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Federico Verna: [email protected] Alessandro Cataldi: [email protected]
19/07/2014
Project 2
Professor Victor H. Mucino Kinematics & Dynamics of machines
Engineering Sciences bachelor degree
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Table of contents Abstract ....................................................................................................................................................... 2
Introduction ................................................................................................................................................. 2
Calculation .................................................................................................................................................. 3
Data set ........................................................................................................................................................ 4
Theoretical Background .............................................................................................................................. 5
Reciprocating Mass ..................................................................................................................................... 6
Force analysis .............................................................................................................................................. 6
Comparison 5 Cylinder Engine ..................................................................................................................... 9
Appendices ................................................................................................................................................ 10
Conclusion .................................................................................................................................................. 13
References ................................................................................................................................................. 14
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Abstract. The selected 1 cylinder IC engine is designed and produced by KTM. It has been developed for KTM 690 Duke. The aim of the following report is to conduct a study of the forces, velocities and accelerations of a radial 5 cylinder mechanism and to make a comparison with a standard S C mechanism.
Introduction
The cylinder used for this analysis comes from the engine of the motorbike above. Since the bore is greater than the stroke, the engine is called over squared. These engines are very performing and they are used a lot in the competitions and the production of powerful cars and motorbikes. They can guarantee high RPM (and this means more power) without increasing too much the speed of the piston. The cylinder is considered rotating at 3000 rpm anticlockwise. In the design activity radial self-aligning ball bearings were chosen in order to ensure an high speed of rotation. They are produced from SKF, the specifications can be found in the Appendices section. The crank length and the disk radius were calculated from the geometry using both operation and a CAD program. Specifications are given in the Calculations section. The conrod length is calculated from the stroke. In this kind of bikes usually the Conrod/stroke = 1.66. The disk thickness comes from the bearing thickness. We made it 4 mm bigger to avoid a misalignment during rotation. Since the disk rotation speed chosen is 3000 rpm ( about 314 rad/s), and since we need a constant contact between the disk and the bearing, we need to put a spring on the top of the cylinder, that will exert a force of the same magnitude but opposite in direction than the one exerted on the bearing by the rotating disk. The calculations for the length and elastic constant are given in the Calculations and Data Set sections. The cylinder, the crank, the disk and the connecting are supposed to have a uniform distribution of mass, so the center of mass will be always considered to be at the geometric center. In this study we are not considering any kind of friction.
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Calculations
Length of the crank: Taking as reference the situation in which the piston lying on the x axis is at the top dead center and bottom dead center we have:
TDC= Lc + Rd + RBEARING +LCR + hPISTON
BDC = (Rd - Lc) + RBEARING +LCR + hPISTON
Calculating the difference between them:
TDC BDC = 2Lc
Where TDC BDC is the stroke, Lc is the length of the crank, Rd and RBEARING are the radii of the disk and bearing, LB is the length of the conrod and hPISTON is the hight of the piston. Elastic constant of the spring: In order to guarantee the contact anytime between the disk and the bearing we can put a spring in opposition to the centrifugal force, which has an elastic constant given from the equation:
Fc = FE Fc= -kx
Since the x is the length of the spring when it is completely compressed, we have that k is:
Fc /x = -k
Torque: At the rotating speed chosen the engine performs 3000 rotations per minute, and erogates 20 hp of power (P). As we can see from the power-torque graph below.
It follows that the corresponding torque is found from:
602
PTRPMp
=
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Data Set
Object Length [mm] Thickness [mm] Material Density kg*m3
Mass [Kg]
Crank 42,25 10 AISI 304 7,93 0,033 Disk 70 (radius) 16 AISI 304 7,93 1,95
Bearing 20 12 0,065 Connecting
rod 140,27 14 AISI 304 7,93 0,15
Piston Stroke: 84,5 Length 26,64
Bore: 102 Alumium 2,7 0,59
Spring (k=640kN)
11,44 (max compression)
/ / / /
Representation of the mechanism in scale made with AutoCAD 2006
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Theoretical Background
Equilibrium
Standard Newtonian equilibrium equations apply throughout the analysis with respect to the each axis (x,y) as follows : 0
0 Doing the summation of the forces we need to sum the forces acting both on X and Y direction for each single link in each configuration. Each force has an equal force acting on the opposite direction.
Above we can see the equivalent mechanism for each cylinder, and below the equivalent configuration for the entire mechanism.
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Reciprocating & Rotating Mass Computation
The masses of the various components of the engine cause the presence of a shaking force. There are two masses to balance: the rotating mass and the reciprocating mass.
