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 Page 2
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
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Dr D R Gordon, Torsional Vibrations, Level 4, 2015/16
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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’
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Dr D R Gordon, Torsional Vibrations, Level 4, 2015/16
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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
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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 Page 6
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
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JAN 2015
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242.3
3.242 x 8 = 25.93
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Dr D R Gordon ED&A4 Session 2015 -16 TORSIONAL VIBRATIONS Page 9
DEC 2011
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JAN 2010
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JAN 2007
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JAN 2005
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