Dynamics of Machinery

Transcription

1 Lab Manual Dynamics of Machinery ( ) Name: Enrollment No.: Roll No.: Batch: Darshan Institute of Engineering & Technology, Rajkot DEPARTMENT OF MECHANICAL ENGINEERING

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3 Certificate This is to certify that, Mr. / Ms. Enroll no. of Sixth Semester Bachelor of Mechanical Engineering has completed the term work satisfactorily in Dynamics of Machinery ( ) for the academic year as prescribed in the GTU curriculum. Place: Date: Subject Coordinator Head of the Department

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5 DARSHAN INSTITUTE OF ENGG. & TECH. Department of Mechanical Engineering B.E. Semester VI Dynamics of Machinery ( ) List of Experiments Sr. No. Title Date of Performance Sign Remark 1. Study the longitudinal vibration of helical spring and to determine the frequency and time period of oscillation, theoretically and actually by experiment. 2. To study the torsional vibrations of single rotor system. 3. Study the free vibration of two rotor system and to determine the natural frequency of vibration theoretically & experimentally. 4. To study the damped torsional oscillation & to determine the damping co-efficient ζ. 5. To verify the Dunker ley s Rule. 6. To study the undamped free vibration of equivalent spring mass system. 7. To study the forced damped vibration of equivalent spring mass system. 8. To study the forced vibration of the beam for different damping. 9. To study the Static & Dynamic Balancing system. 10. To study the effect of whirling of shaft. 11. To find out natural frequency of undamped free vibration of a single degree of freedom system a cylinder rolling in another cylinder. 12. To find out natural frequency of vibration of double pendulum system. 13. To study jump phenomenon in the cam.

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7 Longitudinal Vibration of Helical Spring Experiment No Aim: Study the longitudinal vibration of helical spring and to determine the frequency and time period of oscillation, theoretically and actually by experiment. 2. Description: One end of open coil spring is fixed to the nut having a hole which itself is mounted on a MS strip fixed on one side of the main frame. The lower end of the spring is attached to the platform carrying the weights. The stiffness of the spring can be finding out by varying the weights on the platform and by measuring the deflection of the spring. The time period of vibrations can be calculated by measuring the nos. of oscillation and time taken by them. Figure 1.1: Longitudinal Vibration of Helical Spring 3. Experimental procedure: Fix one end of the helical spring to upper screw. Determine free length. Put some weight to platform and note down the deflection. Stretch the spring through some distance and release. Count the time required in Sec. for say 10, 20 oscillations. Determine the actual period. Repeat the procedure for different weights. Darshan Institute of Engineering and Technology, Rajkot Page. 1.1

8 Longitudinal Vibration of Helical Spring 4. Nomenclature: f act Actual natural frequency sec -1 f theo Theoretical natural frequency sec -1 g Acceleration due to gravity m/sec 2 k Stiffness of the spring N/m L Length of spring m n No. of oscillations. t Time taken by n oscillation sec T act Actual time period sec T theo Theoretical time period sec W Weight applied N Deflection of the spring. m 5. Observation & calculation: 5.1 Data: Free length of spring, 5.2 Observation table: L f = Sr. No L (m) m (Kg) (m) k (N/m) Time for 10 Oscillations, t (sec) t 1 t 2 t 3 t 5.3 THEORETICAL CALCULATIONS: Weight attached, W m g Newton Spring stiffness, W k Natural frequency, ftheo T theo 1 sec f n 1 k 2 m Darshan Institute of Engineering and Technology, Rajkot Page. 1.2

9 Longitudinal Vibration of Helical Spring 5.4 PRACTICAL CALCULATIONS: t Time for oscillation, Tact sec n Frequency, f act 1 Hz T act 5.5 Result Table: Sr. No W (N) N t (sec) T theo. (sec) T act. (sec) f theo. (Hz) f act. (Hz) 6. Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 1.3

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11 Torsional Vibrations of Single Rotor System 1. Aim: Experiment No. 2 To study the Torsional Vibrations of single rotor system. 2. Description: In this experiment, one end of the shaft is gripped in the chuck & heavy flywheel free to rotate in ball bearing is fixed at the other end of the shaft. The bracket with fixed end of the shaft can be clamped at any convenient position along lower beam. Thus, length of the shaft can be varied during the experiments. The ball bearing support to the flywheel provided negligible damping during the experiment. The bearing housing is fixed to side member of the main frame. 3. Experimental Procedure: Figure 2.1: Undamped vibration of single rotor system Fix the bracket at convenient position along the lower beam. Grip one end of the shaft at the bracket by chuck. Fix the rotor on the other end of shaft. Twist the rotor through some angle & release. Note down the time required for 10, 20 oscillations. Repeat the procedure for the different length of shaft. Darshan Institute of Engineering and Technology, Rajkot Page. 2.1

