Published: 25 June 2019

Loss of stability of open two-link mechanisms

Leonid Kondratenko1
Lubov Mironova2
Vladimir Dmitriev3
1Central Research Institute of Machine Building Technology, Sharikopodshipnikovskaya Str., Moscow, 115088, Russia
2, 3Moscow Aviation Institute (State National Research University), Volokolamskoe sh., Moscow, 125993, Russia
Corresponding Author:
Leonid Kondratenko
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The problems of the loss of stable operation of two-link mechanisms consisting of the driving and driven members connected by long force lines are considered. The problem was solved on the basis of a new approach to the study of the dynamics of these systems with straight rods, based on the use of the equations of momentum (for longitudinal oscillations) and angular momentum (for torsional oscillations) in differential form. Stability was estimated by the first Lyapunov method, which consists in solving the resulting differential equations. As a result of modeling the work of a volumetric hydraulic drive, modes of self-oscillations are revealed. The areas of stability and instability are determined.

1. Introduction

Two-link mechanical systems contain a leading (input) link-energy source (engine, pump) and a driven (output) link-executive body (cutting part of the borer, mill, chisel, etc.). These links are connected by a force line. Such systems are widely used in industry. Moreover, in a number of processes (drilling, drilling aperture, drilling wells), feedback is not used. Loss of stability leads to self-oscillations, when loads on system components increase significantly, results of work changes, a sudden stop can occur. Usually such modes are highly undesirable.

The problem of the stability of motion by mathematicians and mechanics has been studied since the 19th century. To solve such problems are used the criteria and theories of Routh, Gurwitz, Lyapunov, Chetaev, Mikhailov, Nyquist, Bolotin, Popov [1-6] and etc.

In a number of publications, such problems are solved by an approximate method with the replacement of the corresponding functional equations by suitable finite-dimensional difference schemes [7, 8]. In the general case, the conditions for the stable operation of a system with distributed parameters are considered in [9-11].

2. Statement of the problem

Usually, for solving such problems, the well-known wave equations or equations based on the Lagrange equations of the 2nd kind are used [11].

This approach allows us to estimate the change in the displacements of the elementary volume of the rod and other parameters in time and space. However, such information, if it is necessary to take into account the interrelationship of a large number of factors, is on the one hand redundant, since it is often enough to know under what conditions do self-oscillations arise of the rotation frequency of a technological object (executive body), i.e. stability is lost, and when what is not. On the other hand, due to the lack of information about the stresses developed in the dynamic process, it is difficult to estimate the probability of part failure or disruption to the work process.

In the stated approaches to solving problems, a linear dependence of the stress on the movement of particles in the material is assumed and a corresponding recalculation is made. However, this is not always justified, since it is known from rheology that the modulus of elasticity may depend on the oscillation frequency [12]. In addition, depending on the stresses, the stem of the gun drill or the drill string may bend and thereby change the characteristics of the formation of force factors on the executive body and the results of work of the mechanism. At the same time, various non-linear effects, including significant ones, have a great influence on the functioning of the system.

3. Basic equations

To solve assigned problem when considering longitudinal oscillations in a straight solid rod in the absence of mass forces, the equation of momentum in differential form was used and the equation of longitudinal vibrations [11, 13, 14]:

ρυt=-σx, 2ut2=Eρ2ux2,

here υ is velocity of the longitudinal displacement of the point in the considered cross-section (υ=u/t); u is displacement along the x axis; σ is longitudinal (normal) stress; ρ is density of the material; E is elastic modulus; t is time.

For E= const, ρ= const from Eqs. (1) and (2) we obtain the expression σ/x=(E2u)/(x2), which, after integration with respect to x, and then differentiation with respect to t, is reduced to:


Eqs. (1), (2), first derived in [13], make it possible to explicitly describe the oscillations of velocities and stresses in a rod. Torsional vibrations can be described in a similar way [14, 15]. It should be noted that the process of motion transmission in systems with hydraulic lines is characterized by the equations [16]:

υt=-1ρPx-2ρ0τprp, Pt=-κυx,

here ρ0 is initial density of the medium; P is line pressure; κ is reduced modulus of elasticity of the highway; τp is shear stress on the pipe wall; rp is the radius of the pipe section.

Thus, the transfer of motion in solid and liquid media can be described by similar equations.

