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

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1 22/3/28 ENGD38 Dynamics, M. Goman Experiment 3 Kinematics of a Mechanism Warwick Shipway Mechanical Engineering, Year Introduction A slider crank mechanism is analysed and the displacement at varying input angles is measured. The velocity and acceleration of the slider is hence determined, and is compared with that obtained analytically. A large value of velocity and acceleration is considered and a vector diagram is constructed and compared with the other attained values. The merits and detriments of each technique is thus evaluated discussed. 2. Background A simple mechanism that transforms an angular motion into a linear motion, like a connecting rod and crankshaft, can be diagrammatically represented and the displacement at any angle can thus be analytically determined. If values like length and angle at any one point at replaced with variables, the analytical formulae can determine the velocity and acceleration by differentiating the displacement once and twice respectively. However for experimental data it is not possible to replace the angles and retain the experimental validity, hence another method is used. Since the derivative of a function is the rate of change of said function, if the function is a plotted line on Cartesian co-ordinates then the rate of change of the line is by definition the gradient. The gradient is defined as δy = y 2 y 1 ; δx x 2 x 1 Figure 1. The slope of the line is found The gradient cannot found for a curve directly (by using the linear equation), and hence the instantaneous slope will be different at each point. This is the basis for finding the derivative and 2 nd order derivative of displacement, i.e. velocity and acceleration respectively. By taking the incremental change in displacement over the change in angle the slope can be defined at each change in angle. A line diagram of the mechanism with all significant dimensions is shown in Figure 2. Figure 2. Line diagram of the mechanism (not to scale). 3. Apparatus Vernier scale Protractor (attached) 4. Procedure The slider crank mechanism is measured for significant dimensions, as needed to calculate the displacement and hence velocity and acceleration. The crank is rotated to the starting point of degrees. It is then rotated in 1 degree increments, and the displacement of the slider is observed and noted. Page 1

2 Displacement (mm), V (mm/s per rad/s), (mm/s 2 ) per (rad/s 2 ) Acc. (mm/sec) 2 per (mm/sec) 2 Velocity mm/s per rad/s 22/3/28 The slider displacement is noted for all 36 increments, resulting in a full rotation of the mechanism. Velocity and acceleration plots are constructed from this data, as shown in the proceeding sections. 5. Results 5.1 Experimentally Obtained Curves The results from the procedure are shown in table 1. The velocity can be determined by finite difference method. Since the change in angle is 1, δx is constant. Such an example by using table 1 is shown: - δy x 2 x 1 x 2 x mm/rad δx 2π 1 π π Where x 1 and x 2 are displacement values at 1 and 2 degrees of the input crank. The acceleration is again found by using the above method, but replacing x 1 and x 2 with V 1 and V 2. The empirical solution is plotted in Figure 3, as a function of input angle used visa versa to find an unknown input. This is explained in more detail in section 6. The input angle alpha is simulated to be incrementally changed by using an array with its initial value of and finishing at 2π, simulating one rotation. The increment of the array from to 2π is set at.1. This could be set for 1 increments as per the experiment, but by simulating smaller incremental changes in angles the accuracy will be higher. As is demonstrated in the experimental procedure to find the velocity and hence acceleration, the displacement array is used to find the velocity via finite difference method, so a for loop is implemented. This loop executes 628 times and at each cycle calls the displacement element(x) and the proceeding element (x+1). The loop is assigned to i, and stops at 628 as this is 2π 1, because the incrementing calculations are.1. Therefore the array length is 2π/ The reason that it is not assigned to exactly 2π is that the array can of course only hold an integer amount of elements. It then adds an extra element to the velocity array by creating an element v(629). This is because there are 629 elements for the displacement including no displacement for no input angle. However, by definition the rotation is one full cycle, and as such the final velocity can be found finitely by taking the first element from the last element. The acceleration is thus found exactly the same way, by taking each element of the velocity array, and then adding a final element The displacement, velocity and acceleration array can consequently be plotted as a function of the input angle array of 1 rad/s, or approximately 1 RPM. The plot is shown in Figure Velocity, mm/s per rad/s Acc, (mm/s)2 per (rad/s)2 5-2 Angle of Input Rotation, degrees Figure 3. The dynamical displacement, velocity and acceleration, as a function of the input angle, for the experimental results. 5.2 Theoretically Obtained Curves The displacement x can be found analytically using Eq. (1). x = E + D (E cosψ + D cosφ) 1 This is implemented using Matlab, with the fundamental aspects of the code being explained here. The majority of the code is inputting the algebraic functions (mostly trigonometric) as defined in section 2, to be manipulated to yield an output for any given input angle. This is similar to a transfer function, which converts a known input into an unknown output. It is important to define this similarity, as it can be Velocity, mm/s, mm/s 2 pi/2 pi 3/2pi 2pi One Rotation (radians) Figure 4. The dynamical displacement, velocity and acceleration, as a function of the input angle, for the analytical solution. 5.3 Velocity and Acceleration Vector Diagrams Using the plots in Figures 3 and 4 an appropriately high velocity value is taken at Φ = 23 o. The vector diagram can be found at the of the paper, and shows to have a velocity of approximately 7mm/s for an arbitrary angular Page 2

