# Russian Impact-Vibration Pile Driving Equipment: Chapter 2, Theory of Operation

CHAPTER II
THEORY OF OPERATION

Compared to many machines, impact-vibration hammers are not especially complex; however, because of their interaction with the pile, their mechanics rival those of the diesel hammers in complexity. Thus, impact-vibration hammers are more complex to design, and attention to all theoretical details must be made even before the practical problems are addressed. First, however, we must establish some basic concepts. The calculation parameters for these machines are discussed by Erofeev (1966, 1985) and Erofeev and Warrington (1991).

5 Clearance in Impact-Vibration Machines

To understand the concept of clearance, Figure 9 shows an impact-vibration hammer with the clearance shown. When not powered, the exciter of an impact-vibration machine will assume a neutral position where the net spring force is zero (effects of gravity excluded.) This is true no matter if the springs are made of steel or other material, or whether or not where are only one or two tiers of springs. The clearance of an impact-vibration hammer is the distance between the ram point of the exciter and the anvil when the exciter is at the neutral position, or thus at rest. There are three options for this clearance:

Negative Clearance: A negative clearance is when, in the neutral position, a bias force is placed on the exciter when the machine is at rest. This can be used on a wide scale; however, in this case the working regime of the impact-vibration hammer substantially depends on controlling the coefficient of stroke speed restoration coefficient R’. Consequently, as the pile depth, soil characteristics and/or pile weight change, the system tuning will deteriorate and the machine will not work properly.

Positive Clearance: A positive clearance is when there is a gap between ram point and anvil in the neutral position. If the clearance exceeds the amplitudes of the system’s forced vibrations without a limiter, vibrations without impact are possible, and for this reason the clearances must be limited.

Zero Clearance: With zero clearance, the exciter is at rest on top of the anvil. This seems to be the most acceptable clearance. At this clearance the changing conditions of the driving system will affect the hammer the least. Also, the maximum velocity generated by the exciter’s eccentrics is at its maximum at anvil impact at zero clearance.

Since spring selection for impact-vibration hammers with clearances difference from zero clearance, the methods for vibratory hammers with positive or negative clearances are omitted for brevity.

6 Frequency Ratios

The exciter/spring system constitutes at its simplest a single degree of freedom vibrating system; however, the mechanics of the system are complicated by the impact of the exciter on the anvil or frame. At the beginning of the development of impact-vibration equipment, Tsaplin (1953) observed that the exciter did not impact the anvil or frame every rotation, but skipped impacts in a regular pattern. The ratio between the rotational frequency and the frequency of impacts is given by the equation

(1)

where
i = Impact Frequency Ratio
= Rotational Frequency, Hz
= Impact Frequency, Hz

The second ratio that must be defined at this point is the natural frequency ratio. The natural frequency of the exciter-spring system is given by the equation

(2)

where
= Natural Frequency of System, rad/sec
C = Spring Constant of exciter-frame springs, N/m
m = mass of exciter, including eccentrics and motor, kg

The maximum exciter mass can be limited, if the machine size must be limited. Since by definition

(3)

where = Frequency of exciter vibrations, rad/sec

we define the natural frequency ratio as

(4)

where i = Natural frequency ratio

The relationship between $\Gamma$ and i assumes zero clearance.

7 Stroke Speed Restoration Ratio

The next important parameter is the coefficient of stroke speed restoration, which is defined as

(5)

where
R’ = coefficient of stroke speed restoration
v1‘ = Exciter velocity just after impact, m/sec
v1 = Exciter velocity just before impact, m/sec (maximum exciter velocity)

Using classical Newtonian impact mechanics (similar to those used to derive the dynamic formulae,) the stroke speed restoration coefficient can be calculated with reasonable certainty by the equation

(6)

where
e = coefficient of restitution of impact between exciter and frame-anvil-pile system1
M = mass of pile and frame, kg
v2 = frame speed before stroke, m/sec

Generally speaking, the velocity of the pile and frame when it is struck again is zero, i.e. v2 = 0, so in this case the formula above can be reduced to

(7)

8 Basic Operating Parameters

Now that we have established the basic operating parameters of the machine, we can proceed to calculate the actual sizes of the components. In doing this, the following items need to be kept in mind:

(8)

(9)

9 Exciter Parameters

To establish the optimum spring stiffness, Equations (1), (2), and (4) can be solved to yield

(10)

where
= coefficient, considering shift of optimal impact speed with negative clearance = 1.1 to 1.2

By zero clearance in the exciter, we can have at our disposal one or two tiers of springs, one of which is situated between the exciter and the frame. Choosing the number of springs in each tier depends upon the total hardness of the spring assembly. Usually in every tier four springs are used. Spring sizes are calculated by normal spring equations, keeping in mind the strength requirements of the steel under fatigue loading.

