CA6 Prof. H. H. Jeffcott on the Whirling Speeds of a 

 deflexion and slope due to the whole system o£ loads. We 



then substitute X = moru, M = mlrco 2 -^ for each load. 



dz 



We then have an expression for the deflexions and slopes at 

 the load points and known constants. By taking the point z 

 to coincide in turn with each of the load points we get the 

 series of equations <j)ii 1 = a 1 n 1 + a' 2 u 2 + . . .. etc., as given in § 3. 



§ 10. Numerical example. 



A simple example will illustrate the procedure. 



Let the length of the first span be double that of the 

 second, i. e. 1 = 21'. Let there be loads of equal mass m at 

 the centre of each span. Let the moments exerted by these 

 loads be zero. Let there be section changes in the first span 

 of fju = ^ at £ = 5, and of fi = J at t = j. Also there be a change 

 of section in the second span of /u. — i at t' = 0, and of fi = ^ 



We first proceed to determine the values of the constants 



as follows : 



C« C ' 



We have T - = ^-7-,. .*. Ci' = JC. 



ti -Li 



Also Gi = 0; G/=0. 



Ordinarily we follow through the constants step by step 



along the spans for each load separately. We may, 



however, take the two loads Xi and X/ together, and for 



., ri £ 6C ri p 6G T ^ p 6K 

 convenience we write u tor - /2 , (jr tor ^—- } K tor ~w •> 



TT , 6H _,., 6C nlf 6G' T „p 6K' , 6H' 

 H for -p-, C tor ^, G for -p-, K for *-^-, H for -^. 



C /■ C ' i 2 

 We then have -4- = . , " , or Ci' = 2Ci, and we find, 

 li j-i 



from § 9, 



d= Ci, Gi=0 ? 



^ 1 n 21 V _i_ 9 Y P 5 V * Y 



^2—2^1 — 32 X ^ 32 A l> ^2— 64 - 1 ffl A l» 



15 AT XT 5 a- , 3 



K 2 =iCi~- rr 2 Y ~ g|Xi, Ho — ^ 1 -f 32 Xj, 



T - , r i 111 v 15 V TT _ 59 ^ i 3 V 



J^3 =4^1— riT X "" 04 A 1? ^3- 128 "L +64 A 1J 



(Y = 2Ci, Gi'=0, 



r , , _ p 21 V 'i 9 v ' C± ' — 5 V * Y ' 



2 — Oi— 32 * + 32 X l > Lt 2 — 61 - 1 32 Al ' 



K 2 ' = Ci— |Y' — IX/, H 2 '= ^Y'+|Xi'. 