3 22/3/28 velocity of 1 rad/s. This is selected as it is used for the analytical solution. To produce a vector diagram it is necessary to find the resulting angular velocity of the slider, which is rotating about a fixed position. This is shown in the diagram as V OA and V O A with accompanying calculations shown. 6. Discussion 6.1 Experimental Analysis Figure 3 shows the results of the experiment and differentiation using the finite difference method. Whilst the displacement has good agreement with the analytical plot the velocity and particularly acceleration plots are quite erroneous. For example a particular value at approx. 12 o contains a large error. If the acceleration is considered as a vector value, then the plot suggests that the slider suddenly changes direction (hence a negative acceleration), which is of course untrue. Generally the velocity plot follows the analytical plot, except again at this location of 12. This suggests an error in the displacement reading during the experiment, even though the plot of displacement does not reveal any error. By differentiating the plots errors can be highlighted, and indeed the higher the order of differentiation the more obvious the errors. This is a standard mathematical tool to highlight errors, and is perfectly represented in Figure 3. This is the main detriment of using experimental data to determine the velocity and acceleration of the slider crank, although the maximum (minimum) values are consistent with that found analytically. Although it is generally considered that the data found analytically is more accurate, one advantage of experiments is that they are of course physical. Taking the error at 12 o, it may well be that this highlights an inherent deficiency with the mechanism. In other words the error discovered by differentiating could suggest that at this angle the mechanism could be stiff. It could be that at this angle the slider got stuck, and so even though the angle read correctly, the slider was not displaced fully at this point. This could be due to bing of the slider arm. Of note, the displacement at some measurements is only approx..1m, a small amount, and so it is entirely possible this could be the case. Note: The velocity and acceleration plots are not functions of time, as would be the normal procedure when dealing with slider crank mechanisms, as it is manually rotated as opposed to an automated influence. However if an arbitrary rotation of 1 RPM were used then the finite difference method would be multiplied by 6, as 1 (36/6) = 6 degrees per second. This is true for acceleration as well as velocity. This could be applied purely for kinetical understanding, as the current values are somewhat abstract. (This is not true for the plots, as they show the points of maximum velocity and acceleration with regards to the slider crank geometry.) 6.2 Analytical Analysis Figure 4 shows the plots when the displacement is calculated, and hence differentiated twice. It is shown that the plots are much smoother, and no erroneous can be immediately observed. This does not however assume that the plot is exact, and error can and indeed will arise. This is mainly due to the measurement of values A, C, D, E and R. It is impossible to accurately measure the slider crank to an infinite degree, and hence there will always be errors in this respect. However, the overall plots are considered comparable with that found experimentally, and as such are accurate. Other disadvantages of using a computer simulation include, - good mathematical modelling skills are needed - computer literate skills - can be easy to put a bug in the program, that may result in inaccurate data The main advantages include, - simulation can be very accurate - no need to make a physical model, hence less capital costs - provides good visual representations (plots) - easy to change values to optimise the mechanism 6.3 Vector Diagram Analysis The vector diagram is shown at the of the paper. It produces a velocity of 72 mm/s at an angle of Φ = 23 o. There is an error of approx. 55% if the experimental data is compared (and at this point the computational data is comparable to the experiment). This is quite considerable, and is considered inaccurate. The acceleration vector diagram for Φ = 23 o shows to be α c = 3 mm/s 2. Comparing with the analytical and experimental plots the errors is thus approx. 6%, which shows roughly the same error as the velocity vector diagram, suggesting they are constructed correctly. The main disadvantages of using vector diagrams include - only applicable to simple mechanisms, otherwise it becomes too complex The main advantages include, - if the slider is not complex it is relatively simple - there is not as much mathematical analysis - no need for a computer to theoretically predict velocity and acceleration - provides a visual representation of the mechanism 6.4 Adjusting the Mechanism As has been previously described, the program provided takes known inputs and calculates unknown outputs. This has been compared to a transfer function, which is a fundamental principle in control engineering. As with a transfer function, the coding can be manipulated to determine either a desired output, or manipulated to experiment with optimisation Page 3

