Mm 203 Lab Manual

2003 Dublin City University

MECHANICS OF MACHINES 1 LABORATORY MANUAL

2nd Year – Semester 4

Computer Aided Mechanical and Manufacturing Engineering, CAM2 Medical Mechanical Engineering, MEDM2

Mechatronic Engineering, ME2

Laboratory Co-ordinator Dr. Paul Young Room: S374 Phone: 01 700-8216 E-mail: [email protected]

MM203 - Mechanics of Machines 1

Practical Manual

2004 Paul Young, Dublin City University 2

Demonstrators Paul Young Julfikar Haider

Technician

Chris Crouch

Experiments

1. Gears 2. Clutch 3. Planar Mechanisms 4. Gyroscope 5. Static & Dynamic Balance

Safety Notices A) Eating, drinking and smoking are prohibited in the

laboratories at all times. B) Extreme care must be exercised operating lab

equipment. Examine equipment before use, and ask demonstrators for assistance when necessary. Use the lab handouts as a guide for start up and shut down procedures.


Practical Manual


Introduction There are five practicals available for this module, although each student will only do four. As a result, at each session, all five will be available. Laboratory manuals are only available online. It is the responsibility of the student to download the laboratory sheet and bring it with them. Hardcopy is essential. Students must purchase a hardback laboratory book with graph paper and each report will be written within the 3 hour scheduled session. The books must be left in the classroom at the end of the session and will be graded before the next scheduled session.

Report Guidelines Comprehensive guidelines for reports were given in MM101 & MM102, and those still apply here. The first page of each report should be a title page and should include page numbers. Contents should be listed in the same manner as a text book. Take all rough measurements on paper first and keep the book for the report only. As outlined in the lectures, laboratory reports should be kept relatively short, but should include some essential components:

• Introduction - a brief description of the subject matter

• Aims/Goals - bullet point statements of what you intend to achieve

• Procedure - Just putting 'as per handout' is not quite enough. Describe the procedure in minimalist terms and include any relevant diagrams, particularly those which assist in understanding the measurements recorded. It is important to clarify any methods which are not explicitly stated in the instructions.

• Results - The information recorded and, if not in the procedure, any relevant information about the manner in which ambiguous results were decided in a consistent fashion reported. (i.e. enough information for others to repeat the experiment and arrive at the same results)

• Calculations - Any subsequent calculations based on the results

• Discussion - A discussion of the contents of the results and calculations. If there are three graphs, then some comment on each graph is necessary.

• Conclusions - These should relate directly to the goals/aims. Bullet point format is fine.

There is enough information in the handouts to complete the experiments. Any apparent lack of information may be overcome through consulting reference books in the library, asking the demonstrators or discussion with the lecturer.


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Brief Description The numbers shown here relate to the experiment number in the timetable.

1. Gears Using a simple configurable arrangement this laboratory seeks to provide an understanding of simple and compound gear trains before gaining experience with the calculation of gear ratios for epicyclic gear systems.

2. Clutch Using a simple arrangement of axial load and applied torque the limits of friction are determined for three different clutches. The readings are the used to validate the theory.

3. Planar Mechanisms This laboratory uses linkages attached to drawing boards to demonstrate the kinematic properties of three mechanisms - Scotch Yoke, Elliptic Trammel and Single/Double Toggle

4. Gyroscope This experiment demonstrates the relationship between precession and the gyroscopic couple. Using a series of balance readings the relationship between the speeds of rotation/precession and the gyroscopic couple is investigated.

5. Static & Dynamic Balancing Static and dynamic imbalance/balance are demonstrated initially in a simplified arrangement. By determining the out-of-balance moment for each of four masses an analytic solution to balancing the system is undertaken. Proof of the solution is found through application of the solution on the apparatus.


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Gear Train Experiment

Aims To gain an understanding of the speed ratios and directions of rotation of simple and compound gear trains

Equipment The apparatus consists of four spur gears, a frame, three axle pins and a vernier callipers. In addition, several spacers and washers are included to allow alignment of the gears and ensure tight meshing.

Figure G1 – Gear Train Apparatus

Procedure

Initial Inspection The four gears should be inspected and the measurements of the pitch circle diameter and the number of teeth taken for each gear. These can then be used to determine the module of this gear system. The pitch circle is shown in the figure below where it can be seen that the two pitch circles intersect at a single point. The module is the Pitch Circle Diameter divided by the number of teeth. Record your results in table G1 below:


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Label Description DG Pitch Circle Diameter Gear DP Pitch Circle Diameter Pinion DRG Root Circle Diameter Gear DRP Root Circle Diameter Pinion DOG Outside Circle Diameter Gear DOP Outside Circle Diameter Pinion hk Working Depth ht Whole Depth a Addendum b Dedendum c Clearance p Circular Pitch (measured along

Pitch Circle) t Tooth Thickness (measured

along Pitch Circle) C Centre Distance

Figure G2: Diagram of Gear Train

Table G1: Gear Measurements

Gear # Diameter (mm) # of teeth Module (mm)

