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TORSIONAL VIBRATION (Un-damped)
Aim: To determine the natural frequency of Torsional vibrations and compare with
theoretical value.
Apparatus: Stand, Torsional wire with rotar, scale, vernier caliper, recorder drum with
motor
Theory: Torsional (Angular) Vibration:
Vibration is a dynamic phenomenon observed as an oscillatory motion around an
equilibrium position. Vibration is caused by the transfer or storage of energy within
structures, resulting within structures, resulting from the action of one or more forces. This
obviously applies equally well to translational vibrations (in one or several linear degrees of
freedom) as it does to Torsional vibrations (in one or several angular degrees of freedom). Inthe latter case, the forcing function is one or more moments instead of linear force acting
on the test structure.
Accurate analysis of torsional plays an increasingly important role when
troubleshooting or designing rotating machinery, yet Torsional vibrations remain, without
modern day laser Doppler-based techniques, notoriously difficult to measure. It is important
to be able to measure and analyse Torsional vibration accurately because the vibrations in
rotations in rotating shafts are well-known sources of numerous vibration problems.
I
dL
Procedure:
Fix the one end of the shaft to the above bracket, and the other end is attached withthe rotor disc.
The length of the shaft can be varied by moving the bracket to any convenientposition along the frame, and then clamping it.
Note down the diameter, & length of the shaft with the help of vernier caliper andscale.
Note down the diameter of the disc, mass of the disc, and rigidity modulus of the disc.
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Attach the graph sheet to recorder drum with marker arrangement, this recorder drumis attached to the motor. Before twisting on the motor.
Twist the rotor through some angle and then release. The amplitude of vibration can be seen on the graphs, take on complete revolution. Measure the distance Y in the graph, as shown in figure
Figure:
Observation: Diameter of disc D = mm Mass of disc M = kg Diameter of shaft d = mm Length of the shaft L = mm Rigidity modulus of shaft material G = N/m2 Diameter of recorder drum dr = mm Speed of drum n = rpm
Calculations: Theoretical frequency of Torsional vibrations is given by:
fn(the) =
= Hzwhere,
J = Polar moment of inertia of the shaft = = m
4
I = Mass moment of inertia of the disc =
= kg.m2
Experimental frequency for Torsional vibrations is given by:fn(exp) =
where,
T = where, Y = distance/cycle on graph (m)
V = velocity of the recorder drum =
m/secSl. No fn(the) Distance(Y) % error =
X 1001
2Results:
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TORSIONAL VIBRATION (DAMPED VIBRATION)
Aim:To determine the logarithmic decrement, Damping ratio and damping coefficient for
the given Torsional pendulum.
Apparatus: Stand, Torsional wire with rotar, scale, vernier caliper, recorder drum with
motor, oil container.
Theory:
In general all physical systems are associated with one or the other type of damping.
Damping is the resistance to the motion of a vibration body. Types of damping are:
i) Viscous damping (for small velocities in lubricated sliding surfaces, dashpots etc.,and in case of eddy current damping). The damping resistance is proportional to
the relative velocity. Analysis is simpler as the differential equation is linear
ii) Dry friction damping is constant and independent of velocityiii) Solid or structural damping is due to internal friction of moleculesiv) Slip or interfacial damping is due to microscopic slip on the interfaces of machine
parts in contact under fluctuating loads.
Differential equation of motion for a damped free vibration for degree of freedom system
figure a is,
where c = damping coefficientGeneral solution of above equation can be obtained as,
where,
c1 and c2 can be evaluated by applying boundary condition
Critical damping coefficient (Cc):
For critical damping, condition is (C = Cc)
or
Damping factor or Damping ration ():
It is the ratio of the damping coefficient to critical damping coefficient
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Over damping : > 1
Critical dampig : = 1
Under damping : < 1
Damped Natural Frequency for the above type is,
Logarithmic Decrement:
It is the ration of any two successive amplitude for an underdamped system vibrating
freely and is a constant and function of damping only.
( )Also,
Fig (a) Free vibration with viscous damping
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1x
2x
x3
Under damped vibrations(Viscous Damping)
I
dL
I
dL
I
dL
Oil
Procedure:
Fix the one end of the shaft to the above bracket, and the other end is attached withthe rotor disc.
