Joints Parameters Identification in Numerical Modeling of Structural Dynamics

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∼3Hz deviations may cause resonance especially in engineering optimization issues; hence it is necessary to identify joints parameters for structural dynamics investigation. In presentwork, 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 thinwalled 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.Themethodology provides reliable guarantee for dynamic analysis and the design of structure.


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][9][10][11][12][13][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][21][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.Section 2 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 thinwalled part of BIW as examples, respectively, to conduct structural actual dynamic tests and make some discussions.And in Section 5 summaries are drawn.1(a).The whole seam-welding structure damping is set to , which is also unknown parameters in numerical modeling.

Theory
Another example description is spot-welding parameters as uncertainty in numerical modeling of thin-walled BIW parts.The uncertainty parameters are diameter , Young's modulus , Poisson's ratio , density , structural whole damping , and local damping  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.

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 is   and vibration mode is {  }, which obtained system dynamic experiment corresponding   as   and its vibration mode as {  }.If simulation dynamic characteristics are needed to express actual ones, the simulation value where []  , []  , and [0]  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 quantities Δ[], Δ[], and [] 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: In order to analyze easier, frequency characteristic functions can be defined as Substituting ( 5) to (4), If the mass of element between system nodes  and  needs to be updated by   , then, Δ[] =   {V ()   }{V ()  }  , where {V ()   } = (0, 0, . . ., 1  , . . ., 0)  .Similarly, if the joint stiffness of element between system nodes  and  needs to be updated by   , then, Δ[] =   {V ()  }{V ()  }  , where {V ()  } = (0, 0, . . ., 1  , . . ., 0)  .If damping between system nodes  and  Assume that the total numbers of identification mass, rigidity, and damping elements are , , and  in system, separately.Equation ( 6) can be transformed to the following: (1) For the situation that  joints rigidity are updated only, the parts of identification rigidity elements in vibration modes {  } corresponding to certain order circle frequency   can be described as { *   12 ,  * 34 , . . .,  *  , . . .,  * 2−12 }  , total of  variables.The nodes matrix is converted to elements matrix.According to [23], the concrete forms of vibration mode transformation vectors {  } and frequency characteristic transformation functions  , (  ) are as follows: Frequencies characteristic functions [(  )] can be concrete described by the form of elements: Equation ( 7) can be described as follows through a series of changes: When rigidity between  and  is updated only, then  12 = 0, . . .,  −1−1 = 0,  +1+1 = 0, . . .,  2−12 = 0. Equation ( 11) can be converted to (2) Similarly, the general solution of local mass matrices with  elements identification can be derived as follows: When element mass between  and  is updated only, then (3) Similarly, the general solution of local damping matrices with  elements identification can be derived as follows: When element damping between  and  is updated only, then The updated quantities of rigidity, mass, or damping through such pair of - 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.

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: where   ,   are the th simulation and test frequencies of automotive frame separately., , ,  are Young's modulus, thickness, Poisson's ratio, and density of local seam-welding. and  represent damping of the whole structure and local seam-welding.The high and low corner mark represent the lower and upper bounds separately: where   ,   are the th simulation and test frequencies of automotive thin-walled BIW parts, separately., , , , and  represent spot-welding diameter, Young's modulus, Poisson's ratio, density, and damping in this spot-welding example.

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 Figure 2(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  computer-operated acquisition system through power amplifier and the experimental modal parameters are recognized by fitting gathered data.The actual modal nature frequencies   and vibration modes {  } are obtained as optimization targets for numerical modeling with joint parameters identification.
All orders modal frequencies   and vibration modes {  } 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, 29.30Hz; 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.
The MAC value of each order frequency is shown in Figure 4, frequency order is in horizontal and vertical coordinates, and the MAC value is in upright coordinates of Cartesian coordinate system.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.Diameter , Young's modulus , Poisson's ratio , density  of spot-welding parameters, and structural damping  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 lowest 3-order frequencies and vibration modes of two modeling methodology are illustrated in Figures 12 and 13.The concrete comparison of two methodologies are in Table 4.
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.

Figure 1 :
Figure 1: Some example descriptions for joints parameters.
(b).The signals of exciting force and acceleration are transported to Non-damping elastic rope (a) Suspend mode Some test points (b) Test points

Figure 8 :
Figure 8: The overall comparison and trends graph.

Table 2 :
Comparison of concrete frequency error.