Engineering science / 3. Branch mechanical engineering

Nikolay D. Andriychuk, Inna K. Nasonkina, Anatoliy A. Guschin,

Aws Mohammed Abdul Jaleel

East Ukrainian National University, School of Technical Âuilding

MATHEMATICAL SIMULATION AND ANALYSIS OF THE VARIABLE-DISPLACEMENT VANE-PUMP DYNAMICS

Variable-displacement vane-pumps, modulated by a pivoting cam, are used extensively in modern automotive transmissions. This paper presents a study of the dynamic and steady-state characteristics of such pumps. Their dynamics concern issues such as pressure fluctuations due to the dynamics of the internal mechanism and transient response-time. Their steady-state characteristics concern limits on the operating envelope and the relationships among the operating variables. These analyses required the development of several models for this type of pump at different levels of detail.

Growing energy costs continue to motivate the use of variable-displacement pumps in hydraulic systems with varying flow requirements. Variable- displacement pumps are already used widely in various industrial and automotive applications [1, 2]. To fully realize their economic potential, these pumps should be carefully designed and controlled. Unlike in fixed-displacement pumps, the mechanisms of variable-displacement pumps respond to the internal forces which develop during operation. These forces determine the dynamic nature of the pump, and can affect the dynamics of the entire hydraulic system.

With no analytical models available for these pumps, their regulation and the related system design are often conducted experimentally. Since many operating variables and design parameters influence the pump, this approach may lead to rather frustrating results. Solutions which are applicable to one set of parameters at a given operating point may worsen performance under different conditions, and minor design changes may lead to unpredictable results. Moreover, the lack of analytical models for the VDVP impairs the engineer's ability to estimate the dynamic behavior of a new system. Consequently, the purpose of this study was to develop models for the VDVP which would facilitate analysis of the pump dynamics and its steady-state performance characteristics.

Figure 1 displays a schematic of the VDVP analyzed in this study. The pump consists of: the housing, the cam, the vanes, the seal, the priming spring, shaft and rotor, pivot.

Fig. 1 Schematic of the variable-displacement vane-pump

Its displacement varies by motion of a circular cam which pivots about a point fixed with respect to the housing, thereby varying the pump offset. This offset, known as eccentricity, can be measured by either the linear distance between the centers of the cam and the shaft, or by the angle θ shown in Fig. 1. In this study, the eccentricity will be measured by the angle θ only [3]. The main parts of the variable-displacement vane-pump are rotor which with eccentricity is put into the fixed ring of the cam. The vanes are into the rotor's groove. The vanes move radially at the rotation of the rotor. Its outside ends slide on an internal circle of the stator. The ports are gashed into the stator. The ports are connected with intake line and line port. The working volume of the displacement machine is limited by the stator's radius and the active rotor's radius.

The pump regulates due to mobile stator and the presence pressure-regulation valve in an external contour of circulation of the working liquid. The function of the pressure-regulation valve is to support a constant pressure despite of change of capacity. When the capacity changes the overfall of the pressure increases on the throttle. The pressure-regulation valve passes the part of flow into the pump's regulation chamber. The stator overcomes the force of the spring and rotates around the pivot under the pressure in the regulation chamber. The output flow depreciates and the discharge pressure balances because of a shortening eccentricity.

The "Exact" model - ¹1. Disregarding leakage effects (which depend on pressure), the eccentricity of the VDVP uniquely determines its capacity. For a given load, the pump capacity determines the other operating variables. Consequently, analysis of the eccentricity dynamics corresponds to analysis of the pump dynamics. Equation (1) models the eccentricity dynamics as a single degree of freedom motion, governed by the torques applied to the cam about its pivot (neglecting dry friction):

(1)

where:

, (2)

where:

J_p - moment of inertia of the VDVP cam about its pivot; - angular eccentricity of the VDVP cam;B_p - viscous-damping coefficient of the VDVP cam ; ζ_p - damping ratio of the cam;; K_pr - priming-spring rate; R_p - distance between the pivot of the cam and the priming-spring center axis; T_ep - torque applied to the cam by fluid external to the cam; T_ip - torque applied to the cam by fluid internal to the cam; T_sp - preset torque applied to the cam by the priming spring; W - axial width of the cam; R_s - distance between the pivot of the cam and its sliding tip; P_d - control pressure.

