# Dynamics Analysis of a Linear Shale Shaker

A drilling fluid shale shaker is a vibrator used for solid/liquid separation. The purpose of using a shaker is to recover drilling fluid and remove large solids as most as possible. With rapid development of drilling new technology, the demand for drilling fluid solids control is getting higher and higher. Much research work on vibration principle, vibration pattern, kinematics, dynamics, screen selection and vibration testing of a shale shaker has been done by many researchers [1–5]. When drilling fluid containing drilled solids flows onto the vibrating screen, the liquid/

solid separation process begins. The drilling fluid gradually passes through the screen, the solids convey toward the discharge end. The liquid along the screen disappears through the screen cloth as it reaches a point, called liquid endpoint. At first, the solid particles are submersed by drilling fluid, then gradually become uncovered, at last, drier solids will be conveyed off the screen. From academic analysis and experimental research, in submersed condition, the height of free flight motion of solids is not larger than the thickness of drilling fluid. After the solids pass the liquid endpoint, they have four types of kinematic modes, sliding backward or forward, free flight motion, remaining in contacting with the screen and staying stationary. During the working of a shale shaker, solids will produce various nonlinear acting forces such as inertial force, impacting force, frictional force. These nonlinear acting forces will influence the dynamic performance and lifetime of a shaker, this paper will address this problem.

## Motion of Solid Particles

The type shaker of bi-axial self-synchronous with translation linear trace is shown in Fig. 1, it is composed of vibration exciter, sieving box and supporting devices. The two vibrating electric motors are installed on the sieving box directly. Two eccentric blocks with same mass moment are installed on each vibrating axis, when the axes rotate, they will produce inertia forces. When the force center of the vibrating forces coincides with the quality center of the vibrating system, the translation linear trace will be realized.

*m _{1} = m_{2} =m m_{1}r_{1} = m_{2}r_{2} = mr.*

Where *m _{1}*,

*m*are masses of eccentric blocks, respectively,

_{2 }*m*,

_{1}r_{1}*m*are mass moments of eccentric blocks, respectively, ρ is vibration angle.

_{2}r_{2}Strictly speaking, the motion of solid particles should be discussed on the whole screen surface. But, once a steady state is achieved, the solids conveying velocities before and after the liquid endpoint are matched. For sake of

simplicity, we only discuss the motion of solid particles after they pass the liquid endpoint.

After the solids pass the liquid endpoint, they have four types of kinematic modes, sliding backward or forward, free flight motion, remaining in contacting with the screen and staying stationary. As shown in Fig.2 the forces acting on

the solids are, gravity, adhesive force of drilling fluid, force of inertia, counterforce of screen.

*F=m _{m}(Ÿ + ΔŸ)*

Where F is force of inertia, *m _{m}* is mass of solid,

*Ÿ*is acceleration component of screen normal to screen surface,

*ΔŸ*relative acceleration between screen and solid, g is acceleration of gravity, α is screen angle, R is adhesive force of drilling fluid, N is counterforce of screen.

### Sliding Backward

When a solid remains in contacting with the screen and stays stationary, it has the same kinematic trail as the vibrating screen. When it begins its sliding backward motion, the direction of friction force is contrary to that of relative velocity. Solids enter sliding backward mode as soon as the following expression is satisfied:

-m_{m}gsinα+f_{d}N<m_{m}Χ

Where f_{d} is dynamic friction coefficient, Χ is acceleration component of screen parallel with screen.

When a solid particle slides backward, its parallel velocity is not equal to that of the screen, and it has a personal acceleration component. At the beginning of sliding backward, its dynamic equation in x direction is:

-m_{m}gsinα+f_{d}N<m_{m}a_{hx}

Where a_{hx} is acceleration component of solid parallel with screen.

### Term of Flight Motion

If N = 0; solids will leave the screen cloth, and enter a flight mode. At the beginning, *ΔŸ =* 0; so the term of flight motion for solids is: N = -m_{m}Ÿ + m_{m}gcosα + R ≤ 0.

After entering a flight motion mode, a solid has a free fall motion.

### Remaining in Contacting with Screen and Staying Stationary

If the following expression is satisfied, a solid will remain in contacting with shaker screen and stay stationary:

|-m_{m}g sina -m_{m}X|=f_{s}N

Where f_{s} is static friction coefficient.

### Sliding Forward

When a solid enter a sliding forward mode, the direction of friction force is contrary to that of relative sliding velocity. The solids enter the sliding forward mode as soon as the following expression is satisfied:

When a particle slides forward, it has a personal acceleration component parallel with screen surface:

Where a_{zx} is acceleration component of solid parallel with screen.

## Dynamics Equations of the Shale Shaker

In order to make a precise analysis of the linear shale shaker, we should consider the effect of solid particles on the vibrating system. When a solid enter a flight motion mode, it will produce various nonlinear acting forces such as inertial force, impacting force, frictional force:

**Dynamics Analysis of a Linear Shale Shaker**

Where F_{y} is nonlinear acting force of solid normal to screen surface, F_{y}(Ÿ, Ý, Y, t); Ý and y are velocity and displacement components of sieving box normal to screen surface, respectively; m_{m} (Ÿ + gcosα)is acting force of solid when having same motion as screen; m_{m}(Ým -Ýz)/Δt is instantaneous impacting force of solid when touching screen; u is phase angle of sieving box; ud and uz are phase angles of initializing flight and ending flight, respectively; Ým and Ýz are velocity components of solid and screen normal to screen surface when solid touching screen, respectively; Ýd is velocity component of screen normal to screen surface when solid departing from screen; x is circular frequency of vibration; Δt is time of impacting.

Nonlinear acting forces of a solid parallel with shaker screen surface:

Where M is mass of sieving box; ky and kx are stiffness coefficient components of spring normal to and parallel with screen surface, respectively; cy and cx are damp coefficient components normal to and parallel with screen surface,

respectively.

## Solution of the Dynamics Equations

Equation is complicated nonlinear differential equations with two freedoms. According to the theory of nonlinear vibration, as for the problem of weak nonlinear, we can perform equivalent linearization treatment on the nonlinear acting forces . Suppose the first order approximate solutions of Eq. are:

Where λy and λx are vibration amplitudes of sieving box normal to and parallel with screen surface, respectively; φy and φx are phase differences between sieving box and exciting force normal to and parallel with screen surface, respectively.

According to the Fourier series, expand the nonlinear acting forces, ignore static force and force of quadratic harmonic wave and upwards, so the nonlinear acting forces may be written as:

## Calculating Example

## Conclusions

After the solids pass the liquid endpoint, they have four types of kinematic modes, sliding backward or forward on the screen, free flight motion, remaining in contacting with screen and staying stationary. During the working process of a shale shaker, solid particles will produce various nonlinear acting forces such as inertial force, impacting force, frictional force. These nonlinear acting forces will influence the dynamic performance and lifetime of the shaker.

By the method of translating nonlinear acting forces into inertial force and damping force, the nonlinear vibration equations for the linear shale shaker are established. The equivalent linearization method is used to get the first order approximate solutions of the equations. The calculating formulas of combination coefficient and equivalent damp coefficient of solid particles are obtained.

Combination coefficient and equivalent damp coefficient of solid particles are important parameters for dynamics analysis of a shale shaker. From the actual calculating results of the linear shale shaker, the values of a1x and b1x are smaller than those of a_{1y} and b_{1y}; so they can be ignored.

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