Inertia Bending of a Connecting
Rod Experiment

15th November 2015

Contents

Abstract----------------------------------------------------------------------------------------------------2 * 1. Introduction-------------------------------------------------------------------------------------------2 1.1 Objective----------------------------------------------------------------------------------------------------2 1.2 Background------------------------------------------------------------------------------------------2 1.3 Theory-----------------------------------------------------------------------------------------------2 1.3.1 Bending moment-----------------------------------------------------------------------------2 1.3.2 The Right Angle Assumption--------------------------------------------------------------------------5 2. Apparatus----------------------------------------------------------------------------------------------6 * 3. Data-----------------------------------------------------------------------------------------------------7 * 4. Procedur-----------------------------------------------------------------------------------------------7 * 4.1 Recording-------------------------------------------------------------------------------------------7 * 4.2 Calibration------------------------------------------------------------------------------------------7
5. Results--------------------------------------------------------------------------------------------------8 * 5.1 Determination of the calibration constant----------------------------------------------------8 5.2 Calculation of the Bending Moments----------------------------------------------------------9 5.2.1 Analysis of the oscilloscope traces----------------------------------------------------------9 5.2.2 Calculation of experimental bending moments----------------------------------------- 12 5.2.3 Calculation of theoretical bending moments-------------------------------------------- 12 5.3 Comparison between theoretical and experimental results----------------------------- 13 5.4 Validation of the 90 degree assumption-----------------------------------------------------14 5.4.1 Theoretical Result---------------------------------------------------------------------------14 5.4.2 Experimental Result-------------------------------------------------------------------------15 5.4.3 Percentage Difference----------------------------------------------------------------------16 * 6. Discussion--------------------------------------------------------------------------------------------16 * 6.1 Error Analysis-------------------------------------------------------------------------------------17 * 6.1.1 Experimental Errors-------------------------------------------------------------------------17 * 6.1.2 human Errors--------------------------------------------------------------------------------17 * 7. Conclusion-------------------------------------------------------------------------------------------17 * 8. References-------------------------------------------------------------------------------------------17 *

Abstract
-------------------------------------------------
This is an experiment using a mechanical device of an engine and a measurement of the peak dynamic bending moment diagram of the connecting rod of the strain measurement. The moment diagram is used to determine the magnitude and position of the maximum peak dynamic moment, and then the value is compared with the theoretical value. In the survey, the right angle hypothesis was found to be successful. Finally, the peak value of the dynamic moment is verified by calculation.

1. Introduction
1.1 Objective

1. To produce the peak dynamic bending moment diagram for the connecting rod of a simple engine mechanism and compare it with that obtained from theory. 2. To determine, and compare with the calculated value, the position and magnitude of the maximum peak dynamic bending moment. 3. To confirm that the maximum peak dynamic bending moment occurs when the crank and connecting rod are at right angles.

1.2 Background

As a part of engine, the connecting rod transfers motion from the piston to the crankshaft and acts as a lever arm. the connecting rod is made of a cast aluminum alloy and its main function is to bear the dynamic stress derived from combustion and piston movement.

Bending moment is a kind of internal moment in the cross section of the force member, the resultant force of the internal force of the cross section.(1) The bending moment of the beam is the algebra of all external forces on the cross section. because of the high strength of the connecting rod in the course of movement , it is a very important factor to analyze the bending moment , and can calculate the peak value of dynamic bending moment by connecting rod.

1.3 Theory

1.3.1 Bending moment

A bending moment is the reaction induced in a structural element when an external force or moment is applied to the element causing the element to bend.(2) It is expected that the maximum bending moment occurs when the crank and connecting rod is at the right angle. In this right angle position, the transverse acceleration of the big end ω2r and that of the small end is negligible. Therefore the intensity of transverse loading varies linearly from zero at the small end to mω2r at the big end as shown in figure 1.(3)

Figure 1. Schematic representation of distributed inertia loads along the connected rod.

The connecting rod is in equilibrium state, hence the reaction forcesand are equal to the transverse loading:

(1)
Balance of torque around point B:

(2)
By integrating equation (2) back into equation (1), the equation for calculating is : (3)
From figure 1, the bending moment at section x :

(4)
At point A and B on the connecting rod, the bending moment will be zero. The bending moment can be determined at any point across the rod according to equation (4),when the value is zero, the position of the point is where the maximum bending moment occurs:

(5)
Substituted into equation (4) to get the equation of the maximum bending moment:

(6)

By the equation (5) and (6), the magnitude and position of the peak dynamic moment is defined, and the peak dynamic bending moment diagram of the connecting rod can be obtained as shown in figure 2.

