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Research ArticleDOI: 10.1002/jst.62
Time-dependent behavior of ropes under impact loading: adynamic analysisIgor Emri, Anatoly Nikonov, Barbara Zupancic and Urska Florjancic
Center for Experimental Mechanics, University of Ljubljana, and Institute for Sustainable Innovative Technologies,
Ljubljana, Slovenia
In this paper, we present new methodology based on a simple, non-standard
falling-weight experiment, which allows for the examination of the
functionality and durability of ropes beyond the findings from Union
Internationale des Associations dAlpinisme experiments. The
experimentalanalyticalnumerical treatment allows for the examination of
the time-dependent viscoelasto-plastic behavior of ropes exposed to arbitraryfalling-weight loading conditions. Developed methodology allows for the
prediction of the impact force and the jolt (the derivative of the acceleration/
deacceleration acting on the climber); the viscoelastoplastic deformation of
the rope; stored, retrieved, and dissipated energy during the loading and
unloading of the rope; and the modification of the stiffness of the rope within
each loading cycle. By means of parametric error analysis, we showed that
the relation between the error of calculated data and the error of input data is
extremely non-linear. This demands careful and precise experiments. It was
shown that the accuracy of prediction of all sought-after physical quantities
could be obtained within the acceptable limits, which confirms that the
proposed experimentalanalytical methodology may be used for analyses of
the functionality and durability of ropes and the safety of climbers. &2008
John Wiley and Sons Asia Pte Ltd
1. INTRODUCTION
Climbing is becoming one of the fastest growing extreme
sports. In this sport, ropes are probably the most critical part
of the equipment. Climbing ropes are designed to secureclimbers, and for that reason, they are dynamic; this means
that they are designed to stretch under a high load so as to
absorb the shock force. This protects the climber by reducing
fall forces. In comparison, static ropes are more durable and
resistant to abrasion and cutting, but they lack the necessary
protection against shock loads produced in a climber fall. For
that reason, they are used only in situations where such shock
loads would never occur (e.g. rappelling, canyoneering, and
spelunking) [1].
Ropes should have good mechanical properties, such as
high-breaking strength, large elongation at rupture, and goodelastic recovery. The Union Internationale des Associations
dAlpinisme (UIAA) has established standard testing proce-
dures to measure, among other things, how ropes react to se-
vere falls [2,3]. The international standard test for climbing
ropes is based on a standard dynamic drop test . Ropes are
drop tested with a standardized weight and procedure simu-
lating a climber fall. This tells us how many of these hy-
pothetical falls the rope can withstand before it ruptures.
Different rope categories have different norms, but the stan-
dard requires climbing ropes to withstand a minimum of five
such test falls. Virtually all of the ropes on the market can
withstand the minimum number of test falls, while some are
*Center for Experimental Mechanics, Faculty of Mechanical Engineer-
ing, University of Ljubljana, Pot za Brdom 104, 1000 Ljubljana,
Slovenia
E-mail: ie@fs.uni-lj.si
Keywords:. ropes. impact. viscoelasticity. time-dependent behavior
. jolt
. energy dissipation
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rated to withstand a much higher number. The second thing that
a standard drop test measures is the amount of force that is
transmitted to the falling climber. For all of the tests, these forces
must stay within a certain range. The standard also rates factors,
such as rope stiffness, sheath slippage, and rope stretch under
body weight. Different simplified testing procedures are pre-
scribed for each of these properties. For example, rope stiffness
in a standardized test is measured by tying an overhand knot,exposing the knot to a 10-kg load, and then measuring the size of
the hole in the knot. This test is known as the knot-ability test,
which indicates the handling and suppleness of a rope. Ac-
cording to the procedure prescribed by the standard, the hole
must measure less than 1.1 times the rope diameter. By all
means, these are very practical ways for rapid testing; however,
they provide no information on the underlying mechanisms that
govern the time-dependent behavior of ropes.
The standard says little about the durability of ropes,
which is more difficult to define or assess with simplified pro-
cedures commonly used by rope manufacturers. Durability in
this case does not mean just failure of the rope, but rather,
deterioration of its time-dependent response when exposed toan impact force. The experiments prescribed by the existing
standard are not geared to analyze the time-dependent de-
formation process of the rope, which causes structural changes
in the material, and consequently affects its durability.
In this paper, we present a comprehensive dynamic analysis
of a simple, non-standard falling-weight experiment, which
allows for the examination of the time-dependent viscoelas-
toplastic behavior of ropes exposed to arbitrary falling-weight
loading conditions. Developed analytical treatment is subse-
quently examined by using the synthetic experimental data.
By means of the parametric error analysis, we determine the
required precision of all measured physical quantities used in
the derived analytical equations for physical quantities thatdetermine the durability of ropes and the safety of climbers.
2. THEORETICAL TREATMENT
The time-dependent response of a rope under dynamic
loading generated by a falling mass may be retrieved from the
analysis of the force measured at the upper fixture of the rope.
This force is transmitted through the rope and acts on the
falling weight (mass), as schematically shown in Figure 1. In
such experiments, a mass is dropped from an arbitrary height,
hp2l0, where l0 is the length of the tested rope.
Force measured as function of time, Ft, may be expressedas a set ofNdiscrete data pairs:
Ft fFi; ti; i 1; 2; 3; ; Ng 1
An example of such measured force is schematically shown in
Figure 2. The diagram is subdivided into three distinct phases:
A, B, and C.
