Kinetodynamic Model and Simplified Model of X-Type Seat System with an Integrated Spring Damper

A simplified analytical model with transformation coefficients of X-type seat system with an integrated spring damper for control strategies development is proposed on the basis of a kinetodynamic model. Firstly, based on a commercial seat of trucks, the relationship between the suspension support force and the spring force was created by using virtual work principle. +e analytical formulae of the equivalent stiffness Ke and the stiffness transformation coefficient ρk were deduced. Based on the principle of conservation of energy, the analytical formulae of the equivalent damping coefficient Ce and the damping transformation coefficient ρc were deduced. +en, the motion equation of the simplified model was created. Secondly, the nonlinear dynamic equation of a complex seat model including the kinematic characteristics was established. +irdly, the road test was conducted using a heavy truck to collect the seat vibration signals. Finally, the simplifiedmodel was validated by the tested data and compared with the complex model. +e results show that the accuracy of the simplified model is acceptable. Moreover, the influence laws of kinematic parameters on ρk and ρc were revealed. +e proposed simplified model provides an accurate and efficient tool for designing controllable seat suspension system that minimizes a necessary tuning process.


Introduction
Trucks play an important role in transportation and logistics industry. As products of human civilization and technology progress, they have made great contributions to the social development [1]. Because of the complex and changeable road conditions, the vibration caused by the road excitation and the power assembly not only leads to the damage of the vehicle parts but also reduces the ride comfort [2]. More seriously, it damages the driver's physical and mental health and endangers the driving safety [3].
With the rapid development of the truck industry, improving ride comfort has become a major concern of modern truck designers [4][5][6]. e seat suspension system is an important part of modern trucks [7]. e seat suspension system directly affects the NVH (noise, vibration, and harshness) performance [8]. us, it becomes a key link to improve ride comfort of trucks [9]. e seat suspension system can be mainly divided into three categories: passive seat suspensions [10], semiactive seat suspensions [11], and active seat suspensions [12]. e vibration isolation capability of passive seat suspensions is limited. In order to improve drivers' ride comfort as much as possible, the controllable seat suspension is becoming a research hotspot [13]. e development and implementation of most of the control strategies depend on the dynamic model of the seat suspension system. us, the dynamic model of the seat suspension system is an important foundation for control strategies development and electric control. To facilitate control strategies development, the seat dynamic model should be simplified as much as possible under the premise of meeting the model accuracy. Generally speaking, it is necessary to compromise the model accuracy and the solution time.
e existing dynamic models considering the seat suspension structure are mainly used for the vibration characteristics analysis. For example, Feng and Hu optimized the vibration acceleration transmissibility for seat suspension system with a parallel mechanism [14]. Wang et al. made a hierarchical optimization for scissor seat suspension [15]. Shangguan et al. optimized vehicle-specific seat suspension systems using a seat kinetodynamic model [16]. ese complex models are useful for analyzing the internal and fundamental characteristics of seat mechanisms. ey also benefit parameter analysis and optimization. However, from the point of view of controller design, their equations of motion are of no practical use because they cannot satisfy the real-timeliness for control strategies realization in practical engineering.
In most cases, the classical single DOF (degree of freedom) model of seat suspension system is used for control strategies development. e model is composed of one lumped mass, a linear spring, and a damper [17,18] [22]. Although the classical single DOF model can be used for exploring various kinds of control strategies, it does not establish the relationship between the model equivalent parameters and the suspension structure. us, it cannot be directly used in the design of the seat suspension controller for actual vehicles [23,24]. Actually, the seat suspension kinematic mechanism and the layout form have great influences on the dynamic responses of the seat system [25,26]. Zhang and Zhao researched the nonlinear stiffness and vibration isolation characteristics of scissor-like structure with full types [27]; however, they did not provide the analytical formula of the equivalent damping coefficient of the seat damper. Available studies provide useful references for the design, control, and modeling of seat suspension system. However, there are few studies on providing an accurate simple model of seat suspension system with complex scissor mechanisms for control strategies development. Creating a simple seat model with reasonable accuracy is the primary motivation of this paper.
To facilitate control strategies development, this paper proposed a simplified analytical dynamic model with transformation coefficients of seat suspension system. In Section 2, according to a commercial seat of trucks, the nonlinear relationship between the suspension support force F e and the spring force F s was created by using virtual work principle. Based on the relationship, the analytical formula of the equivalent stiffness K e and the stiffness transformation coefficient ρ k was deduced. Based on the principle of conservation of energy, the analytical formula of the equivalent damping coefficient C e was deduced, and the damping transformation coefficient ρ c was proposed. e motion equation of the simplified model was created. In Section 3, to facilitate comparison, the nonlinear dynamic equation of the seat kinetodynamic model is established by using Lagrange modeling method. In Section 4, the simplified model was validated by the tested data and compared with the kinetodynamic model. In Section 5, the influence laws of kinematic parameters on ρ k and ρ c were revealed.