Figure 2 theoretical prediction for the bending moment Mxx in different cross sections of the connected rod.

1.3.2 The Right Angle Assumption

There are two equations can be used to calculate the theoretical value for the angle between the crank and mid-line when the crank and connecting rod are at right angles. (7)

Figure 3. Representation of the right angle assumption

Using time values measured during the experiment to calculate an actual value for the angle equation.

(8)

and can be calculated by the horizontal distance between the peak of the ripple of the oscilloscope displayed.

Figure 4. Graph of strain gauge showing the difference between t1 and t2 2. Apparatus (1) Simple engine mechanism rig with strain-gauge connected rod(Fig.5a,b) (2) Strain gauge bridge (3) Tachometer(Fig.4a) (4) Calibration level and weights(Fig.4b) (5) Storage oscilloscope(Fig.4a) * * a) b) * Figure 5. Simple engine mechanism rig with strain-gauge connected rod * * a) b) * Figure 6. Tachometer, storage oscilloscope and calibration level and weights * * 3. Data * Mass of connecting rod M=1.01kg. Length between centres L=720mm.Crank radius r=80mm. * * 4. Procedure * Using the centre of the big end of the connecting rod as a datum to measure the position of the gauge before turning on the engine mechanism. * * 4.1 Recording * * Connecting the output of one of the five strain gauges to the oscilloscope to measure the voltage at each strain gauge, the output of each gauge is inserted into the order of the oscilloscope. After that, turn on the engine mechanism rig and adjust the oscilloscope trace sizes, then save it. Then analyse the oscilloscope traces and determine the time between maximum strains of opposite sign and the time for one complete cycle. * * Measure the peak-to-peak height of the trace, and the height is proportional to the peak-to-peak dynamic bending moment experienced by the connecting rod. Then measure the peak-to-peak voltage of the other pairs of gauges in turn without altering the gain of the strain gauge bridge. * * 4.2 Calibration * * In order to calibrate the system, the connecting rod has been converted into a horizontal simply supported beam using the locking pin in the crank. Using a minimum of six different weights with the lever system built into the rig to apply central concentrated loads. Note the trace’s voltage deflection on the oscilloscope for each of the six loads. Then plot a calibration curve of mass(ordinate) vs voltage and allowing the calibration constant to be obtained by the gradient of the graph in Nm/volt * Determine the peak dynamic moment of each gauge station, and from the peak dynamic moment to plot the peak dynamic bending moment diagram for the connecting rod. And draw the curve produced by calculation from theory on the same diagram. * * According to the time measurements, the fraction of the cycle time was determined, which allowed the angular rotation of the crank between peak strains of the opposite sign also been determined. * 5. Results * 5.1 Determination of the calibration constant * * Using the graph obtained during the experiment, the voltage values against time can be obtained from the Excel of gauge 3 calibration. * * * Figure 7. Graph of voltage(V) Against Time (s) during gauge 3 calibration * * After producing the graph of voltage against time during gauge 3 calibration, the mean voltage can be determined for each load. Each increment on the graph shows a mass being added on the beam and each decrement indicates a mass being taken off. According to the voltage values from Excel of gauge 3 calibration, the mean results of the loading values and unloading values were calculated and showed in table 1. * * Table 1. Calculation of Voltage Relative to Mass * * Mass(kg) | * Load mg(N) | * Loading * Voltage(V) | * Unloading * Voltage(V) | * Mean Voltage (V) | * 0 | * 0 | * 0 | * 0 | * 0 | * 1 | * 9.81 | * 1.68 | * 2.08 | * 1.88 | * 2 | * 19.62 | * 3.20 | * 3.60 | * 3.40 | * 3 | * 29.43 | * 4.72 | * 5.12 | * 4.92 | * 4 | * 39.24 | * 6.32 | * 6.56 | * 6.44 | * 5 | * 49.05 | * 7.84 | * 7.84 | * 7.84 | * * According to the table 1, the bending moment for each load can be calculated with the mass(m), the gravitational constant(g) and the distance from the big end of the connecting rod(X3), the equation is showing below: * (9) * Table 2. Calculation of bending moment for each mass added to gauge 3 during calibration * Mass(kg) | * Distance(m) | * Mean voltage(V) | * Bending Moment(Nm) | * 0 | * 0.305 | * 0 | * 0 | * 1 | * 0.305 | * 1.88 | * 2.99 | * 2 | * 0.305 | * 3.40 | * 5.98 | * 3 | * 0.305 | * 4.92 | * 8.98 | * 4 | * 0.305 | * 6.44 | * 11.97 | * 5 | * 0.305 | * 7.84 | * 14.96 | * * Using the values from table 2, plot a graph about the connection between the mean voltage and the bending moment as shown in figure 8. * * Graph of Bending moment(Nm) against Voltage(V) * * Figure 8. Graph of bending moment against mean voltage during gauge 3 calibration * * By the graph of bending moment against mean voltage, the constant Kcal value can be obtained from the curve: * *

