Analysis and Optimisation of Ride Vibration of a Heavy-Duty Truck Based on a Vertical-Pitch-Roll Driver-Vehicle Coupled Dynamic Model

The ride comfort and the cargo safety are of great importance in the vibration design of heavy-duty vehicle. Traditional ride comfort design method based on the response of components of vehicles or interaction between human and seat overlooks the most direct criterion, the response of occupants, which makes the optimisation not targeted enough. It will be better to conduct the ride comfort design with the biodynamic response of driver. To this end, a 17-degrees-of-freedom (DOFs) vertical-pitch-roll vehicle dynamic model of a three-axle heavy-duty truck coupled with a 7 DOFs human model is developed. The ride comfort of human body under the vertical, the pitch, and the roll vibrations can be evaluated with the weighted root-mean-square (r.m.s.) acceleration of the driver in multiple directions. The ﬂexibilities of chassis and carriage are also considered to improve the accuracy of the prediction of the ride comfort and to constrain the mounting optimisation of cab and carriage. After validation, the sensitivity analysis of the mounting system, the suspensions, and arrangement of sprung masses is carried out and signiﬁcant factors to ride vibration are identiﬁed. The optimal combination of design parameters is obtained with the objective of minimizing the vibration of the driver and carriage simultaneously. The optimisation result shows that the weighted driver vibration is reduced by 27.9% and the carriage vibration is reduced by 31.8% at various speeds.


Introduction
As the requirements for the ride comfort performance of vehicles increase, the human factor to the vibration design is attracting more and more attention especially for a longdistance transport [1]. Among various types of ride vibration, the rotational motions, that is, the pitch and roll motions, affect the comfort sensation of driver and passengers in an even greater manner than the vertical vibration. Apart from the highway, the heavy-duty truck also travels on rough roads connecting the highway and the destination, which means the cargo safety also matters and needs compromise in the vehicle design [2].
Vehicle is a highly coupled dynamic system consisted of subsystems and components with different levels of mass and size. Dynamic simulation is considered as an effective approach in vibration design of vehicles. Abdelkareem and Xu [3] conducted a bounce and pitch simulation of a half suspension of a heavy truck with 6 DOFs and performed vehicle-level simulations using a 23-degree-of-freedom full truck semitrailer mathematical model with random road surface model [4]. It was also illustrated that the loading condition had considerable effect on vibration modes of vehicle, and the ride performance could be adjusted through the design of seat suspension [5]. From the systematic point of view, some subsystems can be integrated as a lumped mass but some should be considered as a rigid body with shape, continuous body with continuous mass, and flexible distribution or other models. For a heavy-duty vehicle, the chassis should be a flexible body in the dynamic simulation considering its vibration modes greatly affect the vibration transmission between the sprung and unsprung masses [6]. Reference [7] built a multibody dynamic model of a tractor-semitrailer with the flexible cab and chassis. e rigid-flexible coupled model shows good performance of prediction of the ride quality. Reference [8] also showed the flexible-rigid combined dynamic model could represent the rigid motion and the complex shaped deformation simultaneously. An appropriate consideration of subsystem flexibilities of the vehicle helped the ride simulation in both precision and efficiency [9,10]. Among all the subsystems, the flexibility of the chassis is of the greatest importance due to its low resonance frequency and large deformation when the sprung masses and their elastic connections are working supported vibration system. Taking the design of the powertrain mounts as an example, the chassis of the heavyduty truck should be considered as a flexible part to constrain the mounting positions during the truck modelling [11]. In the dynamic simulation, the flexible chassis is always incorporated as a Reyleigh beam, which shows good accuracy and efficiency [12].
In most vehicle-level dynamic studies, the ride vibration is evaluated indirectly through the vibrations of the cab floor, the seat rail, or the interactions between human and seat such as seat surface or backrest [13][14][15][16]. e actual dynamic behaviours of the occupants are not deeply considered. is reduces the model complexity but ignores the effects of dynamic response of the human body on the ride sensation, especially for those resonant responses that may cause great discomfort [17,18]. Taking the driver, seat, and cab as one inseparable system, a 4 DOF model was developed to simulate the ride vibration under measured excitations from cab mounts [19]. A driver-seat-cab model considered the interaction caused by seat suspension was developed to represent the vertical vibration of the driver. Limited by the dimensions of the models, only the vertical accelerations of the quarter car and the driver were described. Reference [20] developed a simple 3 DOFs model to study the drivercushion-car dynamics, and [21] presented a passengers-bus coupled model with a 2 DOFs human model and a 3 DOFs bus model to evaluate the ride comfort of passengers as the bus travelled on different roads at different speeds. It was shown that a passenger-vehicle model was superior when taking the response of occupants directly into optimisation to the traditional ride comfort design method with the vibration of vehicle components.
Most present studies took the vertical vibration as the prior ride index ignoring rotational vibrations that matter in a more considerable extent. It has been proved that a lowamplitude rotational vibration leads to equivalent discomfort caused by a high-amplitude vertical vibration [22,23]. To consider the rotational vibration, the present driver-seatvehicle model needs expansion in rotational DOFs. Under this consideration, a seated human model considering vertical and pitch motions at the same time would help understanding the complex ride comfort caused by transportation vibration.
In this paper, a coupled driver-vehicle dynamic model in vertical-roll-pitch directions is developed to comprehensively analyse the ride comfort in translational and rotational directions. e first two orders of vibration mode of the chassis are considered to simulate the vibration transmission from unsprang mass to the cab and carriage precisely. After the model validation, the effects of design variables on the ride vibrations are discussed and the sensitivities of them are identified. With a surrogate model of ride prediction, the driver comfort and the acceleration r.m.s. of the carriage are optimised together under various travelling speeds.