1

2

3

4

Convert the module to equivalent teeth/inch and using the information in table G2 of metric and imperial gear sizes determine whether the gears are metric or imperial. Hint: 1 inch = 25.4 mm

Table G2: Standard Metric & Imperial Modules Metric Imperial Metric Imperial

Module mm

Equivalent Diametral Pitch

teeth/inch

Closest Standard Imperial Gear

teeth/inch

Module mm

Equivalent Diametral Pitch

teeth/inch

Closest Standard Imperial Gear

teeth/inch 0.3 84.667 80 4 6.350 6 0.4 63.500 64 5 5.080 5 0.5 50.800 48 6 4.233 4 0.8 31.750 32 8 3.175 3 1 25.400 24 10 2.540 2.5

1.25 20.320 20 12 2.117 2 1.5 16.933 16 16 1.588 1.5 2 12.700 12 20 1.270 1.25

2.5 10.160 10 25 1.016 1 3 8.467 8


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Simple Gear Train - Part 1 The simplest gear train involves only two gears. Select any two gears and fix them to the arm using the pivots provided. Mark the mesh point on both gears. Rotate the gear (not the pinion) anticlockwise one revolution and count the number of revolutions (including any fractional rotation) and direction of the pinion. Note the gear ratio for this pair of gears and repeat using the other two gears.

Figure G3

Table G3: Two Gear System Results

Gear Rotation Pinion Rotation Assy Teeth Gear

Teeth Pinion Dir Revs Dir Revs

Tooth Ratio

Rotation Ratio

1 Anti-Clock 1

2 Anti-Clock 1

Simple Gear Train - Part 2 Using the last pair of gears from Part 1 above, insert a third gear as an 'idler' between the two gears and ensure a good meshing between the gears. Mark the mesh points and again rotate the gear through one anticlockwise rotation. Count the rotations and direction of the final gear. Replace the idler gear with the remaining gear and repeat the experiment.

Table G4: Idler Gear System Results

Gear Idler Pinion

Ass

embl

y

Teet

h G

ear

Teet

h Id

ler

Teet

h P

inio

n

Dir Revs Dir Revs Dir Revs Toot

h R

atio

G

ear:P

inio

n

Rot

atio

n R

atio

G

ear:P

inio

n

1 Anti-Clock 1

2 Anti-Clock 1

Note the effect of the insertion of the idler gear and the change in size of the idler gear on the overall ratio.


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Compound Gear Train The term compound gear train refers to systems that involve four or more gears where at least two gears rotate together on the same shaft. The gear train should be built as shown in the diagram below. The two smaller gears can be fixed together with a pin forming the central compound idler. Place this with the smaller gear uppermost first.

Assembly 1 Assembly 2

Figure G4: Compound Gear System

As before, rotate the Input gear through one anticlockwise revolution noting the direction and rotation of the other gears. Rearrange the meshing of the gears so that the larger gear on the idler is uppermost as shown below. Rotate the input gear again and repeat the observations. Determine the gear ratios between the input and output gears from these observations.

Table G5: Compound Gear System Results

Gear Compound Pinion

Ass

embl

y

Toot

h R

atio

G

ear:C

ompo

und

Toot

h R

atio

C

ompo

und:

Pin

ion

Dir Revs Dir Revs Dir Revs

Rev

Rat

io

Gea

r:Pin

ion

1 Anti-Clock 1

2 Anti-Clock 1

Again note how the gear ratios are affected by the introduction of the compound gear and the effect of changing the gear on the system. Find the relationship between the tooth rations and the rev ratio.

Epicyclic Gear System The epicyclic gear system consists of at least three gears – sun gear, planet gear & ring gear - and a rotating carrier (or link). The example system allows you to see the operation of a complete assembly – a mechanism at the back allows either the link or the ring gear to be fixed and the handwheel drives the sun gear. An epicyclic gear system allows concentric input and output shafts. There are three possible combinations of input and output available … sun : link (lock ring), sun :


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ring (lock link), ring : link (lock sun). Note the direction of rotation of each gear pair on the sample system with the link locked … the sun & planet behave as seen from the previous part of the experiment, but the internal teeth on the ring gear mean that it rotates in the same direction as the planet gear.

Link

Sun Planet

Ring

The behaviour of the individual gears may be simulated using the same equipment as before. The arm on which the gears are mounted can rotate simulating the link. Any gear positioned at the centre of rotation will operate as a sun gear. A gear meshing with the sun gear will behave like a planet gear. Simulation of the ring gear is achieved by placing a third gear on the arm and holding it so that it cannot rotate. The goal of this is to gain an understanding of the relationships between the movement of the gears so that the relationship between the movement of the sun and the link for the supplied example system may be calculated.

Step 1 – Effect of Rotating Link Place the second largest gear on the shaft at the end of the arm to act as the sun gear. Add the smallest gear so that it meshes with the sun acting as a planet gear. The largest gear should be placed to mesh with the planet gear. Mark all the gears so that their orientation is clearly visible. Unlock the arm from the frame and, ensuring that the sun gear rotates with the arm rotate the arm through one revolution anticlockwise. Stop the movement at quarter cycles to help keep track of the movement. Record the rotation in the first row of table G6.