12 Torsional Vibrations of Single Rotor System 4. Nomenclature D Diameter of disc m d Diameter of shaft m f theo Theoretical frequency sec -1 f act Actual frequency sec -1 G Modulus of rigidity N/m 2 g Acceleration due to gravity m/sec 2 J Polar Moment of system m 4 I Moment of inertia of disc kg m 2 K t Torsional stiffness N-m/ radian L Length of shaft m n Nos. of oscillations T act Actual time period sec T theo Theoretical time period sec t time taken for n oscillations sec W Weight of disc Kg 5. Observation & calculation: 5.1 Data: D = 0.19 m d = G = 0.8 x N/m 2 g = 9.81 m/sec 2 W = kg 5.2 Observation Table: Sr. No L(m) Time for 10 Oscillations, t (sec) t 1 t 2 t 3 t Darshan Institute of Engineering and Technology, Rajkot Page. 2.2

13 Torsional Vibrations of Single Rotor System 5.3 Theoretical Calculations: J d 32 4 Mass moment of inertia of a circular disc of radius r and mass m is mr mr Ixx I yy and Izz 4 2 W D I 8 K t G J L T 2 theo I K t f theo 1 T theo 5.4 Experimental Calculations: T f act act t n 1 T act Darshan Institute of Engineering and Technology, Rajkot Page. 2.3

14 Torsional Vibrations of Single Rotor System 5.5 Result Table: Sr. No L (m) K t T theo. (sec) T act. (sec) f theo. (Hz) f act. (Hz) 6. Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 2.4

15 Free Vibration of Two Rotor System 1. Aim: Experiment No. 3 Study the Free Vibration of Two Rotor System and to determine the Natural Frequency of Vibration Theoretically & Experimentally. 2. Description: In this experiment, two discs having different mass moments of inertia are clamped one at each of the shaft by means of collect and chucks. Attaching the cross lever weights can change Mass moment of inertia of any disc. Both discs are free to oscillate in ball bearing. This provides negligible damping during experiment. Figure 3.1: Free vibration of two rotor system 3. Experimental procedure: Fix two discs to the shaft and fit the shaft in bearing. Deflect the disc in opposite direction by hand and release. Note down time required for particular number of oscillations. Fit the cross arm to one of the discs says B and again note down the time. Repeat the procedure with different equal masses attached to the ends of cross arm and note down the time. Darshan Institute of Engineering and Technology, Rajkot Page. 3.1

16 Free Vibration of Two Rotor System 4. Nomenclature: d Diameter of shaft m D A Diameter of disc A m D B Diameter of disc B m f act Actual frequency sec -1 f theo Theoretical frequency sec -1 G Modulus of rigidity of shaft N/m 2 g Acceleration due to gravity m/sec 2 I A Moment of inertia of disc A kg m 2 I B Moment of inertia of disc B (with weight on cross arm) kg m 2 J Polar Moment of inertia m 4 k t Torsional stiffness N-m/radian L Length of shaft m n Nos. of oscillations R Radius of fixation of weight on the arm m t time required for n oscillation sec T act Actual time period sec T theo Theoretical time period sec W 1 Weight attached to the cross arm kg W A Weight of disc A kg W B Weight of disc B Kg 5. Observation & calculation: 5.1 Data: Dia of disc A = m, Dia of disc B = m, Weight of disc A = kg Weight of disc B = kg Radius on the arm R = m, m Dia of shaft d = G = 0.8 x N/m 2 Darshan Institute of Engineering and Technology, Rajkot Page. 3.2

17 Free Vibration of Two Rotor System 5.2. Observation table: Sr. No 'n' t (sec) W 1 (kg) R (m) 5.3 Theoretical Calculations: d J 32 k t 4 GJ L, Mass moment of inertia of a circular disc of radius r and mass m is mr mr Ixx I yy and Izz 4 2 I I A B theo WD A 8 2 A, 2 2 W D 2W R 8 8 T 2 f Theo 2 2 B B 1 1 T theo IA IB k t( I A I B ), 5.4 Experimental Calculations: T f act act t n, 1 T act Darshan Institute of Engineering and Technology, Rajkot Page. 3.3

18 Free Vibration of Two Rotor System 5.5 Result Table: Sr. No I A (kg/m 2 ) I B (kg/m 2 ) T theo. (sec) T act. (sec) f theo. (sec -1 ) f act. (sec -1 ) 6. Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 3.4

19 Damped Torsional Oscillation 1. Aim: Experiment No. 4 To study the Damped Torsional Oscillation & to determine the damping co-efficient. 2. Description: This experiment consists of a long elastic shaft gripped at the upper end by chuck in the bracket. The bracket is clamped to upper beam of the main frame. A heavy steel flywheel clamped at the lower end of the shaft suspended from bracket. Damping drum is fixed to the lower face of the flywheel. This drum is immersed in water, which provides damping. Rotor can be taken up and down for varying the depth of immersion of damping drum. Recording drum is mounted to the upper face of the flywheel. Paper is to be wrapped around the recording drum. Oscillations are recorded on the paper with the help of specially designed piston of dashpot. The piston carries the attachment for fixing sketch pen. Figure 4.1: Damped Torsional Vibration 3. Experimental procedure: With no water in the container allow the flywheel to oscillation & measure the time for say 10 oscillations. Put thin mineral oil (no. 10 or 20) in the drum and note the depth of immersion. Put the sketch pen in its bracket. Allow the flywheel to vibrate. Darshan Institute of Engineering and Technology, Rajkot Page. 4.1

