The kinematical and dynamical investigation of the jaw crumb machine

 KAUNAS UNIVERSITY OF TECHNOLOGY DEPARTMENT OF ENGINEERING MECHANICS T210B446 study module course project THE KINEMATICAL AND DYNAMICAL INVESTIGATION OF THE JAW CRUMB MACHINE Done by: Ua3/1 gr. Stud. Petras Petraitis ________ ________ (signature) (date) Supervisor: assoc. Prof. Dr. Rymantas Tadas Toločka ________ ________ (signature) (date) Grade __________________ (in numbers and in words) Kaunas 2005 Content Introduction 4 1.2.Structural Analysis of Linkage 5 1.3.Mechanism synthesis 6 1.3.1Kinematic synthesis of linkage 6 1.3.2.Defining crank angular velocity 7 1.3.3.Defining Mechanism Performances 7 1.4.Preparing Data for Calculations by PC 9 Input Data for Four-Bar Mechanism 10 1.5.Dynamic Analysis and Synthesis of Machine 12 1.5.1.Objectives and Methodology 12 1.5.2.Description 13 1.6.Kinematic Analysis of Linkage 18 1.6.1. Objectives and Methodology 18 1.6.2 Description 20 1.7.Linkage Force Analysis 22 1.7.1 Objectives and Methodology 22 1.7.2.Description 24 References 27 appendices 28 1.2 Structural Analysis of Linkage The main mechanism of the machine under investigation is the planar linkage. It is a closed kinematic chain consisting of rigid links joined by revolute kinematic pairs. The structural analysis if this linkage is to be performed. It is the investigation of mechanism composition. 1. Number of degrees of freedom is determined as follows: W=3n-2p1, where n is the number of movable links and p1 is the number of kinematic pairs of one degree of freedom. W=3n-2p1=3*3-2*4=1 The obtained mobility shows the number of input parameters, which must be independently controlled in order to bring the mechanism into a particular position. 2. The number of redundant constrains in the linkage is determined as follows: q=W-6n+5p1=1-6*3+5*4=3 3. Also it’s necessary to decompose mechanism into structural groups and define the class of mechanism in Assur classification. The mechanism belongs to the second class as it contains two-armed group. O1B1= O1B2= ml= l1= l2= l0= 1.3 Mechanism synthesis 1.3.1 Kinematic synthesis of linkage The purpose of the kinematic synthesis of the linkage is to define the lengths of links in accordance with the needed mechanism performance. Quite often the given necessary stroke if the output link and the value of advance-to-return-time-ratio K are known. Graphical method: Given the necessary stroke of the output link and the value of advance-to-return-time-ratio K are known. The second quantity is the ratio of average velocities of the output link in its direct and return strokes. Expression of K in four-bar mechanism: K=, The stroke of output link is laid out and straight triangle OB’B’’ is drawn using the angle which may be found as follows = The hypotenuse of the triangle is to be divided into two equal parts and point O obtained in this way is to be used as a centre for drawing the circle. Every point of this circle can be used for the location of kinematic pair O1 of the synthessed mechanism when any point is chosen, the lengths of crank 1 (l1) and coupler 2 (l2) can be calculated using the equations, which are valid for the mechanism in the limit positions O1B1 and O1B2 are measured from the drawing and ml is the scale Analytical method: The following relationships may be used to perform the mechanism synthesis: 1.3.2 Defining crank angular velocity It is necessary to define the magnitude and direction of crank angular velocity if they are not given as assignment data. In my case n is given so angular velocity is calculated easily: . n=140 so 1=14.65 rad/s. 1.3.3 Defining Mechanism Performances The main performances of a mechanism are its output link stroke H or , advance-to-return-time K and maximum value of or pressure angle max. The mechanism performances must be established when the lengths of links are defined. They will differ from the given initial data if lengths of links were rounded of. ml= Necessary performances can be obtained following the diagram. The stroke of output link is where arcos 0.46874 = 62.43o and The value of stroke is acceptable. Ratio K with accordance to equation is