Introduction
The isolation of seated operators from vibration and shock in various ground vehicles, including on-road and off-road vehicles, industrial trucks, agricultural tractors, and railway vehicles, is crucial for safety and comfort Research indicates that prolonged exposure to vibration can harm the human body, potentially leading to permanent injuries and increased errors in work performance due to impaired perception Additionally, whole-body vibrations can induce dynamic stresses in the spine, resulting in micro-fractures in vertebral bodies Therefore, preventing fatigue during vehicle operation is essential, and enhancing passenger comfort is a key objective in transport design Addressing the impact of vibration on the human body remains a significant area of interest.
The passenger seat is a crucial interface between the human body and the vehicle, categorized into conventional foam cushion seats and suspension seats Suspension seats utilize a spring and damper beneath a firm cushion, while conventional seats are made with foam on a rigid or sprung pan Modern automotive seats feature open-cell polyurethane foam supported by a metal structure and trimmed with upholstery The industry standard is a full-depth open-cell polyurethane foam seat, which is mounted rigidly to the vehicle floor Seat design is influenced by cost, weight reduction, and eco-friendliness, with a focus on disassembly for recycling Foam serves as the primary means of vibration reduction and provides both static comfort—related to posture and pressure distribution—and dynamic comfort—related to vibration isolation Evaluating seat comfort is complex due to the human body's soft tissue characteristics, and objective assessments of foam properties remain challenging The stiffness of the seat, which impacts comfort, is determined by foam properties like hardness, thickness, damping, and density, with variable stiffness options available for personalized seating experiences.
12 personalization Special elements with variable stiffness can be added to the car, train or aeroplane seat, sanitary lounger, hospital bed or other equipment
A patented solution has led to the development of a functional prototype of an adjustable stiffness seat, featuring an innovative active vibration isolation component known as the pneumatic spring element (PSE) This enhanced seat design significantly improves the comfort of individuals while seated.
Objectives of the thesis are:
Derivation of the analytical model of the pneumatic spring system with lumped parameters
Analysis of the system using a multidisciplinary approach
Numerical simulation of the model for different working conditions (constant stiffness and constant pressure mode, quasi-static load, dynamic load)
Investigation of the dynamical behavior of the system Numerical simulation of transmission of acceleration
Finding a solution to improve the system from the point of view of faster regulation of pressure inside the PSE
Providing a theoretical basis for the idea of improvement and solution
Carrying out the numerical simulation of the improved system
Investigation the influence of the PSE on transmission of acceleration
Comparison of simulation results between original and improved systems
Comparison of experimental results between original and improved systems
Assessment of quality of the system improvement
Creation of FEM model of the seat with adjustable pressure profile in interaction with a simplified model of the human body
Determination of suitable constitutive models for materials of seat’s parts
Modelling of interaction between a foam block with a PSE inserted and a mass to simulate the deformation of PSE and foam under static conditions
Modelling of interaction between the car seat cushion with a PSE inserted and a simplified human body
Calculation of the pressure profile in the contact zone between the seat and simplified human body Comparison with the experimental results.
Seat with adjustable pressure profile
Mechanical subsystem
The pneumatic spring element (PSE), a crucial component of the mechanical subsystem, is integrated within the car seat cushion This PSE, composed of a latex tube filled with foam and partially wrapped in fabric adhesive tape, maintains its shape due to the tape's constraining effect When compressed air is introduced into the PSE, the uncovered ends of the latex tube stretch and bulge, allowing for enhanced comfort and support in car seating.
Electro-pneumatic control subsystem
The pressure inside the PSE is controlled and regulated by a control system consisting of pneumatic actuators and a controller A simplified scheme of electro-pneumatic control circuit is in Fig 2.4
Figure 2.4 Scheme of the control system
The compressor C supplies compressed air into the reservoir R Compressed air from the reservoir
The pneumatic spring element (PSE) is fed by fluid from the reservoir (R) through a sophisticated valve system, which includes two proportional valves (V1, V2) and two discrete solenoid valves (V3, V4) Pressure measurements are taken by two sensors (PS1, PS2) that monitor the reservoir and the PSE, with data relayed to a controller This controller manages both the compressor and the valve system, allowing for precise adjustments to the pressure within the PSE, which can be set within a range of 0 to 25 kPa based on user-defined parameters or control software settings.
Figure 2.5 Scheme of the electro-pneumatic system in detail
The system's detailed configuration is illustrated in Fig 2.5, showcasing the distribution of compressed air in and out of the PSE through a system of valves organized into two pairs Each pair comprises one proportional valve and one discrete valve; the first pair (V1, V3) facilitates the distribution of compressed air from the reservoir into the PSE, while the second pair (V2, V4) manages the release of air from the PSE to the outlet.
Valves V1 and V2 regulate small airflow when the control error—defined as the difference between the desired and actual values—remains below a predetermined threshold In contrast, discrete valves V3 and V4 activate for increased airflow when this difference exceeds the threshold Pressure sensor PS1 monitors the reservoir pressure (p cr), while sensor PS2 tracks the pressure within the PSE (p s) These sensors convert pressure into an electrical voltage signal, which is transmitted in analog form to the controller.
Figure 2.6 The control software in Labview
The compressor and discrete valves (V2, V4) utilize an on/off control method, while the orifices of proportional valves (V1, V3) are managed through proportional control Digital signals are transmitted from the controller to the compressor and discrete valves, whereas analog signals are sent to the proportional valves The control software, developed in the LabVIEW environment, facilitates the visualization of system parameters over time, including desired pressure in PSE, unfiltered and filtered pressure measurements, and control error values.
The software is utilized to configure system parameters, including the upper and lower pressure limits in the reservoir, PID controller constants, and the choice of operational mode—either constant pressure or constant stiffness Additionally, it allows for the selection of desired pressure functions during automated operations, which can be set to constant, harmonic, or triangular forms with specified amplitude and frequency.
There are two primary modes of system operation: constant stiffness mode and constant pressure mode In the constant stiffness mode, the pressure inside the PSE is initially set to a desired value, and the inlet and outlet valves remain closed regardless of mechanical loading on the seat This mode enables the control system to maintain fixed stiffness characteristics of the seat cushion Conversely, in the constant pressure mode, the control system actively regulates the pressure within the PSE to match the desired pressure value.
Mathematical model
Mathematical model of the original system
This section describes a mathematical model of the original system, which is divided into two main components: the electro-pneumatic control subsystem and the mechanical subsystem The focus begins with the modeling of the electro-pneumatic control part.
