Research Article  Open Access
Matching, Stability, and Vibration Analysis of Nonlinear Suspension System for Truck Cabs
Abstract
To improve comfort, a nonlinear suspension system is proposed on the basis of the nonlinear vibration isolation theory and the installation space of the cab suspension system for trucks. This system is suitable for allfloating cabs. For easy matching and design, the static and stability characteristics of the suspension system were analyzed, respectively, and the boundary condition for the stability of the system was given. Moreover, the cab simulation model was established, and the dynamic simulation was conducted. The stability analysis shows that the smaller the vibration excitation of the cab system, the higher its stability is. The dynamic simulation results show that the acceleration of the cab with the nonlinear suspension system is effectively suppressed; the dynamic deflection of the suspension is kept within a certain range, and the design space of the suspension stroke can be effectively utilized. Compared with the traditional linear suspension system, the nonlinear suspension system has better vibration isolation characteristics and can effectively improve ride comfort.
1. Introduction
Transportation industry plays an important role in the national economy of any country. As the main body of the transport industry, the stability and ride comfort of trucks have attracted much attention [1, 2]. The suspension system of truck cabs, as one of the main vibration isolation systems, directly affects the ride comfort of drivers. Scholars have conducted a large number of theoretical and experimental studies on this issue [3–5].
The vibration isolation system for cabs has various forms. With the continuous improvement of driving comfort requirements, the allfloating cab has become gradually popular [6]. The air suspension system is only used in the cab of the highend vehicles because of its high price. The common coil springs or rubber springs are usually used in the suspension system of the cab for ordinary vehicles. However, their high dynamic stiffness and poor lowfrequency vibration isolation result in poor ride comfort. Previous studies on cab vibration of commercial vehicles mostly focused on the optimization of passive suspensions [7, 8]. In order to improve vehicle comfort, some scholars have adopted active or semiactive control methods [9, 10], but they have not been applied in practice for many years. For passive suspensions, if their stiffness is larger, the ride comfort will be worse; if their stiffness is smaller, the suspension deflection will be larger. The high static stiffness and lowdynamic stiffness suspension is an effective solution to the above contradictions. The design of many systems involves stability issues [11, 12]. In the design of such suspension systems, attention should be paid to the stability of the system.
Many scholars have conducted a lot of research on the vibration isolation system with high static stiffness and low dynamic stiffness. The most typical structure consists of two oblique springs and one vertical linear spring [13]. Some scholars also applied active control method [14] and new materials [15] to generate negative stiffness, so as to enhance the isolation performance of the vibration isolation system. Zhou et al. [16] analyzed the nonlinear dynamic characteristics of a quasizerostiffness vibration isolator. Huang et al. [17] proposed a method to design a quasizerostiffness system by the parallel connection of the linear spring and the Euler buckling beam with negative stiffness. Meng et al. [18] proposed a design of a quasizerostiffness isolator based on a disk spring and studied the influence of system parameters on the force transfer rate by means of the average method. Le et al. [19] designed a quasizerostiffness vibration isolation system by parallel connection of two connecting rods and springs, applied it to the vibration isolation of automobile seats, and achieved good vibration isolation effect. Xu et al. [20] designed a low frequency vibration isolator with quasizerostiffness characteristics by using the magnetic spring. Carrella et al. [21] designed a vibration isolation system with high static stiffness and low dynamic stiffness by combining the linear spring and the magnet spring. In [22], a method for obtaining negative stiffness by using the nonlinear interval of hyperelastic materials was proposed. A method of generating negative stiffness by using the magnetic nonlinearity was proposed in [23]. Zhou et al. researched a novel quasizerostiffness strut and its applications [24, 25]. In [26], a Stewart isolator with negative stiffness magnetic springs was analyzed.
Previous studies have mostly focused on the form, principle, and characteristics of the quasizerostiffness isolators and rarely applied them to the cab vibration isolation system. Based on the nonlinear vibration isolation theory and the installation space of the suspension system, this paper applied the nonlinear suspension system to the allfloating cab. Moreover, its static and stability characteristics were analyzed, respectively. The simulation model of the truck cab was established by using MATLAB software, and the comfort simulation was carried out.
