Scaling Single-Pole Telescopes |

September 23, 2006

__Background__: For maximum stiffness, you can’t beat a truss or tube scope. However, both suffer from the

drawbacks of bulk and complexity. Many telescope makers have attempted designs with from one to four

cantilevered poles to simply their scopes and make them more portable. The telescope maker much choose the

number, diameter, length, and cross-section of the pole(s). What are the considerations?

__Method__: I used beam equations in an Excel file to estimate the flexure of various cantilevered poles and

configurations.

The maximum flexure of a cantilevered pole (when the pole is horizontal and loaded at the free end) is given by the

equation

1 WL^3

Ymax = --- -------

3 EI

where W is the load in lbs, here equivalent to the weight of the UTA; L is the unsupported length of the pole in

inches; E is the modulus of elasticity of the material in psi, here assumed to be 10,000,000 psi for aluminum; and I

is the section modulus for the pole in in^4.

The section modulus for a thin-walled tube with a circular cross-section is

I = π r3 t

where r is the “average” radius of the tube in inches (approximately equal to the average of the outer and inner

diameters) and t is the wall thickness in inches. This can be re-expressed in “average” diameters as

π D^3 t

I = -------------

8

An alternative formula is

π (D^4 – d^4)

I = ---------------

64

where D is the outer diameter and d is the inner diameter. For a thin-walled square section tube

W^4 – w^4

I = ------------

12

where W is the outer width and w is the inner width of the tube.

__Conclusions__

1. Choose a large-diameter pole. Wider poles are much better than narrower ones. The stiffness of a pole

increases with the cube of the diameter. If you double the diameter of the pole (maintaining the same wall

thickness and length) the flexure for a given load is reduced by a factor of 8. Even a modest increase in diameter

of only 25% cuts the flexure in half. It is important that poles be as wide as possible.

__2. One pole is best.__ One pole is much better and two or more. Comparing poles of same length and wall

thickness, one pole 2” in diameter has the same weight as two 1” poles or four 0.5” poles. Even though the load is

spread over more members, the single pole has a huge advantage: it is four times as stiff as the two pole

configuration and over twenty times as stiff as the four pole arrangement. One big pole is much stiffer than two or

more poles. The “Alice” and Plettstone scopes would both be stiffer and lighter with single poles.

__3. Keep the length as short as practical.__ Shorter poles are much stiffer than longer poles. The flexure of a pole

increases with the cube of its length. Thus, a slight increase in the length of the pole results in a large increase in

flexure: if you double the length, the pole will deflect eight times as much. This is true even for modest increases in

length: an increase in length of only 25% results doubles the flexure at a given load. In my designs it is best to

keep the bracket that holds the tube as long as practical to keep the tube run short.

__4. Use round tubes.__ Round tubes are stiffer in bending and twisting than square tubes. For a given cross-

sectional area and wall thickness, a square tube bends about 20% more than a circular tube. It also is far less

resistant to torsion: a circular tube is about 1.9 times as resistant to twisting.

Excel table:

Single-tube telescopes: beam calculations

No. tubes OD ID L Total W Supported W E I Ymax Factor

Comparison of tubes of different diameter

1 3 2.875 36 4 4 10000000 0.6224 0.009994855 28.71

1 2.5 2.375 36 4 4 10000000 0.3557 0.017489903 50.24

1 2 1.875 36 4 4 10000000 0.1787 0.034812073 100

1 1.5 1.375 36 4 4 10000000 0.073 0.085165352 244.6

1 1 0.875 36 4 4 10000000 0.0203 0.306243551 879.7

1 0.5 0.375 36 4 4 10000000 0.0021 2.966187541 8521

Comparison of different numbers of tubes with same total cross-sectional area and weight

1 2 1.875 36 4 4 10000000 0.1787 0.034812073 100

2 1 0.875 36 4 2 10000000 0.0203 0.153121776 439.9

4 0.5 0.375 36 4 1 10000000 0.0021 0.741546885 2130

8 0.25 0.125 36 4 0.5 10000000 0.0002 4.325690163 12426

Comparison of tubes of different length

1 2 1.875 18 4 4 10000000 0.1787 0.004351509 12.5

1 2 1.875 36 4 4 10000000 0.1787 0.034812073 100

1 2 1.875 45 4 4 10000000 0.1787 0.06799233 195.3

1 2 1.875 72 4 4 10000000 0.1787 0.278496584 800

1 2 1.875 90 4 4 10000000 0.1787 0.54393864 1562

Comparison of round and square tubes with same cross-sectional area

Round tube

1 2 1.875 36 4 4 10000000 0.1787 0.034812073 100

Square tube

1 1.5985 1.4735 36 4 4 10000000 0.1512 0.041130616 118.2

Comparison of round and square tubes in torsion

A seamless round tube is about 1.87X as resistant to twisting

as a square tube of the same weight and cross-sectional area

(James Ambrose (2002) Simplified Mechanics and Strength of Materials, Wiley, P. 45)

Copyright 2009 Ross Sackett

drawbacks of bulk and complexity. Many telescope makers have attempted designs with from one to four

cantilevered poles to simply their scopes and make them more portable. The telescope maker much choose the

number, diameter, length, and cross-section of the pole(s). What are the considerations?

configurations.

