list of experiments - web viewthe planetary gear train thus formed is over closed and thus there...

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DYNAMICS OF MACHINERY LABORATORY

LIST OF EXPERIMENTS

1. Determination of influence co-efficient for multi degree freedom suspension system(– Bifilar & Trifilar).

2. Determination of torsional frequencies for compound pendulum and flywheel system with connecting rod.

3. Whirling of shaft - Determination of critical speed of shaft with concentrated loads.

4. Balancing of rotating masses & reciprocating masses.

5. Vibrating system (Spring Mass system) – Determination of damping co-efficient of single degree of

freedom system.

6. Motorised Gyroscope-Verification of laws -Determination of gyroscopic couple.

7. Governors - Determination of sensitivity, effort, etc. for, Hartnell governor.

8. Cam - Study and drawing profile of the cam.

9. Determination of transmissibility ratio - vibrating table.

10. Transverse vibration – Determination of natural frequency and deflection of cantilever beam and Fixed

beam.

11. Study on types of Gear Trainsmission

12. Study on types of Dynamometers


INDEX

Sl.No. List of Experiments Page No.

1 MULTI DEGREE FREEDOM SUSPENSION SYSTEM – BIFILAR & TRIFILAR SUSPENSION 4

2 COMPOUND PENDULUM & FLYWHEEL AND CONNECTING ROD 8

3 WHIRLING OF SHAFT 12

4 BALANCING OF ROTATING MASSES & RECIPROCATING MASSES 14

5 FREE VIBRATION OF SPRING MASS SYSTEM 17

6 DETERMINATION OF GYROSCOPIC COUPLE 19

7 HARTNELL GOVERNOR 21

8 CAM ANALYSIS 23

9 VIBRATING TABLE SETUP 25

10 TRANSVERSE VIBRATION OF CANTILEVER BEAM & FIXED BEAM 27

11 STUDY ON TYPES OF GEAR TRAINS 32

12 STUDY ON TYPES OF DYNAMOMETERS 38

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Experiment No. : 1MULTI DEGREE FREEDOM SUSPENSION SYSTEM

(a) BIFILAR SUSPENSION

Aim: To determine the radius of gyration of a given rectangular plate using bifilar system

Apparatus required: Main frame, Bifilar plate, Weights, Stopwatch, Threads, etc.

Formula used:Time period, T = t/nNatural frequency, fn = 1/T Hz

Radius of gyration, k = 12 π

bf n √ g

Lwhere, b - distance of string from centre of gravity

L - length of the string n - number of oscillationst - time taken for N oscillations

Procedure:1. Select the bifilar plate.2. Fix the plate with the nylon string to the hook on the frame. 3. With the help of chuck tighten the string at the top.4. Adjust the length of string to desired value.5. Give a small horizontal displacement about vertical axis passing through its center.6. Start the stop watch and note down the time required for ‘n’ oscillation.7. Repeat the experiment by adding weights and also by changing the length of the strings.8. Do the model calculation

Observation & Tabulation:b = 10.15 cm

Tabular column:

Sl.No.

Weight addedm (kg)

Length of string

L (m)

No. of oscillations

n

Time taken for n osc.

t (sec)

Time periodT (sec)

Natural frequency

fn (Hz)

Radius of gyration k (mm)

123

Graph: A graph is plotted between weights added and radius of gyration

Result: Radius of gyration = ……….



Bifilar Suspension

Viva Questions1. Explain the term Suspension.2. Why do we need suspension in a system?3. What do you infer from this experiment?4. What are the factors affecting the oscillation?5. Why does the system come to a stop when oscillated?

(b) TRIFILAR SUSPENSION

Aim: To determine the radius of gyration of the circular plate using trifilar suspension

Apparatus required: Main frame, Chucks 6 mm diameter, Circular plate, Strings, Stop watch, etc.

Formula used : Time period, T = t/n



Natural frequency, fn = 1/T Hz

Radius of gyration, k = 1

2πbf n √ g

Lwhere, b - distance of a string from center of gravity of the plate,

L - Length of string from chuck to plate surfacen - number of oscillationst - time taken for N oscillations

Procedure:1. Hang the disc from chucks with 3 strings of equal lengths at equal angular intervals (1200 each).2. Give the disc a small twist about its polar axis.3. Measure the time taken for ‘n’ no of oscillations with the help of stop watch.4. Repeat the experiment by changing the lengths of strings and adding weights.

Observation & Tabulation:b =

Tabular column:Sl. No.

Length of string

L (m)

Addedmass

m (kg)

No. of oscillation

n

Time for n oscillations

t (sec)

Time periodT (sec)

Radius of gyration

k (m)

Natural frequency

fn (Hz)1234

Graphs: Plot the graph between weight added vs radius of gyration.




