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Whirling of shaft - Technical symposium

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Whirling of shaft - Technical symposiumWhirli

ME 2307 DYNAMICS LABORATORY

    Class : V Semester Mechanical Sections : A & B

    LIST OF EXPERIMENTS

    1. Free Transverse Vibration I Determination of Natural Frequency 2. Cam Analysis Cam Profile and Jump-speed Characteristics 3. Free Transverse Vibration II Determination of Natural Frequency 4. Free Vibration of Spring Mass System Determination of Natural Frequency 5. Compound Pendulum Determination of Radius of Gyration and Moment of Inertia 6. Bifilar Suspension Determination of Radius of Gyration and Moment of Inertia 7. Trifilar Suspension Determination of Radius of Gyration and Moment of Inertia 8. Whirling of Shaft Determination of Critical Speed

    9. Balancing of Rotating Masses

    10. Determination of Gyroscopic Couple

    11. Turn Table

    12. Hartnell Governor

    13. Free Vibration of Spring Mass System Determination of Natural Frequency

Beyond the Syllabus

14. Speed Ratio of Epi-cyclic Gear Train

    15. Speed Ratio of Worm and Worm Wheel

EX NO:1: TRANSVERSE VIBRATION - I

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

    Apparatus required: Displacement measuring system (strain gauge) and Weights

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 milli-

    volts. Reading of the digital indicator will be in mm.

Formulae used:

    1. Natural frequency = 1/2;?(g/;) Hz 2 where g= acceleration due to gravity in m/sand ; = deflection in m.

     32. Theoretical deflection ;= Wl/3EI

    Where, W= applied load in Newton, L= length of the beam in mm 24 3 E= young‟s modules of material in N/mm, I= moment of inertia in mm=bh/1233. Experimental stiffness = W/; N-mm and Theoretical stiffness = W/; =3EI/l N/mm

Procedure:

    1. Connect the sensors to instrument using connection cable.

    2. Plug the main cord to 230v/ 50hz supply

    3. Switch on the instrument

    4. Keep the switch in the read position and turn the potentiometer till displays reads “0”

    5. Keep the switch at cal position and turn the potentiometer till display reads 5

    6. Keep the switch again in read position and ensure at the display shows “0”

    7. Apply the load gradually in grams

    8. Read the deflection in mm

    Graph:

    Draw the characteristics curves of load vs displacement, natural frequency

    Draw the characteristics curves of displacement vs natural frequency Result:

    Observation: Cantilever beam dimensions: Length=30cm, Breadth=6.5cm and Height=0.4cm

Tabulation:

Sl. Applied Deflection Theoretical Experimental Theoretical Natural

    No. mass deflection Stiffness Stiffness frequency ; (mm)

    m (kg) k (N/mm) k (N/mm) fn (Hz) ; (mm) T

EX NO:2 CAM ANALYSIS

Aim:

    To study the profile of given can using cam analysis system and to draw the displacement diagram for the follower and the cam profile. Also to study the jump-speed characteristics of the cam & follower mechanism.

    Apparatus required: Cam analysis system and Dial gauge

Description:

    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 = 25mm

    Diameter of follower shaft = 20 mm, Diameter of roller = 32mm, Dwell period = 180

    Type of follower motion = SHM (during ascent & descent)

Procedure:

    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 rod

    2. Rotate the can through 10? and note down the projected length of the pull rod above the

    guide

    3. 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.

    Graph:

    Displacement diagram and also the cam profile is drawn using a polar graph chart.

    The Force Vs Jump-speed curve is drawn.

    Result.

    Tabulation:

    1.Cam profile

    Sl. Angle of Lift in mm Lift + base circle radius (mm)

    No. rotation

    (degrees)

2. Jump-speed.

    Sl. Load on the Jump-speed

    No. Follower, F (N) N (RPM)

EX NO:3 TRANSVERSE VIBRATIONS - II

Aim: To study the transverse vibrations of a simply supported beam subjected to central or offset

    concentrated load or uniformly distributed load.

    Apparatus Required: Trunnion bearings, beams, weights.