The maximum flexure of a cantilevered pole (when the pole is horizontal and loaded at the free end) is given by the

equation

1 WL^3

Ymax = --- -------

3 EI

where W is the load in lbs, here equivalent to the weight of the UTA; L is the unsupported length of the pole in

inches; E is the modulus of elasticity of the material in psi, here assumed to be 10,000,000 psi for aluminum; and I

is the section modulus for the pole in in^4.

The section modulus for a thin-walled tube with a circular cross-section is

I = π r3 t

where r is the “average” radius of the tube in inches (approximately equal to the average of the outer and inner

diameters) and t is the wall thickness in inches. This can be re-expressed in “average” diameters as

π D^3 t

I = -------------

8

An alternative formula is

π (D^4 – d^4)

I = ---------------

64

where D is the outer diameter and d is the inner diameter. For a thin-walled square section tube

W^4 – w^4

I = ------------

12

where W is the outer width and w is the inner width of the tube.

1. Choose a large-diameter pole

increases with the cube of the diameter. If you double the diameter of the pole (maintaining the same wall

thickness and length) the flexure for a given load is reduced by a factor of 8. Even a modest increase in diameter

of only 25% cuts the flexure in half. It is important that poles be as wide as possible.

thickness, one pole 2” in diameter has the same weight as two 1” poles or four 0.5” poles. Even though the load is

spread over more members, the single pole has a huge advantage: it is four times as stiff as the two pole

configuration and over twenty times as stiff as the four pole arrangement. One big pole is much stiffer than two or

more poles. The “Alice” and Plettstone scopes would both be stiffer and lighter with single poles.

increases with the cube of its length. Thus, a slight increase in the length of the pole results in a large increase in

flexure: if you double the length, the pole will deflect eight times as much. This is true even for modest increases in

length: an increase in length of only 25% results doubles the flexure at a given load. In my designs it is best to

keep the bracket that holds the tube as long as practical to keep the tube run short.

sectional area and wall thickness, a square tube bends about 20% more than a circular tube. It also is far less

resistant to torsion: a circular tube is about 1.9 times as resistant to twisting.

Excel table:

Single-tube telescopes: beam calculations

No. tubes OD ID L Total W Supported W E I Ymax Factor

Comparison of tubes of different diameter

1 3 2.875 36 4 4 10000000 0.6224 0.009994855 28.71

1 2.5 2.375 36 4 4 10000000 0.3557 0.017489903 50.24

1 2 1.875 36 4 4 10000000 0.1787 0.034812073 100

1 1.5 1.375 36 4 4 10000000 0.073 0.085165352 244.6

1 1 0.875 36 4 4 10000000 0.0203 0.306243551 879.7

1 0.5 0.375 36 4 4 10000000 0.0021 2.966187541 8521

Comparison of different numbers of tubes with same total cross-sectional area and weight

1 2 1.875 36 4 4 10000000 0.1787 0.034812073 100

2 1 0.875 36 4 2 10000000 0.0203 0.153121776 439.9

4 0.5 0.375 36 4 1 10000000 0.0021 0.741546885 2130

8 0.25 0.125 36 4 0.5 10000000 0.0002 4.325690163 12426

Comparison of tubes of different length

1 2 1.875 18 4 4 10000000 0.1787 0.004351509 12.5

1 2 1.875 36 4 4 10000000 0.1787 0.034812073 100

1 2 1.875 45 4 4 10000000 0.1787 0.06799233 195.3

1 2 1.875 72 4 4 10000000 0.1787 0.278496584 800

1 2 1.875 90 4 4 10000000 0.1787 0.54393864 1562

Comparison of round and square tubes with same cross-sectional area

Round tube

1 2 1.875 36 4 4 10000000 0.1787 0.034812073 100

Square tube

1 1.5985 1.4735 36 4 4 10000000 0.1512 0.041130616 118.2

Comparison of round and square tubes in torsion

A seamless round tube is about 1.87X as resistant to twisting

as a square tube of the same weight and cross-sectional area

(James Ambrose (2002) Simplified Mechanics and Strength of Materials, Wiley, P. 45)

Copyright 2009 Ross Sackett

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