سلسلي

A chain hanging from points forms a catenary.
Freely-hanging overhead power lines also form a catenary (most prominently visible with high-voltage lines, and with some imperfection near to the insulators).
الحرير في بيت عنكبوت يشكل العديد من السلسليات المرنة.

منحني سلسلي إنگليزية: catenary أو كاتينا في الفيزياء والهندسة الرياضية هو شكل هندسي نظري يتشكل عند تدلي سلسلة أو سلك عندما تكون طرفيها مثبتين في الأعلى وذلك تحت تأثير قوة الثقالة الأرضية (وزن الجسم المعلق).

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الوصف الرياضي

Catenaries for different values of a


المعادلة العامة

تعطى معادلة شكل المنحني الكاتينري بتابع جيب التمام للقطع الزائد بالشكل:

حيث cosh is the hyperbolic cosine function, and where x is measured from the lowest point.[1] All catenary curves are similar to each other, since changing the parameter a is equivalent to a uniform scaling of the curve.

The Whewell equation for the catenary is[2]

Differentiating gives

and eliminating φ gives the Cesàro equation[3]

The radius of curvature is then

which is the length of the line normal to the curve between it and the x-axis.[4]

العلاقة مع المنحنيات الأخرى

When a parabola is rolled along a straight line, the roulette curve traced by its focus is a catenary.[5] The envelope of the directrix of the parabola is also a catenary.[6] The involute from the vertex, that is the roulette traced by a point starting at the vertex when a line is rolled on a catenary, is the tractrix.[5]

Another roulette, formed by rolling a line on a catenary, is another line. This implies that square wheels can roll perfectly smoothly on a road made of a series of bumps in the shape of an inverted catenary curve. The wheels can be any regular polygon except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.[7]

الخصائص الهندسية

Over any horizontal interval, the ratio of the area under the catenary to its length equals a, independent of the interval selected. The catenary is the only plane curve other than a horizontal line with this property. Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the perpendicular segment connecting the centroid of the curve itself and the x-axis.[8]

العلوم

A moving charge in a uniform electric field travels along a catenary (which tends to a parabola if the charge velocity is much less than the speed of light c).[9]

The surface of revolution with fixed radii at either end that has minimum surface area is a catenary revolved about the x-axis.[5]

التحليل

نموذج السلاسل والعقود

In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the chain is parallel to the chain.[10] The analysis of the curve for an optimal arch is similar except that the forces of tension become forces of compression and everything is inverted.[11] An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium.[12] Equations which define the shape of the curve and the tension of the chain at each point may be derived by a careful inspection of the various forces acting on a segment using the fact that these forces must be in balance if the chain is in static equilibrium.

Let the path followed by the chain be given parametrically by r = (x, y) = (x(s), y(s)) where s represents arc length and r is the position vector. This is the natural parameterization and has the property that

where u is a unit tangent vector.

Diagram of forces acting on a segment of a catenary from c to r. The forces are the tension T0 at c, the tension T at r, and the weight of the chain (0, −λgs). Since the chain is at rest the sum of these forces must be zero.

A differential equation for the curve may be derived as follows.[13] Let c be the lowest point on the chain, called the vertex of the catenary.[14] The slope dy/dx of the curve is zero at c since it is a minimum point. Assume r is to the right of c since the other case is implied by symmetry. The forces acting on the section of the chain from c to r are the tension of the chain at c, the tension of the chain at r, and the weight of the chain. The tension at c is tangent to the curve at c and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written (−T0, 0) where T0 is the magnitude of the force. The tension at r is parallel to the curve at r and pulls the section to the right. The tension at r can be split into two components so it may be written Tu = (T cos φ, T sin φ), where T is the magnitude of the force and φ is the angle between the curve at r and the x-axis (see tangential angle). Finally, the weight of the chain is represented by (0, −λgs) where λ is the mass per unit length, g is the gravitational field strength and s is the length of the segment of chain between c and r.

The chain is in equilibrium so the sum of three forces is 0, therefore

and

and dividing these gives

It is convenient to write

which is the length of chain whose weight is equal in magnitude to the tension at c.[15] Then

is an equation defining the curve.

The horizontal component of the tension, T cos φ = T0 is constant and the vertical component of the tension, T sin φ = λgs is proportional to the length of chain between r and the vertex.[16]

اشتقاق المعادلات للمنحنى

The differential equation given above can be solved to produce equations for the curve.[17]

من

the formula for arc length gives

Then

and

The second of these equations can be integrated to give

and by shifting the position of the x-axis, β can be taken to be 0. Then

The x-axis thus chosen is called the directrix of the catenary.

It follows that the magnitude of the tension at a point (x, y) is T = λgy, which is proportional to the distance between the point and the directrix.[16]

This tension may also be expressed as T = T0 y/a .

The integral of the expression for dx/ds can be found using standard techniques, giving[18]

and, again, by shifting the position of the y-axis, α can be taken to be 0. Then

The y-axis thus chosen passes through the vertex and is called the axis of the catenary.