Reciprocating mass = mass piston+ mass conrod = 0.59 + 0.15 = 0.74 kg Rotating mass = mass disk + mass crankshaft = 1.95 + 0.033 = 1.983 kg
The rotating mass is the summation of the mass of the crankshaft and the mass of the disk, while the reciprocating mass is composed by the mass of the connecting rod and the piston. This sharp distinction can be operated because the bearing permits a separation of the two parts. In a classical slider-crank engine half of the mass of the connecting rod contributes to the rotating mass and the other half to the reciprocating mass. The equation of the shaking force is the follow:
2( ) [cos( ) cos(2 )]2rotating reciprocatingRF M M R t tL
w w w= + +
Where W is the angular velocity, R is the length of the crank and L is the length of the connecting rod. In order to balance this force a counterweight can be used. Due to the composition of the engine the rotating mass can be viewed as it is at the end of the crankshaft and the reciprocating as it is on the piston. They can be balanced adding a counterweight at a distance equal to the length of the cranckshaft in the direction opposite to it. The mass of the counterweight will be equal to the sum of the rotating and reciprocating mass.
It can be seen that the acceleration is composed by two harmonics 2 cos( )R tw w and 2
cos(2 )2
R tL
w w .
The counterweight balances efficaciously both the harmonics for the rotating mass, while only the first for the reciprocating mass. Force Analysis
The force analysis was conducted with a Matlab program and also examining one by one the links and the cylinders. Matlab was used also to compute the standard S-C force analysis. The scripts are both in the Appendices section. Following there are the equivalent mechanisms used for the cylinders.
Cylinder 1
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Cylinder 2
Cylinder 3
Cylinder 4
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Cylinder 5
Using this sketches, performing the analysis (for example) for the cylinder number 2, we get the following equation. Anticlockwise direction of momentum is taken positive.
Link 2 00
42,5 0
x Ox Ax
y Oy Ay
in Ay
F F FF F FM T F
S = - - =
S = - - =
S = + + =
Where 42,5 is the length of the crank
Link 3
00
(90, 45 cos82 ) (90, 45 8 ) 0
x Ax Dx
y Ay Dy
Dy Dx
F F FF F FM F F sen
S = - + =
S = - + =
S = + + =
Where 90,45xcos82 is the projection along Y axis and 90,45xsen8 is the projection along X axis.
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Link 4
00
28cos 28
x Dx Ex elastx
y Dy Ey elasty
elastx elast
elasty elast
F F F FF F F F
F F senF F
S = - + - =S = - + - =
= -
= -
Where 90,45 mm (to be transformed in meter) is the length of the bearings radius + disk radius, and the angles are found from 90-72, since the piston axis is at 72from the first one.
Comparison with a 5 cylinder engine
The mechanism chosen is a 5 in-line cylinder IC engine from Audi TT RS 2.5TFSI. It is a very powerful engine, it is able to erogate the maximum torque (about 480 Nm) at only 1600 rpm. Here follows a comparison of the geometries of the two engine Characteristic Advantages Disadvantages
KTM 690 Duke Radial
mechanism
Bore:102 mm Stroke:84,5 mm Conrod length:140,27mm Conrod bearing diameter:40mm Unitary cylinder size:690cm3 Power: 49 kW Compression Ratio: 12,6:1
-It is an over squared engine, so it reaches the max power at higher rpm
-Radial mechanism are very difficult to place due to the dimensions -Higher compression ratio can cause engine knocking (in gasoline engines)
Audi TT RS 2.5 TFSI
S-C inline
Bore: 82,5 mm Stroke:92,8 mm Conrod length: 144 mm Conrod bearing diameter: 47mm Unitary cylinder size:500cm3 Power: 250kW, 50 for each cylinder Compression Ratio: 10:1
-It reaches the maximum torque at lower rpm -they have almost the same cylinder dimension, but -Lower friction -Less vibrations due to a more
In addition, we can say also that a radial mechanism is very difficult to place due to a very big size, and vibrations created are difficult to manage.
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Appendices
SKF Bearing, 6203 series. Width:12mm Bore diameter: 17mm Outside diameter 40 mm
Matlab program
Selecting the length of crank(r2), disk radius(r3) and conrod(r4), and entering the number of the piston chosen, we can start our analysis.
