ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 131 



For instance, at the clamped end the tangent is = o ; at the free end the shear 

 and the moment are both = o. At the supported end the deflection itself is 3/ = o; 

 also the moment = o; but the shear of course is not =0; etc. 



IV. RITZ'S METHOD.* 



I. Although the delicate mathematical structure underlying Ritz's principle is 

 very elaborate, indeed, the actual working of his method is quite easy, especially in 

 its application to the comparatively simple problems in which we are here interested. 

 In the main, its philosophy is as follows : — 



As has been mentioned above, any problem of elastic equilibrium, or motion, can 

 be reduced to some variational equation, in which a variation of some integral is 

 = o. This can either be derived from the principle of virtual work or from Ham- 

 ilton's principle. Now if a certain variation equals zero, this is precisely the same 

 as to say that some integral must be maximum or minimum ; that is what the cal- 

 culus of variations was invented for. And variations must be used only for the 

 reason that the integrand contains dependent functions, such as y, and their deriva- 

 tives, so that plain calculus rules of maximum and minimum cannot very well be 

 applied. 



Now here is what Ritz proposes. In place of the dependent function (such as 

 3;), let us substitute the following tentative expression: — 



where «i, ^2, etc., are certain constant coefficients; ^1, ^2, • • • • are certain functions, 

 of X only (or of whatever the independent variable happens to be) ; and n only means 

 the number of terms we wish to assume: for rough results n may be 2 or 3 so 

 that, for instance, the assumed value may be simply— 



or, for greater precision, n may be as high as we please. The form of the functions 

 ^ (or 1^ {x') as they are sometimes denoted) is rather immaterial, so long as they 

 answer the end conditions in the same manner that 3; itself would ; although some 

 forms may be preferable to others, from the standpoint of labor saving. 



As soon as the expression (i) has been substituted into the definite integral, 

 which is generally of some such form as 



6" = J /(x, y, y , y")dx (2) 



the latter, as regards the variables of the problem, becomes merely a function of its 

 limits. The only quantities which we can adjust to the end of having such integral 

 become a maximum or a minimum will be the coefficients ^1,02,... «>.) regarding 



♦The following articles have been drawn upon, or consulted, in compiling this chapter: — W. Ritz, 

 "Oeuvres," Paris. 1911. Kryloff, Sur.. . la methode Ritz, C. R. 1917, p. 853. H. Lorenz, Physik. 

 Zeitschr., xiv, 1913, p. 71. T. Poschl, Physik. Zeitschr., xiv, 1913, p. 410. Professor Timoshenko, "Elas- 

 ticity" (Russian), 1916. 