With the analytical expressions for T_ip [3], equation (2) becomes:

; . (3)

Where: R - the inner radius of the cam; D- distance between the center of the cam and its pivot; P_i - the pressure in chamber; Ψ₂, Ψ₁ - the instantaneous angular position of the leading and trailing vanes of chamber i with respect to the cam center; =1,5- constant coefficients in the VDVP's equation of motion; n_c - number of VDVP chambers.

This model will be referred to as the "exact" model, or model ¹1. It is most suitable for studying the dynamics of the fluctuations of the internal mechanism of the pump. However, this model, when used in a numerical simulation, requires small and speed-dependent time-steps. In addition, it does not lend itself readily to control analysis. For such applications, i.e., numerical integration, model expansion, and control analysis, this model must be simplified.

The "Approximate" model - ¹2. Simplification of the "exact" model can be achieved based on the periodic nature of T_ip. T_ip is a sum of two periodic components [3]:

• A continuous component due to the exposure of chambers to the line port which appears as a sawtooth waveform at twice the vane frequency (the frequency of vane passage at the ports).

• An intermittent component due to fluid compression in the dead volume which appears as a pulse train at vane frequency.

When the vane frequency is large relative to the natural frequency of the cam, T_ip can be approximated by the mean level of its periodic components.

The mean of the periodic components of the torque was calculated as a sum of the means of the continuous and intermittent components, for a 100 mm diameter VDVP. It resulted in the following model for the pump dynamics:

. (4)

This model will be referred to as the "approximate" model ¹2.

Here: F₀ - force applied to the VDVP cam by the priming spring at Θ=0°;

P_t - line pressure; Ω - shaft rotational speed (r/min).

The simplified formulation of the VDVP dynamics (equation (4)) facilitates the study of the influence of design and operating parameters on the pump's performance:

• b₀ represents torque applied to the cam by the dead volume at 0° eccentricity due to the offset of the cam center with respect to the shaft center.

• b₁ represents a combination of two effects associated with the line pressure:

(a) a negative net contribution of the sawtooth waveform to the mean torque, and (b) a positive contribution related to the leakage from the dead volume - the higher the line pressure, the smaller is the pressure difference between the dead volume and its adjacent line-pressurized chamber, thereby reducing leakage through this path. Reduced leakage maintains a higher dead-volume pressure, with subsequent higher positive contribution to the mean torque. However, the first effect clearly dominates over the second one, resulting in a negative b₁ value.

• by regrouping b₂θ and b₃P_t with K_pθ, the priming-spring stiffness transforms into an "effective stiffness" constant which includes the effects of line pressure and eccentricity. In this expression, b₂ represents the dead-volume effect which may oppose or support the priming-spring action, depending on the leakage and the fluid bulk modulus. b₃P_t represents the line pressure effect introduced by the change in the lever arm of the resultant pressure vector with eccentricity. The higher the line pressure, the larger this reduction becomes. At high enough pressures, it may cancel the priming-spring action, or even change its direction. The reduction of the "effective stiffness" rate, as line pressure increases, lowers the natural frequency of the cam and increases its damping ratio. Such a reduction in the cam natural frequency provides an additional attenuation to the periodic components.

• b₄ represents the effect of shaft speed on leakage. At higher rotational speeds, less time is available for leakage while pressure builds up in the dead- volume zone. As less fluid leaks out of the dead-volume zone, the pressure drop due to leakage is reduced, leading to a larger positive contribution to the mean torque. This positive contribution approaches an asymptotic level at higher shaft- speeds. An exponential formulation generated a satisfactory representation for this effect.

The "Approximate" model - ¹3. Excluding the effect of the dead-volume pressure allows for an additional simplification of the mode. With this approach, only the exposure of the chambers to the line port contributes to the mean of T_ip. It results in the following model:

, (5)

which will be referred to as the "approximate" model ¹3.

Model evaluation and validation. The three models described above were evaluated for steady-state and transient operation using a numerical simulation program.

Results for both steady-state and dynamic operation are presented as spatial histories of:

• The "exact" instantaneous torque applied to the cam.

• The eccentricity as predicted by an "exact" model and an "approximate" model.

Results are plotted against shaft angle (as opposed to time) to eliminate the shaft-speed dependence in the presentation of the data. Each case presents five to six periods of pump operation.

