Kinematics of Piston Mechanism With the Use

III International Conference Industrial Engineering and Environmental Protection 2013 (IIZS 2013) October 30th, 2013, Zrenjanin, Serbia

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KINEMATICS OF PISTON MECHANISM WITH THE USE

ROTATING TRANSFORMATION MATRIX

Danilo Mikić1, Eleonora Desnica2, Aleksandar Ašonja1, Živoslav Adamović 2 1 The Serbian Academic Center, Novi Sad, Serbia

2 Technical faculty “Mihajlo Pupin”, Zrenjanin, Serbia e-mail: [email protected]

Abstract: The paper presents results of application of matrix transformations that can be applied to solve kinemtaike mechanisms of machines, as well as their application in a variety of cases in resolving the kinematics of mechanisms, such as for: determination of the vector segments in different coordinate systems, determination of the position vector of joints, and the focus points of segments, determining the angular velocity and the characteristic points of the segment, determining the angular acceleration of the segment, and the characteristic points. The work is part of the scientific work that is being implemented at the Technical University in Zrenjanin for the development of industrial systems and its results are suitable in solving mechanisms (reciprocating mechanism) and can be used for other purposes in the different branches. Key words: matrix transformations, kinematics, dynamics of motion, crankshaft, connecting rod, piston, piston mechanism. INTRODUCTION Piston mechanisms can be to present the system of the body (segments) that are interconnected rotary and translational relationships, moving under the action of forces and moments. The segments that are interconnected are the kinematic chain consisting of three members: the crankshaft, connecting rod and ukrsne head slider, whose purpose is to convert linear motion into rotary and vice versa. The application of this type of mechanism is large in practice, especially in motor vehicles with internal combustion engines [4]. In a reciprocating compressor, a model that is presented in the paper, a cylinder with a reciprocating charging the suction line and empty the water pressure at the top of the piston. When walking down the piston, the cylinder is filled through the inlet valve, and when moving in the opposite direction, compressed air (compressed) through the relief valve. Pan piston, piston compressor, through the mechanism of the crankshaft turns in the rotation (Fig. 1), [5]. To control mechanism, it is necessary to know the mechanics or its kinematics and dynamics. The problem consists in solving the mechanics of the relative motion of the system of the body, which, due to the complexity of the model and the large number of degrees of freedom of movement is quite complicated. Taking into account the real needs in the design of mechanisms, the problem can be generalized and simplified using matrix transformations. Given that so far there is no generally applicable method for this solution are listed here the results of some studies in this area carried out for the piston rotating models using matrix transformations. Special technique of mathematical modeling in application transformation matrix are determined by individual problems moving the piston, connecting rod and crankshaft [1]. GENERAL MATRIX ROTATING OF TRANSFORMATION The kinematics of mechanisms studied the geometry of its motion relative to the adopted fixed (absolute) coordinate system, without taking into account the forces and moments that cause the motion. Analysis of piston compressor is very complex and gives results strictly related to the specific geometry of the simulated machine. The purpose of this paper is to describe a simple model is easily adaptable to different piston mechanisms. Therefore, the general model of the compressor consists of several sub-models which are related to the different elements that influence its behavior [6], [7], [8]. Scheme of the system resulting from such categorization is presented in Figure 1.

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Figure 1. Components of the piston compressor

Components of a reciprocating compressor crankshaft and connecting rods are modeled as rigid bodies, piston displacement is modeled as a material point and the main shaft is modeled as an elastic body. To determine the kinetic characteristics of segments compressors it is necessary to know the number of segments, the coordinates of their centers of mass and how their mutual connection, it is of kinematic pairs-joints. As the basic kinematic variables analyzed issues of the position, velocity and acceleration segments mechanism. On the basis of this analysis, there is a closed condition vector outline, reflecting the geometric relationships of the movable joints segments. For example, to schedule a piston mechanism shown in Fig. 1 (crankshafts and connecting rods), defined to be stationary and moving coordinate systems. To define a general matrix is necessary to observe the inertial reference coordinate system IXYZ and identified four reference position system (Figure 2). Inertial reference coordinate system is set up for the center bearing (point A), while the observed reference positions B1, B2 and B3 connected to the crankshaft and the moving reference system B4 is related to the piston rod. System B1 (X1Y1Z1) is obtained by rotating, the angle β about the axis X; system B2 (X2Y2Z2) is obtained by rotating B1 angle γ around the axis Y1; system B3 (X3Y3Z3) is obtained by rotating the B2, the angle θ around the axis of Z2, and system B4 (X4Y4Z4) is obtained by rotating and I angle α around the Z axis, [2], [13].

