500 - 16 .- 



for both Type I ana Type II motion. This equation gives on differentiation 



which is the same as equation (45). Thus, on the assumption (66), we obtain the same equation for 

 W In Doth Types I and M motion. While the particular cases of load to De considereO later can Be 

 sclved explicitly without making the assumption that (T^^ - T^)/T^ is small, the simplicity introduced 

 by only 3ne equation for W during all the moticn is very gre^.t. Since this assumption will be 

 found reasonable, a posteriori, in application to the Box Model, and since any attempt at great 

 accuracy is meaningless for such application in view of the approximate nature of both method of 

 analysis and physical assumptions, we shall adopt the assumption (66) that (T^j - '^^)f'^fj '^ small in 

 the remainder of our analysis. 



We shall thus use equation (68) f=r both Types I and II and for any given type of load we 

 can then evaluate W as a function of time. Thence from (29) we can obtain J as a function of time 

 and the various stages can then easily be identified by virtu* ?f the following geometrical properties 

 of the curves for S and Sj^ age.lnst time, 



(a) During Type I motion, (U6) indicates that S is a straight line of positive slope 

 Y which must lie below the S curve to satisfy (U3). 



(b) During Type II motion, (52) indicates that the S curve is coincident with the S 

 Curve which must have a slcoe-^ Y by (56). 



(c) During rest, S = 3 = so that both curves become coincident with the time axis 

 and will remain so provided (6l) is satisfied. 



Since S 5> for Types I, I I or rest, which have alone been pr.ved possible, we must not 

 continue the solution of (68) for negative values of Wand S, .i.e., we first consider only the portion 

 of the S curve up to where it cuts the axis of time. Immediately prior to this time, the motion will 

 be of Type II since in Type I motion Sj^ increases steadily and at the end of Type I motion we must 

 therefore have S = S, > 0; this is also geometrically obvious from (a) above. But if the motion Is 

 of Type II and the S curve is cutting the time axis, whence S = W = 0, then from (57) this can only 

 happen if A P{t) - T W is becoming negative and this in turn Implies from (6l) that a state of rest 

 is pcssible. This state of rest will then continue indefinitely unless P(t) subsequently Increases 

 to violate (61) and further motion will then ensue. Such further motion can then Be treated in the 

 sane way by using equation (68) and considering the S curve, the conditions at the recommencement of 

 motion being the same as initially save that W » but is equal tc the value of W at the first 

 cessation of motion. In the two specific cases of lead which we will new consider, P(t) does not 

 increase after the first cessation of motion, so that no recoranencement of motion Is possible and it 

 is only necessary to consider the S curve up to the time when it first cuts the time axis. 



A. 11 Summary of procedure for solution . 



Summing up from the preceding section, our procedure for solution with any given form of 

 load P{t) is as follows: 



(1) Equation (68) is solved with initial conditions W = W - 0, 



(2) The curve for S versus t is then obtained up to the time at which It first cuts the time axis, 



(3) If the slope of the S curve is alvays ^ Y, then Type I motion Is not possible and^the motion 

 is thus jf Type II throughout. In this case the S, curve is identical with the S curve 

 throughout the motion. 



(«) If the slope of the S curve exceeds Y over a finite portion of the curve then Type I motion 



will commence directly the slope exceeds Y and the curve for S, versus t will become a line 

 of slope Y. Type I motion will cease and Type II ensue when this line cuts the S curve. 

 In this case the S, curve follows the s curve until S first exceeds Y and S, then proceeds 

 along a chord of slope Y until this cuts the S_curve when the S^ curve again follows the 

 S curve; this process may be repeated If the S curve is oscillatory so that S later again 

 exceeds Y. 



(5) 



