Dynamics questions

PR202 KINEMATICS AND DYNAMICS OF MACHINES
ASSIGNMENT
Balancing of Rotating Masses
1. Four masses m1, m2, m3 and m4 are 200 kg, 300 kg, 240 kg and 260 kg respectively. The corresponding
radii of rotation are 0.2 m, 0.15 m, 0.25 m and 0.3 m respectively and the angles between successive
masses are 45°, 75° and 135°. Find the position and magnitude of the balance mass required, if its
radius of rotation is 0.2 m.
2. A shaft carries four masses A, B, C and D of magnitude 200 kg, 300 kg, 400 kg and 200 kg
respectively and revolving at radii 80 mm, 70 mm, 60 mm and 80 mm in planes measured from A at
300 mm, 400 mm and 700 mm. The angles between the cranks measured anticlockwise are A to B 45°,
B to C 70° and C to D 120°. The balancing masses are to be placed in planes X and Y. The distance
between the planes A and X is 100 mm, between X and Y is 400 mm and between Y and D is 200 mm.
If the balancing masses revolve at a radius of 100 mm, find their magnitudes and angular positions.
3. A, B, C and D are four masses carried by a rotating shaft at radii 100, 125, 200 and 150 mm
respectively. The planes in which the masses revolve are spaced 600 mm apart and the mass of B, C
and D are 10 kg, 5 kg, and 4 kg respectively. Find the required mass A and the relative angular settings
of the four masses so that the shaft shall be in complete balance.
4. A shaft carries four masses in parallel planes A, B, C and D in this order along its length. The masses
at B and C are 18 kg and 12.5 kg respectively, and each has an eccentricity of 60 mm. The masses at A
and D have an eccentricity of 80 mm. The angle between the masses at B and C is 100° and that
between the masses at B and A is 190°, both being measured in the same direction. The axial distance
between the planes A and B is 100 mm and that between B and C is 200 mm. If the shaft is in
complete dynamic balance, determine : 1. The magnitude of the masses at A and D ; 2. the distance
between planes A and D ; and 3. the angular position of the mass at
Inertia Forces in Reciprocating Parts
5. In a slider crank mechanism, the length of the crank and connecting rod are 150 mm and 600 mm
respectively. The crank position is 60° from inner dead centre. The crank shaft speed is 450 r.p.m.
(clockwise). Using analytical method, determine: 1. Velocity and acceleration of the slider, and 2.
Angular velocity and angular acceleration of the connecting rod.
6. Find the inertia force for the following data of an I.C. engine. Bore = 175 mm, stroke = 200 mm,
engine speed = 500 r.p.m., length of connecting rod = 400 mm, crank angle = 60° from T.D.C and
mass of reciprocating parts = 180 kg.
7. The crank-pin circle radius of a horizontal engine is 300 mm. The mass of the reciprocating parts is
250 kg. When the crank has travelled 60° from I.D.C., the difference between the driving and the back
pressures is 0.35 N/mm2
. The connecting rod length between centres is 1.2 m and the cylinder bore is
0.5 m. If the engine runs at 250 r.p.m. and if the effect of piston rod diameter is neglected, calculate :
1. pressure on slide bars, 2. thrust in the connecting rod, 3. tangential force on the crank-pin, and 4.
turning moment on the crank shaft.
8. A vertical double acting steam engine has a cylinder 300 mm diameter and 450 mm stroke and runs at
200 r.p.m. The reciprocating parts has a mass of 225 kg and the piston rod is 50 mm diameter. The
connecting rod is 1.2 m long. When the crank has turned through 125° from the top dead centre, the
steam pressure above the piston is 30 kN/m2
and below the piston is 1.5 kN/m2. Calculate the effective
turning moment on the crank shaft.
9. The crank and connecting rod of a petrol engine, running at 1800 r.p.m.are 50 mm and 200 mm
respectively. The diameter of the piston is 80 mm and the mass of the reciprocating parts is 1 kg. At a
point during the power stroke, the pressure on the piston is 0.7 N/mm2
, when it has moved 10 mm
from the inner dead centre. Determine : 1. Net load on the gudgeon pin, 2. Thrust in the connecting
rod, 3. Reaction between the piston and cylinder, and 4. The engine speed at which the above values
become zero.