Step 2 – Effect of Fixing Link

Fix the arm and rotate the sun gear. Note how the system behaves and the relationship between rotation of the gears and their teeth.

Step 3 – Simulation of Locked Gear To simulate the operation of the epicyclic system with the ring gear locked the outer gear must be prevented from rotating. Holding this gear in the same orientation (with respect to the room) by hand while rotating the link through one revolution. Record the movement of the other two gears in the second row of table G6. This gives the gear ratio between the link and the sun gear.


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Table G6: Epicyclic Gear system

Arm Sun Gear Planet Gear Ring Gear

Revs Direction Revs Direction Revs Direction Revs Direction

1 Anti-Clockwise 0 none

1 Anti-Clockwise 0 none

Analysis of Example System There is a fully assembled sample system on the bench. The gear ratio in an epicyclic gear system may be analysed using the following steps.

1. Rotate whole system together one revolution anti-clockwise 2. Fix the Link and rotate the gear which is normally fixed one

revolution clockwise recording the movement of the other gears 3. Add the movements from steps 1 & 2

Example: In an epicyclic gear system the sun gear has 30 teeth, the planet gear has 20 teeth and the ring gear has 70 teeth. Calculate the gear ratio between the Ring and the Link with the sun gear fixed. Revolutions Procedure Sun Planet Ring Link 1 Rotate whole system 1 rev anticlockwise +1 +1 +1 +1

2 Fix Link and Rotate Sun Gear 1 rev clockwise -1 + 2030 + 70

30 0

3 Sum steps 1 & 2 0 +2½ +1 73 +1

The gear ratio between the ring and the link is +1 73 :+1

Count the number of teeth on the example system and calculate the ratio between the sun and the link with the ring gear fixed.


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Clutch Plate Experiment

Aim The purpose of the experiment is to verify the application of the expression for maximum torque

T = µWRmean Where T = torque, µ = coefficient of friction, W = axial load & Rmean is

the mean radius and in particular that

a) torque at slip is proportional to the normal force and b) the torque at slip is proportional to radius of the friction ring

Apparatus A wall bracket with a stationary horizontal plate supports three rings of brake friction material concentrically to a central ball bearing that locates a rotating upper plate of aluminium alloy. The rings are held by pairs of pegs that fit holes on a diameter of each ring. They lift off the pegs, and are meant to be used one at a time. The periphery of the upper plate is grooved to guide two cords over a pair of diametrically opposed pulleys and thence down to two equal load hangers. This system, when loaded with equal weights, produces a pure torsional drive for the plate that acts as a turntable on the friction rings. By loading the turntable a normal force or pressure acts between the upper plate and the ring in use.

Table 1 - Weight Set for Clutch Experiment

No.

Weight (N)

No.

Weight (N)

No.

Weight (N)

No.

Weight (N)

4 50 2 10 4 2 2 0.5

2 20 2 5 2 1

Procedure The apparatus must be kept free of oil or grease (the bearings are sealed), and the turntable must NEVER be put in place without a friction ring in position to keep the turntable off the ring location pegs.


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Figure C1: Key Dimensions for Clutch Apparatus

1. Check the mass of the turntable using the weighing scales: MT kg.

2. Calculate the self load due to gravity of the turntable: WT Newtons

3. Check the mass of the hangers using the weighing scales: MH kg.

4. Calculate the self load due to gravity of the hangers: WH Newtons

5. Measure the mean diameter at which the torque is applied: D mm. 6. Measure the internal and outer diameters of each of the friction rings while the

turntable is removed (push the central locating spigot upward to help lift it).

Ring Inner Diameter

I.D. (mm) Outer Diameter

O.D. (mm) Mean Radius

Rmean = (OD+ID)/4

A: Small

B: Medium

C: Large

7. Remove the two larger friction rings, leaving the smallest (A) in position, and replace the turntable with the torque cords taken over their pulleys. Hook on the load hangers. With only the self-weight of the turntable as its load add equal value weights to the load hangers until, with a slight start, the turntable very slowly revolves at a constant speed. Record the load per hanger in the table on the following page. Repeat the procedure adding increments of 50N onto the turntable up to 200N.


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8. Unload and remove the turntable. Exchange the friction ring for the next larger one and repeat the above procedure. Finally, use the largest friction ring in a similar way.

Results A: Small B: Medium C: Large

Rmean

Nominal Load

Actual W: Self Load + Added Load

Load on 1 Pulley, L =WH + WA

Torque L x D

Load on 1 Pulley, L =WH + WA

Torque L x D

Load on 1 Pulley, L = WH + WA

Torque L x D

Self-Load

50 N

100 N

150 N

200 N

9. At the end of the experiment, position all three friction rings on the stationary plate and replace the turntable.

On a single graph plot the Torque vs Load for each ring. Note the nature of the graphs and comment on the relationship between the three sets of readings.

On a second graph plot Torque vs Rmean for a load of 150 Newtons on the friction rings. Again comment on the form of the graph and its relation to the theory.