3.1.1 Model of the electro-pneumatic control subsystem
The components of this part are proportional and discrete valves, compressor and PID controller The relationship between the characteristic quantities of electro-pneumatic elements is presented by the equations
3.1.1.1 Mathematical model of the valves
The SMC-PVQ13-6M-08-M5-A proportional valves, designated as V1 and V2, exhibit a flow rate (q sj) that is influenced by the pressure difference (Δp = p inlet - p outlet) and the coil current (i j) supplied to control each valve These characteristics are illustrated in the datasheet, highlighting the relationship between flow rate, pressure, and coil current.
[12] and it is transformed by means of data-fit algorithm into a two-parametric function (3.1)
The set of coefficients obtained by calculation in Matlab is:
0895852796e-14 5.555058587281294e-21 k in case of the process of the increasing the flow rate through a valve, and
09778673627654e-14 -2.275862040123769e-21 k in case of the process of decreasing the flow rate through a valve a) From the datasheet b) By a fitting algorithm
Figure 3.1 Characteristics of the proportional valve From Fig 2.5, we have p inlet = p cr , p outlet = p s for valve V1 and p inlet = p s , p outlet = p atm for valve V2 where p atm is the atmospheric pressure ( p atm =0.1 MPa)
To effectively manage the system, we define the pressure error \( e \) as the difference between the desired pressure \( p_d \) within the PSE and the current internal pressure \( p_s \) Additionally, we introduce a threshold \( e_t \) that indicates the control system's insensitivity When the absolute value of the control error is below \( e_t \), the control system refrains from activating the proportional valves \( V1 \).
In certain scenarios, the flow rate through valve V1 is determined using formula (3.1) when the condition e < -e_t is met, while it is set to 0 if e ≥ -e_t Similarly, the flow rate for valve V2 is also calculated using formula (3.1), applicable when e > e_t, and it is set to 0 if e ≤ e_t.
The PID controller is used for control of the coil current which is supplied to proportional valves
V1, V2 Using the equation of PID controller the coil current is a function of the pressure error e and it is presented in the form:
, (3.4) where: i j (t) is the current supplied for controlling the proportional valve j (j=1,2), e is the pressure error given by the expression:
Proportional valves V1 and V2 are limited to small flow rates, which may be insufficient for certain applications To achieve significantly higher flow rates and enhance system performance, it is necessary to utilize discrete valves V3 and V4, specifically the SMC-S070B-6A type, which facilitate step-change flow rate increases.
According to [13] flow characteristics of the discrete valve include sonic conductance C dv =0.083 l/(s.bar) and critical pressure ratio b dv =0.28 The formula for the flow rate calculation of discrete valve is:
273 d nstream dv sj dv upstream dv upstream upstream p b q q p C p p T
21 in case of choked flow, and
1 273 d nstream dv sj dv d nstream upstream upstream d nstream dv upstream dv upstream p b q q p p p p b
In subsonic flow conditions, where j equals 3 or 4, the upstream and downstream pressures of the valve are represented as p upstream and p downstream, respectively For valve V3, the upstream pressure is equal to p cr, while the downstream pressure is p s In contrast, for valve V4, the upstream pressure is p s and the downstream pressure is p atm Additionally, the temperature within the PSE is assumed to remain constant at 297 K.
The discrete valves V3 and V4 play a crucial role in regulating airflow by responding to control errors that exceed a defined threshold (e t) combined with a pressure sensitivity parameter (Δp s) set in the control software Valve V3 activates when the error (e) is less than the threshold minus the pressure sensitivity (e < -Δe t p s) and deactivates when the error meets or exceeds this limit (e ≥ -Δe t p s) Conversely, Valve V4 engages when the error surpasses the threshold plus the pressure sensitivity (e > +Δe t p s) and turns off when the error is equal to or below this value (e ≤ +Δe t p s).
, if 0 s dv cr dv s cr s s dv cr s dv s cr s t t t q p b and p p q q p b and e p p if p e e p p e e e p
4 2 if if 0 atm dv s d v t s s atm s dv s dv t s s s t q p b and p p q q p b an p e e p e e e e p d p p if
The total flow rate q s is then given by the addition of flowrates of all individual valves q sj as it is expressed by:
3.1.1.2 Mathematical model of the compressed air supply
Compressed air is produced by the compressor and stored in a reservoir, where the internal pressure significantly affects the flow rate to the PSE The transportation of air from the compressor to the reservoir, along with the distribution of compressed air through inlet valves to the PSE, is illustrated in the accompanying diagram.
Figure 3.2 The scheme of the process of transmitting compressed air to PSE
According to [14] the elementary equation for an ideal compressed air reservoir is
dt , (3.11) where : q c is the flow rate from the compressor to the reservoir,
C f is the compressed air capacitance found experimentally The capacity of a given volume of fluid in a rigid container is given by: exp f
V exp is the volume of the rigid container (V exp = 0.63 l),
23 β is the bulk modulus of air (β2 kPa)
The characteristics of the compressor in the time and pressure domain are derived from experimental data, illustrating the relationship between compressed air pressure and time during the filling of a rigid container, as well as the flow rate's dependence on pressure (refer to Fig 3.3).
Based on the scheme (Fig 3.2) the total air flowrate q cr through the reservoir is then given by:
The equation \( q = q_s1 + q_s3 - q_c \) (3.13) illustrates the relationship between the flow rates of valves V1 and V3, denoted as \( q_s1 \) and \( q_s3 \), and the supplied flow rate from the compressor, \( q_c \) This relationship highlights the dependence of compressed air pressure over time and the correlation between airflow rate and pressure Figure 3.3 depicts the characteristics of the compressor, emphasizing these key dependencies.
When the compressor is working, the flow rate q c can be expressed in form of a linear function of the pressure inside the reservoir p cr :
The constants a 1 0.062, b 1 275.92 are experimentally determined to correspond to the results in Fig 3.3b
The pressure p_cr is monitored by sensor PS1 and maintained within the defined range of [p_cr1, p_cr2] by the control system The values for p_cr1 and p_cr2 are established as user-defined parameters in the control software The compressor activates when p_cr falls below p_cr1 (State=ON) and deactivates when p_cr exceeds p_cr2 (State=OFF).
“State” is an internal variable which is automatically set by the controller Normally, the values of p cr1 and p cr2 are set to 90 kPa and 110 kPa respectively
The function describing the flow from the compressor to the reservoir is presented as follows:
The article discusses the conditions for activating and deactivating a specific state based on a series of parameters It states that if certain conditions (p p b q) are met, the state should be turned ON, while if different conditions (p p p p q a) are met, the state should be turned OFF The text emphasizes the importance of these parameters in determining the state’s status.
