1.5092052

Numerical studies on influence of friction
coefficient in multi-leaf spring on suspension
basic characteristics
Cite as: AIP Conference Proceedings 2078, 020049 (2019); https://doi.org/10.1063/1.5092052
Published Online: 04 March 2019
Wieslaw Krason, Zdzislaw Hryciow, and Jozef Wysocki
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AIP Conference Proceedings 2078, 020049 (2019); https://doi.org/10.1063/1.5092052
© 2019 Author(s).
2078, 020049
Numerical Studies on Influence of Friction Coefficient
in Multi-Leaf Spring on Suspension Basic Characteristics
Wieslaw Krason1, a), Zdzislaw Hryciow2, b) and Jozef Wysocki2, c)
1
Military University of Technology, Faculty of Mechanical Engineering,
Department of Mechanics and Applied Computer Sciences.
2
Military University of Technology, Faculty of Mechanical Engineering, Institute for Motor Vehicles and Transport.
a)
Corresponding author: [email protected]
b)
[email protected]
c)
[email protected]
Abstract. The results of numerical analyses for a 3D model of a dependent vehicle suspension equipped with a multi-leaf
spring (with a shackle) are presented in the paper. Friction between the feather surfaces and the impact of the sprung and
the unsprung mass of a vehicle suspension are taken into account during tests. The influence of elastic and damping
properties of tires as well as the impact of viscous damping are omitted in the considerations. This approach considerably
simplifies the analysis of the calculation results. The kinematic excitation originating from the micro-profile of road
irregularities is assumed to be a sinusoid with the given parameters (amplitude and frequency). Tests of the force impact,
for the assumed values of static and kinetic friction coefficients, on the quality of suspension work are included in the
multivariate numerical tests. The simulation results are presented in the form of vibrations time courses of the sprung and
unsprung mass as well as tabulations of the vibration transmission factor. This parameter is used to assess the suspension
quality (the degree of vibration isolation). The tests are carried out in the LS-DYNA system.
INTRODUCTION
A wide application of multi-leaf springs in the dependent suspension of trucks and trailers causes the focus of the
research effort on determining the elastic and damping characteristics. The influence of these characteristics on
vibrations and dynamic loads of its basic assemblies and systems (body, suspension system, carrier etc.) is also
investigated. Discrete car vibration models describing its vertical vibrations [3, 9, 10] are frequently used in the
analysis of physical phenomena occurring in the selected elements, parts, assemblies and the entire vehicle. For
example, the effects of generalized forces for the flat system as well as vertical and angular vibrations are considered
for the analysis of dynamic loads of the suspension in the vehicle longitudinal plane [3, 9, 10]. Frequently, a series
of simplifications in the construction of models, for example small vibrations of the body and road wheels, are
considered. A fixed and constant contact between the tire wheel and the surface, constant rolling resistance
coefficient, linearity of the elastic and damping characteristics with omitting significant friction sources in vehicle
suspension, etc., are introduced in order to facilitate the vibration analysis. It was also found that for a large group of
construction solutions for two-axle vehicles, the dependence is quite frequently met that the coefficient of
distribution of sprung mass (ε ≈ 1) is close to or equal to one. The coefficient of sprung masses distribution is
described by the dependence (1) according to [9, 10].
ε = ρ 2 /(d r ⋅ d f )
where:
ρ
– radius of the car sprung mass inertia,
df (dr) – distance between the sprung mass centre and the front (rear) axis.
Computational Technologies in Engineering (TKI’2018)
AIP Conf. Proc. 2078, 020049-1–020049-6; https://doi.org/10.1063/1.5092052
Published by AIP Publishing. 978-0-7354-1806-6/$30.00
020049-1
(1)
When the above condition is met, the front or rear suspension can be analyzed independently. The detailed
spring models developed using the finite element method can be found in addition to the discrete models of the
entire vehicle. Different approaches to modeling the vehicle suspension as well as elastic and damping elements are
presented in literature [3–7]. The results of numerical analyses of multi-leaf springs from a simple three-part model
to multi-object models are presented in works [5–7]. The presented models have different levels of complexity and
accuracy, and consequently a different amount of work in their development and the required calculation time.
