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vibration
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5/22/2018 Solved Problems in Vibration
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Turning moment diagram of a
multi cylinder engine
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A three cylinder engine has its crank set equally at
1200 and runs at 700 rev/min. the turning moment
diagram for each cylinder is a triangle and
maximum torque is 80 Nm at 60 0from top deadcentre of the corresponding crank. The torque on
the return stroke is zero. Determine,
1. Power developed
2. coefficient of fluctuation of speed if the mass of the fly
wheel is 10 kg and the radius of gyration is 100 mm.
3. Coefficient of fluctuation of energy
4. the maximum angular acceleration of the flywheel
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The turning moment diagram
0 60 120 180 240 300 360 420
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Solution
The work done per cycle = are of three triangles
Nm120
802
1
3
Mean torque= Nm602
120
2
cycleperdoneWork
m
T
Power = kW4.4100060
607002
100060
2
x
x
x
NTP
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The combined turning moment
diagram
0 60 120 180 240 300 360
A
B
C
D
E
F
G
H
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(ii) Coefficient of fluctuation of speed.
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(iii) Coefficient of fluctuation of
energy
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(iV) Angular acceleration
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problems in vibration
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From the free body diagram, The
equation of motion is given by.
The solution take the form
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s is a constant. Upon substitution into the differential equation,
which is satisfied for all values of t when
The roots are,
--------(18)
--------(19)
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Hence the general solution is
.
--------(20)
--------(21)
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overdamped (c/2m)2>k/m,
The general solution is given by,
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underdamped
(c/2m)2
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critical damping,
The general solution is given by,
tneBtA )(
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Problems in free vibration
Problem 1
Determine the natural frequency of a vibrating system
shown in the figure
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Let the mass m be displaced by a distance x
Applying Newtons law of motion,
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A cylinder partially immersed in water is dipressed
slightly and released. Find its natural frequency
assuming that it stays upright all the time.
L t
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Let
X - displacement of the cylinder
Aarea of the cylinder
mmass of the cylinder
rdensity of water
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Find the natural frequency of the system shown in
the following figure
Let us first find an equivalent position B for replacement of spring positioned at C
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Equivalent stiffness of two springs is given by,
Natural frequency,
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Determine the natural frequency of
the mass pulley spring system
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solution
m - mass of the block
M - mass of the pulley
r -radius of the pulley
x - displacement of the block
total kinetic energy is:
kinetic energy of mass + kinetic energy of pulley
The polar moment of inerta of the pulley (= Mr2/2)
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The polar moment of inerta of the pulley (= Mr/2)
we can assume that linear displacement x = r
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A uniform stiffrod of length l is restrained to move vertically by means of
both linear and torsional springs as shown in Figure. The stiffness of
linear spring is k N/mm and that of torsional spring is k Nm/rad. Calculate
the frequency of oscillation of the rod.
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Let the rod is displaced by angle .
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Calculate the natural frequency of a spring connected
pendulum system shown in Figure. The mass of
pendulum is mand spring stiffness is k. Neglect the
mass of the rod.
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A springmass--damper system consists of a spring of
stiffness 343 N/m. mass is 3.43 kg. The mass is
displaced by 20 mm beyond the equilibrium position and
Find the equation of motion for the system if the dampingcoefficient of the damper is
(1) 137.2 Ns/m and
(ii) 13.72 Ns/m.
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(1)
(2)
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This is an under-damped system and therefore,
the equation of motion is
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Differentiating the above equation,
Substituting initial conditions
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Solving above equations we get,
The equation of motion for under damped system is
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A single pendulum is pivoted at a point 0 as
shown in Figure. If mass of the rod is
negligible for small oscillations, find thedamped natural frequency of pendulum.
Solution
Solution
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Solution
Consider the position of the pendulum when it
rotates by small angle . The forces acting on the
pendulum rod are shown in the Figure 2. Taking
moments about point 0, we get
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Tutorial 9 Q3
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Taking mass A as the reference plane, the data may be tabulated as follows.
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First of all, the angular setting of masses C and D is obtained by drawing
the couple polygon from the data given in table. Assume the position of
mass B in the horizontal direction
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Now draw OC parallel to vector oc and OD parallel to bc. Measure
the angular position of C position of C
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Measure the angles and
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In order to find the required mass A and its angular setting, draw the
force polygon to some suitable scale. ( column 4) since the closing side
of the force polygon ( vector do) is proportional to 0.1 mA. Therefore
by measurement
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Now draw OA parallel to vector do
By measurement,
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