    Set-up:

Procedure:

    1. Fix the beam into the slots of trunnion bearings and tighten.

    2. Add the concentrated load centrally or offset, or uniformly distributed.

    3. Determine the deflection of the beam for various weights added.

Formulae used: 3Defection at the center, ;= Wl/48EI for central concentrated load. T22Defection at the load point, ;= Wab/3EIl for offset concentrated load. T4Defection at the center, ;= 5wl/384EI for uniformly distributed load. T

     3I = bd/12; b = width of the beam, d = depth of the beam, l = length of the beam. Natural frequency of transverse vibrations, f= 1/2;?(g/;) Hz n2 where g= acceleration due to gravity in m/sand ; = deflection in m.

Observations: b = , d = , l = , E =

Tabular column:

    Sl. Mass added Experimental Theoretical Theoretical Experimental Theoretical

    No. m , kg Deflection Deflection Nat. freq. Stiffness Stiffness

     , Hz K, N/m K, N/m f;, m ;, m nT

Graphs:

    1. Deflection Vs. load (N) from this get stiffness (graph) 2. Deflection Vs. Natural frequency

    3. Load in N Vs. natural frequency

Stiffness experimental, K = load/deflection =W/δ = mg/δ N/mm 3 Stiffness theoretical, K = W/ δ = 48EI/lfor center load, T22 = 3EIl/ab for offset load, 3 = 384EI/5lfor uniformly distributed load,

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

EX NO:4 FREE VIBRATION OF SPRING-MASS SYSTEM

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

    Apparatus required: Weights, Thread, Ruler, Stopwatch

    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 excitational 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/m

    Experimental natural frequency, f =1/t Hz n(exp)

    Theoretical natural frequency, f = 1/2?(g/;) Hz n(the)

Procedure

    Determination of spring stiffness

    1. 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 form

    3. Note down the reading corresponding to the plat form

    4. Add the weight and observe the change in deflection

    5. With this determine spring stiffness

    Determination of natural frequency

    1. Add the weight and make the spring to oscillate for 10 times

    2. Note the corresponding time taken for 10 oscillations and calculate time period

    3. From the time period calculate experimental natural frequency

Calculation:

    Graph:

    Load vs Deflection

    Load vs Theoretical natural frequency

    Load vs Experimental natural frequency

    Result:

    Tabulation:

    Sl Weight Deflection Stiffness Time for 10 Time period Experimental Theoretical

    no added m k (N/m) oscillation T (sec) natural natural ; (mm)

    (kg) t (sec) frequency, frequency

    f, Hz fHz n(exp)n(the),

EX NO:5 COMPOUND PENDULUM

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

    experimentally.

Apparatus required: 1. Vertical frame, 2. Rectangular rod, 3. Stop watch and 4. Steel rule etc

Procedure:

    1. Suspend the rod through any one of the holes

    2. Give a small angular displacement to the rod & note the time taken for 5 oscillations

    3. Repeat the step by suspending through all the holes.

Formulae used: 22 Time period T= t/N sec and also Experimental time period T = 2;?((K+L)/gL) 11222 Where K= experimental radius of gyration and K = ((gLT/4)-L), 11

     L= distance from point of suspension to centre of gravity of rod and L= total length of the rod 1

     Theoretical radius of gyration, K = L/12=0.2866L t22 Natural frequency fn = 1/T (Hz) and Moment of inertia I = mk kg-m m

Result:

Tabulation:

Sl. Distance Time for 5 Time period T Natural Experimental radius

    No. L (m) oscillations (sec) frequency of gyration 1

    t (sec) fn (Hz) (K) exp

Calculation:

EX NO:6 BIFILAR SUSPENSION

Aim: To determine the radius of gyration and the moment of Inertia of a given rectangular plate.