Development of a Simplified Analytical
Model of Seat Suspension System

Physical Structure and Model of the Seat Suspension
System. e seat suspension with an integrated spring damper has been gradually applied in trucks in recent years because of its compact structure [28]. e integrated spring damper is composed of a hydraulic damper, a spiral spring, and two bushings. e hydraulic damper and the spiral spring are connected in parallel. On the basis of the physical structure of X-type seat, the kinetodynamic model is created as shown in Figure 1(a). e moving parts contains seven rigids: the damper rod, the damper tube, the guide wheel at point P, the guide wheel at point R, the seat pan, the linkage PS, and the linkage OR. e coordinate system is shown in Figure 1(a) and its origin is at S point. In the model, the lengths of the linkages RQ and PS are equal, so denote as L. For simplicity, denote the lengths of AS, BR, and BQ as a, b, and c, respectively. e lengths a and c are called as the lower installation distance and the upper installation distance, respectively. e angle between the linkage PS and the seat floor is θ. Table 1 shows the topological relations among moving parts. is kinetodynamic model is rather complex, and it is inconvenient for control strategies development of the seat suspension.
us, in Figure 1(b) a simplified analytical model with single DOF is proposed, which includes three parameters: m is the effective mass of the driver body and the seat pan supported by the seat suspension, K e is the vertical equivalent stiffness of the seat suspension, C e is the vertical equivalent damping coefficient of the seat suspension. For conveniently translating the complex model to the simplified DOF model, the key problem is how to obtain the analytical formulae of K e and C e . In the following, the kinetodynamic model is denoted as the complex model. e simplified model is the equivalent model.

e Analytical Formula of the Equivalent Stiffness K e .
If the seat pan produces an upward displacement u from the static equilibrium position and q � 0, the point P 0 moves to the new point P. In this condition, u � z − q � z. us, u is defined as the vertical deformation of the seat suspension system. According to the geometric relationship, u can be expressed as Substituting z P � L sin θ and z P0 � L sin θ 0 in Equation (1), we obtain the following: 2 Shock and Vibration From Equation (2), we obtain the following: At the static equilibrium position, the distance l 0 between the spring installation points can be expressed as Substituting x A0 a cos θ 0 , z A0 a sin θ 0 , x B0 c cos θ 0 , and z B0 b sin θ 0 in Equation (4), we obtain the following: 2 , and η sin θ 0 , and Equation (5) can be expressed as When the spring is deformed from the static equilibrium position, the distance l between the spring installation points can be expressed as (7), we obtain the following: Equation (8) can be further expressed as e deformation Δl of the spring can be expressed as According to the principle of virtual work, the following can be obtained: where F e is the vertical external force acting on the seat pan, F s is the spring force, and δ is the virtual displacement. From Equation (10), δ(Δl) can be expressed as  Substituting Equation (12) in Equation (11), we obtain the following: Because the elastic element of the seat suspension is a spiral spring, thus, the spring force F s can be expressed as where c is the precompression deformation of the spring and K s is the spring stiffness. Substituting Equation (14) in Equation (13), the force F e can be further expressed as e vertical equivalent stiffness K e of the seat suspension can be solved by where F n is the vertical elastic restoring force of the seat suspension and F n � F e + (−F e + (m b + m 0 )g), m e is the effective mass of the driver body supported by the seat suspension, and m 0 is the effective mass of the seat pan supported by the seat suspension. According to Equations (15) and (16), K e can be expressed as where the stiffness transformation coefficient ρ k can be expressed as In addition, at the static equilibrium position, let F e � mg. at is According to Equations (15) and (19) and u � 0, we obtain the following: From Equation (20), at the static equilibrium position, the precompression deformation c can be determined by