Final Challenge:

From your graphs calculate the coefficient of friction between the clutch material and the underside of the turntable and suggest why using the second graph for this calculation might improve the result.


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Planar Mechanisms Experiment

Introduction The motion of complex assemblies is determined by the configuration of links and joints. Here we have reduced the complexity by limiting the movements to be planar and controlling the scope of the links and the joints. Using the three configurations supplied the operation of rotational and sliding joints may be examined and explored. Care should be taken in the execution of the experiment as the equipment is designed to disassemble easily, allowing for insertion of the drawing paper. The information recorded during the experiment will be used as the basis for for calculations later on, so errors in the 'operational' element of this laboratory will affect the end results and possibly marks. The design and use of machines demands a knowledge of the overall motion and the relative motion of all parts of the machine. Another issue is the concept of how specific motions can be generated. In some cases it is self evident, but in others some ingenious mechanisms only careful analysis will show the precise nature of movements. Machines are made up from a number of parts connected together in various ways to produce the required movements. Two parts of a machine with are in contact, and which undergo relative motion are called a pair. The types of relative motion commonly required are sliding, turning (rotation) or screwing. The pairs which permit these motions are known as lower pairs. All other motions (partial turning, partial sliding) are called higher pairs.

Goals of the Experiment • Understand the movement of interconnected rotating links

• Understand the behaviour of pinned and sliding joints

• Practice velocity and acceleration diagrams

• To investigate the loci of pairs on a selection of mechanisms. Two mechanisms are available on the drawing boards -

• Slotted Link (Scotch Yoke)

• Single & Double Action Toggle


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Slotted Link (Scotch Yoke)

Introduction The slotted frame A is fixed and also carries the fixed axis 0 of the revolving link OP. The slotted link BC contains the slider pinned to the revolving link at P. As OP revolves the element BC reciprocates in the slotted frame with a pure simple harmonic motion whose amplitude is governed by the radius OP. Two positions for P at 50 and 25 mm radii are provided. This mechanism is always used where it is required to produce perfect simple harmonic motion in a line. One application is for driving the paddle of a wave making machine in hydrological models. The link OP is commonly part of a complete disc which then acts as a flywheel to even out variations in the forces and driving torque.

Figure PM1 - Slotted Link Apparatus

Procedure To plot the locus of Q it will be necessary to use a base line related to the angular position of OP. To assist in this P is on a circular protractor with a centre 0. Place a sheet of graph paper under the link ensuring, by eye, that the graoh paper is aligned with the movement of the end of the link. The vertical lines are now perpendicular to OQ. Choose a line outside the range of travel of the point Q and set out points along this line, at a scale of 1 cm = 20°, providing points at 20° intervals from 0 to 80° and then from 90° to 350° with a final point at 360°.


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With the protractor 0° on the line OQ (Q at its leftmost position) and P pinned at 50 mm radius mark the position of the point Q at the intersections of the line visible through Q and the lines representing 0° and 360°. Advance the protractor by 20° and mark the intersection of the position of Q and the 20° line. Repeat this procedure every 20° up to the 80° point and then advance 10° to 90°. Carry on as before plotting the locus or Q as the link OP revolves. Repeat the whole procedure with P at 25 mm radius. Using a French curve, or by eye, draw in the two loci of P through the plotted points. The loci should be cosine curves. Add a base line through the 90° and 270° points and check the symmetry of the loci and the amplitudes. If we assume that the disk is rotating at a constant speed of 1 rad/s mark this base line in seconds.

Reporting Include the graph, or a copy, in your log book. Comment on your observations and indicate how you would generate a sine wave from the equipment. Find the slope of the line at 90° and comment on your finding. If the general equation for a cosine function is y = x Cos(θ - φ), where φ is a constant and θ the angle of rotation, describe how the equipment could be used to generate this graph.


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Single & Double Action Toggle

Introduction In the single toggle mechanism Figure PM3 the links AB and BC are pivoted at B to the connecting rod BD which is operated by a crank OD. The link CB pivots about the fixed point C, and A is attached to a sliding member constrained to move in a straight line. Any force applied to BD by the crank OD is greatly magnified at A due to the relatively slight movement produced at A compared with that of D. The force at A attains maximum value when the links AB and BC are in line. The pivot B does not travel beyond the line of action drawn between A and C. Consequently its use is normally restricted to machines where only a single stroke is required to complete the operation.

Figure PM3 - Single Action Toggle

Where two strokes are required it is usual to employ the double action mechanism shown in Figure PM4. The location of the crank is such that the links AB and BC are straightened before the crank has completed one half of its revolution. Thus, after the first stroke the links are carried beyond the centre line and, as the crank continues to revolve, a second working stroke is produced. The first and second strokes may be of equal or unequal lengths dependent upon the position of the crank relative to the line of the straightened toggle. Nevertheless the rightmost travel of A will be the same for the two strokes.


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Figure PM4 - Double Action Toggle

As the maximum force coincides with the limit of travel, this kind of mechanism could be used for upsetting the head of a nail on round wire rod, or in stone crushing or embossing machines.

