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Research on optimal matching of vehicle suspension parameters for improving vehicle ride comfort on bump road

Research on optimal matching of vehicle suspension parameters for improving vehicle ride comfort... A rigid-flexible dynamics model is built. In view of established 7-DOF vibration model, the expressions of the output indexes are derived and the influence mechanism of suspension parameters on ride comfort is revealed. The effects of different vehicle speeds on ride comfort on bump road are studied in the frequency and time domains by analyzing the suspension dynamic disturbance by taking the vertical acceleration of the foot floor and seat rail, the wheel dynamic loads, and the suspension dynamic deflections of the suspension as the objects of analysis. The effects of dynamic changes in the suspension spring and damper parameters on the time and frequency domain response indices are ana- lyzed. Two window functions are used on the bump road to process and analyze the time domain charts of the front and rear of foot floors and seat rails in the vertical direction, and the Rms values of the global and local vibrations of the foot floor and seat rail obtained are used as one of the evaluation indicators of ride comfort. The evaluation index of the total vehicle considering local and global vibration is determined, and the suspension parameters are collaboratively optimized by genetic algorithm to improve vehicle ride comfort. Keywords Bump road, vehicle speed, suspension parameters, ride comfort, collaborative optimization Date received: 4 January 2023; accepted: 26 April 2023 Handling Editor: Chenhui Liang vehicle ride comfort is the suspension, and the reason- Introduction able matching of suspension structure parameters has When the vehicle is running, the unevenness of the road 1,2 an important impact on the ride comfort. And its and the excitation of the transmission system and other good and bad directly affect the vehicle ride comfort factors will cause the vibration of the vehicle, which will performance and the driving process of the advantages affect the ride comfort. As the vehicle speed increases, and disadvantages. the demand for vehicle ride comfort is also increasing. The vehicle ride comfort is mainly to keep the vibration and shock generated by the vehicle in driving within a Faculty of Transportation Engineering, Kunming University of Science and certain range on the comfort of the occupants. Road Technology, Kunming, China surface unevenness and vehicle speed form the excita- Corresponding author: tion of the vehicle vibration system, this excitation Jin Gao, Faculty of Transportation Engineering, Kunming University of through the tires, suspension, seats and other elastic Science and Technology, No. 727 South Jingming Road, Chenggong and damping elements finally act on the human body. District, Kunming 650500, China. In this process, the most important part that affects the Email: 906845822@qq.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 Advances in Mechanical Engineering Uys et al. aimed at optimizing the spring and shock hybrid control strategy was more able to balance the absorber parameters for the Land Rover Defender 110 vehicle ride comfort and vehicle handling stability. model for optimal ride comfort on different surfaces Bello et al. designed a control method for a suspension and at different speeds. The Dynamic-Q algorithm is structure by first constructing a quarter vehicle state used to optimize the vehicle speed from 10 to 50 km/h, space model with two degrees of freedom and a com- and it is concluded that the rear shock absorber is as plete state feedback controller. Simulations using soft as possible, and the front shock absorber is the Simulink showed that the active suspension system pro- most comfortable from soft to hard according to the duces a good dynamic response even at resonant fre- road condition and speed, and the vehicle ride comfort quencies and had better ride comfort compared to the is related to the elastic stiffness of the rear shock absor- passive suspension system. Arslan et al. designed a ber. Qi et al. proposed a novel suspension structure fuzzy logic sliding mode controller and conducted with hydraulic linkage suspension and electronically dynamics simulation experiments on the performance controlled air springs in order to balance the vehicle of this controller to analyze the biodynamic response handing stability and vehicle ride comfort. By simplify- of the body to the vehicle vibration system. Numerical ing the whole vehicle system with the 9-DOF, hydrauli- results showed that the controller overcomes the short- cally linked suspension model and electric air springs, comings of previous control methods and greatly the relationship between handling stability and comfort reduced the vehicle vibration. Park et al. compared at different suspension heights with different suspen- the vehicle ride comfort of passenger vehicles with sus- sion structures is analyzed. Shi et al. investigated the pension systems equipped with two different magnetor- effect of negative stiffness on the performance of sus- heological dampers, with the difference between the pension systems. This negative stiffness structure con- two MRs being the presence or absence of a bypass sists of a passive magnetic negative stiffness spring and hole in the piston. Simulations and experiments have a modified locating mandrel, and it was experimentally proved that MR dampers with bypass holes provide demonstrated that the increase in negative stiffness better ride comfort than dampers without bypass resulted in better vehicle ride comfort. For semi-active holes. Seifi et al. pit forward a multi-objective optimi- suspensions, negative stiffness mitigated the effect of zation method using a nonlinear damper model to damping in the semi-active oscillator and improved improve the vehicle ride comfort and vehicle handing vehicle ride comfort. Trupti et al. adopted a two- stability. A simulation analysis of the suspension sys- degree-of-freedom quarter virtual prototype model tem under sinusoidal road excitation was carried out with Simulink module to analyze its ride comfort and using a five-degree-of-freedom model of the vehicle. dynamic response of the road surface in order to The results showed that the average ride comfort of the enhance the ride comfort for passenger. And the effects front and rear suspension improved by 3.58% and the 13 14 of suspension damping coefficient, stiffness, spring load handling stability improved by 0.74%. Silveira et al. mass and speed on the smoothness of the virtual proto- researched the effect of damper damping asymmetry on type system were analyzed. Cebon designed a test vehicle ride comfort based on the one-fourth vehicle method for the hysteresis characteristics of leaf springs vibration model and found that the vehicle ride com- to investigate the response of the hysteresis characteris- fort was better considering the damper damping asym- tics of a vehicle model using two degrees of freedom metry than not considering its effect. under random vibration. By conducting random vibra- The above scholars have conducted local studies on tion tests on different forms of multi-piece leaf springs, suspension and vehicle performance from the perspec- the hysteresis charts of leaf springs at different excita- tives of structural parameters, suspension form, virtual tion frequencies were obtained, and the test results prototype modeling, and construction of approximate obtained can be consistent with the empirical equations modeling. However, the vehicle ride comfort perfor- of the leaf spring hysteresis model. Abood and Khan mance when the vehicle vibrates on a bump road has researched the effect of vertical secondary suspension not been systematically studied in depth, and most of stiffness on railroad. The simulation results of the vehi- the scholars have conducted studies based on random cle ride comfort using a railroad vehicle simulation road surfaces. The systematic effect of vehicle speed on model as an example. A combination of linear the vehicle ride comfort is also not explored in combina- Kalk theory and nonlinear heuristic models was used tion with the frequency and time domain perspectives. to optimize the suspension spring stiffness and improve Secondly, when the vehicle is disturbed by external the vehicle ride comfort. Zhang et al. analyzed a vari- excitation, the local vibration and global vibration of ety of the suspension structures based on the vehicle the vehicle are not considered. Finally, based on the one-fourth suspension model to compare the specific premise that the vehicle is subject to local vibration and operation of semi-active suspensions under skyhook global vibration, this is no research on the collaborative control, ground-hook control and hybrid control stra- optimization and matching of vehicle suspension para- tegies, respectively. And the results showed that the meters and vehicle driving ride comfort based on the Gao and Han 3 whole vehicle rigid-flexible coupling model for a specific front and rear suspension structure. To address the unresolved issues, this research paper considers the following aspects: (1) The effect of differ- ent vehicle speeds on vehicle ride comfort of driving on bump road is investigated from the frequency domain and time domain respectively; (2) The output index expression is derived from the 7-DOF vehicle vibration model, and the influence mechanism of spring stiffness and damper damping parameters on vehicle ride com- fort is revealed; (3) The systematic effects of suspension spring stiffness and damper damping on the response indices of the vehicle in the time and frequency domains are systematically analyzed; (4) Based on the premise of considering the local and global vibration of the foot floor and seat rail of the front and rear, the window function is designed to process and analyze the time domain charts of the foot floor and seat rail on the ver- Figure 1. 