kul-24 4410 torsion vibrations.3

Pentti Häkkinen Chapter 3.3Wärstilä New Professionals 3.-7.11.2008

Pentti HäkkinenShaft vibrations -chapter 3.3

Vibrations of a rotating shaft

Torsional vibration

Longitudinal vibration

Transverse vibration (whirling) discussed separately



Vibrations

F

Periodic excitation force f = F sin t acts on the mass

Frequency f = 1 / T [1 / second]

Natural frequency f = 2 k / m

Angular velocity = 2 f [radian / second]

Force transmitted to foundation Ftr has

same frequency and same sinusoidal form as the excitation force

Period T

Ftr



Dynamic magnification at the resonance

Ratio of excitation force and transmitted force F / Ftr

Frequency

F

Ftr

Ftr/ F

1.0

Opposite phase

Same phase

Internal damping of the spring rounds-off the resonance peak and 'softens' the phase shift



Vibrations

Pure sinusoidal force excites vibration of the same frequency. Magnification is smaller or larger than 1.0 depending on how close the resonance lies.

Excitation Frequency Amplitude

LOW Below natural frequency

RESONANCEEqual to natural frequency

HIGH Above natural frequency



Vibrations

Non-sinusoidal force excites vibration of many various frequencies. Amplitude can be smaller or larger depending on the pulse form and the location of resonances.

Excitation Pulse frequency Amplitude

VERY LOW

MEDIUM HIGH

VERY HIGH



Vibrations

Non-sinusoidal excitation (‘pulses’) consists of endless number of sinusoidal curves, so called harmonic components. Fourier analysis is the mathematical method to calculate components.

Period = T

Frequency = 1/T

1/T

1/2T

1/3T

1/4T

1/5T

1/6T1/7T1/8T1/9T1/10T1/11T

etc.



Torsional vibration

When any harmonic component is equal to the natural vibration form, resonance occurs. The responses in resonance are typically sinusoidal. Even when basic excitation has strange pulse form. Outside resonance the vibration is not sinusoidal.

If strain gauges are attached on shaft to record torque variation

Typical signal in resonance is sinusoidal:

Outside resonance signal is irregular, but still periodical:



Shaft vibrations - general

Operation in resonance is avoided. Often only major resonances can be avoided and minor ones must be accepted.

Sometimes stress is critically high even outside resonance.

Damage is seldom shaft fracture. More often problem is wear in attached components.

Every system has natural vibration modes and frequencies. Periodic excitation comes from diesel engine or propeller.

If excitation frequency = natural frequency, resonance exists. Responses are magnified. Free end amplitude may be 3.0 mrad (0.17 degrees), shear stress amplitude 25 MPa (25 N/mm2)



Crankshaft stresses , 6-cyl. low speed engine

Shear stress amplitude MPa

40 60 80 100 120 140 160 180 rpm

80

60

40

20

Permitted Order 6.0 shear stress without damper

Order 6.0 shear stress with damper varustettuna

Nominal speed



Crankshaft stresses, MS Dredge Queen


200 250 300 350 400 450 500 550 rpm

30

25

20

15

10

5

Permitted by LR for 315mm shaft

Order 6.0 shear stress between cyl. 4 and 5 without damper.

Order 6.0 with damper

Nominal speed



Coupling vibratory loads

MS Dredge Queen

Torque amplitude kNm

200 250 300 350 400 450 500 550 rpm

80

60

40

10

Orders 0.5 and 1.0 in misfiring condition.

Order 1.0 normal firing

Nominal speed, nominal torque

20

Mean torque



Torsional vibrations

Basic modes - low speed engine

II mode I mode

Vibration form with free end amplitude +1.0. Other inertias get positive / negative values.

+1.0 +1.0

-1.0



Torsional stiffness and inertia figures

Torsional stiffness of a cylindrical piece (shaft part) follows equation:k = πD4G/32b, where D = diameter, b = length,G = material shear modulus.

Equations are available for conical pieces and sectional changes. Often equations result in too high stiffness.

Inertia of a cylindrical piece I = πD4bρ/32, where D = diameter, b = lenght (thickness), ρ = material density.

Complicated geometrical forms inertia and stiffness figures are supplied by the component manufacturer.

Stiffness grows in 4th power of diameter, and linear to length inverter

Inertia grows in 4th power of diameter, and linear to length



MT Parita mass elastic system

Nro

1

2

3

4

5

6

7

8

9

10

Item

Flange

Cylinder 1

Cylinder 2

Cylinder 3

Cylinder 4.