2. The Cab with the Nonlinear Suspension System
Figure 1 shows the nonlinear suspension system for truck cabs, which consists of the front suspension and the rear suspension. “l_{ij}” denotes the distance from the mounting point of the suspension to the cab mass center. “i” denotes “f” and “r,” respectively; “j” denotes “R” and “L,” respectively; “f,” “r,” “R,” and “L” denotes front, rear, right, and left, respectively. Each suspension uses a pair of transverse linear springs to generate negative stiffness in the vertical direction, and a quasizerostiffness vibration isolation system is formed in parallel with the vertical positive stiffness linear springs. At the static equilibrium position, the negative stiffness produced by the negative stiffness adjusting mechanism and the positive stiffness produced by the vertical spring cancel each other, and the quasizerostiffness characteristic can be obtained [27].
In order to facilitate engineering application, the original cab system is equivalent to four singledegreeoffreedom systems, and matching research is carried out separately. The equivalent system is shown in Figure 2, where m_{i} is the equivalent spring mass of the suspension system; K_{H} is the stiffness of the transverse spring; L_{H} is the installation space of the transverse spring; K_{V} is the stiffness of the vertical spring; C is the damping coefficient of the suspension system; z_{i} is the vertical displacement of the equivalent spring mass; and q_{i} is the vertical displacement excitation of the suspension system.
3. Matching and Static Characteristic Analysis of the Nonlinear Suspension System
The matching and the static characteristics of the nonlinear suspension system are important considerations in the cab design. After the suspension space structure and the performance parameters are determined, the dynamic characteristics of the system are determined accordingly. Therefore, first of all, it is necessary to match the structure and performance parameters of the mount system.
3.1. Matching of the Nonlinear Suspension System
When the cab weight is applied to the suspension system, the mass supported by each suspension can be obtained by force analysis:
In practical engineering applications, it can be approximately considered that the center of the cab mass coincides with the geometric center, thus
After static balance, the transverse spring is just in the horizontal position (recorded as the equilibrium position). The relationship between the compression value L_{V}, the equivalent mass m_{i}, and the vertical spring stiffness K_{V} can be expressed as follows:
According to the vertical motion space of the cab, L_{V} can be determined. According to equation (3), K_{V} can be determined as
According to the lateral mounting space of the transverse spring, L_{H} can be determined. According to the geometric relationship, the original length L_{0} of the transverse spring can be determined by comparing before and after static balance:
According to the geometric relationship, when the relative motion between the frame and the cab occurs, the relative displacement u can be expressed as
According to the principle of virtual work, when the relative displacement u occurs, the relationship between the elastic vertical force F and the relative displacement u of the nonlinear suspension system cab be expressed as follows:
For the traditional linear suspension system, the relationship between the vertical elastic force F and the displacement u of the linear suspension system is as follows:
The stiffness characteristics of the nonlinear suspension system can be obtained by calculating the derivative of F to u by equation (7):
Let z = 0, the stiffness K_{e} of the nonlinear suspension system at the static equilibrium position can be expressed as
According to the natural frequency f_{i} required by the suspension system, the value of K_{e} can be determined as
According to the value of K_{e} and equations (10) and (11), the stiffness K_{H} of the transverse spring can be determined as
In special cases, in order to achieve zero stiffness at the equilibrium position, K_{e} = 0, so the stiffness of the transverse spring can be further expressed as
From equation (9), we obtain the following:
Let K − K_{V} ≤ 0 and based on equation (14), the interval that the stiffness of the nonlinear suspension is less than that of the linear suspension can be obtained as
The interval that the stiffness of the nonlinear suspension is larger than that of the linear suspension can be obtained as
For example, in order to improve the comfort of a heavy truck, the nonlinear suspension system for the cab can be matched according to the method provided in this study. The cab mass center and the geometric center approximately coincide. The cab mass m equals 880 kg. According to the installation space requirement of the suspension system, we can determine L_{V} = 45 mm and L_{H} = 60 mm. In order to achieve zero stiffness at static equilibrium position, K_{H} = 95.82 N/mm and K_{V} = 47.91 N/mm are determined, respectively. Figure 3 is the relationship curve between the vertical elastic force F and the displacement u of the suspension system. From Figure 3, it can be seen that near the equilibrium position, the stiffness of the nonlinear suspension system is close to zero. When u ∈ [−24, 24] mm, the dynamic stiffness of the nonlinear suspension system is lower than that of the traditional linear suspension system, which can attenuate the cab vibration better. When u < [−45, −24] mm or u < [24, 45] mm, the dynamic stiffness of the nonlinear suspension system is greater than that of the traditional linear suspension system, which can better limit the largescale vibration of the cab.
