8 Critical Speeds Of Shafts.pdf

Critical speeds or whirling of Shaft S K Mondal’s

8.

Chapter 8

Critical speeds or whirling of Shaft Theory at a glance (IES, GATE & PSU)

Critical Speed Critical speeds are associated with uncontrolled large deflections, which occur when inertial loading on a slightly deflected shaft exceeds the restorative ability of the shaft to resist. Shafts must operate well away from such speeds. Rayleigh’s equation

m

KY

e O, s

2

O

G

Fig. Two Position of the rotor

Critical Speed of Shafts

s

G

MG (e + y)

Critical speeds or whirling of Shaft S K Mondal’s

Chapter 8

The speed at which the shafts starts to vibrate violently in the direction perpendicular to the axis of the shaft is known as critical speed or whirling speed. 1.

Critical speed of shaft carrying single rotor (without damping): 2

y

§ Z · ¨ ¸ e © ZC ¹ 2 § Z · 1¨ ¸ © ZC ¹

y = deflection of geometric centre due to C.F e = eccentricity of the rotor ZC = critical speed of shaft. K g G m Critical speed of shaft carrying single rotor (with damping). ZC

2.

2

y =

§ Z · ¨ ¸ e © ZC ¹ 2

2 ª § Z ·2 º ª Zº «1  ¨ ¸ »  «2[ » «¬ © ZC ¹ »¼ ¬ ZC ¼ Let 0 = point of intersection of bearing centre line with the rotor. S = geometric centre of the rotor. G = centre of gravity of the rotor. I = phase angle between e and y.

ª § Z· º « 2[ ¨ ¸ » ZC ¹ » 1 « © I = tan « 2» «1  § Z · » ¨ ¸ «¬ © ZC ¹ »¼ G I

s

S

O

O

Z<< Zc

Z< Zc

G I=

90

°

G

I S

S O

O

Z= Zc

Z> Zc

G

Critical speeds or whirling of Shaft S K Mondal’s

Chapter 8 I S 0.4

Z>>Zc

Multi Rotor

Fig. Typical multi-rotor system 1. Rayleigh’s Method: wC =

g 6mi y i 6mi y2i

2. Dunkerley’s Method: 1 1 1 1 =   ........ 2 2 2 ZC ( ZC1 ) ( ZC2 ) ( ZC5 )2

Objective Questions (IES, IAS, GATE) Previous 20-Years GATE Questions GATE-1. An automotive engine weighing 240 kg is supported on four springs with linear characteristics. Each of the front two springs have a stiffness of 16

Critical speeds or whirling of Shaft S K Mondal’s

Chapter 8

MN/m while the stiffness of each rear spring is 32 MN/m. The engine speed (in rpm), at which resonance is likely to occur, is [GATE -2009] (a) 6040 (b) 3020 (c) 1424 (d) 955 GATE-2. For lightly damped heavy rotor systems, resonance occurs when the forcing ω is equal to [GATE-1992]

(a) 2Zcr

(b) 2Zcr

1 ( d ) Zcr 2

(c) Zcr

Where ωcr is the critical speed GATE-3. A flexible rotor-shaft system comprises of a 10 kg rotor disc placed in the middle of a mass-less shaft of diameter 30 mm and length 500 mm between bearings (shaft is being taken mass-less as the equivalent mass of the shaft is included in the rotor mass) mounted at the ends. The bearings are assumed to simulate simply supported boundary conditions. The shaft is made of steel for which the value of E is 2.1 x 1011Pa. What is the critical speed of rotation of the shaft? [GATE-2003] (a) 60 Hz (b) 90 Hz (c) 135 Hz (d) 180 Hz

Previous 20-Years IES Questions IES-1.

Which one of the following causes the whirling of shafts? (a) Non-homogeneity of shaft material (b) Misalignment of bearings (c) Fluctuation of speed (d) Internal damping

IES-2.

Critical speed of a shaft with a disc supported in between is equal to the natural frequency of the system in [IES-1993] (a) Transverse vibrations (b) Torsional vibrations (c) Longitudinal vibrations (d) Longitudinal vibrations provided the shaft is vertical.

IES-3.

Rotating shafts tend of vibrate violently at whirling speeds because (a) the shafts are rotating at very high speeds [IES-1993] (b) Bearing centre line coincides with the shaft axis (c) The system is unbalanced (d) Resonance is caused due to the heavy weight of the rotor

IES-4.