Vehicle
Model. e dynamic model of the three-axle heavy-duty commercial vehicle consists of the cab, the carriage, the chassis, the suspensions and mounts, the axles, and the tires. e cab and the carriage are supported on the chassis by elastic mounts, and the axles are connected to the chassis with suspensions. All suspensions and elastic mounts are modelled with paralleled springs and dampers with equivalent stiffness and damping coefficients. e schematic of the vehicle model is shown in Figure 1. e vehicle is modelled as a left-right symmetric structure according to the longitudinal axis. e left and right suspensions and mounts are identical, but the left and right tracks are different. So, the axles, the chassis, the cab, and the carriage still can vibrate rotationally around the longitudinal axis.
In total, the vehicle dynamic model has 15 DOFs including the vertical, the pitch, and the roll motions of the cab, the carriage, and the chassis, the vertical and the roll motions of the front, and the middle and the rear axles. [24] is employed to study the dynamic responses of the buttock and thorax of the driver. Seen in Figure 2, the up-right seated human model consists of the thigh, the pelvis, three lower lumbar spinal levels (L3, L4, and L5), the viscera, the upper body from head to L2, and the rotational joints with stiffness and damping properties between them. e thigh and the viscera are assumed to move only in the vertical direction. e remaining parts of the model achieve the bending movements of spine thorough the rotational motion around their joints alone; that is, there is no translational degrees of freedom given to these segments. e seated human body is connected to the seat by spring-damper parallels and the seat is connected to the cab floor in the same way.

Chassis Model.
e chassis of the heavy-duty vehicle is the most flexible substructure whose vibration modes may have significant effect on the response of the superstructures of vehicle. A finite element (FE) model of the chassis with 750525 elements and 1460365 nodes is developed. e constrained modal analysis of the chassis is carried out according to the real connection position of suspensions. Two bending modes at 28.71 Hz (the first-order bending mode) and 35.45 Hz (the second-order bending mode) are found within the ride frequency, 0 to 50 Hz. e mode shapes of the constrained chassis are given in Figure 3.
To consider the two bending modes in the vehicle model, the chassis is simplified as an elastic beam with the corresponding mode shape functions. e bending vibration of the elastic-beam chassis is (1) e nth mode shape function of the bending vibration is where β n � (n + (1/2))(π/l). As the external force f(x i ) loads on x i , a time-dependent Ψ n (t) differential equation can be obtained as where m n is the nth order modal mass of the chassis, ω n is the nth order modal frequency of the chassis, and ξ is the nth order damping ratio of the chassis. ese parameters can be obtained from the modal analysis of the chassis.

