ENGD3008 Dynamics, M. Goman Experiment 1 Inertia Bending of a Connecting Rod Warwick Shipway Mechanical Engineering, Year

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1 ENGD3008 Dynamics, M. Goman Experiment 1 Inertia Bending of a Connecting Rod Warwick Shipway Mechanical Engineering, Year Introduction The bending due to the velocity of a connecting rod is simulated for measuring purposes and, using strain gauges, the maximum dynamic bending is found. Theoretical bending moments are calculated and compared with the measured values. The position of the maximum bending moment is also measured.. Background When an engine is in operation the rotational motion necessary to run the gear box and subsequently wheels of the car is translated from a linear motion by what is known as the connecting rod (axial) and crankshaft (radial). Due to the change in motion there are forces acting on the connecting rod, in both compression and tension. Maximum tensile stress occurs at top dead centre, and maximum compressive stress occurs at bottom dead centre. Of note, at both of these locations the instantaneous acceleration is maximum, and hence the respective forces are maximum. However, instantaneous velocity at these points by definition is zero, due to the change in linear motion. Hence maximum linear velocity of the connecting rod occurs at some point along its downward or upward stroke. The force can be considered as a centrifugal vector (i.e. acting away from the centre of rotation), and as a magnitude only, due to direction change being in phase with the direction change of the motion of the rod. This vector force is hence shown to be equated using Newtons nd Law, and is maximum when the rod is transverse to crankshaft. The force (and consequently stresses) exerted on a beam due to the transverse loading is shown to be calculated below, and it is considered applicable to the beam if the main forces applied due to inertial transverse loading are considered only. Hence if Newton s nd Law states F = d(mv), where both the force F, dt and velocity V, are vectors, then the inertial force can be calculated as, F = d(mv) = ma = mrω 1 dt Where m mass product or mass per unit length (kg) r radius of crank (m), ω angular velocity of crank (rad/s). Bending Moment of Applied Masses The bending moment of the connecting rod can be determined theoretically from simple bending moment theory. If the connecting rod is in its horizontal position and constrained either end by the crank and pins (i.e. locked), it can be assumed comparable with that of a simple beam on knife supports. Such a beam is shown in Figure 1. If this beam is loaded with weight W, the maximum bending moment can be calculated as follows Figure 1. Simple beam with an applied load, W. It can be shown that the bending moment at any location along the beam is found (taking moments about A) as Bending Moment at B, M xx = xw This calculation can be used to produce a calibration curve for the measured strain gauges. As previously described the load (force) is considered maximum when the rod is transverse to the crankshaft, and by measuring the cycle time between maximum bending (magnitude only) the angle can be determined. Page 1

2 If moments are taken about any distance x along the beam when in operation, then it can be calculated using Eq. (1) and () that the bending moment is M xx = mrω 6L (L x x 3 ) 3 3. Apparatus 1. Engine mechanism rig with strain gauged connecting rod. Strain gauge bridge 3. Counter timer 4. Weights 5. Metre rule 6. Storage oscilloscope Mass of connecting rod: 1.01 kg Length between centres: 70 mm Crank radius: 80 mm 4. Procedure 1. The strain gauge locations were measured from the centre of the big-end using the metre rule. 5. Results The results from running the test are shown in table 1, with the oscilloscope images shown in Figure. The oscilloscope image for the applied loads is shown in Figure 3. Table 1. Results from the connecting rod Experiment Position Hz of Strain of Gauge Max. to Peak to Hz. Gauge from Big Min. Peak Location End (mm) Strain Voltage The strain gauge bridge was connected for gauge 3, to measure the bending strain. The strain was observed by using an oscilloscope connected to the bridge output. 3. The connecting rod was powered up and allowed to rotate about the big-end. 4. The size of the oscilloscope trace was adjusted for gauge 3, to measure the resultant voltage. The wave was measured peak to peak using the oscilloscope, and the image was stored. Figure.1, Strain Gauge The time between maximum to minimum strains (that is between maximum positive and maximum negative strain) was recorded, as was the time for one rotation of the crank. 6. The other 4 gauges were measured using the same procedure, but the oscilloscope was not adjusted to ensure the measurements were the same for each gauge. 7. The connecting rod was locked into its horizontal position using the locking pin. A weight was loaded onto the connecting rod using the lever system at the location of strain gauge 3. This gauge was connected to the bridge, and the voltage (deflection) was measured as before, for loads of, 4, 6, 8 and 10 kg. The oscilloscope wave is stored. Figure., Strain Gauge. 8. A calibration curve is produced from this data, and is used to plot the peak dynamic bending moment for each gauge location. This is compared with theoretical values. 9. The time between max. and min. strain (from step 5) is used to determine the angular rotation of the crank at maximum bending (max. deflection). Figure.3, Strain Gauge 3. Page

