On 4-degree-of-freedom biodynamic models of seated occupants: Lumped-parameter modeling

Journal of Sound and Vibration 402 (2017) 122–141

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On 4-degree-of-freedom biodynamic models of seated occupants: Lumped-parameter modeling Xian-Xu Bai n, Shi-Xu Xu, Wei Cheng, Li-Jun Qian Laboratory for Adaptive Structures and Intelligent Systems (LASIS), Department of Vehicle Engineering, Hefei University of Technology, Hefei 230009, China

a r t i c l e in f o

abstract

Article history: Received 17 September 2016 Received in revised form 20 April 2017 Accepted 9 May 2017 Handling Editor: J. Macdonald

It is useful to develop an effective biodynamic model of seated human occupants to help understand the human vibration exposure to transportation vehicle vibrations and to help design and improve the anti-vibration devices and/or test dummies. This study proposed and demonstrated a methodology for systematically identifying the best configuration or structure of a 4-degree-of-freedom (4DOF) human vibration model and for its parameter identification. First, an equivalent simplification expression for the models was made. Second, all of the possible 23 structural configurations of the models were identified. Third, each of them was calibrated using the frequency response functions recommended in a biodynamic standard. An improved version of non-dominated sorting genetic algorithm (NSGA-II) based on Pareto optimization principle was used to determine the model parameters. Finally, a model evaluation criterion proposed in this study was used to assess the models and to identify the best one, which was based on both the goodness of curve fits and comprehensive goodness of the fits. The identified top configurations were better than those reported in the literature. This methodology may also be extended and used to develop the models with other DOFs. & 2017 Elsevier Ltd All rights reserved.

Keywords: Biodynamics Seated human Lumped-parameter model Parameter identification Multi-objective optimization

1. Introduction Transportation vehicles such as automobiles, ships, and aircrafts generate vibration when in motion. Passengers may feel uncomfortable and even suffer injuries [1] when the vertical vibration frequency falls within 4–8 Hz [2]. A thorough study on dynamic comfort [3] of vehicle occupants is an important approach to improve the riding experience of the passengers. Experiments evaluated by the feelings of the tested occupants are often implemented to study the ride comfort performance of transportation vehicles. As a result, however, experimental methods are not only time-consuming and costly, but also highly subjective. In addition, it’s hard to quantify and standardize. Another approach is to use simulation methods to improve engineering design, shorten product development cycles, and reduce cost. Simulations should also make it easy to investigate the dynamic comfort of the occupants (or the test dummies) and to help design and improve the anti-vibration devices, such as vehicle seats, suspension systems of ground vehicles, and anti-vibration gloves. Therefore, in order to achieve the goals of comfortable ride for the passengers and high-efficiency anti-vibration devices, it is important to establish simple-while-effective biodynamic models of seated occupants. n

Corresponding author. E-mail address: [email protected] (X.-X. Bai).

http://dx.doi.org/10.1016/j.jsv.2017.05.018 0022-460X/& 2017 Elsevier Ltd All rights reserved.

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Human bodies are highly complex multi-body systems. The mechanical properties of each part of the human body are different. Human bodies also vary from person to person. Dynamic responses of seated occupants are an important criterion to evaluate dynamic comfort of passengers and also the effectiveness of anti-vibration devices. Since 1962, quit a few biodynamic models have been proposed to describe the dynamic responses of seated occupants and further to help optimize the anti-vibration devices. According to modeling methods, these models can be categorized as the finite element models, the multi-body models, and the lumped-parameter models. Finite element models for analyzing the biodynamics of seated human use numerous small units to describe human body parts [4–10]. They are often used to deal with impact issues and local stress-strain issues. Finite element models have the advantages of accurate description and broad application scope, but the modeling procedure is very complicated, timeconsuming, and costly. On one hand, finite element models require extremely high computing capability. Although the researches on local structures of human body systems such as neck [4], spine [5–8], pelvic [9], and joints [10] are sound and helpful, accurate finite element analyzing on the entire human body is still not easy to realize, due to the limitation of the computing power. On the other hand, the finite element models require large amount of actual experimental data of human bodies which are hard to collect. Corpses were dissected to obtain the data of human bodies. However, the mechanical properties of the corpses and living human bodies are not the same [1]. Multi-body models deal with issues of multiple directions and motions by simplifying the body into several articulatedrigid body [11–16]. Multi-body models can be used to investigate the mechanical responses of each part when the system is excited in the way of the kinetics. They can also be used to study the kinestate description of each part in the motion state in the way of the kinematics. Lumped-parameter models describe human body systems by using ordinary difference equations or ordinary differential equations containing mechanical system parameters such as mass, stiffness, and damping. Lumped-parameter models use simple mechanical components, which actually implies that the models could not describe the performances of the human bodies in the way of the human anatomy. In other words, the system parameters of the lumped-parameter models will not be completely consistent with the actual parameters of human anatomy and biodynamics. Lumped-parameter models are designed to describe and predict the human responses when under a variety of excitations. The particular advantages of these models are that the models can be easily established, and the analysis, the parameter identification, and the experimental verification are easy to implement. They are low cost as well. The disadvantage is, due to the necessary simplification for the lumped-parameter models, the lumped-parameter models may not fully reflect the responses of each part of human bodies. The responses of each part of human bodies can be described by increasing the degrees of freedom of the models. But the increase of degrees of freedom will inevitably increase the complexity of the models and the practicability weakens in turn. It is noted that, the lumped-parameter models are often used to describe a uni-directional (vertical or lateral) dynamic responses of the human bodies because of compact expression and effective performance of the models with only parameters of mass, stiffness, and damping. Early lumped-parameter models are not suitable for analysis of multidirectional problems. Researchers studied multi-directional biodynamic responses through analyzing in different directions separately incorporating rotating bushing model or human hand-arm system. In 1962, Coermann [17] measured the mechanical impedances of eight people with different heights, weights, and ages, and presented a 1-degree-of-freedom (1DOF) linear model. From then on, the lumped-parameter model has become an important and attractive means to investigate the dynamic responses of seated occupants. Stech and Payne [18] proposed a 1DOF nonlinear model and applied it to the analysis of dynamic responses of human bodies during helicopter landing. They considered that the connection between human body and the seat is rigid and the seat was not included in the model. Using variable damping devices, Muksian and Nash [19] proposed a 2DOF nonlinear model to describe dynamic responses of human bodies under different excitation frequencies. Allen [20] and Wei and Griffin [21] proposed two different 2DOF linear models considering hip-seat rigid body. The seat mass was not taken into account in Allen’s model, and Wei and Griffin considered that the model excitation comes from hips and legs. Suggs et al. [22] proposed a 3DOF linear model and fabricated a dummy based on the model’s parameters to simulate the dynamic responses of seated occupants. Based on this model and the dummy, Feng modified the model parameters for Chinese passengers and employed the modified model to improve ride comfort of transportation vehicles [23]. Hou and Gao came up with a 2DOF parallel linear model and a 3DOF series-to-parallel linear model with better performance based on the test of 28 Chinese adults’ apparent masses (AMs) [24,25]. To evaluate the effects of the polyurethane seats on human bodies, Kang [26] studied and presented a 3DOF parallel linear model and a 3DOF series-to-parallel linear model. Considering friction between the back and torso, related muscle contraction and ballistocardiographic, and diaphragm muscle forces, a 6DOF nonlinear model proposed by Muksian and Nash [27]. Patil et al. [28] presented a 7DOF nonlinear model based on Muskian and Nash’s 6DOF nonlinear model. The friction was neglected while the elasticity and damping between hip and seat were taken into account. Modifying Patil et al's. model, Qassem et al. presented an 11DOF linear model later, which neglected the nonlinear element in Patil et al's. model. The model divided the torso part in Patil et al's. model into three subparts as the lower limbs, upper limbs, and torso [29]. It also divided the back part into cervical, thoracic, and lumbar vertebras. Later, an 11DOF linear model for pregnant women based on Qassem et al's. model considering fetus mass as a part of abdomen was further presented by Qassem and Othman [30]. In 1995, Wan and Schimmels used a 4DOF series-to-parallel linear model to describe the dynamic responses of seated occupants under vibration excitation [31]. Liu et al. [32] and Abbas et al. [33] implemented nonlinear optimization and model parameters optimization using the weighted genetic algorithm (GA) respectively. Singh and Wereley [34]

