Abstract Wear on the local area of steel wires’ surface is attributed to torsional fretting on the working process of stranded-wire helical spring.
A mathematical model to calculate normal contact force and angular displacement amplitude among the wires is established first when the spring is impacted. With the experimental parameters obtained from the model, the torsional fretting test, which stimulates torsional fretting among the wires in the working process of the spring, is realized successfully on a newly developed fretting tester. Torsional fretting behaviors are strongly dependent upon normal contact force, angular displacement amplitude, and number of cycles. There are three basic types of T-θ curves (short for torque), angular displacement curves during the process of torsional fretting, namely, parallelo-gram, elliptic, and linear T-θ curves. To analyze the damage mechanisms, distribution maps of oxygen in the wear scar of spring wires under different working conditions are revealed. The damage gets slight in the partial slip region, mainly with the abrasive wear and the slight oxidative wear, whereas the wear mechanisms are mainly the abrasive wear, the oxidative wear, and the delamination, accompa-nied with obvious plastic deformation in the mixed fretting and slip regions.
A stranded-wire helical spring is normally made up of sev-eral steel wires that are tied up to form a multilayer and coax-ial strand in the same helical direction. It is then twisted to form a helical spring. Stranded-wire helical springs, compared with conventional single-wire helical springs, exhibit more damping and longer fatigue life in certain dynamic applica-tions [1-6]. They have been used successfully in machine guns, small weapons, and stapling guns to dampen high-velocity displacement of coils. The spring is also broadly used in such fields as vibration-working systems (industrial vibration shak-ers, vibration grinding equipment), high precision platforms, and smooth-ride automobiles. A stranded-wire helical spring acts as reciprocative spring, which mainly bears impact load in to-and-fro movements. Impact load can make the spring com-press steadily or, otherwise, release. After several stress cycles, a small piece of metal delamination appears in the local con-tact area, which results in pitting and speeds up wear damage. From the failed spring, we find that the wear on the local area of the wires’ surface is very serious. This phenomenon is the
most significant reason for spring failure. Fretting, which is a special type of wear, is defined as small
amplitude oscillation along the contact interface between two objects. Waterhouse [7] classified the fretting phenomenon into fretting wear, fretting fatigue, and fretting corrosion. Four basic fretting modes, namely, reciprocating or tangential, ra-dial, rotational, and torsional modes, particularly reciprocating mode, has been widely investigated in literature [8]. Recently, fretting wear was classified into three main regions, namely, partial slip regime, mixed fretting regime, and gross slip re-gime [9-10]. Until now, research on fretting mainly focuses on the mode of reciprocating movement along radial or tangential contacting surface, but little research has been done on tor-sional fretting under rotational working conditions. Fretting wear behaviors of steel wires and the effects of fretting on their fatigue fracture behaviors were analyzed with point con-tact hoisting rope as research object [11-13]. The upper wire was rigidly attached to the bed of the platform and the under wire vibrated reciprocatively in the simplified mode, which belonged to the radial fretting wear area field. Z .B. Cai [14-20] took torsional fretting tests of such materials as LZ50 steel, 7075 aluminum alloy, adopting sphere/flat contact mode.
To predict the fatigue life of a stranded-wire spring, it is necessary to investigate torsional fretting behaviors and wear mechanisms of spring wires under torsional fretting conditions.
† This paper was recommended for publication in revised form by Editor Dae-Eun Kim
2138 S. Wang et al. / Journal of Mechanical Science and Technology 25 (8) (2011) 2137~2147
Based on an analysis of a large number of references on fret-ting, torsional fretting behaviors are strongly dependent upon normal contact force, angular displacement amplitude, and number of cycles. However, the above-mentioned three pa-rameters are hard to measure when the spring bears impact load. As such, a mathematical model to calculate normal con-tact force and angular displacement amplitude among the wires is established. Velocity and weight of the mass corre-spond to actual working conditions in the mathematical model. Normal contact force and angular displacement amplitude are obtained via rigorous formula derivation, so the parameters used in the fretting experiments, which are worth realistic contact press distribution, and the experimental results better reflect actual torsional fretting behaviors of spring wires. With the experiment parameters obtained from the mathematical model, a torsional fretting test adopting cylinder-cylinder con-tact mode is realized successfully on a newly developed fret-ting tester, which stimulates torsional fretting between wires in the course of the spring working. Torsional fretting behaviors and wear mechanisms of a spring under torsional fretting con-ditions are researched.