Viva Questions:1. What is Moment of Inertia?2. Define Mass and Weight.3. Define Centre of Gravity.4. What do you understand by Centre of percussion?5. Name a few applications of Suspension system.



6. Experiment No. : 2(a)COMPOUND PENDULUM

Aim: To determine the radius of gyration and mass moment of inertia of the given rectangular rod experimentally

Apparatus required: Vertical frame, Rectangular rod, Stop watch, Steel rule, etc.

Procedure:1. Measure the length of the rod using steel rule.2. Suspend the rod through any one of the holes.3. Measure the distance from point of suspension to the centre of gravity of the rod. 4. Give a small angular displacement to the rod & note the time taken for 10 oscillations5. Repeat the step by suspending through different holes.

Formula used:Time period theoretically, T = t/n sec (n = 10)Experimental time period T = 2((k2 + L1

2)/gL1)where, k = experimental radius of gyration and k = ((gL1T2/42) – L1

2)L1 = distance from point of suspension to centre of gravity of rod L = total length of the rod

Theoretical radius of gyration, kt = L/12 = 0.2866L

Observations & Tabulation:L = ……….

Tabular column:Sl.No.

Distance L1 (m)

Time for 10 oscillations t

(sec)

Time period T (sec)

Natural frequency

fn (Hz)

Experimental radius of gyration

k

Theoretical radius of gyration

kt




Compound Pendulum

Viva Questions:1. What is Oscillation?2. Name of a few applications of oscillation.3. What is a simple harmonic oscillator?4. What is acceleration dude to gravity?5. What is radius of gyration?

Experiment No. : 2(b)FLYWHEEL AND CONNECTING ROD



Aim: To determine the Moment of Inertia by oscillation of flywheel and connecting rod.

Procedure:1. Measure the centre to centre distance of connecting rod. Also measure inner dia of both side of

connecting rod. 2. Measure the weight of connecting rod and flywheel.3. Attach small end of connecting rod of shaft.4. Give oscillation of connecting rod.5. Measure time taken for 5 oscillations and calculate time period tp1.6. Remove the connecting rod from the shaft and again attach the big end of the connecting rod of shaft.7. Again measure time taken for 5 oscillations and calculate time period tp2.8. Calculate Moment of Inertia of connecting rod.9. Repeat the procedure and take mean tp.10. Attach flywheel to other side of shaft and repeat the same procedure as above.

Formula used: Moment of Inertia is given by,

I = mk2

where, m – Weight of connecting rodk – Radius of gyration of connecting rod

Radius of gyration is given by,k2 = h (L - h)h = (h1 + h2) / 2h1 + h2 = d1/2 + L + d2/2L = (L1 + L2) / 2L1 = g(tp1/2)2 and L2 = g(tp2/2)2

where, L – Centre to centre distance of connecting rodd1 – Diameter of big end of connecting rodd2 – Diameter of small end of connecting rodL1 – Distance of big end from the center of connecting rod



L2 – Distance of small end from the center of connecting rodtp1 – Periodic time of oscillation at big endtp2 – Periodic time of oscillation at small end

Observation & Tabulation:m = 3.2 kgd1 = 0.062 md2 = 0.033 mL = 0.228 m

Tabular column:

Sl. No.

End Position

Time taken for n oscillations

t (sec)

Periodic timetp (sec)tp = t/n

Moment of Inertia

123

Big end

123

Small end

Result: Thus, the Moment of Inertia by oscillation of flywheel and connecting rod was found to be ………Viva Questions:

1. Why is the flywheel connected to the crankshaft?2. How are the torsional vibrations reduced in the system?3. Explain how the energy is stored in the flywheel.4. What is big eye and small eye?5. Where can you find the flywheel and the connecting rod in an automobile?

Experiment No. : 3WHIRLING OF THE SHAFT

Aim: To determine theoretically the critical speed of the given shaft with the given end conditions.

Apparatus required: Disc, Shaft, Motor, Tachometer, Whirling measurement device, etc.

Description: The speed at which the shaft runs so that additional deflection of the shaft from the axis of rotation becomes

infinite is known as critical speed. Normally the centre of gravity of a loaded shaft will always displace from the axis of rotation although the

amount of displacement may be very small. As a result of this displacement, the centre of gravity is subjected to a centripetal acceleration as soon as the Shaft begins to rotate. The inertia force acts radially outwards and bend



the shaft. The bending of shaft not only depends upon the value of eccentricity, but also depends upon the speed at which the shaft rotates.