Dynamic properties of the multi-leaf spring used in a suspension of the light commercial vehicle in terms of
driving comfort are investigated in work [1]. An influence of friction, frequency and amplitude of vibrations on
spring dynamic stiffness is assessed. The work proved that the reduction of friction between leaf springs is the most
effective way to improve the stiffness of the dynamic multi-leaf spring.
An estimation of the total force generated by a leaf spring as the sum of force vectors derived from elastic force
and dry friction is proposed in paper [2]. The spring model, which does not take into account an impact of the
change in relative velocities on the friction surfaces of leaf springs is adopted. The hysteresis in this model is
strongly non-linear. The results of numerical analyses are verified with the experimental tests carried out on a truck.
Noteworthy is also article [3]. The authors focus on the “multibody” approach and the finite element method
(FEM) using to find a compromise between accuracy and a computational speed. A variant of a multibody
numerical model as a multibody model built in LMS Virtual. Lab Motion [3] is proposed. Each leaf spring is
discretized in the form of several rigid bodies joined by linear elastic beams. A contact between individual leaves is
simplified to a numerical model of interaction between the balls. The approach with contact areas known a priori [4]
is advantageous for it significantly improves a computational efficiency. Friction between the leaf springs has a very
large impact on the vibration parameters of the vehicle suspension, especially on the dry friction effect [3, 8–12].
A necessary condition for the analysis of vibrations and dynamic loads of the suspension system (vehicle) [3–8, 11]
is, therefore, taking into account the friction in modern methods for designing and diagnosing the multi-leaf springs.
A 3D model of a dependent vehicle suspension equipped with a multi-leaf spring (with a shackle), which takes
into account the friction between the feather surfaces and the impact of the sprung mass and the unsprung vehicle
suspension, is developed in the work. The impact of elastic and damping properties of the tires and the effect of
viscous damping are omitted in the model. The impact of force, for the assumed values of static and kinetic friction
coefficients, on the quality of suspension work is determined as a result of the tests. The level of dynamic loads of
the vehicle body and selected suspension units is additionally determined
AIM AND SCOPE OF TESTS
The paper focuses on developing a 3D model of a vehicle dependent suspension according to FEM. The
numerical model includes a detailed solid model of a multi-leaf spring with a shackle and a part of the body mapped
in the form of a rigid body. An influence of tires and viscous damping was omitted in the considerations. Values of
the friction coefficients are adopted on the basis of literature data [8]. The excitation of the tested suspension model
is a function of time describing road irregularities in the form of a sinusoid with given parameters (amplitude and
frequency). A determination of the load impact with a fixed surface condition of the leaf springs (coefficient of
friction between the leaves) on the ability of the suspension to reduce the level of vibration transmitted on the
vehicle body is the purpose of this work. FE numerical analyses of the motion of vibrating mass representing the
truck body for various variants are performed with changing conditions of the leaf springs interaction. The changes
concerned three variants of static and kinetic friction coefficients associated with the basic friction pair of leaf
springs. Selected simulation results are presented in the form of time courses of the sprung and unsprung mass
vibrations as well as tabulations of a vibration transmissibility coefficient which is used to assess the suspension
quality (the level of vibroisolation).
MODEL RESEARCH
On the basis of the working conditions analysis of the multi-leaf spring, it can be clearly stated that conditions of
cooperation between individual leaves change during the spring operation [12]. The graphite grease layer is usually
between the leaves in the initial state [11]. It is gradually removed during work and the nature of work approaches
the conditions of dry friction. A layer of oxides appears in addition on the surfaces of the leaves.
Therefore, the paper attempts to estimate an influence of static and kinetic friction coefficients on the amount of
energy dissipated in the spring, as well as the quality of the vibration isolation system during the vehicle motion on
020049-2
an uneven ground. A simulation model consisting of a quadrant of the car’ body modeled as a rigid body, a multileaf spring and connecting elements – an eye and a shackle (Fig. 1) is used in the numerical analysis. The considered
spring consists of seven leaves. 8-node solid elements are used to build the spring model. In total, the numerical
model consists of about 12.000 elements. A contact with friction forces has been defined between the leaves.
