Basics of Ship Vibration

Ship Vibration• SOURCES of SHIP VIBRATION

Internal Sources [Unbalanced machinery forces]

(1) Main & Auxiliary Machines

• Main Propulsion Engine (esp. 4 or 5 cylinder engine) generating large unbalanced force at high frequency close to hull’s natural frequency.

• Rotary Machines (Electric motors, auxiliary machines etc) generate high frequency but low amplitude

• Reciprocating Engines – Usually low frequency

(2) Unbalanced Shaft ( frequency = shaft RPM)

Ship Vibration• SOURCES of SHIP VIBRATION

Internal Sources [Unbalanced machinery forces]

UNBALANCE: occurs when centre of mass is different from centre of rotation. Can be caused by improper assembly, material buildup, wear, broken or missing parts

Detection: High level radial vibration

MISALIGNMENT: is a condition when two coupled machines have shafts whose center lines are not parallel and intersecting, or where one or more bearings are offset or cocked. Mis-alignment can be caused by improper assembly and adjustment, foundation failure, thermal growth, or locked coupling

Detection: High level axial vibration

Ship Vibration• SOURCES of SHIP VIBRATION

External Sources [Hydrodynamic loading by direct action or induced by the ship motions]

(1) Hydrodynamic loading on Propellers

Blades in non-uniform flow (freq. = RPM x No. of blades). More pronounced for low propeller submergence and in shallow water

(2) Unstable cavitation of blades

(3) Vortex induced forces (not on propeller)

Eg. Brackets that holds the propeller

(4) Slamming Load – short duration forces but give rise to high frequency forces.

Ship Vibration• SHIP RESPONSE

• In response to excitation forces, the ship execute elastic vibrations, some of which are observed only locally and some are observed throughout the hull. Local Vibration: Usually high frequency and lower

amplitude Difficult to predetermine Amended easily post-construction (common practice to

overlook during design stage) Hull Vibration: Lower frequency and higher amplitude

(Compared to local) Must be carefully considered and avoided in the design

stage itself

Hull Girder Vibration Distribution of Weights

Source: MUN Notes

The weight will not equal the buoyancy at each section along the ship The weights are combination of lightship and cargo weights The buoyancy forces are determined by the shape of the hull and the position of the vessel in the water (draft & trim) Local segments of the vessel may have more or less weight than the local buoyancy. The difference will be made up by a transfer of shear forces along the vessel.

Hull Girder Vibration• SHIP as a UNIFORM BEAM

• Vibrations that exist throughout the hull are of the same type that may exist in a beam free in space

• Surrounding water plays an important role but it does not destroy their beamlike characteristic and it is helpful to consider the vibrations of the ideal solid beam free in space (free-free beam)

l = 2L

l = L

l = 2L/3

L

Hull Girder Vibration• Types of Elastic Deformation

• A beam free in space can undergo FOUR principal types of Elastic Deformation:

1) Bending

2) Twisting

3) Shearing

4) Extensional

• Elastic deformations that play a significant role, in the case of ship are: Bending and Shearing in both vertical and horizontal

planes through its longitudinal axis (Flexural) Torsion about the longitudinal axis (Twist)

Hull Girder Vibration• Types of Elastic Deformation

Flexural: Bending like a beam Horizontal bending mode Vertical bending mode (usually more of a concern than

the horizontal mode)

Torsional: Twist of a beam More likely for container ships

Hull Girder Vibration• MODES and NODES

• Mode: the pattern or configuration (shape) which the body assumes periodically while in vibratory conditions

• Node: is a point in the body which has no displacement when the vibration is confined to one particular mode.

• Normal Modes: are patterns in which the body can vibrate freely after the removal of external forces

Connecting nodes, give corresponding mode

Hull Girder Vibration

In both Flexural and Torsional vibrations, a natural frequency is associated with each pattern of vibration and the natural frequencies increase as the number of nodes (points at which curves cross x-axis).

If a free-free beam is unsymmetric w.r.t either the vertical or horizontal planes through its longitudinal axis, it will be found that the natural modes of vibration involve Torsion, Bending and Shearing simultaneously.

Hull Girder Vibration

A hull is much more complicated structure than a solid beam and therefore it behaves like the free-free beam ONLY in its lower modes of vibration.

These are called beam-like modes and may be excited by either:

(a) Transient disturbances (due to wave or slamming impact)

(b) Steady-state disturbances (rotating unbalanced engine or machine elements, unbalanced propellers, unbalanced shaftings)

Hull Girder Vibration

How to avoid dangerous vibrations of the ship’s hull?

Avoid exciting forces at frequencies close to the natural frequencies of the ship’s hull.

How to determine the natural frequencies of the hull girder?

Basic concepts are developed from the simple notions of a uniform beam vibration.

It’s then extended to the vibration of a ship with some more added complexities that would reflect the realities of a ship in the way that a ship differs from a uniform beam

Natural Frequency of Hull Girder

Hull Girder Vibration

Minimum number of nodes = 2

Fundamental Mode of Flexural Vibration

Frequency (in cpm) corresponding to this 2 noded vertical vibration (fundamental mode) is denoted by N2V or NV2 (number of cycles per minute in 2-noded vertical vibration)Otto Schlick:

32 L

IN V

I = Imidship of the cross-section of the ship (beam)D = Weight displacement of the ship (beam)L = Length of the ship (beam)

Z

X

q (x, t)

q – the driving force / unit length in the z-direction

Hull Girder VibrationUniform Beam Vibration EquationJust as a S-DOF system provides basis for understanding vibrating characteristics of many mechanical systems, similarly, a uniform FREE-FREE BEAM provides the basis for understanding the essential vibratory characteristics of ship. Free-Free Beam is a continuous system Beam is assume to have a mass/unit length, = A and Bending stiffness – EI in x-z plane BM due to normal internal stresses acting at any cross-section is related to the mean radius of curvature

;R

EIM R – radius of curvature

Hull Girder VibrationUniform Beam Vibration Equation

Mdx

zdEI

2

2

For small deflections in z-direction, the approximation that curvature (1/R) is equal to 2nd derivative of z w.r.t x can be used

2

2

dx

zdEIM

The Euler-Bernoulli equation describing the relationship between beams deflection and the applied load

OR

qdx

zdEI

dx

d

2

2

2 )(4

4

xqdx

zdEI OR

Equation relating BM and deflection in simple beam theory.

Hull Girder VibrationUniform Beam Vibration Equation

Inertia effect of surrounding water The relative high density of water makes the inertial effect a serious concern Apparent increase in mass of a body vibrating in water

),(4

4

2

2

txqx

zEI

t

zA

Inertia effect Restoring force as a

result of elasticity

Loading on the beam