Abstract: On
this page I want to find out the specific parameters which are
essential to describe a climbing rope with viscoelastic models. At
first the Harmonic Oscillator Model will be described and compared
to the Solid Linear State Model of a climbing rope. This model
will be modified to consider the free flight of the fall mass
when it bounces back.
Experiments were made with one of the securest climbing ropes
worldwide: The Rope "Apollo" from the manufactor Beal.
Stretch tests with a 4t crane from ABUS were made to measure
the hysteresis of the rope as well as fall tests with a 80kg
fall mass and high speed filming of the experiments. The analysis of the fall data enabled to find typical average rope parameters for the climbing rope Apollo 11, which may be used for simulating climbing rope falls or rope swings in future.
Max
Bigelmayr, Sept. 2013, April 2014 (V31)
1.
Viscoelastic theory of climbing ropes
1.1
Harmonic Oscillator Model (HO)
In physics nearly everything starts with the harmonic oscillator. A climbing rope may be described very easy by the harmonic oscillator. In mechanics the harmonic oscillator
contains only a simple spring with the spring constant k and a mass m.
The Lagrange equation of the system shown in [Fig. 1]
is the difference between the kinetic energy T and the potential energy U:
By using the EulerLagrange equation we get the differential equation of the system:
This is a standart differential equation of 2th order with the solution
. [1.1]
The frequency w of the oscillator is
.
Now we have to find the parameters C1 and C2. At the starting point t=0, the elongation y must fit the boundary condition
.
Differentiation of [1.1] gives us the boundary condition with the start speed vo:
By putting C1 into this equation we find the parameter C2.
Now we may write down the expression for the elongation y(t):
When you play
around with this expression a little bit, you may get:
[1.2]
Fig. 1: Hamonic Oscillator modell with spring constant k to describe a climbing rope fall experiment
Fig. 2: Phases of the Hamonic Oscillator modell describing a climbing rope fall experiment: The simulation shows a rope with: E=174MPa, l=2,60m, fall height= 2,60m, m=80kg, A=95mm^2. Above the "sero line" the path of the fall mass describes a parabolic curve.
One may seperate three phases of a fall experiment in the harmonic oscillator model: Before the stretching of the spring k the mass will be accelerated with g=9.81m/s^2 (phase 1). The start speed vo in [1.2] is the speed of the fall mass after falling down the lenght lo:
The second phase is described by [1.2] and includes the time intervall when the spring is stretched by deceleration and acceleration of the fall mass m.
While the third phase the mass is flying upwards without stretching the rope. Equ. [1.3] summarises the equations of these three phases:
[1.3]
1.2
Solid Linear State Model (SLS)
In solid state physics different methods were developed to model the behavior of viscoelastic materials. The most famous models are the Maxwell Model, the KelvinVoigt Model and the Wiechert Model [1]. These models use different combinations of springs and damping elements as the viscosity element n. Pavier [3] used a special SLSmodel [which is called Zener kbody] to describe the stretch and damping behavior of climbing ropes during the fall of a mass m. The Zener kbody [Figure 3] contains 3 parts: Sping k1 is connected parallel to a viscosity element c1. This so called KelvinVoigt Model is in series with a second spring k2 with a mass m on the bottom end. The kinetic term of the system is
while the potential energy is
The lagrange function of the system is L=TU:
with the dissipation term
The Euler Lagrange equation with a dissipation function is in general:
[2.0]
The first differential equation is
[2.1]
[2.2]
Differentiation to y2 gives the second differential equation
[2.3]
Fig. 3.: Zener k body as a model to describe a fall into a climbing rope. The spring k1 is parallel to the viscosity element c1. The elongation y1 is invisible and only a theoretical construct, but important to describe the energy dissipation of the system. The elongation of spring k2 is represented by y2, which is the movement of the fall body with mass m.
between them.