3.1.2 Model of the mechanical subsystem
The PSE significantly alters pressure distributions across all contact zones between the human body and the seat cushion For my model, I focus on a smaller area where maximum pressure changes are anticipated, specifically a contact area of 100x100 mm², which aligns with our laboratory equipment's compatibility (see Fig 3.4a) This subsystem is treated as a mechanical model with lumped parameters.
Figure 3.4.The foam block with area (100x100) mm 2 and a PSE inserted inside
The model assumes that the mechanical subsystem consists of a combination of a 100x100x50 mm foam block and a PSE, as illustrated in Fig 3.4b To simplify the model, foam parts (1) and (3) are represented as a single cuboidal foam block arranged in parallel with a latex tube (2) that is filled with compressed air.
To enhance alignment with the actual mechanical subsystem, the internal volume of the latex tube is reduced by the volume of the foam solid material initially inserted Assuming the central hole in the foam is negligible, we can apply the equivalent subsystem model, which consists of a foam block measuring 100x100x50 mm and a latex air spring The forces acting on the mass are depicted in the free-body diagram, highlighting the kinematic displacement, velocity, and acceleration excitations, alongside the contact forces between the foam block, latex air spring, and the mass.
Figure 3.5 Simplified scheme of the mechanical subsystem and it is possible to set up the equation of motion of mass m in the form:
Mathematical model of the improved system
3.2.1 Introduction of the improved system
To enhance efficiency and precision, an improvement was implemented to optimize pressure regulation within the PSE Experimental observations reveal that the relationship between internal pressure changes and volume variations indicates that larger PSEs experience a slower rate of pressure change compared to smaller ones Additionally, the supplied air flow rate significantly impacts the rate of pressure change The theoretical relationship among pressure, volume, and supplied flow rate is articulated in formula (3.28), which is reiterated in the subsequent equation.
In equation (3.42), the original latex air spring volume, denoted as V ls, is substituted with V ils, representing the total volume of the enhanced latex air spring in the improved system.
Following the analysis, a proposal for enhancement was developed, resulting in the complete taping of the PSE An extra latex tube was attached to the PSE (refer to Fig 3.28), indicating that any volume change observed is attributed to the additional latex tube rather than the PSE itself.
When compressed air is supplied, the volume change in the improved PSE is significantly larger than in the original PSE, leading to enhanced pressure regulation.
` Figure 3.28.The additional latex tube connected to PSE
To enhance the flow rate of compressed air, modifications were made to the electro-pneumatic subsystem A vacuum pump, which generates output pressure below 0 kPa, is connected to the outlet valves, while an additional air compressor provides higher inlet pressure in a separate reservoir The pressure sensor in reservoir R1 was upgraded to an SMC PSE540A-R06, capable of measuring up to 1 MPa, and a PSE543A-R06 sensor was added to monitor vacuum pressure in reservoir R2 The control circuit system was also redesigned, incorporating software that operates in two modes—constant pressure and constant stiffness—while displaying the system's pressure characteristics The entire improved system was constructed as a mobile device to streamline the experimental process.
Figure 3.29 The real improved system
Figure 3.30 Detailed scheme of the improved system
The improved system's mathematical model builds upon the original system's framework, incorporating additional equations to reflect changes in design Each new component, such as the additional latex tube and vacuum pump, is represented by these supplementary equations Additionally, certain parameter values are updated to align with the newly integrated components.
3.2.2 Mathematical model of the vacuum pump
The vacuum pump VP is connected to the reservoir R2 to create vacuum inside this reservoir
The R2 reservoir enhances the flow rate through the outlet valve system (V2 and V4), effectively improving efficiency The mathematical model for the vacuum pump can be developed similarly to that of an air compressor due to their shared operating principles As illustrated in Fig 3.31, the flow rates of compressed air transition from the PSE to reservoir R2 via the outlet valve system and subsequently from the reservoir to the vacuum pump.
Figure 3.31.The scheme of the process of releasing compressed air in the improved system
The equation of flow rate under the assumption of an ideal vacuum in reservoir is:
dt (3.43) where p vr is the internal pressure of reservoir R2, q vp is the flow rate from the reservoir R 2 to the vacuum pump,
C f is the vacuum capacitance found experimentally and is given by (3.11)
The total flow rate through the reservoir is then given by:
2 4 vr vp s s q q q q , (3.44) where q s2 , q s4 are flow rates of valves V2, V4 respectively
In this case q s2, q s4 is calculated by (3.2) or (3.3) with p upstream = p s and p downstream =p vr a) Pressure of vacuum pump – time diagram b) Flow rate – pressure diagram
Figure 3.32.The characteristics of the vacuum pump
When the vacuum pump works, the flow rate q vp can be expressed in the form of a linear function of the pressure p r2 inside the reservoir R2:
The constants a 2 0.23778, b 2 -62.754 are experimentally determined in correspondence with the results in Fig 3.32b
The pressure p vr is monitored by sensor PS3 and maintained within the range of [p vr1, p vr2] by the control system User-defined parameters in the control software determine the values of p vr1 and p vr2, typically set at -60 kPa and -40 kPa, respectively The vacuum pump operates when p vr exceeds p vr2 (State=ON) and shuts off when p vr falls below p vr1 (State=OFF) The "State" is an internal variable automatically managed by the controller.
The function describing the flow from the reservoir to the vacuum pump is expressed by:
The system operates based on specific conditions: it will activate (State ON) if the criteria of p, p, b, q are met, while it will deactivate (State OFF) if p, p, p, p, q is true Additionally, the status of the system can change depending on the inputs provided, ensuring a responsive and dynamic operation.
The mathematical model of the compressor is updated with the values of p cr1 and p cr2 which are set to 190 kPa and 210 kPa respectively
3.2.3 Mathematical model of the combination of the latex air spring and the additional latex tube
The additional latex tube is linked to the PSE, which is entirely wrapped in tape, indicating that any volume change in the PSE due to internal pressure is insignificant Consequently, the overall volume change is solely attributed to the additional latex tube This section will concentrate on calculating the volume change of the latex tube.
Based on our analysis, we assume the volume of PSE remains constant, allowing us to calculate its volume using a straightforward method This calculation mirrors the approach applied to the original system.
V V V V V (3.47) in this case V 1 ,V 2 ,V 3 are calculated by:
The total volume of latex air spring is rewritten into the form:
This section discusses the calculation of the volume of the additional latex tube, illustrated in Fig 3.33 Designed to minimize the volume requiring compressed air, the tube is filled with foam, maintaining a foam volume identical to that of the original PSE at 0.15 liters Notably, the structure of the additional latex tube mirrors that of the original PSE.
