OUTLINE
Introduction
When a vehicle is in motion, the suspension system is the most important part This component is critical in ensuring the movement of the vehicle's structure and keeping the vehicle's connection to the road surface Also, the vibrations sent from the road surface to the vehicle are absorbed thanks to the damper, so it plays a essential role in defining the passenger’s driving experience and comfort
In Vietnam's economic development progress, the automobile is the best kind of transportation in many aspects compared to other types of transport due to its mobility In fact, people's travel need comes mainly from the intercity bus because of their economy and speed In the past, passengers mainly needed to go to their desired destination without caring about the quality of the vehicle, but now with socio-economic development, improving service quality is mandatory for every travel company, if they want to score points with customers In particular, vehicle comfort is a key factor because many passengers easily get motion sickness when the bus runs on a bad road, especially children and the elderly
All types of traffic participants are affected by vibration, especially users of vehicles (both drivers and passengers) on all means of transportation In general, drivers of public transport are considered to be in dangerous working condition, which could threaten their health [1] When compared to passenger vehicle drivers, long-distance bus drivers are impacted by higher intensity vibrations during their 8- hour working shifts [2] As a consequence, vibration leads to a large number of serious effects (physiological and psychological illnesses), which are more noticeable, when it influences someone for a long period of time Because of this harmingful reasons, automotive manufacturers should pay a lot of attention to reducing these kinds of discomfort Due to fast economic and social growth, modern vehicles have to bring a satisfactory level of comfort for passengers by
Instructor: Trần Hữu Nhân, Ph.D minimizing the movements as well as the vertical accelerations imposed, and perceived by passengers
Fig 1 1 An intercity bus Samco Primas
In the past, passenger cars were primarily manufactured abroad (CBU- Completely Built-Up) and then imported to Vietnam Many joint-ventures companies have successfully localized these passenger cars, and a large number of domestic manufacturing enterprises could develop buses in Vietnam based on imported chassis and engines with increasingly enhanced quality and standards such as SAMCO, THACO, HYUNDAI THANH CONG,… Because intercity bus mainly comes from the truck chassis, there are still many problems in designing progress that need to be studied and solved, such as the layout of the powertrain, distribution of loads on the axles, and especially the ride comfort when a vehicle is moving on the road.
Literature Review
In recent years, several research programs have been focused on enhancing the ride comfort of vehicles that is ranging from simulation to getting signals from realistic models for analysis and evaluation, as then individuals may increase the suspension system's quality Typical examples are the actual signal measurement, analysis, and evaluation based on standards For example, Hassan Nahvi (2009) evaluates the influence of vibrations on the comfort of passengers sitting on the bus in the frequency domain Based on the signal obtained from the vehicle-mounted sensor, the assessments were conducted [3] Another example is that Hong Zhao
Instructor: Trần Hữu Nhân, Ph.D and colleagues (2016) received a signal from passengers' smartphones via Wi-Fi and transmitted it to the server After the signal is processed and evaluated by ISO 2631-1997, the results will be displayed on LCD screens along the bus, and the system will alarm if the value exceeds the allowable threshold [4]
One of the most crucial systems in a vehicle that is responsible for achieving comfort, stability, and safety characteristics is the suspension system The primary goal of this component is to improve the comfort level for vehicle’s occupants, maintain tire-to-road surface contact, and decrease dynamic forces pressing on the vehicle's structure Due to the nonlinear characteristics, it is evident that the damper is one of the most complicated suspension's components that it influences braking, steering, cornering control, and overall stability Asymmetric dampers with a larger rebound damping coefficient than compression [5], are commonly equipped in almost all vehicle suspensions Many researchers have sought to determine appropriate damper values to achieve better trade-offs between characteristics like as ride comfort, suspension deflection, and road-holding stability [6-9] Despite the fact that these investigations have offered sufficient insight into suspension’s damper design, the majority of the conclusions were based on limited performance metrics, while asymmetric aspects were mostly disregarded.
