Optimal Stroke and Kinematics Parameter Calculations

4.2   Optimal Stroke and Kinematics Parameter Calculations

From the linearised piston working velocity diagram it is also clear that as α changes, piston stroke S also changes. In other words, given fixed v_m and T, stroke (power stroke) S is a function of α, i.e. S = f(α).

From velocity diagram 4-1:

S = ½ v_mT₁

S = ½ v_moT₂

T₁ = T − T₂

α = T₁ / T                                                                             (4.7)

Rearranging Eq. (4.7), the piston stroke is:

S = ½ αv_mT                                                                          (4.8)

Once the optimised α = α_u has been selected, the optimal stroke of the designed hydraulic rock breaker can be calculated from Eq. (4.8). Therefore the piston optimal stroke is:

S_u = ½ α_uv_mT                                                                        (4.9)

In Eq. (4.9), the parameter α_u is discussed in later chapters.

From:

½ v_mT₁ = ½ v_moT₂ = ½ v_mo(T − T₁)                                                 

After rearranging, the maximum return-stroke velocity is:

v_mo = αv_m / (1 − α)                                                               (4.10)

Expressing T₂ in terms of the known α and T, the return-stroke time is:

T₂ = (1 − α)T                                                                     (4.11)

From:

T₂^″ / T₁ = v_mo / v_m                                                                         

After rearranging, the return-stroke braking time is:

T₂^″ = α² / (1 − α) · T                                                            (4.12)

All other relevant kinematics parameters can now be found one by one.

Return-stroke acceleration time:

T₂^′ = (1 − 2α) / (1 − α) · T                                                   (4.13)

Return-stroke acceleration distance:

S_j = α(1 − 2α) / [2(1 − α)²] · v_mT                                           (4.14)

From Eq. (4.8):

S_j = (1 − 2α) / (1 − α)² · S                                                    (4.15)

S_j / S = (1 − 2α) / (1 − α)²                                                    (4.16)

Return-stroke braking distance:

S_s = α³ / [2(1 − α)²] · v_mT                                                      (4.17)

Or:

S_s = α² / (1 − α)² · S                                                            (4.18)

Power-stroke acceleration:

a₁ = v_m / (αT)                                                                      (4.19)

Return-stroke acceleration:

a₂ = α / (1 − 2α) · v_m / T                                                      (4.20)

The accumulator charge and discharge times during the power stroke can be derived from the accumulator design theory. For the sake of completeness of the kinematics calculation formulas, they are given here.

Accumulator charging time during the power-stroke acceleration phase:

T₁^′ = α² / 2 · T                                                                    (4.21)

Accumulator discharging time during the power-stroke acceleration phase:

T₁^″ = (α − α² / 2) T                                                              (4.22)