Figure 2. Kinematic mechanism scheme and vectors

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Where: I- refers to the observed position of the inertial reference system X, Y - refers to the X and Y A, B, C - refer to points A, B, C, respectively Bi - refers to the i-drive system reference ω - speed rotation of the crankshaft, [rad / s] θ - angle of rotation of the knee on the Z-Z axis [rad] α - angle rotating rod, [rad] β - angle rotation around the X-X axis [rad] γ - angle rotation around Y1 Y1-axis [rad] φ - angle position [rad]. Transformation matrix moving coordinate system, relative to fixed, determined by taking into account the possible movements of the segments with respect to one another, that is types of kinematic pairs of entering given segment (taking into account the mobility of the kinematic pairs). Thus, the appearance of each matrix is determined by the choice of the coordinate system, placing and mutual orientation of the axes in space, so that a selected axis can be changed in further analyzes and transformations. Discuss some features of the matrix form of records in the process of converting vector coordinates given in different coordinate systems. Some were given two rectangular coordinate systems with a common beginning [09], [10]. In reciprocating mechanisms are often moving segments drawn so that their rotation is not performed simultaneously around all three axes. More often, these developments made gradually, and some structural mechanisms kinematic couples allow one or two rotation. Therefore, the practical importance of a rotation matrix with respect to a separate axis. Figure 3 presents the coordinate systems that are capable of uniaxial rotation around the x axis by angle β, y axis by angle γ , z axis and the angle θ. Then we get the rotational transformation matrix.

a) b) c) d) e)

Figure 3. The uniaxial rotation of the coordinate system: а) Immovable {N} and moving {P} coordinate system, b), c), d ), e) rotation about the axis of the fixed angle

Transformation matrix are determined using the equation: Transformation matrix of the inertial coordinate system I the drive moving system for the case B1

rotation around Ox-axis for angle β, as shown in Fig. 3 has the form: 1 0 00 cos sin0 sin cos

Tβ β ββ β

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦

( )1

1 0 0; 0 cos sin

0 sin cosxN

BT β β ββ β

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦

(1)

The transformation matrix of a stationary inertial coordinate system into a mobile system B1 to B2 for the case of rotation about Oy-axis for angle γ, as shown in Fig. 3 has the form:

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cos 0 sin0 1 0

sin 0 cos

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Tγ

γ γ

γ γ ( )1

2

0 sin; 0 1 0

sin 0 cos

cosgyB

BTγ

γγ γ

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(2)

The transformation matrix of a stationary coordinate system B2 a mobile system for case B3 rotation around the Oz - axis by angle θ, as shown in Fig. 3 has the form:

cos sin 0sin cos 0

0 0 0

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

Tθ

θ θθ θ ( )2

3

cos sin 0; sin cos

0 0 0z 0

⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

BBT

θ θθ θ θ (3)

The transformation matrix of the inertial coordinate system moving system B4 for the case of rotation around Ox-axis by angle α, as shown in Fig. 3 has the form:

cos sin 0sin cos 0

0 0Tα

α αα α

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦1

01

( )4

cos sin 0; sin cos

0 0x

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

NBT

α αα α α (4)

Rotational transformation matrix around the axis of the B1 to N, B2 to B1, B3 to B2, B4 in N, according to Figure 2, written in the form:

( )( )

( )( )

( )( )

( )( )

1

1 1 22 1

2 3 43 2 4

; ;

; ;

x ;β , y;γ

z;θ , x;α

T TN B B BB N B Bx y

TB B N BB B B Nz x

T T T T

T T T TT

β γ

θ α

⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦

(5)

where the sign [ ]T denotes matrix transposition. Rotational transformation matrix given (1) - (4) are a rotational transformation matrix. For the case when the segments between two segments allow two or three rotations, then the total rotational transformation matrix can be obtained by multiplying the elementary matrix in a specific order. Determining the position vector and the equations connections Equations connections are established in only one case that has accentuated the work. In Fig. 2 shows the vector diagram that illustrates how they relate to the main parts of the system considered. Equation connections and the position vector for analysis of crankshaft and connecting rods is obtained from equation (6), [2], [3], [11], [12], [13], [14]. Figure 4 is a schematic representation of the mechanism with the parameters required for the calculation of the other components.

Figure 4. Kinematic mechanism with rotating segments

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Case I: Ignoring the lateral displacements and crankshaft angle oscillation is given by: I I Ipx l+ = r (6) where,

3

TI Br T rθ= ⋅ ; { }

30

T

B c pr -hr = ; ; { }0 TI -lcos lsinl α α= { }0

T

I p B px -hx = hp - the length of crankshaft rotation, [mm]. l - length of the connecting rod, [mm], r - radius, [mm], I - relative to the inertial reference frame XYZ I - moment of inertia tensor.

Kinematic relations determining the velocity and acceleration Angular velocity of each of the moving reference system can be written as in Equation (7).