Formula used:

Deflection, considering the weight of the shaft only, δg = 5384 ( m1 g d4

EI ) Deflection, considering the weight of the disc only, δd = 1

192 ( m2 g l3

EI ) where, m1 = mass of the shaft per unit length (consider as uniformly distributed load)

m2 = mass of the discd = diameter of the shaftl = length of the shaftE = modules of elasticity of the shaftI = moment of inertia of shaft in m4

Frequency of whirling due to weight of shaft = fns = 0.5623√δ g

Frequency of whirling due to weight of disc = fnd = 0.4977√δ d

Resultant frequency of whirling,1f n

2 =1

fng2 + 1

fnd2

Natural frequency, fn = K√(EgI/wl4) Hz (Theoretically)Whirling speed, N= fn x 60 RPM

where, w = weight/unit length in N/ml = effective length of the shaft between supports in mK= constant (2.45)

Procedure:1. Mount the shaft at fix end and couple with motor.2. Switch on the motor so as to rotate the shaft and hence disc at a particular speed.3. As the speed is increased the axis of rotation of the disc will deviate from the shaft axis.4. The deflection is tabulated at different speeds.5. Repeat the same procedure with shafts of different diameters.

Calculation:1. Moment of inertia2. Weight of solid shaft3. Natural frequency4. Critical speed

Result:Thus the critical speed of the given shaft with the given end conditions were determined.



Whirling of Shaft

Viva Questions1. Define a forced vibration in general terms.2. Define and use a phasor.3. What is harmonic distribution force?4. What is moment of inertia?

Experiment No. : 4(a)BALANCING OF ROTATING MASSES

Aim: To balance the given rotor system dynamically with the aid of the force polygon and the couple polygon.

Apparatus required: Motor, Rotor system, Slotted discs (4 nos.), Steel rule, etc.

Procedure:1. Mount the discs on the rotating shaft as per the given conditions: location, angular position and plane of

masses.2. Select the plane A as reference plane.3. Find out the balancing masses and angular positions using force polygon, and couple polygon using the

reference plane.4. Fix the balancing masses (calculated masses) at the respective radii and angular position.5. Run the system at certain speeds and check that the balancing is done effectively.6. If the rotor system rotates smoothly, without considerable vibrations, means the system is dynamically

balanced.Tabulation:

Sl.No.

Planes of mass

Massm (kg)

Radiusr (m)

Centrifugal Force

ω2mr (kg-m)

Distance from Ref. Plane

l (m)

Coupleω2mrl (kg-m2)

1234

A (ref)BCD

mA

mB

mC

mD

rA

rB

rC

rD



Diagrams:1 Plane of the masses2 Angular position of the masses3 Force polygon4 Couple polygon

Result: Thus the given rotor system has been dynamically balanced with the aid of force polygon and couple polygon.Viva Questions

1. What is static balancing?2. What is dynamic balancing?3. What is an unbalanced system?4. How to make a system to be completely in balance?



5. What is axis of rotation?



6. Experiment No. : 4(b)BALANCING OF RECIPROCATING MASSES

Aim: To balance the given rotor system dynamically with the aid of the force polygon and the couple polygon.

Apparatus required: Rotor system, Weights, Steel rule, etc.

Procedure:1. Initially remove all weights, bolt from the system2. Start the motor, give different speeds. Observe vibration on the system, note down the speed.3. Repeat it for different speeds, note them down.4. Add some weights on piston top, either eccentric or co-axial. Start the motor, fix at earlier tested speed.5. If vibrations are observed, one of the following has to be done to remove the unbalance

- Either remove some of the weights from Piston, run at tested speed and observe.- Add weights in opposite direction of crank, run and observe vibrations at tested speed.- Combination of both the above.

Tabulation:Sl. No.

Crank Speed N (rpm)

Mass (gms) Angular velocity,

(rad/sec) = 2N/60

m1 m2 = (m1 + m2) N

Result: Thus the given rotor system was dynamically balanced with the aid of the force polygon and the couple

polygon.



Experiment No. : 5FREE VIBRATION OF SPRING-MASS SYSTEM

Aim: To calculate the undamped natural frequency of a spring mass system

Apparatus required: Weights, Thread, Ruler, Stopwatch, etc.

Description: The setup is designed to study the free or forced vibration of a spring mass system either damped or

undamped condition. It consists of a mild steel flat firmly fixed at one end through a trunnion and in the other end suspended by a helical spring, the trunnion has got its bearings fixed to a side member of the frame and allows the pivotal motion of the flat and hence the vertical motion of a mass which can be mounted at any position along the longitudinal axes of the flat. The mass unit is also called the exciter, and its unbalanced mass can create an excitation force during the study of forced vibration experiment. The experiment consists of two freely rotating unbalanced discs. The magnitude of the mass of the exciter can be varied by adding extra weight, which can be screwed at the end of the exciter.