Scope
This study seeks to elucidate the different reactions of linear symmetric and linear asymmetric dampers of Samco Primas bus subjected to a random road excitation in range of the common working velocity, and also the single bump road profile cases with triangular and sine-squared bumps The input parameters are determined based on the Samco Primas bus, which is relatively popular in Vietnam
The quarter car model with 2 DOFs is employed to investigate the dynamic behaviors Various researchers have different opinions on vehicle comfort evaluation, and there are a large number of arguments but all essentially adhere to International Standard ISO 2631-1:1997 This guideline employs root mean square
Instructor: Trần Hữu Nhân, Ph.D (RMS) of weighted acceleration as an indication of vibration comfort [10], and the obtained results are mostly evaluated based on these criteria.
Research methodology
The research includes theory, and simulation method using “M-files” in Matlab as a compiler.
Research findings and contributions
The thesis is organized as follows: Chapter 2 introduces background theory includes both the models of linear symmetric and linear asymmetric dampers employed for the simulations The quarter car model is also demonstated that the dynamic equation of system is expressed as a matrix form In the section 2.5, the method for creating random time-domain road profiles, useful for computation and analysis, is provided Next, the ISO 2631-1:1997 and the calculation flowchart are also demonstrated The model’s performance is then analyzed by using a large number of recommended evaluation indexes These details are employed in chapter
3 to examine the dynamic responses of linear symmetric and linear asymmetric dampers based on the data obtained Additionally, an extended result with a transient road input are investigated for deeper understanding in the section 3.2 Finally, the work’s highlights are summarized in the last chapter
In summary, this work is a helpful resource for enhancing ride comfort, safety, and handling control on a random road impact excitation as well as a transient road in some specific cases.
THEORY AND SIMULATION MODEL
Damper Configuration
A shock absorber is an integral part of any suspension system ranging from simple to complex It is responsible for controlling the vibration of the body and wheels thanks to the force generated by the friction of the hydraulic oil in the shock absorber In fact, this force is nonlinear in the operating range of the shock absorber
Instructor: Trần Hữu Nhân, Ph.D
A standard damper must include the following specifications:
- Overal size: Piston journey, overal length (L)…
- Other factors affecting the operation: limited working temperature, power dissipation, cooling requirements…
Fig 2 1 A rebound phase of a shock absorber [11]
The most critical parameter of the shock absorber is the damping ratio, which is a scalar quantity that indicates the damper's ability to absorb vibration For passenger cars, the damping ratio is usually in the range of 0.2-0.4, which will give the best performance In addition, the characteristic curve is quite crucial when it comes to damping, which will be mentioned in the following section
In fact, there are three most common types of shock absorbers: Twin tube, Monotube và Monotube with compression piston
Instructor: Trần Hữu Nhân, Ph.D
Fig 2 2 Types of shock absorbers and basic structures [5]
The essential components of shock absorbers, and their functions are as follows [5]:
- Main piston: contains primary valving to allow oil to flow through
- Compression piston: generates compression force depending on the displacement of the compression piston's rod
- Gas separator piston: keeps the oil and gas separate
- Main piston tube: this tube contains the main piston's workings
- Reservoir tube/Outer tube: generates a space for additional oil and gas pressure
The shock absorber's working principle is dependent on the pressure difference of hydraulic oil when it passes through the valves on the piston In the case of Twin tube, we have the following dynamic calculation expressions:
Instructor: Trần Hữu Nhân, Ph.D
Compression total Compression Compression rod
Fig 2 3 Paramerers of Twin tube shock absorber [5]
Instructor: Trần Hữu Nhân, Ph.D Due to this structure, the damping force generated during compression and rebound is not equal Moreover, the damping force will also be different at working velocity ranges
In fact, the damping force during compression will be smaller than during expansion, which can be answered for by two key factors:
- The compression damping force controls the unsprung mass’s vibration while the rebound damping force controls the sprung mass, Obviously, the sprung mass is heavier by approximately from 5 to 10 times than the unsprung mass, so the force generated during rebound will be larger than the force generated during compression
- In addition, the damping force and the elastic force have the same direction during compression, so less damping force is needed to prevent the object from moving up when a car goes through the bumpy road Conversely, in the rebound phase, the elastic force and the damping force will be in opposite directions, so more damping force is needed.
