Accepted Manuscript Dynamics calculation with variable mass of mountain self-propelled chassis R.M. Makharoblidze, I.M. Lagvilava, B.B. Basilashvili, R.M. Khazhomia PII:
S1512-1887(16)30106-3
DOI:
10.1016/j.aasci.2016.10.006
Reference:
AASCI 70
To appear in:
Annals of Agrarian Sciences
Received Date: 20 July 2016 Accepted Date: 10 October 2016
Please cite this article as: R.M. Makharoblidze, I.M. Lagvilava, B.B. Basilashvili, R.M. Khazhomia, Dynamics calculation with variable mass of mountain self-propelled chassis, Annals of Agrarian Sciences (2016), doi: 10.1016/j.aasci.2016.10.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Annals of Agrarian Science vol. 14, no. 4, 2016
Dynamics calculation with variable mass of mountain self-propelled chassis
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Corresponding author: M.R. Makharoblidze,
[email protected] Agricultural University of Georgia
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R.M. Makharoblidze, I.M. Lagvilava, B.B. Basilashvili, R.M. Khazhomia Agricultural University of Georgia 240, David Aghmashenebeli Ave.,Tbilisi, 0131 Georgia
[email protected];
[email protected];
[email protected]:
[email protected] Received: 20 July 2016; Accepted: 10 October, 2016
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Abstract Many technological processes in the field of agricultural production mechanization, such as a grain crop, planting root-tuber fruits, fertilizing, spraying and dusting, pressing feed materials, harvesting of various cultures, etc. are performed by the machine-tractor units with variable mass of links or processed media and materials. In recent years, are also developing the systems of automatic control, adjusting and control of technological processes and working members in agriculture production. Is studied the dynamics of transition processes of mountain selfpropelled chassis with variable mass at real change disconnect or joining masses that is most often used in the function of movement ( m(t ) = ct ) m(t ) = ct . Are derived the formulas of change
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of velocity of movement on displacement of unit and is defined the dependence of this velocity on the tractor and technological machine performance, with taking into account the gradual increase or removing of agricultural materials masses. According to the equation is possible to define a linear movement of machine-tractor unit. According to the obtained expressions we can define the basic operating parameters of machine-tractor unit with variable mass.The results of research would be applied at definition of characteristics of units, at development of new agricultural tractors. . Keywords: Dynamics, Unit, Variable mass, Resistance force, Driving force, Nominal torque. -------------------------------------------------------------------------------------------------------------------1. Introduction Academician V.P. Goryachkin, developing the problems of agricultural machinery theory, has been formulated the mechanics of working environment as the mechanics of bodies and variable mass media, but the relevant mathematical apparatus is not proposed and is not developed by them. Later, the problem of the dynamics of mechanisms with variable masses of 1
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units and processed medium is beginning to occupy an important place in the modern theory of machines. [1, 2]. Many technological processes in the field of agricultural production mechanization, such as a grain crop, planting root-tuber fruits, fertilizing, spraying and dusting, pressing feed materials, harvesting of various cultures, etc. are performed by the machine-tractor units with variable mass of links or processed media and materials. In recent years, are also developing the systems of automatic control, adjusting and control of technological processes and working members in agriculture production. The separate parts of this system have variable mass or moment of inertia that have a significant impact on the dynamics and stability of the system as a whole. Thus, issues of mechanics of bodies of variable mass for agricultural machinery are very relevant, and in spite of this, to the development of engineering methods of dynamic processes calculating in agricultural machinery has been given short shrift. While in the monograph [3] were developed issues of optimization method of dynamic processes in some agricultural machines with variable reduced inertia moment by the variable gear ratios. In short is studied the dynamics of agricultural machinery, where the variability of the mass, moment of inertia and center of gravity is achieved by physical separation or addition of mass at movement of machinery. 2. Basic part
At real change of disconnecting or joining masses, which is often accepted as a function of time t, the equation of motion would be written down by momentum equation: [4, 5, 6]
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d m ( t ) ( v - v* ) = Pg − Pc ,
(1)
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where m ( t ) –is the reduced mass of unit: v – is the speed of executing link; v* - is the velocity of elementary mass connection to the basic mass; Pg -is the driving force;
Pc - is the resistance force.