Figure 3.33.The scheme of the additional latex tube
An unloaded latex tube initially has a cylindrical shape with an internal pressure of 0 kPa As the internal pressure increases, the tube deforms symmetrically due to its rotational symmetry The volume change of the latex tube is dependent on the pressure, which can be determined through experimental methods In the experimental setup, the latex tube is first filled with compressed air until the desired internal pressure is achieved Once the target pressure is reached, the tube is held in a stable position while its surface is scanned using a 3D scanner.
Figure 3.34.The setup of the experiment of the scanning of surface of the additional latex tube
Figure 3.35 Example of scanned surface of the additional latex tube
The data is illustrated as a cloud of points on the surface of the additional latex tube, as shown in Fig 3.35, reflecting changes in internal pressure within the range of 0 to 25 kPa at specific values.
The calculation of the center line position and cross-sectional area from 0 to 25 kPa is essential for determining the volume of a solid of revolution This can be achieved through integration techniques, as illustrated in Fig 3.36.
The comparison between the original system and the improved system
This section evaluates the impact of enhancements on pressure regulation within the PSE and acceleration transmission by comparing calculated and experimental results Key system characteristics, including response time and acceleration transmission, are analyzed for both the original and improved systems through numerical and experimental methods The study investigates pressure regulation under static conditions while assessing acceleration transmission under dynamic conditions.
3.3.1 The comparison of calculated results
This calculation regards the determination of pressure response inside the PSE without external load The desired pressure is set in the form of a rectangular function which is given by:
The original system operates with an inlet pressure between 90 and 110 kPa and an outlet pressure at atmospheric level (0 kPa) In contrast, the improved system features a higher inlet pressure range of 190 to 210 kPa, with an outlet pressure ranging from -40 to -60 kPa Pressure responses for both systems are illustrated in Fig 3.38.
Figure 3.38 Pressure responses without external load
In dynamic conditions, calculations can be categorized into two types: the first involves assessing the pressure response within the PSE under a periodic excitation function, while the second focuses on the transmission of acceleration.
In the first type the loading mass is set to 10 kg, the excitation function is
(t) 10.sin(2 ) z ft mm and the desired pressure is set to 15 kPa The comparison of results is shown in Fig 3.39
Figure 3.39 Comparison of pressure responses under periodic excitation
The transmission of acceleration was analyzed in section 3.1.3.2, revealing similar results for both the original and improved systems (refer to Fig 3.40) With a 10 kg load, the transmission curves indicate that as the desired pressure increases, the peak position shifts to higher frequencies, accompanied by an increase in peak values Notably, the improved system exhibits slightly lower peak values at smaller frequencies compared to the original system, except at 0 kPa These transmission peaks occur within the frequency range of [6,8] Hz, while the peak values range from [2.5, 4.0].
Figure 3.40 Comparison of transmission of acceleration
3.3.2 The comparison of the experimental results
The original system is distinguished from the improved system by the PSE and electro-pneumatic control subsystems The original PSE lacks tape coverage at both ends, while the improved PSE is fully taped and includes an additional connecting tube Detailed descriptions of the original and improved electro-pneumatic control subsystems can be found in sections 2.2 and 3.2.1, respectively Experiments are conducted to validate the accuracy of the calculated results and to assess the impact of the improvements on pressure regulation within the PSE and acceleration transmission.
There are two primary types of experiments conducted on pressure response within the PSE The first type measures pressure response without any external load, while the second type assesses pressure response under a dynamic external load.
In the first type of experiment, the original PSE is integrated with an electro-pneumatic control subsystem, utilizing LabVIEW software for management National Instruments hardware connects to a laptop via USB, enabling the creation of a LabVIEW controller with a user-friendly interface for adjusting control parameters and regulating pressure within the PSE The experimental setup is illustrated in Fig 3.41, where both the original and improved electro-pneumatic subsystems are tested sequentially, maintaining the same desired pressure set by a rectangular function.
Figure 3.41 Setup of the experiment under static conditions without external load
The responses of pressure inside the PSE are measured by pressure sensors and are recorded by the Labview control software
Figure 3.42 Experimental result of pressure responses
The results (see Fig 3.42.) show that the improved electro-pneumatic subsystem makes the pressure response faster with the pressure error smaller
The second type of experiments focuses on measuring pressure response in various control modes, specifically constant pressure and constant stiffness, under both static conditions—without external kinematic excitation of the lower platen—and dynamic external loads These tests are conducted using the Instron E3000 machine, as illustrated in Fig 3.43.
In a controlled experiment under static conditions, a foam block containing a PSE is positioned between two square platens, with the base platen below and the upper platen applying compression The movement of the upper platen is precisely defined to facilitate the testing process.
54 exciting function:z(t) 10.sin(2 ft mm) Exciting frequency f is in range [0,3] Hz with values of (0.1, 0.3, 0.5, 0.7, 1, 1.5, 2, 2.5, 3) Hz (see Fig 3.44)
The initial experiments were conducted using the original PSE and electro-pneumatic control subsystem, followed by tests that replaced the original PSE with an improved version and the electro-pneumatic control subsystem with its enhanced counterpart This approach allowed for an analysis of the impact on pressure regulation within the PSE Data collected during these experiments were processed using Matlab software to evaluate pressure errors, defined as e = p_s - p_d, where p_d represents the desired pressure and p_s is the measured pressure within the PSE The experimental results, illustrating the pressure error in relation to the exciting frequency, are presented in the subsequent figures, with results for the constant pressure mode detailed from Fig 3.45 to Fig 3.48.
Figure 3.45 Original PSE - original control subsystem - constant pressure mode (case 1)
In the constant pressure mode measurements depicted in Figures 3.45 to 3.48, Figure 3.45 illustrates the experimental results from case 1, utilizing the original PSE and electro-pneumatic control subsystem When the desired pressure is set to 0 kPa, the pressure error remains consistent across exciting frequencies within a narrow range of 1.4 to 2.2 kPa Increasing the desired pressure within the range of (0, 25] kPa leads to a reduction in pressure error, with the exception of the case where the desired pressure is 0 kPa Additionally, raising the exciting frequency within the range of [0, 3] Hz results in an increase in pressure error.
Figure 3.46 illustrates the outcomes of case 2 in constant pressure mode, comparing the original PSE with the enhanced electro-pneumatic control subsystem Notably, there is a reduction in pressure error, approximately 1 kPa, when the desired pressure is set to 0 kPa However, when the desired pressure exceeds 0 kPa, the pressure error remains consistent with the results obtained from the original electro-pneumatic control subsystem.