5.2 Calculation of the Bending Moments

5.2.1 Analysis of the oscilloscope traces

After oscilloscope traces had been obtained from each of the five gauges, the certain values can be inquired from the Excel form of each gauge. Then determined the amplitude of each trace by measuring the peak-to-peak height of trace.

Figure 9. Analysis of oscilloscope traces

The amplitude can be calculated by the wave height equals to two times of amplitude and the results are following below:

Table 3. Calculation of the Amplitudes at each gauge

Gauge | Max Value(V) | Min Value(V) | Amplitude(V) | 1 | 7.92 | 1.6 | 3.16 | 2 | 7.44 | -6.8 | 7.12 | 3 | 7.12 | -7.12 | 7.12 | 4 | 6.32 | -5.6 | 5.96 | 5 | 3.28 | -2.88 | 3.08 |
The data of time between peaks of opposite sign (and) and time for one complete cycle as shown in table 4.

Table 4. Data taken for t1 and t2 Gauge | t1(s) | t2(s) | Time period T (s) | 1 | 0.050 | 0.050 | 0.100 | 2 | 0.055 | 0.040 | 0.095 | 3 | 0.059 | 0.041 | 0.100 | 4 | 0.051 | 0.053 | 0.104 | 5 | 0.056 | 0.044 | 0.100 | mean | 0.055 | 0.47 | 0.102 |

5.2.2 Calculation of experimental bending moments

As the calibration constant had been obtained, the bending moments can be calculated by the following equation: (10)
Table 5.Calculation of Experimental Bending Moment

Gauge | Length(m) | Amplitude(V) | Bending moment(Nm) | 1 | 0.101 | 3.16 | 6.07 | 2 | 0.200 | 7.12 | 13.68 | 3 | 0.305 | 7.12 | 13.68 | 4 | 0.451 | 5.96 | 11.45 | 5 | 0.605 | 3.08 | 5.92 |

5.2.3 Calculation of theoretical bending moments

Mass per unit length for the connecting rod can be obtained by equation: (11)

The angular velocity of the crank can be obtained by equation:

(12)

Calculated the theoretical bending moments at each gauge of the rod using equation:

Gauge 1:

Gauge 2:

Gauge 3:

Gauge 4:

Gauge 5:

Table 6. Theoretical Calculations of the Bending Moment at each gauge

gauge | Periodic time(s) | Angular speed(rads-1) | X(m) | Bending moment(Nm) | 1 | 0.100 | 62.83 | 0.619 | 8.57 | 2 | 0.095 | 66.14 | 0.52 | 14.63 | 3 | 0.100 | 62.83 | 0.415 | 14.7 | 4 | 0.104 | 60.42 | 0.269 | 11.36 | 5 | 0.100 | 62.83 | 0.115 | 5.95 |

5.3 Comparison between theoretical and experimental results

When comparing the theoretical and experimental results, the bending moment of each gauge are presented in table 7 , both theoretical and experimental and difference between this two values.

Table 7. Analysis of Theoretical and Experiment Bending Moment of Each Gauge

gauge | Theoretical Bending Moment(Nm) | ExperimentalBending Moment (Nm) | Difference(%) | 1 | 8.57 | 6.07 | 41.19 | 2 | 14.63 | 13.68 | 6.94 | 3 | 14.7 | 13.68 | 7.46 | 4 | 11.36 | 11.45 | 0.78 | 5 | 5.95 | 5.92 | 0.51 |

After calculating the percentage difference between the theoretical and experimental bending moment of each gauge, there is a big difference between gauge 1 and others. Therefore the gauge 1 was removed from the peak dynamic bending moment diagram.

