By Jack L. Johnson, P. E..
Because it is relatively inexpensive to determine the suitability of a given system design before any hardware is acquired and before prototype systems are assembled, simulation is one of the most important implementations of mathematical models. Furthermore, competing designs and hardware choices can be evaluated and compared. To illustrate the steps in conducting a simulation, a hydrostatic transmission has been modeled using the proposed Type 1 models for the pump and motor. Simulation will be used to study the dynamic details of the transmission during cyclic life testing. The hydrostatic transmission model, as simulated, is shown in Figure 1.
In this simulation we want to determine the dynamic reaction of the transmission differential pressure and motor output speed in a planned life test. The test plan has the pump and motor plumbed with about 6 ft of 1-in. tubing. The motor is to be loaded with a flywheel that is sized to require about 3 sec to accelerate from 0 to 2400 rpm at a transmission differential pressure of 2400 psi. No pressure relief valves will be used. Instead, pressure will be controlled by varying the pump’s displacement from maximum in one direction to maximum in the other direction in about 6 sec.
We are interested in learning how the transmission pressure varies as acceleration and deceleration progress through time, and the motor’s output speed. Because proper life testing requires reaching targeted pressures and targeted speeds, the ability for the test system to reach 2400 psid and 2400 rpm is to be determined. In short, will the planned test set up do the job? The simulation is intended to provide some answers.
Exploring the setup
Figure 1 is an analytical schematic of the linearized hydrostatic transmission as constructed using the Type 1 pump and motor models. It shows that there are two shaft speeds, four torques, seven flows, and seven parameters, plus a time-varying pump displacement. Pump displacement is to be cycled to achieve targeted differential pressure and targeted motor output speed. Hydraulic capacitance is characterized with an electrical capacitor symbol, labeled CH in Figure 1. The inertial load is characterized with a curli-cue (similar to that for an electrical coil) in the output circuit of the motor and labeled JL in Figure 1.
|Figure 1. Analytical schematic of a linearized Type 1 model of a hydrostatic transmission showing algebraic symbology for the pertinent variables and parameters. |
The reason for borrowing electrical symbols is four-fold:
1. The electrical circuit laws of both Ohm and Kirchoff apply equally to linearized hydraulic circuits.
2. Borrowing of symbols is precedented, notably in the use of relay contacts for pneumatic logic circuits.
3. As early as 1950, Warren Wilson wrote of hydraulic capacitance in describing the effects of fluid compressibility.
4. It seems completely redundant to create a whole new circuit methodology for hydraulics when well-established electrical methods are so widely used — and if judiciously applied, are completely workable.
All the variables and parameters for the hydrostatic transmission are listed and evaluated in two tables that space contstraints prohibit from showing here. However, if you click on the March 2009 issue archive on the H&P home page, they are included in a downloadable PDF.
|Pressure and speed responses of the linear HST with a trapezoidal variation in pump displacement as a command input. |
It is necessary in simulation to know all the parameters and one of the variables. Then the other variables can be calculated. The pump and motor in our model have identical displacements, rated speeds, and rated pressures. This is not necessary, but it does simplify calculations. Also, the pump and motor in our model are not of any particular type or manufacturer. They are simply generic machines of quite conventional sizes, characteristics, and performances.
The simulation begins with the writing of defining circuit equations. Readers interested in viewing the mathematical equations can also find them in the PDF on the H&P website. A complete understanding of these equations requires knowledge of calculus, differential equations, and transform methods for their solutions.
Recall that it was desired to simulate the testing of the hydrostatic transmission while the motor was loaded with a flywheel. The flywheel inertia was sized to require 3 sec to accelerate the motor from 0 rpm to 2400 rpm while the pump and motor operate at rated, 2400 psid, differential pressure. Cyclic operation was to be achieved by controlling the variable displacement of the pump from full displacement in one direction to full displacement in the other, and in the form of a trapezoidal waveshape, shown as the Command signal in Figure 2.
Certainly, two of the questions to be asked of the simulation are: Did the differential pressure reach the rated value of 2400 psi, and did the speed reach the targeted of value of 2400 rpm? The answers can be found in the time response graph. The pressure reached a peak value of almost 2700 psid, while the speed did not quite reach its target.
Two more questions arise: Will the higher-than-desired pressure and the lower-than-desired speed constitute a valid cyclic life test for the transmission? If the answer is no, then what can be done to make the test valid? The answer to the first question is both legitimate and important, but it will be relegated to another forum. We would need much more information about the hardware to adequately answer it. The answer to the second question, however, is much more intriguing. Let’s assume that indeed, the test does not meet the requirements. What can be done?
Many things could be changed: Pump displacement could be decreased, because the maximum is 0.5 in.3/radian. Motor displacement (which is variable) could be decreased. The flywheel inertia could be re-sized. Pump shaft speed could be increased. The pump displacement profile shape could be changed; for example, the rate of change could be reduced.
All of these possibilities can be reasonably and approximately explored with the very simple hydrostatic transmission model introduced here. The entire study, summarized here and detailed in the PDF, took only about 2 hr of my time to see the graphical results. Writing this summary report took about 20 times that long. Each of the above “what if’s” would require less than an hour. It is a small price to pay for the insights gained, and the results might surprise you, but they are more than can be covered here. However, there can be little doubt about the value of math modeling and simulation.
Click here to view PDF of original manuscript containing equations.
Introducing the fourth edition of popular handbook
The fourth edition includes subject matter that has been requested by readers demanding more insight into this growing technology. Readers requested more methodologies for analyzing hydraulic circuits. The response is a whole new Chapter One — Physics of Hydraulic Fluid Power. Nowhere else can you find the wealth of methods and example problems for handling the twists and bends of plumbing and manifolds and the like, allowing better design-time control over pressure losses and the design of more efficient hydraulic circuits.
Three chapters were added to deal with methodologies and example problem solutions for mechanical loads. These three chapters show you how to reduce loads to their four essential elements, reconcile them through gears, pulleys, or whatever, and then shows how to solve problems with triangulated loading, cylinders, and their linkages, which are common in hydraulic applications, especially mobile equipment. More methods were requested for control systems and modeling, so a chapter was added on how to make a completely linear, yet utilitarian, analysis of a motion control servo system having an unequal area cylinder. That chapter is followed by one that shows how to include the dynamic limitations (frequency response) of the control valve in a hydraulic system. The result is a unique approach to an old problem, yet solutions are eminently practical. No other publication shows you this kind of analytical power. For the first time, a new chapter has been included on the optimal sizing of hydraulic motor circuits.
Finally, three new chapters have been added to teach you about the ever-increasing role of electronics in hydraulic control systems. There is a chapter on basics of electricity and electrical measurements; another is a veritable catalog of tutorial explanations of the common electronic devices needed in the modern electrohydraulic system, including all the popular transducers and signal conditioners. All of it is demystified for you.
Readers also wanted more on mobile equipment electronic systems, so that has been added in a special chapter. You will have answers to how batteries work their seeming magic (and what you need to do to help them), how vehicle charging systems work, and how joysticks work and interface with the rest of the electrical system.
In all, nearly 250 pages have been added, creating an indispensable reference of 784 pages. It is the musthave technical reference for anyone wanting to get on board the electrohydraulic technology juggernaut.
The book sells for $149.95, plus $5 shipping and handling from the H&P Bookstore. Visit www.HydraulicsPneumatics.com and click on the Bookstore button to order. Once at the Bookstore page, scroll down to the hyperlink to download a PDF order form. Fill out the form and fax it to the number indicated.