Objectives • To determine the motion of point A on the mechanism as link OD rotates at

constant angular velocity.

• To understand space and velocity diagrams

• To compare mathematical, graphical and experimental answers

Procedure The crank OD revolves on a removable peg which can be located in one of two holes in the drawing board. For accuracy a 360o disk is provided to determine the orientation of the crank, the 0o position is in line with the crank when the point A is at the left end of its travel. Place the peg in the lower hole insert a sheet of A3 paper and position the crank on this centre using the shorter radius OD1 = 40 mm. Mark the position of Point C and draw the loci of A, B and D1 for one revolution of the crank. Divide the right hand locus of D1 into approximately 30° intervals and for each position of D1 mark and number the corresponding positions of B and A. Then find the position of the mechanism at the mid-travel of A. Remove the A3 paper carefully and insert and orient a piece of graph paper such that point A tracks along one of the short lines. Clamp the paper carefully in place. Insert the disk marked in 10o angles under the crank and orient it carefully - the zero degree line should be visible through the hole at point D. Mark the graph


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paper along the long axis (outside the travel of point A) at 20°/10mm. With the crank at 0° mark the position of A on the graph paper at the 0° and 360° lines. Repeat this at 20° intervals (if required use 10° or 30° increments) to mark the intermediate positions of point A on the graph paper. Draw the curve for the locus of A. Move the crank centre peg to the upper hole and change the crank radius to OD2 = 65 mm. Insert a new A3 sheet and draw the loci of A, B and D2 for one revolution of the crank. Mark and number sufficient corresponding positions of D2, B and A to show how the mechanism performs. Replace the A3 sheet, mark and draw the position of the mechanism with the crank in the 50° position. Remove the A3 paper and insert a new sheet of graph paper, aligning it as before and marking the long axis in degrees (10mm = 20°). As before use your judgement to mark enough points to allow you to draw the graph showing the position of A with respect to the crank angle.

Analysis • If the crank is rotating anti-clockwise at 30 rpm.

• Mark the long axis of the graph of A vs θ with the appropriate time in seconds.

• Determine from the graph the velocity of the point A at the 50o position.

• From the space diagram, using graph paper draw the velocity diagram for the mechanism, with the crank at the 50o position

Reporting Discuss the differences in the two mechanisms and the reasons for the difference. Explain how the system magnifies the force available at point A. The two strokes of the double toggle have different amplitudes. Determine those amplitudes and suggest where the centre O should be positioned for equal strokes of A? What further modification is needed to produce two equal strokes of A of the same length as the single toggle did? Include all graphs and diagrams in your report.


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Gyroscope

Introduction A gyroscope is made up of a rotor spinning about an axis. Any attempt to change the axis of rotation requires work as it is an attempt to change the angular momentum. As long as the system keeps spinning the orientation of the axis of rotation will remain constant unless an outside force or couple is applied. The rotor is often kept spinning by driving it with an electric motor. Mounting the spinning rotor in gimbals which isolate it from any external couple results in a system capable of providing a constant reference direction which is used for example as a gyro-compass in moving vehicles. Alternatively, fixing the axis of rotation with respect to the vehicle generates forces and couples at the mounting points when the vehicle accelerates. These can form the inputs of an electronic system which controls actuators on the vehicle to provide gyro-stabilizers

Goals • Investigate the direction of the gyroscopic couple

• Determination of the Moment of Inertia of the Rotor

• Investigate the relationship between the gyroscopic couple, angular velocity of the rotor and the precession of the axis of rotation

Background Whenever the direction of the axis of rotation of a body is changed gyroscopic action occurs. The angular momentum of the rotating body causes the axis of rotation to remain in the same direction so long as no external couple acts on the body. If a couple is applied about an axis normal to the axis of rotation, a torque reaction is produced which results in the axis of rotation turning in the plane in which lie the axis of rotation and the axis of the applied couple. This movement is termed Precession. Conversely, if the axis is precessed a couple will result about the orthogonal axis. This is the Gyroscopic Couple. It should be noted that, for the same rate of precession, the Applied Couple (CA) and the Gyroscopic Couple (CR) will have the same magnitude but the opposite sense. The diagram below shows the directions involved. In this case the axis of rotation is the Z axis, and the couple is applied about the X axis. The resultant motion of the axis of rotation therefore lies in the X-Z plane, indicated on the diagram as a rotation about the Y axis.


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Important Comments:

CA : Applied Couple

ω : Angular Velocity

Ω : Rate of Precession

ω >> Ω

Figure GE1

When the axes are orthogonal as shown above, it should be noted that the direction of the precession does not change over time. If this was not the case then the movement of the axis of precession would also generate a gyroscopic couple. The equipment used in this experiment has been designed to ensure that there is no change in the axis of precession.

Procedure

The equipment is covered by a plexiglass dome for your safety. Before removing the dome ensure that all controls are switched off and that the system is at rest. After any adjustments, the cover must be replaced and

securely fastened before the controls are switched on.

Failure to observe this rule will result in zero marks being awarded to the group for this laboratory, a penalty which will affect the final mark for this

module.

