Developing a model for motor leakage

A simulation of hydraulic motor model was conducted on an Interdata Model 14 minicomputer. Using a consistent set of metric units, the program calculates values of shaft speed, shaft position, and motor inlet pressure, and outputs these values with the corresponding time increment.

The program consists of three sections:

  • Simulation of leakage resistance,
  • Simulation of instantaneous motor displacement, and
  • state equation implementation.

Leakage resistance
Referring to Figure 1, equations were developed to simulate a single cycle of the trapezoidal waveform. T1 was chosen as an arbitrary reference point. The remaining values, T2, T3, and T4, were used to derive a set of four straight-line equations involving minimum and maximum flows, Qmin and Qmax, respectively.

Because the wave form is periodic, only one cycle needs to be simulated. Therefore, the position can be calculated and translated back to this fundamental cycle.

The actual flow, Qq , is based on the angular position, , of the shaft. This value must be transformed into time to be used by the simulation.

Instantaneous motor displacement
The instantaneous motor displacement is based on Tp(q), the torque obtained during the low-speed test and P, the pressure during the test.

Because the waveform for Tp (q) and Q (q) are similar, the same set of equations can be used to produce the trapezoidal model of Figure 1. The maximum and minimum torque can be obtained from input data.

State equation simulation
The properties of static and kinetic friction were modeled in this portion of the program. Every time a calculation produces an angular velocity of zero — or the result of the calculations for angular velocity changes sign — a test is made to ensure that the torque produced at the current pressure can overcome the static friction. If the torque is insufficient, the angular velocity is set to zero until sufficient pressure builds up. The angular velocity then is allowed to proceed, and the frictional term is reduced to the kinetic value.

The remaining portion of the simulation implements the hydraulic motor state variable diagram shown in Figure 2. A straight-forward Euler numerical integration process was employed to solve the state equations.

To carry out the simulation, certain assumptions were drawn about the circuit in which the motor is to be used. The first deals with the nature of the load torque. Except for breakaway friction, the load is considered to be absolutely constant for simulation purposes. Admittedly, this is not what we would encounter in reality. However, we can assess the effects of motor parameter variations without contaminating them with load variations. Hydraulic capacitance and load inertia also had to be considered.

The hydraulic capacitance used in the model was based on the compressibility of the oil in a pipe of 1.06 gal capacity using a compressibility factor of 0.5% per 1000 psi. The value of hydraulic compressibility, CH, was found to be 5.283 x 10-4 gal/psi.

The value selected for the moment of the load was based on a flywheel with a 12-in. diameter, a thickness of 3 in., and a weight of 40 lb. The static and kinetic friction terms were chosen to be 100 lb-in. and 50 lb-in., respectively. Although the load torque was selected as a variable in the subsequent runs, it varied from 0 to 66.38 lb-in.

Next month, we will begin investigating results of the simulation.

Figure 1. Internal leakage of a motor varies with shaft rotational position. This is caused by the changing volume of the pressure chamber as the shaft rotates.

Figure 2. A straightforward Euler numerical integration process was employed to solve the state equations from the state variable diagram of the hydraulic motor.

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