In phase A, the weight (mass) is dropped at t50, and it
falls freely until t t0ffiffiffiffiffiffiffiffiffiffi
2h=gp
, whereh indicates the height
from which the mass was initially dropped. Here the rope
becomes straight, which is indicated in Figure 2 as point T0. If
we neglect the air resistance, the velocity of the mass at point
T0is v0 ffiffiffiffiffiffiffiffi2ghp . Point T0represents the end of the free-falling
phase of the mass, and the beginning of phase B, which is the
beginning of the rope deformation process.
At point T0 in phase B, where t t t0 0, the falling
mass starts to deform the rope. Neglecting the air resistance,and the wave propagation in the rope, the equation of motion
of the moving mass between points T0 and T7 may be written
as:
m xt mg Ft 2
Here, m is the mass of the weight, and g is the gravitational
acceleration; xt denotes the second derivative of the weight
displacement, xt, measured from point T0. Thus, xt re-
presents the time-dependent deformation of the rope. The
solution of equation 2 gives the displacement of the weight as
the function of time, which is equal to the viscoelastoplastic
deformation of the rope:
xt gt2
2 1
m
Z t0
Z l0
Fudu
dlC1tC2 3
Constants C1 and C2 may be obtained from the initial condi-
tions at point T0:
xt 0 0; and _xt 0 v0 ffiffiffiffiffiffiffiffi
2ghp
4
Therefore:
C2 0 5
and
C1 v0 1
m
Z t0
Fldl
t0
v0 6
l0
mg
mm
F(t)
F(t)
F(t)
F(t)
m
h
t = t0
l0
mg
mm
F(t)
F(t)
F(t)
F(t)
mm
h
t = t0
Figure 1. Schematics of the rope exposed to the falling mass. Here m
is the mass of the falling weight,his the height from which the mass is
initially dropped. F(t) is the measured force that is generated in the
rope,l0is the initial length of the rope, t5 t0is the time when the rope
becomes straight,g is gravitational acceleration. Reproduced from [3]
by kind permission of Taylor & Francis.
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Displacement of the weight, which is equal to the deformation
of the rope, may be expressed now as:
xt gt2
2
1
m
Z t0
Z l0
Fudu
dlv0t 7
Since the deformation of the rope and the displacement of the
weight are the same, we may now calculate the velocity, the
acceleration/deacceleration, and the jolt acting on the weight,
that is, the climber, respectively:
vt _xt gt 1
m
Z t0
Fldlv0 8
at xt g Ft
m 9
jt _xt 1
m
dFt
dt 10
At point T1, where t t1, the force acting on the rope
becomes equal to the weight of the mass, Ft1 mg. At this
point, the velocity of the weight reaches its maximum value:
vmax vt1 gt11
m
Z t10
Fldlv0 11
The location of T1, where t t1 may be found numerically
from
dvt
dt g
Ft1
m 0 12
At T2, the jolt will reach its negative extreme value,
t t2 tj jmin, where:
jmin jt2 MIN 1
m
dFt
dt
13
The force acting on the rope and on the weight has its max-
imum at T3, where: t t3 tFFmax, and
Fmax Ft3 MAXfFi; i 1; 2; 3;. . .;Ng 14
The deformation of the rope at this point is:
st3 xFFmax xt3
gt23
2
1
m
Z t3
0
Z l
0
Fudu
dlv0t3 15
If the properties of the rope would be elastic, the location of
the maximum force should coincide with the location of the
maximal deformation; however, because of the viscoelastic
nature of the rope, its maximal deformation, s max, will be de-
layed and will take place at t t4, that is, at point T4, where
the velocity of the weight is equal to zero:
v4 vt4 gt4 1
m
Z t40
Fldlv0 0 16
The time, t4, may be retrieved numerically from equation 16.
The maximum deformation of the rope is then:
smax xt4 gt24
2
1
m
Z t40
Z l0
Fudu
dlv0t4 17
Now we can calculate the viscoelastic component of the rope
deformation by subtracting equation 15 from 17:
svesmaxst3 xt4 xt3
gt24t
23
2 v0t4t3
1
m
Z t4t3
Z l0
Fudu
dl
18
The unloading phase of the rope starts at point T4. The elastic
component of a ropes deformation will be retrieved and will
Time - t
Force-
F(t)
T4
T6
t90 t0 t4t1
T1mg
First loading cycle Second loading cycle
CA B
t6
T0
T4
T7
Fmax
T3
T9
t2 t7
T2 T5
T8
0
t3 t5 t8
1
2
3
4
5
6
7
8
9
Figure 2. Schematics of the force measured during the falling mass experiment (phases AC). ti, absolute time of individual events in deformation
process of the rope; ti, relative time of individual events in deformation process of the rope; Fmax, maximum force in the rope;m, mass of the falling
weight;g, gravitational acceleration;T0, beginning of the loading phase of the rope; T1, the moment when the force in the rope is equal to the weightof the mass; T2, the moment of the extreme negative value of the jolt; T3, the moment of the maximum force in the rope; T4, the moment of the
maximum deformation of the rope when the velocity of the weight is equal to 0; T5, the moment of the positive extreme value of the jolt; T6, the
moment when force in the rope is equal to the weight of the load; T7, the moment when the force in the rope is equal to 0 and the weight starts to fly
in upwards;T8, the moment when the weight reaches the maximum upper point of its free fly in the vertical direction; T9, the beginning of the second
loading cycle.