This equation solved to dy2/dt^2 and y1 gives::
[2.4]
Differentiation gives
Now let´s substitude dy1/dt with [2.2]:
With
y1 from [2.4] we get:
With the substitutions
we get:
Fig. 4.: Typical Simulation Results for a standart drop test UIAA 101: The used parameters E1= 272MPa, E2= 479MPa, n = 124MPas, A= 95 mm^2 where found while this work by fall drop experiments with the climbing rope Apollo from the manufactor Beal. The graph represents the simulation of equation number 0 from table 1 with interrupt at the sero line. Therefore three numerical simulations give the graph under the sero line with the parabolic curves "2. parabolic flight" and "2. parabolic flight" between them.
Such a 3th order Differential equation may be solved numerically by mathematica, matlab, labview etc. by defining a differential equation system:
Equation Number
Boundary Condition
Differential Equation
Solution
Name
0
hight
1
speed
2
acceleration
Table 1: Differential equations of the SLS model of a climbing rope
with
1.3 Comparison between HO and
SLSModel
Booklets about climbing ropes and security advices often present the harmonic oscillator as "physical background knowledge". Based on the "Standart Equation of Impact Force" [9] some people try to calculate the forces of climbing ropes with the harmonic oscillator model. In this model the climbing rope is only described by the modulus of elasticity, the lenght and it´s cross section. With the harmonic oscillator model it´s in generall not possible to fit the impact force given by the manufactor together with the dynamic elongation. Whenever you found a perfect value for the modulus of elasticity to fit the maximum imopact force, the dynamic elongation does not fit well. If the dynamic elongation is well described the impact force is wrong. As shown in fig. 2 the harmonic oscillator system may not absorb energy, so that the fall mass will be accelerated back to it´s original start height. It´s obvious that this does not happen in real. Leuthäusser overcame this problem by fitting the data of many different climbing ropes [4] with the SLS model of a climbing rope. In his work he provided typical values for a standart climbing rope as E1=174MPa, E2=478MPa, n=86MPas, A=75mm^2.
If you do simulations with these parameters you will realize that the values do fit the technical data (impact force, dynamic elongation) provided by the manufactors perfectly. But there is still a problem with these parameters: In general the fall body bounces back and describes a parabolic flight while a drop test (I will present measurements in chapter 3.4). Simulations with the values from Leuthäusser give no parabolic flight phase. Thus, these parameters are not perfect at all. So if you really want find perfect parameters
according to the SLSmodel you have to fit:

impact force  dynamic elongation
 end elongation
 maximum parabolic flight height
To become even moore accurate one could also regard the loops of the rope on it´s endings, which do represent a twin rope. In the next chapter the equations of such a loop SLSclimbing rope system will be derivated.
1.4 SLSModel with loops at the climbing rope ends
In practise a climbing rope is used with two loops at the endings(fig. 6a). In general the loops are kinked with a figure eight knot or the yosemite bowline knot. In fall experiments it´s also necessary to create loops at the endings to fix on end of the rope at the ceiling. At the other end of the rope the fall mass is fixed. Le´ts assume a climbing rope with a lenght lo of 300cm between the knots. The lenght of the loops s1 (when they are elongated) are typical around 20cm (distance from the knot to the loop end). In this case the two fix points of the rope have got a distance of 340cm. Then, around 88% oft the rope is a single rope, while 12% of the rope is a dopple rope created by the loops. So in a theoretical describtion of climbing ropes it´s essential to consider "these 12% loopendings". In fig. 6a you see the schematic diagramm of the loop model (model1). In a next step we may combine the two ropes of each loop to one "double rope" (model 2). Then model 2 may be redistributed, so that we get finaly the simulation model (fig. 6b).
Fig. 6a: Viscoelastic model of a climbing rope with two loops at the ends.
Fig. 6b: Simplified Viscoelastic model of a climbing rope with two loops at the ends.
