Shaping the vibro-isolation properties of horizontal seat suspension

Abstract

In this paper, an effective procedure of shaping the vibro-isolation properties of seat suspension system is presented. The simulation model is created to determine the dependence of evaluation criteria to the design parameters by means of which the system characteristics are formed. The developed optimisation procedure allows to find Pareto-optimal system configuration for the conflicted vibro-isolating criteria, i.e. the frequency weighted transmissibility factor used to evaluate dynamic seat comfort based on acceleration signals and the suspension travel. In order to optimise both the conflicted vibro-isolation criteria, a minimising of the transmissibility factor (primary criterion) is proposed taking into account the suspension travel that is transferred to a non-linear inequality constraint. The correctness of proposed procedure is evaluated using experimental research of the best solution of horizontal seat suspension. This research is performed using the passive system with optimal visco-elastic characteristics, which are selected especially for the well-defined input vibration. The satisfactory agreement of experimental and numerical results is obtained for the analysed vibration reduction system.

Introduction

The design process of seat suspension systems is difficult due to the conflicted criteria for evaluating their effectiveness. For example, it is desired to limit the vibrations transmitted from the cabin floor to the machine operator of the earth moving machinery. ¹ Then the vibration isolation of operator can be realised by using a seat suspension. ² On the one hand, the seat motion regarding the inertial reference point should be minimal to protect the machine operators against harmful vibrations.^3,4 In order to control the seat vibration, the transmissibility factor can be calculated as a percentage ratio of the frequency-weighted acceleration on the seat surface to the frequency-weighted acceleration on the floor. ⁵ As the risks to health from whole-body vibration is not the same at all frequencies, the frequency weightings are used to represent the varied sensitivity of the human body in different frequency ranges. ⁶ Unfortunately, only this kind of vibration control contributes to the increase of the suspension travel. ⁷ A machine operator can lose contact with control gears mounted to the cabin, so the controllability of such machines becomes difficult. On the other hand, the suspension travel has to be minimised as well. Both these objectives conflict, therefore an improvement in one objective requires a degradation of another.^8,9 Consequently, the design of seat suspension systems can be treated as an optimisation problem.^10,11

The main goal of the multi-criteria optimisation approach is to find the compromising solutions between several conflicted criteria. Such compromising solutions guarantee the best system performance for the opposite objectives. There are some works presented in the existing literature wherein seat suspension systems are considered in relation to the conflicted system requirements. In the papers,^12,13 the most ‘comfortable’ trade-off between the effective acceleration of isolated body and the relative displacement of suspension system is selected with the help of weighted sum method. In the paper, ¹⁴ a constraint method is employed to solve the problem of the multi-objective optimisation of passive seat suspension vibro-isolating properties. However, an application of the Pareto-optimal approach has been used for selecting dynamic characteristics of seat suspension systems, and it has been employed only for improving vertical whole body vibration isolation. The research presented in the papers^15–17 clearly shows that the horizontal seat suspensions are ineffective in many cases. Although they provide energy dissipation at the sufficiently high frequencies, the low-frequency vibrations are amplified due to the resonance effect. As the consequence of this undesirable effect, it is difficult to achieve the minimum health and safety requirements regarding the exposure of workers to the risks arising from the horizontal vibrations. ¹⁸

The objective of this paper is to formulate and verify an appropriate method for selecting the vibro-isolation properties of horizontal seat suspension. At first, a model of the seat suspension system and the isolated body is created in order to determine the values of conflicted vibro-isolation criteria. Secondly, the optimisation procedure is proposed that ensures finding a set of the compromise solutions (Pareto-otimal solutions). The proposed procedure allows to adjust the vibro-isolation properties of horizontal seat suspension for the individual requirements defined by the driver. The required configurability of passive systems is obtained by proper selecting their non-linear visco-elastic characteristics.