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accelerate the weight in the opposite (upward) direction. At
t t5, indicated as point T5, the jolt will reach its positive
extreme value: t5 tj jmax, where
jmax jt5 MAX 1
m
dFt
dt
19
At T6, where t t6, the force acting on the rope again be-
comes equal to the weight of the load, Ft6 mg. At thispoint, velocity will obtain its extreme value in the opposite
(negative) direction:
vmin vt6 gt6 1
m
Z t60
Fldlv0 20
Time t6 may be again easily determined numerically from
Ft6 mg. At point T7, where the force acting on the rope
becomes equal to zero, Ft7 0, the weight will start its free
fly in the upward (vertical) direction. The velocity of the weight
at point T7 may be calculated with equation 8:
v7 vt7 gt7 1
m
Z t70
Fldlv0 21
We can also calculate the elastic part of rope deformation, sel,which is equal to the weight displacement during the unloading
of the rope that takes place between points T4 and T7:
sel xt4 xt7
1
m
Z t7t4
Z l0
Fudu
dl
gt27t24
2 v0t7t4 22
Furthermore, we can calculate the viscoplastic deformation of
the rope, svp, by subtracting the recovered elastic deformation,
sel, from the ropes maximum deformation, smax. Therefore:
svpsmaxsel xt7 gt27
2
1m
Z t70
Z l0
Fudu
dlv0t7
23
By subtracting the viscoplastic (equation 23) and the viscoelastic
(equation 18) components, we can calculate the plastic compo-
nent of rope deformation:
spl svpsve xt7 xt3 xt4 24
In phase C, point T7 represents the beginning of phase C, in
which the weight has no interaction with the rope, that is,
Ft7 0, and starts to fly upwards with the initial velocity: v7,
v7 vt7 gt7 1
m
Z t70
Fldlv0 25
It then returns back at point T9 to start the second cycle of the
rope deformation process. From the velocity,v7, we can calculate
the time of the weight vertical flight:
tu v7
g t7
1
mg
Z t70
Fldlv0
g 26
Furthermore, we are also able to calculate the height, sb, to which
the weight will be bounced:
sb v7tugt2u
2 27
At point T9, the second loading cycle of the rope starts, which
may be analyzed with the same set of equations derived for
phases B and C.
2.1. ForceDeformation Diagram of the Rope Deformation
Process: Energy Dissipation
Energy dissipation during the rope deformation process,
that is, between points T0and T7, is one of the most important
rope characteristics, and should be used for comparing the
quality of ropes. Force,Ft, measured during the loading and
unloading of the rope in phase B, may be expressed as the
function of the rope deformation, FFs, as schematically
shown in Figure 3. Notations used in the Figure are later ex-
plained.
The discrete form of FFs interrelation may be
obtained by calculating the isochronal values of the ropedeformation corresponding to each discrete value of the
measured force between points T0 and T7:
Fi Fti; sixti gt2i
2
1
m
Z ti0
Z l0
Fudu
dlv0ti; 0ptipt7; i 1; 2;. . .;M
28
Here, M is the number of measured force data points within
the time interval 0; t7.
Fo
rce
F(s)
T0
Fmax
T1 T6
T7
smax
selsvp
Wdis
kend
kinit
T3
T4
mg
s1
s6
Deformation - s
Figure 3. Force deformation diagram of the rope loading and
unloading phase (phase B). Fmax, maximum force in the rope; m,
mass of the falling weight;g, gravitational acceleration;T0, beginning
of the loading phase of the rope; T1, the moment when the force in therope is equal to the weight of the mass; T3, the moment of the
maximum force in the rope; T4, the moment of the maximum
deformation of the rope when the velocity of the weight is equal to
0;T6, the moment when force in the rope is equal to the weight of the
load;T7, the moment when the force in the rope is equal to 0 and the
weight starts to fly in upwards; s1 and s6, deformations of the rope
when the force in the rope becomes equal to the weight of the mass;
smax, maximum deformation of the rope; svp, viscoplastic part of
deformation of the rope; sel, elastic part of deformation of the rope;
kinit, stiffness of the rope at the beginning of loading cycle; kend,
stiffness of the rope at the end of loading cycle; Wdis, dissipated
energy of the process.