The simulation model (fig 4b) may be described by a differential equation system very easy without finding six differential equations for all the hights y1, y2, y3, y4, y5, y6 you would require in the model presented in fig. 6a. But before finding the differential equations let´s think about the new parameters k1´, k2´, k3´, c1´ and c2´:
Parameter
k1´ does represent the movement of the damping part in of both loops. We may write:
The parameters
k1 and k5 are the substitution parameters we got from loops in fig. 6a. With k11=k12we get:
Parameter k5 from the down loop gives us:
Remember the units for calculating the elasticity and viscosity;)
Parameter k11 and k51 are both lenght dependent and we may write
with the "viscoelastic Elasticity modul" EI (equal to k1 in fig. 6a) of the rope. Finally
k1 and k5 are
,
so that we get for the parameter k1´ in the simulation model
The parameter k2´ in does represent the parameter k3 in model 1, so that
Parameter c1´ is simply the combination of c1 and c3. c1 and c3 may be substituted with c11 and c31 so that we get:
Parameter c2´ is quity simple:
At last we have to find
k3´ wich is
After defining the parameters we may find the differential equations to describe the "simulation model". Instead of k1´, k2´ etc. let´s omit the strokes henceforth:
k1´> k1
k2´> k2
k3´> k3
c1´> c1
c2´> c2
Differential Equations of the loop model
The kinetic term of the system is:
The potential energy is:
The dissipation term is
By Using the Euler Lagrange Equation with Dissipation
we get the three differential equations of the system:
Fig. 7: Viscoelastic Model of a climbing rope with loops on both ends
The parameters in fig. 5 may be calculated by these formula:
Note that EI is the elasticity modul of the climbing rope, which is parallel to the viscosity (equal to k1 in Zener k body, Fig. 3). EII is the elasticity modul in series with the spring k2. As explained before the strokes are omited in the differential equations on the left (k1´> k1 etc.).
As you see the equation (4.1) has got the differential dy2/dt in the term, equation (4.2) includes the term dy2/dt. In a differential equation system these differentials have to be substituted. Elimination of dy2/dt in (4.1) and (4.2) gives the follwing differential equation system:
Equation Number
Boundary Condition
Differential Equation
Solution
Name
0
invisible hight y1
1
invisible hight y2
2
hight y3
3
speed v3
4
acceleration a3
Table 2: Differential equations of a climbing rope with loops on both ends.
2.
Stretch Tests with Beal Rope "Apollo"
2.1 Setup of the Stretch Experiments
While this project I was specially interested in the parameters k1, k2 and c [Fig. 3] of the climbing rope Apollo II from the manufactor Beal. This rope is one of the worlds strongest multi fall climbing ropes. Beal gives the following the informations about the rope:
Performances
Performance
UIAA Norm
Euro Norm
• IMPACT FORCE
7.70 kN
≤ 12 kN
• DIAMETER
11 mm
• NUMBER OF UIAA FALLS
16
≥ 5
• DYNAMIC ELONGATION
35 %
≤ 40 %
• ELONGATION UNDER 80KG
9,5 %
≤ 10 %
• SHEATH SLIPPAGE
0 mm
≤ 20 mm / 2m/2m
• WEIGHT PER METER
75g
• MATERIAL
Polyamid (PA)
Table 1: Datas about the Climbing Rope Apollo II from the manufactor Beal
Fig. 8: Beal Rope Apollo II, 50m, d=11mm
In the following tests a brand new climbing rope was used [Fig. 6] to find the wanted parameters. At first the parameters k1 and k2 has to be found. But sadly it´s not possible to find the exact values of k1 and k2 directly by stretching the rope. If you stretch the rope very slowly the viscosity c may be neglected. In this special case the Zener kbody becomes a simple setup with the two springs k1 and k2 in series. The spring rate then is
.
The value of K may be measured by a climbing rope strech test:
2.2 Hysteresis Measurements
To find the parameter K a large 4 tone crane from Abus was used. Sadly we haven´t got any wight which is heavier than 2000kg. Therefore it was necesarry to constructed a special strech setup with woodn beams and threaded rods. Together with a friend we builded the following testing bench:
Fig. 9a: climbing rope testing bench
Fig. 9b: climbing rope testing bench
Fig. 9c: 3000 kg crane scale
Fig. 9d: bottom hook with climbing rope
While a test a piece of climbing rope is cutted to a lenght around 3m. On both ends figureeight loops will be done. One loop of the rope will be thread into the hook of the beam on the floor, the other loop will be thread into the hook of the weighing scale from steinberg systems.
2.3 Exponential Decay of Force
While the experiment the crane drags the rope at a constant speed upwards. Thus the
force is created because of a certain lenght, the crane defines.
