emm3504 - lab manual 1st half (sep 2014)

January 2013

EMM 3504 MECHANICS OF MACHINES

LAB MANUAL

Department of Mechanical & Manufacturing Engineering Faculty of Engineering, Universiti Putra Malaysia

EMM 3504 – MECHANICS OF MACHINES

TABLE OF CONTENTS

No. Title Page

No.

1. Free Vibration of an Undamped Simply Supported Beam 1

2. Determination of Momentum of Inertia of a Gyroscope

Disk through Measurement of Angular Acceleration 3

3. Undamped Spring-Mass Systems 5

4. Damped Spring-Mass Systems 7

5. Spur Gear Lifting Machine 9

6. Static Balancing of a Shaft with Two/Four Masses 13

7. Universal Mechanisms 17

8. Forced, Damped and Undamped Vibration of a Simply

Supported Beam 20

9. Determination of Gyroscope Momentum of Inertia by

Measurement of Gyro-Frequency and Precession

Frequency

24

10. Dynamic Balancing of a Shaft with Two/Four Masses 26

11. Free Torsional Vibration 32

12. Whirling of Shafts 35

Appendix A LVDT Output Conversion 38



1. FREE VIBRATION OF AN UNDAMPED SIMPLY SUPPORTED BEAM

Objectives:

1. To determine the frequency and period of vibration of an undamped simply-supported beam subjected to free vibration.

2. To determine the stiffness of an undamped simply-supported beam subjected to free vibration. Introduction: Determining the natural frequency of any system helps to find how the system will behave when just disturbed and left (free vibration), and to find what kind of excitation frequency to be avoided in the system. Ignoring the effect of self weight, for a simply supported beam loaded at the centre, the beam stiffness, k, is given by,

k = 48 EI / L3 (1)

where; E is the modulus of elasticity of the beam material I is the second moment of area of the beam Theoretically, the fundamental natural frequency of oscillation, ω (rad/s), of a simply-supported beam is given by; ω = √ (k / m) (2)

where;

k is the beam stiffness m is the loaded mass Procedure:

1. Set up the apparatus with consideration of the following: a. Length of beam b. Masses to be fixed to beam

2. Affix the low-voltage displacement transducer (LVDT) appropriately for measurements to be taken.

3. Run the data recorder (quickDAQ software) and follow instructions as given by the demonstrator for recording of experimental values.

4. Attach a predetermined mass to the loading rod centered between the ends of the beam. 5. Initiate the data recorder. Displace the beam slightly and release to allow the beam to oscillate

naturally. Stop the recorder once oscillation dies. 6. Save all data into a MS Excel spreadsheet for later reference. 7. Repeat the experiment for different loadings.

Results/Observations:

1. Experimental Frequency and Period of Oscillation for Each Mass

a. Plot graphs of the displacement of the beam vs. time for each experiment. (Refer to Appendix A of this manual to convert LVDT voltage output to displacement).

b. From the graphs, determine the frequency of vibration of the beam under the different loadings.

Repeat the above procedures for all loadings. Obtain the average angular frequencies and average periods of oscillations of the beam for all the loading conditions. Tabulate the results appropriately.

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For all the loadings selected, compare the experimental angular frequencies obtained to those calculated using Eq. (2). Tabulate appropriately.

2. Theoretical and Experimental Stiffness of the Oscillating Beam

Taking the log of Eq. (2) gives,

log (ω) = ½ log ( k ) - ½ log ( m ) (3)

Equation (3) is a linear graph in the form of, y = ax + c

where,

y = log (ω) ax = - ½ log (m) c = ½ log (k) is the intercept on the y-axis.

Using the experimental values obtained, complete the following table: Table 1: Experimental results

Angular frequency, ω (rad/s)

Loaded mass, m (kg) log (ω) log (m)

From the data in Table 1, plot the graph of log (ω) vs. log (I). From the plot, locate the intercept on the y-axis to determine the beam stiffness, k (N/m). Compare the experimental value obtained from the plot with that obtained from Eq. (1).

