
⋯
α
i
j
,
i
j
ω
t
⋯
α
i
j
,
2
n
-
12
n
ω
t
⋮
⋮
⋮
⋮
⋮
⋮
α
2
n
-
12
n
,
12
ω
t
α
2
n
-
12
n
,
34
ω
t
⋯
α
2
n
-
12
n
,
i
j
ω
t
⋯
α
2
n
-
12
n
,
2
n
-
12
n
ω
t
×
δ
k
12
δ
k
34
⋱
δ
k
i
j
⋱
δ
k
2
n
-
12
n
X
12
∗
,
X
34
∗
,
…
,
X
i
j
∗
,
…
,
X
2
n
-
12
n
∗
T
.Its general solution is(11)
-
α
i
j
,
12
ω
t
δ
k
12
X
12
∗
-
α
i
j
,
34
ω
t
δ
k
34
X
34
∗
-
⋯
-
α
i
j
,
i
j
ω
t
δ
k
i
j
X
i
j
∗
-
⋯
-
α
i
j
,
2
n
-
12
n
ω
t
δ
k
2
n
-
12
n
X
2
n
-
12
n
∗
=
X
i
j
∗
.When rigidity betweeni and j is updated only, then δ
k
12
=
0
,
…
,
δ
k
i
-
1
j
-
1
=
0
,
δ
k
i
+
1
j
+
1
=
0
,
…
,
δ
k
2
n
-
12
n
=
0. Equation (11) can be converted to(12)
δ
k
i
j
=
-
1
α
i
j
,
i
j
ω
t
.(
2
) Similarly, the general solution of local mass matrices with m elements identification can be derived as follows:(13)
ω
t
2
α
i
j
,
12
ω
t
δ
m
12
X
12
∗
+
α
i
j
,
34
ω
t
δ
m
34
X
34
∗
+
⋯
+
α
i
j
,
i
j
ω
t
δ
m
i
j
X
i
j
∗
+
⋯
+
α
i
j
,
2
m
-
12
m
ω
t
δ
m
2
m
-
12
m
X
2
m
-
12
m
∗
=
X
i
j
∗
.When element mass betweeni and j is updated only, then(14)
δ
m
i
j
=
1
α
i
j
,
i
j
ω
t
·
1
ω
t
2
.(
3
) Similarly, the general solution of local damping matrices with l elements identification can be derived as follows:(15)
j
ω
t
-
α
i
j
,
12
ω
t
C
12
X
12
∗
-
α
i
j
,
34
ω
t
C
34
X
34
∗
-
⋯
-
α
i
j
,
i
j
ω
t
C
i
j
X
i
j
∗
-
⋯
-
α
i
j
,
2
n
-
12
n
ω
t
C
2
n
-
12
n
X
2
n
-
12
n
∗
=
X
i
j
∗
.When element damping betweeni and j is updated only, then(16)
C
i
j
=
-
1
α
i
j
,
i
j
ω
t
·
1
j
ω
t
.The updated quantities of rigidity, mass, or damping through such pair ofi-j element can be solved by (12), (14), or (16). The updated quantities of rigidity matrices, mass matrices, or damping matrices through some pairs of local elements can be solved by general solution (11), (13), or (15). This is the theoretical interpretation for property parameters selecting for identification.


