Checking the Soviets: Finding the Frequency and Establishing Basic Parameters

In our last post Checking the Soviets: A Tale of Two Cities we described the underlying institutions that designed and built impact-vibration equipment. Henceforth, based on this post we will refer to the two groups as the “Moscow” and “Leningrad” groups, to save the reader having to deal with the Cyrillic alphabet soup that Soviet institutions used.

Finding the Frequency

In Inclusion of Rotational Inertial Effects in Power Consumption Calculations for Vibratory Pile Equipment we discussed the various frequencies that exist for the analysis of vibratory equipment. Understanding these is crucial for the design and analysis of the equipment. Impact-vibration equipment is no different even though the operation is more complicated due to the existence of the spring system (shown at the top.) There are four basic frequencies to consider, in ascending order of their usual value:

• ω0 = the pendulum frequency of the eccentrics. The pendulum frequency was computed for the S-834 machine in the post Checking the Soviets: Determining the Eccentric Moment. It is not used in this analysis. It is interesting to note that, although both groups noted the variance of eccentric angular velocity could reach 40-45%, much of the development used analytical models (and doubtless physical scale models) which assumed constant angular velocity.
• ω1 = natural frequency of the spring-mass system
• ω2 = impact frequency, which is usually different from the angular frequency/velocity of the eccentrics
• ω3 = average rotational frequency/velocity of the eccentrics

The Moscow group had a very specific scheme for the relationship of the last three frequencies with each other. First we should define

$i={\frac {\omega_{{3}}}{\omega_{{2}}}}$ (1)

where i is the ratio of the eccentric frequency to the frequency of impacts. As the Moscow group noted:

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 practical effect of this can be seen in the figure below.

The parameter i is an integer to insure repeatable and stable operation of the machine for each cycle. Based on this the ratio of the natural frequency to the rotational frequency of the eccentrics can be defined thus:

$\xi={\frac {\omega_{{1}}}{\omega_{{3}}}}=1/2\,{i}^{-1}$ (2a)

The Moscow group made an empirical correction to this, and so

${\frac {\omega_{{1}}}{\omega_{{3}}}}=1/2\,{\frac {\sqrt {\beta}}{i}}$ (2b)

The factor β varied from 1.1 to 1.3, generally using 1.2. In any case, combining Equations (1) and (2b) yields

$\omega_{{1}}=1/2\,\sqrt {\beta}\omega_{{2}}$ (3)

In the case of the S-834, i = 2 and the frequency of rotations is 950 RPM, thus by Equation (1) the hammer has an impact frequency of 475 RPM. This translates into ω3 = 99.5 rad/sec and ω2 = 49.7 rad/sec. Substituting the latter value into Equation (3) yields ω1 = 27.2 rad/sec.

Some Basic Parameters

With the three frequency relationship established, we can turn to some basic parameters. These are as follows:

• The rated striking energy of the machine Er. This will be smaller than most comparable reciprocating impact hammers due to the higher frequency of blows.
• The maximum impact velocity v. Both Leningrad and Moscow groups were univocal that this should not exceed 2 m/sec. By comparison, a Vulcan #0 series of hammers has nearly a metre stroke, which indicates a rated striking velocity of 4.4 m/sec and a net striking velocity (assuming the WEAP 65% efficiency) of 3.6 m/sec.
• The range of acceptable maximum vibrating accelerations of the machine. This is based on free-hanging theory for the vibrator, and is generally expressed in g’s of acceleration. This parameter comes from the Leningrad group, and their recommended ranges of this are from 2 to 6 g’s.

The equation for rated striking energy is

${\it E_r}=1/2\,M_{{1}}{v}^{2}$ (4)

From this, if we know the maximum value of v and the value of Er, we can determine the hammer vibrating mass M1.

The peak acceleration of the vibrating mass (from free-hanging considerations) is given by the equation

$n={\frac {v\omega_{{3}}}{{\it g_c}}}$ (5)

where gc is the acceleration due to gravity.

In the case of the S-834, the maximum velocity was determined to be 1.83 m/sec and the vibrating mass 650 kg. Substituting these values into Equation (4) yields a rated striking energy of 1088.4 J = 1.1 kJ. By comparison the Vulcan #1 has a rated striking energy of 20.3 kJ. A more interesting comparison is the Vulcan DGH-100. In the free hanging mode the DGH-100 has a rated striking energy of 0.52 kJ at 303 blows/minute, and in the mounted mode a rated striking energy of 0.87 kN at 505 BPM, just slightly above the S-834.

Using Equation (4), the peak acceleration of the machine is n = 18.6 g’s. This is well above the Leningrad specifications and probably indicates a difference in design philosophy.

Impact Clearance

One issue that deserves brief mention is the issue of clearance at impact. This is discussed in Russian Impact-Vibration Pile Driving Equipment: Chapter 2, Theory of Operation as follows:

To understand the concept of clearance, (the figure at the top) 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.

Dynamic Force

One specification that is always important with vibratory hammers is the peak dynamic force of the eccentrics. This is given by the equation

$F=1/2\,v\left (1-{\xi}^{2}\right )\left (1-R'\right )\omega_{{3}}M_{{1}}$ (6)

As mentioned earlier, the variable R’ is the stroke speed restoration coefficient. It is the absolute value of the ratio of the velocity of rebound to the velocity of impact, and is generally assumed to be 0.12. For the S-834, substituting values results in a dynamic force F = 48,814 N = 48.8 kN. By comparison the Vulcan 400/400A vibratory hammer had a dynamic force of 151.3 kN.

Eccentric Moment

The eccentric moment was determined using free-hanging considerations, thus

$K={\frac {F}{{\omega_{{3}}}^{2}}}$ (7)

For the S-834, substituting values yields an eccentric moment K = 4.91 kg-m.

All results for the S-834 correspond with the original calculations within floating point error and the fact that we have opted and will opt to dispense with Those Pesky Kilogram-Force Units.