The length of the shaft can be varied by moving the bracket to any convenientposition along the frame, and then clamping it.
Note down the diameter, & length of the shaft with the help of vernier caliper andscale.
Note down the diameter of the disc, mass of the disc, and rigidity modulus of the disc. Dip that disc into the oil container, until the disc is in the oil Attach the graph sheet to recorder drum with marker arrangement; this recorder drum
is attached to the motor. Before twisting on the motor.
Twist the rotor through some angle and then release.
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The amplitude of vibration can be seen on the graphs, take on complete revolution. Measure the distance Y in the graph, as shown in figure
Observations:
1. Diameter of the disc D= ------ meters2. Mass of the disc M = -------- kg3. Diameter of the wire or shaft d = -------- m4. Length of wire L =---------meters5. Rigidity modulus of wire material (Steel) G = 84 GPa= 84x109N/m26. Diameter of the recorder drum dr= --------- mm7. Rotational Speed of the drum n=6 rpm.
Tabular column (Undamped Vibration)
Sl.No Theoretical
frequency fn(the)
Hz
Ratio of
Successive
amplitudes
Logarithmic
decrement
Damping
ratio
Damping
coefficient
C
Frequency of
Damped
vibrations fd
Specimen calculations:
Theoretical frequency of Torsional vibrations is given by:fn(the) =
= Hz
where,
J = Polar moment of inertia of the shaft = = m
4
I = Mass moment of inertia of the disc =
= kg.m2
Ration of successive amplitudes = = . Circular frequency n= 2 fn(the) = rad/sec
Logarithmic decrement = . Damping ratio = . Damping coefficient = N-sec/m Frequency of damped vibration = HzResults:
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SPRING MASS SYSTEM (UNDAMPED VIBRATIONS)
Aim: To determine the natural frequency of vibrations experimentally & compare with thetheoretical value.
Apparatus: Spring, Weight, Scale, Graph sheet.
Theory:
Degree of freedom (D.O.F):
The number of independent coordinates required to specify completely the geometric
location of the mass of the system in space. Single degree of freedom means only one
coordinate is required to define the geometric configuration of the system.
Differential equation of motion:
Differential equation of motion for a simple spring mass system for undamped freevibration is given by,
0kxxm Or 0xm
kx
Where 2n
m
k
Where n = circular frequency in rad/s
f = n/2 linear frequency in cycles/s (or Hz.)
f
1 Time period in seconds.
m
k k
Dm
k
x
Undamped Free vibrations
Procedure:
Fix one end of the given spring to the upper screw. Note down its free length. Attach the known weight to the lower end of the spring Note down the stretched length of the spring Place a graph sheet on the recorder drum and fix the marker to attached to it On the motor, then pull down the platform gently and then releases, thus setting the
spring into free vibration.
The amplitude of vibration can be seen on the graphs, take on complete revolution. Measure the distance Y in the graph, as shown in figure
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Observation:
Free length of spring lo= meters Mass attached to the spring, m = kg Elongation of the spring l= meters
Diameter of recorder drum dr= mm Rotational speed of the drum n = rpm.Tabular column (Undamped Vibration)
Sl.No Theoretical frequencyfn (the) Hz
Distance Y percycle from graph,
mm
Experimentalfrequency fn (exp)
Hz
% error = X 100
Specimen calculations:
Static deflection of spring = (ll0) Theoretical frequency = Hz , where g = 9.81 m/sec2 Experimental frequency is given by:
fn(exp) =
where,
T = where, Y = distance/cycle on graph (m)V = velocity of the recorder drum =
m/sec
Percentage of error = X 100
Results:
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SPRING MASS SYSTEM (DAMPED VIBRATIONS)
Aim: To determine the logarithmic decrement, damping ratio & damping coefficient for the spring
mass system with viscous damping.
Apparatus: Spring, Weight, Scale, Graph sheet, viscous oil
Observations:
Free length of spring lo= meters Mass attached to the spring, m = kg Elongation of the spring l= meters Diameter of recorder drum dr= mm Rotational speed of the drum n = rpm.