Formula used:Stiffness, k = load/deflection N/mExperimental natural frequency, fn(exp) = 1/t HzTheoretical natural frequency, fn(the) = 1/2√(g/) Hz

Procedure:Determination of spring stiffness1. Fix the top bracket at the side of the scale and Insert one end of the spring on the hook.2. At the bottom of the spring fix the other plat form3. Note down the reading corresponding to the plat form4. Add the weight and observe the change in deflection5. With this determine spring stiffnessDetermination of natural frequency1. Add the weight and make the spring to oscillate for 10 times2. Note the corresponding time taken for 10 oscillations and calculate time period3. From the time period calculate experimental natural frequency

Tabulation:Sl. No.

Weight added m (kg)

Deflection (mm)

Stiffness k (N/m)

Time for 10 oscillation

t (sec)

Time periodT (sec)

Experimental natural frequency,

fn(exp) (Hz)

Theoretical natural frequency

fn(the), Hz

Graph: Plot the graph(i) Load vs Deflection(ii) Load vs Theoretical natural frequency(iii) Load vs Experimental natural frequency

Result: 16



Thus the undamped natural frequency of a spring mass system was calculated.

Spring Mass Damping System



Experiment No. : 6DETERMINATION OF GYROSCOPIC COUPLE

Aim: To determine the active and reactive gyroscopic couples and compare them.

Apparatus required: Gyroscope, Tachometer or Stroboscope, Variable voltage transformer, Rotating disc with a light reflecting sticker for stroboscope speed measurement, etc.

Formula used: Mass moment of inertia of the disc, I = md2/8

where, m - mass of the disc and d - dia of the disc Angular velocity of the disc, ω = 2πN/60

where, N - speed of disc in rpm Angular velocity of precession, ωp = (π/180) x (φ/t) Reactive gyroscopic couple, Cr = I.ω.ωp

Active gyroscopic couple, Ca = W x Lwhere, W - weight added = mg and m = mass of the weight added

L - distance between centers of weight to center plane of disc

Procedure:1. The disc is made to rotate at a constant speed at a specific time using variable voltage transformer. Fix

the rotor at vertical position. Adjust the balance weight and dimmer at zero position.2. The motor is started at 1 – 20 V.3. The speed of the (N) disc is measured using a tachometer or a stroboscope. 4. A weight /mass is added on the extending platform attached to the disc. This will cause an active

gyroscopic couple and the whole assembly (rotating disc, rotor and weight platform with weight) is standing to move in a perpendicular plane to that of plane of rotating of disc. This is called gyroscopic motion.

5. The time taken (t) to traverse a specific angular displacement (φ = 45°) is noted.6. Repeat the procedure for different weighs and rotor speed.

Tabulation:Sl. No.

Speed of disc

N (rpm)

Weight addedm (kg)

Time taken for 60° precision

t (sec)

Active couple Ca (N-m)

Reactive coupleCr (N-m)

Graph: Plot the graph(i) Active couple Vs. Reactive couple(ii) Weight added Vs. Reactive couple

Result: Thus, the active and reactive gyroscopic couples were determined and compared.



Viva Questions1. Name a few applications of gyroscope.2. What is the basic principle behind gyroscope3. How do you make a system to stay in equilibrium along its CG?4. Discuss the effect of gyroscopic couple on a two wheeled vehicle when taking a turn.5. Explain the application of gyroscopic principles to aircrafts.



6. Experiment No. : 7GOVERNOR

Aim: To find the stiffness, sensitivity and effort of the spring using Hartnell governor.

Apparatus required: Hartnell governor setup and Tachometer

Description:Hartnell governor is a centrifugal type spring controlled governor where the pivot of the ball crank lever

is carried by the moving sleeve. The spring is compressed between the sleeve and the cap is fixed to the end of the governor shaft. The ball crank is mounted with its bell and the vertical arm pressing against the cap.

Formula used:Fc1 = m ω1

2r1 (N) and Fc2 = m ω22r2 (N)

ω1 = 2N1/60 rad/s and ω2 = 2N2/60 rad/sS1 = 2Fc1 (X/Y) N and S2 = 2Fc2 (X/Y) Nr2 = r - R(X/Y) (mm)Sensitivity = (maximum speed-minimum speed)/mean speed = (N1-N2)/NEffort = (spring force at maximum speed - spring force at minimum speed)/2Spring stiffness = (S1-S2)/R N/mm

where, m= mass of the ball (m = 0.18 kg)ω1 & ω2 = angular speed of governor at maximum radius and minimum radius respectively in rad/secr1 & r2 =maximum and minimum radius of rotationFc1 & Fc2 =centrifugal forces at ω1 and ω2 in NX = length of the vertical ball arm of lever in mY = length of the horizontal ball arm of lever in mS1 & S2 = spring forces at ω1 & ω2 in N

Procedure:1. Keep the speed regulation in zero position before starting the motor.2. Increase the regulated output gradually till the motor takes the critical speed and immediately control the

speed of the governor3. Maintain the speed for each and every graduation as required to take the direct reading

Tabular column: X = 95 mm, Y = 95mm, r1 = r = 95mm

Sl.No.