So the force is a function of l, as well as a function of the
time t. In the graph on the right one may watch the the force F depending
on the time t [F=9.81N/kg*m]. When the engine of the crane is
stopped at 46s the lenght of the rope does not change. But the
scale shows a decaying of the force in the rope while the time
space t=[46s,...;72s]. I tried to find a fit function for this intervall.
In Fig. 8
the Exponential Decay of the force is shown (black line dashed). Fit 1 (blue line) is a fit function with a standart exponential function, which does not really fit well. Fit 2 (red line) is a a douple exponential function in the form
and fits great the measured data. I am not sure why the douple exponential is the ideal function. The Force Decay seems to be a physical phenomena where two independent mechanisms cause a stress decay in the climbing rope. Maybe this is caused by the sheath and the kernel of the rope which may be different in their behavior under stress. Please note that this force decay is while the entire stretch experiment. Therefore the hysteresis in Fig. 7 is a time dependent measurement which would be different if the experiment would be done with a different wire rope hoist speed.
Fig. 10: Force at the rope in dependence of the time.
In the first 46 seconds the force is growing up nearly linear.
In t=[46s,...,72s] the engine of the cran is off. In this time
the force decays exponential (Fig. 7). After
72 seconds the force is reduced again by driving the crane hook
back again.
Fig. 11a: Hysteresis of the Rope "Apollo" from the manufactor
Beal .
Fig. 11b: Exponential Decay of force of the climbing rope "Apollo"
in t=[46s,...,72s]
2.4 Quasistatic Modulus of Elasticity
In Figure 7 I made two different fits to find the derivation of the force to the lenght of the rope, which is equal to the spring rate K of the rope:
The red fit was made in the intervall l=[43.5cm; 53cm] and the lila fit was made in the intervall l=[43.5cm; 57cm].
The modulus of elasticity is
so that we get the values for both fits:
With the cross section of the used climbing rope
we get the modulus of elasticity:
Now we may use this values to find the possible combinations of E1 and E2 to get Emin or Emax. Figure 9 shows the hyperbolic relation between E1 and E2 to give the constant value of E.
[3.1]
Fig. 12: Hyperbolic relation between E1 and E2 to fit Emin and Emax
3.
Fall Tests with Beal Rope "Apollo"
3.1 Setup of the Fall Experiments
With the stretch experiments I measured the quasi static behavior of the climbing ropes. To understand the real dynamic bahavior I build a special fall test setup for filming the fall movement with a high speed camera:
Fig. 13a: Setup of the Fall eperiments with around 2000 W light power
Fig. 13b: Fall body with bench marks
Fig. 13c: bottom hook with climbing rope
Fall body and fixing of the climbing rope
I build a special fall body with massive steel blocks (Fig 10d). One steel block is vertical screwed with a horizontal steel block, so that the entire mass is exactly 80kg (+0,5kg). On the top of the massive fall body a lifting eye bolt is srewed on. After defining the lenght the climbing rope is cutted (for example 500cm). On both ends of the rope two loops have to be kinked. One end of the rope will be fixed at the double Tgrider of the swivel crane 5m above the ground (fig. 10c). The used swivel crane may lift masses up to 1000kg (Fig. 10c). The other end of the rope is fixed on the eye bolt of the fall mass (Fig. 10d). For security reasons some wooden pallets have to be placed on the cement floor under the fall body.
Bench marks and illumination
Behind the "fall line" I placed some wooden slabs with bench marks up to 220 cm above the ground (Fig. 10b). These marks are important to identify the hight of the fall body with a high speed camera (see next chapter 3.2). Because high speed filming requires a lot of light several halogen spotlights are used to create a great illumination with little shadows (Fig. 10a).
Process while a fall experiment
For a fall experiment the fall body will be hoist with the crane chain up to a high of XY m. While this process the climbing rope is hanging loose as you may watch in Fig. 10d. After activating the high speed camera (which is placed some meters away (see next chapter 3.2)) the fall body will be unclamped with a special horse piton. With a small extension on a string the horse piton opens so that the fall body accelerates downwards with 9,81m/s^2.
