Introduction

At present, Russia has adopted a course towards the creation of competitive machinery and equipment. This also applies to mobile timber harvesting machines of the manipulator type; at the present moment, loading and transport vehicles with manipulators are widely used in thinning forest management. In the Russian Federation, hydraulic manipulators of the brands LV-184A-10, LV-185-14A, LV-190-05, MM-100 are often used, which are installed as part of a road train behind the cab of a car or on the rear overhang of a subframe Hydraulic manipulator LV-184A-10 is the lightest in the class of carrying capacity (52 kN) and is well adapted to our conditions. The manipulator MAYMAN-100S (ММ-100) is equipped with a higher quality and efficient hydraulic pump SUNFAB SC-064 (084) with increased productivity and Swedish or Italian hydraulic distributors. It is possible to complete it with a two-loop control system, which allows to reduce dynamic loads and to save fuel consumption by up to 7%. Basic models of Epsilon manipulators: C70L, M100L and Q150L are equipped with strong metal structures, cast column bases.

The experience of operating forest hydraulic manipulators in the regions of the Komi Republic showed that due to high dynamic loads at low air temperatures, failures of high-pressure hoses are 29.7–56%, and hydraulic cylinders 14.0–24.1% [1]. With the wear of the seals, the initial amplitude of the pressure fluctuations of the working fluid decreases, and the period of the oscillations increases. The logarithmic damping decrement decreases to the limit value d = 0,533 which requires repair of the hydraulic cylinder. Using the developed mathematical model of dynamic processes inside the hydraulic cylinder, a new method for diagnosing hydraulic cylinders has been substantiated.

In work [2], technological operations of a forwarder manipulator were investigated: loaded movements; empty movements and movements of the manipulator when performing operations inside the cargo compartment. It was found that the technological cycle of the manipulator includes 63% of the time of the busy movement, 30% of the time is spent on idle transfers and 7% for the operations performed inside the cargo compartment. As a result of the obtained experimental data, linear regressions of the loading time (sec / m3) were constructed depending on the loading volume and the number of loaded assortments (on average, 47 pieces). The movement time of the manipulator links was recorded by a stopwatch from the start of movement in the cargo space until the jaws of the grabber touched the logs lying on the ground, and until the opening of the jaws, when the assortments were unloaded inside the cargo area of ​​the forwarder. Cases of empty manipulator movements to delete branches were taken into account. sorting and moving of individual log truck.

In addition to studying the influence of various factors on individual elements of the technological cycle, there are a number of works in foreign journals, in which the authors published the results of research on various aspects of automation of manipulators. In the works [3,4,7,8], the concept of an electro hydrostatic drive, applicable to hydraulic manipulators of large carrying capacity, is presented. The drive is designed and analyzed for requirements such as load holding, overload handling and differential flow compensation. Numerical analysis is carried out with the system connected to a constant load mass. Load weight 30,000 kg, accumulator precharge gauge pressure 5 bar. The cylinder moves the mass vertically. It is accepted that the force acting on the cylinder consists of three parts: hydraulic force, gravity force and friction force. The friction force is modeled by the viscous friction coefficient and the Coulomb friction force. Comparison with traditional valve operated actuators showing efficiency gains and energy recovery capabilities.

The work [5, 6] is devoted to the study of a promising scheme of a recuperative hydraulic drive of a timber truck-manipulator. On the basis of mathematical and simulation modeling of the operation of the pneumatic recuperation system, the dependences of the influence of time on the amount of compressed air, pressure and temperature in the pneumatic accumulator were obtained. The pressure after three strokes of one and the other pneumatic cylinders is 2.3 MPa. With continuous consumption of compressed air, pressure pulsations do not exceed 0.4 MPa. It has been established that with intensive periodic compression of air in pneumatic cylinders, high temperatures arise above 1000 K, and the average air temperature is about 200-300 °C. High air temperatures in pneumatic cylinders for energy recovery are a disadvantage of the proposed pneumatic systems.

Investigations of the kinematic and dynamic characteristics of hydraulic manipulators with the combination of the movements of the boom and stick are devoted to works [9, 10]. The graphs of the changes in the pressure values of the working fluid in the hydraulic system with the course of time were obtained. When the movement of the two links is combined, increased fluctuations in the working pressure are observed, and when the movement of the arm stops, the pressure is set at a nominal pressure of 12.5 MPa. The graphs of angular velocities were obtained, which at the beginning of the movement have a smooth character; the boom is overloaded.