Figure 3.47 Improved PSE - original control subsystem - constant pressure mode (case 3)
In case 3, depicted in Fig 3.47, the results from the constant pressure mode using the enhanced PSE alongside the original electro-pneumatic control subsystem indicate that an increase in exciting frequency from 0 to 3 Hz leads to a rise in pressure error from 0.5 kPa to 3.8 kPa When the desired pressure is set to 0 kPa, the pressure error varies between 1.5 kPa and 3.2 kPa Conversely, for desired pressure levels above 0 kPa, the pressure error fluctuates between 0.5 kPa and 3.8 kPa Notably, increasing the desired pressure within the range of 0 to 25 kPa results in the pressure error oscillating around a consistent average value at the same exciting frequency.
Figure 3.48 Improved PSE - improved control subsystem - constant pressure mode (case 4)
In case 4, illustrated in Fig 3.48, the implementation of the enhanced PSE and electro-pneumatic control subsystem under constant pressure mode reveals that increasing the exciting frequency from 0 to 3 Hz results in a pressure error escalation from 0.5 kPa to approximately 2 kPa Additionally, raising the desired pressure within the range of 0 to 25 kPa leads to fluctuations in pressure error, which stabilize around an average value at the corresponding exciting frequency.
The pressure error-frequency diagrams presented in Figures 3.44 to 3.48 are simplified using upper and lower envelope curves These envelope curves are compiled in Figure 3.49 to compare the effects of various components, including the original and improved Pressure Subsystem Elements (PSE) and the original and improved electro-pneumatic control subsystems, on pressure regulation within the PSE Specifically, Case 1 (OOC) represents the combination of the original PSE with the original electro-pneumatic control subsystem, while Case 2 (OIC) reflects the original PSE.
Conclusion
This chapter begins with a derivation of the simulation models for both the original and improved systems, providing numerical calculations that detail their characteristics under static and dynamic conditions in various control modes, including constant pressure and constant stiffness The second part of the chapter discusses experiments conducted under conditions similar to those in the numerical simulations, allowing for a comparative analysis of the original and improved systems The comparison focuses on the quality of pressure regulation and the transmission of acceleration between the two systems.
The improved system demonstrates significant positive effects, particularly in regulating pressure under both static and dynamic conditions In constant pressure mode, the regulation quality is assessed through pressure error, revealing that the enhanced system consistently outperforms the original When evaluating dynamic conditions, the regulation quality is measured by the transmission of acceleration, which shows a similar tendency in both systems: as desired pressure increases, the peak position shifts to higher frequencies, accompanied by an increase in peak value Notably, the improved system achieves a reduction in peak values, enhancing comfort for users.
Finite element analysis using MSC Marc software
Finite element model of latex tube
4.1.1 Bulge test of latex membrane
The bulge test was employed to investigate the properties of the latex membrane, utilizing the plane-strain bulge test method to effectively measure the mechanical properties of thin membranes.
The bulge test technique establishes the relationship between stress and strain by analyzing the pressure-deformation behavior of spherical membranes Initially reported by Beams, this method involved developing models for spherical cap geometry and deriving equations for various initial boundary conditions This foundational work ultimately facilitated the creation of straightforward formulas for calculating stress and strain in bulged membranes.
The experimental setup, depicted in Fig 4.1, involves a square latex membrane cut from a tube and secured over a circular window using screws and additional latex sealing, allowing for the pumping or release of compressed air to deform the membrane An OptoNCDT 1402 displacement sensor is positioned above the membrane, focusing its laser beam on the center of the window to measure the displacement at the top point of the spherical cap Additionally, a pressure sensor is utilized to measure the air pressure within the spherical cap of the latex membrane.
The spherical cap geometry illustrated in Figure 4.2 features key parameters: applied pressure (p m), membrane thickness (t m), height of the spherical cap (h m), radius of the spherical cap (r m), and the radius of the circular window (R w).
Experimental setup of the bulge test
Spherical cap geometry and dimensions of the circular window
In accordance with [21] the stress σ m in the spherical cap of the latex membrane can be expressed by the formula:
The thickness t m is given by equation [20]:
(4.2) where t m0 is the initial thickness of the membrane
From the geometry of the deformed membrane the radius r m and angle are presented by:
In this case the engineering strain λ is derived by:
(4.5) where l m0 is the initial length, l m is the instantaneous length of the meridian curve of the deformed membrane, r m 0 , 0 is the initial value of radius r m and angle , respectively
Spherical cap geometry is used for stress and strain calculation of the membrane
From the experiment the obtained result is a relationship between deformation and pressure which is determined by h m =h m (p m )
There are two distinct types of latex membranes characterized by their thickness: the 0.65 mm membrane, utilized in the original PSE's latex tube, and the thicker 1.6 mm membrane, which is employed in the additional latex tube of the enhanced PSE.
The deformation–pressure response of the first type of latex is illustrated in Fig 4.3, depicting the inflation (blue) and deflation (red) curves The strain-stress response is derived from the average of these inflation and deflation paths, with the experimental results presented in Fig 4.4 This data serves as the input for further analysis.
The MSC.Marc software features a function that automatically generates a fit engineering stress-strain curve and coefficients for the selected 2-term Ogden constitutive model, based on 68 experimental data points As illustrated in Fig 4.5, the fit stress-strain curve is represented in red, while the experimental stress-strain curve is shown in blue, highlighting the accuracy of the model.
Deformation-pressure diagram of the bulge test in case of 0.65 mm thickness
Strain-stress diagram of the bulge test in case of 0.6 mm thickness
Experimental (blue) and fit stress-strain (red) curves of the latex membrane
The results for the second type of latex are illustrated in Figures 4.6, 4.7, and 4.8, which depict the deformation-pressure response, the experimental strain-stress response, and a comparison between the fitted stress-strain curve (red) and the experimental stress-strain curve (green).
Deformation-pressure diagram of the bulge test of the latex membrane 1.65 mm thickness (inflating – blue, deflating - red)
Strain-stress diagram of the latex membrane (thickness 1.65 mm)
Experimental (green) and fit stress-strain (red) curves of the latex membrane
The material model for latex is characterized as a hyperelastic material, defined by a stored energy function Conventional hyperelastic models, particularly the Ogden model, effectively represent the behavior of latex Ogden formulated the energy function as a separable function of principal stretch in a generalized form.
are material constants, K is the initial bulk modulus, J is the volumetric ratio, defined as
J Under consideration of incompressible material (J=1) and Ogden model selection for latex, we can use a 3-term formulation with engineering stress- engineering strain curve for biaxial mode with engineering strain 1 2 , 3 2 [23], thus:
According to [23], in the material fitting data option in MSC.Marc the biaxial deformation mode is selected The software automatically calculates results as follows
For the case of the latex membrane (thickness 0.65 mm) the set of coefficients is:
And for case of latex membrane (thickness 1.65 mm) the set of coefficients is:
The bulge test simulation of the latex membrane was conducted using the same boundary and initial conditions as the actual test The deformation of the latex tube subjected to an internal pressure of 25 kPa is illustrated in the simulation results.