Characteristics of linear asymmetric damper
A typical model showing the working principle of a shock absorber is shown below:
Fig 2 4 Working principle of Mono-tube shock absorber [12]
The difference in hydraulic oil pressure between the Rebound Chamber and the Compression Chamber leads to the internal friction between the liquid molecules
Instructor: Trần Hữu Nhân, Ph.D when moving through the valves in the shock absorber, which is the primary reason for creating damping force
In many studies, the damping coefficient is considered constant This implies that the damping force depends on velocity The graph is a linear line where the damping force F D depends only on the velocity However, the actual characteristic curve of the shock absorber is exceptionally complicated depending on the reduction, rebound process, piston position, etc…, as shown in the following figure:
Fig 2 5 The actual damping characteristic curve [12]
From the graph, it is easy to see that the shock absorber will quickly compress when the vehicle suddenly hits a bump on the road surface But, on the contrary, the suspension system must promptly generate great force to absorb the vibration during the rebound phase
The actual characteristic curve of the damper is found in the following experimental model:
Instructor: Trần Hữu Nhân, Ph.D
Fig 2 6 An experimental model to find the characteristics of shock absorbers [13]
Fig 2 7 Displacement of harmonic excitation (u 0 = 10 (mm), f = 0.05 (Hz)) [13]
Instructor: Trần Hữu Nhân, Ph.D
Fig 2 8.Velocity of harmonic excitation (u 0 = 10 (mm), f = 0.05 (Hz)) [13]
A computer-controlled hydraulic mechanism will apply a harmonic displacement
= 0sin(2π ) u u ft to the damper By varying the frequency f , we can get different damping forces F u D ( ) respectively at specific frequency (velocity) values
Fig 2 9 A relationship between F v D ( ) and different velocity values [13]
A set of damping force F v D ( ) values describing the actual damping characteristics is obtained by taking the maximum value of the data obtained at times, u=u 0 , corresponding to the velocity v= 2π fu 0
Instructor: Trần Hữu Nhân, Ph.D
Fig 2 10 The damping characteristic curve obtained from the experiment [12]
The characteristic curve from the experiment shows that the rebound phase will generate more resistance force than the compression process
Instructor: Trần Hữu Nhân, Ph.D.
Linear asymmetric damper model
The suspension damper with linear asymmetric characteristic in the compression and rebound is taken into consideration, as shown in Figure 2.11 [14]
The suspension characteristic curve for the given parameters has the form:
Fig 2 11 The force-velocity characteristics
From Figure 2.11, the bilinear damping coefficient c s can be formulated as [13]:
= s s u s s s u c x x c c x x (2.8) where, c s + and c s − indicate the compression and the rebound damping coefficient, correspondingly
The damper's asymmetric ratio can be defined as:
0 c s refers for the symmetric damping coefficient The compression damping c s − and the rebound damping c s + can be written as:
Instructor: Trần Hữu Nhân, Ph.D
Equations (2.10) and (2.11) are substituted for equation (2.8), the bilinear damping coefficient c s can be rewritten as:
The damping force generated by the bilinear damper can be rewritten as:
From Eq (2.13), various values of the asymmetric ratio β correspond to different damping properties
Instructor: Trần Hữu Nhân, Ph.D.
Quarter car model
The quarter-car model is commonly employed in automotive engineering, as shown in Fig 2.12, because of its simplicity and the qualitatively correct information it convey, at least in the initial design stages [15] In all circumstances, the absolute vertical displacement of the wheel subsystem and the vehicle’s body is represented by the coordinates ( )x u and (x s ), respectively a) Linear symmetric model b) Linear asymmetric model
Fig 2 12 Quarter car models [16] s , u m m : Mass, (kg) s , t k k : Stiffness, (N/m) s , t c c : Damping coefficient, (N.s/m)
, , x x x: displacement, velocity and acceleration, (m), (m/s) and (m/s 2 ) respectively
, y y: excitation displacement, velocity, (m) and (m/s) respectively v 0: vehicle’s velocity, (km/h) f n : natural frequency, (Hz)
Instructor: Trần Hữu Nhân, Ph.D Fig 2.13 shows the equilibrium state and the force diagram of the suspension system
Fig 2 13 Quarter car model and force diagram
Firstly, considering the symmetrical linear model of Fig 2.12(a), the motion equations are simply transposable into matrix form as:
Mx+Cx+Kx= f t (2.14) where ( )x t =(x x u s ) T represents the response vector, while the quantities:
= − describe the system's matrices for mass, damping, and stiffness, respectively In addition, the vector f t( ) represents the external force caused by excitation Particularly, the bus is supposed to drive over a road whose profile is y while maintaining a constant velocity of v 0 The features of this road are determined by a random process As a result, the forcing vector has the form as:
Instructor: Trần Hữu Nhân, Ph.D The key difference between the models, which are shown in Fig 2.12(a) and Fig 2.12(b), is the linear asymmetric damping characteristics, where the suspension damping coefficient c s fluctuates between two unique values.