20 Damped Torsional Oscillation Allow the pen to descend. See that the pen always makes contact with paper & record oscillation. Determine an i.e. amplitude at any position & Entry. Amplitude after r cycle. After complete the experiment drain the water. 4. Nomenclature: n Critical damping factor d Diameter of shaft m D Diameter of disc m G Modulus of rigidity N/m 2 g Acceleration due to gravity m/sec 2 I Moment of inertia of disc kg m 2 J Polar Moment of inertia m 4 K t Torsional stiffness N-m L Length of shaft m n Nos. of oscillations r no. of cycles taken for X ntr. t time required for n oscillations sec T act Actual time period sec T theo Theoretical time period sec W Weight of disc kg X n Amplitude of the vibration at the beginning of the measurement to be found from record X ntr Amplitude of the vibration after r cycles from the record. m n Logarithmic decrement m 5. Observation & calculation: Depth of immersion Sr. No (mm) X 1 (mm) X 2 (mm) X 3 (mm) X 4 (mm) Darshan Institute of Engineering and Technology, Rajkot Page. 4.2

21 Damped Torsional Oscillation 5.3 Calculations: 1 log X 1 n e n Xn1 n 2 n 1 2 n Darshan Institute of Engineering and Technology, Rajkot Page. 4.3

22 Damped Torsional Oscillation 5.4 Result Table: Sr. No Depth of immersion (mm) Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 4.4

23 Dunker ley s Rule 1. Aim: Experiment No To verify the Dunker ley s Rule f f f L b Where: - f = Natural frequency of given beam (considering the weight of beam) with f L central load W. = Natural frequency of given beam (neglecting the weight of beam) with central load W. 1 48EIg 2 L W f L = 3 f b = Natural frequency of the beam. 2. Description: At rectangular bar is supported in trunion fitting at each end. Each trunion is provided in a ball bearing carried in housing. Each bearings housing is fixed to the vertical frame member. The beam carries at its center a weight platform. Figure 5.1: Dunker ley s Rule 3. Experimental Procedure: Arrange the set-up as shown in figure 5.1 with some wt. W clamped to wt platform. Pull the platform & release it to set the system in to natural vibrations. Darshan Institute of Engineering and Technology, Rajkot Page. 5.1

24 Dunker ley s Rule Find periodic time T & frequency of vibration f act by measuring time for some oscillations. Repeat experiment by putting additional masses on weight platform. Plot graph of 1/f act Vs. W 4. Nomenclature: b Width of beam m E Modulus of elasticity of beam material N/m 2 f L Frequency of beam sec -1 f b Natural frequency of beam sec -1 f act Actual frequency sec -1 G Acceleration due to gravity. m/sec 2 H Thickness of beam m I Moment of inertia m 4 L Length of the beam m N Number of oscillations T Time taken for n oscillation sec T act Actual time period sec W Weight of beam per unit length kg W Central load of the beam, OR weight attached. kg 5. Observation & calculation: 5.1 Data: L = 1 m. W = 1.5 Kg/m. b = m. E = N/m 2 h = m. 5.2 Observation Table: Sr. No W (kg) n t (sec) Darshan Institute of Engineering and Technology, Rajkot Page. 5.2

25 Dunker ley s Rule 5.3 Theoretical Calculations: 3 bh I 12 f f 1 48E.I 2 L W L 3 E.I 2 w L B f f f L b 5.3 Practical Calculations: T f act act t n 1 T act Darshan Institute of Engineering and Technology, Rajkot Page. 5.3

26 Dunker ley s Rule 5.4 Result Table: Sr. No T act. (sec) f act. (Hz) f (Hz) 6. Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 5.4

27 Equivalent Spring Mass System Experiment No AIM: To study the undamped free vibration of equivalent spring mass system. To find the frequency of undamped free vibration of equivalent spring mass system. 2. DESCRIPTION: The equipment is designed to study free damped and undamped vibration. It consists of M.S. rectangular beam supported at one end by a trunion pivoted in ball bearing. The bearing housing is fixed to the side member of the frame. The other end of beam is supported by the lower end of helical spring; upper end of the spring is attached to screw, which engages with screwed hand wheel. The screw can be adjusted vertically in any convenient position and can be clamped with the help of lock nut. The exciter unit can be mounted at any position along the beam. Additional known weights may be added to the weight platform under side exciter. Figure 6.1: Undamped free vibration of equivalent spring mass system 3. EXPERIMENTAL PROCEDURE: Support one end of beam in the slot of trunion and clamp it by means of screw. Attached the other end of the beam to lower end of spring. Adjust the screw to which the spring is attached with the help of hand wheel such that beam is horizontal in position. Darshan Institute of Engineering and Technology, Rajkot Page. 6.1