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restructured Liu et al's. model to investigate the dynamic responses of seated occupants in helicopters during landing. In 1998, Boileau et al. [35] systematically collected and analyzed the data of dynamic responses of seated occupants in the profiles of magnitude and phase, and also proposed a 4DOF series linear model [36]. Zhang et al. [37] and Sdjevic and Cveticanin [38] optimized parameters in Boileau and Rakheja's model. The difference is, Sdjevic and Cveticanin used a simple weighted GA while Zhang et al. used an average-weighted GA. Recently, Arslan compared and evaluated Boileau and Rakheja’s model as well as Wan and Schimmels' model in terms of dynamic responses of human bodies under impacts. Arslan thought Boileau and Rakheja’s model is more adaptable than Wan and Schimmels' [39]. To study multi-directional dynamic responses of seated occupants, Kim et al. [40] proposed a 5DOF bidirectional nonlinear model with a rotating bushing. Adding degrees of freedom to the lower limbs and backrest based on Wan and Schimmels’ model, Harsha et al. [41] proposed a 9DOF bidirectional linear model and discussed the dynamic responses of seated occupants for both the situations of considering the backrest and without considering the backrest. Adding degrees of freedom to the upper limbs and backrest based on Wan and Schimmels’ model, Gan et al. [42] proposed a 6DOF bidirectional linear model, studied the backrest influence on dynamic responses of seated occupants. A trust region algorithm was utilized to identify the model parameters. Dong et al. have contributed to the dynamic responses of seated occupants with particular emphasis on the hand-arm system, in two aspects: calibration of models [43–45] and design of anti-vibration devices [46,47]. Based on the proposed calibration method [45], they realized an improved 4DOF model to better describe the dynamic responses of seated occupants, with minor modification of the reported model. The proposed general method provides a fundamental means to easily find or modify-to-realize a more-efficient model for uni- and multi-directional biodynamic responses of human bodies. Vibration isolation effectiveness of anti-vibration gloves was evaluated through developing 5DOF and 7DOF mechanical equivalent models of the glove-hand-arm system based on the measured driving-point mechanical impedances distributed at the fingers and the palm of the hand [47]. Besides, Zhang et al. [48] proposed a 17DOF bidirectional linear model in order to improve the driving comfort and ride comfort of ground vehicles. However, as a matter of fact, increasing degrees of freedom of the models is of little help to improve the model accuracy. Instead, it is more important for describing the dynamic responses of each part and organs of seated occupants [45,49]. According to the literature review of the lumped-parameter models of seated occupants, the characteristics and the development and investigation tendency on lumped-parameter models are listed as follows. (i) The number of degrees of freedom of the models distributes widely. Lumped-parameter models were carried out continuously from 1DOF to multi-DOF and applied to many specific applications. (ii) Linear models are the main research objects, while the nonlinear models are secondary. The early lumped-parameter models in 1960s or 1970s were mostly nonlinear models, and linear models became mainstream after 1990s. However, the nonlinear model proposed by Liu et al. at the end of the 20th century led to a new development of nonlinear models. (iii) Researches on 4DOF models are the majority. Up to date, 4DOF models are the most popular topics. A number of the newly-developed 4DOF and multi-DOF models are based on Wan and Schimmels’ model as well as Boileau and Rakheja’s model. (iv) The accuracy of parameter identification improves much. Because of the progress of algorithms, especially the development of the various evolutionary algorithm theories, new algorithms are able to identify multi-DOF model parameters and further improve the effectiveness of the models with the same structures. (v) Multiple research areas have been developed. Early lumped-parameter models could only solve and analyze the problems of vertical dynamic responses due to the limitations of the structures of the lumped-parameter models. With the development of the lumped-parameter theory and the improvement of computer hardware, multi-DOF models that could describe bidirectional and even three-directional problems were proposed and studied. Bidirectional and threedirectional lumped-parameter models are worth of further study. (vi) Application scope will further expand. The lumped-parameter models have been used in various engineering applications including automobiles, railway vehicles, ships, and helicopters etc. The lumped-parameter models are significant to the development of the transportation vehicles. At the same time, starting with Liu et al's. model, lumped-parameter models are not only applied to analyze dynamic responses of human bodies under vibration excitations, but also they are used to describe and predict the dynamic responses of human bodies under impact excitations, which greatly expands the application of the lumped-parameter models. Liang and Chiang [1] compared and evaluated 12 types of lumped-parameter models of seated occupants with different degrees of freedom, and recommended the 4DOF series-to-parallel model proposed by Wan and Schimmels [31]. After all, more degrees of freedom of the models does not necessarily provide better fitting precision than simple models [45,49]. 4DOF models have appropriate number of parameters and provide reasonable fitting performance. Meanwhile, it is convenient to realize model extension on the basis of 4DOF models. To date, series models such as Wan and Schimmels' model [31], Boileau and Rakheja’s model [36], and Zhang et al's. model [37] and series-to-parallel such as Liu et al's. model [32] and Singh and Wereley's model [34] have been developed. Although these 4DOF series models and series-to-parallel models can be used, systematic and complete analysis for all of structures of 4DOF models cannot be found and referred to yet. In addition, current algorithms used for model parameter identification (or the model evaluation criterion) for the 4DOF models are relatively simple. These algorithms include ordinary GA [34,40,50], average-weighted GA [37], and simple weighted GA [33]. In fact, aiming at understanding the biodynamic responses of seated occupants of transportation vehicles