Speed, N (rpm) Sensitivity Effort (N)

Stiffness (N/mm)Min max mean

Graph: Plot the following graph(i) Mean speed Vs. Sensitivity(ii) Mean speed Vs. Effort

Result:



Thus the stiffness, sensitivity and effort of the spring using various governors were found.

Viva Questions:1. What is the function of a governor? How does it differ from that of a flywheel?2. What is the stability of a governer?3. What is hunt?4. The power of a governer is equal to ----------5. Which of the governer is used to drive a gramophone?6. Define the sensitiveness of a governer.7. Explain the term height of the governor. What are the limitations of a Watt Governor?8. State the different types of governors. What is the difference between centrifugal and inertia type

governor?

Experiment No. : 8CAM ANALYSIS

Aim: To study the profile of given cam using cam analysis system and to draw the displacement diagram for the follower and the cam profile.

Apparatus required: Cam analysis system and Dial gauge

Description:21



A cam is a machine element such as a cylinder or any other solid with a surface of contact so designed as to give a predetermined motion to another element called the follower. A cam is a rotating body importing oscillating motor to the follower. All cam mechanisms are composed of at least there links viz: 1.Cam, 2. Follower and 3. Frame which guides follower and cam.

Specification :Diameter of base circle =150mm, Lift = 18mm, Diameter of cam shaft = 25mmDiameter of follower shaft = 20 mm, Diameter of roller = 32mm, Dwell period = 180Type of follower motion = SHM (during ascent & descent)

Procedure:Cam profile:Cam analysis system consists of cam roller follower, pull rod and guide of pull rod.1. Set the cam at 0° and note down the projected length of the pull rod2. Rotate the can through 10° and note down the projected length of the pull rod above the guide3. Calculate the lift by subtracting each reading with the initial reading.Jump-speed:1. The cam is run at gradually increasing speeds, and the speed at which the follower jumps off is

observed.2. This jump-speed is observed for different loads on the follower.

Tabulation:1.Cam profile

Sl.No.

Angle of rotation

(degrees)

Lift(mm)

Lift + base circle radius (mm)

2. Jump-speedSl. No. Load on the Follower,

F (N)Jump-speed

(RPM)

Graph: Plot the displacement diagram and cam profile is also drawn using a polar graph chart. The Force Vs Jump-speed curve is drawn.

Result: Thus the profile of given cam using cam analysis system was studied.



Circular Arc Cam & FollowerViva Questions

1. What is a track follower?2. What is dynamic bearing capacity?3. What is yoke in this sytem?4. Define dwell region5. Define ststic load capacity.



6. Experiment No. : 9

VIBRATING TABLE SETUP

Aim: To determine transmissibility of forced vibrations and to analyze all types of vibrations with its frequency and amplitude.

Procedure: 1. Attach the vibrating recorder at suitable position with the pen hidden slightly pressing the paper.2. Attach the damp unit to the stud.3. Start the motor and set required speed and start the recorder motor.4. Now vibrations are recorded to the vibration recorder.5. At the resonance speed, the amplitude of the vibration may be recorded as merged over one another.6. Hold the system and max speed little more than the reasonable speed.7. Analyse the recorder frequency and amplitude for both damped and undamped forced vibrations.

Formula used:Transmissibility = FTR / F

= max force on bars / max impressed forceFTR = S x Xmax

where, S - Stiffness of spring = 180 N/mXmax - amplitude of vibration

F = mr2

where, m - Mass of beam = 7.2 kg - motor speed

Result: 24



Thus the transmissibility of forced vibrations was determined.

Viva Questions1. What is law of Transmissibility?2. What are the different types of vibrations3. What is Stiffness?4. Define Stiffness and Toughness.5. What is the use of the absorber in the system?

Experiment No. : 10(a)TRANSVERSE VIBRATION OF CANTILEVER BEAM

Aim: To find the natural frequency of transverse vibration of the cantilever beam

Apparatus required: Displacement measuring system (strain gauge), Weights, etc.

Description: Strain gauge is bound on the beam in the form of a bridge. One end of the beam is fixed and the other end is

hanging free for keeping the weights to find the natural frequency while applying the load on the beam. This displacement causes strain gauge bridge to give the output in mill-volts. Reading of the digital indicator will be in mm.

