Optimae Calculations Parametrorum Impulsus et Kinematicae

4.2 Calculatio Parametrorum Optimalium Cursus et Cinematicorum

Ex diagrammate velocitatis operativae pistons linearis quoque patet quod, cum α mutatur, cursus pistons S etiam mutatur. Alioquin, dato fixo v _m et T , cursus (cursus potentiae) S functio est α , id est S = f (α ).

Ex diagrammate velocitatis 4-1:

S = ½ v _m T ₁

S = ½ v _mo T ₂

T ₁ = T − T ₂

α = T ₁ / T                                                                              (4.7)

Reordinata aequatione (4.7), cursus pistons est:

S = ½ αv _m T                                                                           (4.8)

Postquam optima α = α _u selecta est, cursus optimus pistoni frangendi lapidis hydraulici designati ex aequatione (4.8) calculare potest. Itaque cursus optimus pistoni est:

S _u = ½ α _u v _m T                                                                         (4.9)

In aequatione (4.9), parameter α _u in capitulis posterioribus tractatur.

A:

½ v _m T ₁= ½ v _mo T ₂= ½ v _mo (T − T ₁)                                                 

Post repositionem, maxima velocitas cursus reditus est:

v _mo = αv _m ⁄ (1 − α ) (4.10)

Exprimendo T ₂per quantitates notas α et T , tempus cursus reditus est:

T ₂= (1 − α )T                                                                      (4.11)

A:

T ₂^″ / T ₁ = v _mo / v _m                                                                          

Post repositionem, tempus frenandi in cursu reverso est:

T ₂^″ = α ²⁄ (1 − α ) · T                                                             (4.12)

Omnes alii pertinentes parametri cinematici nunc singillatim inveniri possunt.

Tempus accelerationis in cursu reverso:

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

Distantia accelerationis in cursu reverso:

S _j = α (1 − 2 α ) / [2(1 − α )²)] · v _m T                                            (4.14)

Ex aequatione (4.8):

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

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

Distantia frangendi in cursu reditus:

S _s = α ³/ [2(1 − α )²)] · v _m T                                                       (4.17)

Aut:

S _s = α ²⁄ (1 − α )² · S                                                             (4.18)

Acceleratio in cursu potentiae:

a ₁ = v _m \/ ( αT ) (4.19)

Acceleratio in cursu reditus:

a ₂ = α / (1 − 2 α ) · v _m / T                                                       (4.20)

Tempora incaricandi et descaricandi accumulatorem durante cursu potentiae ex theoria designis accumulatores deduci possunt. Propter integritatem formularum calculationis cinematicae, hic subiiciuntur.

Tempus incaricandi accumulatorem durante accelerationis phase cursus potentiae:

T ₁^′ = α ²/ 2 · T                                                                     (4.21)

Tempus descensus accumulatōris in phāse accelerātiōnis per impulsum potentiae:

T ₁^″= ( α − α ²/ 2) T                                                               (4.22)