2. Analysis of load and deformation of stranded-wire
helical spring
2.1 Normal contact force among the spring wires
Stranded-wire helical spring acts as reciprocative spring, which mainly bears impact load in a to-and-fro movement (Fig. 1). Normal contact force and angular displacement be-tween wires (i.e., corresponding torsional fretting test parame-ters) are obtained via theoretical calculation when the mass impacts the spring with a certain velocity v. Assuming that x and F represent compress deformation and axial load of the stranded-wire spring, respectively, then two expressions are obtained.
2 21 12 2
mv kx
F kx
⎧ =⎪⎨⎪ =⎩
(1)
where m and k represent impact mass’s quality and spring stiffness, respectively.
Carrying out Eq. (1), an expression is given as F v mk= . (2)
Considering that the stranded-wire spring bears axial load F,
the general normal contact force value Fn between any two wires is
cos1n
FFn
α=−
(3)
where α and n are the helix angle and the total number of wires of a stranded-wire spring, respectively. Carrying out Eqs. (2) and (3) together, an expression is given as
2 21 ( )n
v mk DFn P D
ππ
=− +
(4)
where D and P are the diameter and the pitch of stranded-wire spring, respectively.
2.2 Angular displacement between wires
The strand mainly bears torsional effect when the spring ab-sorbs axial load. Therefore, torque value T could be expressed as
2FDT = . (5)
Taking a micro-section strand as research object, then each
wire is equivalent to a single spring and the torque value T＇ is
'2FDT
n= . (6)
For each wire, it is easy to obtain the expressions of the
space coordinates at the Cartesian coordinate oxyz, in which xy plane coincides with the normal cross-section of the strand.
cos2
sin2
tan2
n
n
n
dx
dy
dz
δ
δ
δ δ
⎧ =⎪⎪⎪ =⎨⎪⎪
=⎪⎩
(7)
Fig. 1. Force analysis of stranded-wire helical spring.
S. Wang et al. / Journal of Mechanical Science and Technology 25 (8) (2011) 2137~2147 2139
2 cos , 0 2n
ld
βδ δ π= ≤ ≤ (8)
where β and dn are the twist angle of the strand and the diame-ter of the stranded-wire spring, respectively, and δ and l are the length and the helix angle of the spiral, respectively. Car-rying out Eq. (7), curvature χ and torsional deformation κ of the helical spiral are separately given as
22cos
sin 2
n
n
d
d
βχ
βκ
⎧=⎪
⎪⎨⎪ =⎪⎩
. (9)
From Eq. (9), it is easy to obtain
2
2
4cos 'cos ' '
2cos '
' '
n
n n
d
d d
χ ββ
χ β
∂⎧ =⎪∂⎪⎨∂⎪ = −⎪∂⎩
. (10)
The differential equation of χΔ is
2
2
4cos ' (cos ') 2cos '' ' n
n n
dd d
β β βχ ΔΔ = − Δ . (11)
In a similar way, κΔ can be expressed as
2
2sin 'cos ' '
2cos ' sin ' '
2sin 'cos ' ' '
n
n
n n
d
d
d d
κ ββ
κ ββ
κ β β
⎧ ∂=⎪∂⎪
⎪ ∂⎪ =⎨∂⎪
⎪ ∂= −⎪
∂⎪⎩
, (12)
2
2cos ' (sin ') 2sin ' (cos ')' '
2sin 'cos ' ''
n n
nn
d d
dd
β β β βκ
β β
Δ ΔΔ = +
− Δ, (13)
sin ' (sin ') cos ' (cos ') 0β β β βΔ + Δ = . (14) Combining Eqs. (11)-(14) yields the following expressions:
'sin ' 'cos '(sin ')2 2
n nd dβ ββ χ κΔ = − Δ + Δ , (15)
2'sin ' 'sin '(cos ')2cos ' 2n nd dβ ββ χ κ
βΔ = Δ − Δ . (16)
Assuming that Δdn and Δβ represent variations in basic
parameters of spiral under load, then basic parameters dn and β and under loads 'nd and 'β could be expressed as follows:
' '
n n nd d dβ β β
= + Δ⎧⎨ = + Δ⎩
. (17)
Variations in curvature and torsional deformation of spiral
are related to load applied on strand section and, therefore, three equations are obtained.