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and improving the anti-vibration devices and/or test dummies, parameter identification (or the model evaluation criterion) for 4DOF models should be conducted with consideration of multiple objectives. Using average weights for different objectives, ignoring the dimensions and the orders of magnitude differences among the objectives, and finally using the weight-sum as the evaluation criterion seems lack of theoretical support. Therefore, it is also worth of a further investigation. Consequentially, in this study, a systematic study on 4DOF biodynamics of seated occupants in aspects of model expression, analysis, and parameter identification is studied. A 4DOF biodynamic model with 18 parameters of seated occupants is first deduced and demonstrated as an example. Then, an equivalent simplification expression for 4DOF models is proposed, and all of 23 types of non-reduplicative-while-effective 4DOF biodynamic models of seated occupants are provided. The improved version of non-dominated sorting genetic algorithm (NSGA-II) [51,52] is applied for parameter identification for models. A model evaluation criterion considering both the goodness of fit and comprehensive goodness of fit is established. All of the parameters of the 23 types of models are identified and the corresponding performances of the models are compared, evaluated, and analyzed. Sequentially, two specific obtained models (structural configurations) are compared with the four existing 4DOF biodynamic models of seated occupants reported in the literature. The following sections of this paper are organized as follows. In Sections 2 and 3, we present the 4DOF models and evaluation criteria for the models effectiveness, respectively. In Section 4, parameter identification for the 4DOF biodynamic models is provided, followed by the comparisons of the proposed models and the existing models in Section 5. Concluding remarks are given in Section 6.

2. 4DOF biodynamic models of seated occupants 2.1. Structure and modeling Fig. 1 presents a 4DOF biodynamic model with 18 parameters of seated occupants. As shown in Fig. 1, the seated occupant is a spring-mass-damper system, taking into account the human body as several lumped masses interconnected by springs and dampers. In Fig. 1, m1 is the mass of head; m2 and m3 are the masses of internal organs of torso under indirect excitation from seat; m4 is the mass of torso under direct excitation from seat; cmn is the damping between two body parts (the subscripts m and n denote the sequential numbers of the connected body parts); kmn is the stiffness between two body parts; z0 is the excitation from seat; z1, z2, z3 and z4 are the displacements of the masses centers of the body parts. According to Fig. 1 and Newton's second law, the kinematic model can be expressed as:

Mz¨ + Cż + Kz = fz

(1)

where M is a fourth-order mass matrix; C is a fourth-order damping matrix; K is a fourth-order stiffness matrix; fz is the T T T force vector due to external excitation; z (¼ ⎡⎣ z1, z2, z3, z4 ⎦⎤ ), ż (¼ ⎣⎡ z1̇ , z2̇ , z3̇ , z4̇ ⎤⎦ ), and z¨ (¼ ⎡⎣ z¨1, z¨2, z¨3, z¨4 ⎤⎦ ) are the displacement, velocity, and acceleration vectors of the mass parts, respectively. M, C, K, and fz are given by:

z1

m1 k12

c12

k13

m2 k23

k24

c24

c23

c13 k14

c34

z4

m4 k40

c14

z3

m3 k34

z2

c40

z0

Fig. 1. A 4DOF biodynamic model with 18 parameters of seated occupants.

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⎡m ⎢ 1 ⎢ 0 M=⎢ ⎢ 0 ⎢⎣ 0

0

0

m2

0

0

m3

0

0

0⎤ ⎥ 0⎥ 0 ⎥⎥ m4 ⎥⎦

(2a)

⎤ ⎡ c12 + c13 + c14 −c12 −c13 −c14 ⎥ ⎢ c12 + c23 + c34 −c12 −c23 −c24 ⎥ C=⎢ c23 + c34 + c13 −c13 −c23 −c34 ⎥ ⎢ ⎢⎣ c34 + c40 + c14 + c24 ⎥⎦ −c14 −c24 −c34

(2b)

⎤ ⎡k + k + k −k12 −k13 −k14 13 14 ⎥ ⎢ 12 k12 + k23 + k34 −k12 −k23 −k24 ⎥ ⎢ K=⎢ ⎥ k k k k k k − − + + − 13 23 23 34 13 34 ⎥ ⎢ ⎢⎣ k34 + k 40 + k14 + k24 ⎥⎦ −k14 −k24 −k34

(2c)

⎧ ⎪ ⎪ fz = ⎨ ⎪ ⎪ ⎩ c40z 0̇

(2d)

⎫ 0 ⎪ ⎪ 0 ⎬ 0 ⎪ + k 40z 0 ⎪ ⎭

The Fourier transformation of Eq. (1) is:

( −ω M + jωC + K)·Z( jω) = F ( jω) 2

z

(3)

where j (= −1 ) is the complex phasor and ω is the angular frequency. Z( jω) and Fz( jω) are: T Z( jω) = ⎡⎣ Z1( jω), Z 2( jω), Z3( jω), Z 4( jω)⎤⎦

(4a)

⎡ 0 0⎤ ⎥ ⎢ 0 0 ⎥⎡ 1 ⎤ Fz( jω) = ⎡⎣ 0, 0, 0,( k 40+jωc40)Z 0( ω)⎤⎦ = ⎢ ⎢ ⎥Z ( ω) ⎢ 0 0 ⎥⎣ jω ⎦ 0 ⎥ ⎢k ⎣ 40 c40 ⎦

(4b)

Combining Eq. (3) and (4), Z( jω) can be rewritten as:

⎡ 1⎤ Z( jω) = A−1B⎢ ⎥Z 0( ω) ⎣ jω ⎦

(5)

A = − ω2M + jωC + K

(6a)

⎡ 0 0⎤ ⎥ ⎢ 0 0⎥ B=⎢ ⎢ 0 0⎥ ⎥ ⎢k ⎣ 40 c40 ⎦

(6b)

with

The structure of the 4DOF biodynamic model with 18 parameters, as shown in Fig. 1, considers all of the connections between any two parts. It is only a particular case of 4DOF biodynamic models of seated occupants. It should be noted that, the number of parameters of 4DOF biodynamic models will be different, if the connections of the four masses change. The number of parameters in damping matrix C and stiffness matrix K in Eqs. (2b) and (2c) will be different. According to Eq. (5) and (6), the change of matrix C and K will lead to the changes of the vectors A and B, and the relationship between the dynamic responseZ( jω) and excitation Z 0( ω) of the masses can also be calculated accordingly. 2.2. Equivalent simplification expression method for structures and enumeration of 4DOF biodynamic models of seated occupants To systematically study the 4DOF biodynamic models of seated occupants and improve the analysis and expression efficiency of models, an equivalent simplification expression method for 4DOF biodynamic models of seated occupants is

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1

2

3

4

127

Fig. 2. The equivalent simplification expression of 4DOF biodynamic model with 18 parameters of seated occupants.