K=, where =. Angles and may be defined using equations =arccos0.5814=54.40 and =arcos0.483=61.20 =61.2 - 54.4=6.8 so K ==1.079 (value is acceptable, as K=1.08 given) The maximum pressure angle value can be obtained using equation the value is acceptable, because 3 kgm2, then the steel flywheel parameters are D2=0.80.7=0.56 m b=0.2D1=0.70.2=0.14 6. Determine the crank angular velocity for the linkage i position for which velocity diagram is constructed where Ei=Eo+Ei; Eo=0.5(Irc+Irvo)210 is the kinetic energy in the beginning of the cycle and Ei=Wd(i)-Wl(i) is the value of kinetic energy exchange related to the investigated position. 10 is to be taken from PC data and Ei calculated using curve E(). The constant component of reduced mass moment of inertia is Irc=I’rc+If. Irv0=I’rc. Irv9=1.844 kgm2 Irc=I’rc+If=5.266+19.23=24.496 kgm2. Irv0=I’rc=5.266 kgm2 E0=0.5(Irc+Irvo)210 =0.524.49614.672=2635.8 J E9=2635.8+E=2809.9J 7. Compare all the calculated values with the ones given by PC: Parameter Mrd, Nm Irv9, kgm2 Mrl10, Nm If, kgm2 19, 1/s  PC 94.1 1.8440 -665.57 25 14.75 0.048 Calc. 87.87 1.8449 -705.52 19.23 14.65 0.0478 Error % 6.6 % 0.04% 5.6 % 23% 0.67% 0.4% 1.6. Kinematic Analysis of Linkage 1.6.1. Objectives and Methodology The aim of any mechanism kinematic analysis is to investigate the motion of links when the input link motion is given. The displacements, velocities, and accelerations of links and their points are considered. The motion of links depend on mechanism geometrical properties and given laws of motion of its input links. Let , ,  be the displacement, velocity, and acceleration of any point of any linkage link and , ,  are angular displacement, velocity and acceleration of rotating input link. Assuming the function s=s(). we obtain or and where t is time. Similar relationships can be written when the input link is translating or when it is necessary to define motion of rotating output link. Functions and others of the same origin depend only on the geometrical properties of the mechanism and are called position, the first and the second transfer functions. The first transfer functions were used for reduced parameters Ir and Mrl (Mrd) calculation when performing dynamic analysis of the machine. One can see now again that they are inherent to the mechanism i.e. not dependant on the velocities of links and their points. In the project it is necessary to carry out the kinematic investigation of the linkage, which is the main mechanism of the machine under study. The law of motion 1() of its input link-crank is obtained by dynamic investigation and the only problem now is to construct the necessary position and transfer function. The vector method is based upon vector polygons, which can be brought into correlation with any kinematic chain assuming its links as vectors. Position analysis is based upon a vector constrained equations formed by closed vector loops around the basic kinematic chain. The linkage in a figure below is given and it is necessary to perform its kinematic analysis. The closed vector loop equation is: l1+l2+l3=l0 The equations of vector projections on coordinate axes are used for defining position functions 3(1), 2(1), 1(1). For example, functions 3(1) and 2(1) can be defined using equations Transferring the terms containing 2 to the right-hand side of equations, squaring both sides, and adding equations we can eliminate angle 2 and get where Now we can get , The coordinates of any point of the closed contour mechanism under study can be defined as follows: . For the velocity and acceleration diagrams, the following equations and calculations were used (for the 9th position): Velocity plan: VA = w1 · loA=14.75 ·0.1=1.475 m/s =0; I choose velocity scale: mv = Measuring from the sketch, we obtain: VB = pVb · mv = 56 · 0.0295 = 1.652 m/s VBA = ab · mv = 26 ·0.0295 = 0.46 m/s VS2 = pS2 · mv= 50·0.0295 = 1.475 m/s Acceleration plan: I choose acceleration scale ma: 1.657 / 0.272 = 6.09 mm 0.32 / 0.272 = 1.177 mm 3.9 / 0.272 = 14.34 mm measuring from the velocity diagram we calculate w2 and w3 w2=Vba/l2=0.697 rad/s w3=Vb/l3=2.2 rad/s Measuring appropriate segments from the