Fig. 13d: 80 kg fall body hanging on the swivel crane. The mass is wight by crane scale.
3.2
Geometry at High Speed Filming
If
you want to measure the position of moving objects via high
speed films you should take care about "geometric failures".
Let´s assume a mass m, with
a horizontal distance of a to a recording
camera. Behind the mass is a reference wall (distance
k respectivly to the mass m). In the view of the
camera the mass has got a hight of s(admittance).
This hight is not the real hight of the mass because of the
distance k between the mass and the reference wall in the background.
One may find a equation to describe the real hight h of the
mass depending on the admittance captured by the camera:
Fig. 14: Geometry of the fall experiments. The wall on the left with bench marks allows to identify the hight of the fall mass.
At
first let´s define the lenght h, s and d:
By using the intercept theorem one may write
With (i) and
substitution with (ii) and (iii)
we get:
Finally we may write this equation as:
[6.1]
Fig. 15: Admittance and real hights values of the fall body at different hights.
With
equation [6.1] it´s easy
to calculate the real hight of the fall mass out of the high speed films. You have just to put in the admittance s into the equation to get the real highth(s). The parameters k, o and b have to be measured by a folding metre stick.
To check out the accuracy of this correction term for practical purpose I made "a small experiment" first:
I build a test arrangement with the following conditions, which allowed me to calculate the correction term.
[6.2]
Now I wanted to compare this
linear equation with well defined hights of the fall mass. Therefore I took the swivel crane and hoisted the fall body to certain hights [80cm, 100cm, 120cm, 150cm] above the ground. I made photos of any of this 4 positions of the fall mass and used self built position measurement programms [watch next chapter 3.3] to determine the admittance hight. These admittance values are the hights in the view of the camera and do not represent the real hights. In Fig. 13 the admittance hight positions and the corresponding real hights are shown. The values may be perfectly fitted by the linear equation h(a)=15+0,852*a. Comparing this linear equation (I got by fitting the experimental values) with the geometrically calculated formula [6.2] gives a very well agreement. That´s nice!! :)
3.3
Position Measurement Programms
For Speed
Measurement experiments it´s usefull to mark a reference
wall in the background. I took a large wood as a wall and prepared
it with two black tapes . One tape has got a hight of 30 cm
respectivly to the bottom (a), the other has
got a hight of 180 cm (b). I build a special
LabviewVI where I can measure the hight of the fall mass in each frame (Fig. 20). In most experiments I made films with 600fps, so that I had to identify the hight of the mass in 5000 frames. This is really a hard work and you become crazy after several hours "analyzing the frames"...
Fig. 16: First Version of the Position Measurement Programm. The positions of the whight loop had to be found mechanically by scrolling.
Because I didn´t want to become even more crazy I decided to think about an automatic version of the programm. Somehow it should be possible to do this in an automatic way!!! After searching around a little
I found the so called Lucas Kanade algorithm. This method was invented by B. D. Lucas and T. Kanade in 1981 and allows the position calculation of an moving object via the optical flow [3]. "After programming around" I got a programm with that it´s possible to get the fall mass trajectories of a 5000 frames movie in around 10 seconds (Fig. 21).
Fig. 17: Advanced Measurement Programm. The positions of the whight loop or any other points have to be defined at the beginning. Then a pyramidbased Lucas and Kanade algorithm calculates the positions of the wight via the optical flow in each frame of the high speed movies.
3.4
High Speed Films with position curves
The fall
experiments were filmed with a High Speed Camera from Casio
(Casio Exilim EXF1). Best film results were at a picture
rate of 600fps (frames per second) at a resolution of 432px*192px.
The movies were saved as MOVfiles, which may be converted to
jpg`s by using a jpgconverter. The individual pictures
were synchronized afterwards with the measured position data.
The results are:
Fall Experiment 1
Fall Experiment 2
Fall Experiment 3
Fall Experiment 4
Fall Experiment 5
Fall Experiment 6
4.
