# Shale shaker design -g factor

The g factor refers to a ratio of an acceleration to Earth’s gravitational acceleration. Jupiter has a mass of 418.6*10^25 lb and Earth has a mass of 1.317*10^25 lb. A person on Earth who weighs 200lb would weigh 320 times as much on Jupiter, or 64,000 lb. A person’s mass remains the same on Earth or Jupiter, but weight is a force  and depends on the acceleration of gravity. The gravitational acceleration on Jupiter is 320 times the gravitational acceleration on Earth. The g factor would be 320. (As a point of interest, Mars has a mass of 0.1415*10^25 lb, so the g factor would be 0.107; a 200-lb person would weigh only 21.4 lb on Mars.)

The term ‘‘g force’’ is sometimes used incorrectly to describe a g factor. In the preceding example, the g force on Earth would be 200 lb and the g force on Jupiter would  be 64,000 lb. This is because the acceleration of gravity on Jupiter would be 320 times the acceleration of gravity on Earth.

## Calculation of g Factor

Accelerations are experienced by an object or mass rotating horizontally at the end of a string. A mass rotating around a point with a constant speed has a centripetal  acceleration (Ca) that can be calculated from the equation: Ca=rW^2 (where r is the radius of rotation and W is the angular velocity in radians per second.

This equation can be applied to the motion of a rotating weight on a shale shaker to calculate an acceleration. The centripetal acceleration of a rotating weight in a  circular motion with a diameter (or stroke) of 2r, in inches, rotating at a certain rpm (or W) can be calculated from the preceding equation,
(Ca) = (1/2)(diameter)(w)^2
Ca =1/2(stroke, in inches)(1ft/12inches)*[RPM(2pi/revolution)*(1 minute/60 seconds)]^2
Combining all of the conversion factors to change the units to ft/sec2:
Ca(in ft/sec^2)[stroke, in inches/(RPM)^2]
Normally this centripetal acceleration is expressed as a ratio of the value to the acceleration of gravity:
No. of g’s=Ca/(32.2 ft/sec^2)=[(stroke, in inches)*(RPM)^2]/70490.

Shale shakers are vibrated by rotating eccentric masses. A tennis ball rotating at the end of a 3-ft string and a 20 lb weight rotated at the same rpm at the end of a 3-ft  string will have the same centripetal acceleration and the same g factor. Obviously, the centripetal force, or the tension in the string, will be significantly higher for the  20-lb weight.

The rotating eccentric weight on a shale shaker is used to vibrate the screen surface. The vibrating screen surface must transport solids across the surface to discard and  allow fluid and solids smaller than the screen openings to pass through to the mud tanks. If the weights rotated at a speed or vibration frequency that matched the natural  frequency of the basket holding the screen surface, the amplitude of the basket’s vibration would continue to increase and the shaker would be destroyed.

This will happen even with a very small rotating eccentric weight. Consider a child in a swing on a playground: Application of a small force every time the swing returns to full  height (amplitude) soon results in a very large amplitude. This is a case in which the ‘‘forcing function’’ (the push every time the swing returns) is applied at the natural  frequency of the swing.

When the forcing function is applied at a frequency much larger than the natural frequency, the vibration amplitude depends on the ratio of the product of the unbalanced  weight (w) and the eccentricity (e) to the weight of the shaker’s vibrating member (W); or vibration amplitude=we/W The vibration amplitude is one half of the total stroke length.

The peak force, or maximum force, on a shaker screen can be calculated from Newton’s second law of motion:
force=(force/g)*a
where a is the acceleration of the screen. For a circular motion, the displacement is described by the equation:
x=X(sinNtpi)/30
The velocity is the first derivative of the displacement, dx/dt, and the acceleration is the second derivative of the displacement. This means that the acceleration would
d2x/dt2 be
a=-(N/30)^2(pi)^2Xsin[(pi)^2(Nt/30)]

The maximum value of this acceleration occurs when the sine function is equal to 1. Since the displacement (X) is proportional to the ratio of we/W for high-vibration speeds, the peak force, in lb (from the peak acceleration), can be calculated from the equation: F=weN^2/35200.
So the force available on the screen surface is a function of the unbalanced weight (w), the eccentricity (e), and the rotation speed (N).

