# Mathematical models for electric hydraulic drives

June 1, 2000
Conventional fluid-power laws often cannot adequately model hydraulic systems. They need a method that simultaneously solves for all parameters in the system.

Boris Fikhman
Electrical Engineer Chicago, Ill.

Electric hydraulic drives are a common mechanism for industrial equipment and automation control. The drives still follow general laws of fluid motion discovered by Blaise Pascal, Daniel Bernoulli, and Leonhard Euler in the 17th and 18th centuries. Basic hydraulic mechanisms were originally created for canals and they have evolved over the centuries to become fundamental components of industrial automation.

Scientists typically divide hydraulics into two categories, hydrostatics and hydrodynamics. Hydrostatics describes the behavior of fluids at rest. Hydrodynamics describes the behavior of fluids in motion, including forces that act on fluids.

Hydraulic drives used for factory automation have several interacting qualities. First, fluid flows in confined areas. Second, hydraulic drives are influenced by loads, such as pressure or tool velocity. Hydraulic drives are also influenced by a resistance to change this load. Third, every machine has an elastic coefficient that impacts the volume of fluid compression. For example, fluid is theoretically incompressible and should maintain a constant volume. But based on the interaction of these parameters, researchers have determined the change between calculated and real volumes can be up to 40%.

An interesting paradox becomes apparent after analyzing the working capacity of hydraulic drives in factory automation. On one hand, hydraulic drives follow basic laws of hydrostatics. On the other hand, the machines they work with often operate at high pressures and speeds, placing them in a dynamic regime. So the laws of hydrodynamics don't fully explain the machines' operation.

It's difficult to isolate one component of a hydraulic system in the real world and predict its operation. That's because several processes occur simultaneously and influence one another. These processes include the power that moves hydraulic cylinders or turns hydromotors, the resistance of the load and its momentum, and elastic features of machine components.

Neither the laws of hydrostatics nor hydrodynamics can account for individual parameters of hydraulic components and their interactions. These require a method that simultaneously solves for all parameters in the system.

The diagram at right shows a basic rectilinear drive. Motion for the drive is initiated either by an uncontrolled, uninterrupted, pressure regulator or an automatically controlled, uninterrupted pressure regulator.

The drive consists of a pump that moves fluid from the tank to the cylinder. The selector valve changes the direction of fluid to cavities in the cylinder, which changes the direction of piston motion. The pressure-relief valve limits pressure to a maximum set point. Finally, pipes connect all parts of the drive. A rotary drive has similar components except it uses a hydraulic motor instead of a hydraulic cylinder.

Mathematical models can be used to simulate the action of the hydraulic-system components for real working conditions. The models are similar for rectilinear and rotary drives.

For drives with uninterrupted, uncontrolled pressure regulators, 38 parameters model the mechanical, hydraulic, and electric components in the system. Drives with automatically controlled, uninterrupted pressure regulators need 65 parameters to model the equipment.

Exact methods of calculating electric-hydraulic drive performance is necessary in many industrial systems, such as thermoplastic machinery, foundry equipment, and metalworking tools. The mathematical models can be used to solve a variety of practical problems. For instance, electric-hydraulic drives can be modeled on computers and hydraulic-equipment designs can be optimized by minimizing weight and energy consumption. The precision of machine processes can also be optimized. The models can also be used to automate hydraulic-equipment control systems for parameters such as the load and speed of tools.

As an example of how the models can be used, consider mounting a bushing on an axle using a hydraulic press as shown in the "Bushing press" graphic. In this system elastic deformation takes place when the metal parts of the machine deform and when the fluid deforms. Both deformations increase the volume of the fluid passages. If the pump doesn't compensate for the changing volume, the speed of the press will decrease. The problem is determining the response time of the changing parameters.

The required pressing force F can be characterized by the formula:

The system of equations that models the hydraulic press is:

After determining ΔPb from Equation 2, substituting it into Equation 3, and rewriting the equations using Laplace transforms, the system of equations becomes:

In this system the cross-sectional area of the throttle = 1.7610 -4 m 2 , the piston area = 9.610 -4 m 2 , the ball-valve area = 1.2610 -5 m 2 , the plunger area = 0.158 m 2 , and the elastic coefficient of the hydraulic press = 2.5510 8 N/m.

The next step is to find the deviation of the hydraulic system pressure by calculating the determinants of the Cramer formula:

After calculating this determinant and simplifying the coefficients by renaming them as Ti, the determinant becomes:

The next step is calculating D1 using:

Similar Ti substitutions for this determinant produce:

Substituting into the equation for ΔP produces:

In this case ΔFb = 0, so only the second part of the equation is required. The Ti variables are calculated by substituting values for components in the hydraulic press. After these substitutions, the Ti values are found to be:

Substituting these values into the ΔP equation produces the characteristic equation:

Using a computer, the roots of this equation are found to be:

By evaluating these roots using a detailed mathematical process, the function for ΔP becomes:

Using a similar calculation process, the remaining functions are found to be:

This modeling process shows how elastic deformation takes place during the transient portion of a pressing application. A similar process can be used to determine the response of the pressure regulator connected to the hydraulic pump.

MATHEMATICAL MODELS OF HYDRAULIC DRIVES
The following equations apply to hydraulic drives equipped with uninterrupted, uncontrolled, pressure regulators.

The following equation applies to rectilinear motion.

The following equation applies to rotary motion.

When the hydraulic drive is equipped with an uninterrupted, controlled, pressure regulator, the following equations describe the dynamic process for the drive of the pressure regulator. The equations should be used in addition to the previous ones.

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