### 2.3. Multiobjective Optimization Algorithms in Joint Parameters Identification

Multiobjective optimization algorithms are utilized to update the uncertain joints parameters. What is critical is that the parameters are assumed uncertain and the identification process is approached actual conditions by various multiobjective optimization algorithms. In the seam-welding example in present research, Nondominated Sorting Genetic Algorithm (NSGA-II) as one of the most ideal multiobjective evolutionary algorithms [24] is used in automotive frame dynamics with joints parameters uncertain and Adaptive Multipopulation Genetic Algorithm (AMGA) in the spot-welding example of automotive thin-walled BIW parts dynamics with parameters uncertain is used.The optimization problems of identification in the illustrated samples can be expressed by mathematical model (17) and (18) separately:(17)
target
ω
f
s
i
=
ω
f
t
i
i
=
1,2
,
…
,
6
var.
E
L
≤
E
≤
E
U
t
L
≤
t
≤
t
U
μ
L
≤
μ
≤
μ
U
ρ
L
≤
ρ
≤
ρ
U
G
L
≤
G
≤
G
U
C
L
≤
C
≤
C
U
,where ω
f
s
i, ω
f
t
i are the ith simulation and test frequencies of automotive frame separately. E, t, μ, ρ are Young’s modulus, thickness, Poisson’s ratio, and density of local seam-welding. G and C represent damping of the whole structure and local seam-welding. The high and low corner mark represent the lower and upper bounds separately:(18)
target
ω
p
s
i
=
ω
p
t
i
i
=
1,2
,
3
var.
E
L
≤
E
≤
E
U
D
L
≤
D
≤
D
U
μ
L
≤
μ
≤
μ
U
ρ
L
≤
ρ
≤
ρ
U
G
L
≤
G
≤
G
U
,where ω
p
s
i, ω
p
t
i are the ith simulation and test frequencies of automotive thin-walled BIW parts, separately. D, E, μ, ρ, and G represent spot-welding diameter, Young’s modulus, Poisson’s ratio, density, and damping in this spot-welding example.


## 2.1. Some Example Descriptions for Joints Parameters

The first given description is seam-welding parameters in numerical modeling of an automotive frame. The uncertainty is embodied in parameters of thin interface elements. Young’s modulusE, Poisson’s ratio μ, density ρ, and thickness t of thin interface elements are regarded as uncertainty, which represent local rigidity K
i
j. The uncertainty damping C
i
j of thin interface elements represents local damping. A segment schematic diagram of seam-welding is illustrated in Figure 1(a). The whole seam-welding structure damping is set to G, which is also unknown parameters in numerical modeling.Figure 1
Some example descriptions for joints parameters.
(a)
Local seam-welding parameters description
(b)
Local spot-welding parameters descriptionAnother example description is spot-welding parameters as uncertainty in numerical modeling of thin-walled BIW parts. The uncertainty parameters are diameterD, Young’s modulus E, Poisson’s ratio μ, density ρ, structural whole damping G, and local damping C in connection positions. These parameters, as design variables, are determined by multiobjective optimization algorithm, which are determined by experience or simulating key parts of similar models in conventional modeling. Figure 1(b) shows a schematic diagram of spot-welding with joints parameters uncertainty.