Model of the horizontal suspension with seated human body

In Figure 1(a), a physical model of the horizontal suspension with seated human body is shown. A simple 3-DOF model is employed to describe the biodynamic response of the system – seated human body and cushioned seat. ¹⁹ The object of vibration isolation is assumed as the lumped mass body which consists of three interconnected masses by means of linear springs and dampers. The first mass corresponds to the seat upper part frame ( $m_1$ ), while the subsequent masses represent the sitting part of the human body ( $m_2$ ) in contact with the back support and the sitting part of the human body ( $m_3$ ) not in contact with the back support. The equivalent stiffness $k_{12\text{x}}$ and damping $c_{12\text{x}}$ coefficients describe visco-elastic properties of the human body part in contact with the back cushion. In succession, the coefficients $k_{2\text{x}}$ and c_2x represent the reaction of the human body to the steering wheel (exerted by hands) and the coefficients $k_{23\text{x}}$ and $c_{23\text{x}}$ correspond to the human body part moving without the back support (head movements). In accordance with the presented model, evaluation of the human exposure to whole-body vibration is considered in horizontal x-direction. The mechanical vibrations are generated by using kinematic excitation $q_{\text{sx}}$ of the system.

Figure 1.

Physical model of the horizontal seat suspension with seated human body (a) and laboratory experimental set-up (b).

The corresponding equations of motion are defined in the matrix form as follows where ${\mathbf{q}}_{\text{x}}$ is the displacement vector of the system model, ${\mathbf{M}}_{\text{x}},{\mathbf{C}}_{\text{x}},{\mathbf{K}}_{\text{x}}$ are the corresponding inertia, damping and stiffness matrices, ${\mathbf{F}}_{\text{sx}}$ is the vector of applied forces.

The diagonal inertia matrix ${\mathbf{M}}_{\text{x}}$ includes the masses $m_1,m_2,m_3$ that exist in the system model. The damping ${\mathbf{C}}_{\text{x}}$ and stiffness ${\mathbf{K}}_{\text{x}}$ matrices take the following forms

$${\mathbf{C}}_{\text{x}}=[c_{12\text{x}}-c_{12\text{x}}0-c_{12\text{x}}{c}_{2\text{x}}+c_{12\text{x}}+c_{23\text{x}}-c_{23\text{x}}0-c_{23\text{x}}{c}_{23\text{x}}]$$

(2)

$${\mathbf{K}}_{\text{x}}=[k_{12\text{x}}-k_{12\text{x}}0-k_{12\text{x}}{k}_{2\text{x}}+k_{12\text{x}}+k_{\text{k}23\text{x}}-k_{23\text{x}}0-k_{23\text{x}}{k}_{23\text{x}}]$$

(3)

where: $c_{12\text{x}},c_{2\text{x}},c_{23\text{x}}$ are the damping coefficients and $k_{12\text{x}},k_{2\text{x}},k_{23\text{x}}$ are the stiffness coefficients of the human body model. The model parameters are identified for the seated human body with cushioned seat system and backrest contact in the lumbar region (Figure 1(b)). The numerical values established by identification of many laboratory measurements under well-defined conditions have been presented in the paper. ¹⁹ Numerical values for the model parameters – seated human body and cushioned seat are shown in Table 1.

Table 1.

Numerical values for the model parameters – seated human body and cushioned seat. ¹⁹

Parameter	Symbol	Value	Unit
Mass	m 1	15.1	kg
	m 2	53.4	kg
	m 2	13.2	kg
Damping	c 12x	583.9	Ns/m
	c 23x	208.4	Ns/m
	c 2x	8.2	Ns/m
Stiffness	k 12x	50735	N/m
	k 23x	1549	N/m
	k 2x	92	N/m

The vectors describing the human body displacements and applied forces are set in the following order:

$${\mathbf{q}}_{\text{x}}=[q_{1\text{x}}{q}_{2\text{x}}{q}_{3\text{x}}]\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\mathbf{F}}_{\text{sx}}=[-F_{\text{kx}}+F_{\text{bx}}-F_{\text{cx}}-F_{\text{fx}}00]$$

(4)

where $q_{1\text{x}},q_{2\text{x}},q_{3\text{x}}$ are the displacement of the human body model, $F_{\text{kx}}$ is the mechanical spring force, $F_{\text{bx}}$ is the force from end-stop buffer, $F_{\text{cx}}$ is the force of hydraulic shock-absorber and $F_{\text{fx}}$ is the overall friction force of suspension system. From the practical point of view the vector of applied forces ${\mathbf{F}}_{\text{sx}}$ can be used for predicting the vibrations transmitted from the cabin floor to the human body based on the known kinematic excitation $q_{\text{sx}}$ .