Steady-state operation. Figure 2 displays steady-state operation when the dead-volume effect is excluded. It shows traces which correspond to two different line pressures. In both, the dynamic sawtooth waveform is similar to that shown in Fig.2 of reference [3] under fixed-eccentricity conditions.

The torque amplitude is proportional to the line pressure and cycle-to-cycle variations are minimal. Moreover, the mean level of the "exact" eccentricity oscillations practically coincides with the fixed "approximate" (model ¹3) level.

This agreement repeats whenever the dead-volume effect is excluded. However, as shown later, when included, the dead-volume contribution results in a steady-state bias. The bias is due to the method of calculating the dead-volume contribution in the simplified model, which excluded dynamic effects. In all cases, model ¹3 underestimates the dead-volume effect; however, the bias varies substantially with parameter changes. Figures 3 and 4 compare models ¹1 and ¹2, including the dead-volume contribution. The cases examined in Figs. 2.3 and 2.4 differ in the shaft speed - 1000 r/min and 3000 r/min, respectively. The pump damping ratio is identical in both, ζ_p = 3.

Fig. 2. The "exact" and the "approximate" models at steady-state - no dead-volume effect, moderate shaft-speed

Each of these figures displays eccentricity traces, with and without leakage in the dead-volume zone. Both figures show that no-leakage conditions in the dead volume produce high-amplitude torque-spikes. (The insets magnify the torque trace of one spike along the horizontal scale while compressing it vertically.) The cycle- to-cycle variations of the torque-spikes (amplitude as well as duration) are small. Compared with its fixed-eccentricity level [3], the durations of the dead-volume torque-spikes are lower in all cycles. The decrease in the torque-spike duration under dynamic conditions is a result of the spike pushing the cam toward higher eccentricities, while it is still in the dead-volume zone. With the high bulk-modulus of the fluid, even a small increase in the eccentricity depressurizes the chamber. Consequently, the "approximate" model ¹2 overestimates the effect of the dead- volume torque-spikes even under no-leakage conditions (although this would be the highest possible effect the dead volume can have).

The peaks of the dead-volume torque-spikes shown in fig. 3 are lower than the peaks shown in fig. 4. These lower peaks are a result of the lower shaft-speed. As the frequency of the torque-spikes decreases, the cam has more time to move, thereby decreasing the torque-spike level and duration. With the reduction in the dead-volume effect, the bias grows between the "exact" and the approximate models.

Fig. 3 The "exact" and the "approximate" models at steady-state - with dead-volume effect, moderate shaft-speed

Like the line pressure, the vane frequency has a significant effect on the amplitude of the eccentricity oscillations. Lower vane frequencies increase the amplitude of the eccentricity oscillations, due to the increased transmissibility of the higher harmonics. High-amplitude oscillations, i.e., larger cam displacements, lead to higher fluid-compression in the dead-volume zone. Therefore, lower vane- frequencies result in higher dead-volume torque-spikes.

Fig. 4 The "exact" and the "approximate" models at steady-state - with dead-volume effect, high shaft-speed

Model validation. The previous simulations indicate that a simplified model, based on mean torque, can represent the pump dynamics under conditions of low- amplitude eccentricity-oscillations. However, the model's predictions are invariably biased. The degree of the bias depends on dead-volume effects, and, consequently, will vary between production units and under different operating conditions. Therefore, in evaluating the model's predictions, a range of values must be considered, rather than single-value estimates. The bounds of this range are the predictions of the two simplified models, with and without dead-volume effects.

Consequently, the validation of the model was conducted against such a range. The upper limit represents maximum dead-volume bias, while the lower one represents zero dead- volume bias. Model ¹2 determines the upper limit, assuming no reduction in the dead-volume pressure due to either leakage or cam motion; it is fluid dependent since the predicted pressure is a function of the fluid bulk-modulus. Model ¹3 determines the lower limit, i.e., the dead-volume compression is assumed to be fully relieved. All test results are well within the range established by the limits. Moreover, the slopes of linear-regression lines of the data (not shown) agree well with the predicted slopes. These slopes represent the effect of the continuous component of the torque on the cam due to the exposure of the pump's chambers to the line port. This effect does not include the dead-volume contribution. Similar slope agreement repeated at other shaft-speeds.

References

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