proposed. All of 4DOF biodynamic models of seated occupants are then listed. The proposed equivalent simplification expression method for 4DOF models could be described as follows. (i) Simplify the masses (m1, m2, m3, and m4), as shown in Fig. 1, into dots with sequential numbers (1, 2, 3, and 4). 4DOF biodynamic models of seated occupants have a total of four masses, therefore the number of mass parameters is four. Then the 4DOF biodynamic model with 18 parameters (see Fig. 1) of seated occupants can be equivalently simplified as the graphical expression shown in Fig. 2. (ii) Incorporate the stiffness and damping between any two masses and simplify them into a line. The line represents a combined parameter of stiffness and damping. As shown in Fig. 2, the 4DOF biodynamic model of seated occupants with 18 parameters has six lines in total. (iii) Remove the reduplicative expressions. Four different structural expressions of the 4DOF biodynamic models of seated occupants with 12 parameters are shown in Fig. 3. As shown in Fig. 3, structures ① and ③ are actually the same, and ② and ④ the same too. So the number of the equivalent simplification expressions are reduced from four to two. Equivalent simplification for 4DOF biodynamic models of seated occupants should take the following conditions into account. (i) As shown in Figs. 1 and 2, there are stiffness and damping between m4 and the excitation, and all of the 4DOF biodynamic models have this connection. This connection is omitted. (ii) The combined parameters of stiffness and damping between two masses are always used in the expressions. The stiffness or damping can be appropriately omitted according to the parameter identification results, if further simplification is needed. (iii) For convenience of expression, 4DOF biodynamic models with 12 parameters, 14 parameters, 16 parameters, and 18 parameters are written as 4DOF12, 4DOF14, 4DOF16, and 4DOF18, respectively.

1

1

1

2

3 2

3

3 4

1

3

2

2 4

4

Fig. 3. Rules for removing the reduplicative models.

4

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Table 1 All of the 4DOF biodynamic models of seated occupants before and after removal of the reduplicative models.

Based on the above principles, Table 1 lists all of the structures of 4DOF biodynamic models of seated occupants. For comparison, all of the structures with and without removal of the reduplicative expressions are provided. There are a total of 38 types of structures of 4DOF biodynamic models of seated occupants before removing the reduplicative ones (i.e., 1st round simplification), and only 23 types of non-reduplicative ones remain after.

3. Evaluation criteria Based on the research from Boileau and Rakheja [36], International Organization for Standardization (ISO) incorporated the seat-to-head transmissibility (STHT), driving-point mechanical impedance (DPMI), and AM into the evaluation system [53] and published an international standard ISO5982-2001. The standard ISO5982-2001 improved evaluation criteria for biodynamics of seated human. It is so far the most widely used standard and criterion for the researches on biodynamics of seated human. STHT is defined as the ratio of the response displacement of the head to the forced vibration displacement at the seatbody interface [53], which is a non-dimensional ratio. It can be drawn from Eq. (5) and is expressed as:

STHT( jω) =

⎡ 1⎤ = ⎡⎣ 1, 0, 0, 0⎤⎦A−1B⎢ ⎥ ⎣ jω ⎦ Z 0( ω)

Z1( jω)

(7)

DPMI is defined as the ratio of the applied periodic excitation force to the resulting vibration velocity at the same frequency [53] and it is expressed as:

DPMI( jω) =

F4( jω) v4( jω)

=

⎡ 1 ⎤⎤ k 4+jωc4 ⎡ ⎡ ⎢ 1−⎣ 0, 0, 0, 1⎤⎦A−1B⎢ ⎥⎥ ⎣ jω ⎦⎦ jω ⎣

(8)

where F4( jω) and v4( jω) are the excitation force and the velocity of driving point, respectively. AM is defined as the ratio of applied periodic excitation force to the resulting vibration acceleration at the same frequency [53] and it is expressed as:

AM( jω) =

F4( jω) a 4 ( jω)

=

⎡ 1 ⎤⎤ k 4+jωc4 ⎡ ⎡ ⎢ 1−⎣ 0, 0, 0, 1⎤⎦A−1B⎢ ⎥⎥ 2 ⎣ jω ⎦⎦ −ω ⎣

where a 4( jω) is the acceleration of driving point.

(9)

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4. Parameter identification 4.1. Experimental data The experimental data in terms of STHT, DPMI, and AM of seated occupants when exposed to vertical vibrations within a frequency range of 0.5-20 Hz by Boileau et al. [35], as listed in Table 2, is popular and widely used in the fields of academic researches and engineering applications on multi-DOF biodynamics of seated occupants. We will also use their experimental data to conduct parameter identification and model evaluation. It should be noted that although the employed experimental data is possibly including some measurement errors [43], it is not an issue for the systematic study of the 4DOF models in this study. 4.2. NSGA-II The ordinal optimization method [36], the least square method [49], the ordinary GA [34,40,50], the average-weighted GA [37], the simple weighted GA [33], and the trust region algorithm [42] were applied in parameter identification for lumped-parameter models of seated occupants. However, all three of the STHT, DPMI, and AM objectives should be simultaneously considered when doing parameter identification for 4DOF biodynamic models of seated occupants. It is actually a multi-objective optimization problem in which the objectives always influence and conflict with one another, and trade-off should be made when conducting optimization [54]. For multi-objective optimization problems, there are a set of optimal solutions called Pareto frontier (Pareto surface). Any of the parameter identification results obtained by the abovementioned existing algorithms is only one specific result in Pareto frontier (Pareto surface). GA based on the Pareto optimization principle could deal with multi-objective optimization problems and achieve Pareto frontier (Pareto surface). NSGA-II is a Pareto optimization based GA [51], and has the advantages of a quick search for noninferior solutions and fast convergence rate. Fig. 4 presents the flow chart of the parameter identification for the 4DOF biodynamic models based on the NSGA-II. As shown in Fig. 4, step (i) randomly generates initial population and records the population generation as 0. Step (ii) produces child population after selection, crossover, and mutation. Step (iii) mixes the child population and original parent population to create a new population. Step (iv) non-dominatedly sorts the new population and produces new non-dominated fronts. Step (v) sorts each group of non-dominated fronts based on crowded distance and picks out the optimal population to conduct next generation’s calculation. Step (vi) stops calculation and outputs the optimal solution when the calculation result is less than the preset function tolerance of fitness. 4.3. Multi-objective function and boundary conditions The experimental data in terms of STHT, DPMI, and AM changes with excitation frequency f, as listed in Table 2. A 4DOF biodynamic model is a gene of NSGA-II, and the corresponding simulation results of STHT, DPMI, and AM of the gene can be calculated by Eqs. (7)–(9). The optimization objectives are STHT, DPMI, and AM. It is a three-dimensional row vector and each vector is the variance between the simulation results and experimental data: Table 2 The experimental data in profiles of magnitude and phase of seated occupants when exposed to vertical vibrations with a frequency range of 0.5-20 Hz [35]. Frequency (Hz)