Assuming pure torsion of the coils, the mechanical spring force can be described as a function of the suspension deflection $q_{1\text{x}}-q_{\text{sx}}$ in the following form ²⁰

$$F_{\text{kx}}=\frac{G_{\text{s}}{d}_{\text{s}}^4}{8D_{\text{s}}^3{n}_{\text{s}}}(q_{1\text{x}}-q_{\text{sx}})$$

(5)

where $G_{\text{s}}$ is the shear modulus of elasticity of the material, $d_{\text{s}}$ is the wire diameter, $D_{\text{s}}$ is the mean diameter of the spring coils and $n_{\text{s}}$ is the total number of active coils.

The force of end-stop buffer is described using a combination of the equivalent linear and non-linear springs ²¹ acting only in a specified range of the suspension travel where $k_{\text{b}1}$ is the linear stiffness coefficient, $k_{\text{b}3}$ is the cubic stiffness coefficient, $q_{\text{C}}$ is the free travel of suspension, $q_{\text{T}}$ is the suspension travel corresponding to the transition point.

The shock-absorber force is related to the pressure loss across a simple throttle valve.^22,23 The pressure loss is connected with the volume flows $A_{\text{A}}({\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}})$ and $A_{\text{B}}({\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}})$ between chambers A and B of the shock-absorber

$$F_{\text{cx}}=\{{\zeta }_{(\text{B}\to \text{A})}\frac{{\rho }_{\text{o}}}{2}(\frac{A_{\text{B}}({\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}})}{A_{\text{o}}}{)}^2{A}_{\text{B}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{for}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}}>0-{\zeta }_{(\text{A}\to \text{B})}\frac{{\rho }_{\text{o}}}{2}(\frac{A_{\text{A}}({\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}})}{A_{\text{o}}}{)}^2{A}_{\text{A}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{for}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}}<0$$

(7)

where ${\rho }_{\text{o}}$ is the density of liquid and is assumed as a constant value, ${\zeta }_{(\text{B}\to \text{A})}$ and ${\zeta }_{(\text{A}\to \text{B})}$ are the pressure loss coefficients for the flow from chamber B to chamber A and from chamber A to chamber B respectively, $A_{\text{A}}$ and $A_{\text{B}}$ are the surfaces of shock-absorber piston (side A and side B, respectively), $A_{\text{o}}$ is the cross section area of a throttle valve.

The $\zeta (\text{Re})$ flow characteristic describes laminar as well as turbulent losses and is determined for the actual flow geometry (cross section area and shape). The pressure loss coefficient ζ for the circular orifices (with the cross section areas equal to $A_{\text{o}}=\pi d_{\text{o}}^{2}4$ ) is defined as a function of the Reynolds number $\text{Re}$ ²⁴

$${\zeta }_{(\text{A}\to \text{B})}=\{\frac{64}{\text{R}{\text{e}}_{(\text{A}\to \text{B})}}({\alpha }_{\text{o}}+\frac{l_{\text{o}}}{d_{\text{o}}})\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{for}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\frac{64}{\text{R}{\text{e}}_{(\text{A}\to \text{B})}}({\alpha }_{\text{o}}+\frac{l_{\text{o}}}{d_{\text{o}}})>1.81.8\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{for}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\frac{64}{\text{R}{\text{e}}_{(\text{A}\to \text{B})}}({\alpha }_{\text{o}}+\frac{l_{\text{o}}}{d_{\text{o}}})\le 1.8$$

(9)

where ${\alpha }_{\text{o}}$ is the flow coefficient, $l_{\text{o}}$ is a length of the orifices for a flow between chambers A and B, $d_{\text{o}}$ is a diameter of the orifices for a flow between chambers A and B. The pressure loss coefficient ζ for turbulent losses is assumed as the constant value equal to 1.8.

The Reynolds number ( $\text{Re}$ ), which characterises the ratio of inertia forces to frictional forces in the flow, is calculated as ²⁵

$$\text{R}{\text{e}}_{(\text{B}\to \text{A})}=\frac{4|A_{\text{B}}({\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}})|}{d_{\text{o}}\pi {\nu }_{\text{o}}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{R}{\text{e}}_{(\text{A}\to \text{B})}=\frac{4|A_{\text{A}}({\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}})|}{d_{\text{o}}\pi {\nu }_{\text{o}}}$$

(10)

where ${\nu }_{\text{o}}$ is the kinetic viscosity of liquid and is assumed as a constant value.

