1
Introduction
This report discusses a “Catenary cable” problem, where the equation for the problem is
given. Using the recommended references and given values it is possible to compute
tension in a cable and other values such as potential energy and the cable's lowest
point. The most important tool given is MATLAB, the programme provides what is needed
when calculating and computing the tasks given. Deriving the given equations is also a
central part in the report.
1.1 Problem Statement
This study aims to understand the shape, tension distribution, and mechanical behavior
of a catenary cable hanging under its own weight. The goal is to determine the cable’s
equilibrium shape between two fixed points by solving a nonlinear boundary value
problem (BVP).
To solve this problem, both numerical and analytical methods are used. The numerical
solution is obtain explores the tension distribution of the cable, its support reactions at
A and B, and the total potential energy of the cable.
2
Background
this study will indulge in freely hanging cable, which is more commonly know as a
catenary. A catenary has a important role in engineering applications. it is for example
seen in applications like power lines . However unlike similar parabolic approximation
the type discussed in this report follows a second-order differential equations which
derive from equilibrium conditions. the catenary problem that will be investigated by
reformulating the second-order differential equation to two first order equation. by doing
this it is possible to solve the equations numerically using matlab “bvp4c”. after solving
it the analytical solution is found and compared to the numerical one to confirm
accuracy
another aspect with mentioning is that along the cable the horizontal tension is
constant, however the vertical tension varies throughout the cable, this is caused by the
weight of the cable. the understanding of thece force is key in structural engineering.
these types of calculations are critical for optimizing the safety and material efficiency
of applications.
2.1
Purpose of the lab
In this problem the goal is to investigate the behaviour of a catenary cable’s tension,
support reactions, potential energy, and to determine its equilibrium shape. This is done
by a numerical and analytical calculation.This analysis provides insights into the
structural integrity and performance of cable systems, which are essential in
engineering applications such as bridges, suspension structures, and power lines. This
means the catenary cable is a great introduction into common engineering problems.
3
Resultat
3.1 Boundary value problem
the values for our group (6) was as following:
xA=0 (start point of the cable x value)
yA=0 (start point of the cable y value)
xB = 2.5 (right point x value)
yB = 1.17 (right point y value)
l (m) = 3.45 (length of the cable)
μ (N/m) = 10.0 (weight per meter)
𝑇 0 = 9.7369523753737539 N (horizontal tension in the lowest point of the cable)
we started by determining the second order differential:
𝑑2𝑦 𝜇
𝑑𝑦
= ∗ √1 + ( )2
2
𝑑𝑥
𝑇0
𝑑𝑥
T0 is the horizontal tension in the cable at its lowest point. Now the way to solve this
second- order differential equation is to turn it into two first-order equations. To do that
we can introduce these two new variables
𝑦1 = 𝑦
𝑦2 = 𝑦′
so we have a new system:
𝑑𝑦1
= 𝑦2
𝑑𝑥
𝑑𝑦2
𝜇
= ∗ √1 + 𝑦2 2
𝑑𝑥
𝑇0
we know that y(2.5) = 1.17, however we do not know the initial slope at x = 0 which in our
new system is y2(0). to solve this numerically we need to guess a value for y2(0) then
integrating that to x = 2.5 and see if it matches the value of y at x = 2.5 which is 1.17. if we
do not get that we need to change our initial guess and repeat the process.
after doing so we concluded that the value of y2(0) = -1.0708. now that we have that we
can now check if it meets the length constraint by putting it into this equation:
𝐿 = 𝑥𝑎 , 𝑥𝑏 ∫ √1 + (𝑦2)2
this equation is the equation for curve length. by checking that this is 3.45m we can
ensure that we have satisfied the length constraint and the boundary condition.
3.2 Numerical solution
The numerical solution was obtained using MATLAB’s bvp4c solver, a built-in function
specifically designed for solving boundary value problems for ordinary differential
equations. Since the catenary equation is a second-order differential equation, it was
first reformulated as a system of two first-order equations. This allowed the use of
numerical solvers that require first-order systems as input.
To implement the numerical solution, the boundary conditions were defined at the two
endpoints of the cable, ensuring that the computed curve passed through the given
anchor points. A function defining these conditions was provided to MATLAB's solver.
Additionally, an initial guess for the solution was specified, which is a crucial step in
boundary value problems, as it helps guide the iterative solver toward a physically
meaningful solution.
By using this numerical approach, it was possible to approximate the cable’s profile
accurately without requiring the explicit analytical solution, demonstrating the power of
computational methods in engineering applications. Using the solver, the values
obtained in task one, and the given values the function could be plotted to visualise the
shape of the catenary cable.