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Driving force Pg let’s represented due the mechanical characteristic of the internal combustion engine in the area of characteristic adjusting branch: M g = A − Bω
( 2)
The coefficients A and B are determined from the mechanical characteristics curves: A=
M maxωH − M H ωM M − MH ; B = max , ωH − ωM ω H − ωM
(3)
where M max - is the maximum torque of engine; M H - is the nominal torque;
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rk - Is the rolling radius of the drive wheels.
( 4)
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ωM , ωH – are the angular velocities of engine crankshaft at M max and M H . We obtain that: [7, 8] M g iη m A − Bω iη iη Pg = = iηm = A m − B m ω , rk rk rk rk where ω - is the angular velocity of engine crankshaft; i – is the gear ratio from the engine shaft to the drive axis; ηm - is the transmission efficiency;
Then, the driving force (4) takes the form: rk
−B
i 2η m (1 − δ ) rk2
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Pg = A
iη m
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Let’s represent the angular velocity of crankshaft ω through a linear velocity of v . (1 − δ ) i v . (5) ω= rk
v.
(6)
Let’s assume that the resistance force is constant M c (t ) = M c , whereas the initial equation takes the following form:
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Bi 2ηm (1 − δ ) iη m dv dm(t ) * dm(t ) m(t ) + v +v = v + A − Pc . dt dt rk2 dt rk Hence:
Let’s designate:
(7)
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2 iη dm(t ) Bi ηm (1 − δ ) dm(t ) + v* + A m − Pc 2 dt rk dt rk dv . +v = dt m(t ) m(t )
iη dm(t ) Bi η m (1 − δ ) dm(t ) + v* + A m − Pc 2 dt rk dt rk ; q(t ) . f (t ) = m(t ) m(t )
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The equation (7) can be rewritten in the following form: dv + f (t )v = q (t ) . dt
(8)
(9)
This is a linear equation of the first order, the solution of which at initial condition v(t = 0) = v0 would be written as: [8, 9]
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t
v (t ) = e
t f ( t ). dt v(0) + q (t )e ∫0 dt . ∫0 t
∫
− f ( t ) dt 0
(10)
At initial condition S (t = 0) = S0 for movement of unit we obtain the expression: S (t ) = S 0 + ∫ e
∫
− f ( t ) dt 0
0
t f ( t ) dt S (0) + q (t )e ∫0 dt dt . ∫0 t
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t
S
(11)
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The linear acceleration of unit: dv (12) W (t ) = = q (t ) − f (t )v(t ) . dt Depending on the law of mass variation of machine-tractor unit m and technological resistance Pc are determined the values of f (t ) and q (t ) by formulae (8).
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At study of acceleration of machine-tractor unit the acceleration time would be divided into three periods: the first – time with no-load operation, during that is established certain idle mode, the second – from the start of cargo supply on the working parts of unit prior to the release of cargo, and the third – from the beginning of cargo removal before the steady motion. The first period of acceleration, at that the mass of unit and moment of resistance are constant, we are not considered. We assume that in the second period mass of cargo m* that is under transportation or processing by working bodies of unit depends on the time t , measured from the beginning of period for the reduced to tractor drive wheels, we obtain the expression: 2
r m(t ) = m0 + m (t ) k , i *
(13)
where m0 – is the reduced mass to the driving wheels of all the moving parts of unit;
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m* (t ) – is the mass of cargo being in a given timre in unit for transportation or processing;
i – is the gearbox gear ratio; rk - is the rolling radius of drive wheel.
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Let’s designate as U * the velocity of connected masses. Then reduced to the drive wheels angular velocity v * , corresponding to the U * , would be written down as; U (14) v* = i rk Assume that the moment of resistance is constant ( M c (t ) = M 0 ) , and the cargo is evenly spread to the working bodies, that is m(t ) = ct , where c – coefficient of proportionality. Then, accordingly of (13) and (8): 2
m(t ) = m0 + nt ;
r n = c k ; i 4
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n+a ; m0 + nt
f (t ) =
q(t ) =
v* n + b m0 + nt
(15)
where 2 k
r
;
b=
A ⋅ iηm rk
− Pc .