Formula used:Natural frequency = 1/2(g/) Hz

where, = deflectionTheoretical deflection = Wl3/3EI

where, W = applied load in NL = length of the beam in mmE = young’s modules of material in N/mm2

I = moment of inertia in mm4 = bh3/12Experimental stiffness = W/ N-mmTheoretical stiffness = W/ =3EI/l3 N/mm



Procedure:1. Connect the sensors to instrument using connection cable.2. Plug the main cord to 230v/ 50hz supply3. Switch on the instrument4. Keep the switch in the read position & turn the potentiometer till displays reads “0”5. Keep the switch at cal position and turn the potentiometer till display reads 56. Keep the switch again in read position and ensure at the display shows “0”7. Apply the load gradually in grams8. Read the deflection in mm

Graph: Draw the following graph (i) Load vs Displacement & Natural frequency(ii) Displacement vs Natural frequency

Observation:Cantilever beam dimensions: Length = 30cm,

Breadth = 6.5cm Height = 0.4cm

Tabulation:Sl. No.

Applied mass

m (kg)

Deflection (mm)

Theoretical deflectionT (mm)

Experimentalstiffness

k (N/mm)

Theoreticalstiffness

k (N/mm)

Natural frequency

fn (Hz)

Result: Thus the natural frequency of transverse vibration of the cantilever beam was determined.

Viva Questions1. What are the other types of beams?2. What is natural frequency?3. What is the differential eqn of motion for a transverse beam?4. What are the types of loads?5. Name a few methods of determining a force acting upon a body.



Experiment No. : 10(b)TRANSVERSE VIBRATION OF FIXED BEAM

Aim: To study the transverse vibrations of a simply supported beam subjected to central or offset concentrated load or uniformly distributed load.

Apparatus Required: Truss bearings, Fixed beam, Weights, Stop watch, etc.

Procedure: 1. Fix the beam into the slots of truss bearings and tighten.2. Apply the concentrated load centrally or offset, or uniformly distributed.3. Allow the beam to vibrate. 4. Note the time taken for 10 oscillations, and calculate the time period and hence natural frequency.5. Compare the experimental frequency with theoretical frequency. 6. Repeat the procedure for different loads.

Formula used: Time period of oscillation, T = t/10 Natural frequency found by experiment, fexp = 1/T Hz

where, t = time taken for 10 oscillationsFor concentrated load,

Defection at the center, T = concentrated load. Defection at the load point, T = Wx2y2/3EIl for offset concentrated load. where, W = load applied on the beam in N = Mg

M = mass of the load in kgI = moment of area = bd3/12 b = width of the beamd = depth of the beam l = length of the beam

For uniformly distributed load, Defection at the center, T = 5wl4/384EI for uniformly distributed load.



where, w = load per unit length in N/m = mgm = mass of the load in kg/m

Natural frequency of transverse vibrations, fn = 1/2(g/) Hz

Observations & Tabulation: b = ……….d = ……….l = ……….E = ……….

Tabular column:Sl.No.

Mass addedm (kg)

TheoreticaldeflectionT (m)

Theoreticalnatural freq.

fn (Hz)

Time taken for 10 oscillations

t (sec)

Experimentalnatural freq.

fexp (Hz)

Diagrams: Simply Supported beam with the given load and parameter.

(a) Centrally concentrated load

(b) Offset concentrated load



(c) Unifromly distributed load

Result: Thus the transverse vibrations of a simply supported beam subjected to central or offset concentrated load or

uniformly distributed load was observed.

Viva Questions1. Define truss2. How to you determine the natural frequency of a fixed beam? 3. Define stiffness of a beam.4. Compare and contrast mass, specific mass and effective mass5. How do you represent stiffness and give its formula for fixed beam?



6. Experiment No. : 11STUDY ON TYPES OF GEAR TRAINS

Aim: To study the various types of Gear Trains

Apparatus Required: Models of the various Gear Trains

Description:

A gear train is formed by mounting gears on a frame so that the teeth of the gears engage. Gear teeth are

designed to ensure the pitch circles of engaging gears roll on each other without slipping, this provides a smooth

transmission of rotation from one gear to the next.

The transmission of rotation between contacting toothed wheels can be traced back to the Antikythera

mechanism of Greece and the South Pointing Chariot of China. Illustrations by the renaissance scientist

Georgius Agricola show gear trains with cylindrical teeth. The implementation of the involute tooth yielded a

standard gear design that provides a constant speed ratio.

Some important features of gears and gear trains are:

The ratio of the pitch circles of mating gears defines the speed ratio and the mechanical advantage of the

gear set.

A planetary gear train provides high gear reduction in a compact package.

It is possible to design gear teeth for gears that are non-circular, yet still transmit torque smoothly.

The speed ratios of chain and belt drives are computed in the same way as gear ratios. See bicycle

gearing.

