Lateral Vibration Prediction Issues

Dr. Yuriy BatrakIntellectual Maritime Technologies

Ship Noise and Vibration ConferenceNovember 24, 2010, London

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Do we need to predict the lateral vibration?

The one third of the ship problems by cost

for 1998-2006 is related to machinery

Propulsion claims take 2nd place after main engine claims

¼ by costs of the machinery claims

Unduly operating propulsion system causes…

• complete shafting system destruction;

• shaft and shaft elements fatigue life reduction;

• fatigue cracks at the shaft brackets and foundations;

• stern tube bearing fatigue damage;

• increased wear and damage of the sealing;

• excessive noise, hull and superstructure vibrations

Propulsion system lateral vibration contributes to accident rate!

Lateral vibration destroyed high speed shaftline with cardans

Relation (span/diameter) for one of the shafts was exceeded recommended 20

Shafting damages by vessel types

Why tugs and supply vessels?

frequent switching from ahead to astern operation

fatigue and other vibration related damages

Aft stern tube bearing damages

DNV statistics

Bearing material fatigue

DNV Photo

Fatigue damage of the aft stern tube bearing caused by the propeller shaft lateral vibration

Sealing wearing and sea water pollution

Increased lateral vibration speeds up the sealing wearing and promotes sea water pollution with oil

Partially immersed propeller

• increases shaft vibrations

• affects stern tube bearings significantly

Lateral, bending, transverse or whirling vibration?

Classification Societies terms

Classificaton Society Term used

Bureau Veritas (BV) Bending vibration

Germanischer Lloyd (GL) Bending vibration

Korean (KR) Bending vibration

RINA Bending vibration

Russian Maritime Register (RS) Bending vibration

Indian Register of Shipping (IRS) Lateral vibration

Lloyd’s Register of Shipping (LR) Lateral vibration

Nippon Kaiji Kiokai (NK) Lateral vibration

American Bureau of Shipping (ABS) Lateral (Whirling) vibration

Det Norske Veritas (DNV) Whirling vibration, Lateral vibration

China Classification Society (CCS) Whirling vibration

The variety of used terms is a reflection of a certain ambiguity of this type vibration definition

Lateral vibration Lateral vibration occurs when the direction of beam motion is normal to the centerline of a beam.

However to imagine the propulsion system lateral vibration we never think about solid body lateral motions.

The first that we think about is a beam bending deformations when beam points move from side to side up and down due to the shaft flexibility.

Definition:

Lateral, transverse or bending?

• Terms ‘lateral’ and ‘transverse’ accentuate the beam motions

• Term ‘bending’ emphasizes the beam flexibility phenomenon

When the solid body motions are not impossible or we do not take them into account all terms are the SYNONYMS!

LVA Mathematics

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( ) ( ) ( ) ( ),t t t t MX CX KX F

( )tX

M

C

K

F

- vector of the deflections and slope angles at the system nodes;

- mass matrix;

- damping matrix;

- stiffness matrix;

- excitation forces vector.

where:

Lateral vibration equation(n-degree-of-freedom mechanical system)

Vibration glossaryFree vibration Occurs when F(t) =0 i.e. a mechanical system vibrates freely after an

initial motion was applied

Forced vibration Occurs where an alternating force F(t) ≠0 is applied to a mechanical system. In forced vibration the alternating force F(t) does not disappear when the excited motion is prevented

Self-excited vibrationOccurs where the alternating force F(t) that sustains the vibration motion is created or controlled by the vibration motion itself. When the motion stops the alternating force F(t) disappears

Steady vibration Vibration of a mechanical system caused by a periodic excitation when free vibration oscillations have decayed.

Harmonic excitation Occurs when the periodic excitation force alternates according to the harmonic law: F(t)=Asin(ωt+ψ)

Transient vibration Occurs when a non periodic alternating force F(t) is applied

Parametric vibration Occurs when mass M and/or damping C and/or stiffness K of a mechanical system are variable (M(t), C(t), K(t)), but not depend on vibration motion

Non liner vibration Occurs when mass M and/or damping C and/or stiffness K of a mechanical system depend on vibration motion X(t)

Non damped vibration Occurs when damping in vibrating system is equal to zero C=0

Weakly damped system

For weakly damped propulsion systems free vibration problem can be solved using the equation for non damped vibration

( ) ( ) .t t MX KX 0

Non damped natural frequencies and mode shapes are the main results of the free vibration calculation.

