solved problems in vibration

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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solved problems in vibration

Documents

solved problems

general solution

solutionm mass

blockm mass

natural frequencyassuming

turning moment diagram0

torque onthe

free body diagram