Research on Cab Vibration Control Based on Parameter Hierarchical Interaction Model

/e vibration level of a cab affects the passenger’s ride comfort and safety significantly. It is of great importance to control the vibration level of cabs under various driving conditions. /e associated vibration transfer paths of cabs are studied by using a hierarchical analysis method of a parameter index. /e multiobjective design analysis is carried out by using the multiparameter joint optimisation design. Further, the optimal control of the cab vibration level is obtained from a full-condition simulation environment. Additionally, a multibody vehicle model is established. /e simulation analysis under multiple working conditions is conducted. /e optimal parameter distribution of the cab mounting structure was established by analysing the influence of the design parameters and experimental verification. /is greatly improves the comfort of the cab.


Introduction
Peoples' requirements for the safety and comfort of commercial vehicles are becoming more stringent due to an improvement in living conditions. NVH (noise, vibration, and harshness) [1][2][3] is an important evaluation index to measure the level of vehicle comfort.
is includes the performance impact of vibration, noise, and other related aspects. e vibration of the cab system directly affects the comfort and safety of the vehicle from its direct contact with the body. Excessive vibration will not only directly affect the comfort of driving and the physical and mental health of the human body. It will also strengthen the wear and tear of cab system parts and reduce the service life of the parts. Additionally, commercial vehicles are the main model of logistics transportation and often run for a long distance, at high speeds, and for a long time. erefore, more attention should be paid to the comfort and safety of the vehicle. Longterm and continuous strong vibrations are also likely to affect the driver's physical and mental health and evaluation of the driving state of the car body. is might lead to traffic accidents [4][5][6]. erefore, the research on vibration control of heavy commercial vehicle cabs is important for the engineering application prospect in today's logistics and highspeed era.
Many scholars have studied different design methods [7][8][9][10][11] of structural parameters of cab systems to improve cab vibrations, analysing balance, stability, and safety of cab systems. e most typical methods include the transfer path analysis method [12,13], the fuzzy control method [14][15][16], and modal analysis method [17,18] and so on. Chen et al. [19] proposed a two-level TPA model [20][21][22] to determine the causes of the cab's higher vibration level. Sun and Zhang [23] proposed that researchers consider the absolute displacement of the pitch motion of cabs beyond the traditional absolute accelerations. is improved the indications of the ride comfort for the suspended cab. Wu et al. [24] uses the finite element method and modal analysis method to establish the cab model and study the cab structure attributes. e human operational comfort was improved by optimising the cab structure. Additionally, the equivalent simple model often makes the objective analysis more accurate. At the same time, it can improve the efficiency and stability of the analysis to achieve better design results. For example, Zhao et al. [25] presented a four degree-of-freedom seat-cab-coupled system model and he proposed a new method of hybrid modelling of seat-cab coupled systems to achieve the optimal design of seat suspension and cab suspensions. is further improved the ride comfort of vehicles. Hansson [26] describes a model for the optimisation of parameters describing the characteristics of a passive nonlinear cab suspension. is minimises the total vibration load on the driver.
However, the original diagnosis method of cab vibration is complex and limited in its working conditions. ere is also a lack of research on the effect of parameters on cab vibration. e analytic method of hierarchical index analysis can be proposed to improve the simplicity of the diagnosis analysis. is would simplify the parameters of the vibration damper on the vibration transmission path [27][28][29][30][31]. Additionally, original methods have a good application effect and convergence for local research objects or limited parameters. However, it is difficult to obtain good global convergence and an implementation effect for multisystem objects, complex matching, and the screening of parameters. erefore, it must be organised the coordination between multisystems and multiparameters. e method of index decomposition can decompose the whole system, explore the parameter flow relationship on the local subsystem, and reduce the number of parameters analysed. e rest of this paper is organised as follows. In Section 2, the parameter index method is applied to establish the parameter hierarchical interaction model. en, Section 3 studies the vibration transmission path of cabs. e multibody vehicle model is established in Section 4. Additionally, the vibration coupling characteristics of cabs were analysed. e simulation under multiple driving conditions is conducted in Section 5. Furthermore, the modal analysis of the subsystem related to cabs is analysed in Section 6. Lastly, the parameter cab suspension system is designed in Section 7, and the conclusions are given in Section 8.