7-DOF vehicle vibration system model. tical direction; (5) The evaluation index of total vehicle ride comfort considering local vibration and global vibration is determined, and the parameters of the front According to Figure 1, the force expression F of the and rear suspension are matched and optimized by suspension and body joint is obtained, that is: multi-objective genetic optimization algorithm to improve the vehicle ride comfort. F = K z  z + C z_  z_ ð1Þ i i i + 3 i i + 3 ðÞ i + 3 0 ðÞ i + 3 0 The vertical displacement z , z , z , z of the four 40 50 60 70 7-DOF vehicle vibration system model points where the upper and lower parts of the front and rear wheels are connected to the body are: For studying the dynamic characteristics of the suspension and revealing the systematic influence of the suspension z = z  L z  L z 40 1 1 2 3 3 parameters on the vehicle ride comfort, a 7-DOF vehicle vibration system model is established in this paper, as z = z + L z  L z 50 1 2 2 3 3 ð2Þ showninFigure1. z = z  L z + L z > 60 1 1 2 4 3 The coordinate system is set to oxyz, o is located at z = z + L z + L z 70 1 2 2 4 3 the center of mass of the body, x is the driving direction of the vehicle. First assume that wheel number 1 is the From equations (1) and (2), the motion equation of right front wheel, 2 is the left front wheel, 3 is the right the body in three directions can be derived as follows: rear wheel, and 4 is the left rear wheel. In Figure 1, L , L is the distance from the center of M €z = F ð3Þ 1 2 s 1 i mass to the center plane of the left and right wheels, I €z = M ðÞ F =ðÞ F + F L ðÞ F + F L ð4Þ L , L is the distance from the center of mass to the x 2 x i 2 4 2 1 3 1 3 4 front and rear axles, and M is the body mass. I €z = M ðÞ F =ðÞ F + F L ðÞ F + F L ð5Þ y 3 y i 3 4 4 1 2 3 m (i = 1, 2, 3, 4) is the unsprung mass, K (i = 1, 2, 3, 4) i i is the suspension stiffness, C (i = 1, 2, 3, 4) is the suspension damping, K (i = 1, 2, 3, 4) is the tire stiff- ti The motion equation of the unsprung mass is as ness, q (t)(i = 1, 2, 3, 4) is the road surface unevenness follows: factor, v is the excitation frequency. I is the rotational inertia of the vehicle body around the x-axis, I is the m €z + C z_  z_ + K z  z i i + 3 i i + 3 i i + 3 ðÞ i + 3 0 ðÞ i + 3 0 i i rotational inertia of the vehicle body around the y-axis, + K½ z  q = 0ðÞ i = 1, 2, 3, 4 i i + 3 i z is the vertical displacement of the body M ,z is the 1 s 2 ð6Þ roll angle of body with inertia I , z is the pitch angle of x 3 body with inertia I ,z (i = 4, 5, 6, 7) is the vertical y i Bringing equations (1) and (2) into equations (3)–(6), displacement of four tires m ;m ,z (i = 4, 5, 6, 7) is the 1 4 i0 the vibration equation is obtained as follows: vertical displacement of the upper and lower part of the front and rear wheels at the connection with the € _ ½ M Z +½ C Z +½ K Z =½ K Q ð7Þ body. 4 Advances in Mechanical Engineering 2 3 C  L C + L C  L C  L C  C  C  C  C i 1 1 2 2 3 1 3 2 1 2 3 3 6 7 6 7 i = 1 6 7 6 7 L C + L C L C + L C 1 3 2 4 4 3 4 4 6 7 6 2 2 7 L C + L C L C + L C L L C  L L C L C  L C L C  L C 1 1 2 2 1 1 2 2 1 3 1 2 3 2 1 1 2 2 1 3 2 2 6 7 6 7 2 2 6 7 L C + L C L C + L C  L L C + L L C 1 3 2 4 1 3 2 4 1 4 3 2 4 4 6 7 6 2 2 7 L C  L C L L C  L L C L C + L C L C L C  L C  L C ½C = 3 1 3 2 1 3 :1 2 3 :2 3 1 2 3 1 3 2 4 3 4 4 6 7 6 7 2 2 6 7 L C + L C L L C + L L C L C + L C 4 3 4 4 1 4 1 2 4 4 3 4 4 4 6 7 6 7 6 C L L L C C 00 0 7 1 1 1 3 1 1 6 7 6 7 C  L C L C 0 C 00 2 2 2 3 2 2 6 7 6 7 4 C L C  L C 00 C 0 5 3 1 3 4 3 3 C  L C  L C 00 0 C 4 2 2 4 4 4 In equation (7), M is the mass matrix. ½M = diag(M I I m m m m ) and C is the damping matrix. K is s x y 1 2 3 4 ti 2 3 00 0 0 6 7 00 0 0 6 7 6 7 00 0 0 6 7 6 7 the tire stiffness, ½K  = K 00 0 ;Q is the input vector, Q = ½q q q q  ,Z is the output vec- t t1 1 2 3 4 6 7 6 7 0 K 00 t2 6 7 4 5 00 K 0 t3 00 0 K t4 tor, Z = ½z z z z z z z  and K is the stiffness matrix. 1 2 3 4 5 6 7 Where 2 3 K  L K + L K  L K  L K  K  K  K  K i 1 1 2 2 3 1 3 2 1 2 3 3 6 7 6 i = 1 7 6 7 6  L K + L K L K + L K 7 1 3 2 4 4 3 4 4 6 7 6 2 2 7 L K + L K L K + L K L L K  L L K L K  L K L K  L K 1 1 2 2 1 1 2 2 1 3 1 2 3 2 1 1 2 2 1 3 2 2 6 7 6 7 2 2 6 7 L K + L K L K + L K  L L K + L L K 1 3 2 4 1 3 2 4 1 4 3 2 4 4 6 7 6 2 2 7 L K  L K L L K  L L K L K + L K L K L K  L K  L K ½K = 3 1 3 2 1 3 1 2 3 2 3 1 2 3 1 3 2 4 3 4 4 6 7 6 7 2 2 6 7 L K + L K  L L K + L L K L K + L K 4 3 4 4 1 4 3 2 4 4 4 3 4 6 7 6 7 K L K L K K 00 0 6 7 1 1 1 3 1 1 6 7 6 7 K  L K L K 0 K 00 2 2 2 3 2 2 6 7 6 7 4 K L K  L K 00 K 0 5 3 1 3 4 3 3 K  L K  L K 00 0 K 4 2 2 4 4 4 The output indicators of the 7-DOF whole-vehicle vibration system are body acceleration, dynamic load, and dynamic deflection, respectively. The acceleration includes not only the vertical acceleration, but also the body roll acceleration and pitch accel- eration, as well as the vertical acceleration of the four wheels and the body connection. Similarly, the dynamic tire loads and suspension dynamic deflection include the relative dynamic loads and suspension dynamic deflections at the four wheels. The transfer function of a 7-DOF system is expressed as follows: 2 1 H(jv)=(  v ½M + jv½C + ½K) ½Kð8Þ The output power spectral density matrix is connection to the input power spectral matrix by the following equation: ½G  = ½H½G ½H  ð9Þ z q In equation (9), ½H is the system frequency response function matrix, ½H  is the conjugate transpose of ½H.The relationship between the Rms value of the response s and the power spectral density G is: zi zizi Gao and Han 5 2 2 2 2 2 2 2 E(f )= E(z )+ L E(z )+ L E(z )+ E(z ) s 1 1 2 3 3 4 > 1 D =s = G (f )df ð10Þ zi zi z z i i < 2 2 2 2 2 2 2 E(f )= E(z )+ L E(z )+ L E(z )+ E(z ) s 1 2 2 3 3 5 ð17Þ 2 2 2 2 2 2 2 In equation (10), D is the variance of the response, E(f )= E(z )+ L E(z )+ L E(z )+ E(z ) zi > s 1 1 2 4 3 6 s is the Rms value. 2 2 2 2 2 2 2 zi E(f )= E(z )+ L E(z )+ L E(z )+ E(z ) s 1 2 2 4 3 7 1) Rms value of the second order derivative of the response. 3) Relative dynamic load. The second order derivative Z ðÞ t of the body accel- The static load of the wheels relative to the ground eration as a function of the frequency response to Q (t) is: is: L L 4 2 G = M g > 1 s ZðÞ v 2 > ~ (L + L )(L + L ) H ðÞ jv =  v ð11Þ 3 4 1 2 ij > QðÞ v j L L 4 1 G = M g 2 s (L + L )(L + L ) Z (v) 3 4 1 2 where H (jv)= , so there is ð18Þ ij L L 3 2 Q (v) > G = M g 3 s (L + L )(L + L ) > 3 4 1 2 ~ > H ðÞ jv =  v H ðÞ v ð12Þ ij ij L L 3 1 : G = M g 4 s Then the Rms value of the second order derivative of (L + L )(L + L ) 3 4 1 2 the body acceleration response is: The dynamic load of the tire is s = G (f )df ð13Þ zi z z i i F = F + m €z ð19Þ 0 d i i i + 3 Taking equation (1) into (19) yields equation (20), 2) Rms value of suspension dynamic deflection. that is: From the definition of suspension dynamic distur- F =KðÞ z z +L z +L z +CðÞ z_ z_ +L z_ +L z_ +m €z > d1 1 4 1 1 2 3 3 1 4 1 1 2 3 3 1 4 bance, it can get: F =KðÞ z z +L z +L z +CðÞ z_ z_ +L z_ +L z_ +m €z d 2 5 1 2 2 3 3 2 5 1 1 2 4 3 2 5 f = z  z ð14Þ > s (i + 3)0 (i + 3) F =KðÞ z z +L z +L z +CðÞ z_ z_ +L z_ +L z_ +m €z d 3 6 1 1 2 4 3 3 6 1 1 2 4 3 3 6 i > 3 F =KðÞ z z L z +L z +CðÞ z_ z_ +L z_ L z_ +m €z d 4 7 1 2 2 4 3 4 7 1 2 2 4 3 4 7 Bringing equations (2)–(14) yields the following ð20Þ equation, so that is: The Fourier change to equation (20) is: f = z  L z  L z  z s 1 1 2 3 3 4 f = z + L z  L z  z s 1 2 2 3 3 5 ð15Þ F (jv)=(K v m +jC v)z (jv)(K +jC v)z (jv) f = z  L z + L z  z > d 1 1 1 4 1 1 1 s 1 2 2 4 3 6 > 1 : > f = z + L z + L z  z > +(K L +jC vL )z (jv)+(K L +jC vL )z (jv) s 1 2 2 4 3 7 4 1 1 1 1 2 1 3 1 3 3 > 2 F (jv)=(K v m +jC v)z (jv)(K +jC v)z (jv) > d 2 2 2 5 2 2 2 Squaring both sides of equation (15) yields, that is: > (K L +jC vL )z (jv)+(K L +jC vL )z (jv) 2 2 2 2 2 2 3 2 3 3 2 2 2 2 2 2 2 2 >f =z +L z +L z +z 2L z z 2L z z 2z z F (jv)=(K v m +jC v)z (jv)(K +jC v)z (jv) > d 3 3 3 6 3 3 1 s 1 1 2 3 3 4 1 1 2 3 1 3 1 4 3 > 1 > +2L L z z +2L z z +2L z z +(K L +jC vL )z (jv)+(K L +jC vL )z (jv) > 3 1 3 3 2 3 4 3 4 3 > 1 3 2 3 1 2 4 3 3 4 > 2 2 2 2 2 2 2 2 f =z +L z +L z +z +2L z z 2L z z 2z z >F (jv)=(K v m +jC v)z (jv)(K +jC v)z (jv) d 4 4 4 7 4 4 1 > s 1 2 2 3 3 5 2 1 2 3 1 2 1 5 4 2 > (K L +jC vL )z (jv)(K L +jC vL )z (jv) 2L L z z +2L z z +2L z z 4 2 4 4 2 4 4 4 4 3 2 3 2 3 2 2 5 3 3 5 2 2 2 2 2 2 2 f =z +L z +L z +z 2L z z +2L z z 2z z ð21Þ > s 1 2 2 3 3 6 1 1 2 4 1 3 1 6 > 2L L z z +2L z z 2L z z 1 4 2 3 1 2 6 4 3 6 > F (jv) 2 2 2 2 2 2 2 Defining H (jv)= , and having >f =z +L z +L z +z +2L z z +2L z z 2z z F ;Q s 1 2 2 3 3 7 2 1 2 4 1 3 1 7 d j > i Q (jv) +2L L z z 2L z z 2L z z 1 4 2 3 2 2 7 4 3 7 Z (v) H (jv)= , it follows that: ij ð16Þ Q (v) Since z (i = 1, 2, :::, 7) is an independent random ½H  = ½A½Hð22Þ variable, E(Z )= 0 (i = 1, 2, :::, 7).Therefore, the Rms value of the suspension dynamic disturbance is: where 6 Advances in Mechanical Engineering ½ A = 2 3 K jvC K L +jvC L K L +jvC L K v m +jvC 00 0 1 1 1 1 1 1 1 3 1 3 2 1 1 6 7 K jvC K L jvC L K L +jvC L 0 K v m +jvC 00 6 2 2 2 2 2 2 2 3 2 3 2 2 2 7 6 7 4 5 K jvC K L jvC L K L jvC L 00 K v m +jvC 0 3 3 3 1 3 3 3 4 3 4 3 3 3 K jvC K L jvC L K L jvC L 00 0 K v m +jvC 4 4 4 2 4 4 4 4 4 4 4 4 4 Then the relative dynamic loads are respectively: Based on the virtual simulation environment, road F =G , F =G , F =G , F =G ð23Þ d 1 d 2 d 3 d 4 i 2 3 4 unevenness is often derived by inverse derivation of the road unevenness power spectrum density. Depending From the above derivation process of the output on the road power spectrum density, road unevenness is index of the whole vibration system, the mechanism of segmented into different road levers. There are various the vehicle ride comfort of the influence of the spring methods to generate the frequency domain model of stiffness and damper damping on the can be summar- road unevenness, such as: filtered white noise method, ized as follows: harmonic superposition method, Poisson method, The difference of spring stiffness and damper damp- Fourier inverse variation method, etc. ing of the suspension will affect the suspension stiffness Therefore, when performing ride comfort simula- and suspension damping respectively, which in turn will tion, the corresponding road model file should be cre- affect the transfer function jointly solved by the road ated first. In this paper, it uses the more commonly unevenness coefficient, the road input spectrum deter- used harmonic superposition method to generate a fre- mined by the vehicle speed and the suspension para- quency domain model of the road spectrum. The basic meters, thus affecting the frequency response function idea is to represent the road unevenness function as a matrix of the output index, and ultimately affecting the sum of a number of sinusoidal functions with random vehicle ride comfort. phases. Based on the harmonic superposition method, the Establishment of the vehicle multi-body subject gets the required 3D road unevenness charac- dynamics model and construction of bump teristics by matlab programing simulation, and then road generates pavement files according to the road model criteria, and then imports it into the professional Based on the vehicle three-dimensional model test data dynamic software, so as to construct and get the bump and reference to a real vehicle parameter to establish road. the vehicle rigid-flexible coupling multi-body dynamics Referring to