## 2.2. Joints Parameters Perturbation Theory and Dynamics

In a free vibration system with multiple degrees of freedom, assume that some order circle frequency of system in numerical modeling isω
s and vibration mode is {
φ
s
}, which obtained system dynamic experiment corresponding ω
s as ω
t and its vibration mode as {
φ
t
}. If simulation dynamic characteristics are needed to express actual ones, the simulation value ω
s must be updated to ω
t and corresponding vibration mode is changed from {
φ
s
} to {
φ
t
}. Local corresponding rigidity, damping, and mass matrices are only needed to adjust Δ
[
K
], [
C
], and Δ
[
M
] to achieve that simulation behaviors describe actual ones in a local joints parameters identification problem. Actual stiffness matrix K
t and actual mass matrix M
t can be described as follows:(1)
K
t
=
K
s
+
Δ
K
,
(2)
M
t
=
M
s
+
Δ
M
,
(3)
C
t
=
0
s
+
C
,where K
s, M
s, and 0
s represent simulation rigidity matrix, mass matrix, and damping matrix in conventional rigid numerical modeling, respectively.The nature properties of matrices are used. Only local variable quantitiesΔ
[
K
], Δ
[
M
], and [
C
] are needed to be solved in (1)~(3), and the whole system coefficient matrices are not necessary to be solved. The identification simulation model which can express structural actual dynamic characteristics can be described as follows:(4)
K
s
+
Δ
K
+
j
ω
t
C
-
ω
t
2
M
s
+
Δ
M
φ
t
=
0
.In order to analyze easier, frequency characteristic functions can be defined as(5)
α
ω
t
=
K
s
-
ω
t
2
M
s
-
1
.Substituting (5) to (4),(6)
φ
t
=
α
ω
t
ω
t
2
Δ
M
-
Δ
K
-
j
ω
t
C
φ
t
.If the mass of element between system nodesi and j needs to be updated by δ
m
i
j, then, Δ
M
=
δ
m
i
j
v
m
i
j
v
m
i
j
T, where v
m
i
j
=
0,0
,
…
,
1
i
j
,
…
,
0
T. Similarly, if the joint stiffness of element between system nodes i and j needs to be updated by δ
k
i
j, then, Δ
K
=
δ
k
i
j
v
n
i
j
v
n
i
j
T, where v
n
i
j
=
0,0
,
…
,
1
i
j
,
…
,
0
T. If damping between system nodes i and j needs to be considered, C
=
C
i
j
v
l
i
j
v
l
i
j
T where v
l
i
j
=
0,0
,
…
,
1
i
j
,
…
,
0
T.Assume that the total numbers of identification mass, rigidity, and damping elements arem, n, and l in system, separately. Equation (6) can be transformed to the following:(7)
φ
t
=
α
ω
t
ω
t
2
∑
i
=
1
m
δ
m
i
j
v
m
i
j
v
m
i
j
T
-
∑
i
=
1
n
δ
k
i
j
v
n
i
j
v
n
i
j
T
-
j
ω
t
∑
i
=
1
l
C
i
j
v
k
i
j
v
k
i
j
T
φ
t
=
α
ω
t
ω
t
2
∑
i
=
1
m
δ
m
i
j
v
m
i
j
v
m
i
j
T
φ
t
-
α
ω
t
∑
i
=
1
n
δ
k
i
j
v
n
i
j
v
n
i
j
T
φ
t
-
α
ω
t
j
ω
t
∑
i
=
1
l
C
i
j
v
k
i
j
v
k
i
j
T
φ
t
.(
1
) For the situation that n joints rigidity are updated only, the parts of identification rigidity elements in vibration modes {
φ
t
} corresponding to certain order circle frequency ω
t can be described as X
12
∗
,
X
34
∗
,
…
,
X
i
j
∗
,
…
,
X
2
n
-
12
n
∗
T, total of n variables. The nodes matrix is converted to elements matrix. According to [23], the concrete forms of vibration mode transformation vectors {
