Power Losses in the Engine-Transmission Installation of the Mobile Unit under Probabilistic Load Conditions

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 1, Pages: 196-201

J. Environ. Treat. Tech.

ISSN: 2309-1185

Journal weblink: http://www.jett.dormaj.com

Power Losses in the Engine-Transmission

Installation of the Mobile Unit under Probabilistic

Load Conditions

Ramazan Yusupov *, Vil Yusupov

Russian State Agrarian University – Moscow Timiryazev Agricultural Academy, Moscow, Russia

Goryachkin Institute of Mechanics and Energetics, 127550, Timiryazevskaya st. 49, Moscow, Russia

Received: 27/05/2019 Accepted: 01/10/2019 Published: 30/03/2020

Abstract

The article reviewed a method for assessing the influence of a probabilistic load on the amount of energy loss in an engine-

and-transmission unit (ETU) of a mobile aggregate. The method is based on the usage of a random processes theory. The random

process is a sum of harmonic components with a random initial phase and constant amplitude. The image of the random process

corresponds to an arcsine distribution, in which the normal distribution law is approximated by an arc sine function. A peculiarity

is that the approach allows you to obtain characteristics that reflect the dependence of the energy performance of the engine-and-

transmission installation on the frequency of oscillations of the rotational speed of the crankshaft of an internal combustion

engine. The frequency characteristics of the mathematical expectation of the effective power of the internal combustion engine,

dissipative losses in the dynamic systems of the internal combustion engine and transmission are obtained as a result of all

calculations. We propose a generalized criterion to assess the energy efficiency of a mobile unit under the influence of a

probabilistic load.

Keywords: engine-and-transmission unit (ETU), random processes theory, internal combustion engine, dynamic systems,

mathematical expectation

maximum power received during the operation of the engine

Introduction

with a variable load, to the maximum power received during

standard brake tests. It has been established that the main

reason for reducing the energy performance of the engine is

the fluctuation of the angular velocity of the crankshaft, and

these fluctuations, in turn, are determined by the reduced

moment of inertia, the degree of unevenness of the moment

of resistance, the degree of insensitivity of the regulator, and

the adaptability coefficient of the IC engine.

Following studies of numerous authors were aimed at a

deeper study of the processes occurring in the IC engine

under the unsteady nature of the load (6, 7, 8, 9, 10, 11), to

search for ways to improve engine efficiency (l, 2, 7, 12, 13,

Over recent years through studying the energy indicators

of machine-tractor aggregates, the probability-statistical

characteristics of random processes interacting with both

power transmission and the engine are widely used (l, 2, 3,

). The approach allows us to evaluate the energy

performance of the machine-tractor aggregate (MTA) in

conditions close to real. Common to all tractor vibrations is

that the energy of only one source is expended on their

excitation and maintenance - the engine. So, the level and

intensity of all vibrations accompanying the operation of the

tractor affect its energy performance, and, consequently,

productivity and fuel savings.

Based on (В.Н. Болтинский) V.N. Boltinskiy’s

theoretical and experimental studies, engine power and

economic indicators, due to load fluctuations, decrease,

compared to the indicators when loading with a constant

torque (5). He introduced the concept of engine power

utilization coefficient, which is equal to the ratio the

5, 16, 17), on the methods development for assessing and

forecasting the main technical and economic indicators of IC

engine (l, 11, 13, 18, 19, 29). In most works it is noted that

one of the main reasons for the decline in engine performance

is the non-linearity of its regulatory characteristic (l, 6, 12,

1, 22), and in this regard, negative phenomena are observed

more significantly, the closer the IC engine load to the

nominal, the less adaptability coefficient, the greater the

amplitude of the oscillations, the smaller the reduced moment

of inertia. Due to the fact that the efficiency of the IC engine

Corresponding author: Ramazan Yusupov, Russian State

Agrarian University – Moscow Timiryazev Agricultural

Academy. Email: unoreen@psu.edu.sa.

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 1, Pages: 196-201

under the unsteady nature of the load is largely determined by

its dynamic qualities, scientific research aimed at creating

mathematical models has been developed (11, 23, 24, 25).

With their help, transient processes are studied during faults

and load shedding, and, based on frequency methods,

technical and economic indicators of IC engine in dynamic

modes (4, 11).

Mathematical method in the form of linear and nonlinear

differential equations is widely used in these works. Some

authors focus on the frequency method using statistical

models of random processes as an input signal (4, 18, 23).

The unsteady nature of the load influences the energy

processes in the engine-transmission installation as a whole.