Camshaft drive

Flywheel

Flange

Flange coupling

Propeller

Inertia

420

12500

12500

12500

12500

5666

21650

770

1500

87050

Stiffness

2400

1671

1671

1671

2503

4438

6608

440

480

Inertia kgm2, Stiffness MNm/rad

Propeller inertia includes 18% of entrained water

Engine designer supplies inertia and stiffness figures. Shaft line figures follow the basic equations.



Frequency chart, MS Parita

Vibration frequency, cpm 1000

900

800

700

600

500

400

300

200

100

20 30 40 50 60 70 80 90 100 rpm

9

8

7

6

5

4

3

2

1

Orders 12 11 10

The reddish horisontal lines are the calculted two natural frequnecies – independent on revolution speed

Order are multiplies of the revolution speed. E.g. during one crank shaft revolution the 3rd order vibration makes 3 full cycles (sinus waves)



MT Tervi mass elastic

Nro

1

2

3

4

5

6

7

8

9

10

11

12

Item

Damper sec.

Damper pri.

Cylinder 1

Cylinder 2

Cylinder 3

Cylinder 4.

Cylinder 5

Can shaft drive

Flywheel

Flange

OK -coupling

Propeller

Inertia

32500

1737

17500

17500

17500

17500

17500

7666

21650

770

1500

87050

Stiffness

40

2400

1671

1671

1671

1671

2503

4438

8608

144

132


Damper


Engine designer supplies inertia and stiffness figures. Shafting figures follow the basic equations.



Frequency chart, MS Tervi


900

800

700

600

500

400

300

200

100

20 30 40 50 60 70 80 90 100 rpm

9

8

7

6

5

4

3

2

1

Orders 12 11 10

The reddish horisontal lines are the calculated two natural frequencies – independent on revolution speed




MT Tervi mass elastic diagram

Nro

1

2

3

4

5

6

7

8

9

10

11

12

Item

Damper sec.

Damper pri.

Cylinder 1

Cylinder 2

Cylinder 3

Cylinder 4.

Cylinder 5

Can shaft drive

Flywheel

Flange

OK -coupling

Propeller

Inertia

32500

1737

17500

17500

17500

17500

17500

7666

21650

770

1500

87050

Stiffness

40

2400

1671

1671

1671

1671

2503

4438

8608

144

132


Damper


Engine designer supplies inertia and stiffness figures. Shafting figures follow the basic equations.



Frequency chaft, MS Tervi


900

800

700

600

500

400

300

200

100

20 30 40 50 60 70 80 90 100 rpm

9

8

7

6

5

4

3

2

1

Orders 12 11 10

The reddish horisontal lines are the calculated two natural frequencies – independent on revolution speed




Real vibration form

Simple chain of four lumped inertias

'Node' is the point where angual amplitude reaches minimum – but seldom amplitude is = 0

Real systems have always considerable vibration damping. It lowers the resonance peaks but also affects the vibration phase. System free ends do not reach their extreme position simultaneously but there is phase difference.




Natural vibration forms, MS Dredge Queen

Lowest natural mode at 306 cpm. Node at generator branch coupling.

Second natural mode at 435 cpm. Nodes at generator and main coupling

( CPM= cycles per minute )




Natural vibration forms

Third natural mode at 1152 cpm. Nodes at generator and main couplings and shaftline

Fourth natural mode at 2560 cpm. Nodes at both couplings, shaftline and tors. damper springs.

Fifth natural mode at 2880 cpm. Nodes at both couplings, shaftline, damper and crankshaft.

( CPM= cycles per minute )



MS Dredge Queen vibration model

contents

Nro

1

2

3-11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

Name

Damper sec.

Damper pri.

Cylinders 1-9

Camshaftdrive

Flw + coupl.

Coupling sec.

Clutch

Gear wheel

Pinion

Bullwheel

Shaft mass 1

Shaft mass 2

Propeller

Gear wheel

Coupling pri

Coupling sec

Shaft mass

Generator

Inertia

87

13

110

70

184

11

56

100

43

623

175

435

5140

11

18

3.5

4

272

Stiffness

6.5

419

116 (290)

260

1.4

78

47

140-

-

210

38

29

-

27

0.28

2.5

13


Damper

NOTES:

Figures refer to actual rev.speed of each shaft.

Gear ratios: 16-22: 2.084

17-18: 0.346

Propeller inertia includes 25% water

Gear



Single Cylinder Excitation

Four stroke engine working cycle = 2 crankshaft revolutions

Tan

gent

ial f

orce

0 180 360 540 720Crankshaft angle degrees

Gas excitation

Mass inertia excitation

Resultant

Gas force

Tangential force

Inertia force

+

0

-



Excitation

Harmonic analysis of 4 stroke engine gas excitation

Replacement by Fourier series.

order 1 has one full sinus cycle during one crankshaft revolution.

