(
- angle between banks and links BD, BC and BF, FG
- diameter of the pistons (in meters)
- permanent frequency of rotation the crank AB (in min
Figure 2.1. The mechanism of a W-engine
The origin A of the coordinate system Axy and the banks are connected to the ground link.
Let us analyze below
(1)
The coordinate x
(2)
.
The stroke of the middle piston C is
The coordinates of pivots D and F can be find out form equations of constraints
The equations (3) are nonlinear relative to the coordinates of the pivots D and F. For purpose of solving equations (3) by Mathcad solve block the approximate initial values for these coordinates must be find out. One can determine these values from Fig 2.1
The specified values of coordinates the pivots D and F determines the following two solve blocks
(4)
(5)
The following approximate initial values coordinates of the pivots E and F one can find from Fig 2.1
The specified values of coordinates the pivots E and G determines the following two solve blocks
.
(7)
Whereas the pivot A is at the origin of considered coordinate system (Fig. 2.1) and the pivot C on the y-axis then
For determination the diagram of the mechanism positions the following elements of matrixes X, Y, X', Y', X'', Y'' must be defined
Let us image graphically every
which allow to decrease the dimensions of the graphical representation the diagram of the mechanism's positions in Fig. 2.2
Fig. 2.2. The diagram of positions the mechanism of W-engine
The distance between pivots A and E and A and G is
Then the displacements of the right pivot E and left pivot G one can determine by formulae
The maximum numerical values of the strokes of the pistons C, E, G are (in meters)
Thus the strokes of the pistons are equal on high accuracy
The angular velocity of the crank AB can be determined by the formula
By differentiation the formulae (1) and (2) on time one can get the following formulae for determination of the velocities the pivots B and C
Linear system of equations relative to the velocities
(9)
Here and below the arrow above the formula indicate the vectorizing in the Mathcad environment
The system (9) has the solution
Linear system of equations relative to the velocities
(10)
The system (10) has the solution
Linear system of equations relative to the velocities
The system (11) has the solution
Linear system of equations relative to the velocities
(12)
The system (12) has the solution
The projections of the velocities the left piston (pivot E) and the right piston (pivot G) on vectors
.
By differentiation the formulae (8) on time one can get the following formulae for determination of the velocities the pivots B and C
Linear system of equations relative to the accelerations
where
The system (13) has the solution
Linear system of equations relative to the velocities
where
The system (14) has the solution
Linear system of equations relative to the velocities
(15)
where
The system (15) has the solution
Linear system of equations relative to the velocities
where
The system (16) has the solution
The projections of the accelerations the left piston (pivot E) and the right piston (pivot G) on vectors
The coordinates of the centers of mass S
By differentiation these formulae twice on time the following formulae for determination of the accelerations the centers of mass can be find out
The modules of these accelerations are
From Fig. 1 one can find that
(17)
where
(18)
By differentiation the formulae (18) one can get the following formulae for determination the angular accelerations of the links 2, 4, 6.
Fig. 2.4.
Fig. 2.5.
Fig. 2.6.
links 2, 4, 6
Fig. 2.7.
Fig. 2.8.
links 2, 4, 6
After determination the accelerations of the link's centers of mass there is possible to find out the dynamical reactions in the pivots of the mechanism of W-engine. Let the masses the links are
According to the handbook of Lepikson (1968 - 1971) the moments of inertia can be specified as
Let us suppose that the pistons are loaded by pressure according to the typical indicatordiagram
given numerically by the following vectors
The graphical illustration of these vectors is
Fig. 3.1. The numerical values of the indicator diagram
In Fig. 3.1 the subscripts indicate the following cycles: 1 - intake, 2 - compression, 3 - power,
The scale factor is
Let
For purpose of using the values of the pressures and displacements between given discrete values on indikatordiagram let us make the linear interpolation of these discrete values
Fig. 3.2. The interpolated indikatordiagram. trace 1 - intake, trace 2 - compression,
The analysis of strokes of the left, middle and right pistons one can conclude, that there is several possibilities of cyclograms. One of these possibilities is shown below
The dependencies of the pressure, applied to the pistons, on the angle of rotation
These formulae are composed on the base of the considered indikatordiagram and cyclogram.
The forces, applied to the pistons, are specified by the following formulae
.
.
Let us compose the equations of motion according to the following diagrams of dynamic reactions.
Fig. 3.3. Diagrams of dynamic reactions
According to the diagrams in Fig. 3.3 one can compose the following equations of motion
for links 2 and 3, equations
for links 4 and 5, equations
for links 6 and 7 and equations
for link 1.
The equations (19) - (21) have the solutions
The equation (22) gives
.
.
The power of the cylinders E. C, G one can find out by following formulae
,
Fig. 3.4. Dependence of the pressures P
on angle of rotation
Fig. 3.5. Dependence of the powers W
on angle of rotation
Fig. 3.6. Dependence of the dynamic reactions R
Fig. 3.7. Dependence of the dynamic reactions R
Fig. 3.8. Dependence of the dynamic reactions R
Fig. 3.9. Dependence of the dynamic reactions R
Fig. 3.10. Dependence of the dynamic reactions R
Fig. 3.11. Dependence of the dynamic reactions R
Fig. 3.12. Dependence of the dynamic reactions R
Fig. 3.13. Dependence of the source torque, which must be delivered from the work machine (ground) to the crank AB, on the angle of rotation
Fig. 3.14. Hodographs of the dynamic reactions in the pivots D, D, F
Fig. 3.15. Hodographs of the dynamic reactions in the pivots C, E, G
1. Heinloo, M., Mägi, M. Mathcad-Aided Analysis of a Bar Mechanism, Mathcad Files, Civil and
2. Heinloo, M, Aarend, E, Mägi, M. The Experience on Mathcad-Aided Analysis of Planar
3. Norton, R. L. Design of Machinery. An Introduction to the Synthesis and Analysis of
4. Lepikson, H. Handbook for Mechanical Engineers, Tallinn, 1968, 1971, 820 p.