b 'cos '
'sin 't
M TT T
ββ
=⎧⎨ =⎩
, (18)
'cos '
'sin '
p
TEI
TGI
βχ
βκ
⎧Δ =⎪⎪⎨⎪Δ =⎪⎩
, (19)
'cos ''
'sin ''p
TEI
TGI
βχ χ χ
βκ κ κ
⎧Δ = − =⎪⎪⎨⎪Δ = − =⎪⎩
, (20)
where E and G are the modulus of elasticity and the shear modulus of steel wire, respectively, and I and Ip are the mo-ment of inertia and the polar moment of inertia, respectively. Angular displacement (i.e., variations in spiral angular) be-tween two wires is calculated via a derivation of Eq. (8).
2
(cos ') cos '2' ' n
n n
l dd d
β βϕ δ⎡ ⎤Δ
= Δ = − Δ⎢ ⎥⎣ ⎦
(21)
Considering the deformation of spring as relative negligible
dn＇≈ dn , β＇≈β, an expression on φ is given by carrying out Eqs. (15)-(21) together.
2 2sin cos( )
2 cosn
p
FDdn GI EIπ β βϕ
β= + (22)
Introducing Eq. (2) into Eq. (22), the angular displacement
between two wires is given as
2 2sin cos( ) .2 cos
n
p
v mk Ddn GI EI
π β βϕβ
= + (23)
3. Experiment
3.1 Experimental material and method
This experiment is a study on the wire of a “3+9” stranded-wire spring (twisted by three inner wires and nine outer wires). The diameter of the spring carbon steel (T9A) wire is 1.5 mm. Its major compositions (by mass fraction) are 0.89% C, 0.3% Mn, 0.0085% S, 0.16% Cu, 0.017% P, 0.26% Si, 0.08% Cr, and 0.01% Ni. Its material mechanical properties are HV=468 HV, E=220 GPa, and σb=1550 MPa. To correspond with ac-tual working conditions, two wires are grounded with each other at 90 degrees. The model adopts cylinder-cylinder con-
2140 S. Wang et al. / Journal of Mechanical Science and Technology 25 (8) (2011) 2137~2147
tact mode, in which the wire above is fixed to the bed of the tester platform and the wire below is twisted back and forth, following the motion of the motor with a constant rotary ve-locity (Fig. 2). A sensitive six-axis torque/force (F-T) sensor recorded in real time six components (Fx,, Fy, Fz, Tx, Ty, and Tz) with the change versus test cycle, and analyzed the results using a software called ATI-DAQ. The tests are performed in ambient air (relative humidity: RH=50%~60%) and in room temperature of 20±3℃. Wear morphologies are observed using a scanning electron microscope (SEM) Quanta 200.
3.2 Specific experimental parameters
The normal force measurement of the torsional fretting tester ranges from 0.5 to 580 N. Combining actual working conditions, the rotary velocity of the tester is 10°/s, and the amplitudes of the mass impact velocity (recoil velocity of reciprocative spring of automatic weapons) v are 10 m/s, 20 m/s, and 30 m/s, respectively. Introducing expressions that take D= 28 mm; P= 35 mm; dn= 4.76 mm; β= 19°; n= 12; m= 1.3 kg; and k= 1250 N/m into Eqs. (4) and (14), specific ex-perimental parameters of the test on stranded-wire spring wires are obtained via theoretical calculations, as shown in Table 1.
4. Results and discussions
4.1 Friction torque versus angular displacement amplitude curves
Torsional fretting kinetics behaviors can be described through the T-θ curve of torque variables with angular dis-placement amplitudes. Figs. 3-5 show the T-θ curve of the wire of a stranded-wire helical spring under different load conditions, and the number of cycles. Torsional fretting test parameters are separate (Fn=34 N, θ=4°), corresponding to a 10 m/s mass impact velocity. From the first cycle up to 1000 cycles, the T-θ curve maintains an elliptic shape. As the num-ber of cycles exceeds 1000, the T-θ curve gradually becomes narrower and narrower and transforms into a linear shape by the end of the test. Torsional fretting under this load condition runs in the partial slip regime because T-θ curve evolution only occurs between linear and elliptic shapes during the en-tire test according to running condition fretting maps (RCFMs). Relative torsional angular displacement amplitude is accommodated by elastic deformation between the contact-ing surface, of which the center is stuck without any relative slip but the boundary appears a little micro slip. Curve trans-formation from initial elliptic shape to final linear shape is mainly attributed to the existence of the surface film of the two contact wires, which reduces friction between contact surfaces and makes it easy for contact surfaces to enter
*θ is the angular displacement amplitude of the tester.