Figure 1: Graph of log ω vs. log m Discussion

1. Discuss the differences (if any) between the theoretical and experimental results. 2. Describe the shortcomings of the experiment performed. 3. Although there is no dashpot attached to the system to provide damping, why did the system come

to rest after a period of time?

θ

½ log k

log

ω

log m

Graph of log ω vs. log m

• k is the stiffness of the beam • The slope θ represents the

variation of m with respect to ω

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2. DETERMINATION OF MOMENTUM OF INERTIA OF A GYROSCOPE DISK THROUGH MEASUREMENT

OF ANGULAR ACCELERATION Objective: To determine the momentum of inertia of a gyroscope disk by measurement of its angular acceleration Theory: If a gyroscope disk is set to rotate by means of a falling mass m (Fig. 1), the following relation is valid for its angular acceleration:

P

R

IM

dtd

== αω (1)

where; ωR = angular velocity α = angular acceleration IP = polar moment of inertia M = rF • = torque

Figure 1: Schematic representation of the experimental setup to determine the momentum of inertia of

the gyroscope disk

According to the law of action and reaction, the force which causes the torque is given by the following relation: ( )agmF −= (2) where; g = terrestrial gravitational acceleration a = trajectory acceleration The following relations are true for the trajectory acceleration, a, and the angular acceleration, α:

2

2

Ftha = ;

ra

=α (3)

where; h = dropping height of the accelerating mass tF = falling time r = radius of the thread drum Introducing (2) and (3) into (1), one obtains:

3



hmgr

mrIt PF 2

22 22 += (4)

In general, the following is valid for the momentum of inertia of a disk:

ρπ dRMRIP42

221

== (5)

where; R = radius of circular disk d = thickness of circular disk ρ = specific weight of plastic (= 0.9 g/cm3) Procedure:

1. Fix the gyroscope with its axis directed horizontally and positioned on the experimenting table in such a way that the thread drum projects over the edge of the table (Fig. 1).

2. Wind the thread around the drum and the accelerating mass m (m = 60g; plate with 5 slotted weights) fastened to the free end of the thread

3. Commence the dropping of the mass from several equally-spaced intervals of heights, h. Results/Observations: Plot a graph of tF

2 versus h. Determine IP from the slope of the curve. Show all calculations for determination of IP using the analytical method. Discussion: Compare the experimental value of IP to that obtained from (4). Discuss any possible errors in the experimental works.

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3. UNDAMPED SPRING-MASS SYSTEMS Objective: To compare the theoretical natural frequency of an undamped spring-mass system calculated for various masses to the values obtained by measurement. Theory: Refer to Fig. 1:

Figure 1: Free-body diagram of forces acting on the mass

Figure 2: Description of apparatus

Setting-up the equation of motion involves establishing equilibrium of forces at the mass:

∑ +−== mgFxmF c

The spring load Fc is calculated from deflection x and spring constant c:

cxFc =

This results in the following 2nd order differential equation as equation of motion:

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gxmcx =+

Solving the equation gives harmonic oscillations with natural angular frequency ωo or natural frequency f:

( ) ( )txtx oo ωcos=

mc

o =2ω , mcf

π21

=

The period is:

cmT π2=

As can be seen, the period/natural frequency can easily be adjusted by altering the mass. Procedure:

1. Start the recorder. 2. With no mass attached to the system, deflect carriage downwards by hand

and allow it to oscillate freely until it comes to rest. 3. Stop the recorder. 4. Repeat experiment with other additional masses.

Results: Fill in the following table: Experiment

No. Additional

masses (kg)

Total mass (kg)

Experimental natural frequency

(Hz)

Theoretical natural frequency

(Hz) 1 0 1.250 2 2 3 4 4 6 5 8 6 10

*Note: spring constant = 1710 N/m Do not forget to include SAMPLES of your calculations in the report. Refer to Appendix A of this manual to convert LVDT output voltage to displacement readings. Discussion: Compare the calculated and theoretical natural frequencies obtained and discuss the variation between them, if any.