Tabular column (damped Vibration)
Sl.No Static
deflectionDMeters
Ratio ofSuccessiveamplitudes
Logarithmicdecrement
Damping ratio
Damping coefficient
C
Frequency of Dampedvibrations fd
Specimen calculations:
Static deflection of spring = (ll0) Theoretical frequency = Hz , where g = 9.81 m/sec2 Ration of successive amplitudes = = . Circular frequency n= 2 fn(the) = rad/sec Logarithmic decrement = . Damping ratio = . Damping coefficient = N-sec/m Frequency of damped vibration = Hz
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1x
2x
x3
Under damped vibrations(Viscous Damping)
k
m
C
x
Damped Free vibrations
Results:
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R.P
100mm 100mm 100mm
A B C D
rA
Problem 2: A rotating shaft carries four equal masses A, B, C and D of 50 gms each which
are placed in planes as shown in figure. The radius of rotation of masses B is 70 mm and the
angle between B & C is 900 and between B & D is 2400. Find the radii of rotation of masses
A, C & D and also the angular position of masses A for complete dynamic balance. Check for
the balance graphically.
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CRITICAL SPEED OF SHAFT (WHIRLING OF SHAFT)
Aim: To determine the critical speed of the given shaft theoretically and verify the same
through experiment
Apparatus: shaft, vernier caliper, scale, tachometer, Dimmer stat,
Theory:Whirling of shafts
In many practical applications such a turbines, compressors, electric motors and
pumps, a heavy rotor is mounted on a light weight flexible shaft that is supported between
bearings. The mass centre of rotor does not coincide with the centre line of the shaft. Thus
there will be unbalance in the rotor due to manufacturing errors. When the shaft rotates
centrifugal force is induced on the shaft, which makes it to bend in the direction of
eccentricity of rotor.
In addition to this other effects such as stiffness and damping of the shaft, hysteresis
damping, gyroscopic effects, and fluid friction in bearings also cause the shaft to bend. This
bending further increases eccentricity and hence the centrifugal force. This effect is
cumulative and ultimately the shaft may even fail. The extent to which the shaft bends
depends upon the eccentricity of the rotor mass and speed of the shaft. At certain rotational
speeds the shaft tends to vibrate violently in transverse direction. At these speeds the shaft
has a tendency to bow-out and whirl in a complicated manner.
This phenomenon is called whirling or whipping of shafts and the corresponding
speeds are referred as whirling or whipping or critical speeds of shafts. These critical speeds
are found to coincide with the natural frequencies of lateral (transverse) vibrations of the
shaft.
Figure:
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Procedure:
The experimental set-up is as shown in figure. Measure the diameter (d), length (l), and mass per unit length (m) of the shaft. Make the shaft to rotate (using motor) about its axis. Increase the speed slowly with
the help of the dimmer stat.
Immediately whirling of shaft is observed. During whirling, when the shaft deflects into single bow (1 st mode) of maximum
deflection. Note down the value of critical speed Nc1.
On further increasing the speed of the shaft, when the shaft deflects into two bows(2nd mode) of maximum deflection, note down its corresponding experimental value
of critical speed for that mode.
Compare the above values with that of theoretical critical speed for 1 st and 2nd modeby calculating with the help of equations or formulae.
Observation:
Diameter of the shaft d = mm Length of the shaft L = mm Density of the shaft material = kg/m3 Elastic modulus of shaft material E = GPa
Calculations:
I mode:
Critical frequency for first mode (Single loop) is given by:fc1 = 2.45 = Hz
where,
I = Moment of inertia of circular cross section of the shaft = = m
4
m =Mass per unit length of shaft =
= kg/m
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Theoretical critical speed of shaft for first mode is given by:Nc1 = 60 X fc1 = rpm
From experimentally the critical speed of shaft for first mode is given by:Nc1 = rpm
II mode
Critical frequency for second mode (Two loops with a node) is given by:fc2 = 7.95 = Hz
where,
I = Moment of inertia of circular cross section of the shaft =
= m
4
m =Mass per unit length of shaft =
= kg/m Theoretical critical speed of shaft for first mode is given by:
Nc2 = 60 X fc1 = rpm
From experimentally the critical speed of shaft for first mode is given by:Nc2 = rpm
Result:
For I Mode:
Theoretical critical speed of shaft Nc1 = rpm Experimental critical speed of shaft Nc1 = rpm
For II Mode:
Theoretical critical speed of shaft Nc2 = rpm Experimental critical speed of shaft Nc2 = rpm
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FRINGE CONSTANT USING PHOTOELASTIC
Aim: Determination of fringe constant of photoelastic material using:
a) Circular disc subjected to diametral compressionb) Pure bending specimen (four point bending)
Apparatus: Vernier calipers, circular polariscope and its accessories including light
source, polariscope and loaded model.
