Abstract: On
this page I want to discuss the theory that lies behind rope
swings. On Youtube one is able to discover lotīs of rope swing
videos. The most famous video is "World's largest Rope Swing",
filmed by Devin Graham: http://www.youtube.com/watch?v=4B36Lr0Unp4 Itīs
really an incredible feeling rushing downwards without feeling
any force for about 35m. However what about the g-forces you
have to withstand during such a swing? Is the rope damaged after
a few jumps and what maximum forces are we deeling with?
Is it better to use static ropes instead of standard climbing
ropes?
These and further questions I want to answer here...
Max
Bigelmayr, Sept. 2012, March 2013
1.
Viscoelastic Model of a Rope Swing
1.1
Lagrange
Equation and Dissipation Function
The pendulum is easily calculated with the help of Polar Coordinates.
Letīs start with by discribing the mass-point, dependent on
the angle P and the elongation y2. The derivation of
the mass-point in realtion to time, gives us the speed vector of the mass-point as well as
its absolut value squared:
The potential energy is seperated into Spring Energy
and Hight Energy. The elongation of spring 2 with the
factor D2, equals the realtiv movement y2-y1:
Relaigh suggested a modification of the standard Euler Lagrange
Equation which included a Dissipation Function D:
The Dissipation Function D is defined by Stokes-Friction as follows:
,
While the dissipation energy is discribed as:
.
c is a constant which discribes the viscosity of the entire rope. Its
unit is defined as [N/ms].
Now we can formulate the three differential equations for y1,
y2 and P.
Letīs start with the inner expansion y1:
equ.(1)
The boundery condition for the given equation is:
and
______________________________________________________________________________________________
The differential equation for the outer expansion y2
is:
equ. (2)
The boundery condition in regards to this equation has to be found concerning the freefall, which occurs prior to the experience of rope force exerted on the person jumping. Thus the
velocity vector of the freefall has to be smooth at this critical
point. If you define the precise time at which the mass-point reaches its critical
point, tcritical point=tcp=0,
the boundery condition of y2(t)
becomes apparent:
But you also have to bare in mind the velocity vector.
Remember the mass-point-movement-vector:
At this critical point the following boundary condition is given,
where the speed must be smooth:
So we are provided with the boundary condition for the start speed and start
angle:
_____________________________________________________________________________________________________
The derivation of Phi is as follows:
The acceleration a(t) of the rope equals d^2y/dt^2 [equ.(2)].
The function da(t)/dt is discribed by the following:
The boundary condition for da/dt is:
1.3
Table of equations
In the following table all the required equations
and boundary conditions are summarized.
2.
Numerical Simulations
2.1
Elasticity
moduls and typical rope-climbing parameters
So now letīs look at
some numerical simulations. For calculations itīs usefull to
work with lenght independent constants.
Therefore let`s use the elasticity modul E1
and E2 in place of D1
and D2:
In respect to the Viscoelastic
Theory of Climbing Ropes [Ulrich Leuthäusser] one
is able to discover the typical parameters of a climbing rope.
These are:
Within the Graphs on the right you are able to view the oscillations
of a climber [m=80kg] falling into the rope from
falling height of 5m. The curves were numerically
calculated with a Cash Carp 5th order system.
Select one of the tabs bellow in order to view the graph you`re
interested in:
But now letīs come back to the physics behind the rope-swing.
If you´re lucky [or bad luck??? ;-)] you know a very large bridge
like this:
Some people enjoy falling a long way down before experiencing
the ropes tension. This brings up the interesting question as to the
maximum g-force that a jumper has to withstand during this stage of
the rope-swing.
I programmed a tool with that it´s possible to analyse
the fall paths in detail. All the seven differential equations may be
solved by the programm numerically by a Cash Carp Algorithm. At first
you have to put in the precise, simulation time, intial time step. After
defining the rope parameters [lenght, cross section, elaticity modul
E1, E2, constant n (Pas)] and the jumper mass you may start the simulation.
Screenshot
of the Rope Swing Simulation programm.
In the next step you may wath the falling paths and all
the forces, speeds etc. in realtime. Annother option is to analyse specific
points of the fall path in detail. [f.e. ]
2.3
Fall Path simulations
The following Simulations show the paths of a 75kg person
jumping with a 27m long standart climbing rope [E1=174MPa,
E2=478MPa, n=86MPas, A=7,57*10^-5m^2] at different
jump distances:
Jump Distance 27m
Jump Distance 24m
Jump Distance 20m
Jump Distance 15m
Jump Distance 10m
Jump Distance 5m
2.4
Analysis of characteristic points
While a rope swing the g-force maxima are interesting.
These graphs show the values for a 75kg person jumping with a 27m long
standart climbing rope [E1=174MPa, E2=478MPa,
n=86MPas, A=7,57*10^-5m^2] at a jump distance of 24m:
2.5 g-Factor calculations
By analysing the maximum g-forces at certain jump distances
one gets the following distributions:
It´s quite interesting that there is a certain jump
distance beween 24m and 25m where the jumping person has
to withstand minimum g-forces during the swing.
So it´s better for dorsal to jump at around 24m.
3.
Rope live-time approximation
Martyn
Pavier [University of Bristol, Department of Mechanical Engineering]
did some Experimental and theoretical simulation of climbing falls.
He found a logaritmic dependence of the maximum tension and
the number of falls to failure [graph on the right hand sight]:
Graph1:
Numerical claculation of the maximum rope force while a rope swing
at different jump distances
Graph2: Maximum tension vs. the
number of falls to failure measured by Martyn
Pavier
Let´s assume a rope swing jumper which has got
a jump distance of around 23-27m while his jumps[rope lenght 27m].
As Graph1 shows the maximum rope force while the
swinging is around 2kN. Comparison of this value with the number of
fails M. Pavier found one may assume that a standart climbing rope
will fail after 50-80 jumps. Depending on the "history" of the rope the number of jumps
to failure may also be even less than 50 jumps. So it is allways
a must to use a redundance rope!
4.
Real jumps
After the simulations it´s time for real jumps.
We had the luck to found a 36m hight bridge...
Three
rope system.
Preparing
of the "Anker point" 36m above the ground
First
jump of my friend Markus [click here to watch
the gif.]
AVI-Film
of my first jump [you may watch with "click"..
]
5.
Discussion
The
calculations and simulations which are presented here are a usefull
to get a first approximation of the forces a rope swing jumper has to
withstand during the swing.
Air resistance and rope mass are not respected in the simulations yet.
Next steps will include:
- Finding the differential equations including air resistance effects
- Multi Rope Analysis - Static Rope Analysis
- Rope Failure Simulations [redundance rope behavior after
a failure of the main rope]
- Experimental studies and fall experiments with steel mass
falling into climbing ropes to get the exact rope parameters