Due to fluctuations in the rotational speed of the shafts,

additional power losses due to friction and slipping in

kinematic pairs are formed. The nature of the losses has

received a fairly complete interpretation in the works (20,

systems composed of components of a different physical

nature (electrical, hydraulic, mechanical).

In general, from a review of analytical methods for

studying the dynamic properties of power transmission, it can

be concluded that the scope of existing methods is limited to

one specific physical nature (mechanical, hydraulic,

electrical). It can be noted as a drawback that they do not

consider complex dynamic systems consisting of a set of

heterogeneous elements.

The features of power transmissions of various physical

nature and the related features of the methods do not allow

the use of the latter as universal ones in substantiating the

rational type of continuously variable transmission (33-37).

The analysis of trends in modern tractor engineering

allows us to conclude that one of the main goals of improving

tractors as the main energy tools for mobile crop production

is to increase their productivity. Therefore, the industry has

developed a direction of research work related to the search

for ways to more fully load the diesel engine, to improve the

dynamics of the tractor in operating conditions and, in

general, to increase the fuel economy and productivity of the

MTA. Currently, the following paths have been identified:

the introduction of progressive power transmissions, the

correction of the static and dynamic characteristics of internal

combustion engines. It implies stepless hydrodynamic,

hydrostatic, electric transmissions and an increase, within the

limits of the steepness of the corrector, of the speed

characteristic of the internal combustion engine.

According to experts, the situation in the field of

improving the manufactured equipment and developing

promising modifications of tractors and agricultural machines

can be seriously improved by providing designers and

researchers with universal engineering methods for

calculating the energy and dynamic indicators of MTA,

which are invariant to the physical heterogeneity of the

mechanisms of the unit, allowing for their static and. dynamic

characteristics, speed and load conditions of the machine-

tractor unit and ultimately, giving the opportunity at the

design stage of new technology to obtain reliable

comprehensive information about the MTA indicators of

interest in the probabilistic nature of the external load.

6).

The next group of theoretical methods is associated with

the study of mathematical models composed of equations

describing equivalent circuits of dynamic systems. The first

theoretical works devoted to the study of the dynamic

properties of mechanical systems, based on the principles and

principles adopted in electrical engineering, appeared in the

0s (27, 28, 29, 30, 31). These studies substantiate the

possibility of using electromechanical analogies when

considering issues related to the study of oscillatory processes

in mechanical systems. This approach made it possible to

transfer to the ready-made mechanical system a number of

conclusions obtained in some well-developed branches of

electrical engineering (for example, in the theory of four-

terminal networks, in the theory of electric filters, etc.). In the

papers under consideration, the concept of equivalent circuits

is given and methods for constructing them are shown. The

equivalence of two systems - electrical and mechanical - is

justified by the identity of the Lagrange equations presented

in generalized coordinates. From them equations are derived

that describe mechanical systems and are similar to

Kirchhoff's equations. So, for example, an electric circuit

composed of inductance L, capacitance C and resistance R is

described by the equation of the second Kirchhoff law.

Similarly, a mechanical system is described by equations

obtained on the basis of the de Alambère principle. These

works give an idea of the fundamental possibilities of an

approach based on the formation and study of equivalent

circuits of mechanical systems.

The analysis of technological processes carried out both

on industrial and agricultural machine-tractor aggregates

suggests that the maximum spectral density of load

fluctuations falls precisely on the low-frequency region.

In (32), there is a classification of processes occurring in

a dynamic system. High-frequency vibrations seem to be

associated with individual mechanical links, which constitute

a separate type element through hard links. It is argued that

low-frequency oscillations occur in a system of typical

elements and cause fluctuations in the speed of rotation of the

shafts. A common drawback of all the theoretical works

discussed above is that they do not reveal at all or weakly

reveal the questions of studying the dynamic properties of

Let us give a physical interpretation to the procedure for

calculating the energy indicators of a machine-tractor unit

using the spectral density of a random process. As is known,

spectral density is a function that characterizes the frequency

distribution of elementary dispersions (38). From here, in

accordance with (39), one can proceed to the frequency

distribution of the amplitudes of harmonic processes:

ΔА(ɷ) = √ꢀꢁ ꢂꢃꢄ .