Each component has different amplitude and phase angle.

Note half orders!

0.5

1.0

1.5

2.0

2.5

Mean indicated pressure = positive work

Working cycle = 2 revolutions



Excitation

Harmonic analysis of 2 stroke engine gas excitation

No half orders because engine cycle

= crankshaft revolution

1.0

2.0

3.0

4.0

5.0

Working cycle = 1 revolution



Frequency chart, MS Dredge Queen


4500

4000

3500

3000

2500

2000

1500

1000

500

100 150 200 250 300 350 400 450 500 550 rpm

9.08.58.07.57.06.5 6.05.5 5.0

4.54.03.5 3.02.52.0 1.51.0

0.5

Orders 12 11.5 11 10.5 10 9.5

Horizontal red lines are the 5 natural

frequencies



Excitation

Excitation from single cylinders are summed with phase shift related to the firing order. Example: 4 cylinder four stroke cycle engine

0 180 360 540 720Crankshaft angle degrees

Obviously cylinders balance each other and decrease torque variation. Accurate analysis requires look on the harmonic components.

Phase shift is similar to firing interval 180o.



Excitation

Order 1, 4 cylinder four stroke cycle engine


Phase shift 180o is such that excitation from cylinders completely cancel each other. Phase shift



Excitation



Phase shift 180o is such that excitation from cylinders completely cancel each other. RESULT



Excitation


For this order phase shift is 360o = such that excitation from cylinders pile up = cumulate.

Phase shift




Excitation


RESULT


For this order phase shift is 360o = such that excitation from cylinders pile up = cumulate.



Excitation


1.0

Order

1.0

Vector sum indicates how strong excitation comes from the engine compared with single cylinder. In this calculation the strength of single cylinder is multiplied with the cylinder relative amplitude. Opposite phase in the vibration form is observed by negative amplitude value.



Excitation


Order

1.0

Vector sum indicates how strong excitation comes from the engine compared with single cylinder. In this calculation the strength of single cylinder is multiplied with the cylinder relative amplitude. Opposite phase in the vibration form is observed by negative amplitude value.

calculated vector sum here is 0.7, not 0.Similarly vector sum of order 2.0 is 2.3, not 4.0

RESULT


Calculation of vector sums

Assume simple system consisting of 6 cylinder engine, short shaftline and generator, Fig. 1.

Fig. 2 shows the single node torsional vibration mode (1st mode).

The node point is on intermediate shaft. There is significant twist deformation also in crankshaft

1.0

Fig. 2 single node vibration

Fig. 1 Propulsion plant



Crankshaft of 2 stroke cycle 6 cylinder engine is in Fig. 3. Firing order is 1-5-3-4-2-6.

Crankshaft of 4 stroke cycle 6 cylinder engine in Fig.4. Firing order is 1-3-5-6-4-2.

There are alternative firing orders. They give even firing interval, but bring external engine unbalance

Fig. 3 2 stroke engine crankshaft

Fig. 4 4 stroke crankshaft

1

65

3 2

4

1 6

3 4 5 2



When highly elastic coupling is added between the engine and shaftline, vibration form changes, Fig. 5.

This natural frequency is much lower than in fig. 2. Crankshaft and shaftline behave as rigid bodies.

Amplitudes at free ends are similar to fig. 1.

1.0

Fig. 5 single node vibration form with elastic coupling

Elastic coupling


Torsional vibrations - Calculation of vector sums

Engine excitation is split in harmonic components.

For each harmonic order the phase angles between cylinders are modified, too. Angle is multiplied by the order. In Fig. 6 order is 3.

Fig. 7 shows phase diagrams for 6 cylinder 2 stroke cycle engine. Fig 8. for 6 cylinder 4 stroke cycle engine.

5

1

Angle between cylinders 1 and 5 is 60o. 3 * 60o = 180o . Angle between cylinders 1 and 4 is 180o. 3 * 180o = 180o.

Fig. 6 phase angle calculation

4

1



Orders 1 7 2 8 3 9

65

3 2

4

2 5 4 6

3 5 6

4 6 2 5

56

2 3

4

1 3 1 1 2 3 4 5 6

1 1 3 1 2 4

Orders 4 10 5 11 Main 6 12

Fig. 7 Main and side harmonic orders, 2 stroke engine

Cylinders



Orders 0.5 2.5 3.5 5.5 6.5 1 2 4 5 7 1.5 4.5 7.5

65

3 2

4

2 5 4 6

3 5 6

1 2 3 4 5 6

1 1 3 1 2 4

Main orders 3 6 9Fig. 8 Main and side orders, four stroke cycle engine



Fig. 9 shows vector sums for 6 cylinder four stroke cycle engine plant, calculated separately for each harmonic order.

