The annular cross-section of hollow cylindrical elements reveals the possibility of using longitudinal and rotational oscillations when sinking these elements into the ground.

Early experience showed that, in order to achieve breakthrough into the adjacent ground with pipes 146 and 273 mm in diameter, the amplitudes required if rotational oscillation is used were an average of 1.5 times less than what was required if longitudinal oscillations were used. At the same time, sinking a pipe using rotational oscillations required that the mass of the system being sunk be considerable, enough for the pipe to exert a specific pressure of 0.8 – 1.0 MPa on the ground.

More efficient were longitudinal and rotational oscillations, which made it possible to sink pipes at a specific pressure of 0.15 – 0.2 MPa and with a power requirement 1.6 times less than under the same conditions using longitudinal oscillations (M. G. Tseitlin, 1958). The results of these experiments showed the practicality of using multidirectional oscillations to sink hollow cylindrical elements into the ground.

Considering the known experimental facts about the vibrational pile sinking process, the first thing necessary is to study and compare the conditions under which breakthrough of a pipe into the adjacent ground can be achieved for oscillations in different directions. This problem was solved by conducting experimental field research using an experimental vibrational sinking machine that generates longitudinal, rotational, and longitudinal and rotational oscillations.

As a result of these experiments, experimental data were obtained on the process by which a casing breaks through into the adjacent ground, as were comparative data on breakthrough conditions for oscillations of various directionalities.

Analysis of the experimental data obtained on the development of the breakthrough process over time shows that, in the final stage of full breakthrough the amplitude of the pipe oscillations exceeds the amplitude of ground oscillations by a factor of several hundred. As breakthrough develops and the pipe begins to slip more and more relative to the adjacent ground, the phase angle between the direction of the coercive force and the oscillatory displacement of the vibrator increases.

During the initial stage, the pipe and ground oscillate together when longitudinal coercion is used, and the amplitude of the oscillations, both of the pipe and of the ground, increases; when rotational oscillations are used, the pipe and ground are not observed to oscillate together, since there is an abrupt increase in the oscillation amplitude of the pipe.

During the second stage of longitudinal and rotational coercion, the oscillation amplitude of pipe and ground increases.

The third (final) stage of breakthrough for longitudinal and rotational coercion is characterized by extreme oscillations of the pipe, while the surrounding ground undergoes insignificant displacement, the amplitude of which also stabilizes.

Figure 13 shows a graph of the amplitude and power at which breakthrough occurs when longitudinal, rotational, and longitudinal and rotational oscillations are used. It is evident from the graphs presented that longitudinal and rotational and rotational oscillations are superior to longitudinal in light of the minimal amplitudes and powers at which breakthrough occurs.

Analysis of the results of this cycle of experiments shows that, when longitudinal and rotational and rotational oscillations are used, a hollow cylindrical element breaks through into the adjacent ground at amplitudes 1.25 to 1.6 times smaller than when longitudinal oscillations are used, and at powers that are correspondingly 1.1 to 1.5 times lower.

During the slippage stage, the lateral resistance of the ground is 1.1 to 1.25 times less for longitudinal and rotational oscillations than for longitudinal. As the frequency of the oscillations increases, the advantage of longitudinal and rotational oscillations diminishes.

The experimental research carried out shows that, for multidirectional oscillations, the displacement of the hollow cylindrical element exceeds the displacement of the surrounding ground by several dozen to several hundred times during the slippage stage.

The resistance of the ground to the displacement of the hollow cylindrical element is different for longitudinal, rotational, and longitudinal and rotational coercion. In this regard, longitudinal 4-rotational oscillations are the most efficient, both in terms of overcoming lateral resistance and in terms of the energy used.

The diagram used in calculating for longitudinal and rotational oscillations is shown in Figure 14. In developing the mathematical model, the same assumptions were used as were used for a sinking process utilizing longitudinal oscillations. The vibrational sinking process corresponding to the calculation diagram in Figure 14 can be presented in the form of a piecewise system with the following stages:

The raising stage:

(19)

where *r _{1}* is the arm of the coercive force being applied; r is the outside radius of the element being sunk.

The phase angle can be defined by the formula

(20)

It is possible for the system to stop at the top only if dx/dt = dφ/dt = 0 and the resultant of the active forces is less than the force of friction.

(21)

In this case the downward motion of the system begins at moment in time *t’ _{1}*, at which

*P(t)*=

*F*.

