635 



VIBRATION. 



VIBRATION. 



623 



objectionable ; but in either case the construction of a viaduct for the 

 purpose of obviating the necessity for the incline must depend upon 

 the number of the carriages likely to resort to it. In railway structures 

 viaducts are more commonly employed than in common roads, on ac- 

 count of the greater infiuence of the inclinations of the roadway upon 

 the traction ; but even in them the use of viaducts can only be justified 

 by considerations of economy. 



In the article BRIDGES will be found a summary description of the 

 dimensions of the most celebrated viaducts hitherto constructed. The 

 principles upon which they are built have been discussed either in that 

 article, or under ARCHES. It may suffice, therefore, here to add, that 

 the skill of the engineer may be as usefully displayed in the execution 

 of an embankment for the purposes of the roadway as in the construc- 

 tion of a monumental viaduct ; and that the repairs of the former 

 would be in all probability much less than those of the latter. The 

 Highgate Archway, the North Bridge at Edinburgh, the Dee Viaduct 

 near Chirk, the Crumlin Viaduct, the Aricia, Barentin, Chaumont, 

 Dinting Vale, Elsterthal, Goeltzchthal, Malaunay, Ouse Valley, Port- 

 age, Tyne, viaducts, may be cited as the most remarkable works of this 

 description. 



VIBRATION. We have had in many articles to consider the effects 

 of vibratory motions, but we have not yet given the explanation of 

 the simple vibration, so as to enable a student with no very extensive 

 knowledge of mathematics to form some conception of its character. 

 The theory of the vibrations of the particles of an elastic fluid is the 

 key to what is known of the phenomena of sound and light [ACOUSTICS ; 

 UNDDLATORY THEORY] ; and there is some reason to suspect, or at 

 least those whose opinions are worthy of attention have suspected, 

 that the causes of the sensible phenomena of heat, electricity, and 

 magnetism will also be found in the vibrations of matter of some kind. 

 All the particles of material bodies, even when solid, are probably in 

 continual vibration ; and it is certain that very slight disturbances will 

 communicate sensible amounts of vibration to considerable distances, 

 and this through all manner of different substances, from loose earth 

 to compact stone, and through those in every kind of state, from the 

 aeriform to the solid. 



Little as may be known of most of the vibrations which are per- 

 petually occurring, nothing is more certain, from the fundamental 

 laws of mechanics, than that every such vibration in every individual 

 particle is either made up of one or several motions of one particular 

 kind, or of an exceedingly close approximation to such simple motion 

 or combination of motions. It is not merely swinging backwards 

 and forwards which constitutes a vibration ; such a motion might 

 certainly be so called, at the pleasure of any one, but another name 

 must then be invented to designate that particular sort of vibration of 

 which, and of no other, we have to speak in the first instance. The 

 piston of a steam-engine, for example, when it is forced upwards with 

 continually accelerated velocity until it strikes the top of the cylinder, 

 and is then forced downwards in the same manner, does not show what 

 is mathematically called a vibration ; but take one of those more 

 recent constructions, in which the steam is checked as soon as the 

 piston has acquired momentum enough to carry it to the top of the 

 cylinder, so that the force i nearly spent before it begins to return, 

 and we have something to which the term vibration ia much more 

 nearly applicable. 



The simple vibration, of which we have said all others may be com- 

 pounded, is best imagined as follows: Let a point Q revolve uniformly 

 round a circle A Q a f>, and from <J draw Q r perpendicular to A a. 

 Then P moves over A a in the manner of a simple vibration ; the whole 

 vibration being from A to A again. At A and a the velocity of p is 



extinct, the whole motion of Q being perpendicular to A a but at o 

 the velocity is greatest, p then moving as fast as Q. If we measure the 

 time ( from the epoch of Q being at B, and suppose the motion of Q to 

 be in the direction B o. A, and n to be the angular velocity of Q, we have 

 (o P = x, o A = a) x = a sin n I , while the velocity of p is n o cos t, the 

 acceleration of p is n'a sin n I, or n-x, and if w be the weight of a 

 particle at p, the pressure necessary to maintain it in this state of 

 vibration is always directed towards 0, and is, in unite of the same 

 kind I 



ARTS A2ID JICI. DIV. VOL. VIII. 



