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Dynamic performance of a nonlinear non-dimensional two stage electrohydraulic servovalve model

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Abstract

Electrohydraulic servosystems are commonly used in various engineering applications due to their high power-to-weight ratio, good performance, and ease of control. An electrohydraulic servovalve is an essential item of fluid power servomechanism where fast response, high power output and working fidelity are necessary. A full description of the dynamic performance of a two stage electrohydraulic servovalve is presented. Nonlinear Non-dimensional mathematical model has been developed for that purpose. The system main equations could be derived in minimal symbolic forms; which facilitates a subsequent numerical simulation in order to investigate the static and dynamic behaviors. In addition to a step response, ramp and sinusoidal inputs responses are investigated. The model has been coded in the software package Matlab/Simulink (Liuping, Modern control systems analysis and design using Matlab and Simulink, Springr-Verlag, London, 2009). The use of the non-dimensional form enables designer to expect the valve behavior of the same shape but with different sizes.

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Abbreviations

A p , A n , A o , A s :

Area of load piston, nozzle, orifice, and spool respectively (m2)

\( a_{r} ,\bar{a}_{r} \) :

Dimensional and non-dimensional oil film inertia coefficients respectively (N, –)

\( b_{r} ,\bar{b}_{r} \) :

Dimensional and non-dimensional oil film damping coefficients respectively (N, –)

\( B_{a} ,\bar{B}_{a} \) :

Dimensional and non-dimensional viscous damping coefficient of armature pivot respectively (N m rad−1, –)

B p :

Viscous damping coefficient of load piston (N s m−1)

C dn , C do , C ds :

Discharge coefficient of nozzle, orifice, and spool respectively

C lp , C ls :

Leakage coefficient of load piston, spool valves (m3 Pa−1 s−1)

C rd , C rp , C rs :

Radial clearance of squeeze film damper, load piston, and between spools and sleeves respectively (m)

C v :

Spool valve velocity coefficient

d p , d n , d o , d s :

Diameter of load piston, nozzle, orifice, and spool respectively (m)

\( e,\bar{e} \) :

Dimensional and non-dimensional torque motor input voltages respectively (V, –)

E b , eb:

Bias voltage in torque motor circuit, and feedback voltage respectively (V)

F h :

Hydraulic force on flapper (N)

\( F_{sf} ,\bar{F}_{sf} ,F_{sh} ,F_{ext} \) :

Dimensional and non-dimensional squeeze film damping forces on the flapper, Shear damping force on a spool and the external force acting on the load piston respectively (N, –, N, N)

H:

Distance from squeeze film damper center to the pivot (m)

\( i,\bar{i},I_{o} , \) :

Dimensional and non-dimensional torque motor differential current, and, quiescent current in torque motor respectively (A, –, A)

\( J_{a} ,\bar{J}_{a} \) :

Dimensional and non-dimensional polar mass moment of inertia of armature and flapper around the pivot respectively (kg m2)

\( K_{a} ,\bar{K}_{a} \) :

Dimensional and non-dimensional rate of the rotational spring fixed at the armature pivot respectively (N m rad−1)

\( K_{pg} ,\bar{K}_{pg} \) :

Dimensional and non-dimensional potentiometer gain constants (Vm−1)

\( K_{da} ,K_{pa} \) :

Differential amplifier, power amplifier gain constants respectively

\( K_{tg} ,\bar{K}_{tg} \) :

Dimensional and non-dimensional transformer gain constant respectively (Vm−1, –)

\( K_{t} ,K_{m} \) :

Torque constant and magnetic spring constants of torque motor respectively (N m A−1, N m rad−1)

\( K_{an} ,\bar{K}_{an} \) :

Dimensional and non-dimensional net torque motor spring rates respectively (N m rad−1, –)

\( K_{n} ,\bar{K}_{n} \) :

Dimensional, and non-dimensional nozzle diameter parameters (m (m3 kg−1)−1, –)

\( K_{o} ,\bar{K}_{o} \) :

Dimensional and non-dimensional orifice area parameters (m2 (m3 kg−1)−1, –)

Kp:

Equivalent stiffness of load piston springs (N/m)

\( L,\bar{L} \) :

Dimensional and non-dimensional coil inductance of torque motor (H)

\( L_{t} \) :

Distance between supply pressure port and flow output port (m)

M p , M s :

Masses of load piston and spool respectively (kg)

\( \bar{N} \) :