Fit program for Analysing the Fall Experiments(SLSModel with loops at the climbing rope ends)
After measuring the data (presented in chapter 3.4) the curves had to be corrected geometrically by a correction term described in chapter 3.3. In the experiments with the Beal Climbing rope the correction Term was h(a)=11,57+0,914*a. With this correction the follwing Graph was obtained:
Fig. 19a: Elongation of the fall mass while the third fall experiments. The red line shows the measured data captured via the measurement programm. The blue line is corrected by the geometric correction term h(a)=11,57+0,914*a. The difference between the two curves proves the necessity of the correction process.
Fig. 19b: Heights of the fall mass. The curves were captured via the Measurement Programm and geometrically corrected with a specific correction formula. Depending on the fall number the fall body stretches the climbing rope moore and moore. Increasing the fall numbers means a larger magnitude of the turning points. The red dashed lines are the ending values of the oscillations.
These corrected position graphs do represent the experimental results of the SLSmodel with loops at the end of a climbing roope I described in chapter 1.4.In Table 2 I showed the differential equations, which have to be solved numerically. By Variing the parameters k1, k2 and k3 it´s possible to find the ideal combination to get curves, which fit the measured lines shown in Fig. 21. To handle this fit process I programmed a special Labview VI for changing the parameters to find the ideal model parameters E1 E2 and n:
Fig. 20:Fit program for numerical simulation of the differential equation system presented in table 2. The measured experimental data (txtfile) may be loaded into the program. Then the specific values of the experiment (table 3) have to be typed in (Seildaten in fig. 22) After defining a certain accuracy (nomally I use 1E10) the differential equation system will be solved numerically by a Cash Carp 5th order method. The variation of the relation between E1 and E2 on the hyberbolic curve (remember equation [3.1], fig. 10) and the viscosity modul gives the optimal curve. The "tension line" in green is the hight where the rope begins to be stretched while the fall process. Therefore the value of the "tension line" is equivalent the the hight, where the free fall ends. When the body accellerates back towards the maximum the simulation of the differential euquation system is interuppted at the tension line. Then the trajectory is described by standart parabolic equation (school physics). The boundary condtions at the tension line have to be equal for the differential equation and the parabolic curve to get a smooth curve. After reaching the tension line again at around 1,25s the differential equation system uses again the boundary conditions of the parabolic curve as initial conditions for the next simulation segment. The red line shows the simulation result without segmented simulation.
5.
Fitings of the Fall Datas
5.1
Fit data with rope lenghts
The climbing rope I used in the fall experiments may be seperated into three parts (Fig. 22): The upper loop with the lenght s1, the middle part rope lo and the down loop s2. All these lenghts do depend on the fall number:
Fall Number
lenght loop s1
[cm]
lenght loop s2
[cm]
lenght rope without wight [cm]
lenght rope with wight [80kg] after exp. [cm]
hight above ground after exp. [cm]
rope lengt with loops [cm]
tension line [cm]
fall deep [cm]
fall factor
1
17,2
18,7
294,7
310,2
122,5
346,1
141,4
333,6
0,96
2
19,6
21,1
300,1
315,9
115,3
356,6
130,9
344,1
0,96
3
19,8
21,3
303,6
319,6
111,4
360,7
126,8
348,2
0,97
4
19,9
21,4
307,1
323,3
108,3
364,6
122,9
352,1
0,97
5
20,0
21,5
307,9
324,1
105,9
365,6
121,9
353,1
0,97
6
20,0
21,5
309,1
325,4
104,3
366,9
120,6
354,4
0,97
Table 3: Length values depending on the fall number. The lengths were measured after every fall event. The values "lenght rope without wight" are not measured. I assumed that the values "lenght rope without wight" are 95% of the "lenght rope with wight aftter exp.". 95% is the typical elongation of the Beal rope without fall body compared to the lenght with the 80kg fall body I measured with the Beal rope before starting the fall experiments. In all experiments the upper ending of the climbing rope has got a hight of 495cm, while the hight of the fall body above the ground was 475cm. The fall body is allway 7,5cm lower than
the down end of the climbing rope because of the carabiner (compare fig. 11b).
Fig. 21b: Distance between marks and knots. The distance at the upper knot is du, the distance at the bottom knot is dd. Before the first fall the distance was 0cm. After the 4th. and 5th. fall experiment I made some measurement failures, which were corrected afterwards (dashed red line).