the relevant provisions of the national model of the front MacPherson and rear torsion beam 17,18 standard GB/T 4970-2009, for the pulse input in suspension, as shown in Figure 2. The main structural the ride comfort analysis process, the bump road should design parameters of the established vehicle model are be used. The corresponding values of length, width and shown in Table 1. height are taken differently for different types of vehi- cles. Since the object of this paper is a passenger car, the length, width and height of the bumps established according to the regulations are 100 mm 3 25 mm 3 60 mm. The simulation model of the bump road built in ADAMS is shown in Figure 3. Effect of different vehicle speeds on ride comfort on bump road The test evaluation vehicle speed is 30–60 km/h, and the ride comfort simulation is carried out on the bump road surface respectively, and the simulated results of the fil- tered of the vertical acceleration of the foot floor and seat rail of the front and rear, dynamic load of front and rear wheels and suspension dynamic deflection of front and rear suspension in time domain response are obtained in Figures 4 and 5. For the purpose of explor- Figure 2. The whole vehicle multi-body dynamics model. ing the frequency domain characteristics of the vehicle, Gao and Han 7 Table 1. Structure parameters of the whole vehicle. Parameters Value Parameters Value Sprung mass (kg) 1097.00 Height from center of mass to ground (m) 0.38 Unsprung mass (kg) 551.98 Distance between front wheels (m) 1.59 Center of mass to front wheelbase (m) 0.71 Distance between rear wheels (m) 1.52 Center of mass to rear wheelbase (m) 1.85 Wheelbase (m) 2.56 4 4 Spring stiffness of front suspension (N/m) 2.13 10 Spring stiffness of rear suspension (N/m) 2.273 10 4 3 Damper stiffness of front suspension (Ns/m) 1.43 10 Damper stiffness of rear suspension (Ns/m) 0.913 10 From Figures 4(a) to (d) and 5(a) to (d), it can be seen that the amplitude of each response quantity of the front or rear suspension system on the bump road tends to increase in general with the increase of the vehicle speed compared to the previous speed. It means that the increase of vehicle speed will increase the amplitude of the bump road excitation, the stronger the system vibration, which is not conducive to the vehicle ride comfort. The amount of system response is affected to Figure 3. Bump road. some extent for both the front and rear suspensions, but does not change the time domain distribution. the time domain system response is transformed into Another comparison shows that the rear suspension is the power spectral density corresponding to the fre- subjected to a slightly larger amplitude of vibration quency domain system response after Fourier variation than the front suspension. and then filtering in Figures 6 and 7. Figure 4. Time domain response of the front suspension after filtering: (a) vertical acceleration of the front foot floor, (b) vertical acceleration of the front seat rail, (c) dynamic load of the front wheel, and (d) suspension dynamic deflection of the front suspension. 8 Advances in Mechanical Engineering Figure 5. Time domain response of the rear suspension after filtering: (a) vertical acceleration of the rear foot floor, (b) vertical acceleration of the rear seat rail, (c) dynamic load of the rear wheel, and (d) suspension dynamic deflection of the rear suspension. From Figures 6(a) to (d) and 7(a) to (d), it can be Effect of spring and damper parameters seen that the power spectral density of each response on time and frequency domain response quantity of the front and rear suspension system on the From actual engineering design experience, take the bump road has a rising trend in the overall amplitude rear sus-pension as an example, so that the spring change with the increase of the vehicle speed, indicating stiffness or damper damping is reduced to one-half or that the increase in vehicle speed will increase the increased to two times of the original value, other amplitude of the bump road excitation, and the parameter values remain unchanged. stronger the system vibration is, which is not conducive The vehicle is driven at a constant speed of 30 km/h to the vehicle ride comfort. It is known that the power on a bump road, and the simulation of the vehicle spectral density vibration of each response quantity of vibration system is performed to obtain a comparison the front or rear suspension system basically corre- of the time domain and frequency domain charts of the sponds to the time domain diagram, and the frequency system response quantities under the variation of domain distribution does not change with the vehicle spring stiffness and damper damping values. Due to speed increase. the limitation of space, only the time domain variation Moreover, it is known that the rear suspension is of the vertical acceleration of the rear seat rail is subjected to a slightly larger amplitude of vibration presented in Figures 8 and 9 (The initial values are than the front suspension. And the physical meaning of noted as Baseline in the figure in order to distinguish the power spectrum density is the energy conversion the different parameter values.). per unit of time, which can be obtained in the rear foot From the results in the time and frequency domains, floor and seat rail, dynamic load of the rear wheel and the working space of rear suspension, the body to it can be seen that when the spring stiffness is increased overcome the energy required for vibration than the to twice the original value, the amplitude of each front suspension on some large. response quantity of the system has increased to some Gao and Han 9 Figure 6. Frequency domain response after filtering of the front suspension: (a) power spectral density of the vertical acceleration of the front foot floor, (b) power spectral density of the vertical acceleration of the front seat rail, (c) power spectral density of the dynamic load of the front wheel, and (d) power spectral density of the suspension dynamic deflection of the front suspension. extent compared to the original value, especially the of the original value is almost the same. When the dam- per damping is reduced to one-half times of the original magnitude of the vertical acceleration of the foot floor value, the change of each response quantity of the sys- and seat rail of the front and rear, the suspension tem from the original value is the opposite. For the fre- dynamic deflection of the front and rear suspension, quency domain results, the power spectral density and the dynamic load of the rear suspension increases vibration of each response quantity of the front or rear more, and dynamic load of the front wheel increases suspension system basically corresponds to the time slightly. When the spring stiffness is reduced to one- domain variation. half of the original value, and the response of the sys- Therefore, the variation of suspension spring stiffness tem is reduced to some extent compared to the original and damper damping will have an impact on the time value, but the magnitude of the reduction is small. For and frequency domain response indicators of the car, the frequency domain results, the power spectral den- thus affecting the vehicle ride comfort, which is coincide sity vibration of each response quantity of the front or with the theoretical analysis in Section 3 above. rear suspension system basically corresponds to the The above analysis shows that: time domain response variation. When the damper damping is increased to twice the 1) In a certain range of stiffness, the appropriate original value, the amplitude of the system response is reduction of spring stiffness in a certain degree to reduced to a certain extent compared to the original reduce the amplitude of fluctuations in the par- value, especially the vertical acceleration of the foot tial response of the system, such as vertical accel- floor and seat rail of the front and rear, and the eration of the front and rear foot floor and seat dynamic load of the rear wheel. The suspension rail, suspension dynamic deflection. While the dynamic deflection of the front and rear suspension is tire dynamic load increases slightly, the overall also reduced to a certain extent, but the magnitude of system vibration intensity is reduced. Therefore, the reduction is slightly smaller. The dynamic load on the spring stiffness is reduced within a certain the front wheel is almost unchanged and the magnitude range to improve the vehicle ride comfort. 10 Advances in Mechanical Engineering Figure 7. Frequency domain response after filtering of the rear suspension: (a) power spectral density of the vertical acceleration of the rear foot floor, (b) power spectral density of the vertical acceleration of the rear seat rail, (c) power spectral density of the dynamic load of the rear wheel, and (d) power spectral density of the suspension dynamic deflection of the rear suspension. Figure 8. Time domain variation of vertical acceleration of the Figure 9. Time domain variation of vertical acceleration of the rear seat rail under the change of spring stiffness. rear seat rail under the change of damper damping. 