exam2007_1

of 6 Dynamic Systems 334 (308814 v1) End of Semester 1 Examinations, June 2007

1. Five railway iron ore wagons are coupled (Figure 1) and are located on a section of track

with a 1 in 250 downgrade (falls 1m for each 250 m travelled horizontally). The masses of the wagons plus the ore they contain are, in their sequence, 41, 40, 43, 39 and 40 tonnes. The elasticity of the four couplings is identical and estimated to be 530 kNm-1.

(a) Use the Holzer method to show that one of the undamped natural frequencies of

longitudinal motion is 0.669128 Hz. {Note: Work with six significant figures.} [10 Marks] (b) Sketch a diagram of the associated mode shape, clearly annotating the magnitudes

of the wagons. Choose the magnitude of wagon 1 to be 1 unit. [3 Marks] (c) State unequivocally with which mode of longitudinal vibration this frequency is

associated (1st = lowest, 2nd=next higher, etc). [2 Marks] (d) The last column of your Holzer table contains numbers that have a physical

interpretation. State clearly the physical interpretation. [5 Marks]

[Total Marks 20]

k k k k

40 t 39 t 43 t 40 t 41 t

Grade = 1 in 250 downward

Figure 1.


2. (a) When a harmonic force of frequency is imposed on an elastically mounted mass, the

steady state amplitude response will be: ( ) = ttx Fst sin)( Show that the transmission ratio (the ratio of the magnitude of the force transmitted to the support frame to the magnitude of the harmonic force applied to the mass) is:

222

2

21

21

+

+

=

NN

Ndtransmitte

FF

[10 Marks] (b) Figure 2 shows a direct coupled motor/centrifugal pump whose rotational speed is 1475

rpm. The motor-pump set is mounted on a rigid sub frame. In order to isolate any vibration from the motor-pump set, the sub frame is mounted on vibration mounts in the form of springs. The designer wishes to achieve a transmission ratio of 0.04. Given that the motor/pump/sub frame assembly has a mass of 75 kg, select the appropriate spring stiffness, k. Note that only two of the four springs are shown in Figure 2. Figure 3, on the next page, may be used. [10 Marks]

[Total Marks 20]

motor

k k

pump

sub frame

Figure 2

Question 2 continued on next page.


Figure 3 3. Three rotating masses are mounted on the same shaft. All have their centres of mass offset

from the axis of rotation. The plane of mass B is 1.34 m from the plane of mass A and the plane of mass C is 3.3 m from the plane of mass A, on the same side as mass B. Two balance masses, M1 and M2, are to be fitted, each with its centre of mass 195 mm from the axis of rotation, in planes midway between A and B and between B and C respectively. Determine the balance mass magnitudes and angular orientation relative to mass A.

Plane Mass (kg)

Eccentricity of mass centre

(mm)

Angular orientation(degrees)

Distance from mass A

(m) A 13 270 0 0

M1 195 0.67 B 12 350 50 1.34

M2 195 2.32 C 18 150 140 3.30

[Total Marks 20]


4. (a) Explain the meaning of steady state frequency response. [3 Marks] (b) Small amounts of viscous damping (say 05.0


FORMULA SHEET

f kmN

= 12 f

kIN

= 12

( )

fk I I

I INA B

A B

= +12

k GJl

=

( )( )

( )fkI

kI

k kI

f I I I k k

I I Ix

x

y

y

x y

z

x y z x y

x y z

42

2 42 20 + + +

+

+ + =

kGJ

Xk

GJYx

xy

y= =;

j j ii

i j

kI=

=

= 1 21

1

i

( )&& & && & sinx qm

x km

x x x x x Ae tN Nt+ + = + + = = +0 2 02 ; ;

2

; ; 1 ; 2

2 ==== DNN fmk

mq

= =qq

qmkcritical 2

=

=

1 21 1

0

2

nxxn

ln

222

21

1

+

==

NN

F FXk

222

2

21

+

==

NN

Ninert rm

Xm

222

22

21

41

+

+

==

NN

Ndisp U

X

re

cm

N

N N N

=

+

2

2 2 2 2

1

2

2

2

22

ty

xyc

=

Q4. (a) The steady state frequency response is the constant amplitude sinusoidal

response that remains, after transients have decayed, when excited with an ongoing sinusoidal input. Also known as the particular integral.

(b) True. (c) a principal inertia axis. (d) Slope and displacement equal zero at both ends and one node in the centre of

the span. Left half out of phase with right half. (e) The force involved in accelerating the mass is very small compared to the

spring force and the spring force dominates. (f) Infinite. (g) Measure its transverse vibration frequencies by mounting on supports placed

in the same axial location as its bearings. (h) The natural frequency of the suspension must be well below the excitation

frequency. (i) True. (j) Eigenvalues are the natural frequencies. Eigenvectors are the mode shapes. (k) Forced vibration is the response to an external excitation whereas natural

vibration is the manifestation of the systems internal energy.

End of Examination Paper

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