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3.2. Influence of System Parameters on Static Characteristics
According to the matching method mentioned above, the static characteristics of the nonlinear suspension system are affected by the natural frequency f_{i}, the compression value L_{V}, and the installation space L_{H} of the transverse spring. In order to grasp the influence of parameters on the static characteristics, Figure 4 was drawn based on equation (9). Figure 4(a) shows that with the increase of the natural frequency f_{i} of the static equilibrium position, the lowfrequency vibration isolation region [−24, 24] mm remains unchanged, but the stiffness of the region increases and that of other regions decreases. From Figure 4(b), it can be seen that increasing the installation space L_{H} of the transverse spring is helpful in reducing the stiffness of the lowfrequency vibration isolation area. From Figure 4(c), it can be seen that increasing the compression value L_{V} is helpful in reducing the stiffness of the lowfrequency vibration isolation region, but the stiffness of the limit region is also reduced.
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4. Analysis of Amplitudefrequency Characteristics and Stability
Let the exact analytical expression of the nonlinear spring force be F(u), then the Taylor expansion of F(u) at u = u_{0} is expressed as
In order to explore the inherent characteristics of the cab with the nonlinear suspension system, the thirdorder Taylor expansion of F(u) is adopted in this study:
Therefore, the relationship between the elastic vertical force F and the displacement u of the nonlinear suspension system can be approximately expressed aswhere and .
A comparison of the curves between the elastic vertical force F and the displacement u of the nonlinear suspension system determined by formulas (7) and (19) is shown in Figure 5. By comparison, the approximate curve can approximate the exact curve better. Therefore, it is feasible to approximate the exact analytical expression of the nonlinear spring force by using the thirdorder Taylor expansion.
From Figure 2, the vibration equation of the equivalent cab system can be obtained as follows:
Substituting equations (6) and (19) into equation (20), we obtain the following:
Equation (21) can be further expressed as
Suppose the excitation of the frame iswhere Y_{0} is the vibration acceleration amplitude of the frame.
Substituting equation (23) into equation (22), we obtain the following:
By the identical transformation of equation (24), we obtain the following:
By the variable substitution for equation (25), we obtain the following:where n is the attenuation coefficient, p is the undamped natural frequency, and 2n = c/m, p^{2} = k_{1}/m, and ε = k_{3}/m.
Suppose the principal resonance response near ω ≈ p for equation (26) is as follows:where A is the amplitude of the relative displacement u under the simple harmonic oscillation.
Equation (27) is substituted into equation (26), and the third harmonic is ignored. Then, let the corresponding coefficients on both sides equal, we obtain the following equation:
Based on equation (28), the equation of amplitudefrequency characteristic curve can be obtained as follows [28]:
Calculating the derivative of equation (29) to A, we get
Simplifying equation (30), we obtain the following:
The stability condition of periodic solutions is as follows:
For example, the amplitudefrequency characteristic curves of the designed cab with the nonlinear suspension system in this paper under different excitation accelerations are shown in Figure 6. To show the advantage of the nonlinear suspension, the characteristic curves of the linear suspension with the vertical stiffness K_{V} = 47.91 N/mm were also plotted in Figure 6. The analysis shows that with the increase of the excitation intensity, the peak of the vibration response shifts to the right, and the unstable region increases gradually. At the same time, the response amplitude A also increases. Thus, for the cab with the nonlinear suspension system, the smaller the vibration excitation, the better the stability is. In addition, it can also be seen that the response amplitude A of the nonlinear suspension is obviously less than that of the linear suspension for Y_{0} = 20 m/s^{2} in the range of 0∼2.5 Hz; for the response amplitude A, there is a slight difference between the two suspensions for Y_{0} = 10 m/s^{2}. The results show that, in the stable region, when the excitation is larger, the advantages of the nonlinear suspension are more obvious.