Theory:
Photoelasticity is an experimental method to determine stress distribution in a
material. The method is mostly used in cases where mathematical methods become quite
cumbersome. Unlike the analytical methods of stress determination, photoelasticity gives a
fairly accurate picture of stress distribution even around abrupt discontinuities in a material.
The method serves as an important tool for determining the critical stress points in a material
and is often used for determining stress concentration factors in irregular geometries. The
method is based on the property of birefringence, which is exhibited by certain transparent
materials. Birefringence is a property by virtue of which a ray of light passing through a
birefringent material experience two refractive indices. The property of birefringence or
double refraction is exhibited by many optical crystals. But photoelastic materials exhibit the
property of birefringence only on the application of stress and the magnitude of the refractive
indices at each.
Procedure:
Set up the apparatus. Measure all the magnitude of the model Switch on the apparatus Apply the load till first fringe formed Note down the value of applied load Repeat the procedure for more fringe lines Calculate the fringe constant
Observation:
Diameter of the disc d = mm Thickness of the disc h = mm Magnification factorm = b/a =
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Tabular column:
SL. No Load at the
end of
lever, FL,
(Newton)
Load on
Model, F
(Newton)
Fringe
order at
center of
the disc, nc
Slope of
calibration
curve,
F/nc
Material
fringe
value,
f
Model
fringe
value of
fm1
2
3
Specimen calculations:
Load on the model F = FL X m = N Slope from calibration curve (Load v/s fringe order) = N/fringe Material fringe value = N/mm-fringe
Model fringe value N/mm
2
-fringe
Calibration curve
Result:
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Calibration of photoelastic specimen subjected to pure bending
Aim: To determine the material fringe constant and model fringe constant for the given
photoelastic specimen under four-point bending
Observation:
Depth of the model d = mm Moment arm x = mm Thickness of the disc h = mm Magnification factor m = b/a= .
Procedure:
Setup the specimen under four point bending condition Measure all the magnitude of the specimen Switch on the main Apply the load until the first fringe is formed Note down the load reading Repeat the experiment for more fringe values Calculate the fringe constant
Tabular Column:
SL. No Load at theend of
lever, FL,
(Newton)
Load onModel, F
(Newton)
Fringeorder at
center of
the disc, nc
Slope ofcalibration
curve,
F/nc
Materialfringe
value,
f
Modelfringe
value of
fm
1
2
3
Specimen calculations:
Load on the model F = FL . m = N Slope from calibration curve (Load v/s fringe order) = N/fringe Material fringe value = N/mm-fringe Model fringe value = N/mm2-fringe
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Calibration curve
Result:
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Determination of stress concentration factor in a plate with a hole using
photoelasticity
Aim: To determine stress concentration factor for the given photoelastic specimen subjected
to tensile load.
Observation:
Width of the model w = mm Diameter of the hole d = mm Thickness of the plate h = mm Magnification factor m = b/a = . Material constant f = N/fringe
Procedure:
Measure all the dimensions in the plate with a hole Set the experimental setup Apply load till the formation of stress concentration around the hole Note down the load applied Calculate the stress concentration factor
Tabular column:
SL. No Load at the
end of
lever, FL,
(Newton)
Load on
Model, F
(Newton)
Fringe
order, nc
Maximum
stress max
Nominal
stress nom
Stress
concentration
factor K
1
2
3
Specimen calculation:
Load on the model F = FL . m = N Maximum stress induced in the neighborhood of the circular hole = N/mm2Nominal stress induced in the specimen based on c/s area at hole section
= N/mm2 Stress concentration factor
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Calibration curve
Result:
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PORTER GOVERNOR
Aim: To determine the frictional resistance at the sleeve, centrifugal forces on the governor
balls & to draw the controlling force diagram for porter governor
Apparatus:Porter Governor, Tachometer, dimmer stat, measuring scale.