Simulation of the latex membrane in the bulge test (Displacement z [m])
Simulation of the bulged latex tube at 25 kPa of internal pressure
Finite element model of tape
4.2.1 Uniaxial tensile test of the tape
The tape specimen measured 0.16 mm in thickness, 48 mm in width, and 220 mm in length A uniaxial tensile test was performed using the Tira test 2810 machine, with results illustrated in the force-tensile deformation response graph Additionally, a comparison of the fitted stress-strain curve derived from experimental data and the actual experimental stress-strain curve is presented.
73 The setup of the tensile test
Force – deformation diagram of the tensile test of the tape
Stress-strain diagram of the tensile test of the tape
Experimental (blue) and fit stress-strain (red) curves of the tensile test of the tape
The Mooney-Rivlin material model is used for the tape in form [23]:
For the case of uniaxial extension the engineering strains are 1 , 2 3 2 [22] Hence:
The results obtained from material fitting data option are as follows:
The tensile test simulation of tape in MSC.Marc software adheres to the identical boundary and initial conditions utilized in the experimental tensile testing of the tape.
Simulated extension [m] of the tape in the tensile test
The results in Fig 4.16 show a good correspondence of experiment and simulation
Simulated and experimental force-displacement diagram of the tensile test of the tape
Finite element model of foam
4.3.1 Uniaxial compression test of foam
The stress-strain behavior of foam under uniaxial stress was analyzed through a compression test, utilizing the strain energy function of finite hyperelasticity to characterize the elastic properties of open-cell soft foams The constitutive phenomenological material model was established using experimental data from these tests The specimen, measuring 100x100x80mm, was sourced from car seat cushioning, and tests were conducted with an Instron loading machine, as illustrated in Fig 4.17 The testing procedure involved displacement control at a consistent loading rate of 1 mm/min, reaching a deformation of 60 mm.
The experimental setup of the compression test of foam
The result of the compressible deformation-force relationship is shown in Fig 4.18
Deformation-force diagram of the compression test The stress-strain data is calculated using Matlab software and is shown in Fig 4.19
Stress-strain diagram of the foam
In accordance with [24] the foam is considered as compressible material and the suitable material model is represented by function:
(4.10) where n , n and n are material constants The second term of this function represents volumetric change, J is the Jacobian measuring dilatancy defined as the determinant of deformation gradient f (f x
, where x and X refer to the deformed and original coordinates of the body) with 1 2 3
L For this deformation state we have: 1 2 3
In the material fitting data option, the uniaxial deformation mode was chosen with a fictive Poisson's ratio of 0.01 The selected material model is Foam with two terms (n=2), yielding the following results.
Viscoelasticity models exhibit a response that is influenced by both the degree of deformation and the rate at which deformation occurs This results in time-dependent behavior, with relaxation being a key phenomenon associated with it Relaxation refers to the reduction of stress while maintaining a constant level of deformation.
An experiment was conducted to model viscoelastic foam under uniaxial stress deformation Foam specimens measuring 100x100x80 mm were subjected to compression and maintained at specific strain levels for extended periods Stress relaxation test data was collected at strain values of 5%, 10%, 15%, 20%, and 25%.
…, 70% strain which was kept constant for time period of 1800 seconds For the sake of completeness the sets of stress relaxation test data are shown in Fig 4.21
Relaxation data from the experiment
Using the experimental data fit menu, we determined the coefficients and achieved a fit using a Prony series with a function order set to 9 The relaxation function in the time domain is represented by this Prony series.
S(t) is the time dependent strain energy function,
S ( ) is the infinity relaxation modulus,
N D is the order of the Prony series, k is the relaxation time of Prony term k,
The coefficients generated are shown below:
4.3.4 Comparison of experimental result and simulation result
Result of experimental (green) and fit stress-strain (blue) curves of the compression test are shown in Fig 4.22
Experimental (green square) and fit (plain green) stress-strain curves of the compression test of foam
Term k Relaxation time k Prony coefficients S k
The simulation of compression test of foam in Marc (Displacement z [m])
The simulation was conducted using identical boundary and initial conditions as those applied in the experiments (refer to Fig 4.23) The force-displacement relationship illustrated in Fig 4.24 demonstrates a satisfactory correlation between the experimental and simulation results.
Force – displacement diagram from experiment and simulation
Finite element analysis of the models of compression test
Two distinct models were developed: the first model pertains to the compression testing of a foam block containing an inserted PSE under static conditions, as detailed in Chapter 3 The second model relates to the design utilized in car seat cushions.
The first model focuses on investigating the contact force between the model and the load, featuring a symmetrical design that incorporates an original PSE and a foam block measuring 100x100x80 mm, as illustrated in Fig 4.25 To optimize computation time, the model is half-built and is meshed with a total of 11,195 four-node tetrahedral elements, comprising 4,540 elements for the foam block, 994 for the tape cover layer, 4,242 for the latex tube, and 1,470 for the foam The material models, which were previously validated in sections 4.1, 4.2, and 4.3, are applied in both real-world and simulation contexts.
The compression test utilizes a model featuring a foam brick derived from a car seat cushion, which encapsulates the majority of the PSE, leaving its two free ends exposed These ends are reinforced by a steel plate, integral to the construction, which restricts the inflation of the PSE's ends under high pressure The arrangement of this model is illustrated in Fig 4.27, while the foam brick is appropriately meshed for testing purposes.
19344 four-node tetrahedral elements and the steel plate is a solid body a) In reality b) In simulation
The second model used in the car seat cushion
The scheme of the model used for car seat cushion
When compressed air is supplied to the PSE, the latex tube expands and makes contact with the sloped sections of the steel plate To understand this interaction, the friction coefficient between the latex material of the spring and the steel was experimentally examined using a "ball on disc" tribometer The experimental setup was designed to accurately measure this frictional relationship.
