1. Viscoelastic theory of climbing ropes
    1.1 Harmonic Oscillator Model (HO) 
    1.2 Solid Linear State Model (SLS)
    1.3 Comparison between HO and SLS-Model
    1.4 SLS-Model with Loops at the Ends of the Rope
2. Stretch Tests with Beal Rope "Apollo"
    2.1 Setup of the Stretch Experiments
    2.2 Hysteresis Measurements
    2.3 Exponential Decay of Force
    2.4 Quasistatic Modulus of Elasticity
3. Fall Tests with Beal Rope "Apollo"
    3.1 Setup of the Fall Experiments
    3.2 Geometry at High Speed Filming
    3.3 Position Measurement Programm
    3.4 High Speed Films with Position Curves
4. Fit program to analyse the Fall Experiments
5. Fitings of the Fall Datas
    5.1 Fit Data with Rope Lenghts
    5.2 Data Fits with Average SLS Parameters
6. Summary and Discussion
7. Literature

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 Euler-Lagrange 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]

In solid state physics different methods were developed to model the behavior of viscoelastic materials. The most famous models are the Maxwell Model, the Kelvin-Voigt 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 SLS-model [which is called Zener k-body] to describe the stretch and damping behavior of climbing ropes during the fall of a mass m. The Zener k-body [Figure 3] contains 3 parts: Sping k1 is connected parallel to a viscosity element c1. This so called Kelvin-Voigt 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=T-U:

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]

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

with

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% loop-endings". 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:

First Differential Equation [i=1]:



(4.1)
____________________________________________________________

Second Differential Equation [i=2]:

(4.2)
____________________________________________________________

Third Differential Equation [i=3]:

( 4.3)
____________________________________________________________

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:

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:

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. 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]

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 hight h(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]

3.3 Position Measurement Programms

Fig. 16: First Version of the Position Measurement Programm. The positions of the whight loop had to be found mechanically by scrolling.

3.4 High Speed Films with position curves

Fall Experiment 1

Fall Experiment 2

Fall Experiment 3

Fall Experiment 4

Fall Experiment 5

Fall Experiment 6

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).

The average of the fit parameters (first spline fit fall 2-6) was calculated so that I got the follwing values for the climbing rope Apollo:

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).