2) In a certain range of stiffness, the appropriate seat rail, suspension dynamic deflection of the reduction of damper damping in a certain degree front and rear suspension, dynamic load of the to reduce the amplitude of fluctuations in the rear wheel. Therefore, the damper damping is partial response of the system, such as vertical increased within a certain range to improve the acceleration of the front and rear foot floor and vehicle ride comfort. Gao and Han 11 Figure 10. Variation charts of local vibration and global vibration: (a) vertical direction of the front foot floor, (b) vertical direction of the front seat rail, (c) vertical direction of the rear foot floor, and (d) vertical direction of the rear seat rail. Analysis of the vehicle ride comfort on between the effects of front and rear wheel. RmsGlobal is the vertical acceleration profile calculated adopting a bump road large calculation window to account for the impact In this paper, transient vibrations are considered by effects of all wheels. using the continuous Rms method, which is calculated RmsLocal is the maximum difference between the by integration over a short period of time. This is the period during which the bumps pass and before the root mean square value of the acceleration in the verti- bumps pass RmsLocal. RmsGlobal is the maximum dif- cal direction after weighting the acceleration, which is ference between the period during which the bumps simplified as Rms. By using two different calculation pass and before the bumps pass RmsGlobal.And windows to calculate the charts of vertical acceleration RmsGlobal, RmsLocal are adopted to study transient Rms, the following can be done: signal changes. Figure 10 displays the time domain signal charts of the vertical acceleration of the foot floor and seat rail t + Mloc 0 2 > RmsLocal = a (t)dt of the front and rear when the vehicle passes the bump > v Mloc road at a speed of 30 km/h. In other words, the change t + MGlob 0 2 chart of local vibration RmsLocal and global vibration 2 ð24Þ RmsGlobal = a (t)dt RmsGlobal. MGloc It is clear from the figure that both the curves of Mloc = 3:6   0:52 both local vibration RmsLocal and global vibration MGlob = 3:6   1:05 RmsGlobal have two significant peaks, corresponding In equation (24), Mloc, MGlob is the vibration time, to the front and rear wheels of the vehicle when passing L is the wheelbase, v is the vehicle speed, t is the initial over the bump road. vibration time, and a is the root mean square value of The value of DRmsLocal in the figure is greater than the weighted acceleration. DRmsGlobal regardless of the front or rear foot floor RmsLocal is the vertical acceleration profile calculated and seat rail. The values of Rms, DRmsLocal, and adopting a narrow calculation window to distinguish DRmsGlobal for the front foot floor in vertical direction 12 Advances in Mechanical Engineering f = min DRmsLocal Table 2. Optimization of design variables. > 1 f foot > f = min DRmsLocal 2 R foot Design variables Lower Initial Upper > > f = min DRmsGlbal 3 f foot limit value limit > f = min DRmsGlbal 4 R foot f = min DRmsLocal > 5 f seatrail Scale factor of front suspension 0.5 1 2 f = minðÞ DRmsLocal spring stiffness > 6 R seatrail Scale factor of rear suspension 0.5 1 2 f = min DRmsGlbal 7 f seatrail ð25Þ spring stiffness f = minðÞ DRmsGlbal > 8 R seatrail Scale factor of front suspension 0.5 1 2 f = min Rms damper damping 9 F df =G Scale factor of rear suspension 0.5 1 2 > f = min Rms 10 F damper damping > dr=G f = min Rms > 11 f > f f = min Rms 12 f are smaller than those for the rear suspension, and the The constraints are as follows: values of Rms, DRmsLocal, and DRmsGlobal for the ver- tical front seat rail in vertical direction are smaller than 0:5ł f ł 2 < i those for the rear suspension. Then it means that for 0:5ł c ł 2 ð26Þ the foot floor or seat rail of front and rear, the vibra- f ł (½f =3) s i tion excitation of the front foot floor is less than that of the rear foot floor when passing over the bump road. In equation (26), f is a constraint on the scaling fac- The vibration excitation of the front seat rail is also less tor of the spring stiffness, c is a constraint on the scal- than that of the rear seat rail, so the ride comfort of the ing factor of the damper damping, f is the standard rear driver is worse than that of the front passenger, deviation of the suspension dynamic deflection, ½f  is and the rear suspension is less comfort than the front the suspension limiting travel i = f , r. suspension. Optimization algorithm and optimization results Optimized matching In this paper, the multi-objective optimization algo- Determination of optimization variables and objective rithm PE’s HMGP sub-algorithm is used for the multi- function objective collaborative optimization of spring stiffness and damper damping of the front and rear suspen- The suspension springs and dampers play the role of sions. The whole vehicle model is driven at a uniform shock mitigation and vibration suppression respectively speed of 30 km/h on a bump road, and the Pareto solu- during the driving process of the vehicle. Based on tion set based on ride comfort obtained through itera- the fact that the above two types of components play a tive optimization calculations after building an decisive role on ride comfort, the spring stiffness and integrated platform. Then all the solutions in the Pareto damper damping of the suspension are used as optimi- solution set are compared, and the optimal solution set zation variables. is selected by coordinating trade-off and compromises To facilitate the simulation calculation, the spring among the objectives to make each subobjective as opti- stiffness and damper damping are multiplied by a scale mal as possible. factor to change the size respectively, and the design Only some of the Pareto solution sets of the optimiza- variables are in Table 2. tion objective in the optimization space are listed due to Optimization objectives include the Rms values of the thelengthofthe article, as showninFigures 11 and12. local f = DRmsLocal , f =DRmsLocal and the 1 f foot 2 R foot The blue dots in the Pareto illustration set are the Pareto global f = DRmsGlobal , f = DRmsGlobal 3 f foot 4 R foot front solutions. The changes of parameters before and vibrations of front and rear foot floor in the vertical after optimization are shown in Table 3. From the results direction, the Rms values of local f =DRmsLocal , 5 f seatrail in Table 3, it can be seen that the evaluation indexes of f = DRmsLocal and global f =DRmsGlobal , 6 R seatrail 7 f seatrail ride comfort have been effectively improved after optimi- f = DRmsGlobal vibrations of front and rear seat 8 R seatrail zation, then the vehicle ride comfort has been improved. rail in the vertical direction, the Rms value of dynamic load of front and rear tires f = Rms , f = Rms 9 F 10 F d =G dr =G Conclusion and the Rms value of suspension dynamic deflection of front and rear suspension f = Rms f = Rms . 11 F 12 F s sr In this paper, the vehicle suspension parameters of the Then the total optimization objective can be whole vehicle rigid-flexible coupling model with front expressed as follows: double wishbone and rear torsion beam suspension are Gao and Han 13 Table 3. Changes of parameters before and after optimization. Parameters Before After Change optimization optimization ratio (%) 4 4 Front suspension 2.103 10 1.753 10 216.67 spring stiffness (N/m) 4 4 Rear suspension spring 2.273 10 2.073 10 28.81 stiffness (N/m) 4 4 Front suspension 1.403 10 0.893 10 236.42 damper damping (Ns/m) 4 4 Rear suspension 0.913 10 0.783 10 214.28 damper damping (Ns/m) f (g) 0.094 0.081 213.82 f (g) 0.071 0.068 24.27 f (g) 0.167 0.141 215.56 f (g) 0.121 0.104 214.04 f (g) 0.194 0.169 212.88 f (g) 0.141 0.125 211.34 Figure 11. Pareto solution set of the Rms values of the global f (g) 0.253 0.236 26.72 vibration of the rear foot floor and the Rms values of the f (g) 0.183 0.167 28.74 dynamic load of the rear wheel. f (kN) 0.248 0.264 + 6.45 f (kN) 0.282 0.306 + 8.51 f (m) 0.198 0.185 27.48 f (m) 0.207 0.196 25.61 suspension is subject to slightly larger vibration than the front suspension. 2) The difference in spring stiffness and damper damping will have an effect on suspension stiff- ness and suspension damping, which in turn will affect the transfer function and thus the fre- quency response function matrix, ultimately affecting the vehicle ride comfort. Moreover, within a certain range, the appropriate reduc- tion of spring stiffness and increase in damper damping is conducive to reducing the fluctua- tion of the partial response of the vibration sys- tem, thus helping to improve the vehicle ride Figure 12. Pareto solution set of the Rms values of global comfort. And the parameter change does not vibration of the rear seat rail and the Rms values of the dynamic load of the front wheel. change the time and frequency domain distribu- tion of the front or rear suspension response. 3) On a bump road, the value of DRmsLocal is analyzed and matched to improve the vehicle ride com- greater than DRmsGlobal for both the foot floor fort on bump road. The main conclusions are as and seat rail of the front or rear suspension. The follows: values of Rms, DRmsLocal, and DRmsGlobal for the front foot floor in the vertical direction are 1) The amplitude of the time domain response and smaller than those for the rear suspension. The the power spectral density of the frequency values of Rms, DRmsLocal, and DRmsGlobal for domain response of the front or rear suspension the front seat rail in the vertical direction are on the bumpy road surface both tend to increase smaller than those for the rear suspension. The in general with the increase of the vehicle speed vibration excitation of both the front foot floor compared to the previous speed. The amplitude and front seat rail is less than that of the rear change of the time domain response and the foot floor and rear seat rail when passing over a power spectrum change of the frequency domain bump road. It can be seen that the ride comfort response are more obvious, but with the increase of rear driver is worse than that of the front pas- in speed does not change the time domain and senger, and the ride comfort of the rear suspen- frequency domain distribution, and the rear sion is worse than that of the front suspension. 14 Advances in Mechanical Engineering 4) Genetic algorithm is used to optimize the match- 6. Phalke TP and Mitra AC. Analysis of ride comfort and road holding of quarter car model by SIMULINK. ing of spring stiffness and damper damping of Mater Today Proc 2017; 4: 2425–2430. front and rear on a bump road with a driving 7. 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Bello MM, Akramin Shafie A and Khan R. Active sonable range, thus the overall vehicle ride com- vehicle suspension control using full state-feedback fort is effectively improved. controller. Adv Mater Res 2015; 1115: 440–445. 11. Arslan YZ, Sezgin A and Yagiz N. Improving the ride Acknowledgement comfort of vehicle passenger using fuzzy sliding mode The authors are greatly appreciated for the financial support. controller. J Vib Control 2015; 21: 1667–1679. 12. Park JH, Kim WH, Shin CS, et al. A comparative work on vibration control of a quarter car suspension system Declaration of conflicting interests with two different magneto-rheological dampers. Smart The author(s) declared no potential conflicts of interest with Mater Struct 2017; 26: 9–15. respect to the research, authorship, and/or publication of this 13. Seifi A, Hassannejad R and Hamed MA. Use of non- article. linear asymmetrical shock absorbers in multi-objective optimization of the suspension system in a variety of road Funding excitations. Proc IMechE, Part K: J Multi-body Dynamics 2017; 231: 372–387. The author(s) disclosed receipt of the following financial sup- 14. Silveira M, Pontes BR and Balthazar JM. Use of port for the research, authorship, and/or publication of this nonlinear asymmetrical shock absorber to improve article: This project was supported by National Natural comfort on passenger vehicles. J Sound Vib 2014; 333: Science Foundation of China (NSFC) (No. 51965026). 2114–2129. 15. Li YG. Reconstruction of 3D road model and its verifica- ORCID iD tion based on adams. J Highway Transport Res Dev 2010; 27: 141–144. Jin Gao https://orcid.org/0000-0001-6375-2865 16. Kane TR, Ryan RR and Banerjeer AK. Dynamics of a cantilever beam attached to a moving base. JGuid References Control Dyn 1987; 10: 139–151. 17. Zhang Y, Mo XH, Zhong ZH, et al. 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Multi-objective optimi- handling stability and ride comfort of a vehicle via zation of a steering system considering steering modality. coupled hydraulically interconnected suspension and Adv Eng Softw 2018; 126: 61–74. electronic controlled air spring. Proc IMechE, Part D: J 21. Deb K, Pratap A, Agarwal S, et al. A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Trans Automobile Engineering 2020; 234: 552–571. 5. Shi X, Shi W and Xing L. Performance analysis of Evol Comput 2002; 6: 182–197. vehicle suspension systems with negative stiffness. Smart Struct Syst 2019; 24: 141–155. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Mechanical Engineering SAGE