S. Wang et al. / Journal of Mechanical Science and Technology 25 (8) (2011) 2137~2147 2141
(a) (b)
(c) (d) Fig. 3. T-θ curves of the spring wires under different cycles (Fn=34 N, θ=4°).
(a) (b)
(c) (d) Fig. 4. T-θ curves of the spring wires under different cycles (Fn=68 N, θ=8°).
2142 S. Wang et al. / Journal of Mechanical Science and Technology 25 (8) (2011) 2137~2147
relative slip status. In the 20 m/s mass impact velocity (i.e., test parameters are
Fn=68 N, θ=8°), a significant change in T-θ curve occurs. The T-θ curve always maintains a quasi-parallelogram during the first 100 cycles, which may be related to the existence of the surface film of the two contact wires, which reduces friction between contact surfaces and makes it easy for contact sur-faces to enter relative slip status. Relative motion state be-tween contact surfaces changes greatly with an increase in fretting wear cycle. As the number of cycles exceeds 100 cy-cles, the T-θ curve transforms into an elliptic shape and the relative motion state changes from gross slip to partial slip. In subsequent cycles, the curve shape gradually changes into a narrow quasi-parallelogram with a significant increment in torque. The evolution of T-θ curve with an increase in the number of cycles shows that torsional fretting runs in the mixed fretting regime and that transformation of the relative motion state appears between partial slip and gross slip during the entire test.
The T-θ curve changes significantly again when the mass impact velocity further increases to 30 m/s (i.e., test parame-ters are Fn =102 N, θ=12°). From the first cycle to the 1000th cycle, the T-θ curve maintains a linear shape. Relative tor-sional angular displacement amplitude is accommodated by elastic deformation between contact surfaces, of which the center is stuck without any relative slip and the edge appears a
little micro slip. As the number of cycles exceeds 100 cycles, the T-θ curve changes into an elliptic shape. The relative mo-tion state between contact surfaces changes from gross slip to partial slip, with apparent increments in torque value. The T-θ curve changes into a quasi-parallelogram for 1000 cycles. The T-θ curve becomes wider and maintains a quasi-parallelogram shape until the end of the wear test at 5000 cycles. Torsional fretting runs in the mixed fretting region as T-θ curve evolu-tion occurs in the linear shape, the parallelogram, and the el-liptic shape, and the transformation of the relative motion state appears between partial slip and gross slip during the entire wear test. At the outset, the existence of the surface film de-creases friction torque between spring wires. With an increase in the number of cycles, the surface film is destroyed and fric-tion torque increases. Work hardening and an increase in fra-gility of the surface material result in small-piece delamination, forming abrasive dust. Three-body contact is formed but does not reach a steady state situation because of the accumulation of debris.
4.2 Analysis of SEM micrographs
4.2.1 Effect of the number of cycles In the condition (Fn= 68 N, θ= 8°), SEM micrographs of
wear scars of spring wires under different cycles are presented in Figs. 6-9. In the 500 and 1000 wear cycles, torsional
(a) (b)
(c) (d) Fig. 5. T-θ curves of the spring wires under different cycles (Fn=102 N, θ=12°).
S. Wang et al. / Journal of Mechanical Science and Technology 25 (8) (2011) 2137~2147 2143
(a) Whole micrograph (b) Center micrograph (c) Boundary micrograph Fig. 6. SEM micrographs of the wear scar of spring wires under different cycles (Fn=68 N, θ=8°, N=500).
(a) Whole micrograph (b) Center micrograph (c) Boundary micrograph Fig. 7. SEM micrographs of the wear scar of spring wires under different cycles (Fn=68 N, θ=8°, N=1000).