3 Bending Moment, Nm Bending Moment, Nm Inertia Bending of a Connecting Rod /03/008 Therefore, the bending moment at strain gauge 3 for a mass of 4kg is, BM 3 = x.w = (4/) 9.8 = Nm 15 This is plotted against the measured voltage (abscissa) produced by the strain gauge bridge, as shown in Figure 4. Calibration Curve Figure.4, Strain Gauge Figure.5, Strain Gauge 5. Figure. The oscilloscope traces for the 5 gauges Voltage, V Figure 4. A plot of strain gauge 3 with applied masses. The resulting straight line equation is found to be y = x. The vertical intercept is zero because with no load there is no strain. Hence the results from table 1 can now be used to plot the dynamic bending moments at the 5 strain gauge locations. And, using Eq. (3) to find the theoretical bending moments, the plots from the measured and calculated results are shown below, in Figure 5. Figure 3. The oscilloscope trace for applied masses of to 10 kg Peak Dynamic Bending for each Strain Gauge Exp. results Theory results The results from the applied loads found in Figure 3 are shown in table. 1 Table. The bending moment at strain gauge mass (kg) V Bending Moment (Nm) By using Eq. the bending moment can be found. As the mass is applied using a rig along which the bar interfaces with the connecting rod at half the bars length, the mass applied to the rod is only half of that applied to the rig (and hence it has been halved in Table ) Strain Gauge Position, m Figure 5. A plot of the theoretical and measured bending moments at varying positions along the connecting rod. Page 3

4 6. Discussion It is obvious from Figure 5 that there is a discrepancy between the empirical values and the theoretical (calculated) values. The average error is indeed 80%, which is rather large. However due to the way in which the theoretical and measured results were calculated (by computational methods, see appendix) it is unlikely that the discrepancy is due to the manipulation of the acquired data, rather the error is either in the data itself, or the formulae stated in section. However, due to the formulae being previously provided, this is unlikely to be responsible. Equally, the data recorded is also measured digitally, as shown in Figures.1 to.5, so either the accountable error is in the measuring equipment, or some mathematical discrepancy in the Matlab Coding. However, manual calculations agree with the plots: - M xx = mrω 6L L x x Nm This is correct to the bending in the middle of the connecting rod, and has the same value as that depicted in Figure 5. It is most likely then, that the error arises in the measured results. Indeed there is greater chance or error in this respect, although, if, say the strain gauges were causing a discrepancy, it would be eliminated by the use of a calibration curve. For example, the transverse loading caused by the applied force (a function of the rate of change of angular velocity, ω) would cause the same proportional bending moment as applying a mass on the connecting rod. This is assuming that the strain gauge deformation is within its elastic limits, and the whetstone bridge uses a dummy gauge to eliminate temperature effects on the voltage. It is hence deduced that the likely errors are either the angular velocity, the measured voltage when loading, or the positions of the strain gauges. The gauge positions are thus re-measured, with the same results as previously attained. The calibration constant (the slope of the straight line found in Figure 4) is found by differentiating the line, and can be found as dm dv with the slope constant designated m. This produces m = , although it obviously uses the measured values, rather than a line of best fit. If a line of best fit is used the constant m = , which compared to the errors between the results is insignificant. The reason for this difference is due to the inevitable errors produced when taking empirical data, for example the accuracy of the equipment used. Hence an absolutely straight line is impossible. Of note is that strain gauge 5 has a slightly lower deflection than that of gauge 1. This is due to their respective distance from the point of peak bending, that is gauge 1 is 05mm from gauge 3, and gauge 5 is 97mm from gauge 3. This is due to the bending moment increasing in a linear fashion; with of course no bending at the centres of the big and little ends due to the constraints. Also strain gauge position 4 (at 451mm from the big end) produces the highest dynamic loading when theoretically calculated, which is inconsistent with the proposed theory and the measured evidence. This inconsistency is considered independent of the discrepancy between measured and theoretical, and is erroneous. The measurements of peak to peak strain was observed and recorded as in Table. It was measured that there was rotations per second (Hz.) and the average peak to peak measurement was taken as 44Hz. This shows that the position of maximum bending is 44 Fraction time, 80% Angular velocity, 604( rpm) 63.5( rad / s) Angle between maximum bending, ( rad / s) That is, the angle of offset between the vertical plane of the crank/con rod connection and the centre of rotation is 1 o, for peak dynamic bending. 7. Conclusions The calculated results prove to be consistent with that of measured, with the plotted curves being similar, but with large discrepancies between the two. It is deduced that either the angular velocity or the measured voltage is inconsistent, as all other parameters and variables have either been double checked or calculated computationally. The maximum peak dynamic bending is calculated to be 14.9Nm at gauge position 4 (451mm from the big end), and measured to be.7nm at gauge position 3 (305mm from the big end). The average error between the results is 80%. Maximum bending is shown to occur at an angle of offset of 1 o between the crank/rod and the centre of rotation. It is unlikely that the theoretical and measured results cannot be compared, rather an erroneous procedure is the cause of divergence. o Page 4