0.5 0.63 0.80 1.0 1.25 1.6 2.0 2.5 3.15 4.0 5.0 6.3 8.0 10.0 12.5 16.0 20.0

Magnitude

Phase (°)

STHT (-)

DPMI (N s m-1)

AM (kg)

STHT

DPMI

AM

1.01 1.01 1.01 1.02 1.03 1.06 1.08 1.1 1.16 1.29 1.45 1.23 1.01 0.96 0.86 0.71 0.63

254 304 359 424 493 627 768 947 1429 2002 2346 2065 1939 1981 2023 1750 1755

61.2 61.4 60.6 59.6 59.2 60.0 60.8 62.6 70.7 79.3 74.5 53.2 38.5 31.5 25.9 17.4 14.1

-0.6 -1.0 -1.2 -1.5 -1.8 -2.9 -4.3 -6.3 -9.7 -15.0 -35.6 -59.8 -66.3 -75.6 -93.2 -119.5 -142.2

86 86 86 86 86 85 84 81 75 61 36 26 25 22 18 15 20

-4 -4 -4 -4 -4 -5 -6 -9 -15 -28 -54 -64 -65 -68 -72 -75 -70

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Start Initial population (Gen = 0)

Selection, crossover, mutation to create child population Combine parent population and child population Front = 1

Identify non-dominated individual

Population classified?

Front = Front + 1

Yes Crowded distance sorting

Gen = Gen + 1

New population Func. tolerance < Preset value?

No

Yes Stop and output Fig. 4. Flow chart of parameter identification based on NSGA-II.

⎡ N ⎢ ∑ STHT f − STHT f 0 i i ⎢ i=1 ⎢ ⎢ N minFcn = ⎢ ∑ DPMI fi − DPMI0 fi ⎢ i=1 ⎢ N ⎢ ∑ AM f − AM f 2 0 i i ⎢⎣ i=1

⎤T ⎥ ⎥ ⎥ 2⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

2

(

( )

( ))

(

( )

( ))

( ()

( ))

(10)

where N is the total number of the experimental data; STHT(fi), DPMI(fi), and AM(fi) denote the STHT, DPMI, and AM simulation results at the ith excitation frequency fi, respectively; STHT0(fi), DPMI0(fi), and AM0(fi) denote the STHT, DPMI, and AM experimental data at the ith excitation frequency fi, respectively. It should be noted that in this study we only set the magnitudes of the terms as the optimization objective, since the phases of the terms are only determined if the parameters of the models for the magnitudes of the terms are identified. Boundary conditions of the parameters in 4DOF biodynamic models are considered to accelerate the convergence rate of parameter identification and also to make the parameters match the real human parts. The boundary conditions are based on basic parameters of seated occupants [55]: the weight of head and neck (m1) accounts for 8.4% of whole body weight (mean weight¼ 75.4 kg based on the experimental data). The weight of seated occupant on the seat accounts for 73.6% of whole body weight, i.e., 55.5 kg. The upper and lower boundaries are expanded to an appropriate range of the actual weight. According to the parameters from anthropotomy [56]: the stiffness of each body part is within the range of 100300,000 N m-1. Based on the experiments, the damping of the body parts is recommended as 500–4000 N m-1 [57]. Then, boundary conditions for the optimization is expressed as:

⎧ 5. 31 kg ≤ m ≤ 6. 49 kg 1 ⎪ ⎪ m1 + m2 + m3 + m4 = 55. 5 kg ⎨ −1 −1 ⎪ 100 N m ≤ k n ≤ 300000 N m ⎪ −1 −1 ⎩ 500 N s m ≤ cn ≤ 4000 N s m

(11)

where k n and cn are the stiffness and the damping of each body part, respectively. 4.4. Results and evaluation Based on objective functions (Eq. (10)) and boundary conditions (Eq. (11)), we set the variable number as 18 (take 4DOF18 as an example), the population type as double precision vector, the population number as 100, the crossover rate as 0.8, the aberration rate as 0.01, the migration rate as 0.2, the Pareto frontier individual coefficient as 0.35, and function tolerance of

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131

fitness as 0.00001. All of 35 groups of Pareto solutions are obtained after 51,401 generations. Part of the results are listed in Table 3 and all of the Pareto solution sets form the Pareto surface as shown in Fig. 5. As shown in Fig. 5, all of the specific solutions for AM, DPMI, and STHT are the optimal results for 4DOF18. Particular objective function performance can be selected according to applications. To quantitatively analyze and evaluate the results of parameter identification of single objective function and the comprehensive objective function, goodness of fit ε [58], comprehensive goodness of fit ε ̅ , and maximal comprehensive goodness of fit εm ̅ ax are used to evaluate the parameter identification. The expressions are given by:

ε = 1−

Σ( τm − τc )2/( N−2) Σ( τm/N )

(12a)

ε ̅ = W1⋅εSTHT + W2⋅ε DPMI + W3⋅εAM

(12b)

εmax = Max( W1⋅εSTHT + W2⋅ε DPMI + W3⋅εAM) ̅

(12c)

where τm is the experimental data; τc is the simulation result; εSTHT , εDPMI , and εAM represent goodnesses of fit of STHT, DPMI, and AM, respectively; W1, W2, and W3 are the weighting factors (W1 þW2 þW3 ¼1), and W1 ¼ W2 ¼W3 ¼ 1/3 in this study. The weighting factors can be set according to specific practical requirements. ε, ε ̅ , and εm ̅ ax can describe the consistency of simulation and experimental results. They range from 0 to 1, and the closer to 1, the better the fitting performance will be obtained. Goodnesses of fit for each Pareto solution of the parameter identification for 4DOF18 are listed in Table 4. According to Table 4, adopting the comprehensive goodness of fit (Eq. (12b)) for STHT, DPMI, and AM as the evaluation criterion, the solution number 4, listed in Table 4, is the optimal solution among the 35 Pareto solutions. the comprehensive goodness of fit is 0.875 and it is the maximal (i.e., εmax = 0.875). And the goodness of fit for STHT (εSTHT ) is 0.778, the goodness of fit for ̅ DPMI (εDPMI ) is 0.914, and the goodness of fit for AM (εAM ) is 0.933. With the same principle, parameter identification for all of the 4DOF biodynamic models of seated occupants could be done based on NSGA-II and experimental data, and εm ̅ ax for each model can also be calculated. Table 5 lists all of structures for 4DOF biodynamic models of seated occupants and the corresponding εm ̅ ax . Table 6 lists all of the parameters for each 4DOF biodynamic model of seated occupants corresponding to εm ̅ ax listed in Table 5. Fig. 6 presents the distribution of εm ̅ ax for all of 23 types of 4DOF biodynamic models of seated occupants. As shown in Fig. 6, the results (εm ̅ ax ) of these models are all within the range of 0.85-0.92. In other words, Pareto optimization principle based NSGA-II can identify parameters of the 4DOF biodynamic models of seated occupants effectively and all of 23 types model are capable, to different extents, to describe dynamic responses of seated occupants when under vibration excitations. In Table 5 and Fig. 6, εm ̅ ax value for 4DOF14-9 is the highest, which means that 4DOF14-9 can provide best performance in fitting the experimental data listed in Table 2. εm ̅ ax values for 4DOF12-6, 4DOF12-8, 4DOF14-6, 4DOF14-9, and 4DOF16-4 are all over 0.9, which indicates that all of these five models can provide excellent estimation on the experimental data. As listed in Table 6, however, εm ̅ ax value of 4DOF18 with the most parameters ranks the 13th among the 23 models. Whereas the models with fewer parameters, such as 4DOF12-6, 4DOF12-8, 4DOF14-9, 4DOF14-6, and 4DOF16-4, show better performance. Especially for 4DOF12-6 with only 12 parameters, its εm ̅ ax value ranks the 2nd among the 23 models. Then the conclusion here is: increasing the number of parameters of the 4DOF models does not necessarily improve the model performance.

