Time Dependent Impact Load on Rope

8/11/2019 Time Dependent Impact Load on Rope

1/12

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: [email protected]

Keywords:. ropes. impact. viscoelasticity. time-dependent behavior

. jolt

. energy dissipation

& 2008 John Wiley and Sons Asia Pte Ltd Sports Technol.2008, 1, No. 45, 20821908

Time-dependent behavior of ropes


2/12

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.

Sports Technol. 2008, 1, No. 45, 208219 & 2008 John Wiley and Sons Asia Pte Ltd www.sportstechjournal.com 2



3/12

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.

www.sportstechjournal.com & 2008 John Wiley and Sons Asia Pte Ltd Sports Technol.2008, 1, No. 45, 20821910

Research Article I. Emri et al.


4/12

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.




5/12

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




6/12

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.




7/12

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.




8/12

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.




9/12

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


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.




10/12

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.




11/12

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


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


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


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




12/12

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 Impact Load on Rope

Documents

Transcript of Time Dependent Impact Load on Rope