(a) Whole micrograph (b) Center micrograph (c) Boundary micrograph Fig. 8. SEM micrographs of the wear scar of spring wires under different cycles (Fn=68 N, θ=8°, N=2000).
2144 S. Wang et al. / Journal of Mechanical Science and Technology 25 (8) (2011) 2137~2147
fretting runs in the partial slip regime; wear damage is very slight. A narrow wear ring with dark mark appears at the boundary of the contact zone, whereas only grind and stick traces are found at the contact surface center. Relative tor-sional angular displacement amplitude is accommodated by elastic deformation between the contact surfaces when normal contact force and angular displacement amplitude are rela-tively small. The contact surface center is stuck without any relative slip and the boundary appears a little micro slip, as depicted in Figs. 6-7. Torsional fretting runs in the mixed fret-ting regime when the number of cycles increases to 2000. The contact surface center is stuck with slight damage, whereas a significant wear appears at the boundary of the contact surface. The wire surface metal is peeled by delamination mechanisms with unapparent spalling pits (Fig. 8). As the number of cycles reaches 5000, torsional fretting runs in the gross slip regime. The wear area significantly increases and relative slip occurs throughout the entire contact regions with obvious delamina-tion, which indicates that local wear is much serious (Fig. 9).
4.2.2 Effect of mass impact velocity and chemistry surface
analysis Torsional fretting test parameters of the spring wires corre-
spond to different mass impact velocities, as shown in Tab. 1. SEM and energy dispersive spectroscope (EDX) micrographs of the wear scar of spring wires under different working con-ditions are sketched in Figs. 10-12. Torsional fretting runs in the mixed fretting regime when mass impact velocity is 10 m/s (Fig. 10). The contact center is slightly damaged, whereas significant wear damage appears at the boundary of the con-tacting surface. The wire surface metal is peeled by delamina-tion mechanisms with unapparent spalling pits. As velocity increases to 20 m/s (Fig. 11), the sticking zone gradually dis-appears with much metal delamination and obvious plastic deformation. Torsional fretting runs in the mixed fretting re-gime when mass impact velocity increases to 30 m/s (Fig. 12). The contact center is still stuck with slight damage, but much
delamination appears at the boundary of micro-slip region, which indicates that local wear is much serious (Fig. 9).
The wear area is much wider when mass impact velocities are 20 m/s and 30 m/s because of the increment in normal contact force and angular displacement amplitude. As impact velocity increases, torsional fretting presents a “partial slip regime-gross slip regime-mixed fretting regime” variation law. Transformation from mixed fretting regime to gross slip re-gime first appears as normal contact force and angular dis-placement amplitude increase. The larger the normal load is, the more difficult relative slip occurs between two wires. When impact velocity increases over a certain value, wire contact center is still stuck, and evolution from gross slip re-gime to mixed slip regime occurs again. In the partial slip regime, damage is slight and wear mechanisms are mainly abrasive wear and slightly oxidative wear. Wear mechanisms are mainly abrasive wear, oxidative wear, and delamination wear. Damage intensifies on the contact surface, with obvious plastic deformation in the mixed fretting and in the gross slip regimes.
To analyze the damage mechanisms, EDX distribution maps of oxygen in the wear scar of spring wires under differ-ent working conditions are revealed. In relative low impact velocity (68 N, 34 N are the Fn), distribution maps of oxygen present a type of “V” distribution. There is higher oxygen in the wear zone and less oxygen in the stick zone. The contact center is stuck and the oxidation reaction is suppressed, sup-plying a rational explanation why oxygen is less in the wear zone and increases from the contact center to the contact boundary. In the 30 m/s mass impact velocity (Fn=102 N), the profile of the distribution maps of oxygen presents a “U” dis-tribution. Oxygen content is much less in the outside ring, where wear is very serious. Owing to high contact force and severe wear, there is shortage of time for metals newly ex-posed to oxidates. Generally, these oxygen distribution maps conform to the T-θ curves, giving useful reference to the study of wear mechanisms.
(a) Whole micrograph (b) Center micrograph (c) Boundary micrograph Fig. 9. SEM micrographs of the wear scar of spring wires under different cycles (Fn=68 N, θ=8°, N=5000).