ASSIGNMENT
Turning Moment Diagrams and Flywheel
10. During forward stroke of the piston of the double acting steam engine, the turning moment has the
maximum value of 2000 N-m when the crank makes an angle of 80° with the inner dead centre.
During the backward stroke, the maximum turning moment is 1500 N-m when the crank makes an
angle of 80° with the outer dead centre. The turning moment diagram for the engine may be assumed
for simplicity to be represented by two triangles. If the crank makes 100 r.p.m. and the radius of
gyration of the flywheel is 1.75 m, find the coefficient of fluctuation of energy and the mass of the
flywheel to keep the speed within ± 0.75% of the mean speed. Also determine the crank angle at which
the speed has its minimum and maximum values.
11. A three cylinder single acting engine has its cranks set equally at 120° and it runs at 600 r.p.m. The
torque-crank angle diagram for each cycle is a triangle for the power stroke with a maximum torque of
90 N-m at 60° from dead centre of corresponding crank. The torque on the return stroke is sensibly
zero. Determine : 1. power developed. 2. coefficient of fluctuation of speed, if the mass of the flywheel
is 12 kg and has a radius of gyration of 80 mm, 3. coefficient of fluctuation of energy, and 4.
maximum angular acceleration of the flywheel.
12. A single cylinder, single acting, four stroke gas engine develops 20 kW at 300 r.p.m. The work done
by the gases during the expansion stroke is three times the work done on the gases during the
compression stroke, the work done during the suction and exhaust strokes being negligible. If the total
fluctuation of speed is not to exceed ± 2 per cent of the mean speed and the turning moment diagram
during compression and expansion is assumed to be triangular in shape, find the moment of inertia of
the flywheel.
13. The turning moment diagram for a four stroke gas engine may be assumed for simplicity to be
represented by four triangles, the areas of which from the line of zero pressure are as follows :Suction
stroke = 0.45 × 10–3 m2
; Compression stroke = 1.7 × 10–3 m2
; Expansion stroke = 6.8 × 10–3 m2
;
Exhaust stroke = 0.65 × 10–3 m2
. Each m2
of area represents 3 MN-m of energy. Assuming the
resisting torque to be uniform, find the mass of the rim of a flywheel required to keep the speed
between 202 and 198 r.p.m. The mean radius of the rim is 1.2 m.
14. The turning moment curve for an engine is represented by the equation, T = (20 000 + 9500 sin 2θ –
5700 cos 2θ) N-m, where θ is the angle moved by the crank from inner dead centre. If the resisting
torque is constant, find: 1. Power developed by the engine ; 2. Moment of inertia of flywheel in kg-m2
,
if the total fluctuation of speed is not exceed 1% of mean speed which is 180 r.p.m.; and 3. Angular
acceleration of the flywheel when the crank has turned through 45° from inner dead centre.
Governors
15. A Porter governor has equal arms each 250 mm long and pivoted on the axis of rotation. Each ball has
a mass of 5 kg and the mass of the central load on the sleeve is 25 kg. The radius of rotation of the ball
is 150 mm when the governor begins to lift and 200 mm when the governor is at maximum speed.
Find the minimum and maximum speeds and range of speed of the governor.
16. The arms of a Porter governor are each 250 mm long and pivoted on the governor axis. The mass of
each ball is 5 kg and the mass of the central sleeve is 30 kg. The radius of rotation of the balls is 150
mm when the sleeve begins to rise and reaches a value of 200 mm for maximum speed. Determine the
speed range of the governor. If the friction at the sleeve is equivalent of 20 N of load at the sleeve,
determine how the speed range is modified.
17. A loaded Porter governor has four links each 250 mm long, two revolving masses each of 3 kg and a
central dead weight of mass 20 kg. All the links are attached to respective sleeves at radial distances of
40 mm from the axis of rotation. The masses revolve at a radius of 150 mm at minimum speed and at a
radius of 200 mm at maximum speed. Determine the range of speed.

ASSIGNMENT
18. All the arms of a Porter governor are 178 mm long and are hinged at a distance of 38 mm from the
axis of rotation. The mass of each ball is 1.15 kg and mass of the sleeve is 20 kg. The governor sleeve
begins to rise at 280 r.p.m. when the links are at an angle of 30° to the vertical. Assuming the friction
force to be constant, determine the minimum and maximum speed of rotation when the inclination of
the arms to the vertical is 45°.
Longitudinal and Transverse Vibrations
19. Derive an expression for the natural frequency of free transverse and longitudinal vibrations by
equilibrium method.
20. Discuss the effect of inertia of the shaft in longitudinal and transverse vibrations.
21. Deduce an expression for the natural frequency of free transverse vibrations for a simply supported
shaft carrying uniformly distributed mass of m kg per unit length.
22. Deduce an expression for the natural frequency of free transverse vibrations for a beam fixed at both
ends and carrying a uniformly distributed mass of m kg per unit length.
23. Establish an expression for the natural frequency of free transverse vibrations for a simply supported
beam carrying a number of point loads, by (a) Energy method ; and (b) Dunkerley’s method.
24. Explain the term ‘whirling speed’ or ‘critical speed’ of a shaft. Prove that the whirling speed for a
rotating shaft is the same as the frequency of natural transverse vibration.
25. Derive the differential equation characterising the motion of an oscillation system subject to viscous
damping and no periodic external force. Assuming the solution to the equation, find the frequency of
oscillation of the system.

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