Non damped natural frequencies and mode shapes

Single span beam excited in two planes simultaneously

For a single span uniform beam, steady non damped forced vibration is a polyharmonical motion which is the sum of the two harmonic motions.

Shaft centerline point orbits

Shaft centerline points have a shape of Lissajou’s figures depending on frequencies relation and phase difference of the motions

Plane-parallel motion

When beam centerline orbit shape is an ellipse or a circle, it moves around geometry axis with a constant angular speed:

This motion is plane-parallel, i.e. shaft sections do not rotate around the cross section center.

x y

Precession

Every point of a flexible rotating shaft moves with angular speed around the tangent to deflection curve of the shaft.

At the same time there are no obstacles for the rotating shaft to vibrate in two orthogonally related planes if correspondent alternating forces are applied.

As a result the shaft axis points will travel in the space along the beam vibration orbits.

In the classical Mechanics the motion of the rotation axis is known as the precession.

Whirling vibration terms

Conventional name for the precession motion of a rotating shaft is whirling.

When the directions of shaft rotation speed ω and circular or elliptical

whirling speed Ω coincide, whirling is referred as a forward whirling.

If the directions are opposite, a circular or elliptical whirling is referred as a backward whirling.

When |Ω|= ω it is the case of a synchronous whirling otherwise it is an

asynchronous whirling |Ω| ≠ ω ).

Whirling of non rotating shaft(lateral vibration in two planes)

0 2 4 6

1

0.5

0

0.5

1

1

1

0 t( )

2 0 t

Ω / ω = 0

Surface point orbit Bending stress

Forward asynchronous whirling

0 2 4 6

1

0.5

0

0.5

1

1 t( )

t

Ω / ω = 5

Surface point orbit Bending stress

Forward synchronous whirling

0 2 4 61

0.5

0

0.5

1

2 t( )

t

Ω / ω = 1

Surface point orbit Bending stress

Backward asynchronous whirling

0 2 4 61

0.5

0

0.5

1

3 t( )

t

Ω / ω = -5

Surface point orbit Bending stress

Main sources of propulsion shafting whirling vibration

• pulsating hydrodynamic forces on propeller;• radial excitation forces in reciprocating engines;• pulsating lubrication pressures in plain bearings;• friction forces in shaft material and couplings.• unbalance;• shaft misalignment due to assembly errors or

manufacture defects;• manufacture defects in gearing.

Summary

1. Vibration of non rotating propulsion shaft system should be referred as a lateral vibration.

2. If the shaft cannot loss contact with the point wise supports lateral vibration may be referred as a bending vibration.

3. Transverse vibration is a bending vibration in the horizontal plane.

4. Simultaneous lateral vibrations of non rotating propulsion shaft system in the vertical and horizontal planes may be considered as a whirling vibration.

5. Vibration of a rotating propulsion shaft system should be considered as a whirling vibration in any circumstances

Gyroscopic effect

Gyroscopic moment

In a propulsion system, equipped with a heavy propeller, gyroscopic effect can influence the whirling vibration. When a huge mass with large polar and diametric inertia moments rotates gyroscopic moment arises:

gyrM

2 ,gyr dM A I 1 p

d

IA S

I

S

pI

dI

- polar inertia;

- diametric inertia;

- whirling factor (positive for forward whirling and negative for backward whirling).

- shaft rotation speed;

- whirling speed;

- shaft slope at the propeller node;

Gyroscopic moment can be positive and negative

• Positive gyroscopic moment increases deflections of a rotating shaft i.e. makes shaft ‘softer’

• Negative gyroscopic moment counteracts shaft deflection i.e. makes shaft ‘stiffer’

Natural frequencies for both types of whirling

Natural frequencies for both types of whirling, calculated depending on shaft rotation speed, usually are shown in the Campbell diagram.