A friction coefficient is calculated based on dependence (2) in the LS-DYNA system [13]:
μ = μ k + ( μ s − μ k ) ⋅ e −γ ⋅v rel
(2)
where:
µs – static coefficient of friction,
µk – kinetic coefficient of friction,
γ – exponent determining the change of coefficients as a function of relative velocity,
vrel – relative velocity of cooperating surfaces.
FIGURE 1. 3D model of the dependent suspension according to FEM: 1 – rigid body part, 2 – conventional multi-leaf spring,
3 – spring shackle, CG – mass center of the body (1), za(t) – kinematic excitation, zb(t) – displacement of a rigid body
(a system response)
The calculations were carried out in two stages. In the initial phase it was necessary to compress the leaves in the
middle part, due to different radii of individual leaves. This ensured the introduction of initial stresses and the
emergence of pressure forces between the leaves (generating, among others, the friction forces). The behavior of the
spring and vehicle body during driving on an uneven ground was determined in the second stage of analysis. They
are mapped by the kinematic excitation applying to the central part of the spring (driving axle) described with
relationship (3):
z a (t ) = z o ⋅ [1 − cos(2 ⋅ π ⋅ f ⋅ t )]
(3)
Excitation frequencies are selected to analyze the behavior of the vibration system in the sub-resonance and
super resonant areas [14]. Therefore, the natural vibration frequency (1.67 Hz) was determined for the suspension
model (Fig. 1) in the static equilibrium position. Two frequencies f = 1 and 5 Hz with amplitude values of 3 and
6 mm were assumed for each of them in the next part of simulation tests.
The quality of suspension work (level of vibroisolation) was determined by T transmissibility coefficient as a
comparison of the root mean square values of vibration accelerations for different associations between the
cooperating spring leaves (different values of the friction coefficient):
T=
RMS b
RMS a
(4)
where: RMSa and RMSb – root mean square values of accelerations at points a (driving axle) and b (vehicle body).
020049-3
To identify the suspension quality, a logarithmic scale to determine E effectiveness of vibroisolation
(effectiveness of the isolator) based on dependence (5) can be applied. The effectiveness of the isolator is expressed
in dB:
1
E = 20 ⋅ log  
T 
(5)
The calculations were carried out for three variants of friction coefficients corresponding to possible friction
associations (surface states) of leaf springs, namely, pure and dry covered with graphite grease and covered with a
thin layer of oxides. The values of a static and a kinetic coefficient of friction used in the numerical studies are
shown in Table 1.
Friction coefficients
Static, µs
Kinetic, µk
TABLE 1. Coefficients of a static and a kinetic friction [8]
The surface condition of leaf springs
Dry, clean
Covered with graphite grease
0.78
0.42
0.11
0.058
Dry, covered with rust
0.27
0.14
RESEARCH RESULTS
The results of the suspension model tests in the form of vertical acceleration graphs of the central spring fragment
(point a – Fig. 1) and vertical accelerations of the vehicle body (point b – Fig. 1) for three surface conditions of the
leaf springs, obtained for the excitation with an amplitude zo = 3 mm and frequencies f = 1 and 5 Hz, are shown in
Fig. 2.
FIGURE 2. The suspension model vertical accelerations determined for excitation with an amplitude of zo = 3 mm
and two frequencies: f = 1 Hz (left), f = 5 Hz (right) and considering the leaves surface conditions
Based on the model vibration analysis for the super resonant frequency (f = 5 Hz), it is found that introducing the
lubricant between the cooperating leaves reduced an effective value of the vehicle body acceleration by
approximately 2.25 and a vibration transmission coefficient T decreased from 0.27 to 0.12, compared to the spring
vibrations with both clean and dry leaf surfaces. However, for the subresonant frequency (f = 1 Hz), the opposite
effect is observed, and introducing the lubricant resulted in the greatest increase in the value of the generated
accelerations.