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4. DAMPED SPRING-MASS SYSTEMS Objective: To investigate the influence of damping on free oscillation of a spring-mass system. Theory: Refer to Fig. 1:

Figure 1: Free-body diagram of forces

acting on the mass

Figure 2: Description of apparatus

The equilibrium of forces at the mass is the basis used for setting up the equation of motion. This time a speed-proportional damper force Fd is additionally introduced. As the constant force due to the weight mg has no influence on oscillation behaviour, it is ignored here

∑ −−== dc FFxmF

The damper force Fd results from the velocity x and the damper constant d:

xdFd = This produces the following homogeneous differential equation for the equation of motion:

0=++ xmcx

mdx

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or, with D as degree of damping and ωo as natural angular frequency:

02 2 =++ xxDx oo ωω

omdDω2

= , mc

o =2ω

Solving the equation gives decaying harmonic oscillations with frequency:

21 Dod −= ωω

It can be seen that at 1≥D oscillation is no longer possible. The angular frequency ωd approaches zero or becomes imaginary. Procedure:

1. Fit 5 additional weights (mg = 11.25 kg) and secure with knurled nut. 2. Use adjuster to align carriage with center of plot. 3. Start recorder. 4. Deflect carriage downwards by hand and allow it to oscillate freely. 5. Stop recorder. 6. Repeat experiment with different damper settings.

Results: Provide the oscillation curves for the following experiments:

Experiment No. Needle valve setting 1 Open 8 turns 2 Open 4 turns 3 Open 2 turns 4 Open 1 turns 5 Closed

*Note: spring constant, c = 1710 N/m Determine the damped natural frequency, ωd for each of the experiments. Do not forget to include SAMPLES of your calculations in your report. Refer to Appendix A of this manual to convert LVDT output voltage to displacement readings. Discussion: Discuss each damped natural frequencies obtained with their respective oscillation curves. Show all calculations in your report. Note:

1. Assume a linear relationship between the no. of turns of the damper knob and the range of damping constants (15 – 300 Ns/m), i.e. a fully-closed knob will apply a 300 Ns/m damping, a fully-opened knob will have 15 Ns/m damping.

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5. SPUR GEAR LIFTING MACHINE Objective: To determine the efficiency of a spur gear lifting machine Introduction: Figures 1 and 2 show the spur gear lifting machine. The gears can be arranged in any combination of sizes. When gears are used for lifting, the efficiency of the gear system can be determined by calculating the mechanical advantage and velocity ratio of the gear drive. The mechanical advantage is given by: Mechanical advantage = (Load raised or lowered )

(Effort load ) (1)

The velocity ratio is given by: Velocity ratio = (Distance moved by effort )

(Distance moved by load ) (2)

If the effort pulley moves by one revolution,

Distance moved by the effort load = erπ2

where re is the radius of the effort pulley and the number of revolutions made by the effort gear is one. The number of revolutions made by the load pulley,

L

e

TT

=

where;

Te = number of teeth of the effort gear TL= number of teeth of the load gear

Therefore, the distance moved by the applied load (i.e. the load to be raised or lowered),

( )( )LeL TTrπ2=

Hence, the velocity ratio

( )( )LeL

e

TTrr

ππ

22

=

Since all pulleys are of the same radius, Velocity ratio

e

L

TT

= (3)

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The efficiency of the gear system is a function of the mechanical advantage of the system and the velocity ratio, i.e.,

Efficiency =(Mechanical advantage)

(Velocity ratio)

