PRANIL KAPADIA MMAN2300 ASSIGNMENT C z34516737 INTRODUCTION The slider-crank mechanism is mainly used to convert rotary motion to a reciprocating or vice versa. In this scenario the kinematics and dynamics of a slider-crank mechanism such as is commonly found in internal combustion engines will be analysed. Figure 1 below illustrates the mechanism of interest. The crankshaft AB rotates counter-clockwise with constant angular velocity wAB. It is connected to the piston C by the connecting rod BC. The geometry of the mechanism is detailed in Figure 2 below. The connecting rod has length L and its centre of mass is located at point D which is a distance H from the pin at B. Crankshaft AB has length R and its rotation is given by the crank angle !, which is measured counter-clockwise from the positive x-axis. The angle ! gives the angle of the connecting rod BC. Notice that ! is measured counter-clockwise from the negative y-axis. Figure 1 Slider-crank mechanism used for analysis The parameters to use for the analysis were determined using Table 1 below and the student number zABCDEFG. Student number used: z3415737 The highlighted parameters were used for this analysis DATA USED IN THIS REPORT Piston Mass (mp) = 0.440 kg Digit of Student No. Parameter0123456789 BPiston mass (g) 400410420430440450460470480490 CR (mm)40424446485052545658 DH (mm)34353637383940414243 EL (mm)135137140142145147150155157160 FConnecting rod mass (g) 400410420430440450460470480490 GI of connecting rod (g m2) 1.301.351.401.451.501.551.601.651.701.75 Figure 2 Geometry of the slider-crank mechanism R = 42 mm H = 39 mm L = 155 mm Connecting Rod Mass (mr) = 0.430 kg Ir of connecting rod = 1.65 gm2 The derivations of question c, d, e, f and g can be found in the appendix. ANALYSIS Part a. Figure 3 below shows the plot of the acceleration of the piston as a function of the crank angle for one full rotation of the crankshaft. To understand the relationship fully Figure 4 is simply an extension of the Figure 3 with two cycles instead of one.