Random road profiles
Roughness features of motorways, secondary roads, and poor roads have been described as zero-mean, and Gaussian distribution in several research According to ISO 8086, the Power Spectral Density (PSD) values provided in Figure 2.14 and Table 2.1 are used for classifying road roughness [17] The PSD function of roads indicates a distinctive decline in magnitude with the wave number, which can be used for constructing the random road profile In particular, it can be determined by:
L in (rad/m) describes the angular spatial frequency, while the wavelength is denoted by L
G Ω in (m 2 /(rad/m)) shows the PSD values at Ω 0 =1 (rad/m) wis the waviness, w=2
Table 2 1 Road roughness values categorized by ISO 8608 [17]
Instructor: Trần Hữu Nhân, Ph.D
Table 2 2 Road roughness in terms of spatial frequency [17]
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Fig 2 14 Classification of roads, classes A to H [9]
G Ω G n : displacement power spectral density, m 2 /(rad/m), m 2 /(cycle/m)
We employ the sinusoidal approximation technique to study the dynamic response by solving the equations of motion at continuous sample times If the vehicle is anticipated to keep a consistent speed v 0 along a given road section of length L, a random profile of a single track may be estimated by using the accumulationN(→ ) sine waves [18]
( ) = sin( ) y t A Ω vt +φ (2.17) where the amplitude A i are defined as follows, i = 2 d ( i) ,i = 1, , N
Instructor: Trần Hữu Nhân, Ph.D in which N - 1
= N -1 Ω Ω Ω (rad/s), then the series of Ω i are determined thanks correspondingly to N equal step intervals of Ω Additionally, the phase angles i,i = 1, , N
are provided as random variables inside the [0, 2 ).
In this study, I take the grade C (G d ( 0 ) = 8 x 10 -6 (m 2 /(rad/m)) as an average- quality excitation [10] throughout the velocity interval considered (from 5 to 120 (km/h)) The nominal parameters of the road are taken to be L0(m), N%6(waves) The frequency is chosen from 0.5 to 50(Hz) that the road roughness has the most considerable influence on the oscillatory behavior, then the series of angular spatial frequencies are calculated within this range [19] Then, Figure 2.15 illustrates a typical road profile produced by the approximation method
Fig 2 15 A typical example of C-Class random road at velocity of 60 (km/h)
Instructor: Trần Hữu Nhân, Ph.D
Considering the influence of frequency and velocity of the vehicle, the combination of the Inverse Fourier Transform (IFT) and the Power Spectral Density (PSD) has a clear idea and algorithm Hence, it gives more accurate results than other methods [20]
Assuming that the spatial frequency of the RSR has an upper and lower bound of Ω 2 and Ω 1 , respectively The power spectral density is determined according to the following formula [17]:
The selection of the spatial frequency range must ensure that it contains a frequency domain that is suitable for the daily vehicle speeds Assuming that the vehicle's vibration has a frequency range of f 2 and f 1 The limited spatial frequency: n 2 = f 2 /v and n 1 = f 1 /v For a distance of length L and the number of divided points N , the distance between two points is l or L= N l The distance between two points in the frequency domain is n= 1 /L
Setting q m with m= 0,1, 2, , N -1 is the series of RSR
Applying the Fourier transform formula:
F n q l e dl n (2.20) where q l m ( ) is RSR at point m in the space domain
Since the random road profile is a Gaussian random process, the signal is symmetric about the vertical axis:
Instructor: Trần Hữu Nhân, Ph.D
The limited length is Lso l[0, ]L The signal q l m ( ) is discrete, then:
According to the relationship between PSD and FFT:
N l = N (2.25) where G n d ( k )is the discrete PSD value
Based on the equation (2.25), there is
The details about the random road profile's amplitude is reflected in Equation (2.26), and phase information is lost while using the PSD description approach, thus when recovering the data in time domain, the phase angle should be added carefully
Instructor: Trần Hữu Nhân, Ph.D [21] Since the random road profile's phase adheres to the normal distribution, we obtain:
The Inverse Fourier Transform (IFFT) can be used for getting the discrete signal of the RSR in the space domain:
Since q l m ( ) is in the space domain, so the RSR in time domain is determined by using the following formula:
Instructor: Trần Hữu Nhân, Ph.D.
