Another Approach to Rope Sag - Rope Sections and Numerical CalculationConsider a rope spanning a gap of known width, X, and attached at the same height on either side. The rope can be treated as divided into equal sections with the weight of each section acting at its center of mass. A rope of known total length, L, and total weight, W, if divided into, for example, five sections, then each section would be of length, L/5, and weight, W/5. If the rope were to be laid flat and the value x = 0 assigned to one end, the centers of mass of these equal sections would be at positions x = L/10, 3L/10, 5L/10, 7L/10 and 9L/10. A weight of W/5 will be placed at each of these positions on this otherwise "weightless" rope. See the first attached picture.

This configuration will be approached as a typical force statics problem summing the vertical and horizontal forces at each point to zero. The origin will be taken as the upper left-hand point of rope attachment. The x-axis will be horizontal and positive to the right. The y-axis will be vertical and positive

downward.

There are some simplifications due to symmetry. For 5 sections, positions #5 and #1 are equivalent as are positions #4 and #2 and provide no additional information or constraints. Odd numbers and even numbers produce slightly different sets of equations that must be solved. The examples below will be for an odd number of sections which also happen to have a center of mass point hanging at the midpoint of the span where x = X/2. Also used is the fact that the distances betwen adjacent points are known. If (x

_{1},y

_{1}) corresponds to point #1 and (x

_{2},y

_{2}) corresponds to point #2, then the distance between them which is equal to L/5 is equivalent to SquareRoot[ (x

_{2} - x

_{1})

^{2} + (y

_{2} - y

_{1})

^{2} ].

The problem will be solved by guessing the x-value of the first point, x

_{1}, and calculating the remaining quantities. The x-value of the point in the middle of the span (point #3 for the 5-point configuration), that is, x

_{3}, will be compared to the value X/2 and the process repeated until they are essentially equal. The Excel spreadsheet program has an Add-in software called "Analysis Toolpak" and "Solver" which does this nicely. In Excel, this is loaded from the FILE > OPTIONS > ADD-INS screen and afterwards is accessed from the Excel "Data" tab.

Sum of ForcesIn reference to the first attached picture, T represents the tensions in the rope, θ, the angles, and as before, W, the total weight of the rope. The sum of horizontal forces at point #1 gives T

_{1}*sin(θ

_{1}) = T

_{2}*sin(θ

_{2}). Similarly, the sum of horizontal forces at point #2 gives T

_{2}*sin(θ

_{2}) = T

_{3}*sin(θ

_{3}).

In general, T

_{1}*sin(θ

_{1}) = T

_{2}*sin(θ

_{2}) = T

_{3}*sin(θ

_{3}) = . . . etc. independent of the number of points.

For the summing of vertical forces, it is best to start at the midpoint, that is, point #3 for the 5-point configuration. Here the sum of forces gives T

_{3}*cos(θ

_{3}) = (W/5)/2 = W/10. We divide the load by two because there is an equal contribution from T4 which by symmetry, exactly equals T3. The summing of vertical forces at point #2 gives T

_{2}*cos(θ

_{2}) = T

_{3}*cos(θ

_{3}) + W/5, but, T

_{3}*cos(θ

_{3}) was just determined. Therefore, T

_{2}*cos(θ

_{2}) = 3W/10. Likewise, T

_{1}*cos(θ

_{1}) = 5W/10.

In summary for the vertical components of force for the 5-point configuration,

T

_{1}*cos(θ

_{1}) = 5W/10

T

_{2}*cos(θ

_{2}) = 3W/10

T

_{3}*cos(θ

_{3}) = 1W/10

In general, for any odd number of points = Npts, T

_{i}*cos(θ

_{i}) = [Npts - 2*(i - 1)]*W/(2*Npts) for point #1 to the midpoint.

Example: 21-Point ConfigurationThe 21-point configuration requires consideration of 11 points. The 11 columns of Excel data fit into a full screen without scrolling. Since the formulas repeat for column 3 and upward, this process should be relatively straightforward to extend to a higher number of points using Excel's auto-fill feature.

First the sheet viewing the formulas will be given followed by the sheet viewing the calculated values. The variable names were defined using the FORMULAS Tab > CREATE FROM SELECTION feature of Excel. The columns data was highlighted before the "CREATE FROM NAME" feature was used so that the header variables to the left would be set up as arrays.

First the formulas. The starting parameters were chosen to equal the parameters in the original post.

NPts | | 21 | | # of rope sections |

WtPerLength | | 0.80 | | wt. per length of the rope |

LRope | | 29.69460 | | total length of the rope |

| | | | |

XWidth | | 20.0 | | distance for rope to span |

Weight | | =WtPerLength*LRope/NPts | | rope weight per section |

| | | | |

T1SinAlpha | | =TensionSinAngle | | tension in first rope section times sin(alpha) |

Index | | 0 | | 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | 7 | | 8 | | 9 | | 10 |

Xpos | | 0.2745 | | =DelX+XposPrev | | * | | * | | * | | * | | * | | * | | * | | * | | * |

Ypos | | =SQRT(Length^2-Xpos^2) | | =DelY+YposPrev | | * | | * | | * | | * | | * | | * | | * | | * | | * |

XposPrev | | 0.0000 | | =INDEX(Xpos,1,Index) | | * | | * | | * | | * | | * | | * | | * | | * | | * |

YposPrev | | 0.0000 | | =INDEX(Ypos,1,Index) | | * | | * | | * | | * | | * | | * | | * | | * | | * |

DelX | | =Xpos-XposPrev | | =SQRT(Length^2/(1+(1/TanAngle^2))) | | * | | * | | * | | * | | * | | * | | * | | * | | * |

