Question

A uniform rod of length L is pinned at both ends. Show that the frequencies of longitudinal vibrations are n = nc/L, where c=√?? is the velocity of longitudinal waves in the rod, and n = 0,1,2,3,4……. Note: You must show all the steps. b) Plot the first three natural modes.

Answer 1

TYCH Y Crit) forfly > X 7 y Co, t)=0 Y CL,t) = 0 Boundary Condition We know that General Solution Y(x, t ) = y (x) x G (+) -> (A Sin(%)x +B. GB ( Qs () ] [cs Das wt Substituting First Boundary condition 0-B x (Csiowtto co wt) Sin wt t D as wt which gives B = So, Y Cry t) = م] A Sincero ) x ]*[Csinw tt O court Scanned by TapScanner

Condition Substituting Second Boundary -> [Asin (WL)], [cs sin wt to cost which gives Sin IWL (Web) = 0 509 WL ПЛ С => لیا NXTX C n=¢,2,3... L For First three Natural modej n=1 O LLLLL Scanned by TapScanner

2 n=2 3 n z Z (ر) LLL A Scanned by TapScanner