Research on optimal matching of vehicle suspension parameters for improving vehicle ride comfort on bump road

Gao, Jin; Han, Peifeng
Advances in Mechanical Engineering , Volume 15 (5): 1 – May 1, 2023
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SAGE
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© The Author(s) 2023
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1687-8140
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1687-8140
DOI
10.1177/16878132231175751
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Abstract

A rigid-flexible dynamics model is built. In view of established 7-DOF vibration model, the expressions of the output indexes are derived and the influence mechanism of suspension parameters on ride comfort is revealed. The effects of different vehicle speeds on ride comfort on bump road are studied in the frequency and time domains by analyzing the suspension dynamic disturbance by taking the vertical acceleration of the foot floor and seat rail, the wheel dynamic loads, and the suspension dynamic deflections of the suspension as the objects of analysis. The effects of dynamic changes in the suspension spring and damper parameters on the time and frequency domain response indices are ana- lyzed. Two window functions are used on the bump road to process and analyze the time domain charts of the front and rear of foot floors and seat rails in the vertical direction, and the Rms values of the global and local vibrations of the foot floor and seat rail obtained are used as one of the evaluation indicators of ride comfort. The evaluation index of the total vehicle considering local and global vibration is determined, and the suspension parameters are collaboratively optimized by genetic algorithm to improve vehicle ride comfort. Keywords Bump road, vehicle speed, suspension parameters, ride comfort, collaborative optimization Date received: 4 January 2023; accepted: 26 April 2023 Handling Editor: Chenhui Liang vehicle ride comfort is the suspension, and the reason- Introduction able matching of suspension structure parameters has When the vehicle is running, the unevenness of the road 1,2 an important impact on the ride comfort. And its and the excitation of the transmission system and other good and bad directly affect the vehicle ride comfort factors will cause the vibration of the vehicle, which will performance and the driving process of the advantages affect the ride comfort. As the vehicle speed increases, and disadvantages. the demand for vehicle ride comfort is also increasing. The vehicle ride comfort is mainly to keep the vibration and shock generated by the vehicle in driving within a Faculty of Transportation Engineering, Kunming University of Science and certain range on the comfort of the occupants. Road Technology, Kunming, China surface unevenness and vehicle speed form the excita- Corresponding author: tion of the vehicle vibration system, this excitation Jin Gao, Faculty of Transportation Engineering, Kunming University of through the tires, suspension, seats and other elastic Science and Technology, No. 727 South Jingming Road, Chenggong and damping elements finally act on the human body. District, Kunming 650500, China. In this process, the most important part that affects the Email: 906845822@qq.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 Advances in Mechanical Engineering Uys et al. aimed at optimizing the spring and shock hybrid control strategy was more able to balance the absorber parameters for the Land Rover Defender 110 vehicle ride comfort and vehicle handling stability. model for optimal ride comfort on different surfaces Bello et al. designed a control method for a suspension and at different speeds. The Dynamic-Q algorithm is structure by first constructing a quarter vehicle state used to optimize the vehicle speed from 10 to 50 km/h, space model with two degrees of freedom and a com- and it is concluded that the rear shock absorber is as plete state feedback controller. Simulations using soft as possible, and the front shock absorber is the Simulink showed that the active suspension system pro- most comfortable from soft to hard according to the duces a good dynamic response even at resonant fre- road condition and speed, and the vehicle ride comfort quencies and had better ride comfort compared to the is related to the elastic stiffness of the rear shock absor- passive suspension system. Arslan et al. designed a ber. Qi et al. proposed a novel suspension structure fuzzy logic sliding mode controller and conducted with hydraulic linkage suspension and electronically dynamics simulation experiments on the performance controlled air springs in order to balance the vehicle of this controller to analyze the biodynamic response handing stability and vehicle ride comfort. By simplify- of the body to the vehicle vibration system. Numerical ing the whole vehicle system with the 9-DOF, hydrauli- results showed that the controller overcomes the short- cally linked suspension model and electric air springs, comings of previous control methods and greatly the relationship between handling stability and comfort reduced the vehicle vibration. Park et al. compared at different suspension heights with different suspen- the vehicle ride comfort of passenger vehicles with sus- sion structures is analyzed. Shi et al. investigated the pension systems equipped with two different magnetor- effect of negative stiffness on the performance of sus- heological dampers, with the difference between the pension systems. This negative stiffness structure con- two MRs being the presence or absence of a bypass sists of a passive magnetic negative stiffness spring and hole in the piston. Simulations and experiments have a modified locating mandrel, and it was experimentally proved that MR dampers with bypass holes provide demonstrated that the increase in negative stiffness better ride comfort than dampers without bypass resulted in better vehicle ride comfort. For semi-active holes. Seifi et al. pit forward a multi-objective optimi- suspensions, negative stiffness mitigated the effect of zation method using a nonlinear damper model to damping in the semi-active oscillator and improved improve the vehicle ride comfort and vehicle handing vehicle ride comfort. Trupti et al. adopted a two- stability. A simulation analysis of the suspension sys- degree-of-freedom quarter virtual prototype model tem under sinusoidal road excitation was carried out with Simulink module to analyze its ride comfort and using a five-degree-of-freedom model of the vehicle. dynamic response of the road surface in order to The results showed that the average ride comfort of the enhance the ride comfort for passenger. And the effects front and rear suspension improved by 3.58% and the 13 14 of suspension damping coefficient, stiffness, spring load handling stability improved by 0.74%. Silveira et al. mass and speed on the smoothness of the virtual proto- researched the effect of damper damping asymmetry on type system were analyzed. Cebon designed a test vehicle ride comfort based on the one-fourth vehicle method for the hysteresis characteristics of leaf springs vibration model and found that the vehicle ride com- to investigate the response of the hysteresis characteris- fort was better considering the damper damping asym- tics of a vehicle model using two degrees of freedom metry than not considering its effect. under random vibration. By conducting random vibra- The above scholars have conducted local studies on tion tests on different forms of multi-piece leaf springs, suspension and vehicle performance from the perspec- the hysteresis charts of leaf springs at different excita- tives of structural parameters, suspension form, virtual tion frequencies were obtained, and the test results prototype modeling, and construction of approximate obtained can be consistent with the empirical equations modeling. However, the vehicle ride comfort perfor- of the leaf spring hysteresis model. Abood and Khan mance when the vehicle vibrates on a bump road has researched the effect of vertical secondary suspension not been systematically studied in depth, and most of stiffness on railroad. The simulation results of the vehi- the scholars have conducted studies based on random cle ride comfort using a railroad vehicle simulation road surfaces. The systematic effect of vehicle speed on model as an example. A combination of linear the vehicle ride comfort is also not explored in combina- Kalk theory and nonlinear heuristic models was used tion with the frequency and time domain perspectives. to optimize the suspension spring stiffness and improve Secondly, when the vehicle is disturbed by external the vehicle ride comfort. Zhang et al. analyzed a vari- excitation, the local vibration and global vibration of ety of the suspension structures based on the vehicle the vehicle are not considered. Finally, based on the one-fourth suspension model to compare the specific premise that the vehicle is subject to local vibration and operation of semi-active suspensions under skyhook global vibration, this is no research on the collaborative control, ground-hook control and hybrid control stra- optimization and matching of vehicle suspension para- tegies, respectively. And the results showed that the meters and vehicle driving ride comfort based on the Gao and Han 3 whole vehicle rigid-flexible coupling model for a specific front and rear suspension structure. To address the unresolved issues, this research paper considers the following aspects: (1) The effect of differ- ent vehicle speeds on vehicle ride comfort of driving on bump road is investigated from the frequency domain and time domain respectively; (2) The output index expression is derived from the 7-DOF vehicle vibration model, and the influence mechanism of spring stiffness and damper damping parameters on vehicle ride com- fort is revealed; (3) The systematic effects of suspension spring stiffness and damper damping on the response indices of the vehicle in the time and frequency domains are systematically analyzed; (4) Based on the premise of considering the local and global vibration of the foot floor and seat rail of the front and rear, the window function is designed to process and analyze the time domain charts of the foot floor and seat rail on the ver- Figure 1. 