A shaft carries a weight W at the centre. The CG of the weight is displaced by an amount e from the axis of the rotation. If y is the additional displacement of the CG from the axis of rotation due to the centrifugal force, then the ratio of y to e (where ωc, is the critical speed of shaft and w is the angular speed of shaft) is given by [IES-2001] (a)

IES-5.

1 2

ª Zc º «¬ Z »¼  1

(b)

re 2

ª Zc º «¬ Z »¼  1

ªZ º

2

(c) « c »  1 ¬Z ¼

(d)

[IES 2007]

Z 2

ª Zc º «¬ Z »¼  1

The critical speed of a rotating shaft depends upon [IES-1996] (a) Mass (b) stiffness (c) mass and stiffness (d) mass, stiffness and eccentricity.

Critical speeds or whirling of Shaft S K Mondal’s

Chapter 8

IES-6.

A slender shaft supported on two bearings at its ends carries a disc with an eccentricity e from the axis of rotation. The critical speed of the shaft is N. If the disc is replaced by a second one of same weight but mounted with an eccentricity 2e, critical speed of the shaft in the second case is[IES-1995 (c) N (d) 2N. ] (a) 1/2N (b) l/ 2 N

IES-7.

A shaft has two heavy rotors mounted on it. The transverse natural frequencies, considering each of the rotors separately, are 100 cycles/see and 200 cycles/see respectively. The lowest critical speed is [IES-1994] (a) 5367rpm (b) 6000rpm (c) 9360rpm (d) 12,000 rpm

IES-8.

Assertion (A): A statically and dynamically balanced system of multiple rotors on a shaft can rotate smoothly even at the 'critical speeds' of the system. Reason (R): Total balancing eliminates all the 'in plane' and 'out of plane' unbalanced forces of the system. [IES-2001] (a) Both A and R are individually true and R is the correct explanation of A (b) Both A and R are individually true but R is not the correct explanation of A (c) A is true but R is false (d) A is false but R is true

IES-9.

The critical speed of a shaft is affected by the (a) diameter and the eccentricity of the shaft (b) span and the eccentricity of the shaft (c) diameter and the span of the shaft (d) span of the shaft

IES-10.

Assertion (A): High speed turbines are run at a suitable speed above the critical speed of the shaft. Reason (R): The deflection of the shaft above the critical speed is negative, hence the effect of eccentricity of the rotor mass is neutralised. [IES-1998] (a) Both A and R are individually true and R is the correct explanation of A (b) Both A and R are individually true but R is not the correct explanation of A (c) A is true but R is false (d) A is false but R is true

IES-11.

An automotive engine weighing 240 kg is supported on four springs with linear characteristics. Each of the front two springs have a stiffness of 16 MN/m while the stiffness of each rear spring is 32 MN/m. The engine speed (in rpm), at which resonance is likely to occur, is [GATE -2009] (a) 6040 (b) 3020 (c) 1424 (d) 955

IES-12.

The critical speed of a uniform shaft with a rotor at the centre of the span can be reduced by [IES-1998] (a) reducing the shaft length (b) reducing the rotor mass (c) increasing the rotor mass (d) increasing the shaft diameter

IES-13.

Assertion (A): The critical speed of an elastic shaft calculated by the Rayleigh's method is higher than the actual critical speed. Reason (R): The higher critical speed is due to higher damping ratio. (a) Both A and R are individually true and R is the correct explanation of A (b) Both A and R are individually true but R is not the correct explanation of A

[IES-2000]

Critical speeds or whirling of Shaft S K Mondal’s (c) A is true but R is false (d) A is false but R is true

Chapter 8 [IES-2005]

IES-14.

A shaft of 50 mm diameter and 1 m length carries a disc which has mass eccentricity equal to 190 microns. The displacement of the shaft at a speed which is 90% of critical speed in microns is [IES-2002] (a) 810 (b) 900 (c) 800 (d) 820

IES-15.

The danger of breakage and vibration is maximum? (a) below critical speed (b) near critical speed (c) above critical speed (d) none of the above.

IES-16.

If a spring-mass-dashpot system is subjected to excitation by a constant harmonic force, then at resonance, its amplitude of vibration will be (a) Infinity [IES-1999] (b) Inversely proportional to damp in (c) Directly proportional to damping (d) Decreasing exponentially with time

IES-17.