Theory: Governors are used for maintaining the speeds of the engines within prescribed
limits from no load to full load. In petrol engines; governors control the throttle valve of
carburetor and in diesel engines; they control position of fuel pump rack.
Most of the governors are of centrifugal type. These governors use flyweights to create
centrifugal force. Depending upon the speed, position of weights change, which is
transmitted to sleeve through governor links. Ultimately, the sleeve operates the throttle or
fuel pump.
The apparatus consists of a spindle mounted in bearings vertically. Three types of governors
can be mounted over the spindle, namely Porter, Proell and Hartnell on the existing
apparatus. A sleeve attached to governor links is lifted by outward movement of balls, due to
centrifugal force. Lift of the sleeve is measured over a scale. The spindle is rotated by a
variable speed motor.
The governors may broadly be classified as:
1. Centrifugal governor2. Inertia governor
The centrifugal governors may further be classified as follows;
1. Pendulum typeWatt governor2. Loaded type
i) Dead weight type governor (Porter governor and Proell governor)ii) Spring controlled governors (Hartnell governor, Hartung governor, Wilson
Hartnell governor and Pickering governor)
In the porter governor, added weight is mounded on the sleeve. As the speed increasesdue to centrifugal force, the sleeve is lifted, and when the speed decreases the centrifugal
force decreases and the sleeve comes down.
In the hartnell governor the spring is used as the loading member. It consists of a two
bell crank levers pivoted to the frame. The frame is attached to the governor spindle and
is rotates with it. Each lever carries a roller at the end of horizontal arm and the vertical
arm carries the balls. A vertical compression spring provides a downward force on the
two collers through a coller on sleeve. The spring force may be adjusted tightening and
loosening the nut.
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In the proell governor the ball is fixed at the extensional of the lower links and upper
links are pivoted on the top. The weight are added at the centre of spindle in order to
obtain more equilibrium speed should be smaller and lower mass balls.
Figure:
Procedure:
Arrange the set up as a Porter or Hartnell governor. This can be done by removing theupper sleeve on the vertical spindle of the governor and using proper linkages
provided.
Make the proper connections of the Motor.
Switch on the control unit and rotate the dimmer stat knob slowly increasing the speedof the governor until the central sleeve rises off the lower stop and aligns with the first
division of the graduated scale.
Increase the speed in steps to give suitable sleeve movement Note down the sleeve displacement on the scale and the corresponding speed using
tachometer.
Repeat the sleeve displacement for different displacement and note down the speed.
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Compute the controlling force and radius of rotation for each speed. A graph with controlling force on y axis and radius of rotation on x axis is plotted.
This is the controlling force diagram.
Observation:
Mass of governor fly balls m = Kg Mass of central sleeve assembly M = Kg Length of upper links = Length of lower links L = meters Offset of links pivots from axis of rotation y = meters Initial vertical distance between top & bottom pivots H = meters
Tabular Column:
TrialNo
SpeedNrpm
SleeveLiftxmts
Distance'c'meters
Distance's'meters
Radiusofrotation
'r' mts
Angle''degrees
Height ofgovernor'h' mts
FrictionalForce 'f'Newton
Centrifugalforce'Fc' Newton
EffortENewton
PowerP, Nm
1
2
3
4
Specimen calculation:
i) Distance c = = mii) Distance s = = miii) Radius of rotation r = = miv) Angle = degv) Height of governor
= m
vi) Frictional force: we know that the equilibrium speed of a porter governor with equallink length and equal offset of the upper and lower links is given by:
{ }
Hence,
vii) Effort of the governor is the mean force exerted at the sleeve for a fractional changein speed, for 1% change in speed.
