1. Foundamental Frequency:

  For a vessel with constant wall thickness, constant diameter, and a fixed base, the natural frequencies are those for a cantilever beam:

  n_i = k_i / H²[ E.I /M]^0.5

  n_i = frequency of mode i (Hertz)

  K_i = constant

    = 0.56 for mode 1
    = 3.51 for mode 2
    = 9.82 for mode 3
    = 19.2 for mode 4

    H = height of vessel(m)
    E = Modulus of elasticity (pa)
    I = moment of inertia of vessel(m⁴),   I = p.d³.t / 8
    M = mass of vessel per unit length. (kg per meter or N.s² /m per meter, = N / 9.81)
    d = inside vessel diameter (m)
    t = vessel wall thickness (m)


2. Tall vessel structural damping:

  It depends not only on the vessel itself, but also on the vessel soil-structure interaction.

    Concrete vessel: 0.0150 – 0.025
    Steel vessel: 0.005 – 0.015
    Unlined steel stack: 0.0016 – 0.006
    Gunite-lined steel stack: 0.003 – 0.012
    Concrete chimney: 0.004 – 0.020

    The lower values are appropriate for foundation on rock or piles, average values are appropriate for foundations on compacted soil,
     higher values are appropriate for vessels supported by elevated structures or soft soils.