Joshua Nava Arts

Further problems in loci

After 1 /6 rev. the position of P «the intersection of the line Pt Pi* and the radius. marked off from 0». This is repeated for the twelve divisions

Fig. 16/1 also shows the beginning of a second cycloid and it can be seen that the change from one cycloid to another is sudden. If any locus is plotted and has an instantaneous change of shape H indicates that there is e cessation of movement Anything that has mass cannot chsnge direction suddenly without first ceasing to move. The point of the circle actually in contact with the line is stationary.

This raises the interesting point that theoretically, a motor car tyre is not moving at all when it is in contact with the roed. This is not true in practice, since the contact between the road and tyre is not a point contact but it does explein why tyres last much longer than would be expected.

At the top of the cycloid, between points 5 and 7. the point P is travelling nearly twice the distance that the centre moves in 1/12 rev. Thus, a jet car travelling at 800 km/h has points on the rim of the tyre moving up to 1,600 km/h—faster than the speed of sound

We now come to an interesting set of curves: the cyctoidal curves, the involute, the Archimedean spiral and the helix.

THE CYCLOID The cycloid is the locus of a point on the circumference of a circle as the drde rolls, without slipping, along a straight Una.

The approach to plotting a cycloid, as with ail problems with loci, is to break down the total movement into a convenient number of parts and consider the conditions at each particular pert Fig. 16/1. We heve found, when considering the circle, that twelve is the most convenient number of divisions. The total disunce that the circle will travel in 1 revolution is FIO, the circumference, and this disunce is also divided into twelve equal parts. When the circle rolls along the line, the locus of the centre will be e fine parallel to the base line and the exact position of the centre will, in turn, be directly above each of the divisions marked off.

If e point P, on the circumference, is now considered, then after the circle has rotated 1/12 of a revolution point P is somewhere along the line P, P,,. The disunce from P to the centre of the circle is still the radius and thus, if the inter-section of the line P, P,, and the radius of the drde, marked off from the new position of the centre O „ is plotted, then this must be the position of the point P after 1 /12 of a revolution.

THE TROCHOID A trochoid is the locus of a point, not on ths circumference of a circle but attached to it when the circle rolls, without slipping, along a straight line.

Again, the technique is similar to that used for plotting the cycloid. The mein difference in this case is that the positions of the line P, PM. P, P,„ etc. are dependent upon the distance of P to the centre O of the rolling circle—not on the radius of the rolling circle as before

This distance PO is also the radius to set on your compasses when plotting the intersections of that radius and the lines P, P«i, P» P,«.etc.

If P is outside the circumference of the rolling circle the curve produced is called a superior trochoid. Fig. 15/4,

SUPERIOR TROCHOID

SUPERIOR TROCHOID

Fig 15/4

If P is inside the circumference of the rolling circle the curve produced is called an infenor trochoid. Fig. 15/5.

The trochoid has relevance to naval architects. Certain inverted trochoids epproximate to the profile of waves and therefore have applications in hull design

The superior trochoid is the locus of the point on the outside rim of a locomotive wheel. It can be seen from Fig. 15/4 that at the beginning of a revolution this point is actually moving backwards. Thus, however quickly a locomotive is moving, some part of the wheel is moving back towards where it came from.

liTERiOR TROCHOlO

liTERiOR TROCHOlO

Fig 15/5

THE INVOLUTE There are several definitions for the involute, none being particularly easy to follow.

An involute is the locus of a point initially on a base circle, which moves so that its straight line distance, along a tangent to the circle, to the tangential point of contact, is equal to the distance along the arc of the circle from the initial point to the instant point of tangeocy.

Alternatively, the involute is the locus of a point on a straight line when the straight line rolls round the circumference of a circle without slipping

The involute is best visualized as tho path traced out by the end of a piece of cotton when the cotton is unrolled from its reel

A quick, but slightly inaccurate, method of plotting an involute is to divide the base circle into 12 parts and draw tangents from tho twelve circumferential divisions. Fig. 15/6. Measure 1/12 of the circumference with dividers. When the line has unrolled 1/12 of the circumference, this distance is stepped out from the tangential point. When the line has unrolled 1 /6 of the circumference, the dividers are stepped out twice. When 1/4 has unrolled the dividers are stepped out three times, etc. When all twelve points have been plotted they are joined together with a neat freehand curve

INVOLUTE (METHOD 1)

INVOLUTE (METHOD 1)

Fig. 15/10 also shows the development of the helix, or TMC HCUX

ROUND SECTION WIRE

Fig 15/11

Fig. 15/10 also shows the development of the helix, or TMC HCUX

Fig 15/11

Drawing a helical spring actually consists of drawing two helices, ooe within another. Although the diameters of the helices differ, their pitch must be the same. Once the points are plotted it is just a question of sorting out which parts of the helices can be seen and which parts are hidden by the thickness of the wire.

For clarity, the thickness of the wire in Fig. 15/11 is 1/4 the pitch of the helix, but if it wasn't a convenient fraction, it would be necessary to set out the pitch twice. The distance between the two pitches would be the thickness of the wire.

