Free undamped linear oscillations of a sagging cable and a discrete chain, uncut
Free undamped linear oscillations of a sagging cable and a discrete chain, uncut
It’s a story of differentiation and first-order approximations - again, and again, and again...
Rauan Kaldybayev
The current computational essay gives details for “Free undamped linear oscillations of a sagging cable and a discrete chain”. While the latter only features the most significant information and omits Wolfram Language code, some of the results, and trivial intermediate calculations, the present essay gives all the details about the work. Because of its size (approximately 190 PDF pages), the current essay hasn’t been finalized and can sometimes contain typos or feature hard-to-read sentences, although the mathematical expressions are correct.
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 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, θ=-Yθ, φ=-Kφ, where and are vectors of dimension and , respectively, for , and and 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 and 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: θ=-Yθ, φ=-Kφ, where and are vectors and and 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.
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Introduction
Why is this interesting?
Why is this interesting?
A general picture
A general picture
Assumptions
Assumptions
Coordinatizing the problem
Coordinatizing the problem
A minute of high-school physics
A minute of high-school physics
Finding the equilibrium
A rope is hung between two poles. What shape does it take in equilibrium? This is the question that this chapter is about - finding [x] and [x]. Since the cable remains in the vertical plane in equilibrium, [x]=0. As for [x], the chapter gives the derivation in two ways: using Newton’s second law and calculus of variations. After that, some supplementary quantities, such as the tension in the cable, are calculated, and a method to fit the curve to match the boundary conditions is presented. Lastly, the chapter gives an interactive visualization of a catenary cable.
y
e
z
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y
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z
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A brief argument why ye[x]=0
A brief argument why [x]=0
y
e
Finding ze[x]
Finding [x]
z
e
Finding supplementary quantities
Finding supplementary quantities
Fitting the constants
Fitting the constants
Visualizing a hanging cable
Visualizing a hanging cable
Curvilinear coordinates
When studying the motion of the cable near equilibrium, instead of it is convenient to use coordinates fitted to the cable’s equilibrium shape. This chapter defines these coordinates and explores their properties.
x,y,z
Definition
Definition
A note about domains of functions
A note about domains of functions
About basis vectors
About basis vectors
Derivatives
Derivatives
Introducing small oscillations
Normal modes of the cable
Oscillations of a discrete chain
Conclusion
References
References
The University of Utah, “Systems of differential equations”, http://www.math.utah.edu/~gustafso/2250systems-de.pdf
John Gregg Gale, Oregon State University, master’s thesis, 1976, https://ir.library.oregonstate.edu/downloads/6395w964x
UCONN, 2018, https://www.phys.uconn.edu/~rozman/Courses/m3410_18s/downloads/hanging-chain.pdf
Kaldybayev Rauan, 2020, https://community.wolfram.com/groups/-/m/t/2140007?p_p_auth=Jae8g07P
Titanic image from section “Introduction/Assumptions” taken from https://www.planetminecraft.com/project/rms-titanic-sinking-at-218-am-breaking-in-half/
John Gregg Gale, Oregon State University, master’s thesis, 1976, https://ir.library.oregonstate.edu/downloads/6395w964x
UCONN, 2018, https://www.phys.uconn.edu/~rozman/Courses/m3410_18s/downloads/hanging-chain.pdf
Kaldybayev Rauan, 2020, https://community.wolfram.com/groups/-/m/t/2140007?p_p_auth=Jae8g07P
Titanic image from section “Introduction/Assumptions” taken from https://www.planetminecraft.com/project/rms-titanic-sinking-at-218-am-breaking-in-half/
Bibliography
Bibliography
John Gregg Gale, Oregon State University, master’s thesis, 1976, https://ir.library.oregonstate.edu/downloads/6395w964x
UCONN, “Oscillations of a hanging chain”, 2018, https://www.phys.uconn.edu/~rozman/Courses/m3410_18s/downloads/hanging-chain.pdf
Saxon and Cahn, “Vibrations of a suspended chain”, 1953, The Quarterly Journal of Mechanics and Applied Mathematics, Oxford Academic
UCONN, “Oscillations of a hanging chain”, 2018, https://www.phys.uconn.edu/~rozman/Courses/m3410_18s/downloads/hanging-chain.pdf
Saxon and Cahn, “Vibrations of a suspended chain”, 1953, The Quarterly Journal of Mechanics and Applied Mathematics, Oxford Academic