Types of gear trains in general:

Simple gear train Compound gear train Epicyclic gear train Reverted gear train

Simple Gear Train:

A gear train is called a simple gear train, if the axes of the gears are connected by revolute joints to the fixed link. The geometric representation of a gear train with one gear pair is shown below. In the figure the solid model of the gear train (and its simulation) is also shown. The circles drawn as centerline is the known as the pitch circle of the gear. The linear velocity of the point of contact P, is the point where the relative velocity between the two links is zero (there is pure rolling and no sliding between the toothed surfaces). The gear ratio R23 is defined as:



http://en.wikipedia.org/wiki/Involute_gear

http://en.wikipedia.org/wiki/Georgius_Agricola

http://en.wikipedia.org/wiki/South_Pointing_Chariot

http://en.wikipedia.org/wiki/Antikythera_mechanism

http://en.wikipedia.org/wiki/Antikythera_mechanism

http://en.wikipedia.org/wiki/Gear

Where n1j is the angular speed of link j with respect to the fixed link 1 expressed in rpm. Velocity of point P is:

Hence:

Geometric Representation of Gears in External Mesh (P in between centers)

Where dj and rj are the diameters and radii of the pitch circles of the gears.

From the law of gearing for the gears to be in mesh, the diametral pitch, which is the ratio of the number of teeth over the pitch diameter must be the same for two mating gears: e.g.:

Diametral pitch

Where Tj is the number of teeth on gear j. In European countries rather than the diametral pitch, Module, m , which is the ratio of the pitch circumference to the number of tooth (pdj/Tj), is used. Unit for diametral pitch is in 1/inch and the unit for module is mm. We define that the gear ratio as positive when the two gears in mesh are rotating in the same direction and negative if they are rotating in the opposite direction. As seen in the figure above, if P is in between the fixed centres, the gear ratio will be negative and such gear pairing we shall call “external mesh”. If the pairing is as shown below., The point of contact is outside the fixed centres and the gear ratio is positive. Such gear pairing will be called “internal mesh” .

Planetary Gear Train:

A simple gear trains the axes of each gear was connected to the fixed link by revolute joints. A gear train is called a planetary gear train if there are some gears whose axes are not fixed. The velocity ratio between the rotating links will be different than the gear ratio. They can produce large changes in speed using very few gears. They are used in differentials, computing devices, in automatic car transmission and in a variety of



instruments where a high speed ratio between the input and output is required within a small space (i.e. in heavy duty tractors, hoists, screw drives etc) and/or when a speed change without disengagement of the input and output (i.e. automatic gear boxes) is required.

The simplest form of a planetary gear is as shown. The arm (link k ) can rotate about A0. Link j (which is called sun gear, if external or ring gear, if internal) is also connected to the fixed link by a revolute joint at Ao. Link i, which is called the planet gear, is connected to the arm at A by a revolute joint, and links i and j are gear paired. (In the schematic representation of planetary gear trains side view is usually preferred.)

Planetary gear trains are also called "Epicyclic Gear Train" from the fact that a point on the planet will describe an algebraic curve which is an epicycloid or hypocycloid. This curve can be represented in parametric form as:

x = a1cosq +a2cosRq y = a1 sinq +a2sinRq

or in complex numbers:

Where q is the variable parameter (usually the angle of rotation of the arm) a1, a2 and R are constants (R is a function of the gear ratio, a1 and a2 are the link lengths of the arm and the position of the point relative to the moving pivot of the planet respectively). Very interesting curves (epi- or hypocycloid curves) can be obtained. An example is given in below. One field of application of such motion is in mixers where the pedal or blade is moved on a cycloidal path..

For large force transmission, there is usually more than one planet located symmetrically around the sun gear as shown in Fig. 6.13. Accurate clearance is required for the operation of such a gear train. The planetary gear train thus formed is over closed and thus there must be special dimensional requirements (such that all the planets being equal). In the kinematic analysis of planetary gear trains, one must consider only one planet..

Gear Trains with Bevel Gear:

If the motion is to be transmitted between non-parallel shafts, bevel gears are often used. Bevel gears in kinematic terms are equivalent to the rolling of two cones without slippage. They can be used in simple,



compound or in planetary gear trains. Their kinematic analysis is exactly similar to the cylindrical gears discussed in the previous sections, however the direction of rotation of the gears does not follow the rules given for the cylindrical gears.

Figure shows two bevel gears in contact, the angle between the axes of the gears being q. It follows that:

Bevel (Conical) Gears

and

The direction of rotation must be determined by noting that the velocity of the two cones at the point of contact (P) will be the same. If link 2 is rotating CW when viewed from point O, link 3 will rotate CCW when viewed from point O. A convenient form of determining the direction of rotation of the gears is to consider an arrow representing the direction of rotation for each gear. If the arrow is piercing into the paper, one will see the end of the arrow which will look like a " +" sign. If it is piercing out of the paper, its tip will be seen and it will look like a "." sign. In the side view of a gear, the arrow showing the rotation of the gear can be represented by the above two signs. At the point of contact of two gears, the arrow will either pierce into (+) or out of (.) the paper



and on the other sides of the gears it will be the opposite. Hence, the direction of rotation of each gear can be determined using this convention.