5. Comparison, analysis, and discussion So far, the majority of studies on 4DOF biodynamic models of seated occupants are based on Wan and Schimmels’ 4DOF series-to-parallel model [31] and Boileau and Rakheja’s 4DOF series model [36], such as the works done by Abbas [33] and Table 3 The corresponding parameters of Pareto solutions for 4DOF18 model (part of results). Solution number

1 2 3 4 5 … 34 35

Stiffness (N m-1,  103)

Mass (kg)

Damping (N s m-1,  102)

m1

m2

m3

m4

k12

k23

k34

k40

k13

k14

k24

c12

c23

c34

c40

c13

c14

c24

6.94 6.16 6.14 6.34 6.31 – 6.84 6.08

6.33 5.31 5.27 16.34 16.32 – 7.07 15.92

5.89 25.44 25.92 8.75 8.69 – 6.89 8.25

16.74 16.78 16.81 24.06 24.18 – 17.79 25.24

36.78 26.70 26.55 33.16 33.16 – 35.71 33.30

21.38 20.31 20.30 21.18 21.18 – 21.37 21.20

6.75 6.37 6.37 6.08 6.08 – 6.50 6.09

76.25 259.96 261.88 54.75 52.87 – 77.12 52.19

21.86 21.76 21.76 20.30 20.30 – 21.68 20.13

13.26 13.82 13.82 13.24 13.24 – 13.29 13.23

27.45 18.60 18.47 11.97 11.97 – 27.00 12.03

9.69 8.45 8.17 11.83 11.83 – 10.21 12.04

11.09 9.35 9.33 10.23 10.23 – 11.05 10.32

18.56 6.58 6.53 25.34 25.39 – 19.89 26.44

24.42 14.13 9.87 18.59 18.62 – 24.23 19.28

8.50 7.76 7.62 11.64 11.69 – 8.50 11.97

8.49 8.64 8.63 16.05 16.05 – 10.37 16.08

27.45 7.51 7.32 29.06 29.17 – 27.47 29.88

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Fig. 5. Pareto surface.

Zhang et al. [37]. According to the results of parameter identification in Section 4, the specific obtained models 4DOF12-6 and 4DOF14-9 are compared with the two typical models and the two models with optimized parameters. The models 4DOF12-6, 4DOF14-9, Wan and Schimmels’ model, and Boileau and Rakheja's model are presented in Fig. 7(a)–(d), respectively. The model structures of the two models with optimized parameters are the same with the two typical models. The parameters of the six 4DOF models are listed in Table 7. The comparisons of the six 4DOF models in terms of STHT, DPMI, and AM are shown in Fig. 8(a)–(c), respectively. As shown in the Figures, all of the six 4DOF models are consistent with experimental data. In other words, all of these models could basically describe the dynamic responses of seated occupants. To quantitatively compare the performance of the models, Fig. 9 shows the comparisons between the goodnesses of fit ε in terms of STHT, DPMI, and AM, and the maximal comprehensive goodnesses of fit εm ̅ ax of the six 4DOF models. As shown in Figs. 8 and 9, 4DOF12-6 and 4DOF14-9 provide better fitting performance than the four models reported in the literature in general. In STHT, DPMI, and AM aspects, 4DOF14-9 improves much and the goodnesses of fit are higher than Abbas et al’s., Boileau and Rakheja’s, and Wan and Schimmels' models. 4DOF14-9 is much better than Zhang et al’s. model in terms of STHT (εSTHT increases by 0.142), while slightly worse in terms of DPMI and AM (εDPMI decreases by 0.013 and εAM decreases by 0.012). As for maximal comprehensive goodness of fit εm ̅ ax , 4DOF14-9 shows the best fitting effectiveness (εmax = 0.919 ). ̅ 4DOF12-6 shows reasonable performance for fitting the experimental data, although it is two parameters less than 4DOF14-9. As shown in Fig. 9 in STHT, DPMI, and AM aspects, the goodnesses of fit of 4DOF12-6 are higher than Abbas et al's., Boileau and Rakheja's, and Wan and Schimmel' models. 4DOF12-6 is much better than Zhang et al's. model in terms of STHT (εSTHT increases by 0.147), while almost the same in terms of DPMI and AM (εDPMI and εAM decrease by 0.002 and 0.033, respectively). As for the maximal comprehensive goodness of fit εm ̅ ax , 4DOF12-6 is the second-highest one in all of these six 4DOF models (εmax = 0.917). ̅ Both 4DOF12-6 and 4DOF14-9 have advantages and disadvantages. In terms of maximal comprehensive goodness of fit, 4DOF14-9 is little higher than 4DOF12-6 (0.002 above). Nevertheless, in terms of goodnesses of fit of STHT and DPMI, 4DOF12-6 is higher (εSTHT increases by 0.005 and εDPMI increases by 0.011), and in term of goodness of fit of AM, it is slightly worse (εAM decreases by 0.021). Boileau and Rakheja’s model and Wan and Schimmels’ model are much worse than 4DOF12-6 and 4DOF14-9. Both of Boileau and Rakheja's model and Wan and Schimmels’ model have deficiencies when compared with these two models. Boileau and Rakheja’s model performs worse than Wan and Schimmels' model for both STHT (εSTHT = 0.768) and AM

Table 4 The goodnesses of fit for STHT, DPMI, and AM and comprehensive goodnesses of fit for corresponding Pareto solutions for 4DOF18 model (part of results). Solution number

εSTHT

εDPMI

εAM

ε̅

Variance

1 2 3 4 5 … 34 35

0.926 0.906 0.916 0.778 0.773 – 0.919 0.779

0.646 0.542 0.458 0.914 0.918 – 0.676 0.909

0.573 0.815 0.796 0.933 0.932 – 0.625 0.930

0.715 0.754 0.723 0.875 0.874 – 0.740 0.872

0.186 0.189 0.237 0.085 0.088 – 0.157 0.082

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Table 5 All of the 23 types of the 4DOF biodynamic models of seated occupants and their maximal comprehensive goodnesses of fit εm ̅ ax .