{ }TI 0=β β& 0 { }1 0 0 TB =γ γ& &; ; { }2 0 0T

B =θ θ& & ; {0 0 TI = − }α α& & (7)

For a given case, the absolute angular velocity of the crankshaft is derived from ω θ= & , since the β& and γ& equal to zero. Thus, in these cases the moving reference frame B3 will be simply obtained by rotating I, the angle θ, around the Z axis. Expressions for calculating the velocity and acceleration of piston compressors ( Bx& , Bx&& ) and connecting rod compressors (α& , α&& ) is obtained by differentiating the equation connections in each case respectively, Equations (8-9), [1], [2], [12]. Velocity: 10

B C

C

l sin x r sinl cos r cos

α θ θα α θ θ

&&&&

ì üé ùì ü ï ï-ï ïï ï ï ïê ú =í ý í ýê úï ï ï ïï ïë ûî þ ï ïî þ (8)

Accelerations: ( )( )

2 2

2 2

10

CB

C

sin cos l cosrl sin xrl cos cos sin l sin

θ θ θ θ α ααα α θ θ θ θ α α

&& & &&&

&& && & &

ì üï ï- -é ùì ü ï ï-ï ïï ï ï ïê ú =í ý í ýê úï ï ï ï- +ï ïë ûî þ ï ïï ïî þ (9)

Equations of motion of the crankshaft and connecting rod compressors for the case is obtained from Equation (10-14). As the piston is not subject to the current analysis, it should be noted that the friction forces are not included in the modeling of the piston, [3], [12], [14]. Compressor crankshaft:

- The equation of force crankshaft { , ,0 T

I C I C I A C C Cm m xf = a fΣ ⋅ ⇒ = && && }y (10) - Moment of crankshaft

( ) ( )3 3 3 3 3 3 3 3 3 3 3B C B B A B B C B B B C B C B C cm B Cd mdt

M r f τ I ω ω I ω r −Σ = × + = ⋅ + × ⋅ + ⋅ × a (11)

where,

3 A IB f T T T fθ γ β= ⋅ ⋅ ⋅ A { }3

0,0,t TB z=τ; ; { }

3,0,0 T

B C cm cer − = Compressor piston rod:

- Compulsory equations of force connecting rod I cr I cr I A I Bmf a f fΣ = ⋅ = + (12)

where,

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( )( )

2

2

cos sin

cos sin0

B cr

I cr I B I I I cr I I cr cr

x r

ra a α α r α r

α α α α

α α α α

⎧ ⎫+ +⎪ ⎪⎪ ⎪= + × × + × = −⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

& &&&&

& & && && & ;

- A torque connecting rod equation

( ) ( )4 4 4 4 4 4 4 4 4 4B B B B A B cr B B B cr B cr B cr B Bd mdt

M l f I α α I α rΣ = × = ⋅ + × ⋅ + ⋅ ×& & & a (13)

where:

4 A IB f T fα= ⋅ A 4; B B Ia T aα= ⋅ B ; { },0,0I B Bx Ta = &&

Equation of piston: - Forced piston equation

I B p I B I B I N I Pmf a f f f= ⋅ = + +∑ (14) where:

{ },0,0T

I P g pP Af = . In each case the equation of motion can be written in matrix form as in Equation (15), where vector b contains the main unknown (that is. Reaction force, torque and acceleration response). Where:

{ }, , , , , , , , , , , ,x y z x y z z

T

B B B y z A A A C B C Cf f f N N f f f f θ x x , yb α= && &&&& && && . (15)

For each case the equations of motion may be written in a matrix form as in Eq. (16), where the vector b contains the main unknowns (i.e., reaction forces, reaction moments and accelerations). A b c⋅ = (16) RESULTS AND DISCUSSION Equations 1-4 show the matrix is 3 × 3 and the interconnection between the three variables. Calculation of the correlation matrix is the initial step in any factor analysis. It is obtained by matrix product more compact standardized data matrix. Its elements include cosines of angles between all possible elements within a standardized column vectors. It is possible to display the geometric relationships between the vectors whose the cosine correlation presented. For example, in Equation 1-4 show the three vectors corresponding correlation matrix. Vectors of species data matrices are stored in three-dimensional space, based on equations representing the segment specific application presented model and Matlab software. A simple review of the real mechanism that serves as the basis for calculating the kinematic scheme. Kinematic scheme piston mechanism, made from actual mechanism. This paper presents a general model of a reciprocating compressor with a cylinder that is used in the industry. This describes the basic approach and the advantages and disadvantages of the presented solutions. The results can be compared, if they used the same input parameters [16]. CONCLUSION To determine the kinetic characteristics of piston mechanism with rotating joints very successfully be used rotational transformation matrix whose theoretical basis and method of calculation presented in this paper. Included in the calculation of the position of certain characteristic points, velocity and acceleration. The above calculation methodology is suitable for computer use, because the kinematic calculations amount to solving matrix equations. The convenience of these equations is that it is propelled movement of the internal structure can be assigned in any form, which enables the use of a realistic model of the motion segment of kinematic chain, ie. complete model of engine.

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REFERENCES

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[13] Dufour, R., Der Hagopian, J. and Lalanne, M., 1995, “Transient and Steady State Dynamic Behaviour of Single Cylinder Compressors: Prediction and Experiments”, Journal of Sound and Vibration, Vol.181, pp. 23-41.

[14] P. R. G. Kurka1, Karen L. G. Paulino2, Jaime H. Izuka, Dynamic Modeling of Reciprocating Compressors with Vertical Axis, 1Faculty of Mechanical Engineering, University of Campinas (UNICAMP) P.O. Box 6122, 13083-970 Campinas, SP, Brazil, 2010.

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Kinematics of Piston Mechanism With the Use