These results can be used to eliminate s giving


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اشتقاق بديل

The differential equation can be solved using a different approach.[19] From

it follows that

and

Integrating gives,

and

As before, the x and y-axes can be shifted so α and β can be taken to be 0. Then

and taking the reciprocal of both sides

Adding and subtracting the last two equations then gives the solution

and

تحديد المتغيرات

Three catenaries through the same two points, depending on the horizontal force TH.

In general the parameter a is the position of the axis. The equation can be determined in this case as follows:[20]

Relabel if necessary so that P1 is to the left of P2 and let H be the horizontal and v be the vertical distance from P1 to P2. Translate the axes so that the vertex of the catenary lies on the y-axis and its height a is adjusted so the catenary satisfies the standard equation of the curve

and let the coordinates of P1 and P2 be (x1, y1) and (x2, y2) respectively. The curve passes through these points, so the difference of height is

and the length of the curve from P1 to P2 is

When s2v2 is expanded using these expressions the result is

so

This is a transcendental equation in a and must be solved numerically. It can be shown with the methods of calculus[21] that there is at most one solution with a > 0 and so there is at most one position of equilibrium.

However, if both ends of the curve (P1 and P2) are at the same level (y1 = y2), it can be shown that[22]

where L is the total length of the curve between P1 and P2 and h is the sag (vertical distance between P1, P2 and the vertex of the curve).

It can also be shown that

and
where H is the horizontal distance between P1 and P2 which are located at the same level (H = x2x1).

The horizontal traction force at P1 and P2 is TH = aw, where w is the mass per unit length of the chain or cable.

Variational formulation

Consider a chain of length suspended from two points of equal height and at distance . The curve has to minimize its potential energy

and is subject to the constraint

The modified Lagrangian is therefore

where is the Lagrange multiplier to be determined. As the independent variable does not appear in the Lagrangian, we can use the Beltrami identity
where is an integration constant, in order to obtain a first integral

This is an ordinary first order differential equation that can be solved by the method of separation of variables. Its solution is the usual hyperbolic cosine where the parameters are obtained from the constraints.

التعميمات مع قوة رأسية

سلاسل غير منتظمة

If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.[23]

Let w denote the weight per unit length of the chain, then the weight of the chain has magnitude

where the limits of integration are c and r. Balancing forces as in the uniform chain produces

and
and therefore

Differentiation then gives

In terms of φ and the radius of curvature ρ this becomes

Suspension bridge curve

Golden Gate Bridge. Most suspension bridge cables follow a parabolic, not a catenary curve, because the roadway is much heavier than the cable.

A similar analysis can be done to find the curve followed by the cable supporting a suspension bridge with a horizontal roadway.[24] If the weight of the roadway per unit length is w and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable (see the figure in Catenary#Model of chains and arches) from c to r is wx where x is the horizontal distance between c and r. Proceeding as before gives the differential equation

This is solved by simple integration to get

and so the cable follows a parabola. If the weight of the cable and supporting wires is not negligible then the analysis is more complex.[25]

سلسلي متساوي القوة

In a catenary of equal strength, the cable is strengthened according to the magnitude of the tension at each point, so its resistance to breaking is constant along its length. Assuming that the strength of the cable is proportional to its density per unit length, the weight, w, per unit length of the chain can be written T/c, where c is constant, and the analysis for nonuniform chains can be applied.[26]

In this case the equations for tension are

Combining gives

and by differentiation

where ρ is the radius of curvature.

The solution to this is

In this case, the curve has vertical asymptotes and this limits the span to πc. Other relations are

The curve was studied 1826 by Davies Gilbert and, apparently independently, by Gaspard-Gustave Coriolis in 1836.

Recently, it was shown that this type of catenary could act as a building block of electromagnetic metasurface and was known as "catenary of equal phase gradient".[27]

سلسلي مرن

In an elastic catenary, the chain is replaced by a spring which can stretch in response to tension. The spring is assumed to stretch in accordance with Hooke's Law. Specifically, if p is the natural length of a section of spring, then the length of the spring with tension T applied has length

where E is a constant equal to kp, where k is the stiffness of the spring.[28] In the catenary the value of T is variable, but ratio remains valid at a local level, so[29]

The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.[30]

The equations for tension of the spring are

and

from which

where p is the natural length of the segment from c to r and λ0 is the mass per unit length of the spring with no tension and g is the gravitational field strength. Write

so

Then

from which

Integrating gives the parametric equations

Again, the x and y-axes can be shifted so α and β can be taken to be 0. So

are parametric equations for the curve. At the rigid limit where E is large, the shape of the curve reduces to that of a non-elastic chain.


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تعميمات أخرى

Chain under a general force

With no assumptions being made regarding the force G acting on the chain, the following analysis can be made.[31]

First, let T = T(s) be the force of tension as a function of s. The chain is flexible so it can only exert a force parallel to itself. Since tension is defined as the force that the chain exerts on itself, T must be parallel to the chain. In other words,

where T is the magnitude of T and u is the unit tangent vector.