S. Wang et al. / Journal of Mechanical Science and Technology 25 (8) (2011) 2137~2147 2145
(a) Whole EDX micrograph (b) Center micrograph (SEM) (c) Boundary micrograph (SEM) Fig. 10. EDX and SEM micrographs of the wear scar of spring wires (Fn=34 N, θ=4°).
(a) Whole EDX micrograph (b) Center micrograph (SEM) (c) Boundary micrograph (SEM) Fig. 11. EDX and SEM micrographs of the wear scar of spring wires (Fn=68 N, θ=8°).
(a) Whole EDX micrograph (b) Center micrograph (SEM) (c) Boundary micrograph (SEM) Fig. 12. EDX and SEM micrographs of the wear scar of spring wires (Fn=102 N, θ=12°).
2146 S. Wang et al. / Journal of Mechanical Science and Technology 25 (8) (2011) 2137~2147
5. Conclusions
(a) With experiment parameters obtained from the mathe-matical model, a torsional fretting model adopting cylinder-cylinder contact mode is realized successfully on a newly developed fretting tester, which stimulates torsional fretting between wires in the course of the spring working conditions.
(b) By analyzing the T-θ curve, the torsional fretting condi-tion was shown to depend strongly on normal contact load values between wires, angular displacement amplitude, and number of cycles. There were three basic types of T-θ curves during torsional fretting, namely, parallelogram, elliptic, and linear T-θ curves.
(c) Based on the analysis of SEM and EDX micrographs of the wear scar, we learned that damage in the partial slip re-gime is slight and wear mechanisms are mainly abrasive wear and slight oxidative wear. Damage intensifies in the contact surface with obvious plastic deformation in the mixed fretting and the gross slip regimes.
Acknowledgment
This research was supported by the National Natural Sci-ence Funds for Distinguished Young Scholars (No. 50925518), the National Science Foundation of China (No. 50775226), the Key Project of the Chinese Ministry of Education (No. 109129), the Chongqing Key Scientific and Technological Project (No. CSTC2009AC3049), the Chongqing University Postgraduate Science and Innovation Fund (No. 200911A1A0020318), and the Innovative Talent Training Project, the Third Stage of the “211 Project”, Chongqing Uni-versity (No. S-09106).
T : Torque value θ : Angular displacement amplitude N : Number of cycles v : Impact mass velocity x : Compress deformation F : Axial load of stranded-wire spring m : Impact mass quality k : Stranded-wire spring stiffness Fn : Normal contact force α : Helix angle of stranded-wire spring n : Total number of wires of a stranded-wire spring D : Diameter of stranded-wire spring P : Pitch of stranded-wire spring β : Twist angle of strand dn : Diameter of stranded-wire spring δ : Helix angle of spiral l : Length of spiral χ : Curvature of helical spiral κ : Orsional deformation of helical spiral E : Modulus of elasticity of wire
G : Shear modulus of wire I : Moment of inertia Ip : Polar moment of inertia
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Shi-Long Wang received his B.S., M.S., and Ph.D degrees in Mechanical Engineering from Chongqing University, China in 1988, 1991, and 1995, respectively. Dr. Wang is currently a professor at the School of Mechanical Engineering, Chongqing University, Chongqing, China. He serves as a
director of the Chinese Journal of Mechanical Engineering. Dr. Wang’s research interests include manufacturing automation, computer-integrated manufacturing, and enterprise informati-zation.
Xiaoyong Li received his B.S. degree in Mechanical Engineering from Chongqing University, China in 2009. Li is cur-rently a Ph.D candidate at the School of Mechanical Engineering, Chongqing University, Chongqing, China.
Song Lei received his B.S. degree in Mechanical Engineering from Chongqing University, China in 2005. He received his M.S. degree in Robotics and Mecha-tronics from Sant’Anna University, Italy in 2007. Lei is currently a Ph.D candidate at the School of Mechanical Engineering, Chongqing University,
Chongqing, China.
Jie Zhou received his B.S. degree in Mechanical Engineering and his M.S. degree in Software Engineering from Chongqing University in 1988 and 2007, respectively. Zhou is currently an associate professor at the School of Mechanical Engineering, Chongqing University, Chongqing, China. Zhou’s
research interests include manufacturing automation and mechanotronics.
Yong Yang received his M.S. degree in Mechanical Engineering from Chongqing University, China in 2007. Yang is currently a Ph.D candidate at Chongqing University, Chongqing, China.