Road Excitation.
e road roughness is measured on a public B-class road according to ISO 8608:1995 [25]. An eighty-meter road section (see Figure 4) is used in the simulation. e tracks are straight and zero-paddings are used to settle down the dynamic model in the early iterations.

Numerical Method.
e equations of motion are solved using the Newmark-β method, the iteration step is 0.00001 s, and the dynamic responses are recorded accordingly.

Effects of Vibration Mode of Chassis.
To investigate the effect of the vibration mode of the chassis, three vehicle models are compared, including (1) the 15-DOF model with a rigid chassis, (2) the 16-DOF model considering the first bending mode of the chassis, and (3) the 17-DOF model considering the first two bending modes of the chassis.
Taking the 60 km/h case as an example, the vibrations of the cab and the carriage of the above three vehicle models are calculated and compared, as listed in Table 1. A common pattern is found for the vibrations of the cab and the carriage in different directions: as the number of DOFs of the model increases, the vibrations decrease except the pitch of the cab.
at is to say, the calculated vibration of the uppers tends to decrease when the vibration mode of the chassis is considered. e roll vibration is the most sensitive component to the first vibration mode of the chassis among the three   e second peak at 27. 4 Hz is also suppressed from 2.23 (15-DOF) to 0.78 (17-DOF), which explains why the r.m.s. acceleration in the roll direction has the most obvious reduction among the three directions. For the pitch vibration, a nearly 50% reduction in carriage is found for the first peak at 8.1 Hz when the bending modes of the chassis are considered.

Model Validation.
e vibration acceleration of the heavy-duty vehicle is measured to conduct the model validation. e vibration at the centre of the carriage floor is measured according to ISO 2631-1 1997. In the experiment, the vehicle travels on a straight B-class road at different steady speeds of 40, 50, 60, 70, and 80 km/h. e vibration acceleration caused by the road roughness is recorded for 10 s duration. e comparisons are carried out between the measured acceleration r.m.s. at different vehicle speeds and the simulation results of 15-, 16-, and 17-DOFs models, as shown in Figure 6. Both the experiment and the simulation show a common pattern that the vehicle vibration increases obviously with increasing travel speed. For the three dynamic models, more DOFs lead to lower calculated vibration acceleration r.m.s. of the carriage and better agreement with the experimental results. Relative error between the calculated acceleration r.m.s. with 17-DOF model and the experimental results is less than 4% for all five speeds, showing the best agreement.

Sensitivity of Design Variables on Ride
Vibration. An optimal Latin hypercube design is carried out using the 20 design variables including the stiffness and damping of the chassis suspensions, the seat suspension, the cab mounts, the carriage mounts, and the locations of carriage mounts. e responses are the overall weighted acceleration r.m.s. of the human body and the acceleration r.m.s. of the carriage. e response of the human body is given as follows [26]:    where a v is the overall weighted acceleration r.m.s. of the human body, k sz of 1.0 and a sz are the weighting factor and the acceleration at the thigh in the vertical direction, k bz of 0.4 and a bz are the weighting factor and the acceleration at the back in the vertical direction, and k bx of 0.8 and a bx are the weighting factor and the acceleration at the back in the fore-and-aft direction. e standardized effects of design variables are calculated and shown in Figure 7. e dashed line is the limit calculated according to the T-statistics. A horizontal bar exceeding the limit means the corresponding variable has statistically significant effect on the response. e blue bar means the effect of the design variable on the response is positive, while the red bar means negative effect.
It can be seen the stiffness of the seat suspension, the damping of the cab mounts, the stiffness of the front chassis suspension, the damping of the seat suspension, and the stiffness of the front cab mount affect the driver vibration significantly in descending order. e damping of the chassis suspensions, the stiffness of the rear suspension, the stiffness of the carriage mounts, and the distance between the middle mount and the centre of gravity (CG) of the carriage have significant effects on the carriage vibration.