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The deformation energy of the rope at any stage of
deformation may be expressed as:
Wt
Z st0
Fxdx
Z t0
Fl@xl
@l dl
Z t0
Fl gl 1
m
Z l0
Fuduv0
dl
29
and should be equal to the sum of the kinetic, Wkt, and the
potential energy, Wpt, of the falling weight at any time:
Wt Wkt Wpt 30
We are particularly interested in the stored energy, which is the
only source of energy absorption (neglecting the air resistance),
and consequently the reduction of the force acting on
the climber:
Wstore
Z smax0
Fxdx
Z t40
Fl@xl
@l dl
Z t4
0
Fl gl 1
m
Z l
0
Fuduv0
dl
31
Since the stored energy must be equal to the total potential
energy of the weight, then:
Wstore mghsmax
mg hgt24
2
1
m
Z t40
Z l0
Fudu
dlv0t4
32
During the unloading phase, the elastic component of the rope
deformation is retrieved and it accelerates the weight in anupward direction:
Wret
Z smaxsvp
Fxdx
Z t7t4
Fl@xl
@l dl
Z t7t4
Fl gl1
m
Z l0
Fuduv0
dl
33
The retrieved energy must be equal to the kinetic energy of the
mass at point T7. Thus:
Wret mv27
2 mg xt4 xt7
m
2 gt7
1
m
Z t70
Fldlv0
2
mg xt4 xt7
34
The dissipated energy within a loading and unloading cycle,
represented as the shaded area in Figure 3, can be expressed as:
WdissWstoreWret
Z t40
Fl gl1
m
Z l0
Fuduv0
dl
Z t7t4
Fl gl 1
m
Z l0
Fuduv0
Z t70
Fl gl1
m
Z l0
Fuduv0
dl
35
Alternatively:
Wdiss mghsmax mv27
2 mg xt4 xt7 36
2.2. Increase of the Rope Stiffness
An important parameter for comparing the performance of
different ropes could be the modification of their stiffness within
each loading cycle. The rope becomes stiffer in each loading
cycle, which means that the performance of the rope is de-
creasing. Thus, an indicator of the quality and rope durability
could be the ratio of the stiffness at the beginning,kinit, and at
the end, kend, of the rope deformation process. Therefore:
w kinit
kendp1 37
Stiffness, kinit and kend, may be calculated from the slope of the
force-displacement diagram Fs at points T1 and T6, as sche-
matically shown in Figure 3:
kinit dFx
dx
xs1
38
Table 1. Physical quantities representing the functionality and durability of ropes.
n Physical quantity Symbol Corresponding equation
1 Maximum force Fmax 14
2 Maximum deformation smax 17
3 Elastic part of rope deformation sel 22
4 Viscoplastic part of rope deformation svp smaxsel 23
5 Viscoelastic part of rope deformation sve 18
6 Plastic part of rope deformation spl svpsve 24
7 Stored energy Wstore 31 or 32
8 Retrieved energy Wret 33 or 34
9 Dissipated energy Wdiss WstoreWret 35 or 36
10 Stiffness of the rope at the beginning of deformation kinit 38
11 Stiffness of the rope at the end of deformation kend 39
12 Ratio of the stiffness w kinit=kend 3713 Jolt j 10
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and
kenddFx
dx
xs6
39
where s1 and s6 are rope deformations at corresponding points
T1 and T6, indicating the beginning and the end of the rope
deformation process beyond the deformation caused by the
weight of the falling mass. The stiffness of both is indicated in
Figure 3.
3. PARAMETRIC ERROR ANAYLIS
Based on the measured force, Ft, acting on a rope and a
climber during the falling-weight experiment, we derived a
variety of different physical quantities that may be used as
criteria in the evaluation of the functionality and the durability
of climbing ropes and the safety of climbers. These physical
quantities are summarized in Table 1.
Preliminary experimental investigations [4,5] showed that
calculated physical quantities (listed in Table 1) are very sen-
sitive to the precision of the input data, that is, the mass of the
falling weight, height from which we drop the weight, length ofthe rope, measured force, time at which measurements
were performed (sampling rate), and number of significant
digits in gravitational acceleration. To evaluate the effect
of the input data precision on the accuracy of the
calculated physical quantities, we will use a synthetic
error free reference signal, Ft, which closely mimics the
measured signals:
Ft F1cos20t Ht H t p
10
h iN 40
where Ht is the Heaviside (step) function, that is,
Hto0 0, and HtX0 1. The reference signal is shown
in Figure 4. In addition, we used
F4000 N; m 80 kg h l 3:263 m
t9 1:2 sec; and g 9:80665 m=s2
41
3.1 Calculation of the Error-Free, Sought-After Physical
Quantities
For the parametric error analysis of the characteristic
physical quantities (Table 1), we will first calculate the
reference error-free values. We will first need to determinethe characteristic times: t3, t4, and t7. From equation 40,
it is easy to see that the maximum force, Fmax F, will
appear at t3 p=20 sec. We determine the location of
the maximum deformation at T4 by combining equations 40
and 16:
gF
m
t4
200
m sin20t4 v0 0 42
whereas the location of T7 may be found directly from the
chosen reference signal, equation 40. Thus:
t3 p=20 sec t4 0:176026 sec andt7 p
10
sec 43
Introducing equation 40 into equations 7 and 8, we obtain the
evolution of the rope deformation process, and the
corresponding velocity of the weight:
xt gt2
2 v0t
F
m
t2
2
1
400cos20t 1
Ht H t p
10
h i
20p
m 20tpH t
p
10
;
44
Figure 4. Synthetic reference signal F(t). Heret idenotes relative time
of individual events in deformation process of the rope; Fmax,
maximum force in the rope; m, mass of the falling weight; g,
gravitational acceleration;T1, the moment when the force in the rope is
equal to the weight of the mass; T2, the moment of the extreme
negative value of the jolt;T3, the moment of the maximum force in the
rope;T4, the moment of the maximum deformation of the rope when
the velocity of the weight is equal to 0; T5, the moment of the positive
extreme value of the jolt; T6, the moment when force in the rope is
equal to the weight of the load; T7, the moment when the force in the
rope is equal to 0 and the weight starts to fly in upwards.
Figure 5. Evolution of the rope deformation process (solid line), and
the velocity of weight (dashed line). Here tidenotes relative time of
individual events in deformation process of the rope; t3, the moment
of the maximum force in the rope; t4, the moment of the maximum
deformation of the rope when the velocity of the weight is equal to 0;
t7, the moment when the force in the rope is equal to 0 and the weight
starts to fly in upwards; smax, maximum deformation of the rope; svp,
viscoplastic part of deformation of the rope; sel, elastic part of
deformation of the rope; sve, viscoelastic part of deformation of the
rope.