φ
t
} and frequency characteristic transformation functions α
p
q
,
r
s
ω
t are as follows:(8)
X
12
∗
=
X
1
∗
-
X
2
∗
,
X
34
∗
=
X
3
∗
-
X
4
∗
,
…
,
X
i
j
∗
=
X
i
∗
-
X
j
∗
,
…
,
X
2
n
-
12
n
∗
=
X
2
n
-
1
∗
-
X
2
n
∗
α
p
q
,
r
s
ω
t
=
α
p
r
ω
t
+
α
q
s
ω
t
-
α
p
s
ω
t
-
α
q
r
ω
t
.Frequencies characteristic functionsα
ω
t can be concrete described by the form of elements:(9)
α
12,12
ω
t
α
12,34
ω
t
⋯
α
12
,
i
j
ω
t
⋯
α
12,2
n
-
12
n
ω
t
α
34,12
ω
t
α
34,34
ω
t
⋯
α
34
,
i
j
ω
t
⋯
α
34,2
n
-
12
n
ω
t
⋮
⋮
⋮
⋮
⋮
⋮
α
i
j
,
12
ω
t
α
i
j
,
34
ω
t
⋯
α
i
j
,
i
j
ω
t
⋯
α
i
j
,
2
n
-
12
n
ω
t
⋮
⋮
⋮
⋮
⋮
⋮
α
2
n
-
12
n
,
12
ω
t
α
2
n
-
12
n
,
34
ω
t
⋯
α
2
n
-
12
n
,
i
j
ω
t
⋯
α
2
n
-
12
n
,
2
n
-
12
n
ω
t
.Equation (7) can be described as follows through a series of changes:(10)
X
12
∗
,
X
34
∗
,
…
,
X
i
j
∗
,
…
,
X
2
n
-
12
n
∗
T
=
-
α
12,12
ω
t
α
12,34
ω
t
⋯
α
12
,
i
j
ω
t
⋯
α
12,2
n
-
12
n
ω
t
α
34,12
ω
t
α
34,34
ω
t
⋯
α
34
,
i
j
ω
t
⋯
α
34,2
n
-
12
n
ω
t
⋮
⋮
⋮
⋮
⋮
⋮
α
i
j
,
12
ω
t
α
i
j
,
34
ω
t
⋯
α
i
j
,
i
j
ω
t
⋯
α
i
j
,
2
n
-
12
n
ω
t
⋮
⋮
⋮
⋮
⋮
⋮
α
2
n
-
12
n
,
12
ω
t
α
2
n
-
12
n
,
34
ω
t
⋯
α
2
n
-
12
n
,
i
j
ω
t
⋯
α
2
n
-
12
n
,
2
n
-
12
n
ω
t
×
δ
k
12
δ
k
34
⋱
δ
k
i
j
⋱
δ
k
2
n
-
12
n
X
12
∗
,
X
34
∗
,
…
,
X
i
j
∗
,
…
,
X
2
n
-
12
n
∗
T
.Its general solution is(11)
-
α
i
j
,
12
ω
t
δ
k
12
X
12
∗
-
α
i
j
,
34
ω
t
δ
k
34
X
34
∗
-
⋯
-
α
i
j
,
i
j
ω
t
δ
k
i
j
X
i
j
∗
-
⋯
-
α
i
j
,
2
n
-
12
n
ω
t
δ
k
2
n
-
12
n
X
2
n
-
12
n
∗
=
X
i
j
∗
.When rigidity betweeni and j is updated only, then δ
k
12
=
0
,
…
,
δ
k
i
-
1
j
-
1
=
0
,
δ
k
i
+
1
j
+
1
=
0
,
…
,
δ
k
2
n
-
12
n
=
0. Equation (11) can be converted to(12)
δ
k
i
j
=
-
1
α
i
j
,
i
j
ω
t
.(
2
) Similarly, the general solution of local mass matrices with m elements identification can be derived as follows:(13)
ω
t
2
α
i
j
,
12
ω
t
δ
m
12
X
12
∗
+
α
i
j
,
34
ω
t
δ
m
34
X
34
∗
+
⋯
+
α
i
j
,
i
j
ω
t
δ
m
i
j
X
i
j
∗
+
⋯
+
α
i
j
,
2
m
-
12
m
ω
t
δ
m
2
m
-
12
m
X
2
m
-
12
m
∗
=
X
i
j
∗
.When element mass betweeni and j is updated only, then(14)
δ
m
i
j
=
1
α
i
j
,
i
j
ω
t
·
1
ω
t
2
.(
3
) Similarly, the general solution of local damping matrices with l elements identification can be derived as follows:(15)
j
ω
t
-
α
i
j
,
12
ω
t
C
12
X
12
∗
-
α
i
j
,
34
ω
t
C
34
X
34
∗
-
⋯
-
α
i
j
,
i
j
ω
t
C
i
j
X
i
j
∗
-
⋯
-
α
i
j
,
2
n
-
12
n
ω
t
C
2
n
-
12
n
X
2
n
-
12
n
∗
=
X
i
j
∗
.When element damping betweeni and j is updated only, then(16)
C
i
j
=
-
1
α
i
j
,
i
j
ω
t
·
1
j
ω
t
.The updated quantities of rigidity, mass, or damping through such pair ofi-j element can be solved by (12), (14), or (16). The updated quantities of rigidity matrices, mass matrices, or damping matrices through some pairs of local elements can be solved by general solution (11), (13), or (15). This is the theoretical interpretation for property parameters selecting for identification.