5. Dynamic Modelling
The cab system is a very complex multidegreeoffreedom elastic structure. In order to compare and verify the isolation effect of the nonlinear suspension, a 3DOF (degreeoffreedom) model of the cab system is considered in this paper. According to Figure 1, the differential equations of the vibration system can be obtained as follows:where for the force F_{i} (i = 1, 2, 3, 4), when determined by equation (7), it is a nonlinear elastic force and when determined by equation (8), it is a linear elastic force.
According to the above dynamic equations, the 3DOF simulation model of the cab system is established by using MATLAB software, as shown in Figure 7. On this basis, the performance analysis of the nonlinear suspension system can be carried out. The model parameters used in the analysis are shown in Table 1.

6. Dynamic Simulation Analysis
In order to verify the validity and applicability of the cab with the nonlinear suspension system designed in this paper, this section focuses on the analysis of acceleration and suspension travel response characteristics by comparing with the linear suspension system. In this paper, the random vibration signals of the frame collected by a heavy truck driving at a speed of 75 km/h on the highway (in [29]) are used as the model excitation.
6.1. Simulation Analysis of the Cab Vertical Acceleration
In order to analyze the comfort improvement of the cab with the nonlinear suspension system, the timedomain response and the power spectral density of the cab vertical acceleration were numerically modeled based on the simulation model. In this paper, the time T = 100 s is chosen to calculate the cab vertical acceleration. The calculation results are shown in Figure 8.
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Figure 8(a) shows that compared with the traditional linear suspension, the vertical acceleration of the cab with the nonlinear suspension system decreases significantly and the RMS (root mean square) acceleration value decreases by 49.1%. Figure 8(b) shows that the power spectral density of the cab vertical vibration acceleration decreases significantly in the lowfrequency stage after the use of the nonlinear suspension. The results show that, in the lowfrequency stage, the nonlinear suspension system has good vibration isolation characteristics, which can effectively reduce the cab vibration and improve ride comfort.
6.2. Simulation Analysis of the Suspension Dynamic Deflection
Figure 9 is the simulation result of the suspension dynamic deflection. From Figure 9, it can be seen that the change trend of the travel of the nonlinear suspension is basically the same as that of the linear suspension. The dynamic deflection of each nonlinear suspension is not more than 45 mm, and the travel is within the allowable range. After using the nonlinear suspension system, the dynamic deflection values of the four suspensions increase slightly, thus making full use of the suspension design space. The results show that the nonlinear suspension system can well limit the largescale motion of the cab and avoid collision with the limit block. Therefore, the proposed engineering design scheme of the nonlinear suspension system for truck cabs is feasible.
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7. Conclusions
Based on the nonlinear vibration isolation principle, a nonlinear suspension system for the allfloating cab was designed in this paper. The static characteristics and stability of the system were analyzed. The influences of the nonlinear suspension on the mechanical properties of the cab were simulated and analyzed by using MATLAB. Through this study, the following conclusions were drawn:(1)With the increase of the natural frequency f_{i} of the static equilibrium position, the lowfrequency vibration isolation region remains unchanged, but the stiffness of the region increases and that of other regions decreases. Increasing the installation space L_{H} of the transverse spring is helpful to reduce the stiffness of the lowfrequency vibration isolation area. Increasing the compression value L_{V} is helpful in reducing the stiffness of the lowfrequency vibration isolation region, but the stiffness of the limit region is also reduced.(2)With the increase of the excitation intensity, the peak of the vibration response shifts to the right and the unstable region increases gradually. Meanwhile, the response amplitude A also increases. For the cab with the nonlinear suspension system, the smaller the vibration excitation, the higher the stability is.(3)Compared with the linear suspension system, the vibration isolation frequency range of the nonlinear suspension system is wider, especially in the lowfrequency stage. It has good vibration isolation characteristics, which can effectively reduce cab vibration and improve ride comfort. Moreover, after using the nonlinear suspension system, the dynamic stiffness of the suspension decreases and the dynamic deflection increases slightly, making full use of the design space of the suspension stroke.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (51575325).
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Copyright
Copyright © 2019 Fuxing Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.