viii)Power = Effort x sleeve lift = E x = Nm
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ix) Angular velocity of the spindle = = rad/secx) Centrifugal force or controlling force = NPlot:
Radius of rotation r
centrifugalforce
Fc
Controlling force curve
Result:
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DETERMINATION OF PRESSURE DISTRIBUTION IN
JOURNAL BEARING
Aim: To determine the load carrying capacity, co-efficient of friction & power lost in
viscous friction and to draw the circumferential pressure distribution curve
Apparatus: Scale, tachometer, oil, apparatus setup model
Theory:
A journal bearing sometimes referred to as a friction bearing, is a simple bearing in which a
shaft, or journal, or crankshaft rotates in the bearing with a layer of oil or grease separating
the two parts through fluid dynamic effects. The shaft and bearing are generally both simple
polished cylinders with lubricant filling the gap. Rather than the lubricant just reducing
friction between the surfaces, letting one slide more easily against the other, the lubricant isthick enough that, once rotating, the surfaces do not come in contact at all. If oil is used, it is
generally fed into a hole in the bearing under pressure, as is done for the most heavily-loaded
bearings (main connecting rod big-end and camshaft) in an automobile engine. Simple oil
slinger in the sump and an appropriate feed hole in the bearing shell are considered
adequate for small single-cylinder engines, such as those used in lawnmowers.
A journal bearing works on the principle that, over an infinitesimally small length of the shaft
circumference, the theory of a lubricant pair can be applied. The convergence as well as the
viscosity and velocity of fluid generate a pressure film. As one surface moves, it drags oil
into the gap that is made between it and the other. As the oil moves forward, the space
decreases. The oil can be considered to be incompressible enough to generate pressure. This
pressure prevents oil from entering the gap created. The oil within the gap reaches a pressure
limit after which it pushes oil through the smaller space.
Procedure:
Fill the oil tank by using SAE-20 grade of SAE-40 grade oil in the reservoir and positionthe reservoir at appropriate height (above the level of the journal bearing)
Drain out the air bubbles from all the tubes on the manometer and check level balancewith supply level
Check that some oil leakage is there, some leakage of oil is necessary for cooling purpose Set that speed and the journal sum for about half an hour. Until the oil in the bearing is
warmed up and check the steady oil level at various tapings
When the manometers levels are settled down, take the pressure reading on manometertubes for circumferential pressure distribution and takes for axial pressure distribution
Repeat the experiment for various speeds After the test is over set diameter to zero position and switches off main supply Keep the oil tank at lower most position so that there will be number leakage in the idle
period
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Observation:
Diameter of shaft (Journal) d = m Diameter of bearing D = m Speed of journal n = rpm Viscosity of oil used (SAE 40) = Pa-sec Length of the bearing L = m Attitude of Eccentricity ratio = .
Tabular Column:
Tube
Number
Angular position
deg
Initial head of
oil hi , cm of oil
Final head of
oil hf, cm of oil
Actual head h = (hf-hi),
cm of oil
1 302 60
3 90
4 120
5 150
6 180
7 210
8 240
9 270
10 300
11 33012 360
Calculation:
i) Linear speed of the journal m/secii) Diametral clearance ratio .iii) Load carrying capacity * + Niv) Frictional force [ ] Nv) Coefficient of friction .vi) Power lost in viscous friction KW
To draw the circumferential pressure distribution
Draw a circle with initial head hi, as radius taking a suitable scale Divide the circle into 12 equal parts (of 300) each and mark the tube numbers 1,2,3 etc
on the divisions in the same sense as the equipment.
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Draw radial lines along the divisions Mark actual head (hf hi) (to the same scale selected earlier) along the respective
radial lines from the divisions 1,2,3 etc
If (hfhi) is positive mark outside the circle and if negative, mark within Join all the points thus obtained by a smooth curve.
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DETERMINATION OF PRINCIPAL STRESSES AND
STRAINS IN A MEMBER SUBJECTED TO
COMBINED LOADING USING STRAIN ROSETTES
Aim: To conduct experiment on a member subjected to combined bending and torsion usingRectangular & DeltaRosettes and to determine;
(iv) Principal Strains & their orientations(v) Principal stressesApparatus required: Strain rosettes, Cantilever shaft with torsion arm, Weights, Wheatstone bridge circuit, and Strain indicator.
Theory:When a machine or structural member is subjected to a system of load, stresses will
be induced in the member, which will result in strain and deformation. Strain gages whenmounted at any point on the surface of the body will enable to measure the strain at that point
in the direction of axis of the gage. If xx & yy are the normal strains along X & Y
direction and xy is the shear strain, then the strain in an arbitrary direction XX is
xx= xx Cos2 + yy Sin2 + xy Sin Cos
Where is the angle between X & Y axis .