The stage in which the system travels downward until it contacts the ground plug is described by equation (19) for the initial condition *t* = *t _{1}* or

*t*=

*t’*(

_{1}*dx/dt*=

*dφ/dt*= 0;

*x = x*), depending on whether or not the system has stopped at the extreme topmost position. This stage ends when

_{1}; φ = φ_{1}*t = t*when the system contacts the ground plug, at which time

_{2}*x = x*= 0;

_{2}*dx/dt = dx*;

_{2}/dt*φ = φ*.

_{2}; dφ/dt = dφ_{2}/dtStage of penetration of the plug (plastic deformations):

(22)

where *r _{f}* = the radius of friction;

*r*.

_{f}≈ rThe system may stop at the bottom position both with and without rotational oscillations.

In the case where *dφ _{3}/dt *≠ 0 at

*t = t*the motion of the system is described by the equation:

_{3}(23)

Initial conditions: at* t = t _{3}* ,

*φ = φ*The integration of equation (23) is carried out until time

_{3}; dφ/dt = dφ_{3}/dt.*t*, when the condition

_{4}= 2π/ω*dφ*t should be fulfilled.

_{4}/dt = dφ_{o}/dThe rotational movement may cease during the time interval from *t ^{3}* to

*t*(at

_{4}*t’*). In this case, comparison of the moment of torque

_{4}*M*sin (

_{t}= P_{o}*ωt’*+

_{4}*α*)

*r*and the moment of resistance

_{1}*M*= {

_{r}*F*+ [

_{r}*m*+

_{o}g*P*cos (

_{o}*ωt’*+

_{4}*α*)]

*f*} defines stoppage |

_{1}r_{1}*M*| < |

_{t}*M*| or instantaneous stoppage |

_{r}*M*| > |

_{t}*M*|.

_{r}Stoppage of the rotational oscillations ends at |*M _{t}*| = |

*M*| (

_{r}*t = t*). When

_{5}*t*<

_{5}*t*, the motion of the system is described by equation (23) until

_{4}*t*=

*t*. In the case where

_{4}*t*≥

_{5}*t*, the next cycle under observation begins at

_{4}*φ*= 0 and the phase angle

_{0}*α*is defined by equation (21) for

*P(t)*=

*F*and

*t*= 0.

_{1}The total power for multidirectional oscillations is:

(24)

where the power expended on longitudinal oscillations of the system is

(25)

where *L _{xi}* is the work done in each stage by longitudinal oscillations.

The power expended on rotational oscillations of the system is

(26)

The computations were done on a computer using dimensionless variables and parameters (9), with the following additions:

(27)

One of the most important findings was that, in the process of sinking a hollow cylindrical element using vibration, the vertical component of the lateral ground resistance can be significantly less than the mean, since this value depends on the ratio of the longitudinal to the rotational component of the velocity of the oscillations. In contrast to sinking using longitudinal oscillations, sinking using longitudinal and rotational oscillations is possible under conditions in which sum of the force of gravity and the full lateral resistance of the ground to the oscillations of the casing exceeds the coercive force, i.e., *q* +* f* > 1.

Analysis shows that submergence per cycle (*y _{3}*) and the rate of penetration (

*dy*) increase in proportion to

_{2}/dt*b*. Furthermore, due to the greater efficiency with which the lateral resistance is overcome when the oscillations used have a rotational component, the initial velocity prior to penetration increases, so that the accumulated energy, needed to overcome the frontal resistance of the ground, is greater than what is available from merely longitudinal oscillations. Thus, as the rotational component of the oscillations increases, both the lateral (Figure 15) and the frontal (Figure 16) resistance of the ground is overcome more efficiently.

Analysis of the results leads to the conclusion that the number of rotations of the pipe per cycle *ψ _{4}* increases as b increases, and decreases as the lateral resistance to sinking increases.

The efficiency of this or any other mode of oscillation is defined as the ratio of the power needed to overcome ground resistance to the submergence per cycle of the hollow cylindrical element.

When sinking relatively heavy casings (*q* > 0.35), the ratio of the power in the vibrational sinking process to the submergence per cycle of the casing is less for longitudinal and rotational oscillations than for longitudinal oscillations over the entire range of relative lateral and frontal ground resistance values. When sinking relatively light hollow cylindrical elements (*q* < 0.35), it is practical to apply longitudinal and rotational oscillations if *f* > 0.6 and *q* + *f* > 0.8; when *f* and *q* + *f* are smaller, longitudinal oscillations are superior, from an energy standpoint, for sinking hollow cylindrical elements.

Successfully overcoming the lateral resistance to the sinking of hollow cylindrical elements by using oscillations with a rotational component makes it possible to reduce energy losses by utilizing the force of gravity efficiently. These advantages increase with the weight of the system and the lateral surface area of the element being sunk.