= (𝑊𝑊𝑎𝑎/𝑊𝑊𝑒𝑒)(𝑇𝑇𝐿𝐿/𝑇𝑇𝑒𝑒)

where; Wa = applied load We = effort load

Figure 1: Spur gear lifting machine

Effort Load

Screw For Fixing The Gear In Position

Load lifted

Three different sizes of gears

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Figure 2: Gear arrangement

Apparatus: 1. A simple arrangement of two spur gears. 2. Set of weights. Procedure: Load raising 1. Choose two spur gears and anchor them to the frame such that the teeth of the gears interlock each

other. 2. Count and note the number of teeth for each gear. 3. Wind a cord around the load pulley of each gear and hang a hanger at the free end of each cord. The

cords should be wound in the same manner for both pulleys, i.e., clockwise or anti-clockwise. 4. Place a load to be lifted on the hanger of the larger gear (applied load). 5. Let the hanger rest on the table but keep the cord taut. 6. Place a load on the hanger of the smaller gear (effort load) and watch the applied load. If the load is not

lifted, remove the effort load from the hanger. Increase the effort load and place it again on the effort hanger. Repeat until the effort load is able to raise the lifting load. Record the smallest load that causes the lifting load to rise. This is the effort load.

7. Increase the applied load on the lifting pulley and repeat step 6. Repeat for at least 5 more load increments.

8. Remove all loads on both pulleys.

Load lowering 9. Increase the winding on the load pulley and reduce the winding on the effort pulley such the load

hanger of the load pulley hangs higher than the hanger of the effort pulley. 10. Commence the loading on the hangers according to the final data set in the previous experiment. 11. Remove weights on the effort pulley until the load pulley hanger starts to drop. Record the effort load. 12. Repeat steps 10 and 11 for incremental reductions of the load on the load pulley. Repeat the experiment with another set of gears.

Anchoring screw. The gear assembly moves along the slot and is fixed in position by this screw

Shaft

Pulley

Gear

Pointer used in counting the number of revolutions

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Results/Observations: Number of teeth (larger gear) = Number of teeth (smaller gear) = Record your observations in Tables 1 and 2.

Table 1: Load raised by the effort Applied load (N) Effort load (N)

Table 2: Load lowered by the effort

Applied load (N) Effort load (N)

Using the data in Tables 1 and 2, plot a curve of the applied load vs. effort load for each set of gears experimented. Obtain the slope of each plot. This represents the average mechanical advantage of the system. Calculate the efficiency of the system. Discussion:

1. Compare the efficiencies between raising and lowering of the load for each set of gears. 2. Compare the efficiencies between different types of gear ratios. 3. Give your comments based on the results obtained.

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6. STATIC BALANCING OF A SHAFT WITH TWO/FOUR MASSES

Objective: To analytically and experimentally determine the static balancing conditions of two and four-mass systems. 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 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.

Figure 1: Simple two-mass system

Fig. 1 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 clockwise, must equal that of mass (2), trying to turn the shaft in the opposite direction. Hence for static balance:

2211 rWrW = (1) The same principle holds if there are more than two masses mounted on the shaft, as shown in Fig. 3. The moments tending to turn the shaft due to the out-of-balance masses are:

Mass Moment Direction (1) 111 cosαrW Anti-clockwise (2) 222 cosαrW Clockwise (3) 333 cosαrW Clockwise

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Figure 2: Three-mass system

For static balance: 333222111 coscoscos ααα rWrWrW += (2) In general, values of W, r, and α have to be chosen such that the shaft is in balance. However, for experiments using the TM102 apparatus, the product Wr 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 Fig. 3(a). The moment vector has a length proportional to the product Wr and is drawn parallel to the direction of the weight from the centre of rotation. 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 Fig. 3(b). The order in which the vectors are drawn does not matter, as indicated by the two examples in 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. For example, mass (4) shown in Fig. 3(b) must be placed in the dotted position shown to agree with the direction of vector W4r4.

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a) three-mass system

b) four-mass system

Figure 3: Moment polygons for static balance of three and four-mass systems

(Note: The symbols I, II, III, IIII denote pairs of parallel lines) Procedure:

1. Remove the Perspex dome and shaft drive belt. 2. Remove the discs from the four rectangular blocks using the smaller

hexagon key*. 3. Unclip the extension pulley and insert it in the pulley end of the motor

driven shaft. 4. Measure the Wr of the blocks (one block at a time) using the steel balls

and weight bucket by recording the number of steel balls it takes to rotate a block 90°.