1. Determination of the Direction of the Gyroscopic Couple 1. Ensure that the rotor assembly is balanced about the pivot point on the

bracket. If necessary move the balance weight on the torque arm by slackening the retaining screw. Take care to re-tighten the screw in the new position. Replace the safety cover.


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2. Set the motor driving the rotor running and check to ensure that the system is still balanced and that no precession of the rotor occurs. Note the direction of rotation of the rotor.

3. Set the precession motor running. The balance of the rotor should be affected by this action. Note the direction of the precession and whether the torque arm rises or fall. The arm will move in the same direction as the gyroscopic couple.

4. Using the plugs on the front of the equipment reverse the polarity of the rotor motor and repeat steps 2 & 3. Then reverse the polarity of the precession motor and repeat. Finally change the polarity of the rotor motor back to its original direction and repeat.

5. Draw schematics of the gyroscope similar to that shown below and indicate the relationships derived from these observations.

Figure GE2

2. Determination of the Moment of Inertia of the Armature 1. Fold out the support arm attached to the casing.

2. Hang the spare armature from the arm using the wires.

3. Carefully twist the assembly about its vertical axis, taking care not to move the axis too much and ensuring that the wires are kept taut, by about 10o and then release. The armature will start to twist about the vertical axis passing the equilibrium position and then oscillating about that position.

4. Use the stopwatch to determine the time for (say) 50 oscillations and calculate the periodic time, T. Repeat until you are satisfied that you have a good result.

5. Measure the length of the wires, l, and the distance between them, d.

6. Given that the mass of the armature is 1.09kg, calculate the moment of inertia of the rotor using the equation below:


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ldmgTI44

2

2

2

π=

where m is the mass of the armature and g is the acceleration due to gravity.

3. Determination of the relationship between the angular velocity of the rotor, the Gyroscopic Couple and the Rate of Precession In this part you will vary two control parameters, Gyroscopic Torque (CR) and Rotor Speed (ω), and measure the resultant rate of Precession (Ω). The values for the control parameters are given in the table below: For each combination of the input parameters:

1. Add the mass onto the end of the torque arm. Calculate the torque on the system and the direction. This is the Applied Couple, CA.

2. Connect the rotor and precession motor controllers so that the gyroscope will tend to raise the torque arm.

3. Set the rotor speed using the indicator. 4. Vary the rate of precession until the torque arm is level (the scribed line on

the arm should line up with the stripes on the bracket). In this state the Gyroscopic Couple is equal to the Applied Couple.

5. Measure the rate of precession by timing a suitable number of revolutions of the assembly (total measuring time should be at least 30 seconds). Record the rate of precession.


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Table GE1 B

alan

ce

Mas

s

Cou

ple

CR

Nominal Rotor Speed

ω

Actual Rotor Speed ω

Rate of Precession Ω 1/ Ω

g Nm RPM RPM rad/s rev/s Rad/s s

3,000

2,500

2,000

1,500

1,000

100

250

3,000

2,500

2,000

1,500

1,000

150

700

3,000

2,500

2,000

1,500

250

1,000

3,000

2,500

2,000 350

1,500


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Analysis of the Results The results obtained from these three experiments can now be used to verify the theoretical relationship between the gyroscopic couple, the angular momentum and the rate of precession.

CR = Iω X Ω

For each of the masses applied to the torque bar, the balance position represents a constant gyroscopic couple, CR. Since the moment of inertia, I, is constant then for a particular mass the equation can be rewritten as

CR/I = K = ω X Ω

where K is a constant.

Since ω and Ω are perpendicular then

ω = K/Ω or ω = Kx(1/Ω)

This means that for a given mass the graph of the reciprocal of the rate of precession (1/Ω) against the rotor velocity (ω) should be a straight line.

1. Using a single graph plot 1/Ω vs ω for each mass.

2. Using the information from this graph, plot CR vs Ω for ω = 300 rad/s.

3. Again from the first graph plot CR vs ω for Ω = 2.5 rad/s.

Report In your report compare the experimental findings with the theory. From graph 1 estimate a value for I and compare this with the results from part 2. Draw a Free Body Diagram of the armature in part 2. If possible show how the restoring torque returning the armature to the equilibrium is a function of the angle of rotation. Explain any sources of errors and estimate their magnitude.


Practical Manual


Static & Dynamic Balance Experiment

Introduction Shafts which revolve at high speeds must be carefully balanced if they are not to be a source of vibration. If the shaft is only just out of balance and revolves slowly the vibration may merely be a nuisance but catastrophic failure can occur at high speeds even if the imbalance is small.

For example if the front wheel of a car is slightly out of balance this may be felt as a vibration of the steering wheel. However if the wheel is seriously out of balance, control of the car may be difficult and the wheel bearings and suspension will wear rapidly, especially if the frequency of vibration coincides with any of the natural frequencies of the system. These problems can be avoided if a small mass is placed at a carefully determined point on the wheel rim.