Second, let G = G(s) be the external force per unit length acting on a small segment of a chain as a function of s. The forces acting on the segment of the chain between s and s + Δs are the force of tension T(s + Δs) at one end of the segment, the nearly opposite force T(s) at the other end, and the external force acting on the segment which is approximately GΔs. These forces must balance so

Divide by Δs and take the limit as Δs → 0 to obtain

These equations can be used as the starting point in the analysis of a flexible chain acting under any external force. In the case of the standard catenary, G = (0, −λg) where the chain has mass λ per unit length and g is the gravitational field strength.

هندسة معمارية

قوس تخت كسرى في المدائن كما تبدو اليوم هي تقريباً وليس بالضبط كاتينا.
Arches under the roof of Gaudí's كازا ميلا, Barcelona, Spain that are close to catenaries.
Gaudi's catenary model at Casa Milà

القوس السلسلي هو محل هندسي للنقط التي كل منها تحمل نفس الوزن من مجموع الوزن الكلي للقوس، ولهذا يُستخدم في إنشاء الهياكل المعماريه. الهياكل التي تُصنع بشكل سلسلي تعاني من قوة الشد فقط، كما يحدث في الحبال التي تدعم الجسور معلقه. أو من قوة الضغط عندما الهيكل يبنى على شكل منعكس بالنسبة إلى خط افقي ، كما هو الحال في هياكل القبب (على سبيل المثال قبة في سانت بول في لندن التي صممها روبرت هوك وفي الأقواس التي صممها لاول مرة المهندس الإسباني انطونيو كاودي. بعد ذلك أُستخدم القوس السلسلي في الهندسه المعماريه وفي بناء الجسور (على سبيل المثال في جسور مايلارت وفي السكك الحديديه لكارابيت ).

The Sheffield Winter Garden is enclosed by a series of catenary arches.[32]

انظر أيضاً

الهامش

  1. ^ Weisstein, Eric W. "Catenary". MathWorld--A Wolfram Web Resource. Retrieved 2019-09-21. The parametric equations for the catenary are given by x(t) = t, y(t) = [...] a cosh(t/a), where t=0 corresponds to the vertex [...]
  2. ^ خطأ استشهاد: وسم <ref> غير صحيح؛ لا نص تم توفيره للمراجع المسماة Lockwood122
  3. ^ MathWorld, eq. 7
  4. ^ Routh Art. 444
  5. ^ أ ب ت Yates, Robert C. (1952). Curves and their Properties. NCTM. p. 13.
  6. ^ Yates p. 80
  7. ^ Hall, Leon; Wagon, Stan (1992). "Roads and Wheels". Mathematics Magazine. 65 (5): 283–301. doi:10.2307/2691240. JSTOR 2691240.
  8. ^ Parker, Edward (2010). "A Property Characterizing the Catenary". Mathematics Magazine. 83: 63–64. doi:10.4169/002557010X485120. S2CID 122116662.
  9. ^ Landau, Lev Davidovich (1975). The Classical Theory of Fields. Butterworth-Heinemann. p. 56. ISBN 978-0-7506-2768-9.
  10. ^ Routh Art. 442, p. 316
  11. ^ Church, Irving Porter (1890). Mechanics of Engineering. Wiley. p. 387.
  12. ^ Whewell p. 65
  13. ^ Following Routh Art. 443 p. 316
  14. ^ Routh Art. 443 p. 317
  15. ^ Whewell p. 67
  16. ^ أ ب Routh Art. 443 p. 318
  17. ^ Following Routh Art. 443 p/ 317
  18. ^ Use of hyperbolic functions follows Maurer p. 107
  19. ^ Following Lamb p. 342
  20. ^ Following Todhunter Art. 186
  21. ^ See Routh art. 447
  22. ^ Archived at Ghostarchive and the Wayback Machine: "Chaînette - partie 3 : longueur". YouTube.
  23. ^ Following Routh Art. 450
  24. ^ Following Routh Art. 452
  25. ^ Ira Freeman investigated the case where only the cable and roadway are significant, see the External links section. Routh gives the case where only the supporting wires have significant weight as an exercise.
  26. ^ Following Routh Art. 453
  27. ^ Pu, Mingbo; Li, Xiong; Ma, Xiaoliang; Luo, Xiangang (2015). "Catenary Optics for Achromatic Generation of Perfect Optical Angular Momentum". Science Advances. 1 (9): e1500396. Bibcode:2015SciA....1E0396P. doi:10.1126/sciadv.1500396. PMC 4646797. PMID 26601283.
  28. ^ Routh Art. 489
  29. ^ Routh Art. 494
  30. ^ Following Routh Art. 500
  31. ^ Follows Routh Art. 455
  32. ^ Hymers, Paul (2005). Planning and Building a Conservatory. New Holland. p. 36. ISBN 1843309106.

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