(16)
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a=
Bi 2ηm (1 − δ )
By substituting (15) into the equation (10), to the expression of linear velocity we obtain: n+ a n+ a t ∫ m0 + nt dt v*n + b ∫0 m0 + nt dt 0 v (t ) = e v (0) + ∫ e ⋅ dt . m + nt 0 0 Let’s preliminary calculate: t
t
t
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−
n+a n + a m0 + nt . ln dt = n m0 0 + nt
0
Then: t
−
e
t
n+a . n
n+ a
∫ m0 + nt dt
e0
0
α
m0 = , m0 + nt
α
m + nt = 0 . m0
Then: t
n+ a
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where α =
n+a
∫ m0 + nt dt
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*
(19)
(20)
α −1
t v n + b ∫0 m0 + nt dt v* n + b e ⋅ dt = ( m0 + nt ) ∫0 m0 + nt m0α ∫0 t
(18)
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∫m
(17)
⋅ dt =
v* n + b α ( m0 + nt ) − m0α . m0 ⋅ n ⋅ α
(21)
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After the substitution of the last expression in (11) for the linear velocity of machine-tractor unit with variable mass we obtain the equality; α
m0 v* n + b α v(t ) = v (0) + m0 + nt ) − m0α . ( α m0 ⋅ n ⋅ α m0 + nt Or after the transformation:
(22)
α
v*n + b m0 v*n + b . (23) v(t ) = v(0) − + n ⋅ α m0 + nt n ⋅α According to the equation (11) is possible to define a linear movement of machine-tractor unit: S (t ) = S (0) +
v* n + b v*n + b m0α 1 ⋅ − v t. 0+ n(α − 1) ( m0 + nt )α −1 n ⋅ α n ⋅α
(24) 5
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The linear acceleration: W (t ) =
m0α dv (t ) = v*n + b − v0 n ) . α +1 ( dt ( m0 + nt )
(25)
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According to the obtained expressions (22) … (25) we can define the basic operating parameters of machine-tractor unit with variable mass. 3. Conclusion
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The dynamics of machine-tractor unit with variable mass, in particular, at real change of disconnecting and joining masses is offered. Are derived the design formulae of basic operational parameter of machine-tractor unit with variable mass, as well as linear movement of velocity and acceleration of machine-tractor unit. The results of research would be applied at definition of characteristics of units, at development of new agricultural tractors
References
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[1] A.P. Bessonov, Basics on the Dynamics of Mechanisms with Variable Masses of Links. Nauka, Moscow,1967 (in Russian). [2] V.P. Stolyarchuk, N.F. Rachinets, B.M. Gladko, The Study Movement and Dynamics of Machines Equipped with an Electric Drive. Publishing of Lviv University. Lviv, 1972 (in Russian). [3] R.M. Makharoblidze, Optimization of Dynamic Processes in the Agricultural Machinery. Agropromizdat, Moscow, 1991 (in Russian). [4] R.M. Makharoblidze, I.M. Lagvilvava,R.M. Khazhomia, B.B. Basilashvili , Theory of soil compaction by running bodies of mountain tandem- wheeled self-propelled chassis. Annals Of Agrarian Science, vol.14, №1 (2016) 51-58. [5] R.M. Makharoblidze, I.M. Lagvilvava,R.M. Khazhomia, B.B. Basilashvili, Interaction of the undercarriage of mountain self-propelled chassis with the soil based on the rheological models. Annals Of Agrarian Science, vol.13, № 3, (2015) 61-69 . [6] R.M. Makharoblidze, I.M. Lagvilvava, O.G. Asatiani, A.B. Kobakhidze, Theory of soil compaction running bodies of self-propelled chassis(In Russian). Academy of agricultural sciences Georgia, 31, (2012) 322-327 (in Rusian). [7] R. M. Makharoblidze, I.M. Lagvilvava, O.G. Asatiani. Stability of front wheel of automated additional load system of tandem-wheeled self-propelled chassis. Problems of mechanics, №1(46), (2012) 49-54. [8] R.M. Makharoblidze, I.M. Lagvilvava, O.G. Asatiani, A.B. Kobakhidze, Stability of front wheel of automated additional load system of tandem-wheeled self-propelled chassis. Problems of mechanics, №1(46), (2012) 21-27. [9] R.M. Makharoblidze, I.M. Lagvilvava, O.G. Asatiani, A.B. Kobakhidze, Passability of tandem-wheeled self-propelled chassis. Problems of mechanics, №1(46), (2012) 43-47. 6