It is even more important to ensure that the shaft and rotors of gas turbine engines are very accurately balanced, since they may rotate at speeds between 15,000 and 50,000 rev/min. At such speeds even slight imbalance can cause vibration and rapid deterioration of the bearings leading to catastrophic failure of the engine.

It is not enough to place the balancing mass such that the shaft will remain in any stationary position, i.e. static balance. When the shaft rotates, periodic centrifugal forces may be developed which give rise to vibration. The shaft has to be balanced both statically and dynamically.

Usually, shafts are balanced on a machine which tells the operator exactly where he should either place a balancing mass or remove material. The apparatus requires the student to balance a shaft by calculation or by using a graphical technique, and then to assess the accuracy of his results by setting up and running a motor driven shaft. The shaft is deliberately made out of balance by clamping four blocks to it, the student being required to find the positions of the third and/or fourth blocks necessary to statically and dynamically balance the shaft.

Aims The goals of this experiment are:

• To gain an understanding of the meaning of the terms static and dynamic balance.

• To demonstrate the dynamic balancing of an unbalanced shaft using two added eccentric masses.

Theory A shaft with masses mounted on it can be both statically and dynamically balanced. If it is statically balanced, it will stay in any angular position without rotating. If it is dynamically balanced, it can be rotated at any speed without


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vibration. It will be shown that if a shaft is dynamically balanced it is automatically in static balance, but the reverse is not necessarily true.

Static Balance Figure SDB1 shows a simple situation where two masses are mounted on a shaft. If the shaft is to be statically balanced, the moment due to weight of mass (1) tending to rotate the shaft anti-clockwise must equal that of mass (2) trying to turn the shaft in the opposite direction.

Figure SDB 1 - Static Balance

Hence for static balance,

m1r1cosα1 = m2r2cosα2

The same principle holds if there are more than two masses mounted on the shaft, as shown in figure SDB2.

Figure SDB 2 - Static Balance of Three masses

The moments tending to turn the shaft due to the out of balance masses are:-


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Mass Moment Direction

1 m1.g.r1cosα1 Anticlockwise

2 m2.g.r2cosα2 Clockwise

3 m3.g.r3cosα3 Clockwise

For static balance,

m1r1cosα1 = m2r2cosα2 + m3r3cosα3

In general the values of m, r and α have to be chosen such that the shaft is in balance. However, for this experiment the product W.r can be measured directly for each mass and only the angular positions have to be determined for static balance.

If the angular positions of two of the masses are fixed, the position of the third can be found either by trigonometry or by drawing. The latter technique uses the idea that moments can be represented by vectors as shown in figure SDB3(a). The moment vector has a length proportional to the product mr and is drawn parallel to the direction of the mass from the centre of rotation.

Figure SDB 3 - (a) MR Vectors from Centre to Mass (b) Rearrangement to

show Closed Polygon For static balance the triangle of moments must close and the direction of the unknown moment is chosen accordingly. If there are more than three masses, the moment figure is a closed polygon as shown in Figure SDB3(b). The order in which the vectors are drawn does not matter, as indicated by the two examples on the figure.

If on drawing the closing vector, its direction is opposite to the assumed position of that mass, the position of the mass must be reversed for balance.

Dynamic Balance The masses are subjected to centrifugal forces when the shaft is rotating. Two conditions must be satisfied if the shaft is not to vibrate as it rotates:

a) There must be no out of balance centrifugal force trying to deflect the shaft.

b) There must be no out of balance moment or couple trying to twist the axis of the shaft.


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If these conditions are not fulfilled, the shaft is not dynamically balanced.

Figure SDB 4 - Dynamic Out-of-Balance for a Two Mass System

Applying condition a) to the shaft shown in Figure SDB.4 gives:

F1 = F2

The centrifugal force is mrω2

Therefore:

m1r1ω2 = m2r2ω2

Since the angular velocity, ω, is common to both sides then for dynamic balance

m1r1 = m2r2

This is the same result for the static balance of the shaft. Therefore if a shaft is dynamically balanced it will also be statically balanced.

The second condition is satisfied by taking moments about some convenient datum such as one of the bearings.

Thus,

a1F1 = a2F2

For this simple case where m1 and m2 are diametrically opposite and F1 = F2 (condition a) then dynamic balance can only be achieved by having a1 = a2 which means that the two masses must be mounted at the same point on the shaft.

Unlike static balancing where the position of the masses along the shaft is not important, the dynamic twisting moments on the shaft have to be eliminated by placing the masses in carefully calculated positions. If the shaft is statically balanced it does not follow that it is also dynamically balanced.


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In order for static balance to be achieved the sum of the vectors representing the couple due to each rotor must form a closed polygon. In the case where there are three rotors, the simplest arrangement to give balance is shown in

Figure SDB 5 - Dynamically Balanced Shaft with 3 Eccentric Masses

In this case, it is clear from the first requirement that:

F2 = 2 x F1

The second criterion then says that:

a2F2 = a1F1 + a3F1

2a2F1 = a1F1 + a3F1

a2 = (a1 + a3)/2

or that the eccentric mass in the middle has twice the m.r value of the two masses on either side and is equidistant from both masses.