5 8. References Hearn, E. J., 1997, Mechanics of Materials, 3 rd Ed., Butterworth-Heinemann, Oxford ISBN otion, last modified B. Ulanicki, lecture notes P. S. Shenoy, 004, Thesis, Dynamic Load Analysis and Optimisation of Connecting Rod, Toldeo University 9. Appendix %{ Warwick Shipway Dynamics - Inertia Bending of a Con Rod %} % Constants g = 9.81; %gravitational constant L = 0.7; %length of con rod, m m1 = 1.01; %mass of con rod, kg Hz1 = e-3; %s r = 0.08; %crank radius, m omega = 604; %rpm % Exp. Results %Concentrated applied loads m = [ ]; %applied mass, kg V = [ ]; %strain gauge reading, V %Dynamic inertia loading BM = 0.305*(m/)*g %Bending Moment at gauge 3, Nm x = [ ]; %strain gauge position 1 to 5, m V1 = [ ]; %strain gauge reading, V Maxstrain = [45e-3 44e-3 46e-3 43e-3 4e-3]; % time between max. strains, s dbmdv = diff(bm)./diff(v); %differentiates the calibration curve to find the slope of the line m = sum (dbmdv)/5 % summates the array to find the average value i.e. the slope, m, of y = mx + c %Exp. Calculations Mxx1 = V1*m % converting strain voltage to Dynamic Bending, Nm % Theoretical Results Mxx = (((m1/l)*r*((omega*(*pi/60))^))/(6*l))*(l^*x-x.^3) %Plots Figure; subplot(,,[1 3]); plot (V, BM, '-ro'), grid on xlabel('voltage, V','FontSize',1) ylabel('bending Moment, Nm','FontSize',1) title('calibration Curve','FontSize',16) subplot(,,[ 4]); plot(x, Mxx1, 'ro', x, Mxx, 'gx'), grid on, hold %Define Bestline Fit Lines P1 = polyfit(x,mxx1,); f1 = polyval(p1,x); P = polyfit(x,mxx,); f = polyval(p,x); plot (x, f1, '-r', x, f, '-g') xlabel('strain Gauge Position, m','fontsize',1) ylabel('bending Moment, Nm','FontSize',1) title('peak Dynamic Bending for each Strain Gauge', 'FontSize',16) legend('exp. results', 'Theory results',1) Page 5

ENGD3008 Dynamics, M. Goman Experiment 3 Kinematics of a Mechanism Warwick Shipway Mechanical Engineering, Year

ENGD3008 Dynamics, M. Goman Experiment 3 Kinematics of a Mechanism Warwick Shipway Mechanical Engineering, Year 22/3/28 ENGD38 Dynamics, M. Goman Experiment 3 Kinematics of a Mechanism Warwick Shipway Mechanical Engineering, Year 3 27.3.28 1. Introduction A slider crank mechanism is analysed and the displacement