Mathematical modeling about nonlinear delayed hydraulic cylinder system and its analysis on dynamical behaviors

In this paper, we study dynamics in delayed nonlinear hydraulic cylinder equation, with particular attention focused on several types of bifurcations. Firstly, basing on a series of original equations, we model a nonlinear delayed differential equations associated with hydraulic cylinder in glue dosing processes for particleboard. Secondly, we identify the critical values for fixed point, Hopf, Hopf-zero, double Hopf and tri-Hopf bifurcations using the method of bifurcation analysis. Thirdly, by applying the multiple time scales method, the normal form near the Hopf-zero bifurcation critical points is derived. Finally, two examples are presented to demonstrate the application of the theoretical results.


Introduction.
A hydraulic cylinder (also called a linear hydraulic motor) is a mechanical actuator that is used to give a unidirectional force through a unidirectional stroke (see Fig. 1). In hydraulic systems, cylinders are crucial component converting the fluid power into linear motion and force. It has many applications, notably in construction equipment (engineering vehicles), manufacturing machinery, and civil engineering. Hydraulic cylinders get their power from pressurized hydraulic fluid, which is typically oil. The hydraulic cylinder consists of a cylinder barrel, in which a piston connected to a piston rod moves back and forth. The barrel is closed on one end by the cylinder bottom (also called the cap) and the other end by the cylinder head (also called the gland) where the piston rod comes out of the cylinder. The piston has sliding rings and seals. The piston divides the inside of the cylinder into two chambers, the bottom chamber (cap end) and the piston rod side chamber (rod end / head end) [7,6].
Prepressing and hot-pressing are two important processes during the particleboard production, which are driven by hydraulic cylinder. Hydraulic drive unit is a small system of servovalve-controlled symmetrical cylinder, and the principle of hydraulic drive unit is shown in Fig. 2. It is a common power machine that where q 1 , q 2 = forward and return flows, respectively, m 3 /sec; p 1 , p 2 = forward and return pressures, respectively, N/m 2 ; x v = valve displacement from neutral, m; K q = valve flow gain, m 2 /sec; K c = valve flow-pressure coefficient, m 5 /(N · sec).
Adding the servovalve flow equations, we obtain the usual form of the linearized flow equation of the servovalve as follows: where q L = (q 1 + q 2 )/2 = load flow, m 3 /sec; p L = p 1 − p 2 = load pressure difference, N/m 2 .
Let us now turn to the motor chambers and assume that the pressure in each chamber is everywhere the same and does not saturate or cavitate, fluid velocities in the chambers are small so that minor losses are negligible, line phenomena are absent, and temperature and density are constant. Applying the continuity equation to each of the piston chambers yields where t = time, sec; v 1 = volume of forward chamber (includes valve, connecting line, and piston volume), m 3 ; v 2 = volume of return chamber (includes valve, connecting line, and piston volume), m 3 ; C ip = internal or cross-port leakage coefficient of piston, m 5 /(N · sec); C ep = external leakage coefficient of piston, m 5 /(N · sec); β e = effective bulk modulus of system (includes oil, entrapped air, and mechanical compliance of chambers), N/m 2 . The volumes of the piston chambers may be written where A p = area of piston, m 2 ; x p = displacement of piston, m; v 01 = initial volume of forward chamber, m 3 ; v 02 = initial volume of return chamber, m 3 . In contrast with the rotary motor, the initial chamber volumes are not necessarily equal for the piston. However, it will be assumed that the piston is centered such that these volumes are equal, that is, v 01 = v 02 = v 0 . Therefore, where v t = total volume of fluid under compression in both chambers, m 3 . Then, the volume and continuity expressions can be combined to yield where C tp = C ip + C ep /2 = total leakage coefficient of piston, m 5 /(N · sec), which is the usual form of the continuity equation. To be specific, A p dxp dt means the flow driving cylinder piston, and C tp p L means total flow, and vt 4βe dp L dt means total compression flow.
Note that there exists a delay, denoted by τ , describing the time that the flow is leaked from cylinder, thus, equation (2) becomes Considering the load characteristics of hydraulic drive unit, the output force and load force of servo cylinder balance equation is determined as follows: where m t = total mass of piston and load referred to piston, kg; B p = viscous damping coefficient of piston and load, (m · N · sec)/rad; 946 YUTING DING, JINLI XU, JUN CAO AND DONGYAN ZHANG K = load spring gradient, (N · sec)/rad; F c = coulomb friction, N ; F = load force, N . Generally speaking, load force is nonlinear, and approximately follows = y p , then according to (1), (3) and (4), a model for hydraulic cylinder in supplying glue system can be described mathematically as follows: It is well known that delay differential equations (DDE) may exhibit higher codimension singularities more frequently than that in ordinary differential equations (ODE) [2,1,10,4,5]. In this paper, by applying the local stability theory, we investigate the existence of several types of bifurcations for this hydraulic cylinder system with delay (5). For example, it can bring fixed point bifurcation, Hopf bifurcation, the stability switches, Hopf-zero bifurcation, double Hopf bifurcation or higher codimension bifurcations into the system, thus makes system more complicated. By applying the multiple time scales (MTS) method [8,3,9], we derive the normal form near Hopf-zero critical point.