In order to simulate the dynamic behaviour of seat suspension system, the Coulomb friction model is characterised by a discontinuous arrangement in the following form ²⁶

$$F_{\text{fx}}={\delta }_{\text{f}}\{F_{\text{fa}}\text{for}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}|{\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}}|\le v_{\text{s}}\wedge \left|F_{\text{fa}}\right|\le F_{\text{fs}}{F}_{\text{fs}}\text{sgn}({\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}})\text{for}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}|{\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}}|\le v_{\text{s}}\wedge \left|F_{\text{fa}}\right|>F_{\text{fs}}{F}_{\text{fk}}\text{for}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}|{\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}}|>v_{\text{s}}$$

(11)

where ${\delta }_{\text{f}}$ is the reduction ratio having an influence on the overall suspension friction, $F_{\text{fa}}$ is the current active force, $F_{\text{fs}}$ is the static friction force, $F_{\text{fk}}$ is the kinetic friction force and $v_{\text{s}}$ is the relative velocity which separates sticking and slipping conditions of the system.

According to the Bouc–Wen model, the kinetic friction force $F_{\text{fk}}$ describes a hysteretic properties of the suspension system ²¹

$${\stackrel{·}{F}}_{\text{fk}}=-k_{\text{f}}({\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}})+{\gamma }_{\text{f}}|{\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}}|F_{\text{fk}}+{\beta }_{\text{f}}({\stackrel{·}{q}}_{1\text{x}}-{\stackrel{·}{q}}_{\text{sx}})\left|F_{\text{fk}}\right|$$

(12)

where $k_{\text{f}}$ is a positive stiffness which defines a height of the hysteresis, ${\gamma }_{\text{f}}$ and ${\beta }_{\text{f}}$ give the hysteresis shape and describe visco-elastic properties of the suspension system when the movement direction is changed.

The stiffness $k_{\text{f}}$ can be described as a linear function of the overall suspended mass as follows ¹⁴

$$k_{\text{f}}=a_{\text{f}}\sum _{i=1}^3{m}_{\text{i}}+b_{\text{f}}$$

(13)

where $m_{\text{i}}$ are the masses of each element contained in the model of seated human body and cushioned seat, $a_{\text{f}}$ and $b_{\text{f}}$ are approximation coefficients. Nominal parameter values used by the seat suspension model are presented in Table 2.

Table 2.

Nominal parameter values used by the horizontal seat suspension model.

Component	Parameter	Value	Unit
Helical	Mean diameter of the spring coils (Ds)	11.9 × 10−3	m
spring	Shear modulus of elasticity (Gs)	121 × 109	N/m2
	Wire diameter (ds)	1.8 × 10−3	m
	Total number of active coils (ns)	19	–
End-stop	Linear stiffness coefficient (kb1)	28 × 103	N/m
buffer	Cubic stiffness coefficient (kb3)	4 × 108	N/m3
	Free travel of suspension (qC)	8 × 10−3	m
	Suspension travel corresponding
	To the transition point (qT)	10 × 10−3	m
Hydraulic	Surface of shock-absorber piston,
shock-absorber	side A (AA)	1.77 × 10−4	m2
	Surface of shock-absorber piston,
	side B (AB)	1.64 × 10−4	m2
	Orifice diameter (do)	1 × 10−3	m
	Orifice length (lo)	13 × 10−3	m
	Flow coefficient (αo)	0.667	–
	Kinetic viscosity of a liquid (νo)	4.6 × 10−5	m2/s
	Density of liquid (ρo)	890	kg/m3
Friction	Reduction ratio (δf)	1	–
	Static friction force (Ffs)	150	N
	Relative slipping velocity (vs)	1 × 10−5	m/s
	First approximation coefficient (af)	2.45 × 103	N/(m ⋅ kg)
	Second approximation coefficient (bf)	161 × 103	N/m
	First parameter regulating the shape
	of a hysteretic friction model (βf)	2 × 103	m−1
	Second parameter regulating the shape
	of a hysteretic friction model (γf)	2 × 103	m−1

Nonlinear shaping of the force characteristics: $F_{\text{kx}},F_{\text{cx}}$ and $F_{\text{fx}}$ is desirable to obtain the best vibro-isolation properties of horizontal seat suspension. The mechanical spring force can be modified by means of the wire diameter $d_{\text{s}}$ , so its stiffness characteristic can be changed (Figure 2(a)). A change of the damping characteristics in velocity domain is also possible by modifying the circular orifice inside the shock-absorber. Changing the orifice diameter $d_{\text{o}}$ allows to adjust the value of damping force (Figure 2(b)) and the orifice length $l_{\text{o}}$ influences the non-linear shape of damping characteristics (Figure 2(c)). Furthermore, the friction force of suspension system can be modified by using the reduction ratio ${\delta }_{\text{f}}$ (Figure 2(d)). A contact with the end-stop buffers occurs only occasionally, and therefore shaping their force characteristics $F_{\text{bx}}$ is omitted in this paper.