Establishment of Parameter Hierarchical Interaction Model
From the design level, the vibration acceleration response at the cab seat can be taken as the design index of the vibration control. A hierarchical analysis method of parameter index is proposed. It will divide the expected value of products into the system model of each subset. en, the interactive properties can be studied among the subsystems. Its main features are given as follows.

Coordination and Dispersion Characteristics of the Subsystem.
e vibration transfer interference and resonance effect between subsystems will affect the vibration comfort and smoothness of the cab. erefore, it is necessary to consider balancing the relationship between the excitation and subsystems, reducing or eliminating the resonance effect, and improving the damping capacity of the subsystem.

Deep Decomposition Characteristics of System.
e index decomposition of a subsystem must study the parameter attributes of subsystem components deeply. It should further analyse the influence trend and effect of the parameters of the component layer on the index quantity of the subsystem decomposition.

Parameter Influence under Multiple Working Conditions.
e influence of the subsystem parameters on the whole vehicle level index under multiple working conditions is studied. e best adjustment trend of parameters under each working condition is analysed, and the best parameter distribution value is obtained.
Considering the above features, the model of parametric decomposition method can be summarized in Figure 1.
e following refers to the parameter hierarchical interaction model shown in Figure 1 above. Among them, I U 1 and I L 1 represent the actual vibration response and expected vibration response between the vehicle level and decomposition level, respectively. Further, I part1 and I part is the systemlevel vibration response and the expected vibration response on the decomposed part level, respectively. Under the initial condition, I U part and I L part are the system-level vibration response and the expected vibration response on the decomposed part level, respectively. Under the multidynamic condition, x is the system parameter design variable. According to analyse the combination relationship of commercial vehicle system components and the decomposition direction of "excitation source transfer path response target". ese sources suggest that vehicle vibration is decomposed into a vehicle system and a vehicle subsystem. To realize the exchange and adjustment of vibration transmission variables between systems, the obtained response model of vehicle level and system-level indicators is shown in Figure 2. is demonstrates the exchange and adjustment of vibration transmission variables between systems where R represents the mathematical model of vibration response, and f and g represent equality and inequality constraints of parameter, variables, respectively. Further, ξ R and ξ V are the deviation of reference optimisation indexes. Based on the established response model, the vibration transmission path of cabs is researched. e influence of parameter variables on cab vibrations is also analysed.

Analysis of Vibration Transmission Path Associated with Cabs
Under the exciting action of the road to tyre, the vibration frequency response is produced at the relevant evaluation point of the cab. As shown in Figure 3, the main vibration transfer path can be expressed as (1) Tyre centre-the suspension upper fulcrum-cab suspended lower fulcrum-cab centre of mass-driver's centre of mass 2 Shock and Vibration (2) Tyre centre-the suspension upper fulcrum-the leaf spring front fulcrum-front mount lower fulcrumfront mount upper fulcrum-cab centre of massdriver's centre of mass e engine is another main exciting source of commercial vehicles. It may cause abnormal vibration of the parts connected to the frame. is results in huge noise and vibration interferences in the cab. e vibration is transmitted through Whole vehicle layer Figure 1: Parameter hierarchical interaction model.

Subsystem layer
Where Shock and Vibration 3 the engine, the engine mount, and the cab mount system until it finally reaches to the cab. is greatly affects the comfort of the cab. It is necessary to study and analyse the system parameter effects on the response transfer path under different working conditions. is would improve the damping effect of the dampers on the vibration transfer path.