The rest of the paper is organized as follows. In Section 2, we consider the stability of the equilibria and the existence of fixed point, Hopf, Hopf-zero, double Hopf and some higher codimension bifurcations in the delayed nonlinear hydraulic cylinder system (5). Then we derive the normal form associated with Hopf-zero bifurcation in Section 3. Two examples and its analysis are presented in Section 4. Finally, conclusion is drawn in Section 5.
2. Stability of the equilibria and the existence of several types of bifurcations. In this section, system (5) is considered. First of all, we determine the equilibria of this system. When Kc+Ctp ) > 0, system (5) has two equilibria: When K 2 − 4K 0 (F c − ApKqxv Kc+Ctp ) = 0, system (5) has one equilibrium: For convenience, we assume the equilibrium of system (5) is (x * p , 0, p * L ). Transferring the equilibrium to the origin, that is, letx p = x p − x * p ,ŷ p = y p ,p * L = p L − p * L , then drop the hat, we obtain the following equations: The trivial equilibrium of (8) is (x p , y p , p L ) = (0, 0, 0). The characteristic equation of (8), evaluated at origin, is given by where Case 1. Fixed point bifurcation.
Note that system (5) has one equilibrium ( Kc+Ctp ) = 0, and K +2K 0 x * p = 0, M 0 = N 0 = 0, that is, the characteristic equation of (8), is given by then the characteristic equation (10) has one zero root λ = 0, and system (5) undergoes a fixed point bifurcation when all the roots of Eq. (11) have negative real part expect one zero root when M 1 +N 1 > 0 and M 2 + N 2 > 0.
is given by (23). When M 1 + N 1 > 0 and M 2 + N 2 > 0, we have ), characteristic equation (10) has one zero root and all the other roots have negative real part.
Next, for the 2 −order terms, we obtain Nonhomogeneous equation (32) has a solution if and only if the so-called solvability condition is satisfied [8]. That is, the right-hand side of nonhomogeneous equation (32) is orthogonal to every solution of the adjoint homogeneous problem. Substituting solution (31) into the right expression of (32), we obtain the coefficient of term e iωT0 , denoted as m 1 , and the constant term, denoted as m 2 . As a matter of fact, finding the solvability conditions is equivalent to finding the conditions resulted from eliminating the secular terms. Let h * 1 , m 1 = 0 and h * 2 , m 2 = 0, where h * j (j = 1, 2) is given by (26). Then ∂G1 ∂T1 and ∂G2 ∂T1 are solved to yield where L 1 = iω 4βeAp vt . Then, the particular solution of X 2 (t) is obtained from the resulting equation of (32) as where c.c. stands for the complex conjugate of the preceding terms, and Next, for the 3 −order terms, we obtain Substituting solution (29), (31) ,(33) and (34) into the right expression of (36), by using solvability condition, then ∂G1 ∂T2 and ∂G2 ∂T2 are solved to yield The norm form associated with Hopf-zero bifurcation is given as follows: Note that ∂G1 ∂Tj and ∂G2 ∂Tj are (j+1)-order linear homogeneous polynomial involving G 1 and G 2 . With the use of backwards scaling → 1/ , the above equation (38) which is the normal form derived using the MTS method, where ∂G k ∂T1 , ∂G k ∂T1 (k = 1, 2) are given by (33) and (37), respectively.
Remark 1. Note that system (5) is a stiff differential equation, and they differ quite a bit in the order of magnitudes among of the parameters. When τ = 0, system (5) is an ordinary differential equation, and we can show the numerical simulations in Matlab by using the command "ode15s", which is used to solve stiff ordinary differential equation (see Fig. 4). However, when τ = 0, there does not exist the command in Matlab to solve stiff delayed differential equation, thus we can not show the simulation results for τ = 0.
Remark 2. Note that normal form truncated up to 2-order, Eq. (41) is not locally topologically equivalent near the origin to the original system (5). Thus, we need to consider the higher order normal form (40). However, we can not omit the highorder term with respect to parameters, such as µ 2 i r, µ 2 i z and µ 1 µ 2 r (i = 1, 2), since there exist large difference in the order of magnitudes for these parameters, and we can not also show the numerical simulations for this special delay differential equation. Moreover, it is difficult to obtain the complete bifurcation analysis for normal form (40), and we can only show some simple analysis for certain parameters.
In accordance with above theoretical analysis, the motion of hydraulic cylinder system can be controlled in new stable states by adjusting control parameters total leakage flow C tp and the leakage delay τ , and we give the regions of parameters in which system may have a stable equalized motion. Therefore, according to the above theoretical analysis, we can choose proper controller parameters in delayed nonlinear hydraulic cylinder system in order to achieve various applications.

5.
Conclusion. In this paper, we have modelled a hydraulic cylinder system with delay, and analysed the stability of the equilibria and the existence of several types of bifurcations. To study dynamical motion, we have derived the normal forms associated with Hopf-zero bifurcation by using the multiple time scales method. Two examples, associated with hydraulic cylinder in glue dosing processes for particleboard, are presented to demonstrate the application of the theoretical results. In accordance with above theoretical analysis, reasonable hydraulic cylinder system with proper parameters can be designed in order to achieve various applications.