Fig. 21a: Rope and loop lenghts.
While the experiments all lenghts in this figure are growing up.
Fig. 21c: Rope lenght l* depending on the fall number. After each fall eperiment the rope lenght l* is growing up. I am not sure whether there is a "quasi asymtotic" behavior or not. The rope lenght where measured between the marks (Fig. 22a) while the 80kg fall mass is hanging on the rope.
Sadly I did not measured the lenghts s1 and s2 while the fall experiments. Therefore I assumed that the distance behavior is su=1/2du and sd=172dd. The lenghts s1 and s2 may be assumed by analyzing Fig. 22a. Let´s assume that the differential stretching per lenght d"strech"/ds after each individual fall is allways 50% of the strechting we see in Fig. 22c, because of the dopple rope inside the loops. Fig. 23a shows the assumed stretch behavior of the distances su and sd.
Fig. 22a: Assumed lenght of the loop rope parts drawn out of the knots sd and su. The lenghts are the half values of the average lenghts in Fig. 22b.
Fig. 22b: Lenght of the loops s1, s2 without 80kg mass depending on the fall number. The values are calculated and assumed with the known end value after 6 falls.
5.2
Data Fits and and average SLSparameters
Finally the data in table 3 may be used to fit the measured hights from Fig. 21b with the fit programm presented in Fig. 22. In Fig. 24af the measured data and appropriate fit curves are shown. It was not possible to find fit parameters for the entire measured data in one individual graph. Therefore I had to split up the fittings. Fit 1 (with parameters E1, E2, n) in blue represents the optimal fit for the first negative and positive peak. Fit 2 (with parameters E1', E2', n') in green represents the optimal fit for the second negative and second positive peak. The curve above the tension line is a parabolic curve [1.3] with adapted boundary conditions from the simulated differential equation system.
Fig. 23a: Data fit of Fall experiment 1
Fig. 23b:Data fit of Fall experiment 2
Fig. 23c: Data fit of Fall experiment 3
Fig. 23d: Data fit of Fall experiment 4
Fig. 23e: Data fit of Fall experiment 5
Fig. 23f: Data fit of Fall experiment 6
The found fit parameters are different for any individual fall experiment. In fig. 25a and Fig. 25b the found parameters are shown
as a function of the fall number:
Fig. 24a: Fit parameters for the first positive and negative peak (Fit 1)
Fig. 24b:Fit parameters for the second positive and second negative peak (Fit 2)
Averages of the fit parameters (fall 2fall 6) are presented in fig. 26. The average values Fit 1 are usefull for further simulations (rope swings etc.) with the climbing rope "Apollo". Fit 2 is not that suitable for simulations because of it´s large standart derivations.
Average Values
E1= 272MPa
E2= 479MPa
n = 124MPas
Fig. 25: Averages of the fit parameters of the climbing rope Apollo from Beal . The values from the 2.6. fall are averaged. E1, E2, E, n are the fit parameters of the SLSmodel for the first negative and positve peak of the fall curves. E1', E2', E', n' are the fit parameters of the SLSmodel for the second negative and positive peak of the fall curves. The values E1', E2', E', n' are not that usefull because of it´s large standart derivations.
6.
Summary and Discussions
In this work I first decribed the harmonic oscillator model for a climbing rope drop test. Based on the problems to find perfect parameters to describe the impact force together with the dynamic elongation of the rope the solid linear state model for a climbing rope (SLSmodel) was presented. The typical SLSmodel for a climbing rope was supplemented with a description of the loops at the ends of the climbing rope.
In the experimental part climbing ropes where streched with a self built testing bench. With a crane (and crane scale) theclimbing rope Apollo from the manufactor Beal was stretched and the hysteresis curve was measured up to an accuracy of +0,5kg and appr. 2mm. On the basis of the gradient the modulus of elasticity was evaluated to a value of E(min)=136MPa, E(max)=158MPa. Futhermoore the exponential decay of the force at the climbing rope stretched to a fixed lenght was analyzed. It was shown that the decay of the force may be described quit well by a double exponential function.