## 2.3. Multiobjective Optimization Algorithms in Joint Parameters Identification

Multiobjective optimization algorithms are utilized to update the uncertain joints parameters. What is critical is that the parameters are assumed uncertain and the identification process is approached actual conditions by various multiobjective optimization algorithms. In the seam-welding example in present research, Nondominated Sorting Genetic Algorithm (NSGA-II) as one of the most ideal multiobjective evolutionary algorithms [24] is used in automotive frame dynamics with joints parameters uncertain and Adaptive Multipopulation Genetic Algorithm (AMGA) in the spot-welding example of automotive thin-walled BIW parts dynamics with parameters uncertain is used.The optimization problems of identification in the illustrated samples can be expressed by mathematical model (17) and (18) separately:(17)
target
ω
f
s
i
=
ω
f
t
i
i
=
1,2
,
…
,
6
var.
E
L
≤
E
≤
E
U
t
L
≤
t
≤
t
U
μ
L
≤
μ
≤
μ
U
ρ
L
≤
ρ
≤
ρ
U
G
L
≤
G
≤
G
U
C
L
≤
C
≤
C
U
,where ω
f
s
i, ω
f
t
i are the ith simulation and test frequencies of automotive frame separately. E, t, μ, ρ are Young’s modulus, thickness, Poisson’s ratio, and density of local seam-welding. G and C represent damping of the whole structure and local seam-welding. The high and low corner mark represent the lower and upper bounds separately:(18)
target
ω
p
s
i
=
ω
p
t
i
i
=
1,2
,
3
var.
E
L
≤
E
≤
E
U
D
L
≤
D
≤
D
U
μ
L
≤
μ
≤
μ
U
ρ
L
≤
ρ
≤
ρ
U
G
L
≤
G
≤
G
U
,where ω
p
s
i, ω
p
t
i are the ith simulation and test frequencies of automotive thin-walled BIW parts, separately. D, E, μ, ρ, and G represent spot-welding diameter, Young’s modulus, Poisson’s ratio, density, and damping in this spot-welding example.


