Abstract
A compliant tower is modeled as a partially dry, partially tapered, damped Timoshenko beam with the superstructure modeled as an eccentric tip mass, and a non-classical damped boundary at the base. The foundation is modeled as a combination of a linear spring and a torsional spring, along with parallel linear and torsional dampers (Kelvin-Voigt model). The superstructure adds to the kinetic energy of the system without affecting the potential energy, thereby reducing the natural frequencies. The weight of the superstructure acts as an axial compressive load on the beam, reducing its natural frequencies further. The empty space factor due to the truss-type structure of the tower is included. The effect of shear deformation and rotary inertia are included in the vibration analysis; with the non-uniform beam mode-shapes being a weighted sum of the uniform beam mode-shapes satisfying the end condition. The weights are evaluated by the Rayleigh-Ritz (RR) method, and verified using finite element method (FEM). The weight of the superstructure acts as an axial compressive load on the beam. Kelvin-Voigt model of structural damping is included. A part of the structure being underwater, the virtual added inertia is included to calculate the wet natural frequencies. A parametric study is done for various magnitudes of tip mass and various levels of submergence. The computational efficiency and accuracy of the Rayleigh-Ritz method, as compared to the FEA, has been demonstrated. The advantage of using closed-form trial functions is clearly seen in the efficacy of calculating the various energy components in the RR method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- x :
-
space variable along the beam length
- y :
-
space variable perpendicular to the beam length
- t :
-
time variable
- u :
-
transverse deflection
- Y :
-
spatial part of transverse deflection of uniform beam
- θ :
-
pure bending slope
- θ s :
-
shear slope
- Θ :
-
spatial part of pure bending slope of uniform beam
- L :
-
length of the beam without tip-mass
- Br :
-
side length of the beam
- α :
-
fraction of length over which the beam is submerged
- α’ :
-
fraction of length over which the beam is strengthened at the base
- ρ s :
-
density of steel
- ρ w :
-
density of water
- N :
-
axial load
- M :
-
bending moment
- M 0 :
-
spatial part of bending moment
- V :
-
shear force
- V 0 :
-
spatial part of bending moment
- Q :
-
external force along y-axis
- A :
-
section area of uniform beam
- I :
-
sectional area 2nd moment of uniform beam
- C A :
-
added mass coefficient
- C b :
-
coefficient of internal damping in bending
- η b :
-
proportionality constant of Cb
- C s :
-
coefficient of internal damping in shearing
- η s :
-
proportionality constant of Cs
- kt ,kr :
-
translational and rotational spring constant of uniform beam
- ct ,cr :
-
translational and rotational damping constant of uniform beam
- η r :
-
proportionality constant of cr
- η t :
-
proportionality constant of ct
- kt1, kr1 :
-
translational and rotational spring constant of non-uniform beam
- ct1, cr1 :
-
translational and rotational damping constant of non-uniform beam
- E :
-
modulus of elasticity of the material
- G :
-
shear modulus
- k :
-
shape factor of the cross-section
- mt, It :
-
tip mass and rotary inertia of tip mass of uniform beam
- ka, ki :
-
empty space factor for the beam
- km :
-
coefficient of empty space factor of tip mass of non-uniform beam
- mt1, It1 :
-
tip mass and rotary inertia of tip mass of non-uniform beam
- Lt, brt :
-
height of tip mass, side length of tip mass
- e t :
-
eccentricity of tip mass
- u t :
-
deflection of tip mass
- θ t :
-
slope of tip mass
- T 0 :
-
time component of uniform beam vibration
- v :
-
poisson’s ratio
- ω 0 :
-
free vibration frequency
- ω 1 :
-
decaying part of ω0.
- ω 2 :
-
oscillatory part of ω0
- β :
-
frequency parameter of the ODE
- g :
-
gravitational acceleration
- A s :
-
solid part of section area of non-uniform beam
- A 0 :
-
total section area of non-uniform beam
- I s :
-
2nd area moment of solid part of section area of non-uniform beam
- I 0 :
-
2nd area moment total section area of non-uniform beam
- q :
-
time component of non-uniform beam vibration
- φ :
-
uniform beam mode-shape
- ϕ:
-
non-uniform mode-shape
- ψ :
-
uniform pure bending slope mode-shape
- Ψ:
-
non-uniform pure bending slope mode-shape
- U :
-
potential energy
- W a :
-
work done due to axial force
- T :
-
kinetic energy
- R :
-
Rayleigh dissipation factor
- M, C, K :
-
mass matrix, damping matrix, stiffness matrix
- CD, CM :
-
drag coefficient, inertia coefficient
- ξ :
-
local space coordinate
- L e :
-
length of the beam element
- u e :
-
nodal displacement matrix
- θ e :
-
nodal pure bending slope matrix
- U e :
-
potential energy of beam element
- \( {W}_a^e \) :
-
work done due to axial force of beam element
- T e :
-
kinetic energy of beam element
- R e :
-
Rayleigh dissipation factor of beam element
- n e :
-
Nth element of the beam
- N e :
-
total number of beam elements
References
Ankit, Datta N (2015) Free transverse vibration of ocean tower. Ocean Eng 107:271–282. https://doi.org/10.1016/j.oceaneng.2015.07.045
Ankit, Datta N, Kannamwar AN (2016) Free transverse vibration of mono-piled ocean tower. Ocean Eng 116:117–128. https://doi.org/10.1016/j.oceaneng.2015.12.055
Auciello, Ercolano (2004) A general solution for dynamic response of axially loaded non-uniform Timoshenko beams. Int J Solids Struct 41(18–19):4861–4874. https://doi.org/10.1016/j.ijsolstr.2004.04.036
Majkut L (2009) Free and forced vibrations of Timoshenko beams described by single difference equation. J Theor Appl Mech 47(1):193–210
Uściłowska, Kołodziej (1998) Free vibration of immersed column carrying a tip mass. J Sound Vib 216(1):147–157. https://doi.org/10.1006/jsvi.1998.1694
Wu, Chen (2005) An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia. J Sound Vib 286(3):549–568. https://doi.org/10.1016/j.jsv.2004.10.008
Wu, Hsu (2007) The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia. Ocean Eng 34(1):54–68. https://doi.org/10.1016/j.oceaneng.2005.12.003
Datta N, Thekinen JD (2016) A Rayleigh-Ritz based approach to characterize the vertical vibration of non-uniform hull girder. Ocean Eng 125:113–123. https://doi.org/10.1016/j.oceaneng.2016.07.046
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ankit, Datta, N. Free flexural vibration of a partially wet tapered Timoshenko beam with intermittent mass and stiffness variations, eccentric tip mass and non-classical foundation. J. Marine. Sci. Appl. 17, 498–509 (2018). https://doi.org/10.1007/s11804-018-0035-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11804-018-0035-3