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vt gtv0F
m t
1
20sin20t
Ht H t p
10
h i
400p
m H t
p
10
45
The two relations are shown in Figure 5 , where the solid line
represents the deformation of the rope, and the dashed line
represents the corresponding velocity of the weight. In thesame Figure, the characteristic components of the rope de-
formation, s max, sve, sel, and svp, are also shown. These quan-
tities and spl may be obtained from equations 17, 18, 2224,
respectively:
smaxt24
2 g
F
m
v0t4
F
400 mcos 20t41 46
sve gF
m
t24t232
v0t4t3
F
400 m
cos 20t4cos 20t3
47
sel gF
m
t24t272
v0t4t7
F
400 mcos 20t4cos 20t7
48
svp smaxsel 49
and
spl svpsve 50
Taking into account values in equations 41 and 43, we find
their true (error-free) values:
smax1:0266 m; sel 0:4968 m;
svp 0:5298 m sve 0:0159 m
spl 0:5139 m spl 0:5139 m
51
Using equations 40 and 44, we can now calculate the corre-
sponding force and displacement data points:
fFi Fti; sixti; 0ptipt7; i 1; 2;. . .;Mg 52
We are also able to express the force as function of deforma-
tion, FFs. This relation is shown in Figure 6, where we
also show the elastic,sel, and the viscoplastic,svp, part of rope
deformation, and the stiffness of the rope at the beginning,
kinit, and at the end of deformation, kend.
The stiffness of both and their ratios, w, may be calculated
from:
kinit dF
ds
tt1
dFdtdsdt
tt1
200F sin20t1
g Fm
t1
200m
sin20t1 v0
53
kenddF
ds
tt6
dFdtdsdt
tt6
200F sin20t6
g Fm
t6
200m
sin20t6 v054
and
wkinit
kend
sin20t1
sin20t6
g Fm
t6
200m
sin20t6 v0
g Fm
t1
200m
sin20t1 v055
where t1 and t6 are given with the relations:
t1 1
20Arccos 1
mg
F
0:03185 sec 56
and
t6 p
10
1
20Arccos 1
mg
F
0:28231 sec 57
Figure 7. Synthetic curve of the deformation energy as a function of
time, t. Here ti denotes relative time of individual events in
deformation process of the rope; T4, the moment of the maximum
deformation of the rope when the velocity of the weight is equal to 0;
T7, the moment when the force in the rope is equal to 0 and the weight
starts to fly in upwards; Wstor, stored energy of the process; Wdis,
dissipated energy of the process;Wret, retrieved energy of the process.
Figure 6. Force acting on the rope as function of its deformation. m,
mass of the falling weight; g, gravitational acceleration; T1, the
moment when the force in the rope is equal to the weight of the mass;
T3, the moment of the maximum force in the rope; T4, the moment of
the maximum deformation of the rope when the velocity of the weight
is equal to 0; T6, the moment when force in the rope is equal to the
weight of the load;T7, the moment when the force in the rope is equal
to 0 and the weight starts to fly in upwards; smax, maximum
deformation of the rope; svp, viscoplastic part of deformation of the
rope;sel, elastic part of deformation of the rope; kinit, stiffness of the
rope at the beginning of loading cycle; kend, stiffness of the rope at the
end of loading cycle.
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The deformations of the rope at these two points are
st1 0:2589 m, and st6 0:6813 m.
Introducing equations 40 and 44 into equation 29, we canobtain the relation describing the evolution of the rope
deformation energy, which is shown in Figure 7. Therefore:
Wt F gF
m
t22
t
20sin20t
F
400g1cos20t
FF
800msin2 20t
v0
20sin20t v0t
n o58
In the same Figure, we show also the corresponding stored,
Wstore Wt4, dissipated, Wdiss Wt7, and retrieved
(elastic) energy,Wret WstoreWdiss. Their true values are
given as:
Wstore F gF
m
t242
t4
20sin20t4
F g
4001cos20t4
FF
800msin 2 20t4
v0
20sin20t4 v0t4
h i;
59
Wdiss F gF
m
t27
2
t7
20sin20t7
F g400
1cos20t7
FF
800msin 2 20t7
v0
20sin20t7 v0t7
h i60
and
Wret F gF
m
t24t272
t4
20sin20t4
t7
20sin20t7
g
400cos20t7 cos20t4
5
msin2 20t4 sin
2
20t7 v0
20sin20t4 sin20t7 v0t4t7
o: 61
The numerical values for the three energies are Wstore
3365:35 Nm, Wdiss 2119:11 Nm, and Wret 1246:24 Nm,respectively. According to the law of conservation of energy,
the sum of the kinetic and the potential energy of the falling
mass,Wmt Wkt Wpt, and the deformation energy of
the rope, Wt, should be constant at all times (neglecting the
dissipation due to the air resistance): Wmt Wt const.
This is demonstrated in Figure 8, where the solid line re-
presents the evolution of the rope deformation energy, Wt,
and the dashed line represents the sum of the kinetic and the
potential energy of the falling mass Wmt Wkt Wpt.
For completeness, we also show, with thinner solid and dashedlines, the kinetic, Wkt, and the potential energy, Wpt, re-
spectively. In the same Figure, the corresponding characteristic
times t t0, t4, and t7, which correspond to t0 0,
t4 t4 t0, and t7 t7t0, respectively are also shown.
