Hi,
Look,
Young's modulus of rod, Y = FL / A

L
So, F = (YA / L)

L
i.e the spring constant for the rod is k1 = YA / L
Now, this rod and the spring are in series.
So, net spring constant, knet = k1k2 / (k1 + k2 )
= kYA / L(k + YA/L )
So, the time period = 2(pie)( m / knet )1/2
Here, m = mass of rod