(1)

So, the input variables in a random process obeying the

arcsine distribution can be represented by a set of amplitudes

of harmonic oscillations distributed over a frequency (Figure

), and then it is legitimate to talk about the frequency

characteristics of the mathematical expectation of the

effective power of the internal combustion engine and other

indicators. M - dependence of the torque on the shaft of an

internal combustion engine (ICE) on the speed of rotation of

the shaft; N is the dependence of the effective power on the

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 1, Pages: 196-201

ICE shaft on the shaft rotation speed; M(M ) and M(N ) are

2 Research Methods

We consider the non-normalized spectral density of the

ICE crankshaft rotational speed as the frequency distribution

of harmonics dispersions (38), use the provisions of (l) to

determine the amplitudes of harmonic vibrations for different

frequencies

the dependences of the mathematical expectations of the

torque and the effective power of the internal combustion

engine on the shaft rotation speed; φ(n) - normal and arcsine

distribution laws of

a random process and harmonic

components with a random initial phase and constant

amplitude; n(t) is a random process of the speed of rotation of

the ICE shaft; n(t)_kare the harmonic components of the

random process of the speed of rotation of the ICE shaft.

вал

ΔΩ(ɷ) = √ꢀꢁꢂꢃꢄ_ꢅ_ꢆ_ꢇ

(2)

Substituting (2) into the expression for calculating the

expectation value of the effective ICE power under the

assumption of the arcsine distribution of a random function of

rotation speed (l), we get

This distinguishes the technique considered in the work

from that described in (l).

ꢋ

ꢏ

ꢋ

ꢏ

ꢖꢗ

[

M (ꢈ )](ɷ) = 0.5[ꢊ ꢌ +ꢎ ꢌ ꢐ ꢑꢒꢓꢎ (ꢔꢌꢄ ꢂꢃ)]-ꢕ х

ꢉ

ꢍ

(

ꢋ

ꢝ

ꢋ

ꢝ

[

ꢘꢙꢅꢚꢛꢜꢙꢅ ꢛꢞꢒꢟꢜꢙꢂꢠꢃꢄ ꢂꢃꢄ]ꢘꢡꢢꢣꢤꢥꢂꢅꢦꢖꢅꢚꢄ

ꢚ

ꢐ

ꢠꢅꢂꢃꢄ

ꢋ

ꢜꢙ

ꢏ

ꢂꢨꢌ_ꢍꢩ ꢌ ꢄ√ꢂꢔꢌꢄ ꢂꢃꢄ ꢩ ꢂꢌ ꢩ ꢌ ꢄ .

ꢦ

ꢍ

ꢏꢧ

Let’s look onto the methodology for determining, based

on the obtained frequency response, the integral value of the

mathematical expectation of the effective power of the

internal combustion engine.

. The relation is determined to determine the frequency

response of the expectation value of power loss

[

M(Δꢈ )](ɷ) = ꢈ ꢩ ꢪꢫꢂꢈ ꢄꢬꢂꢃꢄ.

(4)

ꢉ

. By integrating the frequency response (4), we get the

mathematical expectation of the total power loss

M (Δꢈ ) = ∑^ꢥꢪꢫꢂꢔꢈ ꢄꢂꢃ ꢄꢔꢃꢬꢭ

(5)

ꢉ

ꢍꢮꢗ

ꢉ

ꢍ

K is a harmonic order.

. The expectation value of the effective power of the

internal combustion engine is calculated, corresponding to the

influence of the spectrum of speed fluctuations

M (ꢈ ) = ꢈ – M (Δꢈ ).

(6)

ꢉ

Numerous experimental and theoretical studies (40, 41,

2) suggest that vibrational processes cause additional

dissipative losses in a dynamic system. From the theory of

oscillations (41, 42) it is known that dissipation losses due to

Fig.1: Engine performance under spectrum exposure harmonic

vibrations of the crankshaft rotation speed

the action of harmonic oscillations on

determined from the relation

system are

(7)

We note one more difference, namely, that as an input

function on the cranked shaft of the ICE, we do not consider

the torque, as in (l), but the rotation speed. This step

corresponds to the methodology for the formation of a

mathematical model of the engine-transmission installation as

a whole. And above it is shown that a mathematical model is

formed for the condition of impact on the input of a dynamic

system of rotation speed fluctuations.

ꢔꢈ ꢯ ꢌꢰꢫꢱꢲꢳꢴ,

Ω is rms-speed for one period of oscillation,

М is a same torque.

