Modeling the dynamics of the chassis of construction machines

The article presents the results of a study of the transfer functions of a construction machine as a complex dynamic system. Authors constructed a dynamic model of a construction machine. The paper formulates and solves a system of nonlinear differential equations of motion of the chassis system of a construction machine on the basis of the d'Alembert- Lagrange equation. The numerical values of the transfer function coefficients for the construction machines were determined from the experimentally obtained curves of acceleration, processed by area method. Authors determined the experimental curves of the transition process of chassis system of a construction machine. The results of the study show that the difference of source curves ordinates and calculation of transients is less than 4% on average, which indicates a fairly accurate description of the process. The resulting expressions of transfer system functions of the chassis with sufficient precision can be used for practical purposes in the design and development of new construction machines.


Introduction
A construction machine is one of the most massive earth-moving machinery, employed in road construction. Modern market conditions require high-performance, high-speed and powerful heavy transport machine. Constant increase of available power leads to an increase in dynamic loads acting on both the human operator and the machine.

Construction of dynamic models
For the investigation of dynamics of the construction machine chassis we constructed its dynamic model according to its kinematic scheme [2].
On the basis of the d'Alembert-Lagrange system of equations we made non-linear differential equations of motion of the construction machines chassis system [1]: 1) a bent shaft of the engine and a power shaft of a reducer with the general moment of inertia of J 1 ; 2) output shaft of a distributing reducer and a gear box with J 2 moment of inertia; 3) shaft of running wheels with the given J 3 construction machine moment of inertia.
On the basis of it the system of the differential equations of running system has the following appearance: 2 To whom any correspondence should be addressed. We accept the moment M 1.5 arising in a drive gear of a distributing reducer to constants. Let's lead the system of the differential nonlinear equations (1) relatively ω 1 , ω 2 , ω 3 to a linear look. For this purpose we decompose the components of the equations (1) in ranks and we are limited only to linear elements of increments Δω 1 , Δω 2 , Δω 3 of rather established values 1 ω , 2 ω , 3 ω [5]: (2) (3) Where ΔM св1 , ΔM св2 , ΔM св3 , ΔM 1 , ΔM 1.2 , ΔM 2.3 -increments of the moments; Having substituted expressions (2) -(7) in (1), we will receive the following system of the linear equations of the movement of a running gear of the construction machine: We will enter designations: With the accounting of designations the system of the equations (8) looks like this: Having Δω 2 expressed from the third equation of system, having substituted Δω 2 in the second equation of system (9) we will receive: (   1  21  3  23  3  33  3  3  32   22  3  33  3  3 From the received ratio we will express Δω 1 and also we will find a derivative on time from Δω 1 : Having substituted expressions (10)-(11) in the first equation of system (9) we will receive the differential equation: For the purpose of convenience we enter the following dimensionless sizes: According to (10) transfer function of running system of the construction machine W п (P) has an appearance: a 1, a 2 , a 3 , a 0 , b 2 , b 1, b 0 -the equation coefficients determined by formulas: Numerical values of coefficients of transfer function for the construction machine (11) were determined by experimentally received curves of dispersal [4].
When determining curves of dispersal indignation was carried out by sharp change of entrance size -an angle of rotation of the lever steering productivity of the fuel pump of hydromechanical transmission (GMT) on the established corner Δα.
Speed of progress got out in the range of the working speeds of 1,2 -1,8 m/s [3]. Rotary speed measurement ω 3 was performed the tacho generator and an induction marker [6,9], and the entrance size α -the electrometric sensor of the PLP-21 brand.
Experimental curves of transition process of running system of the construction machine are given in figure 1.
Then transfer function of running system of the construction machine taking into account time of delay will assume an air: where К = 3,13 -coefficient of strengthening of running system; τ = 0,2 с -delay time. Next we define an error of approximation of transfer functions. For this purpose we will find analytical expressions of transition processes ∆V(t)construction machine [8].
We will write down the image of transition process according to Laplace: where L[∆V п (t)] и L[α(t)] -images according to Laplace of a deviation of speed of progress of the construction machine and the revolting influence. The roots of denominators of transfer functions of running system (19) found an iterative method [10] the following: Р 1 =-4,141; Р 2,3 =-0,998±0,479i.
We will write down the original of transition process in a general view: [ ] where R и D -numerator and denominator of transfer functions; Р 1 -valid root of a denominator; А к , φ к -amplitude and phase of fluctuations which to be determined by formulas: where k α and k β -respectively valid and imaginary parts of complex roots of a denominator; δ and σ -respectively valid and imaginary speak rapidly expressions R(Р к )/D I (P k )in a case when R k is a complex root. After substitution of roots in (20), we will receive expressions of transition processes for the construction machine: We will check correctness of calculations by comparison of the transition processes calculated according to (21) and received experimentally.  figure 1 it is possible to say that the toe-out of ordinates of initial curves of 1 and settlement curve 2 transition processes on average doesn't exceed 4% that points to rather exact description of process. 2. Probe of dynamics of chassis of the construction machine showed that its mathematical model is described by the differential equation of the third order. The received expressions of transfer functions of running system are confirmed with pilot studies and with a sufficient accuracy can be used in the form of (21) for practical calculations.