7-DOF vehicle vibration system model. tical direction; (5) The evaluation index of total vehicle ride comfort considering local vibration and global vibration is determined, and the parameters of the front According to Figure 1, the force expression F of the and rear suspension are matched and optimized by suspension and body joint is obtained, that is: multi-objective genetic optimization algorithm to improve the vehicle ride comfort. F = K z  z + C z_  z_ ð1Þ i i i + 3 i i + 3 ðÞ i + 3 0 ðÞ i + 3 0 The vertical displacement z , z , z , z of the four 40 50 60 70 7-DOF vehicle vibration system model points where the upper and lower parts of the front and rear wheels are connected to the body are: For studying the dynamic characteristics of the suspension and revealing the systematic influence of the suspension z = z  L z  L z 40 1 1 2 3 3 parameters on the vehicle ride comfort, a 7-DOF vehicle vibration system model is established in this paper, as z = z + L z  L z 50 1 2 2 3 3 ð2Þ showninFigure1. z = z  L z + L z > 60 1 1 2 4 3 The coordinate system is set to oxyz, o is located at z = z + L z + L z 70 1 2 2 4 3 the center of mass of the body, x is the driving direction of the vehicle. First assume that wheel number 1 is the From equations (1) and (2), the motion equation of right front wheel, 2 is the left front wheel, 3 is the right the body in three directions can be derived as follows: rear wheel, and 4 is the left rear wheel. In Figure 1, L , L is the distance from the center of M €z = F ð3Þ 1 2 s 1 i mass to the center plane of the left and right wheels, I €z = M ðÞ F =ðÞ F + F L ðÞ F + F L ð4Þ L , L is the distance from the center of mass to the x 2 x i 2 4 2 1 3 1 3 4 front and rear axles, and M is the body mass. I €z = M ðÞ F =ðÞ F + F L ðÞ F + F L ð5Þ y 3 y i 3 4 4 1 2 3 m (i = 1, 2, 3, 4) is the unsprung mass, K (i = 1, 2, 3, 4) i i is the suspension stiffness, C (i = 1, 2, 3, 4) is the suspension damping, K (i = 1, 2, 3, 4) is the tire stiff- ti The motion equation of the unsprung mass is as ness, q (t)(i = 1, 2, 3, 4) is the road surface unevenness follows: factor, v is the excitation frequency. I is the rotational inertia of the vehicle body around the x-axis, I is the m €z + C z_  z_ + K z  z i i + 3 i i + 3 i i + 3 ðÞ i + 3 0 ðÞ i + 3 0 i i rotational inertia of the vehicle body around the y-axis, + K½ z  q = 0ðÞ i = 1, 2, 3, 4 i i + 3 i z is the vertical displacement of the body M ,z is the 1 s 2 ð6Þ roll angle of body with inertia I , z is the pitch angle of x 3 body with inertia I ,z (i = 4, 5, 6, 7) is the vertical y i Bringing equations (1) and (2) into equations (3)–(6), displacement of four tires m ;m ,z (i = 4, 5, 6, 7) is the 1 4 i0 the vibration equation is obtained as follows: vertical displacement of the upper and lower part of the front and rear wheels at the connection with the € _ ½ M Z +½ C Z +½ K Z =½ K Q ð7Þ body. 4 Advances in Mechanical Engineering 2 3 C  L C + L C  L C  L C  C  C  C  C i 1 1 2 2 3 1 3 2 1 2 3 3 6 7 6 7 i = 1 6 7 6 7 L C + L C L C + L C 1 3 2 4 4 3 4 4 6 7 6 2 2 7 L C + L C L C + L C L L C  L L C L C  L C L C  L C 1 1 2 2 1 1 2 2 1 3 1 2 3 2 1 1 2 2 1 3 2 2 6 7 6 7 2 2 6 7 L C + L C L C + L C  L L C + L L C 1 3 2 4 1 3 2 4 1 4 3 2 4 4 6 7 6 2 2 7 L C  L C L L C  L L C L C + L C L C L C  L C  L C ½C = 3 1 3 2 1 3 :1 2 3 :2 3 1 2 3 1 3 2 4 3 4 4 6 7 6 7 2 2 6 7 L C + L C L L C + L L C L C + L C 4 3 4 4 1 4 1 2 4 4 3 4 4 4 6 7 6 7 6 C L L L C C 00 0 7 1 1 1 3 1 1 6 7 6 7 C  L C L C 0 C 00 2 2 2 3 2 2 6 7 6 7 4 C L C  L C 00 C 0 5 3 1 3 4 3 3 C  L C  L C 00 0 C 4 2 2 4 4 4 In equation (7), M is the mass matrix. ½M = diag(M I I m m m m ) and C is the damping matrix. K is s x y 1 2 3 4 ti 2 3 00 0 0 6 7 00 0 0 6 7 6 7 00 0 0 6 7 6 7 the tire stiffness, ½K  = K 00 0 ;Q is the input vector, Q = ½q q q q  ,Z is the output vec- t t1 1 2 3 4 6 7 6 7 0 K 00 t2 6 7 4 5 00 K 0 t3 00 0 K t4 tor, Z = ½z z z z z z z  and K is the stiffness matrix. 1 2 3 4 5 6 7 Where 2 3 K  L K + L K  L K  L K  K  K  K  K i 1 1 2 2 3 1 3 2 1 2 3 3 6 7 6 i = 1 7 6 7 6  L K + L K L K + L K 7 1 3 2 4 4 3 4 4 6 7 6 2 2 7 L K + L K L K + L K L L K  L L K L K  L K L K  L K 1 1 2 2 1 1 2 2 1 3 1 2 3 2 1 1 2 2 1 3 2 2 6 7 6 7 2 2 6 7 L K + L K L K + L K  L L K + L L K 1 3 2 4 1 3 2 4 1 4 3 2 4 4 6 7 6 2 2 7 L K  L K L L K  L L K L K + L K L K L K  L K  L K ½K = 3 1 3 2 1 3 1 2 3 2 3 1 2 3 1 3 2 4 3 4 4 6 7 6 7 2 2 6 7 L K + L K  L L K + L L K L K + L K 4 3 4 4 1 4 3 2 4 4 4 3 4 6 7 6 7 K L K L K K 00 0 6 7 1 1 1 3 1 1 6 7 6 7 K  L K L K 0 K 00 2 2 2 3 2 2 6 7 6 7 4 K L K  L K 00 K 0 5 3 1 3 4 3 3 K  L K  L K 00 0 K 4 2 2 4 4 4 The output indicators of the 7-DOF whole-vehicle vibration system are body acceleration, dynamic load, and dynamic deflection, respectively. The acceleration includes not only the vertical acceleration, but also the body roll acceleration and pitch accel- eration, as well as the vertical acceleration of the four wheels and the body connection. Similarly, the dynamic tire loads and suspension dynamic deflection include the relative dynamic loads and suspension dynamic deflections at the four wheels. The transfer function of a 7-DOF system is expressed as follows: 2 1 H(jv)=(  v ½M + jv½C + ½K) ½Kð8Þ The output power spectral density matrix is connection to the input power spectral matrix by the following equation: ½G  = ½H½G ½H  ð9Þ z q In equation (9), ½H is the system frequency response function matrix, ½H  is the conjugate transpose of ½H.The relationship between the Rms value of the response s and the power spectral density G is: zi zizi Gao and Han 5 2 2 2 2 2 2 2 E(f )= E(z )+ L E(z )+ L E(z )+ E(z ) s 1 1 2 3 3 4 > 1 D =s = G (f )df ð10Þ zi zi z z i i < 2 2 2 2 2 2 2 E(f )= E(z )+ L E(z )+ L E(z )+ E(z ) s 1 2 2 3 3 5 ð17Þ 2 2 2 2 2 2 2 In equation (10), D is the variance of the response, E(f )= E(z )+ L E(z )+ L E(z )+ E(z ) zi > s 1 1 2 4 3 6 s is the Rms value. 2 2 2 2 2 2 2 zi E(f )= E(z )+ L E(z )+ L E(z )+ E(z ) s 1 2 2 4 3 7 1) Rms value of the second order derivative of the response. 3) Relative dynamic load. The second order derivative Z ðÞ t of the body accel- The static load of the wheels relative to the ground eration as a function of the frequency response to Q (t) is: is: L L 4 2 G = M g > 1 s ZðÞ v 2 > ~ (L + L )(L + L ) H ðÞ jv =  v ð11Þ 3 4 1 2 ij > QðÞ v j L L 4 1 G = M g 2 s (L + L )(L + L ) Z (v) 3 4 1 2 where H (jv)= , so there is ð18Þ ij L L 3 2 Q (v) > G = M g 3 s (L + L )(L + L ) > 3 4 1 2 ~ > H ðÞ jv =  v H ðÞ v ð12Þ ij ij L L 3 1 : G = M g 4 s Then the Rms value of the second order derivative of (L + L )(L + L ) 3 4 1 2 the body acceleration response is: The dynamic load of the tire is s = G (f )df ð13Þ zi z z i i F = F + m €z ð19Þ 0 d i i i + 3 Taking equation (1) into (19) yields equation (20), 2) Rms value of suspension dynamic deflection. that is: From the definition of suspension dynamic distur- F =KðÞ z z +L z +L z +CðÞ z_ z_ +L z_ +L z_ +m €z > d1 1 4 1 1 2 3 3 1 4 1 1 2 3 3 1 4 bance, it can get: F =KðÞ z z +L z +L z +CðÞ z_ z_ +L z_ +L z_ +m €z d 2 5 1 2 2 3 3 2 5 1 1 2 4 3 2 5 f = z  z ð14Þ > s (i + 3)0 (i + 3) F =KðÞ z z +L z +L z +CðÞ z_ z_ +L z_ +L z_ +m €z d 3 6 1 1 2 4 3 3 6 1 1 2 4 3 3 6 i > 3 F =KðÞ z z L z +L z +CðÞ z_ z_ +L z_ L z_ +m €z d 4 7 1 2 2 4 3 4 7 1 2 2 4 3 4 7 Bringing equations (2)–(14) yields the following ð20Þ equation, so that is: The Fourier change to equation (20) is: f = z  L z  L z  z s 1 1 2 3 3 4 f = z + L z  L z  z s 1 2 2 3 3 5 ð15Þ F (jv)=(K v m +jC v)z (jv)(K +jC v)z (jv) f = z  L z + L z  z > d 1 1 1 4 1 1 1 s 1 2 2 4 3 6 > 1 : > f = z + L z + L z  z > +(K L +jC vL )z (jv)+(K L +jC vL )z (jv) s 1 2 2 4 3 7 4 1 1 1 1 2 1 3 1 3 3 > 2 F (jv)=(K v m +jC v)z (jv)(K +jC v)z (jv) > d 2 2 2 5 2 2 2 Squaring both sides of equation (15) yields, that is: > (K L +jC vL )z (jv)+(K L +jC vL )z (jv) 2 2 2 2 2 2 3 2 3 3 2 2 2 2 2 2 2 2 >f =z +L z +L z +z 2L z z 2L z z 2z z F (jv)=(K v m +jC v)z (jv)(K +jC v)z (jv) > d 3 3 3 6 3 3 1 s 1 1 2 3 3 4 1 1 2 3 1 3 1 4 3 > 1 > +2L L z z +2L z z +2L z z +(K L +jC vL )z (jv)+(K L +jC vL )z (jv) > 3 1 3 3 2 3 4 3 4 3 > 1 3 2 3 1 2 4 3 3 4 > 2 2 2 2 2 2 2 2 f =z +L z +L z +z +2L z z 2L z z 2z z >F (jv)=(K v m +jC v)z (jv)(K +jC v)z (jv) d 4 4 4 7 4 4 1 > s 1 2 2 3 3 5 2 1 2 3 1 2 1 5 4 2 > (K L +jC vL )z (jv)(K L +jC vL )z (jv) 2L L z z +2L z z +2L z z 4 2 4 4 2 4 4 4 4 3 2 3 2 3 2 2 5 3 3 5 2 2 2 2 2 2 2 f =z +L z +L z +z 2L z z +2L z z 2z z ð21Þ > s 1 2 2 3 3 6 1 1 2 4 1 3 1 6 > 2L L z z +2L z z 2L z z 1 4 2 3 1 2 6 4 3 6 > F (jv) 2 2 2 2 2 2 2 Defining H (jv)= , and having >f =z +L z +L z +z +2L z z +2L z z 2z z F ;Q s 1 2 2 3 3 7 2 1 2 4 1 3 1 7 d j > i Q (jv) +2L L z z 2L z z 2L z z 1 4 2 3 2 2 7 4 3 7 Z (v) H (jv)= , it follows that: ij ð16Þ Q (v) Since z (i = 1, 2, :::, 7) is an independent random ½H  = ½A½Hð22Þ variable, E(Z )= 0 (i = 1, 2, :::, 7).Therefore, the Rms value of the suspension dynamic disturbance is: where 6 Advances in Mechanical Engineering ½ A = 2 3 K jvC K L +jvC L K L +jvC L K v m +jvC 00 0 1 1 1 1 1 1 1 3 1 3 2 1 1 6 7 K jvC K L jvC L K L +jvC L 0 K v m +jvC 00 6 2 2 2 2 2 2 2 3 2 3 2 2 2 7 6 7 4 5 K jvC K L jvC L K L jvC L 00 K v m +jvC 0 3 3 3 1 3 3 3 4 3 4 3 3 3 K jvC K L jvC L K L jvC L 00 0 K v m +jvC 4 4 4 2 4 4 4 4 4 4 4 4 4 Then the relative dynamic loads are respectively: Based on the virtual simulation environment, road F =G , F =G , F =G , F =G ð23Þ d 1 d 2 d 3 d 4 i 2 3 4 unevenness is often derived by inverse derivation of the road unevenness power spectrum density. Depending From the above derivation process of the output on the road power spectrum density, road unevenness is index of the whole vibration system, the mechanism of segmented into different road levers. There are various the vehicle ride comfort of the influence of the spring methods to generate the frequency domain model of stiffness and damper damping on the can be summar- road unevenness, such as: filtered white noise method, ized as follows: harmonic superposition method, Poisson method, The difference of spring stiffness and damper damp- Fourier inverse variation method, etc. ing of the suspension will affect the suspension stiffness