Similarly, we can find jolt
jt 20F
m sin20t 62
which is shown in Figure 9.
The absolute values of the minimum and the maximum
jolts are the same:
jjmaxj jjminj 1000 m=s3
63
The calculated true values of the characteristic physical
quantities will now be used in the parametric error analysis to
determine the accuracy of the calculated physical quantities,
which represent the functionality and the durability of the
tested rope and the safety of a climber.
3.2. Error Analysis
The goal of the parametric error analysis is to determine
the effect of the error of the input data on the accuracy of the
calculated physical quantities (Table 1).
Figure 9. Jolt as a function of time. Here tidenotes relative time of
individual events in deformation process of the rope; T2, the moment
of the extreme negative value of the jolt; T3, the moment when the
force in the rope reaches its maximum; T5, the moment of the positive
extreme value of the jolt;T7, the moment when the force in the rope is
equal to 0 and the weight starts to fly in upwards; jmin, the extreme
negative value of the jolt; jmax, the extreme positive value of the jolt.
Figure 8. Evolution of the rope deformation energy in relation to the
sum of the kinetic and potential energy of the falling mass. Heretiand
ti denote absolute and relative time of individual events in
deformation process of the rope; T0, beginning of the loading phase
of the rope; T4, the moment of the maximum deformation of the rope
when the velocity of the weight is equal to 0; T7, the moment when the
force in the rope is equal to 0 and the weight starts to fly in upwards;
W(t), the energy of the rope as a function of time; Wm(t), the energy of
the mass as a function of time; Wk(t), kinetic energy of the mass as a
function of time; Wp(t), potential energy of the mass as a function of
time.
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It is easy to see that Fmax may always be determined di-
rectly from the source data of measured force,
Fmax MAXfFi; ti; i 1; 2; 3;. . .;Ng. Therefore, an error in
the predicted maximum force, Fmax, is given directly with the
accuracy of the force sensor, and the sampling rate of the data
acquisition. Error estimation of the rope deformation com-
ponents, smax, svp, spl, sel, and sve, is much more complex. It
depends on errors in numerical integration in equation 7, er-rors in determiningt3,t4,t7, and errors in the input data ofg,
m, h, and Ft. The same is true for Wstore, Wdiss, and Wret,
where we need to integrate equation 29. The physical quan-
tities, Wret, svp, and spl, are linear combinations ofWstore and
Wdiss, and smax, sve, and sel, respectively. Thus, we need to
analyze the influence of the error of the input data on the last
five quantities only.
Assuming that the accuracy of the measured time,ti, at the
moment when we measure the force, Fi, may be considered as
error free (which is a reasonable assumption), then the mea-
sured force may be expressed as:
Fti Fi Ft 64
where Fi is the measured strength of the force, and Ft is its
error-free time dependency. Consequently, the expressions for
smax, sve, sel, Wstore, and Wdiss may be rearranged as:
smax xt4 gt24
2 t4
ffiffiffiffiffiffiffiffi2gh
p
F
m
Z t40
Z l0
Fudu
dl 65
svext4 xt3 gt24t
23
2 t4t3
ffiffiffiffiffiffiffiffi2gh
p
F
m
Z t4t3
Z l0
Fudu
dl
66
sel xt4 xt7 gt24t
27
2
t4 t7ffiffiffiffiffiffiffiffi
2ghp
F
m
Z t7t4
Z l0
Fudu
dl
67
Wstore Wt4 Fg
Z t40
lFldl Fffiffiffiffiffiffiffiffi
2ghp Z t4
0
Fldl
F
2
m
Z t40
Fl
Z l0
Fudu
dl
68
and
WdissWt7 FgZ t7
0
lFldl F ffiffiffiffiffiffiffiffi2ghp Z t7
0
Fldl
F
2
m
Z t70
Fl
Z l0
Fudu
dl
69
The errors of calculated smax, sve, sel, Wstore, and Wdiss may
now be estimated from the sum of their partial derivatives with
respect to g, m, h, F, t3, t4, and t7. Therefore:
Dsmax @x
@g
tt4
Dg
@x@m
tt4
Dm
(
@x
@h
tt4
Dh
:
@x
@F
tt4
DF
@x
@t
tt4
Dt
) 70
Dsve
@x@g
tt4
@x@g
tt3
Dg
@x@m tt4 @x@m tt3
h iDm
@x@h
tt4
@x@h
tt3
h iDh
@x@F tt4 @x@F tt3
h iDF @x@t
tt4
@x@t tt3 h iDt
8>>>>>>>>>>>>>:
9>>>>>>>=>>>>>>>;
71
Dsel
@x@g
tt7
@x@g
tt4
Dg
@x@m tt7 @x@m tt4
h iDm
@x@h
tt7
@x@h
tt4
h iDh
@x@F tt7 @x@F tt4
h iDF @x@t
tt7
@x@t tt4 h iDt
8>>>>>>>>>>>>>:
9>>>>>>>=>>>>>>>;
72
DWstore
@W@g
tt4
Dg
@W@m tt4Dm
@W@h
tt4
Dh
@W@F
tt4
DF
@W@t
tt4
Dt
8>>>>>:
9>>>=>>>;
73and
DWdiss
@W@g
tt7
Dg
@W@m tt7Dm
@W@h
tt7Dh
@W@F tt7DF @W@t tt7Dt
8>>>>>:
9>>>=>>>;
74
where DFis defined as the maximal error in the measured force
throughout the experiment, and Dt is the maximal error in
determining t3, t4, and t7. Therefore:
DF MAXfjDFij i 1; 2;. . .;Ng 75
and
Dt MAXfjDt3j; jDt4j; jDt7jg 76
0
10
20
30
40
50
60
70
80
90
100
0 0.5 1.0 1.5 2.0
n(
%)
W
W
s
s
s
(%)
Figure 10. Relative error, n, of smax, sve, sel, Wstore, and Wdiss as a
function of the relative error, k, ofg, m, h, Fand tc. The symbols used
in the Figure denote the following physical quantities: smax, maximum
deformation of the rope; Sve, viscoelastic part of deformation of the
rope; sel, elastic part of deformation of the rope; Wdiss, dissipated
energy of the process; Wstore, stored energy of the process.