e Analytical Formula of the Equivalent Damping
Coefficient C e . According to the conservation of energy, the energy dissipated by the damper in simplified model is equal to that in the kinetodynamic model. us, where C d is the damper damping. According to the chain derivation rule, Equation (22) can be expressed as From Equation (23), we obtain the following: From Equation (10), dΔl/du can be expressed as Substituting Equation (25) in Equation (24), the vertical equivalent damping coefficient C e of the seat suspension can be expressed as where the damping transformation coefficient ρ c can be expressed as

e Motion Equation of the Simplified Model.
Substituting u � 0 in Equations (17) and (26), the vertical equivalent stiffness K e and the vertical equivalent damping coefficient C e at the static equilibrium position can be obtained. ey are denoted as K e0 and C e0 , respectively. At the static equilibrium position, the stiffness transformation coefficient and the damping transformation coefficient are denoted as ρ k0 and ρ c0 , respectively. us, K e0 and C e0 can be further, respectively, expressed as Using Newton's second law, the motion equation of the simplified model can be written as

Motion Equation of the Complex Model
e vibration equation of the seat system is created using the Lagrange modeling method in the section. Under the input excitation q of the seat oor, the angle variation is φ, which is designated as the generalized coordinate for the kinetodynamic model [28].
e Lagrange equation of the seat system can be expressed as where, T, V, and D are the kinetic energy, the potential energy, and the dissipated energy of the seat system, respectively. From (30), we obtain the nonlinear motion equation of the kinetodynamic model:

Model Validation and Comparison
In order to verify the e ectiveness of the simpli ed model, the numerical simulations and comparison of the vertical acceleration responses were carried out in this section. e parameters of the seat system are as shown in Table 2. e driver mass is 75 kg.

Veri cation of Random Vibration Response.
To obtain the vibration signals for simulation and validation, according to the national standard GB/T 4970-2009, the road test was carried out and the vibration signals were collected. Based on the complex model and the simpli ed model, the values of the seat pan vertical frequency-weighted RMS acceleration a w were calculated using the measured seat base accelerations at di erent speeds as inputs. e calculated results are shown in Table 3. In order to make it clear to observe the di erences, a histogram is used to provide a comparison of the calculated results, as shown in Figure 2.
From Figure 2, it can be seen that the values of the seat pan vertical frequency-weighted RMS acceleration a w calculated from the tested data are larger than those calculated from the simulated data based on the complex model and the simpli ed model. For the vehicle at lower speeds 50, 60, and 70 km/h, the values of a w calculated from the complex model and the simpli ed model are the same, respectively. However, for the vehicle at higher speeds 80 and 90 km/h, the values of a w calculated from the complex model are slightly larger than those from the simpli ed model. e maximum of the relative error between the simulated values of a w calculated from the complex model and the tested data is 8.3%. e maximum of the relative error between the simulated values of a w calculated from the simpli ed model and the tested data is 10.4%. e results show that the two models can re ect the random acceleration response of the seat pan, and the accuracy of the simpli ed model is slightly lower than that of the complex model. From the point of view of controller design, the accuracy of the simpli ed model   Shock and Vibration 5 can satisfy the control strategies realization in practical engineering.

Veri cation of the Vibration Transmissibility.
In this section, the transmissibility T of the seat system was measured through the test of sine wave sweep. Before the test, the seat was xed on the test bench. Moreover, the driver was replaced by a weight block with 55 kg which is calculated by (8 + 75) × 9.8 × 0.73. On the basis of the frequency sweep method, the transmissibility of the seat system was measured. In addition, the transmissibility of the complex model was calculated based on the frequency sweep method, and the transmissibility of the simpli ed model was analytically calculated using the transfer function from q to z. A comparison of the transmissibility curves is shown in Figure 3. Based on Figure 3, the values of the natural frequency f 0 and the maximal transmissibility T max were extracted. A comparison of the two characteristic parameters of the seat system is shown in Table 4. From Figure 3, it can be seen that the simulation curves of the transmissibility T coincides with that of the measured curve. ere is a certain deviation in the frequency bands 0.5∼1.5 Hz and 8.0∼10.0 Hz. e transmissibility curve calculated from the simpli ed model can re ect the real transmission characteristics of the seat suspension system. Table 4 shows that the tested f 0 is the same as the simulated f 0 ; the absolute deviation between the tested T max and the simulated T max is 0.01, and the relative deviation is 1.0%. e results further prove that the simpli ed model is e ective. e main reason for the deviations is that the friction and gaps between the moving parts of the scissor mechanism are ignored in the simpli ed model.  Figure 4, it can be seen that the transmissibility curves calculated from the simpli ed model are in good agreement with those calculated from the complex model at di erent values of m b . Moreover, as the value of the mass m b increases, the natural frequency f 0 decreases, however, the maximal transmissibility T max increases.