In a general case we have to measure strain in three directions to obtain xx , yy and
xy in equation (1) . There are two types of gage rosettes:
1. Rectangular Rosette2. Delta Rosette
Observations: Elastic modulus of the material of the beam E = GPa Poisson's ratio v = . Type of strain gauges: Electrical resistance type : Rectangular & Delta arrangement
Procedure:Connect the rosette strain gauges to the strain indicator by using wires. Ensure that dummy resistance are connected properly ON the indicator and keep the indicator for at least 5 minutes, till the gauges heats up
and get stabilized.
Set the gain to 50 for each indicator by using multiturn gain control pot provided overthe indicator.
Balance the indicator. Ensure pressure gauge reading at zero. Tight the check valves of the hand pump and apply the pressure in the pressure vessel
cylinder.
Load the pressure vessel in step like 2/4/6/8/10/12/14 kg / cm2 Read the strain values from the indicator.
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Tabular column: (Rectangular Rosette)
Tabular Column
Specimen calculations:
Principal Strains:
( ) +ve sign for 1 (major principal strain) andve sign for 2(Minor principal strain)
(Note: i f any of the individual strain values are negative, substi tute them as it i s)
Principal Stresses:
(
)
( )
Tria
l No
Load,
Newton
Micro
Strain
''x(10
-6)
Micro
Strain
''x(10
-6)
Micro
Strain
'C'x(10
-6)
Major
Principal
strain
1
Minor
Principal
strain
2
Major
Principal
stress
1
Minor
Principal
stress
2
1 2
1
2
3
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Where, 1and 2 are the major and minor principal stresses.
Principal angle : ( )
and, Maximum Shear stress:
Delta Rosette:
Tabular column:
Specimen calculations:
Principal Strains:
+ve sign for 1 (major principal strain) andve sign for 2(Minor principal strain)
(Note: i f any of the individual strain values are negative, substi tute them as it i s)
Principal Stresses:
Trial
No
Load,
Newton
Micro
Strain
''x(10
-6)
Micro
Strain
''x(10
-6)
Micro
Strain
'C'x(10
-6)
Major
Principal
strain
1
Minor
Principal
strain
2
Major
Principal
stress
1
Minor
Principa
l stress
2
1 2
1
2
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( )
(
)
Where, 1and 2 are the major and minor principal stresses.
Principal angle:
and, Maximum shear stress:
Result:
1) For Rectangular RosettePrinciple Strains: 1 =
2 =
Principle Stresses: 1 =
2 =
Principle angle: 1 =
2 =
Maximum shear stress: max =
2) For Delta Rosette:Principle Strains: 1 =
2 =
Principle Stresses: 1 =
2 =
Principle angle: 1 =
2 =
Maximum shear stress: max =
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STRESSES IN A CURVED BEAM
Aim:To determine the stresses in inner and outer fibers of a curved beam of square crosssection using strain gauges.
Apparatus: Curved beam, Strain gauges (Electrical resistance type) mounted at inner,middle and outer radii of the semicircular beam, Strain indicator, Dead weights.
Theory:The apparatus consists of a main frame which is mounted on a working table & fixed to
the table top. The frame is placed on the table in vertical position with the shorter side
vertical. Five types of curved beams are provided which can be clamped to the vertical sides
of the main frame. Brackets are provided to fix one end of the curved beam to the main frame& the free end is provided with a hook, so that load may be attached in either vertical
position. A magnetic dial gauge stand is provided which can be attached to the main frame &
will measure the deflection of the desired point in mm.
Weight pan with suitable is provided for applying the load to the curved beam. Loads can
be placed in loading pan. Standard specimen with rectangular cross section of the curved
beam supplied with the apparatus is:
Semicircle Quadrant Curved Davit Angle Davit Circular ring.
Strain gauges are mounted on the beam at proper location to measure the strain by
using a digital strain indicator.
In mechanical engineering, crane hooks, proving ring and chain links are typical examples of
a curved beam.
The apparatus includes different beams, borne on statically determinate supports: a
circular beam, a single davit (curved & straight) etc.
All the beams have the same cross section and so the same 2nd moment of area. This
enables test results to be compared directly. Castiglianos theorem rule is employed
Procedure:
Fix one end of the curved beam as shown in the figure, to the curved beam as shownin the main frame, with the help of brackets provided.