Result:

The study on Gear trains and its various applications was satisfactorily completed in this experiment.



Experiment No. : 12STUDY ON TYPES OF DYNAMOMETERS

Aim: To study the various types of dynamometers.Apparatus Required:Models of dynamometer.Description:

The dynamometer is a device used to measure the torque being exerted along a rotating shaft so as to determine the shaft power.Dynamometers are generally classified into:1) Absorption dynamometers (i.e. Prony brakes, hydraulic or fluid friction brakes, fan brake and eddy current dynamometers)2) Transmission dynamometers (i.e. Torsion and belt dynamometers, and strain gauge dynamometer)3) Driving dynamometers (i.e. Electric cradled dynamometer)

PRONY BRAKE: The prony and the rope brakes are the two types of mechanical brakes chiefly

employed for power measurement. The prony brake has two common arrangements in the block type and the band type. Block type is employed to high speed shaft and band type measures the power of low speed shaft.

BLOCK TYPE PRONY BRAKE DYNAMOMETER: The block type prony brake consists of two blocks of wood of which embraces

rather less than one half of the pulley rim. One block carries a lever arm to the end of which a pull can be applied by means of a dead weight or spring balance. A second arm projects from the block in the opposite direction and carries a counter weight to balance the brake when unloaded. When operating, friction between the blocks and the pulley tends to rotate the blocks in the direction of the rotation of the shaft. This tendency is prevented by adding weights at the extremity of the lever arm so that it remains horizontal in a position of equilibrium.

Torque, T = W*l in NmPower P = 2πN* T/60 in N-m/s



= 2πN * W*l/60* 1000 in kWWhere, W= weights in Newtonl = Effective length of the lever arm in meter andN = Revolutions of the crankshaft per minute.

BAND TYPE PRONY BRAKE DYNAMOMETER: The band type prony brake consists of an adjustable steel band to which are

fastened wooden block which are in contact with the engine brake-drum. The frictional grip between the band the brake drum can be adjusted by tightening or loosening the clamp. The torque is transmitted to the knife edge through the torque arm. The knife edge rests on a platform or communicates with a spring balance.

Frictional torque at the drum = F*rBalancing torque = W*lUnder equilibrium conditions, T = F*r = W*l in Nm.Power = 2πN* T/60 in N-m/s

= 2πN * W*l/60* 1000 in kW

ROPE BRAKE DYNAMOMETERS: A rope brake dynamometers consists of one or more ropes wrapped around the fly wheel of an engine whose power is to be measured. The ropes are spaced evenly across the width of the rim by flywheel. The upward ends of the rope are connected together and attached to a spring balance, and the downward ends are kept in place by a dead weight. The rotation of flywheelproduces frictional force and the rope tightens. Consequently a force is induced in the spring balance.Effective radius of the brake R = (D+ d)/2Brake load or net load = (W-S) in NewtonBraking torque T = (W-S) R in Nm.Braking torque =2πN* T/60 in N-m/s

= 2πN * (W-S)R/60* 1000 in kWD= dia. Of drum



d = rope dia.S = spring balance reading

FLUID FRICTION (HYDRAULIC DYNAMOMETER): A hydraulic dynamometer uses fluid-friction rather than friction for dissipating the

input energy. The unit consists essentially of two elements namely a rotating disk and a stationary casing. The rotating disk is keyed to the driving shaft of the prime-mover and it revolves inside the stationary casing. When the brake is operating, the water follows a helical path in the chamber. Vortices and eddy-currents are set-up in the water and these tend to turn the dynamometer casing in the direction of rotation of the engine shaft. This tendency is resisted by the brake arm and balance system that measure the torque.Brake power = W*N/k,

Where W is weight as lever arm, N is speed in revolutions per minute and k is dynamometer constant.Approximate speed limit = 10,000rpmUsual power limit = 20,000kW

BEVIS GIBSON FLASH LIGHT TORSION DYNAMOMETER: This torsion dynamometer is based on the fact that for a given shaft, the torque

transmitted is directly proportional to the angle of twist. This twist is measured and the corresponding torque estimated the relation:T = Ip* C*θ / lWhere Ip = πd4/32 = polar moment of inertia of a shaft of diameter dθ = twist in radians over length l of the shaftC = modulus of rigidity of shaft material

Applications:i) For torque measurement.ii) For power measurement.

Result:



A relative study on the various dynamometers was made and corresponding power and braking capacity

was studied.



list of experiments - web viewthe planetary gear train thus formed is over closed and thus there...

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