DelY | | =Ypos-YposPrev | | =SQRT(Length^2-DelX^2) | | S | | S | | S | | S | | S | | S | | S | | S | | S |

Length | | =LRope/(2*NPts) | | =LRope/NPts | | A | | A | | A | | A | | A | | A | | A | | A | | A |

Angle | | =ATAN(TanAngle) | | =ATAN(TanAngle) | | M | | M | | M | | M | | M | | M | | M | | M | | M |

SinAngle | | =SIN(Angle) | | =SIN(Angle) | | E | | E | | E | | E | | E | | E | | E | | E | | E |

CosAngle | | =COS(Angle) | | =COS(Angle) | | * | | * | | * | | * | | * | | * | | * | | * | | * |

TanAngle | | =Xpos/Ypos | | =T1SinAlpha/(WeightCos/2) | | * | | * | | * | | * | | * | | * | | * | | * | | * |

WeightCos | | =(NPts-2*Index)*Weight | | =(NPts-2*Index)*Weight | | * | | * | | * | | * | | * | | * | | * | | * | | * |

Tension | | =(WeightCos/2)/CosAngle | | =(WeightCos/2)/CosAngle | | * | | * | | * | | * | | * | | * | | * | | * | | * |

TensionSinAngle | | =Tension*SinAngle | | =Tension*SinAngle | | * | | * | | * | | * | | * | | * | | * | | * | | * |

2*TensionCosAngle | | =2*Tension*CosAngle | | =2*Tension*CosAngle | | * | | * | | * | | * | | * | | * | | * | | * | | * |

Below are the calculations.

NPts | | 21 | | # of rope sections |

WtPerLength | | 0.80 | | wt. per length of the rope |

LRope | | 29.69460 | | total length of the rope |

| | | | |

XWidth | | 20.0 | | distance for rope to span |

Weight | | 1.1312 | | rope weight per section |

| | | | |

T1SinAlpha | | 5.0044 | | tension in first rope section times sin(alpha) |

Index | | 0 | | 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | 7 | | 8 | | 9 | | 10 |

Xpos | | 0.2745 | | 0.8714 | | 1.5242 | | 2.2426 | | 3.0382 | | 3.9245 | | 4.9158 | | 6.0247 | | 7.2558 | | 8.5949 | | 10.000 |

Ypos | | 0.6515 | | 1.9334 | | 3.1877 | | 4.4057 | | 5.5746 | | 6.6765 | | 7.6848 | | 8.5622 | | 9.2579 | | 9.7119 | | 9.8708 |

XposPrev | | 0.0000 | | 0.2745 | | 0.8714 | | 1.5242 | | 2.2426 | | 3.0382 | | 3.9245 | | 4.9158 | | 6.0247 | | 7.2558 | | 8.5949 |

YposPrev | | 0.0000 | | 0.6515 | | 1.9334 | | 3.1877 | | 4.4057 | | 5.5746 | | 6.6765 | | 7.6848 | | 8.5622 | | 9.2579 | | 9.7119 |

DelX | | 0.2745 | | 0.5969 | | 0.6528 | | 0.7184 | | 0.7956 | | 0.8862 | | 0.9913 | | 1.1089 | | 1.2311 | | 1.3391 | | 1.4051 |

DelY | | 0.6515 | | 1.2819 | | 1.2543 | | 1.2179 | | 1.1690 | | 1.1018 | | 1.0084 | | 0.8773 | | 0.6957 | | 0.4541 | | 0.1588 |

Length | | 0.7070 | | 1.4140 | | 1.4140 | | 1.4140 | | 1.4140 | | 1.4140 | | 1.4140 | | 1.4140 | | 1.4140 | | 1.4140 | | 1.4140 |

Angle | | 0.3987 | | 0.4358 | | 0.4799 | | 0.5329 | | 0.5976 | | 0.6774 | | 0.7769 | | 0.9015 | | 1.0564 | | 1.2439 | | 1.4583 |

SinAngle | | 0.3883 | | 0.4221 | | 0.4617 | | 0.5081 | | 0.5626 | | 0.6268 | | 0.7010 | | 0.7842 | | 0.8706 | | 0.9470 | | 0.9937 |

CosAngle | | 0.9215 | | 0.9065 | | 0.8871 | | 0.8613 | | 0.8267 | | 0.7792 | | 0.7131 | | 0.6205 | | 0.4920 | | 0.3211 | | 0.1123 |

TanAngle | | 0.4213 | | 0.4657 | | 0.5205 | | 0.5898 | | 0.6806 | | 0.8043 | | 0.9831 | | 1.2640 | | 1.7695 | | 2.9492 | | 8.8477 |

WeightCos | | 23.756 | | 21.493 | | 19.231 | | 16.968 | | 14.706 | | 12.443 | | 10.181 | | 7.9186 | | 5.6561 | | 3.3937 | | 1.1312 |

Tension | | 12.889 | | 11.855 | | 10.840 | | 9.8501 | | 8.8944 | | 7.9846 | | 7.1384 | | 6.3812 | | 5.7482 | | 5.2842 | | 5.0362 |

TensionSinAngle | | 5.0044 | | 5.0044 | | 5.0044 | | 5.0044 | | 5.0044 | | 5.0044 | | 5.0044 | | 5.0044 | | 5.0044 | | 5.0044 | | 5.0044 |

2*TensionCosAngle | | 23.756 | | 21.493 | | 19.231 | | 16.968 | | 14.706 | | 12.443 | | 10.181 | | 7.9186 | | 5.6561 | | 3.3937 | | 1.1312 |

The second picture shows a comparison of the 21-point configuration fit to a catenary. The parameter "a" was 6.25 in the original post.

The third picture shows a comparison between 21-point and 5-point configurations.