However, in our opinion, studies of the dynamic and kinematic characteristics of energy-saving devices as applied to the lifting mechanisms of forest manipulators have not been carried out enough, therefore, additional research in this direction is urgent.

Purpose of the study is to reduce the dynamic loading and energy intensity of the working processes of the boom lifting mechanism of a forest manipulator by substantiating the parameters of an energy-saving hydraulic drive based on solving a mathematical model.

Materials and methods

A serial hydraulic manipulator LV-184 A -10, which is mounted on the chassis of short log trucks, was chosen as the object of research. We propose a new energy-saving hydraulic drive for the boom lifting mechanism of a forestry manipulator [11], which includes an additional recuperation hydraulic cylinder and a hydropneumatic accumulator, which accumulates braking energy when lowering the load and returns it during subsequent lifting of the load (Fig. 1).

Рисунок 1. Энергосберегающий гидропривод механизма подъема стрелы лесного манипулятора

1-бак; 2- насос; 5; 6; 7; 10; 14- гидролинии; 4- гидроцилиндр привода стрелы 5; 8-дополнительный гидроцилиндр рекуперации; 11; 13; 15; 17; 21; 23- обратные клапаны 12-гидроаккумулятор; 19-предохранительный клапан; 18-разгрузочное устройство; 22-демпфер; 25- плунжер

Figure 1. Energy-saving hydraulic boom lift for forestry manipulator hydraulic tank 1; pump 2; hydrolines 5; 6; 7; ten; fourteen; hydraulic cylinder 4 of the boom drive 5; additional hydraulic cylinder for recuperation 8; check valves 11; 13; 15; 17; 21; 23; hydroaccumulator 12; safety valve 19; unloading device 18; damper 22; plunger 25

Источник: собственная композиция автора

Source: author’s composition

To study the process of loading operations, taking into account the connection of a hydropneumatic device to the hydraulic drive of the boom lifting mechanism, we considered the calculated case of lifting a bundle of assortments load with a maximum reach of the manipulator on a slope, for example, when the soil subsides under the right outrigger at point A. To substantiate the parameters of the energy-saving hydraulic drive of the boom lifting mechanism a computational scheme was drawn up, which is shown in Fig. 2

Рисунок 2. Расчетная схема манипулятора

на склоне (действующие силы: G_бр - сила тяжести бревен в захвате, Н; G_с - сила тяжести стрелы, Н;

G_р - сила тяжести рукояти, Н; F_цс - усилие

в гидроцилиндре для подъема стрелы, Н)

Figure 2. Calculation diagram of an auto-log truck

manipulator on a slope (acting forces: Gbr

is the gravity of the logs in the grip, N; Gs is the

gravity of the boom, N; Gs is the gravity of the handle, N; Fcs is the force in the hydraulic cylinder for lifting the boom, N)

Источник: собственная композиция автор(ов)

Source: author’s composition

We have developed a mathematical model for lifting the boom of a manipulator with an energy-saving hydraulic drive when working on a slope, when the accumulator is fully charged and gives up the accumulated energy when lowering the load to raise the boom:

Jбр+Jр+Jсd2φdt2=πdc2l sinβ4∙p--GбрL+Gрlц.р+Gсlц.сcosφ-α,qнnн=πdc2l sinβ4∙dφdt-kакPA-p  ++ayp-VсумEпр⋅dpdt,dPAdt=-Eж kакV01-PоPA1К+EЖKPAP0PA1К∙PA-p  ,     (1)

where  J_бр, J_р, J_с –   moments of inertia of a bundle of logs, a stick, an arrow relative to a point О₁, kg∙m²;

φ  – boom angle, radians;

 α – slope angle, radians;

 t – time, s;

 G_бр – gravity of the bundle of logs in the gripper, Н;

d_с – inner diameter of the hydraulic cylinder, m;

q_н – pump displacement, м³/ turnover;

n_н – pump speed, s^-1;

 p – current pressure in the discharge line, Pa;

 P_A – current value of pressure in the accumulator , Pa;

Pо – accumulator pre-charge pressure, Pa;

Vо – working volume of the accumulator, m3 ;

Eпр – reduced modulus of elasticity of the working fluid and elastic elements of the hydraulic drive, Pa;

Eж-  modulus of elasticity of the liquid, Pa

K – gas adiabatic index in the accumulator equal to 1,41;