Apparatus required: Main frame, bifilar plate, weights, stopwatch, thread

Formula used:

    Time period T=t/N

    Natural frequency fn = 1/T hz

    Radius of gyration k =(Tb/2)(g/L) (mm)

    Where, b=distance of string from centre of gravity, T= time period

    L= length of the string, N= number of oscillations

    t= time taken for N oscillations

Procedure:

    1. Select the bifilar plate

    2. With the help of chuck tighten the string at the top.

    3. Adjust the length of string to desired value.

    4. Give a small horizontal displacement about vertical axis.

    5. Start the stop watch and note down the time required for „N‟ oscillation.

    6. Repeat the experiment by adding weights and also by changing the length of the strings.

    7. Do the model calculation

Graph:

    A graph is plotted between weights added and radius of gyration

Calculations:

    Result:

Observation:

    Type of suspension = bifilar suspension

    Number of oscillation n=10

    b =10.15 cm d = 4.5 cm b=21.5 cm 1

Tabulation:

    Weight Length of Time taken Natural Radius of Sl. added string for N osc. frequency gyration No. m (kg) L (m) T sec fn (Hz) k (mm)

EX. NO. 7 TRIFILAR SUSPENSION

Aim: To determine the radius of gyration of the circular plate and hence its Mass Moment of Inertia.

    Apparatus required: Main frame, chucks 6 mm diameter, circular plate, strings, stop watch.

Procedure: 01. Hang the plate from chucks with 3 strings of equal lengths at equal angular intervals (120

    each)

    2. Give the plate a small twist about its polar axis

    3. Measure the time taken, for 5 or 10 oscillations.

    4. Repeat the experiment by changing the lengths of strings and adding weights.

Formulae used :

    Time period, T = t/N, Natural frequency, fn = 1/T Hz Radius of gyration, K = (bT/2Л) (g/l) m.

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

     l- Length of string from chuck to plate surface. 2 2 2 Moment of inertia of the plate only, I=(Rx W) / (4πfnx l) p122 2 Moment of inertia with weight added ,I=R x (W+ W) / fnx l) t1

    Where, R- Radius of the circular plate and W-Weight of the circular plate = mg in N m= 3.5 kg 111

     W- Weight of the added masses = mg in N Moment of inertia of weight, I= I- I w t p

Result: The radius of gyration of the plate and moment of inertia of the weights were determined

    and tabulated.

    Graphs:

     Weight added vs radius of gyration

     Weight added vs moment of inertia

    Observations:

    Type of suspension:…………………, No. of oscillations …………………….

    Radius of circular plate, R=…….m, mass of the plate, m = ……..kg 1

Sl. Length Added, Time for N Time Radius of Natural Moment of

    No. of string mass, oscillations, period gyration, frequency inertia of

    l, m m, kg t, sec T, sec k, m f, Hz weight n

    Iw,kgm

EX NO: 8 WHIRLING OF SHAFT

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

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: 4f =K?(EgI/wl) and N= f X 60 nn

    Where, f= natural frequency of vibration in Hz n2g= acceleration due to gravity, (9.81m/s), E= modules of elasticity of the shaft 4I=moment of inertia of shaft in m, w= weight /unit length in N/m

    l=effective length of the shaft between supports in m. and N= speed of the shaft in RPM

    K= constant (2.45)

Result:

Calculation:

    1. Moment of inertia

    2. Weight of solid shaft

    3. Natural frequency

    4. Critical speed

EX NO:9 BALANCING OF ROTATING MASSES

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

Apparatus required: rotor system, weights, steel rule, etc.

Procedure:

    1. Fix the unbalanced masses as per the given conditions: radius, angular position and plane of

    masses.

    2. Find out thee balancing masses and angular positions using force polygon, and couple

    polygon

    3. Fix the balancing masses (calculated masses) at the respective radii and angular position.

    4. Run the system at certain speeds and check that the balancing is done effectively.

    5. If the rotor system rotates smoothly, without considerable vibrations, means the system is

    dynamically balanced.

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

     22Sl. Planes Mass Radius C.Force / ω Distance from Couple / ω 2No. of mass m, kg r, m mr, kg-m Ref. Plane mrl, kg-m

    l, m

    1 A

    2 B

    3 C

    4 D

Diagrams:

    1 Plane of the masses 2. Angular position of the masses 3. Force polygon

    4 Couple polygon

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