84 c) Dependence of friction coefficient on velocity Determination of friction coefficient between steel and latex
An experiment was conducted using a tribometer machine that operates on the "ball on disc" principle, as illustrated in Figures 4.28a and 4.28b The setup involved a load of Fz = 5 N, with varying combinations of the disk's angular velocity (ωd) and radius (Rd) to achieve a range of tangential velocities from 0.01 m/s to 3 m/s The radius of the steel ball used in the experiment was rb.
The friction coefficient, as shown in Fig 4.28c, exhibits a clear dependence on velocity up to 1 m/s, after which it stabilizes at approximately 0.43 In this simulation, the deformation speed of the latex membrane is considered very low, leading to the selection of a friction coefficient value of 0.8 between the latex tube and the steel plate for accurate modeling.
4.4.3 Simulation results and experimental results
Simulations were conducted with internal pressures maintained at constant values of 0 kPa, 5 kPa, 10 kPa, 15 kPa, 20 kPa, and 25 kPa For each internal pressure scenario, an external force was applied by an indentor pressing down on the foam block, resulting in varying displacements.
The indentor is centrally positioned along the longitudinal axis of the foam block, with a displacement range of 0 to 30mm The displacement over time exhibits a triangular pattern, occurring at a rate of 2 mm/s.
Simulation of model 1 with p s = 20 kPa and displacement of the indentor at 30 mm
Force-displacement results of experiment and simulation (model 1)
Figure 4.29 illustrates the deformation observed in the initial model during the compression test, with an internal pressure of 20 kPa and an indentor displacement of 30 mm Additionally, Figure 4.30 presents a comparison of experimental and simulation results regarding the force-displacement relationship during both the loading and unloading phases of the compression test, conducted under constant pressure mode in static conditions.
In the second model of the compression test, as illustrated in Fig 4.31, the deformation occurs with an internal pressure of 20 kPa and an indentor displacement of 10 mm The comparison of experimental and simulation results for the force-displacement relationship is presented in Fig 4.32.
Simulation of model 2 with p s = 20 kPa and displacement of the indentor at 10 mm
Force-displacement results of experiment and simulation (model 2)
Finite element analysis of a seat cushion with a simplified human body
This analysis focuses on simulating the distribution of contact pressure between a simplified human body and a car seat cushion featuring an inserted PSE, operating in constant pressure mode under static conditions The model is designed with symmetry in mind, resulting in a half-built representation for all components involved.
The complete model represents the interaction between a simplified human body and a seat cushion featuring an inserted PSE Initially, a finite element model of the car seat cushion was created, incorporating a cutout for the foam brick, as illustrated in Fig 4.26 The 3D solid geometric model was developed using Inventor and subsequently imported into MSC.Marc, where it was meshed with 288,327 four-node tetrahedral elements, as shown in Fig 4.33 The cushion is supported by a fixed square plane.
Geometry and mesh of the car seat cushion
The foam brick model featuring an inserted PSE was simplified by eliminating the steel plate and the ends of the PSE, as depicted in Fig 4.34 This simplification is assumed to have no impact on the pressure distribution in the contact zone between the seat cushion and the human body The model was meshed using four-node tetrahedral elements, comprising 19,344 elements for the foam block, 994 for the tape cover layer, 1,380 for the latex tube, and 1,640 for the foam within the latex tube The materials were modeled consistently with the methodologies outlined in sections 4.1, 4.2, and 4.3.
Geometry and mesh of the foam brick with a simplified PSE inserted inside
The simplified human body model features a half-build design that includes the muscle layer comprising the abdomen, waist, and thigh The skeletal structure consists of half of the sacrum, pelvis, and femur, as referenced in [25] This bone model is constructed using four-node tetrahedral elements, totaling 5769 for the femur, 5565 for the pelvis, and 1982 for the sacrum (see Fig 4.35) Assuming the bones are solid, the constitutive model is defined as steel, while the mass density is adjusted to match that of bone, in accordance with [26] The material parameters for the bones are thus established.
Mass density: 2000 [kg/m 3 ] Young’s Modulus: 2.1e+11[Pa] Poison’s ratio: 0.3
The geometry and mesh of bones are influenced by standard human body proportions, with muscles simplified into basic shapes; for instance, the thigh is modeled as a cylinder and the abdomen as an oval cylinder.
Geometry and mesh of the muscle layer (without bones)
The constitutive model of muscle tissue is detailed in the article [27], which provides essential parameters for calf and heel muscle tissues Table 2 in the article presents these parameter values, offering crucial insights into the mechanical behavior of muscle tissue.
Table 2 The parameters of the constitutive model of Ogden type of muscle tissue used in the article [27]
The muscle tissue in the complete model, derived from the abdomen and thigh, is softer compared to that of the calf and heel, necessitating quantitative modifications to the parameter values of the constitutive model Consequently, the Ogden-type constitutive model for muscle tissue has been established to reflect these adjustments.
In accordance with [27] the mass density of muscle tissue is set to 985 [kg/m 3 ]
In accordance with [28] the components are assembled into the model as shown in Fig 4.37
A contact table was created to address contact problems, featuring a comprehensive list of entries that represent components within the complete model Each entry specifies a pair of contact bodies that interact, with each entry corresponding to a specific contact interaction.
90 two kinds of contact interaction: glue contact (G) and touch contact (T) which define the properties of the interaction between the two contact bodies
In this study, we calculate the mass of a simplified human body model, assuming a total mass of 75 kg and a height of 175 cm The results, derived from the weight distribution of various body parts, are detailed in Table 4.
Table 4 The weight of parts of the human body
Total weigh : 75 kg Height: 175 cm
Upper part of body 11.9594 Middle part of body 12.3335 Lower part of body 8.39
According to Table 5, the simplified model of the human body considers the mass of the lower body, including the thighs and bones, to be 14 kg The remaining upper and middle body parts, which consist of the head and arms, have a mass of 18 kg for half of this section This weight is assumed to be represented as an equivalent force of 180 N, distributed across the pelvis and sacrum (refer to Fig 4.38).
The equivalent force distributed on pelvis and sacrum
4.5.1.1 Simulation results and experimental result
To simulate the contact pressure distribution between the human body and a seat cushion, the internal pressure of the PSE is designed as a time-dependent function, varying within the range of 0 to 25 kPa.
25 10 15 s kPa if t p t kPa if t kPa if t
The simulated contact pressure distribution results for the designed internal pressure of the PSE reveal significant changes as the pressure increases from 0 to 25 kPa Specifically, the peak contact pressure between the body and the car seat cushion rises from 15.84 kPa to 18.67 kPa, illustrating how variations in internal pressure impact the overall pressure distribution.