Surrogate Model of Ride Analysis.
In ride optimisation, a huge number of iterations will be carried out to find the best combination of design variables. Although the dynamic model is developed mathematically and the numerical solution method is quite efficient, the optimisation process is still time-consuming. So, a surrogate model is developed by the response surface method (RSM) based on the data generated by the design of experiment. e acceleration of the human body and the carriage are compared between the RSM predictions and the dynamic model, as shown in Figure 8. R 2 of the human body is 0.999 and that of the carriage is 0.993. e biggest gap between the surrogate model and the dynamic model is less than 5%, which verifies the availability of the surrogate model in optimisation.

Optimisation of Ride Vibration.
e significant parameters of the model are picked as optimisation variables. To seek better balance between the ride comfort and the cargo safety, two objectives are employed: the overall weighted acceleration r.m.s. of the driver and the acceleration of the carriage. Four constrains are set corresponding to the partial frequencies of the front and the rear suspensions and the static flexibilities of the front and the rear suspensions. e optimisation problem can be described as follows: Find: minimise {a v , a carriage }, subject to: 0.8<f f < 1. 15 Hz, 0.98<f r < 1. 3 Hz, 150<δ f < 300 mm, and 128<δ r < 255 mm.
Except for the location of the middle mount of the carriage, the range of other optimisation variables is set as 80% to 120% of the baseline value. Since the mount arrangement affects the vibration of the carriage a lot, the modal analysis of the carriage is carried out to obtain the mode shape for determination of the mounting positions. As shown in Figure 9, a vibration mode is found below 30 Hz falling into the ride frequency range. According to the mode shape, the distance between the middle mount and the carriage CG is set as (1, 1.5) m.  e multiobjective optimisation is carried out by the particle swarm optimisation algorithm. e optimal design variables are listed in Table 2.
Using the optimised parameters, the vibrations of the driver and the carriage are simulated and compared to that calculated with the baseline parameters, as listed in Tables 3   and 4. It can be seen the weighted acceleration r.m.s. of the driver significantly reduces at all travel speeds from 40 km/h to 80 km/h.
In the original condition, there is "a little uncomfortable" at all those speeds according to ISO 2631: 1-1997. After the optimisation, an average of 27.9% reduction of the weighted Shock and Vibration 7 acceleration r.m.s. of the driver is reached. From the subjective respective, the change of environmental vibration can be identified by human being if the variation is greater than 10% from the baseline. As a result, a perceptible improvement in ride comfort is achieved. e vibration reduction in all the three directions improves the ride quality to a higher level comparing with the original "little discomfort" sensation and the ride assessment becomes "no discomfort" for low speeds (i.e., 40 km/h-60 km/h). e objective vibration at higher speeds, that is, 70 km/h and 80 km/h, are suppressed greatly as well, but the reduction of vibration is not great enough to be perceived by the driver. For the cargo safety, the vibration of the carriage at various speeds is reduced at the same time. In the original case, the vibration at 70 km/h is the worst. After optimisation, the most significant vibration attenuation achieves 34.7%. An average of 31.8% vibration decrease is obtained thanks to the comprehensive consideration of driver and carriage.

Conclusions
(1) A 24-DOF vertical-pitch-roll driver-vehicle coupled dynamic model is developed to describe the comprehensive ride vibration of occupant as well as carriage caused by road roughness. e flexibility of the chassis is considered at the same time. e mechanical properties of the mounts and suspensions and the arrangement of the sprung masses are optimised in order to achieve a compromise between the ride comfort and the cargo safety in multiple directions. e main conclusions are drawn as follows. e pitch motion of upper body of the driver plays an important role in the ride comfort evaluation as well as the optimisation of vehicle vibration. e developed 24-DOF driver-vehicle model can reflect the pitch and vertical vibrations of the seated driver, which provides critical information in ride assessment and optimisation.
(2) e flexibility of the chassis is vital considering the mode shapes greatly affect the dynamic responses of driver, cab, and carriage in terms of resonance peak and frequency. e flexibility of the carriage is also important for the mounting arrangement since the modal deformations constrain the mounting positions.    Data Availability e dynamic model data used to support the findings of this study are included within the article and references.

Disclosure
Peng Guo and Jiewei Lin are co-first authors.

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