Figure 10. Peak Dynamic Bending Moment Diagram

Table 8.Analysis of theoretical and Experimental maximum bending moment of the rod

| Theoretical | Experiment | %Error | Length at max BM | 0.243m | 0.239m | 1.67 | Max BM(Nm) | 15.16Nm | 14.11Nm | 7.44 |

5.4 Validation of the 90 degree assumption

5.4.1 Theoretical Result

A theoretical value can be obtained from equation (7) for the angle between the crank and mid-line when the crank and connecting rod when the peak dynamic bending moment occurs.

5.4.2 Experimental Result

Equation (8) can be used to calculate the values of time recorded between maximum voltages in 5.2 can be input as follows:

Due to the values of t1 and t2 are different from each gauge, the mean value of t1 and t2 can be used to calculate the experiment value:

5.4.3 Percentage Difference

The percentage difference between the theoretical and experimental values can be calculated using the two values above：

* 6. Discussion * By using the values of theoretical and experimental, the peak dynamic bending moment graph is very successful. It can be clearly seen from the graph that the curves of the theoretical and experimental have a similar curve, although the curve of the theoretical is slightly higher than that of the experimental which shows that the experiment may have been subject to some kind of systematic error. The error may caused by the precision of the measuring during the experiment. All of the oscilloscope traces did not follow a perfect sinusoidal curve in the peak that means some adverse factors are influenced the results of the experiment. * * When comparing the experimental values and theoretical value of the peak dynamic bending moment of the position, different degrees of differences have been obtained. The difference in magnitude may be caused by the measurement technique of the oscilloscope traces. * * The maximum bending moment from the verification theory occurs at the right angle between the crank and the connecting rod. The difference between the theoretical values and the experimental values are less than 1%. The result obtained totally supports the theory with the neglect of gauge 1. * * * 6.1 Error Analysis * * There are many different factors that cause errors in the experiment, including experiments and human. * * 6.1.1 Experimental Errors * (1) Strain gauge 1 was broken and therefore the values taken from the gauge may not be accurate. (2) The simple engine mechanism rig in the experiment was very old and may not function well. (3) The machine in the lab had not been running for a long time and may not be as accurate. (4) As a result of wear and tear of the masses and discrepancies during manufacture, the values of masses may not equal to the values stated on the masses. (5) Due to friction, the peaks on the oscilloscope didn’t reach at the maximum very well. (6) The connecting rod was assumed to be a uniform bar which may not be true. (7) The strain gauges were not vertical to the bar. * * 6.1.2 human Errors * (1) The data of the measurements of the oscilloscope traces may not have been read correctly. (2) During the calculation, some errors may have occurred. (3) The values read from the Excel may not have been read properly. * 7. Conclusion * After this experiment, the results obtained support the following conclusions : * (1) The experimental results of the peak dynamic bending moment are similar to the theoretical results. The gauge 1 results were neglected from the results due to the broken of it. (2) A low percentage error occurred when verified the position and magnitude of the peak dynamic bending moment with theoretical values. (3) The assumption of maximum bending moment occurring when the angle between the crank and connecting rod is 90 degree was verified. * * There are some advice to improve the experiment: * (1) The analysis of the oscilloscope traces could be improved by implementing a more accurate method of measuring the oscilloscope traces. (2) A new machine could be used if possible. * * * * * 8. References * (1) Bending moment, baidu baike * http://baike.baidu.com/link?url=GTCcQU-dUbd_duOczw67Gca1insq_zprWD88oOSrVy9odQaE4VIdWvZxjSR9NbTpmMbyxZFngKJzackKCtE9TK(Accessed 9/11/2015) * (2) Bending Moment, Wikipedia , the free encyclopedia, * https://en.wikipedia.org/wiki/Bending_moment(Accessed 9/11/2015) * (3) Goman,M Laboratory Experiment-Slider Crank Experiment * https://vle.dmu.ac.uk/bbcswebdav/pid-3238469-dt-content-rid-5112942_1/courses/ENGD3038_2016_Y/ENGD3038%20-%20Inertia%20Bending%20-%20Lab%20Sheet.pdf(Accessed 9/11/2015) *