28 Equivalent Spring Mass System Weight the exciter assembly along with discs, bearing and weights platform. Clamp the assembly at any convenient position. Measure the distance L1 of the assembly from pivot. Allow system to vibrate freely. Measure the time for any 10 oscillations and periodic time and natural frequency of vibration. Repeat the experiment by varying L 1 and also putting different weights on platform. 4. Nomenclature: f act Actual frequency sec -1 f theo Theoretical frequency sec -1 g Acceleration due to gravity m/sec 2 k Stiffness of the spring kg/m L Distance of spring from pivot. m L 1 Distance of w from pivot. m L i Length of spring. m m Total mass of exciter assembly kg-sec 2 /m m e Equivalent mass exciter assembly kg-sec 2 /m n No. of oscillations. t Time taken by n oscillation sec T act Actual time period sec T theo Theoretical time period sec w Weight attached on exciter assembly kg W Weight of exciter assembly along with wt. platform. kg 5. Observation & Calculation: 5.1 Data: Li = K = 930 kg/m. W = 21 Kg. 5.2 Observation Table: Sr. No W (kg) w (kg) L l (m) L (m) n t (sec) Darshan Institute of Engineering and Technology, Rajkot Page. 6.2

29 Equivalent Spring Mass System 5.3 Theoretical Calculations: W w m g m L m L 2 1 e 2 T 2 f theo theo 1 T theo me K 5.4 Practical Calculations: T f act act t n 1 T act Darshan Institute of Engineering and Technology, Rajkot Page. 6.3

30 Equivalent Spring Mass System 5.5 Result Table: Sr. No T act. (sec) T theo. (sec) f act. (sec -1 ) f theo. (sec -1 ) 6. Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 6.4

31 Forced Vibration of Equivalent Spring mass system Experiment No Aim: To study the Forced damped Vibration of Equivalent Spring Mass System. To find the frequency of forced damped equivalent spring mass system To find amplitude of vibration. 2. Description: It is similar to that described for exp. no. 6. The exciter unit is coupled to D.C. variable speed motor. RPM of motor can be varied with the speed control unit. Speed of rotation can known from the RPM indicator on control panel. It is necessary to connect the damper unit to the exciter. Amplitude of vibration can be recorded on strip chart recorder. Figure 7.1: - Forced damped Vibration of Equivalent Spring Mass System 3. Damping Arrangement: Close the one hole of damper for light damping. Close the two holes of damper for medium damping. Close all the three holes of damper for heavy damping. 4. Experimental Procedure: Arrange the set-up as described for exp. no. 6 Start the motor and allow the system to vibrate. Wait for 1 to 2 minutes for amplitude to build the particular forcing frequency. Darshan Institute of Engineering and Technology, Rajkot Page. 7.1

32 Forced Vibration of Equivalent Spring mass system Adjust the position of strip chart recorder. Take the recorder of amplitude Vs. time on strip chart recorder by starting recorder motor. Press recorder platform on the pen gently. Pen should be wet with ink. Avoid excessive pressure to get good result. Take record by changing forcing frequencies. Repeat the experiment by adjusting the holes on the piston of damper can change different damping. Plot the graph of amplitude vs. frequency for each damping conditions. 5. Nomenclature: f a Frequency. sec -1 K Stiffness of spring kg/m L i Length of spring. m N RPM of motor. RPM n Nos. of oscillations from graph. t Time for n oscillations. sec 6. Observation & calculation: 6.1 Data: Li = 0.126m K = 930 kg/m. 6.2 Observation Table: Sr. No N (RPM) n t (sec) 6.3 Calculations: T act t n f act 1 T act Darshan Institute of Engineering and Technology, Rajkot Page. 7.2

33 Forced Vibration of Equivalent Spring mass system 6.4 Result Table: Sr. No f a sec -1 Amplitude ( mm) 7. Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 7.3

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35 Forced Vibration of Beam Experiment No Aim: To study the forced vibration of the beam for different damping. To find frequency of beam at different damping. 2. Description: In this experiment, a slightly heavy rectangular section bar than used in Expt. No. 6 is supported at both ends in trunion fittings. Exciter unit with the weight platform can be clamped at any conventional position along the beam. Exciter unit is connected to the damper, which provides the necessary damping. Figure 8.1: Forced lateral vibration of the beam for different damping 3. Damping Arrangement: Close the one hole of damper for light damping. Close the two holes of damper for medium damping. Close all the three holes of damper for heavy damping. 4. Experimental Procedure: Arrange the set-up as shown in figure 8.1. Connect the exciter Motor to control panel. Darshan Institute of Engineering and Technology, Rajkot Page. 8.1

36 Forced Vibration of Beam Start the Motor and allow the system to vibrate. Wait for 5 minutes for amplitude to build up for particular forcing frequency. Adjust the position of strip chart recorder. Take the recorder of amplitude vs. time on strip chart recorder by starting recorder motor. Take record by changing forcing frequency for each damping. Repeat the experiment for different damping. Plot the graph of amplitude vs. frequency for each damping. 5. Observation & calculation: 5.1 Observation Table: Sr. No N (RPM) n t (sec) 5.2 Calculations: T act t n f act 1 T act Darshan Institute of Engineering and Technology, Rajkot Page. 8.2

37 Forced Vibration of Beam 5.3 Result Table: Sr. No f a sec -1 Amplitude (mm) 6. Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 8.3