Establishment of Vehicle Model
According to the arrangement of the real vehicle, the cab subsystem, the frame subsystem, the powertrain subsystem, the frame subsystem, and other subsystems are decomposed and established. Concerning the layout of the vehicle, the cab subsystem, frame subsystem, powertrain subsystem, frame subsystem, and other subsystems are established. e real car model is shown in the below figure. Additionally, the corresponding bench structures are established at the bottom of the four wheels of the car body. ey are used to exert vibration and impact excitation on the front and rear tyres, respectively, as shown in Figure 4.
Based on the vehicle model established by ADAMS (Automatic Dynamic Analysis of Mechanical Systems) software, the frequency sweep excitation signal with a frequency of 1∼50 Hz is applied to the four the bench structures [32]. It separately beats at the front bridge, the rear bridge, and the vertical jump. After the complete constraint was applied to the point between the wheel and the bench, the acceleration response of the cab is tested.

Analysis of Vibration Coupling Characteristics of Cab
Under vibration transmission excitation of the frame, the front and back mount arm of the cab will generate a deflection angle between the initial equilibrium position and the initial equilibrium position. is is assuming that the angle between the front suspension motion and rear suspension motion is Δα and Δβ, respectively. e influence of steering constraint of the cab mounting steering mechanism causes the vibration displacement to be generated in other directions. is includes a vertical displacement Δz, horizontal displacement Δx, and lateral displacement. erefore, the vibration response coupling problem exists in cabs in every direction.
is requires detailed analyses and research.      Figure 6, the condition of driving assumes that the exciting force transmitted by the vibration at the bottom of the cab mount is F 1 F 2 F 3 . e response force at the centroid position of the cab can be expressed as F x F y F z M x M y M z following the vibration absorption treatment of the cab mounting damping parameters. Among them, the excitation force is assumed as the excitation that contained the same amplitude and different periodic.
is simulates different vibration response behaviours. Its expression can be set as (i) Pre jump condition and back jump condition. ere exists a signal phase constraint relation between each force: Force F 2 and force F 3 are equal currently, and forceF 1 and force F 4 are identical. Still, there exists a phase angle difference of 180 degrees between force F 1 and force F 3 . At this point, the cab body will maintain a certain delay pitch response.
(ii) Vertical jump condition. Aiming at the vertical vibration response of cab, the signal phase constraint relation between each force can be expressed as: At this time, the excitation vibration signals with equal amplitudes and phases, guaranteeing the stability of the cab's vertical vibration.

Establishment of Dynamic Equation for
Cab. Based on the analysis of mechanical principle, the generalized dynamic equation of vibration dynamics of the cab is as follows: where M represents the mass of the cab mounting coupling system, C and K denote the equivalent damping and stiffness of the cab mounting system, X represents the spatial generalized displacement, and f(t) represents the exciting force. Considering the dynamic response schematic diagram of cabs shown in Figure 1, the parameters are as follows: e vibration responses of a single DOF model can be analysed when considering the inherent frequencies and vibration modes of cab structures.
is is mainly the no damping model or small damping model. On the one hand, equation (4) can be rewritten as follows when considering the no damping model: If pis defined as the attenuation coefficient and ω is the inherent frequency of the cab mount system, then the above formula can be further simplified: Using the formula mentioned above, the formula of the natural frequency of a cab suspension system without damping can be expressed as follows: On the other hand, when considering the small damping effect of the cab suspension system. From this, the vibration response of the cab suspension damping depends on the damping ratio of the coupling system mainly. Its expression can be conveyed as follows:

Shock and Vibration
At this time, the coupling system of the cab suspension mainly vibrates with inherent frequency ������ � ω 2 − p 2 . Its vibration amplitude is attenuated according to e − pt . Its corresponding damping cab mount system can be expressed as follows: In the vehicle engineering field, the natural frequency values of small damping and natural frequencies without damping might approximate (ω ≈ ω r ). erefore, the influence of small damping can be ignored, considering the problem of cab structure resonance. It can further be analysed the structure frequency of the cab's structure.