𝑅 = 𝜇𝑑𝑠
𝑑2𝑦 𝑤
𝜇 𝑑𝑠
𝑤𝑑𝑥 ⇒ 2 = =
𝑑𝑥
𝑇0 𝑇0 𝑑𝑥
3.3 Analytical Solution
The given secondary differential:
𝑑2𝑦 𝜇
𝑑𝑦 2
√
=
∗
1
+
(
)
𝑑𝑥 2 𝑇0
𝑑𝑥
was derived by this derivation:
𝑅 = 𝜇𝑑𝑠
𝑑2𝑦 𝑤
𝜇 𝑑𝑠
𝑤𝑑𝑥 ⇒ 2 = =
𝑑𝑥
𝑇0 𝑇0 𝑑𝑥
by using the (12.12 equation) in Jonas Månssons book Endimensionell Analys.
From previously computing the boundary values by doing a numerical calculation it was
possible to use those in order to plot the shape of the cable. To validate the numerical
approach, the exact analytical solution was computed and plotted alongside the
numerical results. The comparison showed strong agreement between the two
solutions, confirming that the numerical method effectively captures the cable's shape.
Small deviations may arise due to discretization errors in the numerical solver, but these
were within an acceptable tolerance range.
3.4 Tension in the cable
The tension in the cable at any position is givetn by:
𝑑𝑦
𝑇(𝑥) = (𝑇0 2 + (𝑇0 ∗ )2 )1/2
𝑑𝑥
where the tension is a function of position x.
Since the slope varies along the cable length, the tension is not uniform. A MATLAB
script was used to compute and plot , showing how the tension increases with the
vertical displacement of the cable.
The forces exerted by the supports at and were determined by evaluating at these
endpoints. The total reaction forces include both horizontal and vertical components,
given by:
𝑇𝐴 = T(𝑥𝑎 ) 𝑇𝑏 =T(𝑥𝑏 )
This analysis provides insight into the forces that must be counteracted at the mounting
points to maintain equilibrium.
The cable shape is determined with this function:
a*(cosh(a^(-1)*x)-1)+𝒚𝟎 .
In this case the value of a is T0/μ. y_func is what calculates the height of the cable
depending on its position. s_func is the arc length. x0 and y0 is the unknown lowest part
of the cable. the f function is what ensures that the cable total length matches the given
length and that the cable starts at xA and yA and stops at xB and yB. to then numerically
solve this system we have the fsolve functions to get the coordinates of the lowest point
of the cable. We have the results stored in z. Finally, the last part is about plotting the
cable with a smooth curve with the right lowest, start and end points.
The tension in the Cable at TA and TB turnt out to be:
𝑇𝐴 =14.2669 N
𝑇𝑏 =25.9669 N
3.5 Potential Energy
We have the total gravitational potential energy Ep of the catenary cable which was
calculated by integrating the potential energy density along its length. As previously
discussed the cable has a uniform mass distribution per unit length μ and is subjected
to gravity g, thus the potential energy at any small segment dx is given by:
𝑑𝐸𝑝 = 𝜇𝑔𝑦(𝑥)𝑑𝑥
To obtain the total potential energy, we integrate over the entire cable:
𝐸𝑝 = ∫ 𝑦𝑔𝑦(𝑥)𝑑𝑥
Since an analytical solution to this integral is complex, we used a numerical approach.
MATLAB’s trapz function was employed to approximate the integral by summing the
discrete values of y(x) over the range of X.
The results that matlab calculated was that Ep in the cable was:
𝐸𝑝 = -20.3174 J
4
Discussion
The results of this study highlight the effectiveness of numerical solvers in addressing
complex, nonlinear boundary value problems. The close agreement between the
numerical and analytical solutions confirms the reliability of MATLAB’s bvp4c solver in
capturing the behavior of the catenary cable. There were some deviations found due to
discretization and the solver’s iterative nature, but these errors remain within
acceptable limits.
One important thing to highlight is the nonuniform distribution of tension in the cable.
The highest tension occurs at the anchor points, which is logical since that is where the
force and tension is applied. Reasonably the lowest tension level will be at the cable's
lowest point. This behaviour aligns with the physical expectations.
Potential improvements to this study could involve analyzing cases with varying mass
distributions or incorporating dynamic loading effects, such as wind or oscillatory
forces. Additionally, using higher-order numerical methods or refining the discretization
could further improve the accuracy of numerical solutions.
5
Conclusion
This study analyzed a catenary cable using both numerical and analytical methods. The
numerical solution was obtained by MATLAB’s bvp4c solver, and compared to the exact
analytical solution, which showed agreement between both of them. Additionally, the
tension distribution, support reactions, and total potential energy were computed,
providing a comprehensive understanding of the system’s mechanical behavior.
The findings highlight the effectiveness of numerical solvers in addressing nonlinear
boundary value problems, demonstrating their utility in engineering applications such as
bridge cables, suspension systems, and power lines. By comparing numerical and
analytical solutions, this study underscores the importance of validating computational
approaches against known mathematical models.
6 Referenser
[1] Månsson J, Nordbeck P. (Eq 12.12) Endimensionell Analys. 2011.
[2] L.G.Kraige, J N Bolton, James L Meriam. Meriam Engineering Mechanics : Statics.
2020.