The equations that are given above describe the movement of an elementary volume of a substance at relatively low speeds of movement. However, in the case of any abrupt changes caused by either external influences or fast-flowing wave processes, there is a need for a deeper study of the process of motion transmission in mechanical systems.

From the study of the phenomelogical Zener model in [13], it was shown that the elastic properties of matter should be described by a generalized function using complex numbers.

In many cases, the imaginary component is substantially less than the real component.

It is known that the interaction of an elastic wave with the interface of media is largely due to the wave resistance, which is determined by the relation [17]:

-συ=ρa1, a1=ΘEρ12, Θ=1-μ1+μ1-2μ12,

here a1 is velocity of propagation of longitudinal oscillations in the medium; μ is Poisson’s ratio. Writing impedance in an operator form, it is possible for ρ= const, Θ= 1, Eu=E2=E get:

Zbjω=Euωθjωjω, θjω=±jωρE,

here θjω is operator coefficient of waves propagation. After a one-dimensional transformation of Eqs. (1), (2) according to Laplace with zero initial conditions, we obtain:

ρsυs=-dσsdx, Edυsdx=-sσs.

The solution of the system of equations under the corresponding boundary conditions [13] is:


For the case of a consistent load (when there are no reflected waves) and υ2 ≥ 0 from Eqs. (7), (8) with the boundary conditions υs,0=υ1s,υ(s,l)=υ2s,σ(s,0)=σ1(s),σ(s,l)=σ2s, σ2(s)(1-c)=[F(s)+hnυ2(s)+msυ2(s)]/f2, where f2 is sectional area of the main line in the vicinity of the executive body of mass m; c, hn is coefficients of friction loss; F is resistance force acting on the executive body; l is length of the force line, we can obtain the equations:


here ϑns=ϑn0Zns; ϑn0=l/(Ef2); Zns=thA/A; A=θsl.

The equation describing the dynamics of a volumetric hydraulic drive, after similar transformations, is written in the form [18, 19]:



ϑss=ϑs0 ZnsEg, ϑs0=lf1κ1w2, τ=K1w2, Zs=thA1A1, A1=θ01sL, θ01 s=ρ0E12.

In Eq. (8), the notation: τ is hydraulic tightness criterion; ϑ0s is coefficient of elasticity; Egs, Ψs is polynomials obtained during transformations; w is volumetric constant of the hydraulic motor; Q is pump flow rate; θ01s is operator coefficient of wave propagation in the pressure line; J is moment of inertia of rotating parts; c is coefficient of friction loss proportional to the pressure drop; h is coefficient of friction loss proportional to the velocity of motion; ρ0, κ is density and bulk modulus of elasticity of the liquid; Ω is frequency of rotation of the output shaft; Mr is moment of resistance; Kl is fluid leakage rate.

Wherein, the boundary conditions were determined from the balance of the flow of fluid entering the pipelines and flowing from them.

4. Stability of open two-link systems with distributed parameters

In the study of stability Lyapunov [3] used two methods. 1) Covers the methods that lead to the definition of the general and particular solutions of differential equations. 2) Without finding solutions to differential equations, it is required to find a function of generalized coordinates and time, the total derivative of which, by virtue of the equations of motion under consideration, has certain properties (Lyapunov function).

The development of the second method, as the most convenient, led to the use of linearized differential equations with constant coefficients and of special stability criteria proposed by Routh, Gurwitz, Mikhailov, Nyquist, etc. Referring to Eqs. (12), (10), we see that the coefficients of the characteristic equations with variable s depend on the same variable. Since the use of these criteria turned out to be difficult, a new technique for assessing sustainability was developed using the first Lyapunov method. From Eqs. (8), (10), it can be seen that the dynamic properties of the systems significantly depend on the functions ϑns, ϑss. If we consider the oscillations in the system, then such a function, in addition to density, elastic modulus (shear) and line geometry, depends on the oscillation frequency. At the same time for longitudinal vibrations in the rod can be written [14]:

Zkjω=thθkjωlθkjωl=jtgαjα=tgαα, α=ωlρG, chjα=cosα,

here α is a real dimensionless quantity. Therefore, ϑns=ϑnα and cos(α) are not complex functions. These considerations are valid and for torsional vibrations.

From the analysis of the function Znα follows: with α 0, Zn 1; for:

π2+kπ>α>π+kπ, Zn<0,

here k= 0, 1, 2, ..., n.

