Abstract Variable Design Theory for Hydraulic Rock Breakers

The research idea behind abstract variable design theory: no matter how the working parameters of a hydraulic rock breaker change during operation, the two parameters that satisfy the design requirements — impact energy W_H and impact frequency f_H — must not change; as for the other parameters, they are not particularly important to the designer, and especially not to the user. However, the designer should pay special attention to the piston stroke S, because every behaviour of the piston occurs over a fixed stroke S, and the piston stroke S is constrained by the structure — it cannot be arbitrary. Too large a stroke is not allowed by the mechanical structure; too small a stroke cannot satisfy the requirements for impact energy and impact frequency. In other words, it is a constraint on the operation of the hydraulic rock breaker, and there must be an optimal value.

How to treat the design calculation problem of a hydraulic rock breaker — which is in reality a nonlinear system — using linear methods is the core content of this chapter.

3.1   The Equivalent Force Principle

— Theoretical Basis for Converting a Nonlinear System into a Linear System

When a hydraulic rock breaker is operating, the working parameters — such as system pressure p, piston velocity v, acceleration a, and piston load — all change nonlinearly and are functions of time. Calculating such a system is quite difficult and complex. But the design objective in this book is relatively simple: to find the structural parameters and working parameters of a hydraulic rock breaker that can deliver the required impact energy W_H and frequency f_H. The impact energy formula is:

W_H = (m / 2)v²_m                                                                    (3.1)

where: m — piston mass, constant;

       v_m — instantaneous velocity when the piston strikes the chisel tail, i.e. the maximum impact velocity; this is the velocity that must be guaranteed in design.

There are two conditions for ensuring the required impact energy is achieved: the piston must have a certain mass and a certain velocity. For a hydraulic rock breaker, piston mass m cannot change during motion. So ensuring impact energy is achieved means ensuring that the maximum impact velocity v_m is reached.

It must be pointed out that piston motion occurs over a given stroke. In other words, the purpose of design calculation for a hydraulic rock breaker is to ensure that, over a given stroke, a piston of fixed mass is accurately accelerated to the specified maximum impact velocity v_m within the specified cycle time T, hitting the chisel tail and outputting the specified impact energy W_H. The instantaneous changes of a, v, and p during motion are not important to the design calculation objective and can be ignored. Ensuring cycle time T also ensures the specified impact frequency f_H.

Cycle time T and impact frequency f_H satisfy f_H = 60 / T, where T is the piston working cycle time (for simplicity of calculation, the brief pause at the impact point is ignored).

If a simple design calculation method could be found to achieve the above objective, it would be useful for engineering design. As is well known, hydraulic oil pressure drives the piston to do work; based on the law of conservation of energy and ignoring other energy losses, all of this work converts into piston kinetic energy and is output externally, giving the following relation:

(m / 2)v²_m = ∫₀^SF(S) dS                                                           (3.2)

The physical meaning of Eq. (3.2): the right-hand side is the work done by the varying force F(S) over stroke S; the left-hand side is the kinetic energy gained by the piston while moving over stroke S.

To achieve linearised calculation, one can imagine a constant force F_g doing the same work as the varying force F(S) over the same stroke S. So the constant force F_g can replace the varying force F(S) in linearised calculation with equal effect, giving:

(m / 2)v²_m = ∫₀^SF(S) dS = F_g × S                                              (3.3)

Substituting Eq. (3.1) into Eq. (3.3) gives:

F_g = W_H / S                                                                          (3.4)

In Eq. (3.4), the constant force F_g is called the equivalent force; it does exactly the same work as the varying force F(S).

Eq. (3.4) is the formula for calculating the equivalent force. Impact energy W_H = (m/2)v²_m is specified by the design task and is a known parameter. Stroke S can be obtained from kinematics calculations and is also known; therefore the equivalent force required to achieve the needed impact energy can be calculated. The correct selection of design stroke S and the frequency f_H, as well as the optimisation of stroke S, will be introduced gradually in later chapters.

This equivalent force is very useful in hydraulic rock breaker design calculations. Based on the equivalent force, the pressure-bearing area of the piston — i.e. the structural dimensions of the piston — can be found, the working conditions and effective volume of the accumulator can be determined, and kinematics and dynamics calculations for the hydraulic rock breaker can be performed.

The piston pressure-bearing area is:

A = F_g / p_g                                                                           (3.5)

In Eq. (3.5), p_g is the equivalent oil pressure of the system, corresponding to the concept of equivalent force, and is a virtual variable. However, considering that oil motion involves resistance, the actual system working oil pressure must be higher than the equivalent oil pressure, so the rated pressure used in design is:

p_H = Kp_g                                                                              (3.6)

In Eq. (3.6), K = 1.12 to 1.15 is the resistance coefficient for hydraulic system operation. The value of p_H is in practice chosen based on the overall requirements of the system being designed, so the pressure-bearing area of the piston becomes calculable and known. Therefore:

A = KF_g / p_H                                                                         (3.7)

Substituting Eq. (3.4) gives:

A = KW_H / (p_HS)                                                                   (3.8)

It must be pointed out that the kinematics and dynamics results calculated from the above are not fully realistic — they are described as linearly varying, i.e. piston motion is treated as uniformly accelerated and uniformly decelerated. However, the piston cycle time T, maximum velocity v_m, and motion stroke S are real; for satisfying design requirements, they are simple, practical, and accurate.

In fact, the most critical question is whether the impact energy W_H, impact frequency f_H, and flow Q driving the hydraulic rock breaker are real. Because piston pressure-bearing area A is fixed and stroke S is fixed, it follows that the pump flow Q is necessarily also real.

In this way, applying the equivalent force principle can simplify the nonlinear hydraulic rock breaker design calculation into a linear one; both kinematics and dynamics calculations can be greatly simplified and treated as uniformly accelerated and uniformly decelerated motion.

The academic insight of the equivalent force is to ignore the complex process, grasp the essence of the problem, and linearise the nonlinear problem. But the results needed are very real and reliable, and are helpful for deepening the understanding and exploration of the operating patterns of the hydraulic rock breaker.

3.2   Piston Motion Dynamics

Based on the equivalent force principle, the piston velocity and forces are as shown in Fig. 3-1, comprising three stages: return-stroke acceleration, return-stroke deceleration (braking), and power stroke.

(1) Dynamics equation for the piston return-stroke acceleration stage

Let the return-stroke driving force F_2g, velocity v, and acceleration a be defined as [+]. The equivalent driving force that accelerates the piston on the return stroke is:

F_2g = p_gA^′₂ = ma₂                                                                   (3.9)

where: a₂ = [+] — return-stroke acceleration of the piston;

       A^′₂ — effective pressure-bearing area of the piston front chamber;

       p_g — equivalent pressure of the system.

(2) Dynamics equation for the piston return-stroke deceleration stage

The equivalent driving force that decelerates the piston on the return stroke is:

F_3g = p_gA^′₁ = ma₃                                                                 (3.10)

where: a₃ = [−] — deceleration (braking) of the piston on the return stroke.

(3) Dynamics equation for the piston power stroke stage

The equivalent driving force that accelerates the piston on the power stroke is:

F_1g = p_gA^′₁ = ma₁                                                                 (3.11)

where: a₁ = [−] — acceleration of the piston on the power stroke;

       A^′₁ — effective pressure-bearing area of the piston rear chamber.

The concept of effective pressure-bearing area differs depending on the three different working principles of the hydraulic rock breaker described above; it is discussed in detail in the dynamics chapter.