n-x 

 82-1908 



X >, 



if x and a be measured in feet, n in theoretical angular units [ANGLE], 

 and t in seconds [VELOCITY]. If T be the number of seconds in the 

 whole vibration from A to A again, we have n = 2x 3'14159-^T, and the 

 pressure is l'226ixw-^l f . The pressure, it appears, requisite to main- 

 tain a simple vibration must be always in a given proportion to the 

 distance of p from o, and always directed towards o ; and the relation 

 between the pressure at a given value of x and the time of vibration is 

 wholly independent of a, the excursion of the particle. For the 

 mechanical reason of this property, see ISOCHKOXISSI. To form a more 

 convenient expression, let N be the number of vibrations in a second, 

 and let A- be measured in hundredths of inches instead of in feet ; then 

 T = !H-N, and for x we must write a; -e- 1200, which gives for the pres- 

 sure -001022 n'xw. For example, if a particle vibrate only 100 times 

 in a second, which is not much [Acoustics], and have an excursion of 

 one five-hundredth of an inch (N = 100, x=-2), the force of restitution 

 at the extremity of the excursion is more than twice the weight of the 

 particle. By this formula it is easy to get a just idea of the greatness 

 of the molecular forces required to produce those vibrations which are 

 constantly excited in sonorous and other bodies. 



If we suppose a second vibration to be communicated to p, in the 

 same line, and of the tame duration, but whether of the same extent or 

 not does not matter, the compound vibration is only equivalent to 

 another simple vibration. Let a circle move with Q, and in that circlo 

 let a point (R) revolve uniformly, and let R V be perpendicular to o A. 

 Then, while p vibrates about o, v performs a vibration in the sams 

 time relatively to p ; or a spectator who does not see the motion of p, 

 will see no motion in v except a vibration about p. Now it is easily 

 shown that B not only describes a circle about Q, but also actually 

 describes either a circle in space, about the centre o, or an ellipse, in 

 the manner presently explained. And v, vibrating about P, which 

 itself vibrates about o, does, if these vibrations be of the same duration, 

 nothing but vibrate about o. Mathematically, this is easily obtained 

 as follows : Let the angles A o Q and c Q B (Q c being parallel to o A) bo 

 at some one moment a and /3, and let o q = a, Q R = b, and let the time 

 be measured from the instant at which the angles are a and $. Then 

 we have 



x=a cos (n t + o) + 6 cos (n t + ), 



the sign + being used when the circular vibrations are in the same- 

 when they are in opposite, directions. This is equivalent to x = ' 

 cos (n t + A), provided I and A be found from 



I cos A = a cos a + 6 cos , I sin A = a sin a + 4 sin ; 



and the joint vibration U one of the excursion I, and such that the 

 angle is A when the angles of the component vibrations are a and 

 0. It is easy to show in like manner that any number of vibration* 

 whatsoever, made in the same times and in the same lines, are not 

 distinguishable from one single vibration, of the same duration and in 

 the same line. 



Again, it is easily shown that a vibration which is represented in 

 direction and excursion by the diagonal of a parallelogram is the com- 

 pound effect of two vibrations of the same duration, represented in 

 direction and excursion by the two sides of the parallelogram, if th 

 particles of the component vibrations begin to describe the sides at tho 

 same instant as the particle of the resultant vibration begins to de- 

 scribe the diagonal ; and the same thing may be shown of the diagonal 

 of a parallelepiped and its three sides. Hence any number of vibrations 

 of equal times about any lines drawn through one point may each bo 

 decomposed into three in the direction of three given axes passing 

 through that point, and those in the several axes may be compounded 

 together into one. The student who appreciates the similarity of tho 

 laws by which velocities, pressures, and rotations are compounded and 

 decomposed, will see that to the list must be added vibrations. But 

 the only vibrations which bear the application of these rules are thos 

 of equal duration. 



Let us now suppose that any number of vibrations of equal times, 

 and about the same point, are reduced to three, in the directions of 

 three axes of x, y, and 2. When a cos { represents the distance of a 

 vibrating particle from its centre of vibration, let the angle { be called 

 the pluue of the vibration. If the three vibrations be always in tho 

 same phase, the diagonal of the parallelepiped described on the thres 

 excursions represents the direction and excursion of the resulting 

 vibration, which is simple and rectilinear. But if the simultaneous 

 phases be not the same, so that x= a cos (n(-Kt), y=b cos (nt + p), 

 z = c cos (nt + y), represent the simultaneous distances in the threo 

 vibrations, and also the co-ordinates of a point which is affected by 

 them all, the particle, thus triply vibrating, does not move in a straight 

 line, but in an ellipse. Let us consider two vibrations in a plane, and 

 let A a and B 6 be their double excursions about the common centre o. 

 The axes in the figure are drawn at right angles, but any angle will 

 do equally well. 1 )raw the parallelogram w x Y z, which always con- 

 tains the particle, and suppose that p and v are contemporaneous 

 positions in the two vibrations, whence N is one of the positions of the 

 particle. Through N can be drawn two ellipses, having the centre o, 

 and touching all the four sides of the parallelogram WXYZ. The 



R S 