Figure 2.

Force characteristics $F_{\text{kx}}$ of the mechanical spring for different wire diameter $d_{\text{s}}$ (a), force characteristics $F_{\text{cx}}$ of the shock-absorber for different orifice diameter $d_{\text{o}}$ (b), force characteristics $F_{\text{cx}}$ of the shock-absorber for different orifice length $l_{\text{o}}$ (c), friction force $F_{\text{fx}}$ for different reduction ratio ${\delta }_{\text{f}}$ (d).

Optimisation of the vibro-isolation properties

Optimisation of the seat suspension vibro-isolation properties is realised using simulation model of the passive system. Dynamic behaviour of the seat suspension system is modelled in the MATLAB-Simulink® software package. The system equations are programmed using the interactive graphical environment, which allows to simulate and test a variety of the time-varying systems. The non-linear ordinary differential equations (ODE) in the model are solved numerically using the fixed-step (step time of 1 ms) Bogacki-Shampine solver. ²⁷

The wire diameter $x_{\text{d}1}:=d_{\text{s}}$ of mechanical spring, the orifice diameter $x_{\text{d}2}:=d_{\text{o}}$ and length $x_{\text{d}3}:=l_{\text{o}}$ of shock-absorber and also the reduction ratio $x_{\text{d}4}:={\delta }_{\text{f}}$ of friction force are chosen as the set of decision variables which affect the vibro-isolation properties of the suspension system significantly. Then a vector of the decision variables can be defined in the following form:

$${\mathbf{x}}_{\text{d}}=[x_{\text{d}1},x_{\text{d}2},x_{\text{d}3},x_{\text{d}4}]$$

(14)

The frequency weighted transmissibility factor $\text{TF}{\text{E}}_{\text{x}}$ and the suspension travel $s_{\text{tx}}$ are taken as the conflicted optimisation criteria. The frequency weighted transmissibility factor is used to evaluate the effectiveness of vibration reduction system as follows

$$\text{TF}{\text{E}}_{\text{x}}=\frac{({\stackrel{\mathrm{··}}{q}}_{1\text{x}})\text{RMS}}{\hspace{0.17em}({\stackrel{\mathrm{··}}{q}}_{\text{sx}})\text{RMS}}$$

(15)

where: $({\stackrel{\mathrm{··}}{q}}_{1\text{x}}{)}_{\text{RMS}}$ is the frequency weighted root mean square value of the isolated body acceleration for longitudinal x-direction, $({\stackrel{\mathrm{··}}{q}}_{\text{sx}}{)}_{\text{RMS}}$ is the frequency weighted root mean square value of the input acceleration. To determine such a transmissibility factor (equation (15)), the frequency weightings defined in ISO-2631⁶ are used to describe the human body's sensitivity to vibration in specific frequency bands. The $\text{TF}{\text{E}}_{\text{x}}$ factor of value 1 means that a direct seating on the vibration platform would produce the same vibration discomfort. If the $\text{TF}{\text{E}}_{\text{x}}$ factor value is greater than 1, the vibration discomfort is increased by the suspension system. If the $\text{TF}{\text{E}}_{\text{x}}$ factor value is less than 1, the useful vibro-isolation is provided by the system.

The suspension travel is second numerical assessment of the system performance and can be defined in the following form ¹²

$$s_{\text{tx}}={max}_{t\in [0,t_{\text{k}}]}(q_{1\text{x}}(t)-q_{\text{sx}}(t))-{min}_{t\in [0,t_{\text{k}}]}(q_{1\text{x}}(t)-q_{\text{sx}}(t))$$

(16)

where $q_{1\text{x}}(t)$ is the displacement of isolated body for longitudinal x-direction, $q_{\text{sx}}(t)$ is the displacement of input vibration, t is the current time instant and $t_{\text{k}}$ is the observation time. To designate the following vibro-isolation criterion (equation (16)), it is required to measure the relative displacement $q_{1\text{x}}(t)-q_{\text{sx}}(t)$ from which the maximum deflection (rebound) of the suspension system is calculated. The suspension travel should be evaluated for a specific input vibration and the low values of this criterion are required in order to ensure the controllability of a vehicle by the driver.