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Equations 7074 may be expressed in a matrix form as:
Dsmax
Dsve
Dsel
DWstore
DWdiss
2666666664
3777777775
a11 a12 a13 a14 a15
a21 a22 a23 a24 a25
a31 a32 a33 a34 a35
a41 a42 a43 a44 a45
a51 a52 a53 a54 a55
2666666664
3777777775
jDgj
jDmj
jDhj
jDFj
jDtj
2666666664
3777777775
D
jDgj
jDmj
jDhj
jDFj
jDtj
2666666664
3777777775
77
Individual components, aij, of the matrix D are given in
Appendix I.
We still need to comment the errors in estimating the
stiffness at the beginning, kinit, and at the end, kend, of the
impact loading cycle, and the error in the calculation of the jolt(derivative of the acceleration/deacceleration). These require
numerical derivation of the measured force for rope de-
formation and time, respectively. Numerical derivations may
often be troublesome; however, it is a standard, well-known
numerical problem, which has been properly addressed in
commercial mathematical softwares, such as Mathematica,
and does not need any additional comment.
3.2.1 Sensitivity of the error of calculated data to the error of
input data
Let us first assume that the relative error of all input
physical quantities, g, m, h, F, and tc 2 ft3; t4; t7g, is equal:
Dg
g
Dm
m
Dh
h
DF
F
Dti
ti; i 3; 4; 7
78
Of course, this assumption is not realistic. However, it will help
us to understand which of the calculated physical quantities,
Wstore and Wdiss,smax,sve, andsel is most sensitive to the error
of input data. The relative error of the calculated data is de-
fined as:
Z DC
Ctrue 100% 79
where DCrepresents Dsmax, Dsve, Dsel, DWstore, and DWdiss, and
Ctrue is their corresponding error-free values, respectively.
Equivalently, we may define the relative error of the input data
g, m, h, F, and tc as:
k D
true
100% 80
where Drepresents Dg, Dm, Dh, DF, and Dt, whereas true is
the error-free values of g, m, h, F, and tc. Here, tc again
represents t3, t4, and t7. Figure 10 shows the results
of these error analyses, shown as Z Zk for each of the
five sought-after physical quantities. From the Figure, it
can be seen that the accuracy of prediction of the viscoelastic
component of rope deformation, sve, is most sensitive to the
errors of input data, followed by sel, Wdiss, Wstore, and smax.
The most important observation is that the errors of the cal-
culated data are up to 100 times larger than the error of input
data. Thus, in order to utilize the derived theory for analyzingthe durability of ropes and the safety of climbers, we need to
carry out experiments very accurately.
3.2.2 Example for realistic measuring setup
Let us now turn to the analysis of a realistic situation,
which corresponds to the experimental setup used in our
laboratory. The errors of the input data in our experiments
are typically: Dg 0:00001 m=s2, Dm 0:02 kg,
Dh 0:01 m, DF 5N, and Dt 0:0001 s. According
to equation 77, this leads to the following absolute errors of
calculated data: Dsmax 0:002959 m, Dsve 0:001568 m,Dsel 0:00639153 m, DWstore 11:0115 Nm, andDWdiss 27:5671 Nm. The corresponding relative errors are
then: dsmax 0:29%, dsve 9:88%, dsel 1:29%, dWstore
0:33%, and dWdiss 1:3%, respectively.
As predicted previously, the largest error appears
in the prediction of the viscoelastic component of rope
deformation, sve. However, the prediction is still within the
acceptable limit. Predictions of all other physical quantities
are very good, which confirms that the proposed experi-
mentalanalytical methodology may be used for the analyses
of the functionality and durability of ropes and safety of
climbers.
4. CONCLUSIONS
We have presented the methodology based on a simple
non-standard falling-weight experiment, which allows for
the examination of the functionality and durability of ropes
beyond the experimental findings of the UIAA. The experi-
mentalanalyticalnumerical treatment allows for the
examination of the time-dependent viscoelastoplastic
behavior of ropes exposed to arbitrary falling-weight loading
conditions. A developed methodology can be successfully
applied for calculating the following important physicalparameters: the impact force and jolt (the derivative of the
acceleration/deacceleration acting on the climber); the viscoe-
lastoplastic deformation of the rope; stored, retrieved, and
dissipated energy during the loading and unloading of the
rope; and modification of the stiffness of the rope within each
loading cycle.
A developed analytical treatment was subsequently ex-
amined by using the synthetic experimental data. By means
of the parametric error analysis, we analyzed the required
precision of all measured physical quantities used in the cal-
culation of physical quantities that determine the durability of
ropes and safety of climbers.
