The mathematical model is developed based on the physical structure of a single-degree-of-freedom oil-gas spring, as illustrated in Figure 2. The initial position of the hydraulic cylinder piston corresponds to the equilibrium point of the oil-gas spring under a specific preload. When an excitation signal x is applied, the pressure inside the cylinder chamber becomes p1. At the junction between the damping valve and the pipeline, the pressure is p2, while the pressure at the accumulator outlet is p3. Inside the accumulator, the oil pressure is p4 and the gas pressure is pg. The initial oil pressure is denoted as p0. It is assumed that the temperature of the oil and gas spring remains constant during its operation.
In real hydraulic systems, a small amount of gas is typically mixed into the oil, which means that the compressibility of the oil and its low-pressure nonlinearity must be taken into account. According to fluid mechanics principles, the instantaneous density Q of the oil can be expressed as:
Q = Qâ‚€[1 + (p - pâ‚€)/Bâ‚€] (1)
where Qâ‚€ represents the oil's density at standard conditions, Bâ‚€ is the bulk modulus of the oil, and p is the instantaneous pressure of the oil.
According to fluid theory, the mass flow rate m, defined as the product of the volume flow rate Q and the fluid density Q, remains constant, i.e., m = Q * Q (2). During the operation of the oil-gas spring, the force acting on the piston rod consists of two components: the pressure inside the hydraulic cylinder and the friction between the piston and the cylinder wall. This includes both static friction Fsta and dynamic friction Fdyn. Since there is a transition from static to dynamic friction, the dynamic friction force can be modeled as:
Fdyn = [1 - LDmin(x, utr)/utr] * Fsta (3)
where LD = (Fsta - Fdyn)/Fsta, x is the velocity of the excitation signal (positive for extension, negative for compression), and utr is the threshold speed for full dynamic friction. Therefore, the total force exerted by the oil-gas spring is given by:
F = pâ‚Aeff - Fdyn * sign(x) (4)
where Aeff is the effective area of the piston, and sign(x) is the sign function.
The mass flow rate of the oil within the hydraulic cylinder is given by:
m = Qâ‚ * Qâ‚ (5)
where Q₠is the volumetric flow rate of the oil in the cylinder, calculated as Q₠= x * Aeff, with x being the piston rod velocity and Aeff the effective piston area. Q₠also represents the density of the oil in the cylinder. Additionally, the oil density in the pipeline is denoted as Q₃, with Dp representing the pipe diameter, lp the pipe length, Q₃ the volumetric flow rate in the pipeline, Ap the cross-sectional area of the pipe, and K the resistance coefficient along the path, defined as:
K = 2 / [lg(ReK) - 0.8] (9)
For the accumulator performance analysis, since the accumulator outlet is connected to the oil pipeline, there is a partial pressure loss at the outlet. The relationship between the pressures can be described as:
p₄ - p₃ = (1/2) * Q₄ * N * Q₄ / Aa² * sign(x) (10)
where Qâ‚„ is the oil density at the accumulator outlet, N is the partial pressure loss coefficient, Qâ‚„ is the volumetric flow rate at the outlet, and Aa is the cross-sectional area of the accumulator outlet.
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