## 3. Seam-Welding in an SUV Frame

Experimental modal is conducted to an SUV frame to investigate the structural actual dynamics. The test is carried out in a whole automotive production and test workshop of a company in China. Nondamping elastic rope is used to suspend the frame to free state and a schematic diagram is illustrated in Figure2(a). The method of single-point exciting and multipoint collecting signals is used. 60 test points in the area of structural key connections or response positions but not modal order nodes are taken as measurement response points. Some test points are illustrated in Figure 2(b). The signals of exciting force and acceleration are transported to computer-operated acquisition system through power amplifier and the experimental modal parameters are recognized by fitting gathered data. The actual modal nature frequencies ω
f
t
i and vibration modes φ
t are obtained as optimization targets for numerical modeling with joint parameters identification.Figure 2
An SUV frame experiment.
(a)
Suspend mode
(b)
Test pointsAll orders modal frequenciesω
f
t
i and vibration modes φ
t of the SUV frame are acquired by dealing with experimental data. The lowest six-order modal dates and vibration modes are illustrated as follows: the first order is the first torsion, 22.46 Hz; the second order is the first bending, 29.30 Hz; the third order is the second torsion, 44.92 Hz; the fourth order is the second bending, 66.41 Hz; the fifth order is bending-torsion complex, 71.29 Hz; and the sixth order is the third torsion, 82.03 Hz, respectively. The vibration modes are illustrated in Figure 3.Figure 3
Experimental vibration modesφ
t.
(a)
1st mode
(b)
2nd mode
(c)
3rd mode
(d)
4th mode
(e)
5th mode
(f)
6th modeThe MAC value of each order frequency is shown in Figure4, frequency order is in horizontal and vertical coordinates, and the MAC value is in upright coordinates of Cartesian coordinate system.Figure 4
MAC of test modal.Young’s modulus, thickness, Poisson’s ratio, density, damping of local seam-welding thin plate, and damping of the whole structure are variable; the lowest six-order frequencies of experimental valueω
f
t
i are objectives. NSGA-II is used to solve this inverse parameters problem. 242 iterations are conducted. The iteration processes of objectives are shown in Figure 5. The improvement Pareto results are illustrated in Table 1 and the 204th step is the most suitable parameters solution for dynamics simulation of the example structural seam-welding. Horizontal coordinates is iterative steps, and vertical coordinates is frequency. Unit of vertical coordinates is Hz.Table 1
The improvement Pareto results.
Runs
E/GPa
t/mm
μ
ρ/g/cm3
C
G
1
36.13
4.59
0.42
6.10
0.44
0.68
18
370.67
0.77
0.49
6.65
0.63
0.35
22
36.13
4.17
0.20
6.10
0.44
0.68
24
36.13
4.59
0.42
6.93
0.44
0.68
59
36.13
5.15
0.42
6.93
0.44
0.62
186
69.87
5.67
0.14
5.14
0.54
0.43
190
471.21
0.77
0.49
6.65
0.63
0.35
204
370.59
1.27
0.49
5.61
0.69
0.35Figure 5
The iteration processes of objectives.
(a)
ω
f
s
1
(b)
ω
f
s
2
(c)
ω
f
s
3
(d)
ω
f
s
4
(e)
ω
f
s
5
(f)
ω
f
s
6Finite elements analysis solver MSC.Nastran is used for validation. The lowest six-order nonrigid frequencies are selected to investigate the differences of dynamics between two numerical methodologies. The frequenciesω
f
s
i and vibration modes φ
s are obtained. Results of dynamics with rigid joints are illustrated as follows: the first order is the first torsion, 21.32 Hz; the second order is the first bending, 28.66 Hz; the third order is the second torsion, 44.25 Hz; the fourth order is the second bending, 63.46 Hz; the fifth order is bending-torsion complex, 70.47 Hz; and the sixth order is the third torsion, 79.40 Hz, respectively. The vibration modes are illustrated in Figure 6.Figure 6
Vibration modes of rigid modeling methodology.
(a)
1st mode
(b)
2nd mode
(c)
3rd mode
(d)
4th mode
(e)
5th mode
(f)
6th modeThe frequencies and vibration modes generated by multiobjective identification are obtained under the same situation as rigid one, whose results are as follows: the first order is the first torsion, 22.25 Hz; the second order is the first bending, 29.72 Hz; the third order is the second torsion, 45.41 Hz; the fourth order is the second bending, 66.40 Hz; the fifth order is bending-torsion complex, 72.10 Hz; and the sixth order is the third torsion, 82.08 Hz, respectively. The vibration modes are illustrated in Figure7. Multiobjective optimization algorithm modeling methodology is abbreviated to MOD.Figure 7
Vibration modes of MOD modeling methodology.
(a)
1st mode
(b)
2nd mode
(c)
3rd mode
(d)
4th mode
(e)
5th mode
(f)
6th modeFrom the above numerical results we can see that there are differences deviations of 0.93~2.94 Hz for the lowest six-order frequencies in numerical analyses of seam-welding simulations by rigid and MOD modeling methodologies. Vibration modes of the lowest six-order models are in full agreement: the first order is the first torsion, the second order is the first bending, the third order is the second torsion, the fourth order is the second bending, the fifth order is bending-torsion complex, and the sixth order is the third torsion.Figure8 shows the comparison of the lowest six experimental modals and analysis modals obtained by rigid and MOD modeling methodologies as well as the trends graph. The overall comparison, concrete frequencies data, and trends are listed. All order frequencies obtained by MOD modeling methodology are more accurate than rigid modeling within experimental frequencies. The maximum frequency error in rigid modeling methodology is 2.95 Hz which appeared in the fourth order. The error value of the sixth order is 2.63 Hz, also larger than other orders. The error value of the first order is 1.14 Hz. However, in MOD modeling methodology, the error value of the fourth order is reduced to 0.01 Hz, the largest error is 0.91 Hz which appeared in the fifth order, and the error value of the first order reduced to 0.21 Hz. The concrete frequency error of the lowest six orders are shown in Table 2. The correctness and accuracy of MOD modeling methodology to simulate seam-welding are effectively verified.Table 2
Comparison of concrete frequency error.
Error/Hz
Order
1st
2nd
3rd
4th
5th
6th
Conventional
1.14
0.64
0.67
2.95
0.82
2.63
MOD
0.21
0.42
0.49
0.01
0.91
0.05Figure 8
The overall comparison and trends graph.