Simulation under Multiple Driving Conditions.
To simulate the response behaviour of the whole vehicle (such as front jump, back jump and so on), the frequency sweep excitation is applied to the front wheel and rear wheel simultaneously and separately. is simulates the whole vehicle, passing through the brake belt. According to the simulation results, the vibration transfer function of the cab seat is shown in Figure 7.
According to Figure 7, the system transfer rate is the highest ranging from 5.3 Hz to 18.7 Hz. At this time, it is referred to the wheel rotation frequency calculation formula.
is can be expressed as follows: where N represents the modal order, V represents the driving speed, and r indicates the wheel radius. After the calculation, the order frequency of the wheel at each speed is shown in Table 1.
where f 1 represents the first-order tyre rotation frequency, and f 2 represents the second-order tyre rotation frequency. Table 1 shows that the wheel rotation frequencies are 6.25 Hz (first-order) and 18.76 Hz (second-order) when the drive shaft load speed reached 35 km/h and 105 km/h, respectively. ese findings are consistent with the peak frequency values in Figure 7.
e frequency value of the excitation frequency signal might be close to the natural frequency of the system structure. In this case, it is easy to cause a large peak vibration of the system. erefore, the relationship between the frequency of input signal and structure natural frequency needs to be analysed.

Modal Analysis of Cab.
e experimental modal test and analysis of the cab structure are carried out to ensure the accuracy of the cab model. In the modal experiment, there are two kinds of boundary conditions: free boundary and constrained boundary [33][34][35]. Between them, the free boundary means it is completely suspended in the air, does not have any Earth constraint, and has a rigid body mode and elastic mode. is can be realized by a rubber rope, sponge, and airbag during the test. e constraint boundary means that the structure is subject to a certain binding force without a rigid body mode. It only has an elastic mode. It can be fixed by a fixture or constrained according to the actual assembly state during the test. In this paper, the constrained boundary method is used for the experiments. Some measuring points and acquisition instruments are shown in Figure 8.
Combined with the mode value in Figure 9, the modal frequency value of the cab is far from the corresponding rotation frequency value. erefore, the influence of the cab's resonance can be ignored.

Modal Analysis of Additional Subsystems on the Cab Body.
Beyond the modal analysis results of the cab, it is necessary to analyse the structural modes of additional subsystems like the cab mounting system. e constraints between the car body and the excitation bench were set. Following this, the modal solver type is set to the static solver, and the solution step is set to 0.01. e whole vehicle mode is obtained, as shown in Figure 10. Figure 10 demonstrates that the frequency distribution of the front and rear suspension modes of the cab and the corresponding modal modes values ranges from 1 to 5 Hz.
is information is based on the whole vehicle modal analysis. Table 2 shows that there is little difference between the theoretical frequency calculation value and the simulation value of the front and rear cab mounting. is verifies the accuracy of the model. e magnitude of the front and rear suspension frequencies reveals that the resonance response of the suspension structure caused by the frequency offset can be ignored.
is is because there is a significant difference between the bias frequency and the actual wheel rotation

Parameter Design and Verification
e cab suspension decomposition subsystem's design response target, design variables, and constraint relationships can be summarized as follows: Among them, the vibration response acceleration values of the cab seat positions before and after parameter improvement are indicated by R s 0 and R s . I U and I L indicate the acceleration values of upper and lower fulcrum positions of a suspended position, respectively. ξ R 1 , ξ R 2 , ξ R 3 , and ξ R 4 indicate deviations from response acceleration values for four suspended positions at upper and lower pivots. c min and k min indicate lower limit values for stiffness or damping parameters, respectively. Both c max and k max correspond to upper limit values. e optimal design and analysis of the parameters are further conducted based on the determined cab mount parameter variables.

