Don C. Warrington, P.E.
Originally presented at the Colloquium of the Department of Mathematics of the
26 September 2006.
This paper details the mathematical modelling of vibratory pile driving systems using a linear model with the objective of obtaining a closed form solution to estimate either the power requirement of the machine, the torque requirement of the motor driving the eccentrics, or both. It begins by reviewing the system model for the system without a suspension, which is used to enable connection of the vibrating machine with a crane, a mast of a dedicated machine, or an excavator. It proceeds to solve the equations of motion for a system with a suspension, using
transforms and solving the inverse transform using residues and complex integration. The model indicates that, under certain conditions, both the amplitude and the power consumption of the system increase with a suspension, but the results make the practical implications of the result uncertain. Finally a simple set of equations is developed for actual vibratory design which results in the suspension being ignored and the necessary torque of the driving motor computed.
As one remains in an industry for an extended period of time, the time comes all too often to say farewell to colleagues who depart. Such is the present time; this paper is dedicated to the memory of Lev Vikorovitch Erofeev, Chief of the Pile Driving Equipment Design Department at VNIIstroidormash, the All-Union Amalgamation for Construction and Road Machinery in
, who passed away in 2002.
Right: Lev Erofeev (left) with V.S. Warrington, Vulcan's Chairman of the Board (centre) and V.A. Nifontov (right) during a visit to Vulcan's facility in the early 1990's.
I first met Lev in 1988, on my first trip to the
. At the time it was difficult to make contact with Soviet people and institutes, especially for a small concern such as Vulcan Iron Works. The combination of poor communications with the tendency of too many intermediaries to “get in the way” almost ended our relationship. In 1990, while on tour in
with CBN, I sent Lev a letter, which would have received an earlier response had he not been in his dacha, but which re-established contact. With the end of the
the following year, this led to proper relationships and the cooperation between Vulcan, VNIIstroidormash and other Russian organisations that were a bright spot in Vulcan’s last years.
With mutual visits, I found Lev to be a man of integrity, a fine engineer and a good friend. Our contact was a voyage of mutual discovery, with such things as finding his VNIIstroidormash department room piled high with hydraulic hose, or his attentive viewing of the two-hour long Jesus video in Vulcan’s board room. Lev retired just about the time that Vulcan passed out of the
family. For both of us the end came too soon, but for Lev it was especially difficult. In his earlier days he had been involved in work in Siberia, downriver from one of the
’s nuclear sites. Eating mutant crawfish out of the river, he developed radiation poisoning, which combined with Soviet medicine disfigured him and, in his later years, made it difficult for him to speak.
Lev told the story about the Soviet engineer who designed the minipile system for the Ostankino Tower in Moscow. Minipiles were a novelty at the time, which meant that much of the Soviet engineering and construction establishment panned the concept. The minipiles worked, but the engineer passed away, after which time he was honoured for his brilliance. Lev noted that people who are trashed while they are living are frequently lionised after they are gone.
It has been my objective since our first joint article in Pile Buck in 1991 that Lev receive the recognition that he deserves for his contribution to the advancement of the technology of pile driving equipment. Without the information he furnished, the entire effort that started with my OTC paper in 1989 would not have been possible. Let this paper be a final tribute to Lev Viktorovitch, and beyond this let us remember the following: “Urge upon them to show kindness, to exhibit a wealth of good actions, to be open-handed and generous, storing up for themselves what in the future will prove to be a good foundation, that they may gain the only true Life.” (1 Timothy 6:18-19, Positive Infinity New Testament)