In order to solve the optimisation problem, the concept of Pareto optimality is used in this paper. Therefore, non-dominated solutions should be separated from an area of the feasible solutions.^8,9 Each of the Pareto-optimal system configurations ensures the optimality of investigated systems in the conflicted criteria domains. An application of the multi-criteria optimisation procedure for the two-dimensional problem is graphically illustrated in Figure 3. Each of the Pareto-optimal solutions (Figure 3(a)) are represented by the value set of the decision variables (Figure 3(b)) that define a force characteristics of the mechanical spring, hydraulic shock absorber and suspension friction.

Figure 3.

Graphical illustration of minimising both of the conflicted vibro-isolation criteria: (a) criterion space and (b) decision variable space.

The $\text{TF}{\text{E}}_{\text{x}}$ factor is taken as the primary criterion and the suspension travel $s_{\text{tx}}$ is transferred to the non-linear inequality constraint. Such a formulation of the optimisation task produces desired results in minimising the transmissibility factor

$${min}_{{\mathbf{x}}_{\text{d}}}\text{TF}{\text{E}}_{\text{x}}({\mathbf{x}}_{\text{d}})$$

(17)

subject to the previously defined suspension travel

$$s_{\text{tx}}({\mathbf{x}}_{\text{d}})\le s_{\text{txj}},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{j}=1,\dots ,\text{u}$$

(18)

and the bounds of decision variables

$$({\mathbf{x}}_{\text{d}}{)}_{\text{min}}\le {\mathbf{x}}_{\text{d}}\le ({\mathbf{x}}_{\text{d}}{)}_{\text{max}}$$

(19)

where $s_{\text{txj}}$ is the constraint value of suspension travel, $({\mathbf{x}}_{\text{d}}{)}_{\text{min}}$ and $({\mathbf{x}}_{\text{d}}{)}_{\text{max}}$ are the vectors containing the minimum and maximum values of decision variables.

An appropriate selection of the constraint value $s_{\text{txj}}$ (equation (18)) allows to adjust the vibro-isolation properties of seat suspension system. Low value of such a defined constraint contributes to a significant limitation of the suspension travel, but the transmissibility factor achieves appreciable values. A desired reduction of the transmissibility factor may be obtained for higher values of the constraint, nevertheless the suspension travel increases.

The vibro-isolation properties of horizontal seat suspension are optimised for an exemplary excitation signal which is similar to white band limited noise in the range of frequency 0.5–10 Hz. Ten non-dominated solutions are separated from an area of the feasible solutions (Figure 4(a)). Each individual Pareto-optimal solution corresponds to the set of decision variables which defines the different force characteristics of the mechanical spring (Figure 4(b)), hydraulic shock-absorber (Figure 4(c)) and suspension friction (Figure 4(d)). Only selected Pareto-optimal system configurations, which are marked by the point nos. 1, 5 and 10 in Figure 4(a), are discussed in this paper. The first Pareto-optimal configuration (point no. 1) corresponds to the stiff suspension system in which a high limitation of the suspension travel $s_{\text{tx}}$ is obtained. The last system configuration (point no. 10) corresponds to the soft suspension system but the transmissibility factor $\text{TF}{\text{E}}_{\text{x}}$ has decreased significantly. The configuration no. 5 is one of the compromise solutions with respect to conflicted system requirements. Power spectral densities and transmissibility functions of the Pareto-optimal seat suspension at various system configurations are presented in Figure 4(e) and (f).

Figure 4.

Pareto-optimal point distribution (a) and corresponding force characteristics of the mechanical spring (b), hydraulic shock-absorber (c) and suspension friction (d), power spectral densities (e) and transmissibility functions (f) of the Pareto-optimal seat suspension at different system configurations.

As shown in Figure 4, force characteristics of the mechanical spring (Figure 4(b)), hydraulic shock-absorber (Figure 4(c)) and suspension friction (Figure 4(d)) for the 5th and 10th Pareto-optimal solutions are close to each other. Although the low stiffness and friction are required for both system configurations, an application of the progressive damping characteristics increases the system effectiveness in reducing the whole-body vibration. This results in lowering the transmissibility factor $\text{TF}{\text{E}}_{\text{x}}$ that is obtained for the Pareto-optimal solution no. 10 in comparison with the solution no. 5 (Figure 4(a)).