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39 Static & Dynamic Balancing 1. Objective: Experiment No. 9 To study the Static & Dynamic Balancing system. 2. Aim: To balance the masses Statically & Dynamically of a simple rotating mass system. To observe the effect of unbalance in a rotating mass system. 3. Introduction: A system of rotating masses is said to be in static balance if the combined mass centre of the system lies on the axis of rotation. When several masses rotate in different planes, the centrifugal forces, in addition to being out of balance, also form couples. A system of rotating masses is in dynamic balance when there does not exist any resultant centrifugal force as well as resultant couple. Figure 9.1 Static Balancing apparatus 4. Theory: Conditions for static & dynamic balancing: If a shaft carries a number of unbalanced masses such that the center of mass of the system lies on the axis of rotation, the system is said to statically balance. The resultant couple due to all the inertia forces during rotation must be zero. Darshan Institute of Engineering and Technology, Rajkot Page. 9.1

40 Static & Dynamic Balancing These two conditions together will give complete dynamic balancing. It is obvious that a dynamically balanced system is also statically balanced, but the statically balanced system is not dynamically balanced. Balancing of several masses rotating in different planes: When several masses revolve in different planes, they may be transferred to a reference plane (written as RP), which may be defined as the plane passing through a point on the axis of rotation and perpendicular to it. The effect of transferring a revolving mass (in one plane) to a reference plane is to cause a force of magnitude equal to centrifugal force of the revolving mass to act in the reference plane, together with a couple of magnitude equal to the product of the force and the distance between the plane of rotation and the reference plane. In order to have a complete balance of the several revolving masses in different planes, the following conditions must be satisfied: The forces in the reference plane must balance, i.e. the resultant force must be zero. The couple about the reference plane must balance, i.e. the resultant couple must be zero. Let us now consider four masses m 1, m 2, m 3 and m 4 revolving in planes 1, 2, 3 and 4 shown in fig. The relative angular positions of these masses are shown in the end view Fig. The magnitude, angular position and position of the balancing mass m1in plane 1may be obtained as discussed below: Take one of the planes, say 1as the reference plane (R.P.). The distance of all the other planes to the left of the reference plane may be regarded as negative, and those to the right as positive. Tabulate the data as in table. The planes are tabulated in the same order i.e. 1, 2, 3. Darshan Institute of Engineering and Technology, Rajkot Page. 9.2

41 Static & Dynamic Balancing Plane Weight No. Mass (m) Radius (r) Angle Mass moment m x r Distance from plane 1 (L) Couple m x r x L (1) (2) (3) (4) (5) (6) (7) (8) 1(R.P.) 4 m 1 r 1 1 m 1 r m 2 r 2 2 =0 0 m 2 r 2 L 2 m 2 r 2 L m 3 r 3 3 m 3 r 3 L 3 m 3 r 3 L m 4 r 4 4 m 4 r 4 L 4 m 4 r 4 L 4 The position of plane 4 from plane 2 may be obtained by drawing the couple polygon with the help of data given in column no. 8. The magnitude and angular position of mass m1 may be determined by drawing the force polygon from the given data of column no.5 & column no.6 to some suitable scale. Since the masses are to be completely balanced, therefore the force polygon must be closed figure. The closing side of force polygon is proportional to the m 1 r 1. The angular position of mass m1 must be equal to the angle in anticlockwise measured from the R.P. to the line drawn on the fig. parallel to the closing side of force polygon. 5. Description: The apparatus consists of a steel shaft mounted in ball bearings in a stiff rectangular main frame. A set of four blocks of different weights is provided and may be detached from the shaft. A disc carrying a circular protractor scale is fitted to one side of the rectangular frame. A scale is provided with the apparatus to adjust the longitudinal distance of the blocks on the shaft. The circular protractor scale is provided to determine the exact angular position of each adjustable block. The shaft is driven by electric motor mounted under the main frame, through a belt. For static balancing of weights the main frame is suspended to support frame by chains then rotate the shaft manually after fixing the blocks at their proper angles. It should be completely balanced. In this position, the motor driving belt should be removed. For dynamic balancing of the rotating mass system, the main frame is suspended from the support frame by two short links such that the main frame and the supporting frame are in the Darshan Institute of Engineering and Technology, Rajkot Page. 9.3

42 Static & Dynamic Balancing same plane. Rotate the statically balanced weights with the help of motor. If they rotate smoothly and without vibrations, they are dynamically balanced. 6. Utilities required: Electricity Supply: Single Phase, 220 V AC, 50 Hz, 5-15 Amp. combined socket with earth connection. Bench Area Required: 1m x 0.5 m 7. Experimental procedure: Insert all the weights in sequence from pulley side. Fix the pointer and pulley on shaft. Fix the pointer on 0 o (2) on the circular protractor scale. Fix the weight no. 1 in horizontal position. Rotate the shaft after loosening previous position of pointer and fix it on 3. Fix the weight no. 2 in horizontal position. Loose the pointer and rotate the shaft to fix pointer on 4. Fix the weight no. 3 in horizontal position. Loose the pointer and rotate the shaft to fix pointer on 1. Fix the weight no.4 in horizontal position. Now the weights are mounted in correct position. For static balancing, the system will remain steady in any angular position. Now put the belt on the pulleys of shaft and motor. Supply the main power to the motor through dimmerstat. Gradually increase the speed of the motor. If the system runs smoothly and without vibrations, it shows that the system is dynamically balanced. Gradually reduce the speed to minimum and then switch off the main supply to stop the system. 8. Precaution & maintenance instructions: Never run the apparatus if power supply is less than 200Volts and above 230 Volts Increase the motor speed gradually. Experimental set up should be tight properly before conducting experiment. Before starting the rotary switch, dimmer stat should be at zero position. Darshan Institute of Engineering and Technology, Rajkot Page. 9.4