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The parametric error analysis showed that that the errors of
calculated data are up to 100 times larger than the errors of input
data. Thus, in order to utilize the proposed methodology, one
needs to carry out experiments very accurately. When doing so, the
accuracy of prediction of all sought-after physical quantities are
within the acceptable limits, which confirms that the proposed
experimentalanalytical methodology may be used for the analyses
of the functionality and durability of ropes and safety of climbers.
Acknowledgements
We would like to acknowledge the financial support provided by
the Slovenian Research Agency (http://www.arrs.gov.si/en/
dobrodoslica.asp). The contribution of our coworker Pavel
Oblak, University of Ljubljana, Slovenia in standardizing the
experimental procedures is also greatly appreciated.
5. APPENDIX I
5.1 Components of the Matrix D
a11 @x
@g
tt4
t
24
2 t4
ffiffiffiffiffih
2g
s
a12 @x
@m
tt4
Fm2
Z t40
Z t0
Fldl
dt
a13 @x
@h tt4
t4 ffiffiffiffiffig
2hr a14
@x
@F
tt4
1m
Z t40
Z t0
Fldl
dt
a15 @x
@t
tt4
gt4
ffiffiffiffiffiffiffiffi2gh
p
F
m
Z t40
Fldl
a21 @x
@g
tt4
@x
@g
tt3
t24t
23
2 t4t3
ffiffiffiffiffih
2g
s
a22 @x
@m
tt4
@x
@m
tt3
Fm2
Z t4
t3
Z l
0
Fudu
dl
a23 @x
@h
tt4
@x
@h
tt3
t4t3
ffiffiffiffiffig
2h
r
a24 @x
@F
tt4
@x@F
tt3
1
m
Z t40
Z l0
Fudu
dl
1
m
Z t30
Z l0
Fudu
dl
a25 @x
@t
tt4
@x@t
tt3
gt3ffiffiffiffiffiffiffiffi
2ghp
F
m
Z t30
Fldl
gt4ffiffiffiffiffiffiffiffi
2ghp
F
m
Z t40
Fldl
a31 @x
@g
tt7
@x
@g
tt4
t
27t
24
2 t7t4
ffiffiffiffiffih
2g
s
a32 @x
@m
tt7
@x
@m
tt4
Fm2
Z t7t4
Z l0
Fudu
dl
a33 @x
@h
tt7
@x
@h
tt4
t7t4
ffiffiffiffiffig
2h
r
a34 @x
@F
tt7
@x@F
tt4
1
m
Z t70
Z l0
Fudu
dl
1
m
Z t40
Z l0
Fudu
dl
a35 @x
@t
tt7
@x
@t
tt4
gt
7ffiffiffiffiffiffiffiffi2ghp
F
mZ t7
0
F
l
dl
gt4ffiffiffiffiffiffiffiffi
2ghp
F
m
Z t40
Fldl
a41 @W
@g
tt4
F
Z t40
lFldl F
ffiffiffiffiffih
2g
s Z t40
Fldl
a42 @W
@m
tt4
F
2
m2
Z t40
Fl
Z l0
Fudu
dl
a43 @W
@h
tt4
F
ffiffiffiffiffig
2h
r Z t40
Fldl
a44 @W
@F
tt4
g
Z t40
lFldl
2F
m
Z t40
Fl
Z l0
Fudu
dl
ffiffiffiffiffiffiffiffi
2ghp Z t4
0
Fldl
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a45 @W
@t
tt4
Fgt4Ft4 F ffiffiffiffiffiffiffiffi2ghp Ft4
F2
mFt4
Z t40
Ftdt
a51 @W@g
tt7
FZ t7
0
lFldl Fffiffiffiffiffi
h2g
s Z t70
Fldl
a52 @W
@m
tt7
F
2
m2
Z t70
Fl
Z l0
Fudu
dl
a53 @W
@h
tt7
F
ffiffiffiffiffig
2h
r Z t70
Fldl
a54 @W
@F
tt7
g
Z t7
0
lFldl2F
m
Z t7
0
Fl
Z l
0
Fudu
dl
ffiffiffiffiffiffiffiffi
2ghp Z t7
0
Fldl
a55 @W
@t
tt7
Fgt7Ft7 F ffiffiffiffiffiffiffiffi2ghp Ft7
F2
mFt7
Z t70
Ftdt
REFERENCES
1. Jenkins M, ed. Materials in sports equipment. Woodhead Publishing Limited:Cambridge, 2003. ISBN 1 85573 599 7.
2. EN 892:2004 (E). Mountaineering equipment. Dynamic mountaineeringropes. Safety requirements and test methods.The European Committee forStandardization, November 2004.
3. http://www.theuiaa.org/upload_area/cert_files/UIAA101_DynamicRopes.pdf.[15 March 2008]
4. Oblak P. Development of the methodology for dynamic characterization ofropes (Dissertation). University of Ljubljana: Ljubljana, 2007.
5. Emri I, Udovc M, Zupancic B, Nikonov AV et al. Examination of the time-dependent behaviour of climbing ropes. In: Fuss FK, Subic A, Ujihashi S, eds.The Impact of Technology on Sport II. Taylor & Francis: London, 2008; 695700.
Received 1 May 2008
Accepted 9 June 2008
Published online 6 January 2009
Sports Technol 2008 1 No 4 5 208 219 & 2008 John Wiley and Sons Asia Pte Ltd www sportstechjournal com 2
Time-dependent behavior of ropes
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