Theoretical Basis for Design Calculations

2.3 Theoretical Basis for Design Calculations

2.3.1 Analysis of Piston Motion

Hydraulic rock breaker design means calculating the structural parameters that will satisfy the performance requirements set out in the design specification. Under these structural parameters, the hydraulic rock breaker can achieve the required impact energy and impact frequency.

It must be strongly emphasized that the hydraulic rock breaker outputs impact energy and impact frequency through the piston moving back and forth within a fixed stroke S inside the cylinder body. Over this fixed stroke, the piston moves in a continuous cycle: return-stroke acceleration → return-stroke deceleration (braking) → return-stroke speed drops to zero → power-stroke acceleration → hits the impact point at maximum velocity v_m → hits the chisel tail (outputs impact energy) → stops, starts the next cycle. This fixed stroke S is called the piston stroke; it is an important basis for determining the cylinder body dimensions.

The piston moves back and forth inside the cylinder body. Starting from the impact point, it accelerates on the return stroke to reach the maximum return-stroke velocity v_mo, then starts to decelerate due to valve switching; the speed quickly drops from v_mo to zero — the piston stops at top dead center. The stroke the piston travels is called the return stroke. At this point, because the valve is still in its original state, the piston starts to accelerate on the power stroke until it hits the impact point. When the piston contacts the chisel tail, its velocity has reached the maximum — called the piston's maximum impact velocity v_m. The stroke the piston travels from top dead center to hitting the chisel tail is called the power stroke. Clearly, the return stroke and power stroke must be equal.

To study hydraulic rock breaker design theory more deeply, it is helpful first to understand the piston velocity, various chamber pressures, and flow distribution and variation during operation. The reasons for and direction of changes in the working parameters of a hydraulic rock breaker during operation are shown in Fig. 2-4.

p₀ is the nitrogen pre-charge pressure of the accumulator; Q is the flow delivered to the hydraulic rock breaker by the pump; Q₁ is the accumulator intake flow (+) and discharge flow (−); Q₂ is the intake flow (+) and discharge flow (−) of the piston front chamber, with Q = Q₁ + Q₂. Q₃ is the intake flow (+) and discharge flow (−) of the piston rear chamber; p is the system pressure.

Fig. 2-4 shows the piston at the start of the return stroke. The pump flow Q enters the system; one part (Q₂) enters the piston front chamber and drives its return stroke, while the rear chamber discharges oil to the tank (Q₃); the other part (Q₁) enters the accumulator and compresses the nitrogen, so the system pressure p starts from the accumulator pre-charge pressure p₀ and rises continuously as Q₁ flows in. The motion of the hydraulic rock breaker, based on piston working state, can generally be divided into three stages, described as follows:

(1) Piston return-stroke acceleration

The piston starts the return stroke from the impact point. As the pump continuously injects flow, system pressure p↑ → piston velocity v↑ → Q₂↑ → Q₁↓ → Q₃↑, and oil continues to be discharged to the tank. Because piston velocity v↑ → Q₂↑ → Q₁↓, until Q₁ = 0. The characteristic of this period is v↑ and p↑. When Q₁ = 0, a turning point appears: pressure p no longer increases, but piston speed continues to grow (because the driving force for the piston return stroke still exists). After this turning point, because v↑, the pump flow Q can no longer satisfy the flow demand for piston motion, i.e. Q₂ > Q. To satisfy the flow demand of the piston front chamber, the accumulator must now discharge oil to supplement the pump's shortfall. Based on the flow balance principle, Q₂ = Q + Q₁; at this point Q₁ is the flow flowing out of the accumulator and into the piston front chamber, until v↑ to v = v_mo, the valve switches, and the piston enters the return-stroke deceleration phase.

(2) Piston return-stroke deceleration

During the return stroke, because the piston front shoulder has passed the feedback hole, the valve switches and reverses the force direction on the piston; the driving force is applied to the piston in the reverse direction, and the piston starts to decelerate until v = 0. The return stroke is now complete; the piston has reached top dead center and traveled the full stroke S, ready for the power stroke to begin.

