184.9.1 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



107 



forward the railway. He showed that the canal owners had raised 

 the freight of corn' from 6.?. 84. per ton to 124-. 6d., and cotton from 

 6.?. 8rf, to 15.?.; indeed they had done all tliey could to strangle the 

 growing cotton-trade. The freight in 1S22 was thrice what it was 

 in 1795. In April 1822, the corn-dealers of Liverpool had sent 

 to the Duke of Bridgewater's trustees, hegging the freights on 

 corn might be lowered; but this was flatly refused. Tlius the 

 traders of Li\'erpool were quite ready to welcome the proposal of 

 a railway, which moreover made the length between tlie two towns 

 33 miles, instead of .50 by water, which saved water risks and 

 wrecks, and gave greater speed. 



('To he continued.) 



NOTES ON ENGINEERING.— No. XIII. 

 By HOMERSHAM C'ox, B.A. 



THE VIBRATORY STRAINS OF SUSPENSION BRIDGES. 



In the tenth of these Notes of Engineering (p. 258, Vol. XL), an 

 investigation was given of the dynamical strains of girders arising 

 from tiie rapid passage of loads over them. In the present paper 

 it is proposed to continue the subject of the effect of moving 

 loads upon structures, by investigating the strains to which sus- 

 pension bridges are most usually subject when in a state of vibra- 

 tion. 



The greatest uncertainty has hitherto pre^'ailed respecting the 

 effect of vibration in straining the chains of suspension bridges. 

 The almost entire absence of all theoretical or experimental 

 knowledge of this part of engineering has left the engineer with- 

 out the means of forming even a wide conjecture respecting it. It 

 is generally assumed in practice that a suspension bridge ought to 

 have four or five times the strength theoretically required for the 

 greatest load that can rest upon it; and the excess of strength is a 

 provision partly against the effects of vibration, and partly against 

 accidental flaws and imperfections in the metal. But it is clear, 

 that upon a rule so vague, and so entirely empirical, little depend- 

 ence is to be placed — it may in some cases direct an enormous and 

 unnecessary expenditure of metal, and in other cases provide an 

 insufficient degree of strength. 



Before proceeding to the direct consideration of the subject, we 

 shall briefly and partially notice its history. The advantages 

 arising from a historical view of scientific questions are unfor- 

 tunately underrated in England. Continental writers almost uni- 

 versally preface their memoirs by some notice of the efforts of 

 predecessors; and this custom has the threefold advantage of en- 

 listing the confidence of the reader in the labours of one who 

 exhibits a knowledge of the state of the scientific question dis- 

 cussed; of rendering merited homage to earlier essays; and lastly of 

 putting the subject itself in a clear light, by indicating its real 

 difficulties and the errors most naturally incidental to it. 



The laws of vibration of flexible chains have from a very early 

 period attracted attention. Galileo detected the isoehronism of 

 the oscillations of chains hanging from the roof of the Cathedral 

 of Pisa and sustaining lamps, by comparing the time of the swing- 

 ing of the chain with the lieating of his own pulse. In the year 

 1732-3, Daniel Bernouilli published in the sixth volume of the 

 St. Petersburg Transactions, a memoir entitled Theoremata de Os- 

 cillatinnibu.'i Corporum Filo fle.vili conne.rorum et Calfince verticaliter 

 su.ipenste, in which he gives, without proof, the length of the tanto- 

 chronous simple pendulum, corresponding to cases where two or 

 more bodies are attached to a fine thread suspended at one extre- 

 mity : throughout this paper the extent of the oscill.ations is sup- 

 posed to be indefinitely small, and experiments in confirmation of 

 the results are instanced. It is also shown that, generally, osdlla- 

 tiones contmriai, or oscillations where all the bodies vibrate in con- 

 trary ways, are much quicker than oscillationes conspirante.i, or those 

 in which the bodies move all to the right or all to the left at the 

 same time. 



The theory of the oscillation of flexible chains has been inti- 

 mately connected with a principle laid down by Huyghens in his 

 Horologium Oscillatorium, which asserts that the centre of gravity 

 of any oscillating system acquires a descending velocity sufficient 

 to raise it to a similar altitude. In the Phoronomia of Hermann, 

 published in 1716, the principle is laid down that in the bodies 

 forming a compound pendulum the impressed forces of gravity are 

 in equilibrium with the effective forces applied in the opposite 



direction. This principle was generalised by Euler, and em])loyed 

 by him to determine the oscillations of flexible bodies, in a memoir 

 printed in 174.0 in the seventh volume of the St. Petersburg Trans- 

 actions. 