They are obtained by addition of single cylinder vectors...

… which obtain their length from the relative cylinder amplitude (from Fig. 2) and their direction from phase diagram (from Fig. 8).

Resultant vector R shows how big is the engine excitation.



Harmonic 0.5 2.5 3.5 1 2 4 1.5 4.5 7.5 3 6orders 5.5 6.5 5 7 8

Fig. 9 Vector sums: one node vibration mode of Fig. 2, four stroke cycle engine without elastic coupling

6543

2

1

R = 4.18

1

2 4

6

53

R = 1.09

1

63

4

25

R = 0.20

1

4

52

3

6

R = 0.24



Harmonic 0.5 2.5 3.5 1 2 4 1.5 4.5 7.5 3 6 orders 5.5 6.5 5 7 8

Figure 10 Vector sums: one node vibration mode of Fig. 5, 4 stroke cycle engine with elastic coupling

6

5

4

3

2

1

R = 5.78

R = 0.05

1

35

6

4 2R = 0.02

1

6

34

25

R = 0.18

1

4

52

3

6



Fig. 11 Second, two node vibration form with elastic coupling and 6-cylinder, 4 stroke cycle engine

1.0

Elastic coupling



Harmonic 0.5 2.5 3.5 1 2 4 1.5 4.5 7.5 3 6 orders 5.5 6.5 5 7 8

Fig. 12 Vector sums: two node vibration mode of Fig. 11, 4 stroke cycle engine with elastic coupling

3 2

1

6

5

4R

R = 0.02

R = 4.51

654

3

2

1Downscaled to 50%

R=0.01

1

2

3

6

5

4R=0.931 1

6

2

5

34



Vector sums MS Dregde Queen

(zoomed detail area of three lowest vibration forms)

200 250 300 350 400 450 500 550 rpm

0.5

1.0

1.5

2.0

2.5 Orders3.03.54.0

435 cpm

306 cpm

0.34 0.08 1.08 0.154.5

7.0

0.02 0.17

0.08 0.007

1152 cpm

0.014


Effect on engine load, misfiring

When injection to a cylinder fails, pme drops by 90 %. Excitation drops by 50 … 80 %.

If vector sum has been small (cylinders cancel each other), resultant grows 10 … 50 times higher. Stress level grows in same proportion.

Pme

Fig 13 Tangential gas excitation depending on load (selected orders)

ptan

1

1.50.52

46

Pme

Fig 14 Misfiring decreases the excitation

ptan0.5



Vector sums, MS Dredge Queen

When crankshaft vibrates as a solid piece, cylinders cancel each other. Resultant vector is ~ 0 (except for main orders). This is good, as orders 0.5 - 1.5 have high gas excitation.

When balance between cylinders is disturbed (misfiring), resulting vector sum raises to 0.9 causing high responses.

Deformations in crankshaft (twist) grow together with frequency. Vector sum of main orders decrease while other orders increase. Typical for orders 4.5 - 7.5. Crankshaft stress is the critical issue.

Damper is installed at crankshaft free end for protection, to reduce shear stress.



Damper

Damper is tuned to a close frequency. Steel spring deformations consume vibration energy.

Damper is not needed for 6-cylinder engine if large crankshaft diameter keeps the shear stress level low.

Vee-engines are equivalent to in-line engines. Higher cylinder numbers are generally more difficult.

Cranshaft free end flange Inner stern

Inertia ring = ‘seismic mass’

Oil inlet

Steel springs

Outside diameter up to 1200 mm, for low speed engines 2000 mm



Crank shaft stresses , 6-cyl. low speed engine


40 60 80 100 120 140 160 180 rpm

80

60

40

20

Permitted Order 6.0 shear stress without damper

Order 6.0 shear stress with damper

Nominal speed



Crankshaft stresses, MS Dredge Queen


200 250 300 350 400 450 500 550 rpm

30

25

20

15

10

5

Permitted by LR for 315mm shaft

Order 6.0 shear stress between cyl. 4 and 5 without damper.

Order 6.0 with damper

Nominal speed



Coupling vibratory load

Misfiring increases torque amplitude in elastic coupling and reduction gear. Heat may damage rubber couplings.