43 Static & Dynamic Balancing 9. Troubleshooting: If the shaft is not rotate but motor rotate check both the pulleys. If the main light is not ON check the main switch. If the motor is not working check the rotary switch & dimmer stat. 10. Experimental procedure: L Distance between particular weight from weight 1 mm W Mass of particular weight kg θ Angle of particular weight from Reference Point degree 11. Observation & calculation: Observation table: Sr. No Plane Mass, m (gms.) Angle from reference line, (degree) Distance L (mm) Calculations Table: Sr. No Mass, m (gms.) Mass moment m r Couple m r L Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 9.5

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45 Whirling of Shaft Experiment No Objectives: To study the effect of whirling of shaft with: Both ends fixed Both ends supported One end fixed & one end supported 2. Aim: To study the modes of vibration and to measure the frequency of each case. Figure 10.1: Whirling of shaft apparatus 3. Introduction: A rotating shaft tends to bow out with large amplitude at a certain speed of rotation. This phenomenon is known as the whirling of the shaft. The speed at which it occurs is called the critical speed. 4. Theory: In actual practice, a rotating shaft carries different mountings and accessories in the form of gears, pulleys etc. when the gears or pulleys are put on the shaft, the center of gravity of the pulley or gear does not coincide with the center line of the bearings or with the axis of the shaft, when the shaft is stationary. This means that the center of gravity of the pulley or gear is at a certain distance from the axis of rotation and due to this, the shaft is subjected to centrifugal force. This force will bend the shaft, which will further increase the distance of center of gravity of the pulley or gear from the axis of rotation. The bending of shaft not only Darshan Institute of Engineering and Technology, Rajkot Page. 10.1

46 Whirling of Shaft depends upon the value of eccentricity (distance between center of gravity of the pulley and axis of rotation) but also depends upon the speed at which the shaft rotates. The speed, at which the shaft runs so that the additional deflection of the shaft from the axis of rotation becomes infinite, is known as critical or whirling speed. Possible Experiments with Elastic Rods ExperimentNo. End Fixing Mode of Whirl 1 One supported other fixed 1st Mode 2 One supported other fixed 2nd Mode 3 Both end Supported 1st Mode 4. Both end Supported 2nd Mode 5 Both end fixed 1st Mode 5. DESCRIPTION: The apparatus consists of a DC motor as the driving unit, which drives the shaft supported in fixing ends. Fixing ends can slide and adjust according to the requirement on the guiding pipes. Motor is connected to the shaft through flexible coupling. The shafts of different diameters can be replaced easily with the help of fixing ends. A dimmerstat is provided to increase or decrease the rpm of the motor. The whole arrangement is fixed on M.S frame. Guards are provided to protect the user from accident. 6. UTILITIES REQUIRED: Electricity Supply: Single Phase, 220 V AC, 50 Hz, 5-15 Amp. combined socket with earth connection. Bench Area Required: 2 m x 0.5 m Darshan Institute of Engineering and Technology, Rajkot Page. 10.2

47 Whirling of Shaft 7. Experimental Procedure: Fix the shaft to be tested in the fixed ends. Supply the main power to the motor through dimmerstat. Gradually increase the speed of motor until the first mode of vibration is not arrived. Study the first mode of vibration and note down the corresponding speed of the shaft with the help of hand tachometer. Gradually increase the speed of motor again, until the second mode of vibration is not arrived. Study the second mode of vibration and note down the corresponding speed of the shaft with the help of hand tachometer. Reduce the speed gradually and when shaft stop rotating, cut off the main power supply. Repeat the experiment for the shafts of different diameters. 8. Precaution & maintenance instructions: Never run the apparatus if power supply is less than 200Volts and above 230 Volts If the revolutions of an unloaded shaft are gradually increased, it will be found that a certain speed will be reached at which violent instability will occur, the shaft. Deflecting in a single bow and whirling round like a skipping rope. If this speed is maintained the deflection will become so large that the shaft will be fractured. It is advisable to increase the speed of shaft rapidly and pass through the critical speeds first rather than observing the 1st critical speed which increases the speed of rotation slowly. In this process, there is a possibility that the amplitude of vibration will increase suddenly bringing the failure of the shaft. If the shaft speed is taken to maximum first and then reduce slowly (thus not allowing the note the corresponding speed and then by reducing the speed further the next mode of lower frequency can be observed without any danger of rise in amplitude as the speed is being decreased and the inertia forces are smaller in comparison with the bending spring forces hence possibility of buildup of dangerous amplitudes are avoided. It is a destructive test of shafts and it is observed that the elastic behavior of the shaft material changes a little after testing it for a few times and it is advisable to use fresh shafts afterwards. Fix the apparatus firmly on the suitable foundation. Always keep apparatus free from dust. Darshan Institute of Engineering and Technology, Rajkot Page. 10.3

