If w = 1⁄2 GΩ, P1 and P2 are constant throughout the stroke; and the moving force in ascending and descending is the same, if
B = W + R + wS − GΩ (H + Hb) / 2.
Using the values just found for w and B,
P1 = P2 = 1⁄2 GΩ (H − Hb) − F.
Let W + R + wS + B = U, and let P be the constant accelerating force acting on the system, then the acceleration is (P/U)g. The velocity at the end of the stroke is (assuming the friction to be constant)
v = √ (2PgS / U);
and the mean velocity of ascent is 1⁄2v.
| Fig. 172. |
§ 177. Armstrong’s Hydraulic Jigger.—This is simply a single-acting hydraulic cylinder and ram, provided with sheaves so as to give motion to a wire rope or chain. It is used in various forms of lift and crane. Fig. 172 shows the arrangement. A hydraulic ram or plunger B works in a stationary cylinder A. Ram and cylinder carry sets of sheaves over which passes a chain or rope, fixed at one end to the cylinder, and at the other connected over guide pulleys to a lift or crane. For each pair of pulleys, one on the cylinder and one on the ram, the movement of the free end of the rope is doubled compared with that of the ram. With three pairs of pulleys the free end of the rope has a movement equal to six times the stroke of the ram, the force exerted being in the inverse proportion.
§ 178. Rotative Hydraulic Engines.—Valve-gear mechanism similar in principle to that of steam engines can be applied to actuate the admission and discharge valves, and the pressure engine is then converted into a continuously-acting motor.