μ – the flow coefficient is 0,7...0,8;

d_ак – inner diameter of the hydraulic accumulator nozzle, m

d_р – internal diameter of the energy recovery hydraulic cylinder, m;

g – acceleration of gravity, m/s²;

ρ  – working fluid density, kg/m³;

V_sum – total volume of the supply pipeline, m3;

k_ак – battery choke throttling ratio, m³·с·Pa^–1/2

kак= μπdак24 2ρ.                      (2)

The designations of the remaining geometric parameters of the boom lifting mechanism included in equations (1) are clear from Fig. 2. Note that in the triangle O1DC, the cosine theorem implies the relation between the angles β , γ , φ

sinβ=b1sin⁡(γ+φ)l2+b2-2lb1cos⁡(γ+φ).                       (3)

In this work, for the possibility of solving the problem, it is assumed that the angle β  does not depend on t on a separate site, this means that the ratio on the right side of the formula is constant for each t.

On the segment t∈[0;tk]  we consider the Cauchy problem:

φ0=φ0,    p0=p0 ,    PA0=PA0,φ' 0=φ1,    p' 0=p1 ,    PA' 0=PA1.            (4)

Research results. A mathematical model of the boom lifting process is presented, described by a system of second-order nonlinear differential equations. The highest derivative has an irreversible operator - such systems are unresolved with respect to this derivative. Due to the nonlinearity of the system, it is impossible to find a solution in the explicit form of the dependence on t, which entails the need to use approximate methods. The sought functions are calculated at the nodal points ti . We denote

φi=φti,   pi=pti,   PAi=PAti  ,       (5)

where number i varies from 0 to n inclusive. One of the approximate methods is the finite difference method, in which all derivatives are replaced by the corresponding difference analogs:

dφdtti≈φi+1-φih, d2φdt2ti≈φi+2-2φi+1+φih2;dpdtti≈pi+1-pih, d2pdt2ti≈pi+2-2pi+1+pih2;dPAdtti≈PAi+1-PAih, d2PAdt2ti≈PAi+2-2PAi+1+PAih2.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (6)

which leads to a system of second-order recurrence relations. 

Since system (1) is not resolved with respect to the derivative, the Cauchy problem has a solution not for all initial values.

To solve it with respect to the highest derivative, the result obtained in [12, Theorem 2.1] is used, due to which the system splits into equalities in subspaces of decreasing dimensions. This solution is called the cascade decomposition method. This method was successfully applied, for example, in the study of perturbations of a linear algebraic-differential equation caused by the presence of a small parameter in [13]. For systems of recurrence relations, such an approach is applied, in our opinion, for the first time.

We have carried out the necessary calculations.

Theorem. A solution to problem (1), (3) exists if and only if the following equalities hold:

p1-p0=σφ1-φ0++h∙EпрVсум∙-kакPA-p+ay p0-qн nн ,PA1-PA0= -h Eж kакPA0-p0V01-PоPA01К+EЖKPA0P0PA01К ∙    (7)

The following algorithm has been developed for solving problem (1), (3) on a computer.

1. Enter the values of the coefficientsJбр , Jр , Jс , dc2 , l , Gбр , L , Gр , lц.р , Gс , lц.с , α , qн , nн , μ , ay , Vсум , Eпр , Eж , V0 , Pо , К , ρ . Определить kак  формулой (2) и σ  - формулой

                  σ=π dc2 l Eпрsinβi4Vсум. (8)

2. Enter initial values (at time t=0) the required quantities: φ0 ,  p0 , PA0 .

It follows from the theorem above that the values of the rates of change φ1 , p1 , PA1  of the required quantities at the initial moment of time must be entered so that equalities (7) are satisfied.

 3. Enter value tk  the right end of the segment of change of the variable t. Split this segment with equally spaced anchor points ti  with step h:

ti=i∙h,                                                 (9)

where   h=tkn,    n-number of split points

Comment. The larger n, the smaller the error in approximating real functions by their values at the nodal points.

4. Enter the following values

Φi1=π dc2 h2 lsinβ4Jбр+Jр+Jс pi-

- GбрL+Gрlц.р+Gсlц.с h2Jбр+Jр+Jсcosφi-α, (10)

Φi2=π2 dc4 h2 l2Eпр sin2β16 Vсум Jбр+Jр+Jс pi-

-π dc2 h2 Eпр l sinβ GбрL+Gрlц.р+Gсlц.с Jбр+Jр+Jсcosφi-α-

-Eпр h kак μVсум PAi+1-pi+1+