APPENDIX E ENERGY ANALYSES OF WEIGHT-DROP TESTS AND COMPARISONS BETWEEN EXPERIMENTAL AND ANALYTICAL RESULTS

For the mechanical system in Figure 3.7, the following equation of energy balance can be established: 

		(1/2)Mvi2 + (M+mp)gum + mgum/2 = (1/2)Fmum   
                                                              (E1)
where vi and M are the velocity and mass of the dropped steel weight respectively; Fm and um are the maximum force and displacement respectively; m and mp are the masses of the steel rod and the bottom plate respectively; and g is the gravitational acceleration. Substituting the relationship of um = k-1Fm with k being the stiffness of the steel rod, into equation (E1), we obtain the following equation: 

		Fm2 - 2(M+mp+m/2)gFm - Mkvi2 = 0   
                                                              (E2)
From the above equation, the maximum force can be obtained as follows: 

		Fm = (M+mp)g[1+1/(2b)] + {(M+mp)2g2[1+1/(2b)]2 + Mkvi2 }1/2              

                                                              (E3)
where the mass ratio b = (M + mp)m-1. If the two sides of the above equation is divided by the cross sectional area of the bolt shank A, the maximum stress sm in the steel rod can be expressed as: 

		sm = ss + [ss2 + A-2Mkvi2 ]1/2              

                                                              (E4)
where ss is the gravitational stress due to the masses of the dropped weight, the bottom plate and the steel rod with ss = (1+0.5b-1) sst and sst = (M + mp)gA-1. Substituting vo = viM(M + mp)-1 (equation (C1) in appendix C) and the relationships of k = EAL-1 and E = rc2 (where E is the Young's modulus of the steel rod) into the above equation, we have: 

		sm = ss + [ss2 + (1+mp/M)b so2 ]1/2              

                                                              (E5)
where the initial stress so = rcvo = rcvi(1 + mpM-1)-1. If the mass ratio b approaches infinity, the mechanical system is equivalent to a harmonic vibrational system. The circular frequency w of the harmonic vibration is w = [k/(M + mp)]1/2. The half period of the vibration T is: 

		T = p/w (b)1/2              

                                                              (E6)
where t is the time for a wave to propagate the length of the steel rod with t = Lc-1.



(Yi 2019 note) For a simple model vibrational system, of an elastic spring of stiffness, k, with a fixed top end, plus a fixed weight, of mass, M, at the bottom end of the spring, the displacement, x, of the weight is: 

		x = Am cos(w t)               

                                                              (E7)
where, Am, is the displacement amplitude of the harmonic vibration, and, t, is the elaspe of time; w = 2pf = (k/M)1/2. The Taylor series expansion of equation (E7) is: 
		x / Am = 1 -  (w t)2 / 2! + (w t)4 / 4! - (w t)6 / 6! + etc. + (-1)n (w t)2n / (2n)!          

                                                              (E8)
where the power, 2n, is even integer number, and, (2n)!, is its permulation, for integer number n > 0. In comparison, the Taylor series expansion of a sine function would include only a series of odd integer number term of power, (2n+1), and its permulation, (2n+1)!.

One may compare the above formula (E8) for a simple model vibrational system, to the formulae (C14) to (C16) of Appendix C, for a series of wave generation, revolution and superposition. Apparently, the latter wave formulae do not simplify into the former vibration formula, e.g., the above (E8) does contain an infinite series of term, with even number of power, 2n, and its permutation (2n)!, but formula (C14) does show a finite series of term, with the continuous power, n, and permulation, n!. Furthermore, the above formula (E8) is a sum of infinite term at any time moment, but, formulae (C14) to (C16) show a sum in historic time from the very beginning of impact till a certain time moment.

For the above simple vibration model, one may attach a viscous damping element to the spring in parallel connection, to simulate vibration decay with the elapse of time, where the damping force is directly proportional to the velocity of the weight (ref. on the one hand, river dam is like a damping force / power on a river, to slow down or cut off water flow, fish and plant migration, and on the other hand, there is Chinese idiom to praise rain and flower fall but to satirize fast flow of water, simply in PinYin "Luo4 Ye4 Gui1 Geng1" and "Luo4 Hua1 You3 Yi4, Liu2 Shui3 Wu2 Qing2" with English translation "rain fall and leaf fall return to root" and "rain fall and flower fall have will, there is flow water without sentiment"). Such simulation is somewhat similar to the wave reflection coefficient at the top and bottom end fixture point, R = Rt + Rb, in the formulae (C14) to (C19) of Appendix C.



Table E.1 Comparisons between experimental parameters and those from energy analysis

Test No.

Stress amplitude from energy analysis, sm (MPa)

Half period from energy analysis, T (milli-sec.)

Relative difference in sm for top strain gauge

(%)

Relative difference in T for top strain gauge

(%)

Relative difference in sm for middle strain gauge (%)

Relative difference in T for middle strain gauge (%)

1-2

118

1.5

29+3

13+7

10+10

13+7

1-3

198

1.5

68+5

-7+7

17+4

-7+7

2-2

168

2

47+6

5+5

36+11

5+5

2-3

281

2

27+2

10+5

7+5

5+0

3-2

239

2.7

32+1

11+4

13+10

4+7

3-3

399

2.7

17+4

15+4

-1+2

11+0

4-1

127

3.2

24+6

3+3

11+7

6+6

4-2

284

3.2

23+4

6+3

4+3

6+0

5-1

142

3.6

46+5

28+3

27+5

25+3

5-2

317

3.6

37+5

25+3

20+2

31+6

6-1

320

7.9

3+1

10+1

-6+2

8+1

Notes:

A relative is shown in percentage. It is the experimental value minus and then divided by the theoretical value.