If ɷ is an instantaneous state of velocity, the root-mean-

square will be:

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 1, Pages: 196-201

_ꢠ_ꢃꢝ

ꢶ

e₁is the elasticity compliance in the dynamic system of a

ꢗ

ꢶ

ꢞ

ꢶ

ꢞ

ꢠꢅ

ꢻꢏ

ꢏ

Ω = ꢵ ∫ ꢃ ꢷꢸ = ꢵ

∫ ꢳꢹꢺ ꢂꢃ ꢸꢄꢷꢸ =

(8)

(9)

ꢤꢣ

ꢞ

heat engine; Ɣ - mechanical conductivity, characterizes the

ꢶ

speed loss in the same place; Ɣ₂₃- mechanical conductivity,

the reciprocal of the mechanical resistance, characterizes the

Similar for torque

loss of torque in the same place; J is the moment of inertia of

the engine flywheel and the translationally and rotationally

moving masses rigidly connected with it; Ɣ45 - mechanical

conductivity, characterizes the speed loss in the dynamic

systems of the first and second converters; e₂₃— elastic

coupling compliance in dynamic systems of the first and

second transducers; Ɣ₆₇- mechanical conductivity, the

reciprocal of the mechanical resistance, characterizes the loss

ꢗ

ꢶ

ꢞ

ꢂꢠꢽꢄ^ꢝ

ꢶ

ꢞ

ꢠꢽ

ꢻꢏ

ꢏ

M =ꢵ ∫ ꢼ ꢷꢸ =ꢵ

∫ ꢳꢹꢺ ꢂꢃ ꢸꢄꢷꢸ=

ꢞ

ꢶ

T is the period of harmonic oscillation,

ΔΩ, ΔM are the amplitudes of the oscillations of speed

and torque,

m is the instantaneous value of the torque,

ꢃ is the angular frequency of oscillations.

Theoretical studies performed by a number of authors

suggest that the above reasoning can be applied to random

functions with sufficient accuracy. In this case, the standard

deviations can be obtained from the relations

of torque to the dynamic system of the second converter; J is

ꢞ

the moment of inertia of the translationally and rotationally

moving masses rigidly connected with the second transducer;

Ɣ - mechanical conductivity of the engine slipping section;

Ɣ - a parameter reflecting the resistance of an external load.

In order to establish how losses are distributed between

the internal combustion engine and the transmission, we find

the corresponding coefficient and call it the coefficient of the

level of losses in the tractor transmission. We will again use

the equivalent circuit. It is convenient here to turn to some

provisions of the theory of electric circuits (42). In

accordance with them, the ratio of power on any part of the

circuit to the total power supplied to the input of the dynamic

system is equal to the ratio of the resistances (conductivities)

of the corresponding sections.

ꢾ_ꢽ

ꢯ √ꢿꢂꢫꢄ, ꢾ_ꢅꢯ √ꢿꢂꢌꢄ

(10)

and to determine the amplitudes of the oscillations

acceptable ratio

^Δ^M⁼^ꢻ^ꢀ^ꢾꢽ^,

ΔΩ = ^ꢻ^ꢀ^ꢾꢅ

(11)

Under the probabilistic nature of the external load, power

losses in the dissipative elements of the engine-transmission

installation of the mobile unit can be found by the expression

ведущее колесо

For our case, we can write:

ꢠ꣝ꢂꢃꢄ_꣙

꣜ꢉ꣓_꣙

ꢯ

= λ(ɷ),

(15)

ꢠ꣝ꢂꢃꢄ꣙꣚꣛

꣜ꢉ꣓꣙꣚꣛

ΔNꢂꢃꢄ_ꣀ_ꢶ_ꣁ=√ꢿꢂꢫꢄD(Ω) ꢾꢂꢃꢄ_ꣂ_ꣃ_꣄ꢾꢂꢃꢄ_ꢅ_ꣃ_꣄cosφ(ɷ),

(12)

Y - mechanical conductivity of a dynamic ICE system,

Y_e_t_i- mechanical conductivity of the MTA dynamic

system, found when calculating the angle of phase shift from

14)

ꢾ

ꢂꢃꢄ_ꣂ_ꣅ_꣆ꢭ ꢾꢂꢃꢄ_ꢅ_ꣅ_꣆are normalized spectral densities of

processes on the driving-wheel (leading wheel) of the tractor

φ(ɷ) - phase angle between torque and speed at a fixed

harmonic.

Let’s find an expression for calculating the phase shift at

a fixed harmonic. We will use the equivalent circuit of the

engine-transmission installation (Fig.2). Full mechanical

conductivity of the ETI dynamic system.

