Past Exams Tors Vibs and Solns 1516

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Dr D R Gordon, Torsional Vibrations, Level 4, 2015/16

Dr D R Gordon ED&A4 Session 2015 -16 TORSIONAL VIBRATIONS

Jan 2015 (14-15 session)

Q.3 The propeller P of a marine vessel is powered by means of a reciprocating Engine E

and an exhaust turbine T through a branched geared system G as depicted in

Figure Q.3.

(a)

Using the given DATA, develop an equivalent un-geared branched systemREFERRED ABOUT THE PROPELLER SHAFT and derive the appropriate

characteristic dynamic matrix equation [K] relating the system properties to

the amplitude of oscillation [ ] and the periodic torques [T] in the form:

[K][ ] = [T].

[10]

(b) During a resonance test it was found that a free vibration response produced

oscillations of 2 degrees at the gearwheel when subject to a resonant frequency

of 10.72 rad/sec. Using the matrix derived in (a) above determine, the

amplitude of the oscillations of the turbine.

[5]

(b) Determine the amplitude of the periodic torque on the propeller shaft when the

engine produces a periodic torque of the form 50.Cos (25t) Nm.

[10]

DATA: Moment of Inertia Torsional Stiffness

TURBINE: ‘IT’ = 1.5625 kgm

2

‘K T’ = 468.8 Nm/rad

ENGINE: ‘IE’ = 8 kgm2 ‘K E’ = 400 Nm/rad

PROPELLER: ‘IP’ = 400 kgm2 ‘K P’ = 20,000 Nm/rad

GEARS: Gearwheel ‘IG’ = 109.5 kgm2

Turbine Pinion ‘I1’ = 2 kgm2

Engine Pinion ‘I2’ = 2.5 kgm2

IT

IE

IP I1

I2

IG K P

K E

K T

Figure Q.3

Gear ratio

8:1

Gear ratio

5:1

deg92.25

deg242.3

T

T





December 2011 (2011-12)

Q.3 A motor ‘M’ operating at 600rev/min drives two machines ‘A’ and ‘B’ by means of a

speed reduction gearbox assembly ‘G’ which has an effective combined moment of

inertia IG as shown in Figure Q.3. Machine ‘A’ operates at 400rev/min in the opposite

direction to that of the motor, whilst machine ‘B’ operates at 200rev/min in the same

direction as the motor. The moments of inertia of each machine and the torsionalstiffness for each shaft connecting these machines to the gearbox assembly are as

listed in

Table Q.3(a) below.

Table Q.3(a)

Machine

Moment of

Inertia (kgm2) Shaft

Torsional

Stiffness

(Nm/rad)

M 2 MG 200Nm/rad

A 9 AG 675Nm/rad

B 9 BG 900Nm/rad

(a) Refer all components to motor ‘M’ and derive the equivalent Dynamic Matrix

for the referred system.

[10]

(b) Given that the fundamental frequency of torsional oscillation of the system

and the corresponding non-referred normalised mode shape are as shown in

Table Q.3(b) below, verify that the effective moment of inertia of the gearbox

IG is 0.3404kgm2.

[8]

Table Q.3(b) ωn

(rad/s)

Θm

(rads)

Θg

(rads)

Θa

(rads)

Θb

(rads)

9.35 1 0.126 0.507 0.333

(c) It is a design requirement that the system ‘node’ is to occur within the gearbox

assembly ‘G’. Determine the fundamental frequency of this new system and

the required moment of inertia of machine ‘A’.

[7]

M

B G

A

IA

IM

IB

Figure Q.3

Gearbox

Assembly

‘IG’





January 2010 (2009-10)

Q.2 The schematic image of the rotary components of a turbo-compressor is

shown in Figure Q.2. The various mass moments of inertia are to be determined from

the ‘undamped’ experimental measurements of ‘actual normalised response’ as

given in Table Q.2. The gear ratio is known to be 8:1 and the gearwheel is assumed to

singly introduce any damping effects due to oil sump effects. Referring all to themotor shaft and ignoring shaft mass effects, determine:

i) the actual moments of inertia of the component parts IM, IG, IP and IC, given IP

is 10% of IG;

[12]

ii) the amplitude of vibration at the Turbo Compressor IC, if the Electric Motor

introduces an excitation torque of the form TM = 1000cos(315t) Nm, given the

gearwheel introduces a damping coefficient CTG of 500Nms/rad.

[13]

Natural Freq. Actual Mode Shape Measurements

(rads/sec) M

G

P

C

5471.281 n 1 0.6944 -5.5552 1.7512

Table Q.2

Oil Sump

k motor =80,000 Nm/rad

k turbo=781.25 Nm/rad

Figure Q.2

IM

Electric

Motor

Turbo

Compressor

IC

IG

IP

Gear Wheel

Pinion

Damping

CTG





January 2005 (2004-05)

Q.5 A geared and branched torsional vibration system is idealised in the model shown in

Figure Q.5. The information for each element of the system is indicated in Table Q.5

where IG1, IG2 & IG3 are gearwheels.

(a)

Determine the natural frequency ωn1, and the value of the Inertia IA, for thenode to be located at the gears.

[12]

(b) Derive, from either first principles or other means, the characteristic matrix

equation for the entire torsional system referred particularly to the shaft

containing k B , IB1 & IB2. The gear ratio between IG1/IG2 is ‘N’:1, and that for

IG2/IG3 is 1:1.

[8]

(c) Explain, without calculation, the necessary modification to the matrix in (b)

above if damping is to be introduced at IA , IB1 & IB2 only, and the effect this

would have on the natural frequencies and amplitudes of the system.

[5]

INERTIA VALUE (kgm ) STIFFNESS VALUE (Nm/rad)

IA ??? k A -

IB1 4 k B 10,000

IB2 2 k C 10,000

IC1 4

IC2 2

IG1 1

IG2 0.5

IG3 0.5Table Q.5

Figure Q.5

IA

IB1

IC2

IG1

IG2

IG3

40mm

dia.50mm

dia.

60mm

dia.

k B

k C

k B

k C

IB2

IC1

1m 1.5m1m

Data: G= 80GN/m2





JAN 2015





242.3

3.242 x 8 = 25.93





DEC 2011





JAN 2010





JAN 2007





JAN 2005

Past Exams Tors Vibs and Solns 1516

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