Comparison of Harmonic Response.
e vibration response analysis of the seat system under harmonic excitation is one of the e ective methods to investigate the di erences of the models and the real system. Figure 5 provides a comparison of the seat pan acceleration responses for the seat system under harmonic excitations with the amplitude A 0 10 mm and the excitation frequency f 1.0 and 2.0 Hz.
From Figure 5, it can be seen that the simpli ed model can be considered suitable for reproducing the real seat acceleration response.

Influence Analysis of Kinematic
Parameters on ρ k , ρ c , K e , and C e In this section, the in uence laws of kinematic parameters on the sti ness transformation coe cient ρ k , the damping transformation coe cient ρ c , the equivalent sti ness K e , and the equivalent damping C e were revealed on the basis of the values for the seat system in Table 2. e in uences of the kinematic parameters a and c on the sti ness transformation coe cient ρ k and the equivalent sti ness K e are shown in Figure 6 and Figure 7, respectively. e in uences of the kinematic parameters a and c on the damping transformation coe cient ρ c and the equivalent damping C e are shown in Figure 8 and Figure 9, respectively. Figure 6(a) shows that the sti ness transformation coe cient ρ k approximately linearly increases with the increase of the suspension deformation u. Moreover, with the increase of the kinematic parameters a, the sti ness transformation coe cient ρ k decreases. Figure 6(b) depicts the same rules of the kinematic parameters a and c on the equivalent sti ness K e . Figure 7 further illustrates the above rules. From Figure 8, it can be seen that the damping transformation coe cient ρ c also approximately linearly increases with the increase of the suspension deformation u.
e larger the values of the kinematic parameters a and c are, the smaller the value of the damping transformation coe cient ρ c is. Similar change characteristics are also observed to occur in Figure 9.

Conclusions
For the seat suspension system, the development and implementation of most of the control strategies depend on its dynamic model. e primary motivation of this paper is to create a simple seat model with reasonable accuracy. us, a simpli ed analytical model with transformation coe cients of seat suspension with a complex scissor mechanism for control strategies development is proposed. Firstly, the analytical formulae of the equivalent sti ness K e and the sti ness transformation coe cient ρ k were deduced. e analytical formulae of the equivalent damping coe cient C e and the damping transformation coe cient ρ c were also deduced. en, the motion equation of the simpli ed model was created. Secondly, the nonlinear dynamic equation of a complex seat model including the kinematic characteristics is established. Finally, the simpli ed model was validated by the tested data and compared with the complex model and the tested data. Moreover, the in uence laws of kinematic parameters on ρ k and ρ c were revealed.
(1) rough the accuracy veri cation of random vibration response, it can be seen that the accuracy of the simpli ed model can satisfy the control strategies realization in practical engineering. (2) By comparison of the vibration transmissibility, the results show that the transmissibility curve calculated from the simpli ed model can re ect the real transmission characteristics of the seat suspension system. (3) By comparison of harmonic response, the results show that the simpli ed model can be considered suitable for reproducing the real seat acceleration response. (4) Both the sti ness transformation coe cient ρ k and the damping transformation coe cient ρ c approximately linearly increase with the increase of the suspension deformation u. With the increase of the kinematic parameters a and c, both ρ k and ρ c decrease. e proposed model provides an accurate and e cient tool for designing controllable seat suspension system that minimizes a necessary tuning process. In the following study, the nonlinear sti ness transformation and the asymmetric damping transformation would be an interesting topic about getting reasonable simple models for control strategies development.
Data Availability e data used to support the ndings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no con icts of interest.  Shock and Vibration 9