103 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[April, 



parts attnehed. Then, using thfi other letters as before, the moment 

 of impressed forces about A will be 



P rs'ind — N; 

 P r sin e estimates tlie moment of P correctly, because sin e is nega- 

 tive when the rod descends ; a; the vertical height the rod has risen 

 from its lowest position at time ^—excluding the consideration of its 

 lateral motion by the idea of its lengtli being indp6nitely great. Hence 

 by D'Alembert's principle, the equation of moment is 



d- x . , ., ,, d- e ,, > 



P rsin e — N = J» -;^;^rsin e + MA- ^- (1.) 



But by the geometry 



dP 



x-==. r versin ( 



dl' 



dx _ 



dt' 



de 

 (ft 



Hence multiplying (1) by ^ we get 



Integrating 



C — Prcosfl— Ne=:vJ,«^ + ^MZ:='^ ^" 



rf«2 



rfi 



And if when 6=0 — = w, 

 dt 



P versin e — N e = J (m >-2 sin =a + M /;-2) 



CK-) 



(2.) 



Which determines the velocity of the crank for every position. 

 If R denote the reaction at B, 



R = P-M— (3.) 



rfi^ 



But -rT= r Sin 0:77 

 dt dt 



: r sm 9 



jP r versin e — Na 

 'm f- sin ^e + M A 



\ +"=}i 



from (2.) 



DifFerentiating with respect to /. 



, P r versin e — N e , „ \ 



• • TZ. T cos 9 < •^ 5 — -. r . ,, ,., 



dl- \ mr- sin =9 + M k'- 



P)-sin9(mrversin29+MF)— N{MF+»ir=(sin=9— 9sin2fl)} 



-j-^rsin9- 



(mr2 sin 29+ MAT 



Substituing this value of -^ in (3), we have the value of R in 



terras of 0, &c. ; and it will be seen from the exceedingly complicated 

 nature of this expression, even in the simplified form of the problem 

 which we have taken, how remote from the truth is Mr. Pole's as- 

 sumption. 



We have dealt with this first error thus particularly because, being 

 the basis of every one of the calculations which follow, it totally 

 vitiates them all, and, even if no other error occurred, must condemn 

 the whole paper. 



But even if this first objection were not fatal, there would be found 

 many others entirely invalidating the resuKs. In Art. 13, quoted above, 

 it is assumed that the pressure on the bearing A lies along A B. Nothing 

 can be more erroneous. The direction of the pressure is a function 

 of many variables, as we will presently shew. We may, however, 

 previously observe that the double sign in the expression 

 + Pcos9 



is sufficient confutation of the assumption from which it is deduced. 

 When the expression passes from one sign to the other it must pass 

 through zero. Hence when the crank is horizontal there must be no 

 pressure on its extremity A ! That is, the crank will support itself 

 and its concomitant machinery by some virtue of its own! On this 

 supposition, if the end of a beam rest on a table the beam will support 

 itself horizontally ; which case by no means agrees with our usual 

 experience. 



The following problem will, we submit, shew that the direction of 

 the pressure depends on many quantities. We have omitted all the 

 machinery but the crank itself, and exclude also the consideration of 

 gravity. 



An uniform crank A B, 

 revolves about A, acted on 

 at B by a constant vertical 

 force P. A constant re- 

 tarding force acts at the 

 lowest point of B's path 

 tangentially. The end A 

 turns on an axle with fric- 

 tion. To determine the 

 direction of the pressure on 

 the axle. 



The axle will touch the 

 hole in which it works at 

 one point. Let n be the 

 pressure normal to the two 

 circles, and /j. n the friction 

 tangential to the two circles. X,Ythe resolved parts of the pressure 

 along the axes of co-ordinates O .r, O y. Then the resolved parts of 

 the friction are 



-ftX, f.Y. 



Let r be the distance of any particle of AB from A; its co-ordi- 

 nates are 



rcose, rsin9; 



9 being the angle through vrhich AB has revolved from the horizontal 

 at the time t. If m be an unit of the mass of the crank, the resolved 

 effective forces of the particle are 



, d' (r cos 9) 

 ,„<f, ______ 



mdr 



d'' (r sin e) 

 d¥~ 



A B = a, radius of axle = c. 

 The equation of horizontal forces is by D'Alembert's principle 



pa ( J d-(r cos 9) 



F + X- 



mdr- 

 d' cos 9 



df 



d' cos 9 pa , 

 =:ia-?«l — .0039^ — rsin9^ I (1.) 



The equation of vertical forces, 



d^ (r sin fl) 



P-l-Y + ^Y^ 



=j:{ 



mdr 



dt' 



= ia'»«(^-sin9jp-fcose^,^ (2.) 



If the moment of inertia about A be M A' = i M o', the equation of 

 moments is 



d'e 

 Racose — Fa—iivc = i'M.a-jj^ (3.) 



dd- 

 Integrating (3) we get the value of j^,, and substituting in (1) and 



(2) these values of — -^ and — we get two equations independent of t. 

 "• ' dt'-df^ 



And since the inclination of n to the horizontal is 



-'Y 

 tan -; 



we get that angle exhibited as a function of 9, R, F and m. 



The reader may judge hence how near the truth is the assumption 

 above quoted. , r^ , 



We now proceed to other investigations of the paper. Alter what 

 has been said they may be disposed of more briefly. 



" 14. We mav take, as the next simple arrangement, a form of 

 engine which was early used in steam boats, having been patented 

 and so applied by Mr. Aaron Manby, in 1821, namely, 

 "the vibrating engine, 



" Fig. 2, in which the cylinder being made to oscillate upon the gud- 

 geons C, the connecting rod is altogether dispensed with. 



" The friction we have to consider is that upon the three bearings 

 A, B, C. 



" For tie crank pin B. The piston rod acting directly upon this 

 gudgeon, the pressure will necessarily be constant = P, and therefore 

 the loni hj friction in a semi-revolution will be, as in Ait, 12, 

 =:: m P T J. 