Non-dimensional spring rate due to flow striking the flapper

\( \begin{gathered} p_{1} ,p_{2} ,\bar{p}_{1} ,\bar{p}_{2} , \hfill \\ p_{a} ,p_{b} ,\bar{p}_{a} ,\bar{p}_{b} \hfill \\ \end{gathered} \) :

Dimensional, and non-dimensional output pressures of the first, and second stages respectively (Pa, –, Pa, –)

\( p_{L} ,\bar{p}_{L} \) :

Dimensional, and non-dimensional differential pressures across the load piston respectively (Pa, –)

\( P_{s} ,\bar{p}_{t} \) :

Supply pressure and reservoir pressure respectively (Pa)

Q 1, Q 3, Q 2, Q 4 :

Flow through orifices and nozzles respectively (m3/s)

Q 5, Q 6, Q a, Q b :

Flow rates from/to second stage to/from first stage, and flow rates from/to piston to/from second stage (m3/s)

Q lp , Q ls :

Leakage flow through the load piston, through spools from/to reservoir (m3/s)

c :

Distance from pivot to nozzle (m)

\( r_{p} ,R,\bar{R} \) :

Internal resistance of amplifier in each coil circuit, dimensional, and non-dimensional coil resistances of torque motor respectively (Ω, –)

R d :

Radius of squeeze film damper (m)

t, τ :

Dimensional and non-dimensional times respectively (s, –)

\( \bar{T} \) :

Non-dimensional flapper nozzle valve flow force parameter

V 1, V 2 :

Initial volumes of top and bottom spool chambers (m3)

V t :

Total volume of oil cylinder chambers and connecting lines (m3)

W p :

Port width (m)

\( x_{a} ,x_{b} ,x_{f} ,\bar{x}_{a} ,\bar{x}_{b} ,\bar{x}_{f} \) :

Dimensional and non-dimensional displacements of left spool, right spool, and flapper respectively (m, –)

x f0 :

Equilibrium position of the flapper (m)

\( \begin{gathered} x_{i} ,\bar{x}_{i} ,x_{\max } ,x_{p} , \hfill \\ \bar{x}_{p} ,x_{{p_{\max } }} ,x_{s} \hfill \\ \end{gathered} \) :

Dimensional, non-dimensional input displacement, maximum spool displacement, dimensional, non-dimensional load piston displacements, maximum load piston displacement, and squeeze film damper displacement respectively (m, –, m, m, –, m)

\( \bar{\alpha }_{f} \) :

Non-dimensional parameter for effect of feedback spring on the flapper

\( \bar{\alpha }_{p} \) :

Frequency ratio ω p /ω p

\( \bar{\alpha }_{s} \) :

Non-dimensional differential pressure force parameter across spools

\( \beta ,\bar{\beta } \) :

Dimensional and non-dimensional bulk modulus of fluid respectively (N m−3, –)

\( \bar{\delta }_{f} \) :

Non-dimensional parameter for the effect of feedback spring on the flapper

\( \varepsilon \) :

Eccentricity ratio

\( \bar{\gamma }_{p} \) :

Non-dimensional differential pressure force parameter across the load piston

\( \bar{\gamma }_{s} ,\bar{\lambda }_{s} ,\bar{\lambda }_{f} ,\bar{\lambda }_{p} \) :

Non-dimensional shear force, volumetric change, valve dimension, and leakage flow across load piston parameters respectively

μ:

Fluid viscosity (N s m−2)

θ:

Angular deflection of flapper (rad)

ρ:

Fluid density (kg m−3)

ω a , ω p :

Natural frequencies of the armature and load piston respectively (rad s−1)

ω h :

Hydraulic natural frequency (rad s−1)

\( \bar{\xi } \) :

Non-dimensional modified flow coefficient

ζ p :

Damping coefficient of the piston (N s m−1)

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Authors and Affiliations

  1. Faculty of Engineering, Cairo University, Cairo, Egypt

    M. El-Araby

  2. Faculty of Engineering, Suez Canal University, Port-Said, Egypt

    A. El-Kafrawy & A. Fahmy

Authors
  1. M. El-Araby
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  2. A. El-Kafrawy
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  3. A. Fahmy
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Correspondence to A. El-Kafrawy.

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El-Araby, M., El-Kafrawy, A. & Fahmy, A. Dynamic performance of a nonlinear non-dimensional two stage electrohydraulic servovalve model. Int J Mech Mater Des 7, 99–110 (2011). https://doi.org/10.1007/s10999-011-9150-x

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