Fix the dial gauge stands provided to measure the deflection at the required point ofinterest.
Make connection of the strain gauges to the strain gauge indicator. Apply load at the required point of loading and note the, deflection values. Also note the strain values as indicated in the strain gauge indicator.
Observations:
Outer radius of the curved beam ro= mm, Inner radius of curvature of the beam ri= mm Cross section , 15 x 15 mm square
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Length of straight portion L = mm. Elastic modulus of material of the beam E= N/mm2
180 mm
A
R27
.5
A
Section A-A
Load
R42.5
132
Tabular Column:
Trial
No
Load
F,
Newton
Strain
(x10-6)
Total Stress
Mpa
(Experimental
)
Stress ,Mpa
(N/mm2)
(Theoretical)
Total Stress
Mpa
(Theoretical)
(Inner)
(Outer)
(Middle)
i 0 bi bo d i 0
1 1 x 9.81
2 2 x 9.81
3 3 x 9.81
4 4 x 9.81
5 5 x 9.81
Specimen Calculations:
Bending moment about Centroidal axis M=F (L+rc)Where rc= radius of Centroidal axis = (r0+ri)/2
For the given beam, rc =35 m, L=180mm
Total stress at the inner fiber 1 1where is the strain in the inner gaugei E Total stress at the outer fiber 2 2where is the strain at the outer gaugeo E Direct stress
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Where 3 is the strain in the middle gauge
Theoretical bending stress at the inner fiber ibi
i
Mc
Aer
Theoretical bending stress at the outer fiber oboo
Mc
Aer
Theoretical direct stressd
F
A
Theoretical value of total stress at the inner fiber as both areopposite in sign
Theoretical value of the total stress at the outer fiber as both areopposite in sign
15
15
7.5
27.5
centre line
of curvature
180
Load line
r
C
A
N
A
rn
o iC C
NA=Neutral axis
CA=Centroidal axis
c
For the given beam of square section 15 mm 15 mm, from Data hand book,
15( ) 35 (35 34.4575)
42.5loglog
27.5
Eccentricity 0.5424
Also ( ) (34.4575 27.5) 6.9575
a
c n c
oee
i
i n i
he r r r
r
r
e mm
c r r mm
0 0nd ( ) (42.5 34.4575) 8.0425nc r r mm
3 6
1
3
0 3
Load F=2 kg=2 9.81=19.62 N
Bending moment M=19.62(35+65)=1962 N-mm.
, =207 10 5 10 =1.035 Mpa
= 207 10 (
bi
b
E
E
Experimentally
Sample Calculations : (For tr ial No 2)
6
3 6
2
-2) 10 = 0.414 Mpa
Direct stress 207 10 (1) 10 0.207 Mpa
19.6215 15
d
d
E
FA
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GYROSCOPE
Aim: To conduct an experiment on motorized gyroscope and to determine; Gyroscopic couple Applied couple and To demonstrate the effect of gyroscopic couple using right hand screw
rule
Observations:
Mass of the spinning disc M= ------------ kg Diameter of the disc D= ----------- meters Distance of the weights added from the fulcrum of the motor x= -------
meters
Tabular column:
Trial
No
Weight
added,
'W'
Newton
RPM
of the
disc
'N'
Angular
velocity
of spin
''rad/sec
Angle
''deg
Time
taken t
(for
precession
of'') sec
Angular
velocity of
precession
'p' rad/sec
Gyroscopic
couple Cg,
Nm
Applied
couple
Ca , Nm
12
3
4
Specimen calculations:
Mass moment of inertia of the disc = kg-m8
MDI
22
Angular velocity of spin 2 = rad/sec60
N
Angular velocity of precession = rad/sec180
pt
Gyroscopic couple Cg= =pI Nm Applied couple Ca= X =W Nm
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gC
aC
gCaC
Oa
a'
Characteristic curve Spin Vector diagram
As the disc is spinning counterclockwisewhen viewed from the front, theapplied couple acts so as to tilt the spin vector to oa'. The reactive couple tends
to rotate the entire system clockwisewhen viewed from the top, which is
evident from the Right hand screw rule.
DISC
FULCRUM
WEIGHT
WX