ISO 2631-1:1997
The source of vibration appearing on the vehicle can be due to the following reasons:
- The force and torque components acting on the mass elements in the vehicle's dynamics system
- The behavior of the vehicle during movement controlled by the driver
- Random road profile (Primary source)
Factors of vibration that affect people's perceptions, including:
- Posture and body part affected
The variation of acceleration is crucial as it determines the vehicle's vibration, but it is important to carefully analyze the natural frequency of the human body According to VDI 2057 standard, the vertical direction is the main vibrational direction appearing on most road vehicles [22]
The root mean square of acceleration is the most used approach for determining the physiological response to vibration levels, which computed using the following equation [10]:
(2.31) where: w( ) a t – Vertical acceleration in time domain, (m/s 2 )
Instructor: Trần Hữu Nhân, Ph.D
The influence of vibration on passengers based on the root mean square of acceleration is shown as below:
Table 2 3 Comfort reactions to vibration environments [10]
The root mean square of acceleration value that does not cause discomfort level for passengers must be less than 0.315 (m/s 2 ); this is the main basis for evaluating, and selecting appropriate suspension’s structural parameters In addition, the duration time also significantly affects human health, as shown in Fig 2.16 as follows:
Fig 2 16 Relationship between health and duration of acceleration value [10]
Instructor: Trần Hữu Nhân, Ph.D After considering many different criteria, the evaluation strategy is primarily concentrated on the root mean square of the acceleration The simulation calculation process will focus on the ISO 2631-1:1997 to investigate the quality of passenger vehicles (Samco Primas)
Acceleration data obtained from simulation must be multiplied by frequency weighting to properly assess how vibration influences human health at the bands to which people are most sensitive [23]
The ISO 2631-1:1997 standard examines vibrations with frequencies between 0.4 and 100 Hz [10] Accordingly, accelerometer data from measurement or simulation will be filtered by analog filters:
According to ISO 2631-1:1997, when evaluating vertical acceleration on the “z” axis, the frequency filter used is:
Instructor: Trần Hữu Nhân, Ph.D The following table shows the conversion function's specifications:
Table 2 4 Parameters of the conversion function of frequency weighting [10]
Fig 2.17 shows the filter transfer fuctions, which are generally used for evaluating human response to vibration The “Wz” is used for the vertical axis where is the main vibrational direction.
IIR filter design
The first step in using ISO 2631-1:1997 for vibration evaluation is to use analog filters to find the acceleration taking into account the influence of frequency weighting But the acceleration data obtained from the simulation is in the time domain, and using the frequency filter requires complex calculations If these filters are used in the time domain, the computation becomes more straightforward, hence a digital filter should be designed to carry out this procedure
Instructor: Trần Hữu Nhân, Ph.D
In the design process of digital IIR filters, we use the available analog filters to design digital filters with the essential design characteristics Generally, there are many methods of creating digital filters based on analog filters However, this study will use the design method according to the bilinear transform Through the use of this technique, a discrete system is created whose frequency response is identical to that of the equivalent analog system Therefore, finding a conversion from the Laplace domain to the "z" domain is thus crucial for keeping the frequency response's property [24]
Thanks to the application of the digital filter H(z) to the signal x(n), we will receive the signal y(n) according to the following equation:
The design of a digital filter is to find the coefficients: a 0 , a 1 ,…, a N và b 0 , b 1
,…, b M s is substituted by the bilinear relation described by Eq (2.39) [24]:
The analog and digital frequencies, on the other hand, have a non-linear relationship This challenge can be overcome by pre-warping the frequencies [25]:
Instructor: Trần Hữu Nhân, Ph.D
(2.41) where f c : center frequency f s : sampling frequency that follows Nyquist frequency requirement
This enables for the direct calculation of the filter coefficients (a 0 to a 2 , and b 0 to b 2 )
These coefficients are calculated for each analog filter in ISO 2631-1 are as belows:
Table 2 5 IIR coefficients for weighting filter [25]
Instructor: Trần Hữu Nhân, Ph.D b 1 −8Q 1 2 2 ' 2 Q 2 2 4 ' 2 Q 4 5 ' 2 5 5
The weighted accelerations obtained from the simulation is quickly done by using the filter function in Matlab
Syntax: y=filter(b,a,x) với b=[b 0 , b 1 , b 2 ], a=[a 0 , a 1 , a 2 ] where y: is the weighted accelerations x: is vertical accelerations from simulation.