Therefore, when performing ride comfort simula- and suspension damping respectively, which in turn will tion, the corresponding road model file should be cre- affect the transfer function jointly solved by the road ated first. In this paper, it uses the more commonly unevenness coefficient, the road input spectrum deter- used harmonic superposition method to generate a fre- mined by the vehicle speed and the suspension para- quency domain model of the road spectrum. The basic meters, thus affecting the frequency response function idea is to represent the road unevenness function as a matrix of the output index, and ultimately affecting the sum of a number of sinusoidal functions with random vehicle ride comfort. phases. Based on the harmonic superposition method, the Establishment of the vehicle multi-body subject gets the required 3D road unevenness charac- dynamics model and construction of bump teristics by matlab programing simulation, and then road generates pavement files according to the road model criteria, and then imports it into the professional Based on the vehicle three-dimensional model test data dynamic software, so as to construct and get the bump and reference to a real vehicle parameter to establish road. the vehicle rigid-flexible coupling multi-body dynamics Referring to the relevant provisions of the national model of the front MacPherson and rear torsion beam 17,18 standard GB/T 4970-2009, for the pulse input in suspension, as shown in Figure 2. The main structural the ride comfort analysis process, the bump road should design parameters of the established vehicle model are be used. The corresponding values of length, width and shown in Table 1. height are taken differently for different types of vehi- cles. Since the object of this paper is a passenger car, the length, width and height of the bumps established according to the regulations are 100 mm 3 25 mm 3 60 mm. The simulation model of the bump road built in ADAMS is shown in Figure 3. Effect of different vehicle speeds on ride comfort on bump road The test evaluation vehicle speed is 30–60 km/h, and the ride comfort simulation is carried out on the bump road surface respectively, and the simulated results of the fil- tered of the vertical acceleration of the foot floor and seat rail of the front and rear, dynamic load of front and rear wheels and suspension dynamic deflection of front and rear suspension in time domain response are obtained in Figures 4 and 5. For the purpose of explor- Figure 2. The whole vehicle multi-body dynamics model. ing the frequency domain characteristics of the vehicle, Gao and Han 7 Table 1. Structure parameters of the whole vehicle. Parameters Value Parameters Value Sprung mass (kg) 1097.00 Height from center of mass to ground (m) 0.38 Unsprung mass (kg) 551.98 Distance between front wheels (m) 1.59 Center of mass to front wheelbase (m) 0.71 Distance between rear wheels (m) 1.52 Center of mass to rear wheelbase (m) 1.85 Wheelbase (m) 2.56 4 4 Spring stiffness of front suspension (N/m) 2.13 10 Spring stiffness of rear suspension (N/m) 2.273 10 4 3 Damper stiffness of front suspension (Ns/m) 1.43 10 Damper stiffness of rear suspension (Ns/m) 0.913 10 From Figures 4(a) to (d) and 5(a) to (d), it can be seen that the amplitude of each response quantity of the front or rear suspension system on the bump road tends to increase in general with the increase of the vehicle speed compared to the previous speed. It means that the increase of vehicle speed will increase the amplitude of the bump road excitation, the stronger the system vibration, which is not conducive to the vehicle ride comfort. The amount of system response is affected to Figure 3. Bump road. some extent for both the front and rear suspensions, but does not change the time domain distribution. the time domain system response is transformed into Another comparison shows that the rear suspension is the power spectral density corresponding to the fre- subjected to a slightly larger amplitude of vibration quency domain system response after Fourier variation than the front suspension. and then filtering in Figures 6 and 7. Figure 4. Time domain response of the front suspension after filtering: (a) vertical acceleration of the front foot floor, (b) vertical acceleration of the front seat rail, (c) dynamic load of the front wheel, and (d) suspension dynamic deflection of the front suspension. 8 Advances in Mechanical Engineering Figure 5. Time domain response of the rear suspension after filtering: (a) vertical acceleration of the rear foot floor, (b) vertical acceleration of the rear seat rail, (c) dynamic load of the rear wheel, and (d) suspension dynamic deflection of the rear suspension. From Figures 6(a) to (d) and 7(a) to (d), it can be Effect of spring and damper parameters seen that the power spectral density of each response on time and frequency domain response quantity of the front and rear suspension system on the From actual engineering design experience, take the bump road has a rising trend in the overall amplitude rear sus-pension as an example, so that the spring change with the increase of the vehicle speed, indicating stiffness or damper damping is reduced to one-half or that the increase in vehicle speed will increase the increased to two times of the original value, other amplitude of the bump road excitation, and the parameter values remain unchanged. stronger the system vibration is, which is not conducive The vehicle is driven at a constant speed of 30 km/h to the vehicle ride comfort. It is known that the power on a bump road, and the simulation of the vehicle spectral density vibration of each response quantity of vibration system is performed to obtain a comparison the front or rear suspension system basically corre- of the time domain and frequency domain charts of the sponds to the time domain diagram, and the frequency system response quantities under the variation of domain distribution does not change with the vehicle spring stiffness and damper damping values. Due to speed increase. the limitation of space, only the time domain variation Moreover, it is known that the rear suspension is of the vertical acceleration of the rear seat rail is subjected to a slightly larger amplitude of vibration presented in Figures 8 and 9 (The initial values are than the front suspension. And the physical meaning of noted as Baseline in the figure in order to distinguish the power spectrum density is the energy conversion the different parameter values.). per unit of time, which can be obtained in the rear foot From the results in the time and frequency domains, floor and seat rail, dynamic load of the rear wheel and the working space of rear suspension, the body to it can be seen that when the spring stiffness is increased overcome the energy required for vibration than the to twice the original value, the amplitude of each front suspension on some large. response quantity of the system has increased to some Gao and Han 9 Figure 6. Frequency domain response after filtering of the front suspension: (a) power spectral density of the vertical acceleration of the front foot floor, (b) power spectral density of the vertical acceleration of the front seat rail, (c) power spectral density of the dynamic load of the front wheel, and (d) power spectral density of the suspension dynamic deflection of the front suspension. extent compared to the original value, especially the of the original value is almost the same. When the dam- per damping is reduced to one-half times of the original magnitude of the vertical acceleration of the foot floor value, the change of each response quantity of the sys- and seat rail of the front and rear, the suspension tem from the original value is the opposite. For the fre- dynamic deflection of the front and rear suspension, quency domain results, the power spectral density and the dynamic load of the rear suspension increases vibration of each response quantity of the front or rear more, and dynamic load of the front wheel increases suspension system basically corresponds to the time slightly. When the spring stiffness is reduced to one- domain variation. half of the original value, and the response of the sys- Therefore, the variation of suspension spring stiffness tem is reduced to some extent compared to the original and damper damping will have an impact on the time value, but the magnitude of the reduction is small. For and frequency domain response indicators of the car, the frequency domain results, the power spectral den- thus affecting the vehicle ride comfort, which is coincide sity vibration of each response quantity of the front or with the theoretical analysis in Section 3 above. rear suspension system basically corresponds to the The above analysis shows that: time domain response variation. When the damper damping is increased to twice the 1) In a certain range of stiffness, the appropriate original value, the amplitude of the system response is reduction of spring stiffness in a certain degree to reduced to a certain extent compared to the original reduce the amplitude of fluctuations in the par- value, especially the vertical acceleration of the foot tial response of the system, such as vertical accel- floor and seat rail of the front and rear, and the eration of the front and rear foot floor and seat dynamic load of the rear wheel. The suspension rail, suspension dynamic deflection. While the dynamic deflection of the front and rear suspension is tire dynamic load increases slightly, the overall also reduced to a certain extent, but the magnitude of system vibration intensity is reduced. Therefore, the reduction is slightly smaller. The dynamic load on the spring stiffness is reduced within a certain the front wheel is almost unchanged and the magnitude range to improve the vehicle ride comfort. 10 Advances in Mechanical Engineering Figure 7. Frequency domain response after filtering of the rear suspension: (a) power spectral density of the vertical acceleration of the rear foot floor, (b) power spectral density of the vertical acceleration of the rear seat rail, (c) power spectral density of the dynamic load of the rear wheel, and (d) power spectral density of the suspension dynamic deflection of the rear suspension. Figure 8. Time domain variation of vertical acceleration of the Figure 9. Time domain variation of vertical acceleration of the rear seat rail under the change of spring stiffness. rear seat rail under the change of damper damping. 