%project 2 function project2(r2, r3, r4) disp 'Select the cylinder on which you would see the analysis perfomed' cylsel = input('chose now the cylinder'); if cylsel == 1 %thetaX in radiants theta2 = 0 theta3 = 0 theta4 = 0 F3ix = -9876.75 F3iy = 0 F4ix = -759.75 F4iy = 0 end if cylsel == 2 %thetaX in radiants theta2 = 0 theta3 = 1.71 %98 theta4 = 1.26 %72 F3ix = -1374.6
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F3iy = -9780.6 F4ix = -234.77 F4iy = -722.6 end if cylsel == 3 %thetaX in radiants theta2 = 0 theta3 = 2.79 %160 theta4 = 2.51 %144 F3ix = -9281.1 F3iy = -3378 F4ix = 614.7 F4iy = -446.6 end if cylsel == 4 %thetaX in radiants theta2 = 0 theta3 = 3.48 %200 theta4 = 3.77 %216 F3ix = -9281.1 F3iy = 3378 F4ix = 614.7 F4iy = 446.6 end if cylsel == 5 %thetaX in radiants theta2 = 0 theta3 = 4.57 %262 theta4 = 5.03 %288 F3ix = - 1374.6 F3iy = 9780.6 F4ix = - 234.8 F4iy = 722.6 end %A is the matrix of forces and momenti %r3 = disk radius %r2 = crank %r4 = conrod length A=[-1, 0, 1, 0, 0, 0, 0, 0, 0; 0,-1, 0, 1, 0, 0, 0, 0, 0; 0, 0, -r2*sin(theta2), r2*cos(theta2), 0, 0, 0, 0, 0; 0, 0,-1, 0, 1, 0, 0, 0, 0; 0, 0, 0,-1, 0, 1, 0, 0, 0; 0, 0, 0, 0, -r3*sin(theta3), r3*cos(theta3), 0, 0, -1; 0, 0, 0, 0,-1, 0, 1, 0, 0; 0, 0, 0, 0, 0,-1, 0, 1, 0;
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0, 0, 0, 0, 0, 0, -r4*sin(theta4), r4*cos(theta4), 0] % B = [F1x; % F1y; % F2x; % F2y; % F3x; % F3y; % F4x; % F4y; % F5x; % F5y; % Tout] C= [143.74 ; 0; -46; -F3ix; -F3iy; 0; -F4ix; -F4iy; 0] B = pinv(A)*C end
Force analysis of the S-C mechanism
function AnalysiSliderCrank(r2,r3,theta2,theta3,F2i,F3i,F4i,M2i,M3i,Tin) A = [ -1, 0, 1, 0, 0, 0, 0, 0, 0; 0,-1, 0, 1, 0, 0, 0, 0, 0; 0, 0, -r2*sin(theta2), r2*cos(theta2), 0, 0, 0, 0, 0; 0, 0,-1, 0, 1, 0, 0, 0, 0; 0, 0, 0,-1, 0, 1, 0, 0, 0; 0, 0, 0, 0, -r3*sin(theta3), r3*cos(theta3), 0, 0, -1; 0, 0, 0, 0,-1, 0, 1, 0, 0; 0, 0, 0, 0, 0,-1, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0] % B = [F1x; % F1y; % F2x; % F2y; % F3x; % F3y; % Fs; % Fn; % Tout]
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C= [-F2i*cos(theta2); -F2i*sin(theta2); Tin-Mi2; -F3i*cos(tehta3); -F3i*sin(theta3); Mi3; F4i; 0; 0] B = pinv(A)*C end
Conclusion
This kind of slider crank mechanism is more complex if compared to a normal one (inline or radial) and also less efficient. Even if the piston motion is more linear, varying the rpm we will vary also the centrifugal force acting on the bearing-conrod-piston elements, but since the value of the elastic constant of the spring is fixed, we will have a spring not in condition to balance the force, this will not guarantee the constant contact between disk and bearings. Radial mechanism are also less common for the space they require. A possible improvement of this mechanism could be done substituting a cam rotating about the center to the crank-disk. This will ensure a more constant raise of the piston. Below there is an approximate picture for the cam.
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References
Drawings: Made using AutoCAD 2006, CamShaft, Matlab
Bearing Dimensions
http://www.skf.com/group/system/
http://www.skf.com/it/products/bearings-units-housings/ball-bearings/deep-groove-ball-bearings/single-row/index.html?prodid=1050010203&imperial=false
Density of Al for the piston
http://raffmetal.it/pdf/schede-leghe/en/EN-48000.pdf
Power Torque graph
http://missmotorcycle.files.wordpress.com/2013/12/2013-ktm-690-duke-hp-torque-dyno.jpg?w=529
Power Torque graph Audi TT
http://www.jlosee.com/images/TTRS/PDF/TT%20RS%20w%202500cc%205%20cylinder%20TFSI.pdf
KTM Engine Picture
http://www.twowheelmania.com/wp-content/uploads/2012/10/2012-KTM-690-Duke-Engine-Photo_Mitterbauer_H1.jpg
Reciprocating & Rotating Mass Computation
http://worldtracker.org/media/library/Physics/Theory%20of%20Machines/ch-22.pdf