(3) Piston power stroke

When piston velocity drops to v = 0, the force on the piston reverses, so piston velocity v also reverses, changing from '+' to '−'. The piston then starts to accelerate on the power stroke under the reversed force. At the start of power-stroke acceleration, piston velocity starts from v = 0, at which point the piston oil consumption Q₃ = 0; all the pump discharge Q flows into the accumulator, Q₁ = Q, Q₂ = 0. As the power-stroke velocity v↑ → Q₃↑ → Q₁↓ → Q₂(−)↑. It should be noted here that because front-chamber area A₂ is smaller than rear-chamber area A₁, based on the flow balance principle, there must be Q₃ = Q₂ + Q − Q₁, with v↑ and Q₁↓, until Q₁ = 0. This means v↑; at this point all the pump discharge Q is fully injected into the piston rear chamber, i.e. Q₃ = Q, Q₁ = 0, but the piston velocity v has not yet reached maximum velocity v_m. The piston continues to accelerate; the pump flow Q can no longer satisfy the demand, so the accumulator starts to supplement the flow, i.e. Q₃ = Q + Q₁(−), until the piston hits the chisel tail at maximum velocity v_m. At the instant of impact, piston velocity suddenly becomes v = 0, and the piston outputs impact energy W externally, completing one working cycle.

As the accumulator intake/discharge flow Q₁ changes, system pressure p also changes accordingly. When charging the accumulator, Q₁ = '+', system pressure p↑; when the accumulator discharges to the outside, Q₁ = '−', system pressure p↓. In other words, the working process of a hydraulic rock breaker is always accompanied by changes in system pressure. When the most oil has been charged into the accumulator, system pressure is at its highest. When the piston has reached the impact point, the accumulator has discharged the most oil — this is the moment of lowest system pressure. Therefore, from the time the hydraulic rock breaker starts up until it reaches steady operation, its system working pressure p always cycles back and forth between a maximum pressure p_max and a minimum pressure p_min, and it is absolutely impossible for it to be constant and unchanging. Fig. 2-5 shows the variation of all system parameters when the hydraulic rock breaker is operating.

Fig. 2-5 Variation of system parameters during operation of a hydraulic rock breaker [Legend: hatched = accumulator charging; cross-hatched = accumulator discharging; white = piston oil consumption]

The working process described above shows that the variation in working parameters is quite complex — it is a nonlinear system. This creates considerable difficulty for in-depth theoretical analysis and research. In fact, this is one of the main reasons why theoretical research on hydraulic rock breakers has lagged behind product development.

2.3.2 Current State of Theoretical Research

Researchers worldwide have generally taken two different technical approaches to theoretical research on hydraulic impact devices (hydraulic rock breakers): research based on linear system theory and research based on nonlinear system theory.

1) Research based on linear system theory assumes the force on the piston is constant, the piston velocity increases linearly at a uniform rate, and certain influencing factors are ignored; a linear mathematical model is built on this basis for theoretical research. This research method is clearly simple and can solve some practical problems, but it is not very accurate and has considerable errors.

2) Research based on nonlinear system theory uses high-order nonlinear differential equations to describe the motion patterns of the hydraulic rock breaker, and more accurately depicts the kinematics and dynamics of the hydraulic rock breaker piston. This nonlinear research is more accurate than linear research, but still relies on some assumptions. While it can more accurately reveal some physical phenomena of hydraulic impact, it is difficult to solve, not easy to interpret, and can only produce numerical solutions through computer calculation, which makes it inconvenient to use.

In addition to these two approaches, the authors, after many years of dedicated research, proposed the Abstract Variable Design Theory for Hydraulic Rock Breakers (hydraulic impact mechanisms). Using abstract variable design theory, analytical solutions for hydraulic rock breakers can be found, which can deeply reveal the internal patterns of hydraulic rock breaker motion and provide a theoretical basis for technical innovation by users.

The research approach of hydraulic rock breaker abstract variable design theory: acknowledging the nonlinearity of hydraulic rock breaker working parameters, but using equivalent force transformation to linearize the nonlinear system, so that it can be studied using linear system methods to obtain analytical solutions. The working parameters and structural parameters of hydraulic rock breakers obtained with this method are quite accurate and the calculation is simple. Hydraulic rock breaker abstract variable design theory will be covered specifically in subsequent chapters.