Schilick’s formula for Calculating ships Hull frequency The first systematic investigation of ship hull vibration seems to have been made by Otto Schilick who had published the first of a long series of papers on the subject in 1884. He pointed out that vibration is present to some degree in all vessels, and is not due to excessively powerful engines nor with construction, but to the fact that a ship is an elastic structure and therefore, has a definite period in which it will vibrate if once disturbed. The governing factors are the magnitude of distributing force, the restraining force provided by the elasticity of the structure, the mass of the latter, and the damping effect of the water. March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) Let us consider a uniform beam of length L to be simply supported. A load W acts at the centre of the beam as shown in figure below: March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) Let us neglect the weight of the beam. Let the dynamic deflection at the centre at any instant be y. Then we can write kinetic energy of the system: The potential energy of the system is equal to the strain energy stored in the beam. Suppose that a rod of length L and a cross-sectional area A is subjected to an axial tensile force P. The length of the rod is increased by δ. Then the stress and strain are given by the following expressions. March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) Again, δ P Elongation Force Strain energy, March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) Similarly when a simply supported beam is subjected to a bending moment M, the strain energy can be expressed as Thus the static strain energy of a simply supported beam can be expressed as Equatipon (1) March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) But, Therefore, Equation (2) We know that the static deflection of the simply supported beam is given by Equation (3) March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) From equations (2) and (3) we obtain, Eq. (4) From equation (4) we find that strain energy is proportional to square of deflection. Let us now consider the dynamic deflection ‘y’ due to a vibrating load. Then we can write strain energy Eq. (5) March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) Total energy =KE+PE=constant Eq. (6) Now differentiating w.r.t. as ……Eq. (7) March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) Let, Therefore equation (7) becomes This is a linear second order differential equation. The solution of this equation is March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) Therefore, Time period, March 16, 2017 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) Schilick applied this formula to the case of ship vibration. He gave it an empirical form by the use of a coefficient Where, I= Moment of inertia of all longitudinal material in the cross section amidships Δ= Displacement of the ship L= length of the ship September 26, 2007 Dr. M. Rafiqul Islam

Schilick’s formula for Calculating ships Hull frequency (Contd..) coefficient incorporating the constant factors E, g and also an empirical term taking account of the different modes of vibration and of departures of the ship girder and its loading from the basic uniform bar. He obtained the values of the coefficient for several ships. =156850 for vessels with very fine lines (destroyer) = 143,500 for large trans-atlantic passenger liner with fine lines =127,900 for cargo boats with fullness March 16, 2017 Dr. M. Rafiqul Islam

Problem As ship of fine form has a length of 3 60 ft, displacement 3000 tons and a moment of inertia of the midship section of 58500 in2ft2. Calculate the frequency of vertical vibration of the ship using the schiclick’s formula. Use Schilick’s constant 157,000. September 26, 2007 Dr. M. Rafiqul Islam

Problem The following particulars have been given, estimate the 2NV natural hull frequency for the new similar ship. Vessel LBP(m) Displacement (tonnes) 2NV(cycles/min) INA(m4) Basis Ship 120 12,500 76.15 18.00 New Ship 125 14800 20.85 September 26, 2007 Dr. M. Rafiqul Islam