The general case, where the eccentric masses differ on each rotor and the directions are not exactly opposite is shown in Figure SDB6.


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Figure SDB 6 - General Case for Three Out-of-Balance Masses

The method for balancing such shafts requires the addition of two extra eccentric masses to the system at locations chosen by the engineer. These masses are determined in many ways. One method will be outlined in the lectures, while this experiment shows a method that uses preset out of balance forces and determines the orientation and position of the masses relative to the eccentric masses already on the shaft.

Equipment: The Static and Dynamic Balance Equipment consists of a shaft mounted on a plate isolated from the base by rubber bushes. A motor, also attached to the plate, may be used to drive the shaft using a belt. Four identical eccentric mass blocks are provided, together with four individual inserts which may be used to alter the imbalance on each block. An extension shaft and pulley are stored on the base and used in conjunction with the string/buckets and ball bearings to determine the imbalance associated with each block. Two hexagonal keys (Allen keys) are provided to clamp the blocks on the shaft and the inserts into the blocks. Two guides, one on the pulley at the end of the shaft and one on the plate, are used to measure the relative angle and position along the shaft of each block. At either side of the plate two clamps allow the plate to be locked to the base. The plate should be clamped for the static parts of the experiment. If the motor is in use then the clamps should be released and secured away from the plate by tightening the screws.


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Figure SDB 7 - Eccentric Masses and Axial Positioning

Experimental Procedure: The experiment is divided into three parts:

1. Observation of the phenomenon

2. Calculation of a solution to a posed problem

3. Implementation of the solution

Demonstration of Static and Dynamic Balance For this part of the experiment the four blocks will be used without the inserts. This provides four identical eccentric masses for use in demonstration of static and dynamic balance.

Step 1 - Static Imbalance

Lock the plate to the base. Attach one of the blocks securely to the shaft near the pulley. The mark on the protractor should align with zero when the block is positioned against the guide. Slide the guide clear of the block. Rotate the shaft and observe the behaviour. Record your observations.

Step 2 - Static Balance

Attach a second block near the opposite end of the shaft. Align this block such that the protractor reads 180o. Rotate the shaft and observe the behaviour. Record this and compare with step 1.

Step 3 - Dynamic Imbalance

Release the plate and secure the clamps clear of the plate. Attach the belt between the motor and the pulley on the end of the shaft. Ensure that all loose components are removed from the equipment and then place the safety cover over the motor and shaft. Switch on the motor controller and the motor. Slowly increase the speed of the motor and observe the behaviour of the shaft and plate.


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Record your observations. Switch off both the motor & controller and allow the shaft to come to rest before removing cover.

Step 3 - Dynamic Balance

There are three steps to this part of the experiment

1. Preparation of the imbalances Clamp the plate and remove the blocks from the shaft. Insert the four circular imbalances into the blocks and clamp them securely. Attach one of the blocks to the shaft with the protractor reading 0o. Remove the drive belt from the motor and attach the pulley extension to the shaft so that the pulley overhangs the end of the bench. Loop the string and buckets around the pulley so that there is no slip. Add ball bearings to one bucket until the protractor reads 90o. At this point the moment due to the bearings is equal to the eccentric moment of the block. Record the number of bearings and repeat for the other three blocks.

2. Calculation of the solution The problem for solution must be posed carefully if a satisfactory solution is to be found in the time available. The relative axial position and angular orientation of the two largest eccentricities should be selected first and will represent the dynamic imbalance to be corrected. An axial spacing of 100 mm and relative angle of 150o provide a reasonable starting point. The main limitations are the total length of the shaft and the thickness of each block. The solution is calculated graphically in two parts. Initially, the static balance of the system is obtained by

a. Draw vectors representing the two imbalances set above. These have a length proportional to the number of ball bearings and a direction relative to the angular orientation. The vector should go from the centre of the shaft along the block. The drawing should like Figure SDB 3b with the m1r1 and m2r2 vectors only.

b. On the graph, knowing the lengths of the other two imbalances, complete the four sided closed polygon. Record the angular orientation of the two balancing vectors.

The axial position of the two balancing masses needs to be calculated next. In this case, it is simplest to take the largest eccentricity as the reference axial location, eliminating it from this part of the calculation.

c. On a new diagram draw a vector representing the axial turning moment of the second eccentricity. The length is proportional to the number of ball bearings and the axial distance from the reference.

d. Complete the closed triangle using the directions for the balancing eccentricities found in b. From the scale calculate the axial distance associated with each eccentricity.

Assemble all the information into a table indicating the eccentricity (mr) of each block, its axial location (l) and its angular orientation (θ).


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Figure SDB 8 - Graphical Solution to Dynamic Balance

3. Test the result Carefully attach the blocks to the shaft at the locations and orientations in the table. Remove the pulley from the system and reattach the motor drive belt. Release the platform clamps and secure them to the base. Put safety cover in place and run motor. Record whether dynamic balance has been achieved and if necessary revise calculations.

Reporting Your report should give a detailed description of your observations and include all rough work and calculations. The graphs should be clearly labelled and neat.

Documents

Mm 203 Lab Manual