Match List-I with List-II and select the correct answer using the codes given below the lists: [IES-1998] List-I List-II A. Node and mode 1. Geared vibration B. Equivalent inertia 2. Damped-free vibration C. Log decrement 3. Forced vibration D. Resonance 4. Multi-rotor vibration Code: A B C D A B C D (a) 1 4 3 2 (b) 4 1 2 3 (c) 1 4 2 3 (d) 4 1 3 2

[IES-1992]

Previous 20-Years IAS Questions IAS-1.

Whirling speed of a shaft coincides with the natural frequency of its (a) longitudinal vibration (b) transverse vibration [IAS-1995] (c) torsional vibration (d) coupled bending torsional vibration

IAS-2.

Assertion (A): Every rotating shaft has whirling speeds [IAS 1994] Reason (R): Eccentricity of rotors on rotating shafts is unavoidable. (a) Both A and R are individually true and R is the correct explanation of A (b) Both A and R are individually true but R is not the correct explanation of A (c) A is true but R is false (d) A is false but R is true

IAS-3.

Whirling speed of shaft is the speed at which (a) shaft tends to vibrate in longitudinal direction (b) torsional vibration occur (c) shaft tends to vibrate vigorously in transverse direction (d) combination of transverse and longitudinal vibration occurs

IAS-4.

The rotor of a turbine is generally rotated at

[IAS-2002]

Critical speeds or whirling of Shaft S K Mondal’s

IAS-5.

Chapter 8

(a) the critical speed (b) a speed much below the critical speed (c) 3 speed much above the critical speed (d) a speed having no relation to critical speed

[IAS-1999]

Consider the following statements The critical speed of a shaft if affected by the 1. eccentricity of the shaft 2. span of the shaft Of these statements: (a) 1 and 2 are correct (b) 1 and 3 are correct (c) 2 and 3 are correct (d) 1, 2 and 3 are correct.

[IAS 1994] 3. diameter of the shaft

Answers with Explanation (Objective)

Previous 20-Years GATE Answers GATE-1. Ans. (a) K = K1  K 2  K 3  K 4

fn =

1 2S

k m

GATE-2. Ans. (c) GATE-3. Ans. (b) Here, m = 10 kg = mass of rotar d = diameter of shaft = 30 u 105 m l = length of shaft = 500 u 10-3 m E for steel = 2.1u 1011N/m2 mgl 3 ' = deflection of shaft = 4gEI 4 S 4 S I d u 30 u 103 64 64





Critical speeds or whirling of Shaft S K Mondal’s

Chapter 8

3.976 u 10 8 m4 '

mgl 3 48EI



10 u 9.81 u 500 u 10 3



3

48 u 2.1 u 1011 u 3.976 u 10 8 3.06 u 10  5 m g 9.81 ' 3.06 u 105 Zn 2S 566.24 90Hz. 2 u 3.142

Zn fn

566.24rad / s

Previous 20-Years IES Answers IES-1. Ans. (a) IES-2. Ans. (a) IES-3. Ans. (d) IES-4. Ans. (b) IES-5. Ans. (c) 2

§S· Z1 ¨ ¸ ©l¹ IES-6. Ans. (c) 2 §S· Z1 ¨ ¸ ©l¹

IES-7. Ans. (a)

EI m

§S· ¨l¸ © ¹

2

gEI AJ

EI m

§S· ¨l¸ © ¹

2

gEI AJ

1 f n2

1 1  2 2 f1 f2

Z1

§S· ¨l¸ © ¹

IES-8. Ans. (d) IES-9. Ans. (c)

2

EI m

§S· ¨l¸ © ¹

2

IES-10. Ans. (c) IES-11. Ans. (a) K = K1  K 2  K 3  K 4

fn =

1 2S

IES-12. Ans. (c) 2 § S · EI Z1 ¨ ¸ ©l¹ m IES-13. Ans. (c) IES-14. Ans. (a) IES-15. Ans. (b) IES-16. Ans. (a) IES-17. Ans. (b)

k m §S· ¨l¸ © ¹

2

gEI AJ

gEI AJ

Critical speeds or whirling of Shaft S K Mondal’s

Chapter 8

Previous 20-Years IAS Answers IAS-1. Ans. (b) IAS-2. Ans. (b) Z1 IAS-3. Ans. (c) IAS-4. Ans. (c) IAS-5. Ans. (c)

§S· ¨l¸ © ¹

2

EI m

§S· ¨l¸ © ¹

2

gEI AJ