48 Whirling of Shaft 9. Troubleshooting: The main light is not ON check the main switch. The motor is not working check the rotary switch & dimmer-stat. 10. Nomenclature: A Area of shaft m 2 d 1 Diameter of shaft 1 m d 2 Diameter of shaft 2 M d 3 Diameter of shaft 3 M E Young s Modulus of elasticity kg/cm 2 f n Frequency of transverse vibration Hz g Acceleration due to gravity m/s 2 I Moment of inertia of shaft m 4 L 1 Length of shaft 1 m L 2 Length of shaft 2 m L 3 Length of shaft 3 m m s Mass of the shaft kg/m N a Actual whirling speed RPM N t theoretical whirling speed RPM W Weight of the shaft N/m ρ Density of shaft material kg/m 3 δ S Static deflection of shaft due to mass of shaft m 11. OBSERVATION & CALCULATION: 11.1 DATA: Acceleration due to gravity g =9.81m/sec 2 Diameter of shaft 1, d 1 = m Diameter of shaft 2, d 2 = m Diameter of shaft 3, d 3 = m Density of shaft material, ρ = kg/m 3 Young s Modulus of elasticity, E = N/m 2 Length of shaft 1, L 1 = 1 m Length of shaft 2, L 2 = 1 m Length of shaft 3, L 3 = 1 m 11.2 Observation Table: Darshan Institute of Engineering and Technology, Rajkot Page. 10.4

49 Whirling of Shaft Sr. No Shaft 1 Shaft 2 Shaft 3 Sr. No Shaft 1 Shaft 2 Shaft 3 Sr. No Shaft 1 Shaft 2 Shaft When both the ends are fixed: When both the ends are supported: N a R.P.M. 1 st Mode 2 nd Mode N a R.P.M. 1 st Mode 2 nd Mode When one end is fixed and other end is supported: 11.3 Calculations: I D 64 4 W ms g ms A A d 4 When both the ends are fixed: 2 4 WL S 384EI f n S 1.27 N 60 f t n N a R.P.M. 1 st Mode 2 nd Mode Darshan Institute of Engineering and Technology, Rajkot Page. 10.5

50 Whirling of Shaft When both the ends are supported: 4 5W L S 354EI f n t S 1.27 N 60 f n When one end is fix and other is supported: 4 WL S 185EI f n t S 1.27 N 60 f n Darshan Institute of Engineering and Technology, Rajkot Page. 10.6

51 Whirling of Shaft 11.4 Result Table Shaft 1 Shaft 2 Shaft 3 I m s W A Shaft 1 Shaft 2 Shaft 3 δs m Both end are fixed One Fixed other supported Both end are supported f n Hz Nt rpm δs m f n Hz Nt rpm δs m f n Hz Nt rpm 12. Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 10.7

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53 Rolling Cylinder Experiment No Aim: To find out natural frequency of undamped free vibration of a single degree of freedom system a cylinder rolling in another cylinder. 2. Description: In this experiment, two cylinders are used. One big half cylinder and other small cylinder. Small cylinder is roll inside the big half cylinder. Material of big half cylinder is wood, where as material for small cylinder is wood, Aluminum and MS. Dimension of big cylinder is fixed, where as dimensions of small cylinder is vary. 3. Experimental procedure: At first measure the diameter of the rolling cylinder. Also measure the diameter of cylinder surface in which we are interested to roll another cylinder. Then make free the rolling cylinder in cylindrical surface. So, that it can continue with motion. Note the time for 10 oscillations. Repeat the procedure for different diameter roller and different material roller. 4. Nomenclature: r Radius of smaller cylinder m R Radius of bigger cylinder m g Acceleration due to gravity m/sec 2 n Nos. of oscillations ω n natural frequency sec -1 f act Actual natural frequency sec -1 f theo Theoretical natural frequency sec -1 T act Actual time period sec T theo Theoretical time period sec 5. Observation & calculation: Darshan Institute of Engineering and Technology, Rajkot Page. 11.1

54 Rolling Cylinder 5.1 Data: Radius of bigger cylinder, R = m Radius of smaller cylinder, r 1 = Radius of smaller cylinder, r 2 = Radius of smaller cylinder, r 3 = Sr. No 5.2 Observation Table: Material Size of Roller (m) Time for n oscillations (sec) t 1 t 2 t 3 t 1 Wood 2 Aluminum 3 M.S. 5.3 Theoretical Calculations: Natural frequency, n 2g 3( R r ) T f theo theo 2 sec, n 1 Hz T theo 5.4 Practical Calculations: Time for n oscillations Time for oscillation, Tact sec n Frequency, f act 1 Hz T act Darshan Institute of Engineering and Technology, Rajkot Page. 11.2

55 Rolling Cylinder 5.5 Result Table: Size of Roller Sr. No Material (m) T theo. (sec) T act. (sec) f theo. (Hz) f act. (Hz) 1 Wood 2 Aluminum 3 M.S. 6. Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 11.3