Mode shapes of the multi-span beam whirling vibration

Critical speeds concept

• Speed of a rotating propulsion shaft at which severe vibration occurs is known as a critical speed.

• Critical speeds of weakly damped propulsion system coincide with the natural frequencies of the shaft whirling vibration.

• In cases when shaft is supported by the bearings which have different stiffness in horizontal and vertical plane (anisotropic bearings) twice as many critical speeds exist (separate for vertical and horizontal vibration).

Sources of whirling vibration

The sources of propulsion system whirling vibration

• propeller fluctuating loads; • diesel engine excitation; • oil whirl and oil whip;• hysteretic excitation; • unbalance;• alignment-related errors;• manufacture defects in gearing.

Propeller hydrodynamic loads• Transverse hydrodynamic

loads on the propeller are the main source of the whirling vibration of a propulsion system.

• Transverse hydrodynamic loads arise because an effective hydrodynamic force is not directed along the shaft centerline.

• Hydrodynamic loads must be determined as accurately as possible.

Propeller hydrodynamic loads consist of constant and fluctuating component!

Constant and fluctuating hydrodynamic components

Measured bending stresses in two propeller shaft sections

The fluctuating components of the hydrodynamic loads excite intensive whirling vibration of the propulsion shafting at the range 75-95 rpm.

The red line corresponds to the constant component of hydrodynamic loads.

Locus of the effective thrust force

Locus of the effective thrust force travels along the closed path that results in variable vertical and horizontal moments.The thrust force has some variable inclination angle to the propeller axis that results in variable vertical and horizontal forces.

Propeller hub

Fluctuating components of the propeller hydrodynamic loads

Fluctuating bending moment at the thrust bearing

The fluctuating component of the thrust can excite propulsion shafting whirling vibration owing to an alternate bending moment produced by the thrust bearing.

Diesel engine excitation

• excites whirling vibration in the same way as a torsional vibration is excited;

• the most reliable data for radial components of gas pressures are supplied by the engine manufacturers;

• all the problems of radial diesel engine harmonic coefficients determination is similar to the torsional vibration problems.

Oil whirl and oil whip

Oil whirl and oil whip relevant to high speed rotors

Below twice the first critical speed circulatory forces in lubricating film excite shaft whirling with the speed equal to one-half of the rotation speed.

Oil whip starts at twice of critical speed (mainly at low loaded bearings) and continue exists beyond that speed.

Hysteretic excitation

Whirling of a high speed rotors due to:• hysteretic damping in shaft material,• friction in the shaft couplings

is a self-excited vibration.

Hysteretic excitation is specific for the speeds above the critical speed.

Friction excites a backward whirling

Unbalance

• A propulsion shafting system is never perfectly balanced. Due to material inhomogeneity the center of gravity does not coincide with the rotation axis.

• When the shaft rotates, the eccentric masses produce exciting centrifugal force.

• Unbalance excites forward synchronous whirling.• Significant unbalance of a propulsion systems is

prevented by special technological standards and survey procedures.

Alignment-related errors

Alignment-related errors excite first order synchronous whirling vibration.

Such errors must be prevented by adherence to the technological specifications and standards of shaft manufacture and alignment processes.

Manufacture defects in gearing

Manufacture defects in gearing are the possible source of harmful whirling vibration too. It should be taken into account when whirling vibration problems of geared installations are under consideration.

Gearing excited vibration can be easily identified due to high frequency of vibration. But similar to alignment-related errors they never been taken into account in whirling vibration calculation

Shaft Modeling

Propulsion shaft line and its FEM model

FEM model parametersEvery shaft lumped mass has three characteristics, calculated as a half sum of the adjacent elements characteristics:• mass;• diametric inertia moment;• polar inertia moment.

Every flexible shaft element has five characteristics:• length;• cross section inertia moment;• effective section area;• module of elasticity;• Poisson’s ratio. Section and material

characteristics are the same for vertical and horizontal bending.

Lined elements

When the propeller shaft is lined for sea water corrosion protection element characteristics are calculated using liner dimension and material properties

Inertia moment of a lined element

44 164

L LE EI d d t

E E

Added mass

When the shaft element is submerged in a liquid its mass must be increased by added mass .