Experimental investigation of the optimised horizontal suspension with seated human body

Two different seat suspensions, i.e. conventional and optimal system, are measured during the laboratory tests. The optimised system uses lower stiffness, harder damping and reduced friction in order to obtain the vibro-isolation criteria evaluated in the previous section. The system configuration close to the Pareto-optimal solution no. 10 (Figure 4) is experimentally investigated for which the transmissibility factor $\text{TF}{\text{E}}_{\text{x}}$ has the lowest value. The suspension travel $s_{\text{tx}}$ indicates acceptable range of the system motion, therefore undesirable contact with the end-stop buffers should not occur.

The stiffness characteristics of mechanical spring are evaluated by using the force–deflection measurement of suspension system (Figure 5(a)). The motion of the horizontal suspension is excited slowly with the triangle cycling at a frequency of 0.1 Hz and an amplitude of $\pm 0.015\text{m}$ . The force of two different shock absorbers in the velocity domain is estimated by sinusoidal cycling of the damper at various frequencies, i.e. 0.83 and $1.66\text{Hz}$ , and an amplitude of $\pm 0.0125\text{m}$ (Figure 5(b)). The friction characteristics of the conventional and optimal systems are measured once more with triangle cycling of the seat suspension, but at higher frequency of $1\text{Hz}$ and lower amplitude of $\pm 0.002\text{m}$ (Figure 5(c)).

Figure 5.

Conventional and optimal force characteristics of the helical spring (a), hydraulic shock-absorber (b) and suspension friction (c) obtained using simulation model (solid line), force–deflection measurement of the horizontal seat suspension (dotted line) and force–velocity measurement of the damper (circles).

The measured force–deflection characteristics of the helical spring do not agree with the simulation results for displacements more than $\pm 0.01\text{m}$ because the effect of the end-stop buffers is also shown in Figure 5(a). The helical spring stiffness is evaluated by calculating the measured force per the measured deflection only within the range of suspension free travel. The vibrating structure of an external sensor fixation system probably has caused low oscillations of the force characteristics that are graphically presented in Figure 5(c). The averaged value of measured force well corresponds to the results obtained by using the computer simulation.

In Figure 6, power spectral densities and transmissibility functions of the conventional and optimal horizontal seat suspensions are presented. As shown in this figure, the optimised seat suspension considerably reduces human exposure to the harmful effects of vibration in comparison with the conventional system. Lower amplitudes of the human vibration are observed in the considered frequency range between 0.5 and 10 Hz. Numerical values of the transmissibility factor and suspension travel are presented in Table 3. The measured vibro-isolation criteria demonstrate the improved system effectiveness due to optimal characteristics of the mechanical spring, hydraulic shock-absorber and suspension friction. Both vibro-isolation criteria, i.e. the transmissibility factor $\text{TF}{\text{E}}_{\text{x}}$ and the suspension travel $s_{\text{tx}}$ , are lower of about 15% as compared to the conventional system. This would prove that the proposed optimisation procedure is correctly formulated and yields the desired vibro-isolation properties in view of the conflicted system requirements.

Figure 6.

Power spectral density (a) and transmissibility function (b) of the conventional (dashed line) and optimal (solid line) horizontal seat suspension.

Table 3.

Numerical values of the transmissibility factor and suspension travel measured for the conventional and optimised systems.

Conventional		Optimal		Improvement
TFEx	stx,	TFEx	stx,	${\delta }_{\text{TF}{\text{E}}_{\text{x}}}$ ,	${\delta }_{{\text{s}}_{\text{tx}}}$ ,
factor	mm	factor	mm	%	%
1.050	18.2	0.900	14.8	14.2	18.6

Conclusions

An application of the method for shaping vibro-isolation properties of seat suspension system leads to the reduction of harmful vibrations across a wide frequency range of the excitation signal with a slight amplification of the vibration amplitude in resonance. In addition, amplitudes of the system (relative) displacement are lowered to provide a proper vibration isolation of the driver in horizontal direction. The optimisation procedure proposed in the paper assists in selecting between the desired reduction of vibrations transmitted to the driver and the conflicting requirement for limitation of the suspension travel. Such a system design contributes both to the improved driver comfort and the enhanced ability to drive a vehicle.