Figure 3Plot of Vertical Acceleration of Piston against Crank Angle Figure 5 is a plot of the vertical velocity of the piston C against the crank angle for two full rotations. From Figures 3 and 4 it can be shown that maximum value of the vertical acceleration occurs at the very top of the piston stroke, known as Top Dead Centre (TDC). Also during maximum piston velocity, the piston stops speeding up and begins to slow down at which point the acceleration changes from a positive sign to a negative sign (shown in Figure 3) The point of zero acceleration occurs at the point at which the piston velocity is at a maximum, where velocity is reversing direction.Figure 4 Two full cycles of Figure 3 Figure 5 Vertical Velocity of Piston against Crank Angle for two rotations The unintuitive flat section in Figure 3 can be attributed to the fact that the total piston acceleration in the vertical direction (equation 10) is the sum of several orders of acceleration, which includes the angular acceleration of BC (equation 11). This section is dependent on the rod stroke ratio (R/S). For this analysis the rod stroke ratio is given as: !! !!!!""!!!!!"#! !!!"# This is demonstrated by Figure 4.2, which shows the acceleration of a piston with two different rod stroke ratios. Therefore it can be said that increasing the rod stroke ratio will alter the acceleration and produce a greater flat section of the graph. Part b. Figure 6 shows the plot of the Angular Acceleration of the Piston C against the Crank Angle for one full rotation of the slider-crank mechanism. Equation 11 from the index was the relationship to theta used to plot the angular acceleration (the derivation of this relationship is contained in the kinematic analysis handout). Figure 4.2 Plot showing the effect of lowering the rod stroke ratio has on the shape of the acceleration of the piston for one full rotation. http://ftlracing.com/images/rsratio_002.gif From Figure 6 we can deduce that the maximum angular acceleration occurs at the halfway point of the rotation cycle. Also it is clear that the angular acceleration is zero at TDC and Bottom Dead Centre (BDC). From equation 11 it can be shown that changing either L or R slightly will result in a different value for the maximum angular acceleration, this can be taken into when designing a slider-crank mechanism. However altering !AB (The operating speed) will ultimately have the most significant impact on the angular acceleration Part c/d. Figure 6 Plot of Angular Acceleration of connecting rod BC against the crank angle for one full rotation From the derivations of equations 6, 7, 8 and 9, Figure 7 was created. Plotted against the crank angle, it shows the horizontal and vertical components acting on the connecting rod BC. From figure 7 it can be shown that at TDC ("/2 rad) and BDC (3"/2 rad), both Bx and Cx result to zero. Similarly BY and CY have the highest magnitude of force at these points. This relates back to Figure 6 since at TDC and BDC the value of the angular acceleration is zero at both these points, which results in no forces at in the horizontal direction at this instant Figure 7 Force Components acting on the connecting rod BC against Crank Angle Part e. Figure 8 below shows the plot of the magnitude of the forces at B and C as a function of the crank angle for one full rotation. The figure below shows the similarity in shape between the magnitude forces of B and C Part f/g. Figure 8 Magnitude forces of B and C against crank angle for one rotation From figure 9 it can be shown that the maximum angular kinetic energy is reached at TDC and BDC. As a result it is show again in figure 9 that linear and so total kinetic energy are at a minimum at TDC and BDC The angular and linear kinetic energy equations contain constants except for #BC (for which the equation can be found in the kinematics analysis document), which is a function of ! and so produces the sinusoidal feature of the corresponding plots. In terms of design, the rod stroke ratio is a key parameter since it ultimately determines the wear, velocity and acceleration of the pistons, all important features to take into consideration when designing a slider-crank mechanism like this. APPENDIX Figure 9 Kinetic Energy of Connect Rod BC against crank angle for one rotation Drawing the Free Body Diagram Gives figure 10 below Now summing the forces in the horizontal and vertical components ! ! !! From the Free Body Diagram there are horizontal and vertical forces acting at points C and D in the connecting rod as show below !!! ! !!! ! !!! ! !!! ! !!!! ! !!! Separating the above equations into i and j components gives equations (1) and (2) Following on from this, summing moments taking anticlockwise as the positive direction about the centre of mass D produces equation (3). Figure 11 below is a Free Body Diagram labeling the perpendicular distances and forces produced in the system !! !!! !! ! !!!!"#$ !! !!! !!! !!! ! !!!!"%$ Figure 10 Free Body Diagram of the connecting rod BC Using Figure 11, equation (4) can be determined by simple summation of forces about D Note from the Kinematic Analysis it is given that ! !!! !!!"

!!!!"# ! ! !!!!"# ! ! !! ! ! ! !"# ! ! !!!! ! !! !"# ! !!!!"

"&$ !"# ! !!!! !! !"#!!! !"# ! !!! !"#! Figure 11 Force Diagram for summing moments about D Now to calculate the acceleration of the centre of mass D we have Substituting the given variables gives: Simply separating into i and j components results in equations (4) and (5) Now analysing the piston itself from Figure 12 which is the Free Body Diagram of the piston C Once again using ! ! !! The resulting equation (6) is yielded !!! !!! !!"! !!!!! !!"! !!!!

!!! !!!! !!"!! ! ! ! !"#! !! ! ! ! ! !"#!!! ! !!"! !! ! !!!!"#!! ! !"#!!!!! !!" ! !!" ! ! ! !"#! ! !!"!! ! ! !"#! !! !!" ! !!"! !!" ! ! ! !"#! ! !!"!! ! ! !"#! "'$ "($ !! ! !!! ! !!!!"")$ Figure 12 Free Body Diagram of the piston itself And so substituting equation (2) into (6) we get, From these results we can deduce the horizontal component of the force at B, equation (8) is, Now by substituting equation (8) into (3), an equation (9) for the horizontal component of the force at C can be established The following equations (10 & 11) are both derived from the kinematic analysis document and are both used frequently throughout the report !!! ! !!!!"!! !!! !!"#!!! !! !"#!! !!!!"#!!!! !! !"#!!! ! !"#!"#*$ !!" !! !!! !!"#!!! !! !"#!! !!!!"! "##$ !! ! !!! !!! ! !!!!" "+$ !! ! !!! !!!!"",$ !! !!!!"! !!!"#$! ! !!!!" !"#$! ! !! ! ! ! !"#!!"#$! "-$ !"#$%&' !"#$%"& !"#$%& !!!!!!!"! ! "#%$ !"#$%& !"#$%"& !"#$%& !!!!!!!!!! "#&$