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Rauan Kaldybaev

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

Rauan Kaldybaev

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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 Cosh[χ]ψ[χ]=0 when the horizontal coordinate x is used as the spatial coordinate. Maybe in general, every system has its own “natural coordinates”, which are most convenient when studying the system and can be guessed from physical intuition? Having derived the equations of motion, the work explores their behavior in the limiting cases, when the cable is hung very tightly (e.g. its ends are pulled far from each other and the cable straightens out) and when the cable has a large sag (e.g. when its endpoints are brought together, almost touching each other). While the equations are generally complicated and couldn’t be solved analytically, their behavior in these limiting cases is comparatively simple. For a lax cable, the normal modes can be expressed through Bessel functions. For a tight cable, an approximate algebraic formula for the natural frequencies and normal modes is given. The formula is quick to compute numerically but can be difficult to analyze due to the presence of hypergeometric functions. Saxon and Cahn’s 1953 article gives a different estimate for the natural frequencies. Empirically, it was observed that the normal modes have greater curvature and amplitude near the cable’s lowest point, and near the edges, they flatten out and decrease in amplitude. This is because tension is lowest at the bottom of the cable and increases from there. The equations of horizontal and in-plane oscillations of a sagging discrete chain are, respectively, 2 d 2 dt θ=-Yθ , 2 d 2 dt φ=-Kφ , where θ and φ are vectors of dimension N and N-1 , respectively, for N∈ + Z , and Y and K are square matrices of corresponding dimensions. It was empirically observed that a chain with a large number of elements oscillates like a continuous cable – it has the same natural frequencies and normal modes, and the eigenvalues of the matrices Y and K converge to certain limits. In addition, the essay features many visualizations and illustrations (including interactive animations), which make the content more intuitive. It also includes Wolfram Language code, which can often give a clearer picture, for example, when illustrating a numerical method. To study the oscillations of a catenary cable, the essay starts by finding the cable’s equilibrium configuration; in addition to mathematical formulas, it presents Wolfram Language functions to automatically compute the cable’s shape. Curvilinear coordinates fitted to the cable’s equilibrium shape are defined. Two PDEs describing the cable’s small oscillations are derived from Newton’s second law, incompressibility condition, and geometrical identities. The PDEs are non-dimensionalized and solved as a superposition of their normal modes. The normal modes could not be found analytically, so an approach based on Frobenius’s method was used. The solution was calculated automatically using frobeniusDSolve, a Wolfram Language function was written as a part of the project. The function has many possible applications in other problems. To study the oscillations of a discrete chain, Newton’s second law is first written, and the equilibrium is found; Wolfram Language functions are presented alongside mathematical formulas. Newton’s 2nd law is linearized to explore small oscillations. Using boundary conditions, two independent ODEs of motion are then obtained: 2 d 2 dt θ=-Yθ , 2 d 2 dt φ=-Kφ , where θ and φ are vectors and Y and K are square matrices. Natural frequencies and normal modes are computed through the eigenvalues and eigenvectors of these matrices. Empirically it is observed that the normal modes and natural frequencies of a discrete chain with a large number of elements are the same as those of a continuous cable. 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 ∂ y ∂ 2 t = 2 ∂ y ∂ 2 x , the normal modes of which are sine functions, and the dispersion relation is linear - simple. If we have two strings with different densities are connected to each other, things get more interesting. A pulse travelling from one string to another would partially reflect at the sewing point because of impedance mismatch. Things are even more interesting when a string is hung vertically and impedance changes continuously. The normal modes are then written through Bessel functions, and dispersion appears. We can see that from a simple string, we can get a wide range of fascinating behaviors. But what if we go a bit further and hang the string from two points that are not on the same vertical, such that the string would be sagging? How would the string oscillate then? There are many effects we would expect: dispersion due to variable tension, pulses “continuously reflecting” when travelling because of changing tension. We might even expect nonlocality - if we pick such a string at one point, the entire string will have to move immediately because otherwise its length would change. As for the normal modes of this system, even trying to imagine them is difficult. The problem discussed in here is interesting first of all from the theoretical standpoint. It can be curious to people interested in the physics of oscillations and to students learning about the topic. As mentioned above, the system features interesting phenomena like dispersion, “continuous reflection”, and nonlocality; in addition, as will be shown below, one of the PDEs describing the cable’s oscillations actually contains fourth-order position derivatives and mixed position and time derivatives, which is quite exotic for Newtonian mechanics. The results reached in this work can also be applied on practice to estimate the normal frequencies of conductor galloping, which is a very relevant issue in engineering. There exist more accurate models of conductor galloping, however, the model presented here is particularly simple because it’s linear; the theory can be helpful as a first introduction. Cable in equilibrium; Cartesian coordinates A cable is hung between two fixed points - let’s call them O and P . The force of gravity pulls the cable down, while the force of tension keeps it in place. In equilibrium, the cable takes the shape that minimizes its potential energy. This shape is known as catenary: Out[]= To describe the system, we can use Cartesian coordinates where the point O is the origin of coordinates, the z axis goes vertically upward (against the force of gravity), and the x axis goes horizontally such that the points O and P lie in the x-z plane. In these coordinates, the equilibrium shape of the cable is given by the function z e [x] (see derivation in the uncut version): z e [x]=ΛCosh x- x 0 Λ -Cosh x 0 Λ  ( 1 ) Here Λ and x 0 are constants determined from the boundary conditions, e.g. are picked such that the curve would pass through the points O and P . The subscript e in z e [x] denotes that the function describes the cable’s equilibrium. In equilibrium, the cable lies in the x-z plane, so y e [x]=0 ( 2 ) The meaning of the functions y e [x] and z e [x] is that if we pick some point on the cable when it’s in equilibrium, then the coordinates x,y,z of this point would be related by (1) and (2). When the cable is displaced from equilibrium, its shape would of course be different, and the functions y[x] and z[x] describing its new shape would be different from y e [x] and z e [x] . 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 , where σ is stress and E is the Young modulus of the cable’s material, say steel. To estimate the order of magnitude of σ , we can use dimensional analysis, which yields that σ~ρgL , where ρ is the density of steel, g is the free fall acceleration, and L is the cable’s length. For ε , we then have ε~ ρgL E . If we want ε to be less than -3 10 , L must be smaller than approximately 2500 meters (with ρ=8000 , g=10 , E=2· 11 10 , all in SI units). The approximation holds for cables under 2500 meters in length, or basically for all cables we would normally see. 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 , we have q= q e +O[ε] ( 3 ) where q stands for some (or any) generalized coordinate, q e is the value of q in equilibrium, and ε is a small number. The current work assumes that for each generalized coordinate, there is exactly one equilibrium value; there is only one equilibrium configuration for a cable hung between two

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

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

Why is this interesting?

Cable in equilibrium; Cartesian coordinates

Assumptions

Small oscillations