20m r

0

m

m

The increasing of propeller shaft mass leads to the lowering of critical speeds!

m

r

R

shaft radius;

stern tube internal radius;

liquid density.

Added mass influenceThe “Rotterdam” ship has 3-blade propeller so the forward whirling of the 3-rd excitation order is of a main importance.

Without added mass of the propeller shaft Added mass of the propeller shaft is taken into account

Flanged couplings

Geometric characteristics of short shaft elements with abrupt changing of diameters (such a shaft flange) must be reduced, because element’s material is not fully involved in bending deformations:

Crankshafts

• complicate structure for mass and stiffness characteristics modeling;• manufacturers never supply mass elastic system for the lateral

vibration;• for the propulsion shafting vibration calculation 3D detailed modeling

is excessive;• elements inertia moments and element stiffness are dependent on

shaft rotation angle.

Crankshaft 3-D solid model

Main bearing loads with one turn

In whirling vibration calculation mean effective values of mass and stiffness characteristics of the crankshaft are usually used.

It is supposed that due to the relatively short distances between engine bearings whirling vibration amplitudes of the whole propulsion system are not affected significantly.

(J. Bergande)

Bearings Modeling

Bearings are the lengthy supports

In whirling vibration propulsion shafting is subjected to bending deformations in two planes and moves within the bearings’ clearance.

Shaft whirling vibration motions

Red Green Blue Pink

The shaft motion orbits measured ABS within the aft stern tube bearing.

Bearing structure train

• lubricating film;• bearing bush;• bearing case;• bearing stool;• hull structure.

The way to model bearing structure train

Dynamic stiffness coefficients of the shaft support as well as damping coefficients are frequency dependent:

( )

yy yz y y

zy zz z z

by z

y z

k k k k

k k k kK

k k k k

k k k k

( )

yy yz y y

zy zz z z

by z

y z

c c c c

c c c cC

c c c c

c c c c

Each matrix’s element consists of two components: lubricating film component and structural component. The main problem is a determination each of these components.

Static and dynamic structural bearing stiffness

Excitation Point Excitation direction

Static stiffness[MN/m]

Dynamic stiff ness Excitation

frequency: 7 Hz [MN/m]

Dynamic stiff ness Excitation

fre quency: 10.5 Hz [MN/m]

Aft end of stern tube bearing

horizontal 850 2300 930

Intermediate bearing horizontal 970 4700 1800Fore engine main bearing

horizontal 4300 7300 3000

Aft end of stern tubebearing

vertical 910 470 1700

Intermediate bearing vertical 1500 870 3600Fore engine main bearing

vertical 3800 4100 4600

Calculated by L. Murawski for chemical tanker (40000 DWT, 180 m length),

Mega-yacht shaft whirling vibration

www.technofysica.nl

The stiffness of bearing support structural component was not agreed with a shaftingdynamic parameters

Lubricating film stiffness and damping

Revolutions

[rpm]

Dynamic stiffness

[MN/m]

Damping

[MNs/m]

30.0 69500 2900

50.0 31000 9500

76.0 25000 538

86.0 24600 471

95.0 24100 419

Lubricating film stiffness is much higher then those for the structural components.

Calculated by L. Murawski

General algorithm of bearing lubricating film stiffness and damping calculation 

For the current positions of the bearing defined by shaft alignment calculation the Reynolds’ equation:

3 30 0 00 0

( )6

p p hh h R

R R x x R

has to be solved in iterations to find the solution: function, integrals of which satisfy to shaft system static requirements:

( , )siny

F

R p x dF ( , ) cosz

F

R p x dF

R

0 ( , )h x ,

,x

– shaft radius;– gap thickness between shaft and bearing bush;– lubricant’s density and dynamic viscosity; – shaft angular speed;– shaft surface point coordinates

After the equilibrium solution is found additional dynamic lubrication pressure has to be calculated from the following Reynolds’ equation:

3 3 2 20 0* *

2 20 0

6 cos 3 sin 3 sin

6 sin 3 cos 3 sin

12 sin 12 cos ,

y

z

y z

p pp ph h h h u

R R x x R R x x

p ph h u

R R x x

u u

0 ( , )p x ,y zu u

,y zu u

– lubrication pressure for non vibrating shaft;

– vibration motions;

– vibration velocities;

0 ( , )p x

General algorithm of bearing lubricating film stiffness and damping calculation 

Finally dynamic stiffness and damping have to be calculated as the components of dynamic loads:

**( , )siny yy y yz z yy y yz z

F

R p x dF k u k u c u c u

**( , ) cosz zy y zz z zy y zz z

F

R p x dF k u k u c u c u

As can be seen from calculation algorithm, dynamic stiffness and damping coefficients are dependent on bearing shaft alignment parameters. In the case of harmonic excitation dynamic stiffness and damping are to be calculated for each frequency.

Propeller damping

Schwanecke’s formulas derived from unsteady propeller theory calculations:

22

0

0.1536 eF

AD Pc

D A

224

0

0.1128 eM

AD Pc

Z D A

propeller damping coefficients for linear and angular motions

Propeller damping is significant but its influence of the whirling vibration parameters is not as decisive as in torsional vibration case. Propeller is installed in the immediate proximity of the support (aft stern tube bearing) where the motions amplitudes are not considerable.

Whirling vibration parameters as a shaft alignment criteria

Whirling vibration parameters as a shaft alignment criteria

Two screw ship “Rotterdam” has two identical geared installations. Port side shaft line of the ship was run smoothly while the star board shaftline generated the violent vibration.

“Rotterdam” ship alignmentThe measurements performed Machine Support Company by revealed that the shaft lines are aligned differently:

Alignment of the port side shaftline (non vibrating)

Alignment of the star board shaftline (vibrating)

After starboard shaftline was realigned in the same manner as the port side shaftline, harmful vibration disappeared.

Motor and pump misalignment

To work perfect short and rigid motor and pump shafts must be aligned geometrically before mating

Ship shafts misalignment

Propulsion system shafts are always misaligned in the “motor-pump sense”

Shaft flexibility compensates to some extent the misalignment and vibration does not arise

What is a propulsion train shaft alignment?

To align the propulsion train means to find bearings’ space positionsfor which all formulated alignment acceptance criteria are satisfied.

Bearing positions are specified bythe linear and angular offsets:

, , ,x yW W

Currently whirling vibration criterionis not included in shaft alignmentcriteria list! But…

Shaft alignment and shaft whirling vibration are interconnected

Beam spans for vibration calculation purpose are measured between the reaction points. These points are migrating depending on bearings’ positions.

Shaft alignment influences the natural vibration frequencies!

Lubrication pressure distributions

Lubrication pressure distribution and lubricating film properties depends on shaft alignment parameters (bearings positions).

Shaft alignment influences the lubricating film dynamic properties!

Whirling vibration criteria for shaft alignment

• The critical speeds of whirling vibration are not to be in immediate vicinity of shaft nominal speed.

• The sum of the static load and additional dynamic load from whirling vibration shall not exceed the permissible value.

• Bearings are to be reliably loaded. The difference of the static load and additional dynamic load from whirling vibration shall not to be near zero or negative.

If the whirling vibration criteria are not satisfied the shaft alignment must be redesigned.

Conclusions

• Classification Societies require free whirling vibration calculation only. It is quite reasonably because the uncertainties of the excitation loads, dynamic stiffness and damping coefficients make impossible a rigorous prediction of whirling vibration parameters at present.

• Forced whirling vibration calculations, if they are undertaken, should be performed as parametric calculations.

• Propulsion system whirling vibration calculation is in need of further studies and developments.

Recommendations for prevention of the excessive whirling vibration

• Avoid having a long distance between the bearings.

• A bearing placed at the vibration node does not reduce the whirling vibration displacements.

• Too flexible bearings and bearing supports promote whirling vibration

• Couplings should be near a bearing, e.g. a tooth coupling has no bending stiffness.

• Large masses at the shaft free ends are critical. The length of the propeller shaft to the aft of the aft stern tube bearing should be controlled.

Thank you for your attention!

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