Fall experiments were made with a 80kg falling weight to find the parameters E1, E2 and n of the SLSmodel. Therefor the falling weight was droped down appr. 3,50m into the Apollo rope six times by a fall factor of appr.1 to analyse the elongation and damping process. The movement of the weight was filmed by the high speed camera Casio Exilim F1 at a frame rate of 600fps at a resolution of 432px*192px. Afterwards the exact values of the fall paths were captured by the Lucas Kanade algorithm for each individual frame. The recorded data were corrected geometrically by a specific correction term which is appropriate to the experimental scales in the experimental setup.
A comparison of the different fall paths according to the six fall experiments gave a continuous decrease of the height of the turning point (maximum elongation). Each further drop test gave a larger elongation value than the drop test before as well as a higher vertex of the parabola after the phase when the fall weight bounces back (fig. 19b). The end hights after each individual fall tests do also lower from fall to fall. Associated the with this the rope lenght (between the loops) itself increases approx. up to 4% if you compare the lenght after the 6th fall with the lenght after the 1th fall (fig. 21c). While the 1th fall the damping of the oscillation process is dominated by tighting up the figure eight knots at the loops. This effect causes a relativ large difference oft he height curve hfall 1hfall 2 in comparison to the other differences hfall3hfall2 etc. (fig. 19b, 21b, 22a,b).
Analyses of the measured oscillations were made with a self built "Fit program", which uses the differential equation system (ODE= Ordinary Differential Equations) for climbing ropes with loops at the ends presented in capter 1.4. The ODE was solved numerically with a Cash Karp algorithm of 5th order for different values for the parameters E1, E2 and n. With adaptive variation of the parameter values I could found individual "sets of E1, E2,, n) for each fall experiment (fig. 23af). It was shown that it´s not possible to find such a parameter set describing the entire oscillation process while one drop test. Hence it seemed to be practical to subdivide the fits into two splines with smooth boundary conditions at the endings. The first splines (in blue ) are suitable to desribe the first negative peak and to maximum of the parabloic flight, while the second splines (in green) do fit the swing off in fig. 23af.
The average of the fit parameters (first spline fit fall 26) was calculated so that I got the follwing values for the climbing rope Apollo:
Average Values
E1= 272MPa
E2= 479MPa
n = 124MPas
Climbing Rope Apollo 11 Beal, SLSmodel
These values seem to be quit good for fall factors [0,..,1]. Simulations for larger fall factors like 1.7, (which is used in the standart drop test by the UIAA (fig. 26)) do varify in comparison to the data given by the manufactor Beal (table 1).
Fig. 26: Standart Drop Test geometry (source [8])
The experiments do suggest the assumption that the values E1, E2,, n one may get by analysing the fall experiments do depend on the fall factor used in the experiments. Whenever you try to find the fit parameters E1, E2,, n for a specific climbing rope on the basis of the datasheets by the UIAA drop tests, you will find many different sets of E1, E2,, n which fit the impact force, dynamic elongation and end elongation. In my opinion it´s necessary to fit the entire osciallation process with respect to the maximum parabolic flight height, as I did in this publication. Another point is the 300mm path before the carabiner in the standart drop test (fig. 26). This rope section grates at the carabiner, which qualifies a new model including energy dissipation effects when using the UIAA drop tests data to find the parameters of a climbing rope in the SLS model.
Assigning the results of the experiments of this publication to thephysics of rope swings I made path simulations with all found fit parameters from fig. 25a, b. All paths of the six simulations are higher than the path of a typical climbing rope from Leuthäusser [4] with E1= 174MPa, E2= 478MPa, n= 86MPas, A=75mm^2. The rope swing path with the found average parameters from the climbing rope Apollo with E1= 272MPa, E2= 479MPa, n=124 MPas, A=95mm^2 is marked green in the following graph:
Fig. 27:
Swing paths of a
rope swing simulated with the found fit parameters. The
upper blue line is the radius of the rope (27m), the down line in red
represents the swing with a typical climbing rope, with the parameters
found by Leutheusser. Rolling shows the swing path (in green) with the average values presented in fig. 26.