Contact pressure distribution in simulation
An experiment was conducted to compare experimental data with simulation results, involving a participant weighing 75 kg and standing 175 cm tall, seated in a car with a PSE inserted The electro-pneumatic control subsystem regulated the pressure within the PSE, varying it between 0 and 25 kPa The distribution of contact pressure between the participant's body and the seat cushion was measured using an Xsensor pressure mapping system that covered the seat cushion's surface.
Experimental results indicate that increasing the pressure within the PSE from 0 to 25 kPa alters the contact pressure distribution between the body and the car seat cushion, with the peak value rising from 16.78 kPa to 19.36 kPa.
94 a) p s =0 kPa b) p s % kPa Contact pressure distribution in experiment
The simulation and experimental results align closely, with peak values ranging from 15.84 kPa to 18.67 kPa in simulations and from 16.78 kPa to 19.36 kPa in experiments Notably, the difference in peak values at 0 kPa and 25 kPa pressure within the PSE is approximately 3.5 kPa, representing around 20% These findings underscore the significant impact of PSE on pressure distribution.
Conclusion
This chapter explores the distribution of contact pressure using the finite element method (FEM) It details the step-by-step development of finite element models for compression tests, starting with the identification of appropriate constitutive models for the materials involved Two distinct finite element models are utilized for the compression test: one replicates the model from Chapter 3, while the other is designed for a car seat cushion Additionally, a comprehensive finite element model illustrates the interaction between a simplified human body and the seat cushion, incorporating a PSE (Pressure Sensitive Element) for enhanced accuracy.
The study simulated model behavior and compared it with experimental results, focusing on the deformation of the PSE and seat cushion, as well as the contact force during compression tests and contact pressure distribution in the complete model The findings revealed a strong correlation between simulation and experimental data regarding the contact force-deformation relationship and the impact of the PSE on contact pressure distribution.
Summary
This thesis explores an innovative device designed to be integrated into car seat cushions, enabling the adjustment of contact pressure between the human body and the seat Central to this device is a pneumatic spring element (PSE) that modifies the contact pressure, alongside an electro-pneumatic control subsystem that regulates the pressure within the PSE The study investigates how this device affects pressure regulation and aims to enhance its development for improved speed and precision in operation The primary objectives of the research presented in this thesis include examining the device's impact on pressure management and advancing its functionality.
To model the influence of the device on the distribution of contact pressure between the human body and the car seat with PSE inserted inside
To investigate the influence of the device on the transmission of acceleration
The studies are carried out using analytical calculation method and FEM in combination with experimental methods as they are presented in chapter 2, 3 and 4
First, the structure and function of the car seat using the device which was made in the past is presented and analyzed carefully in chapter 2
Chapter 3 presents analytical and numerical calculations of simplified models illustrating the interaction between the human body and a car seat cushion featuring an inserted PSE Two models are analyzed: the original model and an improved version, both consisting of a mass in contact with a foam block containing the PSE The key distinction lies in the PSE and the electro-pneumatic control subsystem The results provide insights into the models' behavior under two control modes—constant pressure and constant stiffness—across static and dynamic conditions The comparison aims to evaluate how improvements affect pressure regulation within the PSE and the transmission of acceleration Experiments aligned with numerical simulations validate the calculated results, demonstrating a strong correlation between experimental and theoretical findings.
The finite element method is employed to simulate the contact pressure distribution between a car seat cushion containing a PSE and a simplified human body, utilizing MSC Marc software for this analysis A comprehensive examination of this study is detailed in Chapter 4, where the constitutive models for all materials utilized in the simulation are clearly defined.
The cushion consists of latex, tape, and foam, with the material parameters determined through optimization-based curve-fitting techniques using stress-strain data from load-deformation tests A finite element model simulating the interaction between a simplified human body and the car seat cushion under static conditions is created, allowing for simulations at pressure levels of 0 kPa and 25 kPa Subsequently, the pressure distribution within the contact zone of the seat cushion is assessed.
This thesis demonstrates that the experimental model device can effectively influence acceleration transmission and alter contact pressure distribution when integrated into a real car seat.
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7 List of papers published by the author
Main publications related to the thesis:
[1] Cirkl, D., TranXuan, T.: Simulation model of seat with implemented pneumatic spring, in Journal Vibroengineering PROCEDIA, Vol 7, ISSN Print 2345-0533, ISSN Online 2538-8479 ,
[2] TranXuan, T., Cirkl, D.,: Simulation model of seat with implemented pneumatic spring with consideration of variable pressure in air reservoir, 32 nd COMPUTATIONAL MECHANICS Conference, 2016
[3] TranXuan, T., Cirkl, D.,: FEM model of pneumatic spring assembly, in Journal
Vibroengineering PROCEDIA, Vol 13, ISSN Print 2345-0533, ISSN Online 2538-8479, 2017
[4] TranXuan, T., Cirkl, D.,: FEM model of pneumatic spring supported by a steel plate,
[5] TranXuan, T., Cirkl, D.,: Modeling of dynamical behavior of pnuematic spring-mass system, in Proceedings EM 2018, Paper #279, pp 865–868, 2018
[6] TranXuan, T., Cirkl, D.,: The effect of system improvement on regulation of pressure inside pneumatic spring element and on transmission of acceleration, , in Journal Vibroengineering
PROCEDIA, Vol 27, ISSN Print 2345-0533, ISSN Online 2538-8479, 2019
Appendix A: Model of Polyurethane Foam for Uniaxial Dynamical Compression
PU foam features a unique pore structure that provides significant resistance to pressure loading, thanks to its inherent buckling strength The force generated by the foam's buckling strength is denoted as F b, while c b represents the coefficient related to the structural buckling strength.
(A.1) where x-z is the relative displacement between the mass and the PU foam block
As cells undergo increasing compression, they eventually come into contact with one another at their walls The force dynamics during this phase closely resemble the behavior of air compression in a sealed container, which can be described using a progressive polytrophic function.
S p , p p , h p , n p are constants of the model, where h p means the vertical asymptote position
The damping behavior of the matrix material is characterized by Maxwell's viscoelastic components, incorporating a nonlinear spring with polytropic characteristics and defined constants S 0i, p 0i, h i, n 0i Additionally, it includes a nonlinear damper with a damping constant c i and an exponent n i, utilizing three Maxwell components in total.
In the context of cell-wall contact and the interaction between cell struts during compression, it is reasonable to conclude that friction plays a significant role in the damping properties of PU foam The model incorporates friction through a friction coefficient, denoted as f f, which is defined as a function of velocity using the arctan function combined with a power function, as outlined in equation (3.23).
, i i n n i i di dc di i i i di i di h h
n i sgn , , 1 3 di i di di di di