2) In a certain range of stiffness, the appropriate seat rail, suspension dynamic deflection of the reduction of damper damping in a certain degree front and rear suspension, dynamic load of the to reduce the amplitude of fluctuations in the rear wheel. Therefore, the damper damping is partial response of the system, such as vertical increased within a certain range to improve the acceleration of the front and rear foot floor and vehicle ride comfort. Gao and Han 11 Figure 10. Variation charts of local vibration and global vibration: (a) vertical direction of the front foot floor, (b) vertical direction of the front seat rail, (c) vertical direction of the rear foot floor, and (d) vertical direction of the rear seat rail. Analysis of the vehicle ride comfort on between the effects of front and rear wheel. RmsGlobal is the vertical acceleration profile calculated adopting a bump road large calculation window to account for the impact In this paper, transient vibrations are considered by effects of all wheels. using the continuous Rms method, which is calculated RmsLocal is the maximum difference between the by integration over a short period of time. This is the period during which the bumps pass and before the root mean square value of the acceleration in the verti- bumps pass RmsLocal. RmsGlobal is the maximum dif- cal direction after weighting the acceleration, which is ference between the period during which the bumps simplified as Rms. By using two different calculation pass and before the bumps pass RmsGlobal.And windows to calculate the charts of vertical acceleration RmsGlobal, RmsLocal are adopted to study transient Rms, the following can be done: signal changes. Figure 10 displays the time domain signal charts of the vertical acceleration of the foot floor and seat rail t + Mloc 0 2 > RmsLocal = a (t)dt of the front and rear when the vehicle passes the bump > v Mloc road at a speed of 30 km/h. In other words, the change t + MGlob 0 2 chart of local vibration RmsLocal and global vibration 2 ð24Þ RmsGlobal = a (t)dt RmsGlobal. MGloc It is clear from the figure that both the curves of Mloc = 3:6   0:52 both local vibration RmsLocal and global vibration MGlob = 3:6   1:05 RmsGlobal have two significant peaks, corresponding In equation (24), Mloc, MGlob is the vibration time, to the front and rear wheels of the vehicle when passing L is the wheelbase, v is the vehicle speed, t is the initial over the bump road. vibration time, and a is the root mean square value of The value of DRmsLocal in the figure is greater than the weighted acceleration. DRmsGlobal regardless of the front or rear foot floor RmsLocal is the vertical acceleration profile calculated and seat rail. The values of Rms, DRmsLocal, and adopting a narrow calculation window to distinguish DRmsGlobal for the front foot floor in vertical direction 12 Advances in Mechanical Engineering f = min DRmsLocal Table 2. Optimization of design variables. > 1 f foot > f = min DRmsLocal 2 R foot Design variables Lower Initial Upper > > f = min DRmsGlbal 3 f foot limit value limit > f = min DRmsGlbal 4 R foot f = min DRmsLocal > 5 f seatrail Scale factor of front suspension 0.5 1 2 f = minðÞ DRmsLocal spring stiffness > 6 R seatrail Scale factor of rear suspension 0.5 1 2 f = min DRmsGlbal 7 f seatrail ð25Þ spring stiffness f = minðÞ DRmsGlbal > 8 R seatrail Scale factor of front suspension 0.5 1 2 f = min Rms damper damping 9 F df =G Scale factor of rear suspension 0.5 1 2 > f = min Rms 10 F damper damping > dr=G f = min Rms > 11 f > f f = min Rms 12 f are smaller than those for the rear suspension, and the The constraints are as follows: values of Rms, DRmsLocal, and DRmsGlobal for the ver- tical front seat rail in vertical direction are smaller than 0:5ł f ł 2 < i those for the rear suspension. Then it means that for 0:5ł c ł 2 ð26Þ the foot floor or seat rail of front and rear, the vibra- f ł (½f =3) s i tion excitation of the front foot floor is less than that of the rear foot floor when passing over the bump road. In equation (26), f is a constraint on the scaling fac- The vibration excitation of the front seat rail is also less tor of the spring stiffness, c is a constraint on the scal- than that of the rear seat rail, so the ride comfort of the ing factor of the damper damping, f is the standard rear driver is worse than that of the front passenger, deviation of the suspension dynamic deflection, ½f  is and the rear suspension is less comfort than the front the suspension limiting travel i = f , r. suspension. Optimization algorithm and optimization results Optimized matching In this paper, the multi-objective optimization algo- Determination of optimization variables and objective rithm PE’s HMGP sub-algorithm is used for the multi- function objective collaborative optimization of spring stiffness and damper damping of the front and rear suspen- The suspension springs and dampers play the role of sions. The whole vehicle model is driven at a uniform shock mitigation and vibration suppression respectively speed of 30 km/h on a bump road, and the Pareto solu- during the driving process of the vehicle. Based on tion set based on ride comfort obtained through itera- the fact that the above two types of components play a tive optimization calculations after building an decisive role on ride comfort, the spring stiffness and integrated platform. Then all the solutions in the Pareto damper damping of the suspension are used as optimi- solution set are compared, and the optimal solution set zation variables. is selected by coordinating trade-off and compromises To facilitate the simulation calculation, the spring among the objectives to make each subobjective as opti- stiffness and damper damping are multiplied by a scale mal as possible. factor to change the size respectively, and the design Only some of the Pareto solution sets of the optimiza- variables are in Table 2. tion objective in the optimization space are listed due to Optimization objectives include the Rms values of the thelengthofthe article, as showninFigures 11 and12. local f = DRmsLocal , f =DRmsLocal and the 1 f foot 2 R foot The blue dots in the Pareto illustration set are the Pareto global f = DRmsGlobal , f = DRmsGlobal 3 f foot 4 R foot front solutions. The changes of parameters before and vibrations of front and rear foot floor in the vertical after optimization are shown in Table 3. From the results direction, the Rms values of local f =DRmsLocal , 5 f seatrail in Table 3, it can be seen that the evaluation indexes of f = DRmsLocal and global f =DRmsGlobal , 6 R seatrail 7 f seatrail ride comfort have been effectively improved after optimi- f = DRmsGlobal vibrations of front and rear seat 8 R seatrail zation, then the vehicle ride comfort has been improved. rail in the vertical direction, the Rms value of dynamic load of front and rear tires f = Rms , f = Rms 9 F 10 F d =G dr =G Conclusion and the Rms value of suspension dynamic deflection of front and rear suspension f = Rms f = Rms . 11 F 12 F s sr In this paper, the vehicle suspension parameters of the Then the total optimization objective can be whole vehicle rigid-flexible coupling model with front expressed as follows: double wishbone and rear torsion beam suspension are Gao and Han 13 Table 3. Changes of parameters before and after optimization. Parameters Before After Change optimization optimization ratio (%) 4 4 Front suspension 2.103 10 1.753 10 216.67 spring stiffness (N/m) 4 4 Rear suspension spring 2.273 10 2.073 10 28.81 stiffness (N/m) 4 4 Front suspension 1.403 10 0.893 10 236.42 damper damping (Ns/m) 4 4 Rear suspension 0.913 10 0.783 10 214.28 damper damping (Ns/m) f (g) 0.094 0.081 213.82 f (g) 0.071 0.068 24.27 f (g) 0.167 0.141 215.56 f (g) 0.121 0.104 214.04 f (g) 0.194 0.169 212.88 f (g) 0.141 0.125 211.34 Figure 11. Pareto solution set of the Rms values of the global f (g) 0.253 0.236 26.72 vibration of the rear foot floor and the Rms values of the f (g) 0.183 0.167 28.74 dynamic load of the rear wheel. f (kN) 0.248 0.264 + 6.45 f (kN) 0.282 0.306 + 8.51 f (m) 0.198 0.185 27.48 f (m) 0.207 0.196 25.61 suspension is subject to slightly larger vibration than the front suspension. 2) The difference in spring stiffness and damper damping will have an effect on suspension stiff- ness and suspension damping, which in turn will affect the transfer function and thus the fre- quency response function matrix, ultimately affecting the vehicle ride comfort. Moreover, within a certain range, the appropriate reduc- tion of spring stiffness and increase in damper damping is conducive to reducing the fluctua- tion of the partial response of the vibration sys- tem, thus helping to improve the vehicle ride Figure 12. Pareto solution set of the Rms values of global comfort. And the parameter change does not vibration of the rear seat rail and the Rms values of the dynamic load of the front wheel. change the time and frequency domain distribu- tion of the front or rear suspension response. 3) On a bump road, the value of DRmsLocal is analyzed and matched to improve the vehicle ride com- greater than DRmsGlobal for both the foot floor fort on bump road. The main conclusions are as and seat rail of the front or rear suspension. The follows: values of Rms, DRmsLocal, and DRmsGlobal for the front foot floor in the vertical direction are 1) The amplitude of the time domain response and smaller than those for the rear suspension. The the power spectral density of the frequency values of Rms, DRmsLocal, and DRmsGlobal for domain response of the front or rear suspension the front seat rail in the vertical direction are on the bumpy road surface both tend to increase smaller than those for the rear suspension. The in general with the increase of the vehicle speed vibration excitation of both the front foot floor compared to the previous speed. The amplitude and front seat rail is less than that of the rear change of the time domain response and the foot floor and rear seat rail when passing over a power spectrum change of the frequency domain bump road. It can be seen that the ride comfort response are more obvious, but with the increase of rear driver is worse than that of the front pas- in speed does not change the time domain and senger, and the ride comfort of the rear suspen- frequency domain distribution, and the rear sion is worse than that of the front suspension. 14 Advances in Mechanical Engineering 4) Genetic algorithm is used to optimize the match- 6. Phalke TP and Mitra AC. Analysis of ride comfort and road holding of quarter car model by SIMULINK. ing of spring stiffness and damper damping of Mater Today Proc 2017; 4: 2425–2430. front and rear on a bump road with a driving 7. 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Journal

Advances in Mechanical Engineering – SAGE

Published: May 1, 2023

Keywords: Bump road; vehicle speed; suspension parameters; ride comfort; collaborative optimization

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