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# Free undamped linear oscillations of a sagging cable and a discrete chain

Posted 10 months ago

Free undamped linear oscillations of a sagging cable and a discrete chain It’s a story of differentiation and first-order approximations - again, and again, and again... Rauan Kaldybayev The work uses Newton’s second law to analyze small oscillations of a catenary cable. Rigidity, air resistance, friction, and stretching or compression due to axial stress are neglected. The aim was to compare the oscillations of a catenary cable to those of a tight straight string. It was expected to find dispersion and nonlocality, and these effects were indeed observed. One unexpected finding was that the partial differential equation governing in-plane oscillations is of fourth-order and includes mixed space and position derivatives. The work also studies the oscillations of a hanging discrete chain; it was found that a chain with a large number of elements oscillates like a continuous cable. To check the correctness of the work, the predicted natural frequencies were compared to those given by John Gregg Gale in his 1976 master’s thesis. The results matched, suggesting that the current computational essay is most likely correct. The work is useful primarily from the theoretical standpoint to better understand the physics of oscillations, as the systems explored here exhibit exotic behaviors like nonlocality. Empirical observations made here lead to a general conjecture regarding oscillatory systems, namely the existence of “natural coordinates” (explained later in the abstract). The essay can also be used to estimate the frequencies of conductor galloping, which is a major question in engineering. The first novelty of this essay is that it uses a unique approach that is based on curvilinear coordinates, as opposed to past works, which used a more visual approach. The present work is more formal: it starts with Newton’s second law and the incompressibility condition and transforms the equations in various ways using geometrical identities. The equations of motion yielded by this method are written in terms of the cable’s perpendicular displacement only, as opposed to other works, which also used quantities like angles. This can make the physical meaning of the equations more transparent. The two PDEs have variable coefficients and include second-order time derivatives. The first equation describes horizontal oscillations and includes second-order position derivatives. The second equation describes in-plane oscillations and involves fourth-order position derivatives and mixed derivatives. The PDEs are independent. Arc length, displacement of the cable from equilibrium, and Cartesian coordinates are the most intuitive coordinates for the problem. Turns out, they also have convenient mathematical properties. The normal modes of horizontal oscillations are orthogonal when arc length is used as the spatial coordinate. The equations of motion of the cable are independent when written in terms of the cable’s horizontal and in-plane displacement from equilibrium. The PDEs describing the cable’s oscillations have polynomial coefficients when the vertical coordinate z is used as the spatial coordinate. The PDE describing the normal mods of horizontal oscillations takes the simple form ψ''[χ]+ 2 ϖ 2 d 2 dt 2 d 2 dt θ φ N N-1 N∈ + Z Y K Y K 2 d 2 dt 2 d 2 dt θ φ Y K Introduction
Why is this interesting? The original motivation for this work came from the YouTube video where a stay cable was hit with a wrench and produced awesome Star Wars blaster sounds. I looked up some articles on this topic and was immediately fascinated - not by the sound, however, but by the oscillations themselves. A vibrating string - a classic example of an oscillatory system. The system is governed by the wave equations 2 ∂ ∂ 2 t 2 ∂ ∂ 2 x
Cable in equilibrium; Cartesian coordinates A cable is hung between two fixed points - let’s call them O P Out[]= To describe the system, we can use Cartesian coordinates where the point O z x O P x-z z e z e x- x 0 Λ x 0 Λ ( 1 ) Here Λ x 0 O P e z e x-z y e ( 2 ) The meaning of the functions y e z e x,y,z y[x] z[x] y e z e
Assumptions This work neglects friction and air resistance, the cable’s stiffness, and stretching and compression due to axial stresses. From physical intuition, we can guess that air resistance is negligible for massive cables. A small grain of dust is easily dragged by slightest movements of air, while a large boulder can only be moved by powerful storms. The stiffness of the cable can also be neglected for large cables. When Titanic capsized and its bow rose into the air, the ship broke in half under its own weight; the force of gravity dominated the structure’s strength. On the other hand, a smaller model of the same ship can easily support itself; elastic forces are much stronger than the force of gravity. This shows that the larger the object, the more is the degree to which gravity dominates elastic forces. Again, for sufficiently large cables, we can neglect stiffness altogether. On the left, you see an illustration of Titanic breaking in half due to the mechanical stresses (taken from https://www.britannica.com/story/timeline-of-the-titanics-final-hours). On the right is shown a smaller model of Titanic; the model can easily support its weight even though it sits on a narrow base. The third assumption is that we can neglect compression and stretching due to axial stresses. To see that this approximation is valid, let us use Hooke’s law, which states that the relative elongation of a small piece of cable is given by ε=σ/E σ E σ σ~ρgL ρ g L ε ε~ ρgL E ε -3 10 L 2500 ρ=8000 g=10 E=2· 11 10 2500
Small oscillations The current work investigates small oscillations, when the displacement from equilibrium is very small. Mathematically, we describe the cable’s shape using some set of variables, or generalized coordinates. For each generalized coordinate q q= q e ( 3 ) where q q e q ε |