Simplified Design of Mounting Model.
e cab is the main optimisation objective. erefore, the parameter variable optimisation design index of the subsystem is the parameter characteristic for the stiffness and damping of the front and rear suspension. It must be considered the uneven distribution of left and right loads influenced by the driver's weight. From this, both stiffness and damping values of the front and rear suspension can be symmetrically and evenly distributed. Further, the damping can be arranged asymmetrically. en, the cab mass is equivalent to a particle and frequency sweep signal. An input signal frequency range of 1 to 50 Hz is applied at the bottom of each mount, as shown in Figure 11. rough the tested and analysed the cab mounting model in the forward jump, back jump, and vertical direction. e vibration response curves of the cab centroid in all directions are as follows: Figure 12 shows that the frequencies with large centroid responses of the cab appear near 5.19 Hz and 18.59 Hz. Further, the Z-direction vibration responses are large, especially in the vertical jump motion.

Parameters Optimal Design of Cab Suspension.
e human body is most sensitive to changes in acceleration. e Z-direction vibration in the vertical direction has the greatest contribution to the cab vibration. After processing the vibration data of the cab centroid, the original timedomain signal can be converted into the cab Z-direction acceleration power spectrum (PSD) curve in the frequency domain environment. is is because the curve defines the peak value of the numerical square of the signal. erefore, it is more convenient to intuitively analyse the variation of the peak value of the cab centroid acceleration vibration response with the parameters of the parts.
By taking the minimum peak value of the acceleration PSD curve of the cab centroid as the design goal. It can further take the low-order frequency of the cab and the whole vehicle as to the constraint condition. By adjusting the stiffness and damping values of cab suspension (shown in Figure 13), the best target design value is found. Based on the  stiffness and damping data of existing models, it can finetune the adjustment range of stiffness and damping to determine the best stiffness and damping ratio gradually. e previous suspension damping and stiffness matching are illustrated in Figure 14.
As stiffness increases, the natural frequency of the mount moves backwards, and the PSD peak value of the vibration acceleration decreases slightly. At the same time, when the damping value reaches 0.4, the PSD peak value of centroid acceleration is the lowest.
is is seen when the original damping value is expanded by 0.2 variable unit value. By adjusting the stiffness and damping values of the front and rear suspension, the optimal ratio of damping and stiffness is finally determined in Table 3.
After the optimal stiffness and damping ratio of the parts is obtained, the corresponding parameters of the ratio are brought back into the whole vehicle model. en, an experiment can be conducted to verify the accuracy of the ratio. Similarly, the sweep signal is applied to the whole vehicle tyre before and after optimisation to simulate the road signal.
Furthermore, the changes of the PSD curve of cab centroid acceleration before and after stiffness and damping optimisation are tested under the forward jump, backward jump, and vertical working conditions. e results are shown in Figure 15. Figure 15 shows that after optimising the parameters of the cab mount, the acceleration response value of the cab centroid is reduced to an extent. is improves the driver's ride comfort. Additionally, the vibration response at the centre of the cab's mass is shown in Figure 16 when the vehicle is running at a separate speed of 35 km/h and 105 km/h.
Optimisation results of Figure 16 show that the vertical vibration acceleration response value of the cab has reduced. Compared with the original Z-direction vibration acceleration response value of the cab, the optimised cab vibration response is significantly improved. is verifies the effectiveness of the model and method.

Conclusion
is paper aims to the problem of cab vibration. Firstly, the index analytic hierarchy process is proposed and used to analyse the angle of the cab vibration index. e vibration of the cab became the vibration control target. e influence trend of the parameters of the cab mounting layer on the vibration target is analysed. en, based on the study of cab associated vibration transmission path, a virtual simulation model of vibration under multiple working conditions is established. Finally, three typical vibration responses are simulated and analysed according to the actual working conditions and the vibration behaviour of the cab. e parameters of the cab mounting layer under multiple working conditions are also studied and designed based on the model. Combined with the analysis of the experimental test results, the vibration response of the whole vehicle under the tyre rotation excitation frequency is significantly reduced. e comfort of the vehicle is improved.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.