accelaration diagram we calculate ab, as2, ε2, ε3 : ] as2=pas2*ma=42 *0.1=4.2m/s2 ab=pab*ma=32 *0.1=3.2m/s2 Parameter vB, m/s vS2,, m/s 2, 1/s 3, 1/s aB,, m/s aS2,, m/s 2, 1/s 3, 1/s PC 1.652 1.47 0.69 2.15 6.88 8.70 13.95 -11.99 Polygons 1.652 1.475 0.69 2.2 6.2 8.70 12 -10.9 Error% 0 % 0.3 % 0 % 2.27 % 9.8 % 0 % 14 % 9.09 % 1.7. Linkage Force Analysis 1.7.1. Objectives and Methodology The linkage of the machine under study is moving under the action of the external loads: technological resistances, driving and gravity forces. Inertial loads are also assumed as external, when the force analysis is carried out using kinetostatic method. We shall focus on the determination of the reactions in its kinematic pairs and the calculation of the trimming moment Ml , which must be applied to the driving gear. The reaction in the kinematic pair is the force by which one link acts upon the joining link. It is used to denote the reaction as Fkl where the first index k indicates the link, which is under the action. and the second / indicates the link. which the force originates. Evidently Fkl= -Flk. The moment, which must be applied, to the arbitrarily chosen link to get the prescribed motion of the linkage input link is called the trimming moment Mr. Example. If the prescribed motion of the link 1 is 1=const then It is necessary to evaluate external load, weight of links and inertia forces, when performing force analysis. Analytically the force analysis will be carried out using equations, representing static equilibrium conditions: where the last equation is the sum of the moments of all the forces around arbitrarily chosen point P. These equations may be written for each separate link and for a group of them. It is necessary to remember that force analysis will be successful only if it is carried out on statically definite kinematic chains. The Assur groups are of such kind. Calculations are performed by PC using equations constructed in the coordinate system XOY. The pressures acting in the revolute kinematic pairs are assumed decomposed into tangential and normal components, the latter acting along the link. The reactions in the sliding pairs are directed as normally to the directions of relative motion of the links. The forces of weight are directed perpendicularly to axis Xo. Force analysis in any position may be performed by graphoanalytical methods as well. The following equations are used in this case where the first equation is as previous and the second is the vectorial sum of forces acting upon any link or a group of them (the vectorial polygon of forces must be closed). The linkage inertia loads and links weight forces can be determined as follows: weight – G=-mk*g: inertia forces - Fik=-mk*ask, inertia couple - Mik=-Isk*k, where k is the index of the link; mk and Isk are mass and mass inertia moments of k link; ask and k are the linear acceleration of the gravity centre of the link and the angular acceleration of the link. Every inertia force Fik passes the link centre Sk and is of opposite direction to its acceleration ask. The direction of inertia couple Mik is opposite to its angular acceleration k.. Link inertia load consisting of Mik and Fik can be replaced by inertia force F’ik only translating it from the mass centre by the distance hk=Mik/Fik in such a way that couple Fik,hk about the center of gravity would be of the same direction as Mik. 1.7.2. Description Equations and calculations for graph analytical force analysis of the linkage in the position N=9. Calculations: The latter equation lets to determine and graphically. For determination of we use the following eguation: h1=0.38(measured) Measuring approapraite segments from the sector diagrams and multiplying by the scale, we get the following values of forces: F10=92*20=1840N F12=185*10=1850N F23=189*10=1890N F30=203*10=2030N Compare results of graphoanalytical analysis with those, which are achieved by analysis method using PC. Parameter F10, kN F12, kN F23, kN F30, kN M1r, kN PC 1880 1885.91 1928.3 2084.5 -75.64 Calculated 1840 1850 1890 2030 -71.3 Error% 2.12 % 2 % 2 % 2.61 % 5.68 %

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