133

134

Table 6 Parameters identification results for the 23 types of the 4DOF biodynamic models of seated occupants.

12

14

16

18

Model

4DOF12-1 4DOF12-2 4DOF12-3 4DOF12-4 4DOF12-5 4DOF12-6 4DOF12-7 4DOF12-8 4DOF12-9 4DOF14-1 4DOF14-2 4DOF14-3 4DOF14-4 4DOF14-5 4DOF14-6 4DOF14-7 4DOF14-8 4DOF14-9 4DOF16-1 4DOF16-2 4DOF16-3 4DOF16-4 4DOF18

Stiffness (N m-1,  103)

Mass (kg)

Damping (N s m-1,  102)

m1

m2

m3

m4

k12

k23

k34

k40

k13

k14

k24

c12

c23

c34

c40

c13

c14

c24

6.960 5.812 5.931 6.936 6.210 6.184 6.952 5.829 6.339 6.617 6.310 6.446 6.035 6.936 6.102 6.307 6.913 6.142 6.314 6.959 6.111 5.890 6.342

3.602 3.591 4.910 3.783 22.943 10.763 5.693 19.994 8.228 8.612 4.827 7.942 15.877 10.688 13.509 10.577 13.182 8.578 16.342 10.288 25.099 30.427 16.345

8.471 7.454 9.859 9.458 8.236 12.650 8.649 5.475 11.555 5.997 6.177 16.055 6.131 5.195 16.079 14.977 19.642 20.593 5.282 2.200 4.347 4.218 8.754

34.947 36.111 32.867 34.796 16.751 21.161 33.928 22.297 25.752 32.370 37.496 24.672 27.216 32.650 18.792 23.640 14.731 17.791 26.327 35.067 19.574 14.414 24.056

21.649 12.850 53.164 – – – – – – 17.937 22.496 13.872 12.916 5.661 22.405 15.228 8.468 – 17.041 17.651 18.854 – 33.158

45.418 – – 17.316 67.223 31.240 – – – 44.812 46.052 – 14.302 8.850 11.074 – – 22.741 19.487 62.682 – 16.714 21.182

– – – – – – 37.098 17.415 – – – 22.447 37.445 21.263 – 14.442 17.612 20.408 19.403 94.165 40.442 19.054 6.084

67.440 55.792 64.861 72.767 182.197 120.711 54.698 95.215 62.803 81.243 59.362 86.325 74.858 57.293 125.691 65.725 200.976 92.753 106.998 48.073 82.180 195.293 54.751

– 7.042 18.914 37.745 41.176 – 63.786 – 46.290 17.515 22.808 5.193 – – – – 23.285 – 6.656 11.626 24.093 10.292 20.301

– 148.366 – 28.073 – 18.673 – 13.427 196.686 40.303 – – 14.839 – 48.103 6.454 23.653 59.186 16.135 – 25.438 8.539 13.245

46.167 – 217.102 – 32.479 41.521 37.109 18.872 59.023 – 41.110 19.468 – 47.100 37.964 13.822 – 22.281 – 232.126 22.980 15.708 11.971

9.097 11.888 14.713 – – – – – – 11.527 18.959 6.485 8.641 13.678 9.696 7.217 9.626 – 8.811 6.473 9.157 – 11.830

7.837 – – 9.118 10.631 8.370 – – – 11.550 12.262 – 12.6437 11.417 11.141 – – 18.513 6.323 11.652 – 10.137 10.226

– – – – – – 9.608 18.154 – – – 9.851 9.712 11.523 – 15.673 7.291 8.456 8.624 6.447 9.816 6.260 25.339

21.115 19.515 22.325 19.574 11.815 18.501 21.022 18.855 24.281 20.946 20.814 24.254 21.197 21.455 24.358 20.127 10.970 23.580 19.233 19.556 24.306 20.074 18.588

– 10.857 9.258 15.054 24.758 – 8.853 – 8.882 14.485 13.281 9.631 – – – – 7.938 – 7.651 10.220 13.029 7.876 11.640

– 9.330 – 13.948 – 11.939 – 16.032 8.531 9.511 – – 16.689 – 11.045 10.758 6.492 9.497 6.347 – 14.915 7.736 16.047

13.812 – 7.104 – 14.679 9.561 14.765 6.542 20.127 – 11.102 7.844 – 8.681 12.144 8.965 – 8.619 – 9.208 9.447 6.226 29.057

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Number of parameters

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Fig. 6. Maximal comprehensive goodnesses of fit εm ̅ ax for the 23 types of 4DOF biodynamic models of seated occupants.

(εAM = 0. 867), while both Boileau and Rakheja’s model and Wan and Schimmels’ model perform poorly for DPMI (εDPMI = 0. 801). This implies that further study is necessary to improve Boileau and Rakheja's model and Wan and Schimmels’ model. Based on Wan and Schimmels’ model, Abbas et al. optimized the model parameters. All of the fitting performances for STHT, DPMI, and AM are improved a lot. εSTHT , εDPMI , and εAM increase by 0.019, 0.120, and 0.064, respectively, relative to Wan and Schimmels’ model. In addition, the maximal comprehensive goodness of fit of the optimized model increases by 0.010 for Wan and Schimmels’ model. Similarly, Boileau and Rakheja’s model optimized by Zhang et al. shows much better fitting performances in STHT, DPMI, and AM and the maximal comprehensive goodnesses of fit. εSTHT , εDPMI , εAM , and εm ̅ ax increase by 0.019, 0.120, 0.064, and 0.068, respectively. Once again, it verifies the effectiveness of the parameter optimization of Abbas et al. and Zhang et al's. work. As a matter of fact, Zhang et al's. improvement to Boileau and Rakheja's model is more than Abbas et al's. improvement to Wan and Schimmels’ model. As analyzed above, the optimization objectives of this study are only the magnitudes of the terms of STHT, DPMI, and AM. In order to see how the optimization results influence the phases of the terms, Fig. 10(a)–(c) respectively present the phases in terms of STHT, DPMI, and AM using the obtained optimal parameters as listed in Table 7. The model-based computed phases are compared with the experimental data as listed in Table 2. It can clearly be seen from the figures, the fitting performance of the specific obtained models 4DOF12-6 and 4DOF14-9 is much better than the other four models reported in the literature. According to the computing principle for the magnitude and phase of the complex expressions and combining Eqs. (7)–(9) and parameter identification, the phase is actually related to the magnitude. It means that if the magnitudes of the terms of STHT, DPMI, and AM can be fitted with precision, the phases can also be high-efficiently obtained, since all of the model parameters for the magnitude are the same for the phase. Based on the comparison and analysis, the two newly-obtained 4DOF biodynamic models of seated occupants (4DOF12-6 and 4DOF14-9) can provide better fitting performance on experimental data. It means that the systematic study of the 4DOF models is of help for better understanding the biodynamic responses of seated occupants and is of engineering significance for improving the anti-vibration devices and/or test dummies. What is more, it is effective to use NSGA-II (the multiobjective optimization) to conduct parameter identification (the new evaluation criterion considering both the goodness of fit and comprehensive goodness of fit) for the 4DOF biodynamic models. Further attentions need to be drawn. (i) We chose the maximal comprehensive goodness of fit as the main evaluation criterion (the average values of the goodnesses of fit of STHT, DPMI, and AM, i.e., W1 ¼W2 ¼W3 ¼1/3) when evaluating the models, with the goodnesses of fit of STHT, DPMI, and AM as an auxiliary reference. Weighting factors (W1, W2, and W3) could be selected according to specific practical engineering requirements or special requirements. (ii) All of the parameters of the 23 types of the 4DOF models expressed by using equivalent simplification expression method are identified. Either of the parameters of the stiffness-damping pairs could be selectively omitted according to the specific applications (the accuracy of the modified models may not be guaranteed then). (iii) Wan and Schimmels’ model as well as Boileau and Rakheja's model, as presented in Fig. 7(c) and (d), are actually the simplification expressions of 4DOF12-5 and 4DOF12-9 as listed in Table 5.