Analyzing Eqs. (8), (10), we see that they are similar. In this regard, we will carry out the following reasoning with the equation for the volumetric hydraulic drive Eqs. (10), with which it was easier to perform an oscillographic recording of the oscillatory process in natural experiments. After the algebraic decomposition of Eq. (10) into two equations and the inverse transition from the Laplace images to the originals, we obtain the equations:


here p is the differential pressure on the hydraulic motor. Wherein, the complex functions Ψs and Es in (10) are replaced by their real parts Ψω and ϑα=ϑ0E-1ωZα, which substantially exceed modulo imaginary parts [15, 19]. The admissibility of using Eq. (12) to study the dynamic properties of the systems under consideration was checked by constructing the frequency characteristics of a volumetric hydraulic drive in various ways [13]: calculating the frequency characteristics of the 4th order upgraded Runge-Kutta method [20, 21] and from a natural experiment.

Next, we introduce the operator Dd/dt, and with Mr= const we transform the system Eq. (12):



2ζT=Jτ+hϑα1-c+τh, T=jϑα1-c+τh12, ζ=0,5τϑα+hj, κ=1-c1-c+τhwΨω.

We assume that the ratio Ωn=Ω+τMr1c+τh is the reduced velocity, and τMr/(1c+τh) is the static error. Then Eq. (13) can be rewritten:


Eq. (14) describes the dynamics of the system in deviations from a certain static mode at Qt=Q0 and the moment of resistance Mrt=Mr0 with a steady rotation frequency Ω=Ω0.

Due to the non-uniformity of friction in the pistons of the axial-piston hydraulic motor, we assume that the moment of resistance oscillates with a small amplitude according to the law:


In expression (15), the introduced factor “a” takes into account the possibility of generating oscillations from the other pistons of the hydraulic motor. Consequently, at a known speed in the linear system there will be stable small oscillations with a frequency ω0=aΩ0. At such a frequency, the coefficients Ψω, ϑ(α) will have quite definite values Ψω0, ϑα0.

We rewrite Eq. (15) in Laplace images for the case when there are no deviations from the steady state operation in the system, and define the transfer function:


With a small jump action Qt=q0·1t, when changes of the coefficients can be neglected, the equation of motion in the images will look like:


Using transition tables [13] from images to originals, we obtain:



α=ζT=0,5τϑhα+hhj, ω=1-ζ2T212, T=jϑhα1-c+τhh12.

From Eq. (18) it can be seen that when α > 0, and therefore ϑ(α) > 0, the value of Ωn tends to some steady value, that is, the process is stable. If ϑ(α) < 0 and α < 0, then the quantity Ωn tends to infinity, i.e. the process will be unstable. Let now the hydraulic drive switched to another Ω=Ω01. In this case, the coefficients of Eq. (17) will have other values and the response to a small change in the impact will be different.

If, at the same pump feed, the new value of the resistance moment due to leakage causes the output motor shaft to rotate with another steady frequency Ω01, then the coefficients of characteristic Eq. (17) and transfer function Ws will also be different which will be determined by the steady state frequency Ω01.

It was shown above that the values of the functional coefficients in the equations depend on the functions Zα, Znα, Zkα. From their analysis it can be seen that, periodically, Z becomes negative. Since, depending on the pump flow or static load, the stationary frequency of rotation of the motor will be different, it is possible that, under certain operating conditions, the coefficients of the characteristic Eq. (17) become negative. In this case, with a small discontinuous change in the disturbing influence, the rotation frequency will either increase indefinitely or decrease to zero, i.e. the system will be unstable. Thus, the stability of an open-loop system is determined by the inequality tgα/α ≥ 0, where α=lωρ/χ1/2; χ is the coefficient of elasticity the material.

In nonlinear systems, for example, in drilling hydraulic drives, loss of stability leads to auto-oscillations.

5. Conclusions

The study showed that open two-link systems, depending on the frequency of disturbing oscillations and the properties of the lines of force, may become unstable. Loss of stability will consist either in stopping the work or in the appearance of self-oscillations.


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About this article

10 May 2019
21 May 2019
25 June 2019
System dynamics in manufacturing system modeling
two-link mechanisms
force lines
distributed parameters
system stability
moment of momentum
straight rod
oscillation frequency
volume hydraulic drive