The power supply requirement of the hammer is based on the difference between the kinetic energy of the exciter just before impact and that just after impact. Taking this and the resonant effect of the springs into account, the power requirement can be computed by the equation

$N_{ud} = \frac{mv_1\theta_{ud}(1-R'^2)}{2}$(11)

where
Nud = Power Requirement of Exciter, W

The eccentric moment is computed, based on the maximum impact velocity,

$K = \frac{mv_1(1-\Gamma^2)(1-R')}{\Omega_{ud}}$ (12)

where
K = eccentric moment of exciter, kg-m

followed by the amplitude of the vibrations of the exciter during its impact on the limiter,

(13)

where
ß = attenuation factor

and finally the dynamic force of the eccentrics

$F = \frac{m v_1 \Omega (1-\Gamma^2)(1-r')}{2}$ (14)

where
F = Dynamic Force of Eccentrics, N

Concerning the bearings and other parts, to properly specify them it is necessary to know the inertial forces created during impact. These forces represent the basic work load, and from this the components of the hammer can be properly specified.

Inertial forces can be defined, if the acceleration of the impacting parts is known. Assuming the impact force-time curve to be sinusoidal, the maximum acceleration for the exciter is given by the equation

(15)

where a1 = maximum deceleration of exciter, m/sec¨
t = impact time (determined to be 20 msec)

Acceleration for the frame and pile is given by the equation

$a_2 = \frac{\pi v_1}{2t}\frac{\frac{m}{M}(1+e)}{1+\frac{m}{M}}$ (16)

where
a2 = maximum acceleration of frame and pile, m/sec¨

Calculation of basic parameters for sheet pile extractors is similar to those used for calculation of vibratory machines.

10 Pile Connection

For stable operation of an impact-vibration hammer, it is necessary for the machine to be stable on the pile, as is also the case with conventional impact hammers. In addition to guiding the exciter, the frame of the machine also stabilizes it on the pile.

Each time the exciter travels upward, it exerts an upward force on the springs. Without some kind of restraint, the frame would travel upward with the exciter, thus adversely affecting the motion of the exciter and possibly interfering with the impact process.

There are two ways of accomplishing this: 1) adding enough weight to the frame to overcome the upward force of the exciters, and 2) clamping the frame to the pile. Option (1) is the same as is used with double acting impact hammers and Option (2) is used with conventional vibratory hammers. So once again the hybrid nature of the machine is emphasized.

The general form of the equation for this condition is

(17)

where
P = Theoretical Clamping Force, N (zero without clamp)
= Preliminary spring delay, m (only applies to machines without an upper tier of springs)
G = Weight of frame, N
ß’ = Factor for hammer racking and frame movement during operation
ß’ = 0.8-0.9 without clamping (this can be higher with heavier machines)
ß’ = 1 with clamped machines

Unclamped Machines (Option 1): For frames that do not need connection to the pile (Erofeev, 1968), the weight of the frame must be sufficient to prevent movement of the machine off of the pile during driving. This is done by either simply increasing the weight to be greater than the maximum upward spring force or by decreasing its weight and facilitating displacement relative to the pile with a given amplitude during operation of the vibrating hammer, according to the nomograms given by Erofeev (1986). In this case the nomograms also facilitate introducing corrections into the parameters of the vibrating hammer in order to assure its stable mode of operation during the displacement of the head relative to the pile. In either case P=0 and Equation (17) solves to

(18)

Clamped Machines (Option 2): The use of devices which connect the head and the pile during the upward movement of the vibration exciter and releasing them at the moment of impact became possible after a theoretical substantiation of the kinematics and dynamics of its operation, making it possible to establish both the mutual displacement of the vibration exciter, the body of the head and the clamping lever and the basic input parameters of the head influencing its operation. In the given case, the size of the gap between the clamp and the pile depends upon the geometric dimensions of the clamp, the rigidity of the springs of the vibrating motor, and the mass of the head

The clamping force must be

(19)

where T = actual clamping force, N
f = coefficient of friction between the clamp and the pile

The diameter of the clamping cylinder can then be given by the equation

(20)

where d = cylinder diameter, m

The design for sheet pile extractors is similar to that for drivers.

Note:

1 This coefficient of restitution e is not to be confused with the coefficient of restitution used in the wave equation program; this is the coefficient of restitution as described in most elementary treatments of dynamics.