Torque amplitude in gear may exceed average torque. Some gear hammering is acceptable at low speed range. Gears are dimensioned typically for +25 or 35 % torque amplitude.

Coupling stiffness and flywheel inertia are utilized to move 0.5 & 1.0 order resonances outside normal speed range. In complicated plants some resonances must be tolerated.

Coupling should have high damping.




MS Dredge Queen

Torque amplitude kNm

200 250 300 350 400 450 500 550 rpm

80

60

40

10

Orders 0.5 and 1.0 in misfiring condition.

Order 1.0 normal firing

Nominal speed, nominal torque

20

Mean torque




MS Dredge Queen

Calculation continued to 600 rpm to show order 0.5 resonance. Amplitude at 520 rpm misfiring is too high. Nominal torque is 77 kNm, amplitude 35 kNm.

Gear hammering is not expected. But if coupling damping is halved, limit is reached at misfiring and 300 rpm.

Nobody intentionally - except in emergency.

But normal unbalance between cylinders causes excitation that is equivalent to complete misfiring.

Who would operate engines in misfiring condition?!



Coupling types

Coupling stiffness may depend on

vibration frequency, average torque, twist angle, ambient temperature, rev. speed etc.

Rubber couplings have low damping, = 0.15 …0.25.

Steel spring couplings can have = 0.7.

Rubber couplings can be connected in series if very low stiffness is required.

Coupling element dismantling must be foreseen. Normal overhaul period is 5000 h or higher.



Low speed engine vibrations

‘Simple vibrations’. With short shaftline only 1st mode is important.

Misfiring condition is usually not interesting.

If shaft generator requires an elastic coupling, misfiring is calculated like in 4 stroke engine plants.

Short shaftline raises the 1st natural frequency so that main order resonance lies at 80 … 100 % of nominal speed. Vibration shear stress in crankshaft, intermediate and propeller shaft are also high.



Torsional vibrations, low speed engines

For 4 and 5 cylinder engines

shaft diameter is increased and main order resonance pushed above nominal speed = undercritical operation.

For 6 and 7 cylinder engines

shaft diameter is decreased; engine pushed forward and flywheel added to front end. The resonance is brought below normal operation speed range = overcritical operation.

If barred speed range / high idle speed is not accepted, torsional vibration damper is required. Dampers are heavy and expensive but operate without trouble.




Under-critical tuning, 5 cylinder 700 mm bore engine

Propeller shaft L 5.2 m, D 825 mm Intermediate shaft L 6.3 m, D 760 mm

1st natural vibration mode

Torque in intermediate shaft

Shear stress in intermediate shaft

Rev. speed

MNm

3

2

1

Order 5

MPa100

50

Permissable 1, 2

Order 5

11500




Over-critical tuning, same 5 cylinder 700 mm bore engine

Prop.shaft UTS 600; L 9.1m, D 580 mm. Interm. shaft UTS 690 L 4.7m, D 460 mm

Natural vibration modes

Torque in intermediate shaft

Shear stress in intermediate shaft

Rev. speed

MNm

3

2

1

Order 5 without with damper

MPa100

50

12900



Longitudinal vibrations

Problems caused with weak foundation of super- long stroke low speed engine in resonant condition.

+ 2 mm

+ 2 mm

+ 1 mm



Longitudinal vibrations

Main excitation comes from engine.

Surprises were caused by super-long stroke engines. Crankshaft longitudinal stiffness was much lower than predicted.

Torsional and longitudinal vibrations are coupled in propeller and in crankshaft.

Integral dampers added in low speed engines.

RLA90

RTA84



Longitudinal vibration excitation

Propeller excitation is today no problem thanks to advance in hydrodynamic design.

Crankshaft elongates and shortens periodically due to combustion and inertia forces.



Longitudinal - torsional coupling

Coupling in propeller

Torsional vibration elongates / shortens the individual cranks.

Coupling in crankshaft

Torsional vibration amplitude in propeller causes longitudinal excitation on same frequency.

Resulting thrust variation may be 40 … 60 % of mean thrust.



Longitudinal vibration damper

Damper is fastened to the 1st main bearing shield.

Damper absorbs vibration energy and decreases amplitude in resonance.

Throttle is adjustable.

Oil supply Adjustable throttle

Forged flange

Main bearing

Free end



Longitudinal vibrations - Criteria

Bending stress level incrankshaft fillet is the criteria.

Crankshaft free end amplitude is monitored. Typical resonant amplitudes without axial damper +1.5mm, with damper + 0.5mm

Axial vibrations excite hull vibrations, causing complaints rather than structural damages.

!

kul-24 4410 torsion vibrations.3

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