Numerous analogous researches by the Bernouillis, Clairaut, and 

 Euler are scattered over the earlier volumes of the Transactions 

 of St. Petersburg, Berlin, and Paris, the works of John Ber- 

 nouilli, and the Opuacula of Euler. The problems proposed are, 

 to determine the movements of heavy bodies attached to cords 

 or rods, and moving by mutual constraint or that of fixed curves, 

 &c. 



Lagrange, in the Mechanique Anahjtiqne, 2de partie, sect, vi., has 

 given a general investigation of the small oscillations of any linear 

 system, by the method known as the Variation of Arl)itrary Con- 

 stants. He includes the celebrated problem of the vibrations of a 

 stretched elastic cord ; but it is not necessary to notice the other 

 investigators of that problem, as it is almost entirely different from 

 the problem of the vibrations of an inelastic chain or cord. With 

 respect to the latter, Lagrange confines himself to the case of very 

 small oscillations. The small oscillations of a cord suspended at 

 one fixed point and charged with any number of equal weights 

 placed at equal distances, he shows to be susceptible of determina- 

 tion; but when the cord is fixed at both extremities, the solution 

 involves a certain general expression which, he says, cannot be de- 

 termined by any known methods. 



It will be seen, then, that the investigations above noticed afford 

 very little assistance in solving the problem here proposed. AVe 

 cannot safely assume, with respect to a suspension bridge, that the 

 oscillations are "very small," in the sense in which that phnise is 

 employed by mathematicians. A single foot passenger passing 

 over the Hungerford Suspension Bridge, will produce sensible 

 vibrations throughout the structure; that is, the passage of a 

 weight of 10 or 12 stones produces considerable motion in a mass 

 of 1,000 tons, or 16,000 times as great. What, then, must be the 

 effect of the marching of a troop of soldiers, the action of a storm 

 of wind, or the transit of a railway train.'' But, although we can 

 not avail ourselves of the simplifications which arise where the 

 oscillations are considered small, we have, on the other hand, this 

 immense advantage — that, in order to find the tension of the 

 chains, there is no necessity to integrate the equations of mo- 

 tion. 



The cases of vibration to be examined in the following investi- 

 gations are those where all the parts of the chain reach the ex- 

 treme extent of their motion simultaneously, and simultaneously 

 begin to return. At the instant of retrogression, the velocity of 

 every part of the chain is zero, and the whole is in a state of 

 instantaneou.^ rest. 



Periodic vibrations, in which the chain at recurring intervals 

 reaches a position of instantaneous rest, are perhaps those most 

 frequently observed. The present investigation will apply, whether 

 the centre of the chain moves vvith vertical or with horizontal mo- 

 tion, or both, and whether the two points of suspension Be, or be not, 

 of the same altitude. The motion is supposed to take place wholly 

 in the vertical plane, and the tension is made to depend on an 

 assigned form assumed by the chain in the position of instantane- 

 ous rest. 



Funicular Polygon. 



In the first instance, we will consider the problem of n bodies of 

 equal mass, connected by n-j-l fine inextensible strings of equal 

 length, of which the first and last are fixed at their extremities, — 

 the whole forming a funicular polygon of n-(-l equal sides. 



Taking one end of the polygon as the origin of co-ordinates, let 



{^iPi), {■^'ij/j), {■^3^^) '••• {■^„,!/n) ^^ tl^e co-ordinates of the 



bodies at the time t ; js being measured vertically downwards, and 

 y horizontally. Also, let a be the length of each side of the poly- 

 gon, and (i, e) the co-ordinates of its further extremity. Then 

 the CO- ordiuates are connected by the following system of rela- 

 tions : — 



•»^i' + yl = a- 



{■^^—■^.y + iii.—y,)- = a- 



&c. &c. 



(* - •^,,)' + (c-2/„)' = a- 



Now, when t becomes t-\-dt, the same relations will hold among 

 the new co-ordinates of the several bodies. Consequently, the dif- 

 ferentials of the above equations with respect to ( express true 

 relations. Hence, differentiating the equations twice with respect 

 to t, and in the resulting equations of second differentials putting 



15* 