When the mass and dimensions of the casing are large and the oscillations of the hollow cylindrical element being sunk are rotational, most of the energy is expended in overcoming ground resistance. As resistance to sinking and the mass of the casing decrease decrease, the energy lost to longitudinal displacement increases and the time the hollow cylindrical element spends stopped at the bottom position is shortened. At *b* = 2 the portion of the power going to rotational movement increases abruptly, at the same time that the useful energy spent on longitudinal oscillations increases insignificantly. This shows the energy advantages of this mode at *b* = 1.0.

The results of experimental research and industrial experience confirm the theoretical conclusions. Thus, in a mode consisting of longitudinal and rotational oscillations hollow cylindrical elements of various dimensions have been sunk efficiently at *q* + *f* values up to 1.8.

The work of V. F. Dmitriev has shown that, when the rotational motion of the casing in the longitudinal and rotational mode is in only one direction, it has been possible, using a cap equipped with teeth, to make progress in tophaceous soil. If the hollow cylindrical element is equipped with a cutting crown, then it has been possible to drill into hard rock, as experiments have shown (V. V. Moskvitin, E. N. Samoylenko, A. S. Ryazanov, 1978).

In *rotational oscillation* around a vertical axis, the velocity vs of every point on the surface of the hollow cylindrical element is the geometric sum of the axial velocity *v _{o}* which is equal to the extraction rate

*v*, a constant, and the tangential velocity

_{w}*v*, which is equal to the velocity of the rotational oscillations

_{τ}(28)

The friction pulse that acts on the element along the axis of velocity vs over time dt is equal to *Fdt*, while the friction pulse that acts along the vertical axis over time *dt* is equal to *F v _{o}/v_{s} dt*. The corresponding friction pulse over the entire period of the oscillation is

*F v*∫

_{o}*. On the other hand, this same pulse can be written as the product of the mean force of friction over a period*

_{0}^{T}dt/v_{s}*F*acting along the vertical axis for that period, i.e., as

_{rot}*F*, whence

_{rot}T(29)

If we represent the ratio of the extraction rate to the amplitude of the oscillation velocity with *χ = v _{w} /φ_{0} r ω*, then after a series of transformations we have:

(30)

Reducing the integral thus obtained to a first order elliptical integral, we obtain

(31)

The graph showing the change in the force of friction for rotational oscillations as a function of *χ* is shown in Figure 17.

For purposes of comparison, Figure 17 also contains the graphs of the extraction force for longitudinal oscillations constructed according to formula (12) for the case in which *A ω = φ _{0} r ω*.

When extracting a hollow cylindrical element that is executing helical oscillations, we have:

(32)

Proceeding in exactly the same way as in the preceding derivation, after transforming we have the following expression for the extraction force in the case of helical oscillations:

(33)

The graph of the change in the force of friction for the helical oscillations of an element was constructed by computing integral (33) on a computer and is shown in Figure 18.

When extracting a hollow cylindrical element that is executing *longitudinal and rotational oscillations*, we have:

(34)

After a series of transformations we obtain an expression for the extraction force under longitudinal and rotational oscillations:

(35)

The graph of the change in the force of friction for a hollow cylindrical element executing longitudinal and rotational oscillations was constructed by computing the integral of (35) on a computer and is shown in Figure 19.

As is evident from an analysis of Figures 18 and 19, at one and the same value of *χ** the greatest drop in the extraction force can be obtained using longitudinal and rotational oscillations in the mode where *j* > 1. Below are experimental data on the extraction forces for hollow cylindrical elements oscillating in varying directions.

Figure 20 shows the characteristic of the change in extraction force over time for longitudinal and longitudinal and rotational oscillations, and also when static force alone is applied. For all of these cases (oscillations in various directions) the first stage is characterized by an increase in the extraction force until the moment the hollow cylindrical element breaks out of the adjacent ground, after which the extraction rate becomes equal to the lifting rate of the pulling winch. The magnitude of the extraction force after breakout and the rate at which it diminishes depend on the vibration parameters and the extraction rate.

The experimental data obtained show that the extraction force for longitudinal and rotational oscillations (*b* ≈ 1) is 1.3 to 1.6 times less than for longitudinal oscillations.

Analysis of the results of experimental research have shown the advantages of the longitudinal and rotational mode of oscillation over the longitudinal mode when extracting hollow cylindrical elements, and also demonstrates the concurrence of the experimental data obtained with the theoretical research.

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