L2

Pi

2

expended on the compression l, is represented by 1⁄2KE,

therefore u =

L1

(KE+KE) P2; or substituting for P its

448. The velocity of the impinging body at any period of the impact, the impact being supposed to take place vertically.

It is evident that at any period of the impact, when the velocity of the impinging body is represented by v, there will have been expended, upon the compression of the two bodies, an amount of work which is represented by the work accumulated in the impinging body before impact, increased by the work done upon it by gravity during the impact, and diminished by that which still remains accumulated in it, or

Representing, therefore, by u the work expended upon the compression of the bodies, we have

W.

· V2 + WI

g

Substituting, therefore, for u its value from equation

Or substituting for its value in terms of P (equation 716),

THE PILE DRIVER.

449. It is evident that the pile will not begin to be driven until a period of the impact is attained, when the pressure of the ram upon its head, together with the weight of the pile, exceeds the resistance opposed to its motion by the coherence and the friction of the mass into which it is driven. Let this resistance be represented by P; let V represent the velocity of the ram at the instant of impact, and v its velocity at the instant when the pile begins to move, and W1, W, the weights of the ram and pile; then, since the pile will have been at rest during the whole of the inter

2

vening period of the impact, since moreover the mutual pressures Q of the surfaces of contact are, at the instant of motion, represented by P-W2, we have by equation (720)

If the value of v determined by this equation be not a possible quantity, no motion can be communicated to the pile by the impact of the ram: the following inequality is therefore a condition necessary to the driving of the pile,

(P – W2)2

KE2 W1 −2(P_W2) } . (722).

:) {

1

After the pile has moved through any given distance, one portion of the work accumulated in the ram before its impact will have been expended in overcoming, through that distance, the resistance opposed to the motion of the pile; another portion will have been expended upon the compression of the surfaces of the ram and pile; and the remainder will be accumulated in the moving masses of the ram and pile. The motion of the pile cannot cease until after the period of the greatest compression of the ram and pile is attained; since the

reaction of the surface of the pile upon the ram, and therefore the driving pressure upon the pile, increases continually with the compression. If the surfaces be inelastic, having no tendency to recover the forms they may have received at the instant of greatest compression, they will move on afterwards with a common velocity, and come to rest together; so that the whole work expended prejudicially during the impact will be that expended upon the compression of the inelastic surfaces of the ram and pile. If, however, both surfaces be elastic, that of the ram will return from its position of greatest compression, and the ram will thus acquire a velocity relatively to the pile, in a direction opposite to the motion of the pile. Until it has thus reached the position in respect to the pile in which it first began to drive it, their mutual reaction Q will exceed the resistance P, and the pile will continue to be driven. After the ram has, in its return, passed this point, the pile will still continue to be driven through a certain space, by the work which has been accumulating in it during the period in which Q has been in excess of P. When the motion of the pile ceases, the ram on its return will thus have passed the point at which it first began to drive the pile: if it has not also then passed the point at which its weight is just balanced by the elasticity of the surfaces, it will have been continually acquiring velocity relatively to the pile from the period of greatest compression; it will thus have a certain velocity, and a certain amount of work will be accumulated in it when the motion of the pile ceases: this amount of work, together with that which must have been done to produce that compression which the surfaces of contact retain at that instant, will in no respect have contributed to the driving of the pile, and will have been expended uselessly. If the ram in its return has, at the instant when the motion of the pile ceases, passed the point at which its weight would just be balanced by the elasticity of the surfaces of contact, its velocity relatively to the pile will be in the act of diminishing ; or it may, for an instant, cease at the instant when the pile ceases to move. In this last case, the pile and ram, for an instant, coming to rest together, the whole work accumu

lated in the impinging ram will have been usefully expended in driving the pile, excepting only that by which the remaining compression of the surfaces has been produced; which compression is less than that due to the weight of the ram. This, therefore, may be considered the case in which a maximum useful effect is produced by the ram. The following article contains an analytical discussion of these conditions under their most general form.

450. A prism impinged upon by another is moveable in the direction of its axis, and its motion is opposed by a constant pressure P: it is required to determine the conditions of the motion during the period of impact, the circumstances of the impact being in other respects the same as in Article 448.

Let f and f represent the additional velocities which would be lost and acquired per second (see Art. 95.) by the impinging prism and the prism impinged upon, if the pressures, at any instant operating upon them, were to remain

W.

from that instant constant; then will W, repre

sent the effective forces upon the two bodies (Art. 103.) or the pressures which would, by the principle of D'Alembert, be in equilibrium with the unbalanced pressures upon them, if applied in opposite directions.

A

B

Now the unbalanced pressure upon the system BP composed of the two prisms is represented by (W1+ W2-P),

2

also the unbalanced pressure upon the prism PQ =W2+Q-P, where Q represents the mutual pressure of the prisms at Q;

2

Let A have been the position of the extremity B of the

impinging prism at the instant of impact; and let a represent the space through which the aggregate length BP of the two prisms has been diminished since that period of the impact, and X2 the space through which the point P has moved; then (equation 716)

Also AB=æ1+x; therefore velocity of point B=d(x, +x2)

dt

Substituting these values off, and Q in equations (723) and (724), and eliminating f, between the resulting equations,

Integrating this equation by the known rules, we obtain

in which expression the value of y is determined by the

and A and B are certain constants to be determined by the conditions of the question. Substituting in equation (724) the value of Q from equation (725), and solving in respect to fa

Substituting for

X1 its value from equation (727), and for f, its

value and reducing,

dt

Integrating between the limits 0 and t, and observing that