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57 Double Pendulum System 1. Aim: Experiment No. 12 To find out natural frequency of vibration of double pendulum system. 2. Description: The system which requires two coordinates independently to describe its motion completely is called a two degree freedom system. In such system there are two masses which will have two natural frequencies. The system at its lowest or first natural frequency is called its first mode; at its next second higher it is called the second mode. If the two masses vibrate at the same frequency and phase, it is called a principal mode of vibration. This experiment consists of a long string in which two masses are attached at some distance. Figure 12.1: Analysis of a Double Pendulum Figure 12.2: Mode Shapes Darshan Institute of Engineering and Technology, Rajkot Page. 12.1

58 Double Pendulum System 3. Experimental Procedure: Fix the both of the balls at X 1 and X 2 distances from centerline respectively. Then note down the time period for n oscillations. Repeat the procedure for different values of X 1 and X 2. Compare the theoretical and experimental values. 4. Nomenclature: X 1 Distance of first ball from center axis m X 2 Distance of second ball from center axis m g Acceleration due to gravity m/sec 2 l Length of each string m n Nos. of oscillations f act Actual natural frequency sec -1 f theo Theoretical natural frequency sec -1 T act Actual time period sec T theo Theoretical time period sec 5. Observation & Calculation: 5.1 Data: Length of string, l = 5.2 Observation Table: Sr. No Mode of Vibration X 1 X 2 Time for n Oscillations (sec) t 1 t 2 t 3 t Theoretical Calculations: Darshan Institute of Engineering and Technology, Rajkot Page. 12.2

59 Double Pendulum System X X 1 When g( 2 2 ) n l T f X X theo theo 2 sec n 1 Hz T theo 1 When g( 2 2 ) n l T f theo theo 2 sec n 1 Hz T theo 5.4 Practical Calculations: Time for n oscillations Time for oscillation, Tact sec n Frequency, f act 1 Hz T act Darshan Institute of Engineering and Technology, Rajkot Page. 12.3

60 Double Pendulum System 5.5 Result Table: Sr. No Mode of Vibration T theo (sec) T act (sec) f theo. (Hz) f act. (Hz) Conclusion: Darshan Institute of Engineering and Technology, Rajkot Page. 12.4

61 Cam 1. Aim: To study jump phenomenon in the cam. Experiment No Description: In a cam-follower system, the contact between the cam surface and follower is maintained by means of a retaining spring. Beyond a particular speed of cam rotation, the follower may lose contact with the cam, because of inertia force acting on the follower. This phenomenon is called as Jump Phenomenon. When the follower re-establishes contact with the cam, it may do so with severe impact loads that can damage the surface of the cam and hammering nose can be heard at this jump speed. During the follower jump, transient vibrations are set up in the follower and these occur only with high speed, highly flexible cam-follower system. With jump, the cam and follower separate owing to excessively unbalanced forces exceeding the spring force during the period of negative acceleration. This is undesirable since the fundamental function of the camfollower system, the constraint and control of follower motion are not maintained. Also, the life of the cam flank surface reduces due to hammering action of follower on cam and hammering noise is generated which further results in vibrations of the system. The jump phenomenon will be avoided by limiting the speed of cam or by increasing the stiffness of the retaining speed. Fig.13.1 shows an eccentric cam follower system, which is analyzed for jump phenomenon. Fig Darshan Institute of Engineering and Technology, Rajkot Page. 13.1

62 Cam From Fig.13.1, we have Lift of follower = y = OS 2 OS 1 = A 2 Q + QY OS 1 = R + e cos (180º θ) (R e) y = e (1 cos θ).. (13.1) Differentiating equation (13.1) with respect to time t, we get Velocity of follower = ẏ = e ω sin θ. (13.2) Differentiating equation (13.2) with respect to time t, we get Acceleration of follower = ÿ = e ω 2 cos θ. (13.3) Where, θ = ω. t = cam angle turned from lowest position. (13.4) Now, consider the arrangement as shown in Fig.13.2, then Fig Let, m = mass of follower e = eccentricity k = stiffness of spring F = constant force between cam and follower F s = total spring force P = Preload in spring ω = Angular speed of cam Darshan Institute of Engineering and Technology, Rajkot Page. 13.2

63 Cam Then, from free body diagram, we have Inertia Force = Σ External Forces m ÿ = F S F F = m ÿ + F S F S = P + k y F S = P + k e (1 cos ωt)... (a)..(b) Substituting from equation and equation and (b) in (a), we get F = m (e ω 2 cos ωt) + P + k e (1 cos ωt) F = (m ω 2 k) e cos ωt + (P + k e)... (13.5) This contact force between cam and follower is maximum when θ = 0 and minimum when θ = 180. It is also dependent upon the square of cam velocity. When this contact force between cam and follower becomes negative (i.e. less than zero.) The follower would lose contact with the surface resulting in jump. This would happen if the speed is increased beyond a particular critical speed ω j (at θ = 180 ). F = 0 = (m ω 2 j k) (e cos 180) + (P + k e) Where ω j = jump speed 0 = (m ω 2 j k) ( e) + P + k e (m ω 2 j k) e = P + k e m e ω 2 j = P + 2 k e j P 2 k e me Therefore, to avoid jump j P 2 k e me To avoid jump P > (m ω 2 2 k) e Darshan Institute of Engineering and Technology, Rajkot Page. 13.3