## 4. Spot-Welding in a Thin-Walled Part of BIW

Thin-walled part of 1 mm with 45 mm space spot-welding and its modal experiment were studied by Gao et al. [25]. The lowest three-order modal frequencies and vibration modes are the first torsion, 16.64 Hz; the longitudinal first bending, 26.42 Hz; the horizontal first bending, 32.23 Hz. Geometric model and experimental modal vibration modes are shown as Figures 9 and 10.Figure 9
Geometric model for spot-welding plate.Figure 10
Vibration modes for experimental modal.
(a)
Torsion
(b)
Longitudinal bending
(c)
Horizontal bendingDiameterD, Young’s modulus E, Poisson’s ratio μ, density ρ of spot-welding parameters, and structural damping G are selected as variables for identification according to the reason in Section 2. The lowest three-order actual frequencies are objectives. AMGA embedded in Isight software is used to approach actual conditions. 501 iterations are carried out. The iteration processes of objectives are shown in Figure 11. Horizontal coordinates are iterative steps, and vertical coordinates are frequency. Unit of vertical coordinates is Hz. The improvement Pareto results are illustrated in Table 3 and the 240th step, which is the 10th in improvement Pareto results, is the more suitable parameters approaching solution in MOD modeling methodology for dynamics of the example structural with spot-welding. So, rigidity variable is calibrated through Young’s modulus E, diameter D, density ρ, and Poisson’s ratio μ. Damping variables are calibrated through structural damping G.Table 3
Improvement Pareto solutions.
Runs
D/mm
E/GPa
μ
ρ/(g/cm3)
G
1
3.5
619.0
0.41
2.7
0.98
4
9.5
133.5
0.18
1.9
0.35
5
2.1
569.0
0.33
1.6
0.28
9
5.5
45.0
0.18
5.7
0.20
11
2.0
974.0
0.25
2.3
0.13
21
1.5
831.0
0.16
7.8
0.044
31
1.2
387.0
0.08
4.1
0.44
91
1.1
828.0
0.37
8.2
0.089
136
1.2
587.5
0.05
5.1
0.13
240
1.1
831.0
0.02
4.1
0.12Figure 11
Iterative processes of objectives.
(a)
ω
p
s
1
(b)
ω
p
s
2
(c)
ω
p
s
3Conventional modeling methodology and MOD modeling methodology are used to establish finite element models and separately to analyze modal. Eigenvalues and vibration modes of 1 mm steel plate with spot-welding are obtained. The lowest 3-order frequencies and vibration modes of two modeling methodology are illustrated in Figures12 and 13. The concrete comparison of two methodologies are in Table 4.Table 4
Comparisons of two spot-welding modeling methodologies.
order
actual
rigid
MOD
1
Vibration mode
torsion
torsion
torsion
frequency/Hz
16.64
16.92
16.92
Error/Hz
-
0.28
0.28
2
Vibration mode
longitudinal bending
longitudinal bending
longitudinal bending
frequency/Hz
26.42
25.81
26.09
Error/Hz
-
−0.61
−0.33
3
Vibration mode
horizontal bending
horizontal bending
horizontal bending
frequency/Hz
32.23
31.84
32.31
Error/Hz
-
−0.39
0.08Figure 12
The lowest 3-order modal of conventional modeling.
(a)
1st mode (16.92 Hz)
(b)
2nd mode (25.81 Hz)
(c)
3rd mode (31.84 Hz)Figure 13
The lowest 3-order modal of MOD modeling.
(a)
1st mode (16.92 Hz)
(b)
2nd mode (26.09 Hz)
(c)
3rd mode (32.31 Hz)The frequencies of two methodologies are both in agreement with test values. The lowest 3-order vibration modes of MOD modeling methodology are more consistent with test modes than conventional modeling. The first and third vibration modes with rigid modeling are different by 180 degrees with test vibration modes. Complex modal considering structural damping is more accurate in solving engineering problems.


## 5. Conclusions

(1)
In present work, multiobjective optimization algorithms (MOD) are used to identify joints parameters by approaching actual test values in each series of structure to decrease unreliability or error for the numerical modeling, and it can also improve the simulation technical capability and achieve more hope of reference value.
(2)
Taking automotive dynamics with seam-welding and spot-welding as examples, the relationship of joints parameters perturbation and structural dynamics is derived to give the selecting reason of parameters for identification.
(3)
In application of local seam-welding within an SUV frame with MOD modeling, all the investigated order frequencies through MOD are more accurate than conventional rigid values. The maximum error is 0.91 Hz and the first order is 0.21 Hz in MOD modeling methodology while the maximum error is 2.95 Hz and the first order is 1.14 Hz in conventional rigid methodology.
(4)
In application of local spot-welding within automotive thin-walled part in BIW, the first and third vibration modes with rigid modeling are of 180 degrees different with test vibration modes. The MOD modeling methodologies are more in agreement with test modes than conventional modeling.



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*Source: 2365759-2018-06-05.xml*