* ALWAYS TAKE THE BLOCKS OFF OF THE SHAFT BEFORE REMOVING OR REPLACING THE DISCS!!

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5. The demonstrator will give you the positions of three of the blocks to be mounted on the shaft.

6. Using the calculation or drawing method, determine the position of the fourth block for the shaft to be in static balance.

7. Verify that the shaft has achieved static balance by mounting the fourth block on the shaft according to your calculations/drawing.

Results/Observations: Upon mounting the fourth block on the shaft, observe whether the shaft achieves static balance or otherwise. Draw the moment polygon for the four-mass system experimented with. Show all your calculation work. Discussion:

• Was the four-mass system indeed in balance? Discuss any shortcomings (if any)?

• Is a statically balanced four-mass system also dynamically balanced? Explain.

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7. UNIVERSAL MECHANISMS Experiment 1 - Whitworth Quick Return Mechanism The Whitworth quick return mechanism (Fig. 1) converts rotary motion into reciprocating motion. Unlike the crank and slider mechanism however, the forward reciprocating motion rate is different with the backward stroke rate.

Figure 1: The Whitworth quick return mechanism

The implementation of a Whitworth quick return mechanism can be useful for applications requiring slow initial motion and a quick reset operation. The Whitworth quick return mechanism features different input duration for the forward motion and return strokes. The time ratio (TR) of a Whitworth quick return mechanism can be calculated by taking the time spent on the backward stroke divided by the time spent on the forwarding motion, i.e.,

Time ratio (TR) = RT/FT

where; RT = time spent on backward stroke FT = time spent on forwarding motion Objectives:

• To understand the Whitworth quick return mechanism • To observe the transformation of rotary to reciprocating motion • To determine the Whitworth quick return time ratio (TR)

Procedure: 1. Calculate the mobility of the apparatus using the appropriate equation. 2. Slowly rotate the rotating disc plate by turning the rotating knob, observe the movement of the link

at the slider. 3. State down the observation from procedure 2. 4. Rotate the rotating disc plate to zero degrees. Note the values obtained from the protractor scale

and the sliding scale. 5. Plot graphs of displacement against rotation angle, θ. 6. From the graph obtained, discuss the movement pattern of this Whitworth quick return. 7. Prepare a stopwatch. Rotate the rotating disc plate (A) by turning the rotating know (B). as the disc

plate rotates (must be a constant speed), use the stopwatch to capture the time taken to complete forward and backward strokes. State down the time taken. (Note: the rotating speed of the disc plate must be constant).

8. Compute the time ratio (TR) for this Whitworth quick return mechanism using the appropriate equation(s).

9. Discuss the working principle of the Whitworth quick return mechanism and state its applications with examples.

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Experiment 2 - Crank and Connecting Rod A crank and connecting rod (slider) mechanism (Fig. 2) is actually a type of four bar linkage with output crank and ground member of infinite length. The slider crank mechanism is used in a variety of machines when there is a need to convert rotary motion into reciprocating motion or to convert reciprocating motion into rotary motion.

Figure 2: Crank and connecting rod mechanism

In the crank and connecting rod mechanism, the input motion to the slider is typically from a motor or other rotating device running at a constant speed. The resulting motion is therefore a reciprocating motion of the slider with every cycle of the crank. Unlike the Whitworth quick return mechanism, the forward and return motion of the slider has the same duration. When the slider is slid along a guide, there is a position where the slider would reverse the motion; this point at which the slider motion reverses is called the dead center. Positions at which slider motions reverse are called dead centers. When the crank and connecting rod are extended in a straight line and the slider is at its maximum distance from the axis of the crankshaft, the position is called the top dead center (TDC); when the slider is at its minimum distance from the axis of the crankshaft, the position is called the bottom dead center (BDC). Objectives: • Understanding of crank and connecting rod mechanism • To determine the relationship between crank angle and stroke • To study the effects of changing the

o Crank radius o Connecting rod length

• To investigate the graphical relationship between different crank and connecting rod combinations Procedure:

1. Calculate the mobility of the apparatus using the appropriate equation. 2. Select the desired crank (10, 20, 30 mm) and connecting rod (100, 110, 120 mm)

distances 3. Place the selected crank to the rotating disc and connecting rod to the crank holder and

the slide holder. Record the length of the crank and connecting rod in an appropriate table.