Stroke length for a given design depends on the amount of eccentric weight and its distance from the center of rotation. Increasing the weight eccentricity and/or the rpm  increases the g factor. The g factor is an indication of only the acceleration of the vibrating basket and not necessarily of performance. Every shaker design has a  practical g-factor limit. Most shaker baskets are vibrated with a 5-hp or smaller motor and produce 2–7 g’s of thrust to the vibrating basket.

Conventional shale shakers usually produce a g factor of less than 3; fine-screen shale shakers usually provide a g factor of between 4 and 6. Some shale shakers can  provide as much as 8 g’s. Greater solids separation is possible with higher g factors, but they also generally shorten screen life. As noted previously (in the Linear Motion Shale Shakers subsection), only a portion of the energy transports the cuttings in the proper direction in unbalanced elliptical  and circular vibration motion designs. The remainder of the energy is lost due to the peculiar shape of the screen bed orbit, as manifested by solids becoming  nondirectional or traveling in the wrong direction on the screen surface. Linear motion and balanced elliptical designs provide positive conveyance of solids throughout the vibratory cycle because the motion is straight-line rather than elliptical or circular. Generally, the acceleration forces perpendicular to the screen surface are responsible for the liquid and solids passing through the screen, or the liquid capacity. The  acceleration forces parallel to the screen surface are responsible for the solids transport, or the solids capacity.

On a linear motion shaker, the motion is generally at an angle to the shaker screen. Usually the two rotary weights are aligned so that the acceleration is 45° to the screen  surface. The higher liquid capacity of linear motion shale shakers for the same size openings in the screens on unbalanced elliptical or circular motion shakers seems  related primarily to the fact that a pool of drilling fluid is created at the entry end of the shale shaker. The pool provides a liquid head to cause a higher flow rate through the screen. The linear motion moves the solids out of the pool, across the screen, and off the end of the screen.

On a linear motion shaker with a 0.13-inch stroke at 1500 rpm, the maximum acceleration is at an angle of 45° to the shale shaker deck. The g factor would be 4.15. The  acceleration is measured in the direction of the stroke. If the shale shaker deck is tilted at an upward angle from the horizontal, the stroke remains the same. The  component of the stroke parallel to the screen moves the solids up the 5° incline. Relationship of g Factor to Stroke and Speed of Rotation Relationship of g Factor to Stroke and Speed of Rotation 2

An unbalanced rotating weight vibrates the screen deck. The amount of unbalanced weight combined with the speed of rotation will give the g factor imparted to the  screen deck (see preceding paragraphs). The stroke is determined by the amount of unbalanced weight and its distance from the center of rotation and the weight of the  shale shaker deck. (This assumes that the vibrator frequency is much larger than the natural frequency of the shaker deck.) Stroke is independent of the rotary speed.

The g factor can be increased by increasing the stroke or the rpm, or both, and decreased by decreasing the stroke or rpm, or both. The stroke must be increased by the  inverse square of the rpm reduction to hold the g factor constant. Examples are given below to hold 5 g’s constant while varying stroke length at different values of rpm:
5 g’s @ 0.44″ stroke at 900 rpm 4 g’s @ 0.35″ stroke at 900 rpm;
5 g’s @ 0.24″ stroke at 1200 rpm 4 g’s @ 0.20″ stroke at 1200 rpm;
5 g’s @ 0.16″ stroke at 1500 rpm 4 g’s @ 0.13″ stroke at 1500 rpm;
5 g’s @ 0.11″ stroke at 1800 rpm 4 g’s @ 0.09″ stroke at 1800 rpm.