The numerical analysis results describing the excitation impact on the vibration transmissibility coefficient (T)
and the effectiveness of the isolator (E) are shown in Table 2 for the considered variants. For super resonance
frequencies, the largest decrease in an effective value of vertical accelerations is observed for the spring with the
leaves covered with a layer of grease and the smallest for a dry, clean, steel surface. A greater improvement in
vibration isolation is observed for a smaller amplitude. The vibration transmissibility coefficient is increased and,
thus, the vibration isolation efficiency is reduced by increasing the amplitudes of vibrations. For the subresonant
frequencies (f = 1 Hz), an increase of an effective value of the vehicle body accelerations in relation to the
020049-4
accelerations of the drive bridge (point a – input of the considered system, Fig. 1) is observed for all the analyzed
variants. In this case, the opposite trend is noted to the one described above. Decreasing the coefficient of friction as
well as decreasing the amplitude of vibrations causes an increase in the values of accelerations.
TABLE 2. Comparison of the test results
T – Transmissibility
E – Effectiveness of the Isolator [dB]
Excitation Parameters
Frequency
f [Hz]
Amplitude
zo [mm]
Covered with
graphite grease
Covered with
rust, dry
Clean,
Dry
Covered with
graphite grease
Covered with
rust, dry
Clean,
Dry
1
1
5
5
3
6
3
6
1.39
1.54
0.12
0.11
1.06
1.26
0.14
0.12
1.05
1.05
0.27
0.17
–2.87
–3.77
18.44
19.13
–0.54
–2.00
16.98
18.67
–0.42
–0.40
11.47
15.42
An influence of a change in the friction coefficient and the vibrations amplitude (zo = 3 mm and zo = 5 mm) on
the values of the vibration transmissibility coefficient (T) for the excitation frequency f = 1 Hz (Fig. 3a) and f = 5 Hz
(Fig. 3b) is presented in Fig. 3 to complement the numerical results.
1,8
0,3
zo = 3 mm, f = 1 Hz
zo = 3 mm, f = 5 Hz
0,25
zo = 6 mm, f = 1 Hz
1,4
Transmissibility, T
Transmissibility, T
1,6
1,2
1
0,8
0,6
0,4
zo = 6 mm, f = 5 Hz
0,2
0,15
0,1
0,05
0,2
0
Covered with graphite
grease
Covered with rust, dry
Clean, Dry
0
Covered with graphite Covered with rust, dry
grease
Clean, Dry
Surface condition
Surface condition
(a)
(b)
FIGURE 3. Vibration transmissibility coefficient (T) for excitation with frequency f = 1 Hz (left) and f = 5 Hz (right)
and amplitude values: zo = 3 and 6 mm for the analyzed frictional pair
Distributions of HMH stresses in individual leaf springs for excitation with 3 mm amplitude operating at 5 Hz
frequency are additionally illustrated in Fig. 4a. They are mainly caused by compressing the leaves through the
pulling screw. The highest stress values are observed in the middle part of the spring, especially the shortest leaf.
Figure 4b presents the distribution of contact force between the leaf spring. Based on the distribution of contact
force, it is found that the main interaction between the leaves occurs at their ends. This is confirmed by observations
of the experimental results [4]. The increased heating of leaves as well as intensive wear of their surfaces was found
in these areas [4, 12].
CONCLUSIONS
Based on the obtained results, it was found that the vibration transmissibility coefficient (T) depends not only on
the leaf springs surface condition, but also on the parameters of the excitation (amplitude and frequency).
There may appear a phenomenon of blocking the leaf springs, in particular for small kinematic excitations from
the road side. Such a phenomenon is observed for roads in a condition with a very small amplitude of the road
micro-profile in real conditions of vehicle operation for the vehicle suspension in a poor condition caused by
negligence of a periodic technical service [9, 10]. As a consequence, the vibrations from the drive axle on the
vehicle body are transmitted directly. The results of numerical analysis also indicates a high impact of the leaf
springs surface condition on the vehicle body acceleration values. Operational negligence can increase the
acceleration values by several times and an undesirable phenomenon, from the point of view of smooth traffic and
passenger ride comfort, occurs.
020049-5
(a)
(b)
FIGURE 4. HMH stress [σmax = 500 MPa] specified in the leaf spring (a) and contact forces [Fmax = 500 N] determined
between the spring leafs (b) for excitation with frequency f = 5 Hz and amplitude zo = 3 mm
ACKNOWLEDGEMENT
The works were carried out as part of the research project PBS 23-937, Military University of Technology,
2016-2018.
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