ROUND SECTION WIRE

Fig. 15/10

Coiled springs

Most coiled springs are formed on e cylinder and are. therefore, helical. They ere. in fact more often called helical springs than coiled springs. If the spring is to be used in tension, the coils will be close together to allow the spring to stretch. This is the spnng that you will see on spring balances in the science lab. If the spring is to be in compression, the coils will be further apart. These springs can be seen on the suspension of many modern cars, particularly on the front suspension. COILEO springs SQUARE SECTION WIRE

Screw thread projection

A screw thread is helical. Unless the screw thread is drawn at a large scale, it is rarely drawn as a helix-except as an exercise in drawing helices I

A SINGLE START SQUARE THREAD

j PITCH

A TRIPLE START SQUARE THREAD

A SINGLE START SQUARE THREAD

j PITCH

Fig. 15/12

A good example is to draw a »crew thread with a square section. This is exactly the same construction as the coiled spring except that the central core hides much of the construction.

A right-hand screw thread is illustrated in Fig. 15/12. To draw a left-hand screw thread merely plot the ascending points from right to left insteed of from left to right.

Sometimes a double, triple or even e quadruple start thread is seen, particularly on the caps of some containers j PlTCM

A TRIPLE START SQUARE THREAD

j PlTCM

Fig. 15/12

where the top needs to be taken off quickly. A multiple start thread is also seen on the starter pinion of motor cars. Multiple start screw threads ere used where rapid advancement along a shaft is required. When plotting a double start screw thread, two helices are plotted on the same pitch. The first helix starts at point 1 and the second et point 7. If a triple start screw thread is plotted, the starts ere points 1, 9 and 5. Fig. 16/13. If e quedruple start thread is plotted, the starts are points 1.10, 7 and 4.

Exercises 16 (All questions originally set in Imperial units) 1. Fig. 1. shows a circular wheel 50 mm in diameter with a point P attached to its periphery. The wheel rolls without slipping along a perfectly straight track whilst remaining in the same plane. Plot the path of point P for one-half revolution of the wheel on the track. Construct also the normal end tangent to the curve at the position reached after one-third of a revolution of the wheel. Cambridge Local Examinations

Fig 1

2. The views in Fig. 2 represent two discs which roll along AB. Both discs start at the same point and roil in the same direction. Plot the curves for the movement of points p and q and state the perpendicular height of p above AB where q again coincides with thelineAB

Southern Universities'Joint Boerd

3. A wheel of 62 mm diameter rolls without slipping along a straight path. Plot the locus of a point P on the rim of the wheel and initially in contact with the path, for one half revolution of the wheel along the peth. Also construct the tangent normal and centre of curvature at the position reached by the point P after one quarter revolution of the wheel along the path. Cambridge Locel Examinations

4. The driving wheels and coupling rod of a locomotive are shown to a reduced scale in Fig. 3. Draw the locus of any point P on the link AB for one revolution of the driving wheels along the track. University of London School Examinations

5. A piece of string AB, shown in Fig. 4. is wrepped around the cylinder, centre 0. in a clockwise direction. The length of the string is equel to the circumference of the cylinder.

(a) Show, by calculation, the length of the string, correct to the nearest 1 mm. taking n=3.14.

(b) Plot the peth of the end B of the string as it is wrapped round the cylinder, keeping the string taut.

(c) Name the curve you have drawn.

Middlesex Regional Examining Board

6. A cylinder is 48 mm diameter and a piece of string is equal in length to the circumference. One end of the string is attached to a point on the cylinder.

(a) Draw the path of the free end of the string when it is wound round the cylinder in a plane perpendicular to the axis of the cylinder.

(b) In block letters, name the curve produced

(c) From a point 66 mm chord length from the end of the curve (i.e. the free end of the string) construct a tangent to the circle representing the cylinder.

Southern Universities' Joint Boerd

7. A circle 50 mm diameter rests on a horizontal line. Construct the involute to this cirde. making the last point on the curve 211 mm from the point at which the circle makes contact with the horizontal line. Cembridge Locel Examinations

8. P. O and Q are three points in that order on a straight line so that PO - 34 mm and 00 - 21 mm. 0 is the pole of an Archimedean spiral. Q is the nearest point on the curve and P another point on the first convolution of the curve. Draw the Archimedean spiral showing two convolutions.

Southern Universities' Joint Boerd

9. Draw two convolutions of an Archimedean spiral such that in two revolutions the radius increases from 18 mm to 76 mm.

Oxford Locel Exeminetions

10. A piece ol cotton is wrapped around the cylinder shown in Fig. 5. The cotton starts at C and after one turn passes through D, forming a helix. The start ol the helix is shown in the figure. Construct the helix, showing hidden detail.

Middlesex Reg'ionel Exemming Boerd

11. A cylinder, made of transparent meteriel. 88 mm 0/D, 50 mm l/D, and 126 mm long, has its axis parallel to the V.P. Two helical lines merited on its curved surface—one on the outside and the other on the inside—have e common pitch of 63 mm.

Draw the elevation of the cylinder, showing both helices starting from the same radial line end completing two turns. Associated Examining Boerd

12. Draw a longitudinal elevation, accurately projected, showing two turns of a helical spring. The spring is of 100 mm outside diameter, the pitch of the coils is 62 mm and the spring material is of 10 mm diameter. Cembridge Locel Exeminetions

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