4. Rotate the crank to 0° by rotating the turning knob. The angle measurement can be read directly from the protractor provided.

5. Once the crank is set at 0°, record the initial position of the connecting rod using the linear scale provided.

6. Slowly rotate the crank of the apparatus. For every 30° increment, record the displacement value for the connecting rod. Take the reading up to 360°. (Note: the displacement value is obtained by using the final value minus the initial value).

7. Repeat the experiment using different values of rotation radii and connecting rod lengths. 8. Plot graphs of displacement against crankshaft rotation angle (θ) for all combinations. 9. State and discuss the findings show in the graph. 10. Discuss the working principles of the crank and connecting rod and state its applications

with examples.

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Experiment 3 - Cam and Follower The cam and follower mechanism is normally used to transform one form of motion such as rotation into another form of motion, linear motion. In general, a cam and follower mechanism consists of a cam and a follower which is mounted on a fixed frame. A typical cam follower system is shown in Fig. 3. This system is a spring closed system containing a cam and a roller follower.

Figure 3: Typical cam follower system

A cam is a mechanical element having a curved groove or outline that is used to drive another element, the follower, through a specified motion by direct contact. The cam plays a very important role in many industries, especially those involving automation. The reason why cams are popular is because they are versatile and almost any arbitrarily specified motion can be obtained. In any of the cam follower system it is important to ensure that the follower is always in contact and following the motion of the cam. This can be achieved through a number of ways including the following:

• Gravity • Spring force • Weight force • Pneumatic or hydraulic force • Mechanical constraint (e.g. groove)

Objectives: • Understanding of cam and follower mechanism • To plot the cam life curve Procedure:

1. Adjust the dial gauge to ensure the pointer of the gauge is just touching the cam. 2. Rotate the cam to 0°. The angle measurement can be read directly from the protractor

provided. 3. Slowly rotate the cam of the apparatus. For every 30° increment (for 180°, use 170°

instead), record the displacement value. Take readings up to 360°. 4. Plot graphs of displacement against cam rotation angle (θ) for all the combinations. 5. State and discuss the findings shown the graph. 6. Discuss the working principles of the cam and follower mechanism and state its

applications with examples.

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Experiment 4 - Scotch Yoke The Scotch-yoke mechanism (Fig. 4) is a four-bar mechanism in which a crank is connected by a slider with another link which, in turn, forms a prismatic pair with the frame. This mechanism is equivalent to a slider-crank mechanism in which the length of the connecting rod is infinite.

Figure 4: Scotch-yoke mechanism

When the crank is driven with a constant speed, the slider moving on the frame can output a simple harmonic motion. For various applications this mechanism can be used in the testing machines for simulating the simple harmonic vibration. It can also be adopted for driving pumps and compressors. In comparison to the slider-crank mechanism, the scotch-yoke mechanism possesses lower mechanical efficiency due to the friction of the sliding motions. Hence, it is only adoptable for small machines operated under light loading. Objectives: • Understanding of scotch yoke mechanism • To obtain the relation curve between angle and displacement for a scotch yoke Procedure:

1. Slowly rotate the rotating disc plate by turning the rotating knob, observe the movement of the link at the slider.

2. State the observation from procedure 1. 3. Rotate the cam to 0°. The angle measurement can be read directly from the protractor

provided. 4. Slowly rotate the rotating disc of the apparatus. For every 30° increment (for 180°, use

170° instead), record the displacement value. Take readings up to 360°. 5. Plot graphs of displacement against rotation angle (θ) for all the combinations. 6. State and discuss the findings shown in the graph. 7. Discuss the working principle of the scotch yoke and state its applications with examples.

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emm3504 - lab manual 1st half (sep 2014)

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