Evaluation indexes
For traditional vehicles, the absolute acceleration of vehicle’s body (x s ) , the distance (x s −x u ) between the wheel subsystem and the vehicle’s body, and a force generated between the wheel and the ground surface are generally chosen as the indexes for evaluating ride comfort, safety, and handling control [16] This study employs some evaluation indexes for analyzing the dynamic responses of model examined [26]: a The root mean square of the vehicle’s body vibration acceleration (BVA) can be expressed as:
(2.42) b The root mean square of the tire relative dynamic load (TDL) is calculated as follows:
Instructor: Trần Hữu Nhân, Ph.D c Following are the formula for the root mean square of the suspension dynamic deflection (SDD):
Runge-Kutta Method
The Runge-Kutta algorithm is a well-known approach for solving differential issues This method may be used for creating high order accurate numerical approaches [27]
When applying the Runge-Kutta method, we need two steps:
Step 1: Build a set of 2*n first-order equations from n second-order ones
Instructor: Trần Hữu Nhân, Ph.D
Step 2: Use numerical algorithm to easily solve these differential equations
Calculation flowchart
According to the calculation flowchart, as shown in Fig 2.19, the random road profiles, the single bump road profile, and system parameters are first generated Then, the output after solving the differential equations of the motion, which are needed to be focused, will be displacement, velocity, and acceleration in the time domain In case of random road profile, the weighted acceleration is obtained Next, RMS values of weighted acceleration are conveniently calculated, and the evaluation indexes are obtained For the case of single bump road profile, dynamic responses in time domain are also obtained to calculate the evaluation indexes Then, the maximum values of evaluation indexes are calculated Finally, the conclusions are drawn, which are based on simulation results between linear symmetric damper and linear asymmetric one
Instructor: Trần Hữu Nhân, Ph.D.
MATLAB Program
The software used in this thesis is MATLAB version R2016a, files are organized as following:
Fig 2 20 Organization of MATLAB program files
Instructor: Trần Hữu Nhân, Ph.D where:
- MAIN.m: contains the main code that calculate the required variables and write data to *.mat files
- INPUT.m: contains input parameters of quarter-car system (suspension system)
- PLOT.m: plots out result from *.mat files
- Others user-defined function files: are presented in Table 2.6
File name Inputs Outputs fn2k.m Natural frequency Suspension stiffness
RANDOM_Road_Profile.m Road number Parameters of excitation
RANDOM_road_generate.m Creating random road profile based on the sinusoidal approximation approach Myrungekutta.m Using Runge-Kutta Method to solve differential equations ISO2631_Wk.m Creating IIR filter & converting acceleration into weighted acceleration
Instructor: Trần Hữu Nhân, Ph.D.
CALCULATED RESULTS AND DISCUSSION
Analysis of evaluation indexes
Figs 3.8-3.10 illustrate the evaluation indexes in the velocity domain with two types of asymmetric ratios β to better illustrate the distinctions among them
Fig 3 8 BVA versus velocity with two types of dampers
With the C-Class random road excitation, at the initial increasing of velocity, the higher velocity the higher BVA is obtained It reaches peak values at about 55(km/h), and then decreases gradually for both cases, as shown in Fig 3.8 The linear symmetric damper yields a slightly better comfort level than that of the linear asymmetric one (only 1.5%) at a regular working velocity range (40