(

꣇

ꢙꢛ꣉ꢃꢉꢙꢪꢢ꣋ꢣ꣌ꢙꢂꢃꢄꢛ꣉ꢣꢤꢥ꣌ꢙꢂꢃꢄꢬ

= _꣇

(16)

ꢙꢛ꣉ꢃꢉꢙꢪꢢ꣋ꢣ꣌ꢙꢂꢃꢄꢛ꣉ꢣꢤꢥ꣌ꢙꢂꢃꢄꢬꢂ꣍ꢙꢛ꣉ꢃ꣎ꢙꢄꢛꢗ

If we know λ(ɷ), the components of losses in the internal

combustion engine and transmission are calculated. So the

losses in the engine are determined by the ratio

ꢔꢈꢂꢃꢄ_ꢉ

17)

ꢯ

ꢔꢈꢂꢃꢄ_ꢉ_꣐_ꢤ

ꢰ

λ(ɷ),

(

transmission losses

ꢔꢈꢂꢃꢄ_꣐_ꢡꢔꢈꢂꢃꢄ_ꢉ_꣐_ꢤꢪ꣞ ꢩ ꣟ꢂꢃꢄꢬ .

Fig.2: Equivalent circuit of the tractor engine-transmission

installation

ꢯ

(18)

꣇

ꢙꢛ꣉ꢃꢉꢙꢪꢢ꣋ꢣ꣌ꢙꢂꢃꢄꢛ꣉ꢣꢤꢥ꣌ꢙꢂꢃꢄꢬ

We express the components of the power loss in the ETI

in relative terms. The relative amount of losses in the ICE

Y = ꣇ ꢐ ꣉ꢃ꣊_ꢏ_꣈ꢐ _꣇

ꢏ꣈

ꢙꢛ꣉ꢃꢉꢙꢪꢢ꣋ꢣ꣌ꢙꢂꢃꢄꢛ꣉ꢣꢤꢥ꣌ꢙꢂꢃꢄꢬꢂ꣍ꢙꢛ꣉ꢃ꣎ꢙꢄꢛꢗ

(

13)

꣓

꣔

꣏

ꢉ꣐ꢤ

ꢯ ꣑ ꢐ ꣑

ꢐ

ꢠ꣝ꢂꢃꢄ꣙

꣒

ꢉꣅ

꣓

꣔ꢂ꣍꣕ꢛ꣉ꢃ꣎ꢝꢄꢛꢗ

꣟ꢗꢂꢃꢄ = 1 -

(19)

꣝

꣙

Phase angle

And in transmission

ꣁ

꣘꣓꣙꣚꣛

꣏

ꢉ꣐ꢤ

ꢯ ꢊ꣖ꢱꢸ꣗ _꣜

ꢭ

(14)

ꢠ꣝ꢂꢃꢄ꣚꣠

ꢉ꣓꣙꣚꣛

꣟_ꢏꢂꢃꢄ = 1-

(20)

꣝

꣙

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 1, Pages: 196-201

To estimate the losses caused by the impact on the

dynamic system of a set of harmonics, we integrate relations

Frequency response of the ICE effective power factor

(

3.64) and (3.65):

ꢪꢽꢂ꣝꣙ꢄꢬ ꢂꢃꢄ

꣟

ꢂꢃꢄ_꣝_ꢉ

(21)

꣝

꣙

_ꢔ_ꢈ_ꢯ_∑ꢥ ꢔꢈꢂꢃ ꢄ ꢔꢃ ,

ꢔꢈ_꣐_ꢡꢯ ∑

(23)

(24)

ꢉ

꣄ꢮꢗ

꣄ ꢉ

ꢥ

ꢔꢈꢂꢃ_꣄ꢄ_꣐_ꢡꢔꢃ ,

꣄ꢮꢗ

The convenience of the obtained coefficients lies in the

fact that with their help it is possible to compare the levels of

all loss components and formulate a generalized criterion that

allows us to evaluate the effectiveness of the engine-

transmission installation as a whole

k is a harmonic order.

Conclusions

. A random process following the normal distribution

law can be described by an arcsine distribution.

. Input variables in a random process following the

krit(ɷ) = ꣡ ꢂꢃꢄ ꣡ ꢂꢃꢄ ꣡ꢂꢃꢄ → max.

꣝ꢉ

(22)

ꢗ

ꢏ

normal distribution law can be represented by a set of

harmonic oscillations with a random initial phase and

constant amplitude, i.e. following the arcsine distribution.

Example of calculating energy losses of tractor engine-

transmission installation

. Under the probabilistic nature of the external load, the

mathematical expectation of the effective ICE power and

power loss in the dissipative elements of the engine-

transmission installation of the mobile unit can be reflected in

the frequency characteristics.

. It is possible to establish the frequency range of load

oscillations on the driving wheels of the mobile unit based on

the analysis of the data of frequency characteristics. It

significantly affects the effective power of the ICE and the

amount of dissipation loss in dynamic systems of the engine

and transmission.

. Modeling the dynamic system of the engine-

transmission installation of a mobile unit can be carried out

on the basis of equivalent circuits, allowing calculations

using methods that have been widely used in the theory of

electrical engineering.

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