4 Displacement (mm), V (mm/rad/s), (mm/(rad/s) 2 ) 22/3/28 without making a physical model. Examples include changing the geometrical parameters (like the length of the sliders). The dimensions specified in Figure 1 can be manipulated for its appropriate application. For example if no offset angle β (i.e. when α = β) is used a more violent velocity curve is plotted. Such a plot is shown in Figure 5. In Figure 4 the maximum velocities in both directions are approx. 12 mm/s and 7 mm/s. However, in Figure 5 the maximum velocities are approx. 12 mm/s and 22 mm/s. The acceleration of the slider is seen to be much more ferocious Velocity, mm/s, mm/s 2 pi/2 pi 3/2pi 2pi One Rotation (radians) Figure 5. A displacement, velocity and acceleration curve for no offset angle β. Note: The plot is manipulation of the code in the appix and simply replaces the code beta = acos(a/r); with beta = alpha;. Whilst this is not the most elegant code for a slider crank mechanism with no offset it is a perfect example of engineering use of computational simulations. 6.5 Applications The main application for slider crank mechanisms is the transport industry, being used in practically every internal combustion engine. This is particularly applicable to the manipulation of the code in the analytical results, where the maximum velocity can be used to determine the power of the engine. Indeed it is more applicable to use with the material strength. For example if the maximum velocity and acceleration can be determined then it can be calculated whether the crank/connecting rod will fail. 6.6 A Note on Coding As has been previously described, the provided code uses for loops to call the elements in the displacement array. However, Matlab has the ability to do this more simplistically, known as vectorising loops. For example, if a for loop is defined for the sine of 11 values: i = ; for t = :.1:1 i = i + 1; y(i) = sin(t); The statement can be coded using a vector loop: t = :.1:1; y = sin(t); A can be seen, it replaces five lines of code with two. IT also gets rid of the assigned variable i, and so reduces the likelihood of reassigning variables, and hence, bugs. 7. Conclusions The results from the experimental and analytical plots show good agreement, suggesting the experimented proved successful (see Figures 3 and 4). The vector diagrams provided a difference of approximately 55% to 6% between those found experimentally and analytically. By differentiating data the errors can be highlighted, in what may initially seem to provide good results (i.e. the plot of displacement). However, the error at 12 o does not necessarily show an error with the data; rather the mechanism may become stiff at this angle, and hence produce very slight bing in the slider arms. This will always be an advantage of producing a physical model. The analytical code provides excellent ability to adjust the design of the mechanism and produce results without actually making anything, hence reducing capital costs (see Figure 5). 8. References B. D. Hahn, D. T. Valentine, Essential Matlab for Engineers and Scientists, 3 rd Ed., 27, Butterworth-Heinemann, Oxford en.wikipedia.org/wiki/finite_difference_method, last modified last modified e3m3performance.com/tech_articles/enginetech/rod-ratio/ Page 4

5 22/3/28 9. Appix % De Montfort University ENGD38 Lab Kinematics of a Mechanism % Quick Return machnism with slider crank output % Analytical Solution % % Mechanism geometrical parameters A=25.4; % mm R=83; % mm D=75; % mm E=25; % mm % one turn twopi=2*pi; % the angle of offset beta, when psi=phi= beta=acos(a/r); % sampling points for computation alpha=:.1:twopi; theta=alpha+beta; % location of input slider C=sqrt(A^2+R^2-2.*A.*R.*cos(theta)); % expressions for trigonometric functions sinphi=(r.*cos(theta)-a)./c; sinpsi=d.*sinphi/e; cospsi=sqrt(1-sinpsi.*sinpsi); cosphi=r.*sin(theta)./c; % slider crank output position x=e+d-e*cospsi-d*cosphi; % plotting results Figure; plot(theta,x,'-g') %DISPLACEMENT title('', 'fontsize',2) xlabel ('One Rotation (radians)', 'fontsize',16) ylabel('displacement (mm), V (mm/rad/s), \alpha (mm/(rad/s)^2)', 'fontsize',16) set(gca,'xtick',:pi/2:2*pi) set(gca,'xticklabel',{'', 'pi/2', 'pi', '3/2pi', '2pi'}) grid on; hold on; % Differentiation using finite difference method % we need some angular rate for input crank omega=1; %rad per second % sampling time dt=2*pi/omega/629; % velocity of slider crank output for i = 1:628 v(i)=(x(i+1)-x(i))/dt; v(629)=(x(1)-x(629))/dt; plot(theta,v,'--b') %VELOCITY %acceleration of slider crank output for i = 1:628 a(i)=(v(i+1)-v(i))/dt; a(629)=(v(1)-v(629))/dt; plot(theta,a,'-.k') %ACCELERATION leg('','velocity, mm/s','\alpha, mm/s^2',3) Page 5