76 
Using the assumed deflected shape of equation B1, equations E3, El and E5 
beoome of the form 
i 
gy 4 at? +Bbn+h0n- 
RE eben” als PARE eeo A irae! 1 CED) 
Sw =PSE +n 
where A, B, C, P, Q and k are constants depending on the beam and spring 
properties and the applied load w. For the two unknowns & and, 
Lagrange's equations are 
Ae) + 5 39 occ aicle eee eee sisle (87) 
28). # = Q SOOM Ora) woche acca) cde (G3) 
which, when applied to equation E6 gives 
Ag+By +5) =P Sekt, Gate: Meac-o! bons MOAN CES) 
” 
BE +07 +k» =Q ooo eee eee coe eee (£410) 
Initially, the system is at rest so that 
Boe 5 an .0 
2 
A 5 when t =0 eee eco eee eee (E41) 
= 2 = 
The ordinary simultaneous differential equations E9 and E10 can be 
solved subject to conditions of equation E11 and hence the method 
determines and 2 