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z1

m1

z1

m1

m3

z3

m2

z2

m3

z3

z4

m4

z4

m4

z2

m2

z0

z0 (b)

(a)

m1

z1

m1

z1

m2

z2

m2

z2

z3

m3

z3

z4

m4

z4

m3 m4

z0

z0

(c)

(d)

Fig. 7. Structures of the models for comparison: (a) 4DOF12-6, (b) 4DOF14-9, (c) Wan and Schimmels’ model and Abbas et al's. parameter-optimized model, and (d) Boileau and Rakheja’s model and Zhang et al's. parameter-optimized model.

Table 7 The parameters of the models. Model

4DOF12-6 4DOF14-9 Abbas et al. Boileau and Rakheja Wan and Schimmels Zhang et al.

Stiffness (N m-1,  103)

Mass (kg)

Damping (N s m-1,  102)

m1

m2

m3

m4

k12

k23

k34

k40

k13

k14

k24

c12

c23

c34

c40

c13

c14

c24

6.184 6.142 4.170 5.310

10.76 8.578 15.00 8.620

12.65 20.59 5.500 28.49

21.16 17.79 36.00 12.78

– – 167.0 –

31.24 22.74 10.00 183.0

– 20.41 20.00 –

120.7 92.75 49.34 90.00

– – – 310.0

18.67 59.19 – –

41.52 22.28 144.0 162.8

– – 3.100 –

8.370 18.51 2.000 47.50

– 8.456 3.300 –

18.50 23.58 24.75 20.64

– – – 4.000

11.94 9.497 – –

9.564 8.619 9.091 45.85

4.170

15.00 5.500 36.00 134.4 10.00 20.00 49.34 –

–

192.0

2.500 2.000 3.300 24.75 –

–

9.091

70.04 310.0 –

111.5

–

26.98 40.00 –

40.00

5.310 10.45

24.14

15.60

–

150.1

–

40.00 –

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137

(b)

(a)

(c) Fig. 8. Performance comparison of the models in different terms: (a) STHT, (b) DPMI, and (c) AM.

6. Conclusions To help understand the biodynamic responses of seated human occupants and to help design and improve the anti-vibration devices and/or test dummies, it is of great importance and engineering significance to establish simple and effective biodynamic models of seated occupants. 4DOF lumped-parameter biodynamic models have drawn attention for a long time. Based on the existing researches, this paper conducted a systematic and in-depth study on 4DOF lumped-parameter linear biodynamic models. A 4DOF biodynamic model with 18 parameters of seated occupants was introduced. A specific equivalent simplification expression method for the 4DOF biodynamic models was proposed and all of the 23 types of non-duplicative and effective 4DOF biodynamic models of seated occupants were enumerated. NSGA-II based on Pareto optimization principle was used for parameter identification for the 4DOF models. A model evaluation criterion taking into account of both the goodness of fit and comprehensive goodness of fit of the magnitudes of the STHT, DPMI, and AM was proposed, and all of the 23 types of 4DOF models were identified and the corresponding performances were compared, evaluated, and analyzed. Lastly, two specific obtained models

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1

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0.929 0.768 0.934 0.915 0.909

0.9 0.787

0.919 0.908

0.921

0.801 0.82 0.801

0.931 0.919 0.898 0.871 0.868 0.867

0.919 0.917 0.869

0.880 0.859 0.812

0.8 0.7 4DOF 0.6

4DOF

0.5

Abbas et al Boileau and Rakheja

0.4

Wan and Schimmels 0.3

Zhang et al

0.2 0.1 0 STHT

DPMI

AM

Fig. 9. The goodnesses of fit in terms of STHT, DPMI, and AM and maximal comprehensive goodnesses of fit of the models.

(4DOF12-6 and 4DOF 14-9 ) and four reported 4DOF models of seated occupants were compared and analyzed. Based on the research results, concluding remarks are drawn as follows. (i) The total number of the 4DOF linear biodynamic models of seated occupants is limited. All of the 4DOF models can be provided by using the equivalent simplification expression method and the parameters of the models can be effectively calculated via parameter identification. The graphical simplification methodology of the equivalent simplification expression could potentially be expanded to other DOFs and fields, such as the analysis for complicated multi-body dynamics. (ii) Increasing the number of the parameters of 4DOF biodynamic model has little (or no) influence on the goodness of fit and comprehensive goodness of fit of the models. (iii) If the magnitudes of the terms of STHT, DPMI, and AM are fitted with precision, the phases will also be high-efficiently fitted with the experimental data. In both profiles of the magnitude and the phase, the specific obtained models 4DOF12-6 and 4DOF14-9 show better performance on consistency with experimental data than the existing models reported in the literature. (iv) Pareto optimization principle based NSGA-II could effectively conduct parameter identification of 4DOF biodynamic models of seated occupants with multi-objective functions. 4DOF linear biodynamic models investigated in this paper establish the foundations for more in-depth researches and engineering applications of the biodynamics of seated human. How the obtained models could be applied to improve ride comfort of transportation vehicles and the realization of the goal of “comfortable ride” and “healthy drive” will be the next topic.

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139

(b)

(a)

(c) Fig. 10. Phase of the models in different terms: (a) STHT, (b) DPMI, and (c) AM.

Acknowledgments The authors wish to acknowledge the Natural Science Foundation of China (Grant Nos. U1564201 and 51675151), Key Research and Development Project of Anhui Province (Grant No. 1704E1002211), and the Fundamental Research Funds for the Central Universities (Grant No. JZ2017HGTB0202), for their support of this research. The authors also acknowledge Mr. Lipeng Alex Liang, the Boeing Company, for his proofreading. Thanks also go to the anonymous reviewers and the editor for their suggestions and comments, which have helped improve the quality of this paper.

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