Page 615 - Mechatronics with Experiments

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JWST499-c07
JWST499-Cetinkunt
ELECTROHYDRAULIC MOTION CONTROL SYSTEMS 601
The only initial condition needed in this case is y(t ). The rest of the initial conditions (p (t ),
P 0
0
p (t ), p (t ), ̇ y(t ), Q (t )) do not need to be specified. Instead, they are calculated directly from the
0
0
0
r
A
B
0
simultaneous solution of the algebraic equations. In this case the differential equations describing
the variation in cylinder pressure and cylinder motion reduce to an algebraic equation where the
cylinder velocity is directly determined by the flow rate into the cylinder. The solution can be
obtained directly by substitution among the algebraic equations since there are as many equations
as there are unknowns. However, a more useful method is to use a numerical method to solve
these equations. The iterative solution of the algebraic equations requires initial guesses (not initial
conditions), p (t ), p (t ), p (t ), ̇ y(t ), Q (t ), which are used as the starting point for the numerical
P 0
A
0
r 0
0
B 0
search algorithm for finding the roots. In order to simulate this case, simply set the left hand side of
the differential Equations 7.776–7.778 to zero, turning them into algebraic equations, and solve the
whole equation set (1)–(10) numerically.
Case 2 – Take into account cylinder and load inertial dynamics, and fluid compressibility
on both pump and load side of the line: In this case we have the second-order differential equation
as Equation (1), that defines the net force–acceleration relationship due to inertia and the differential
Equations (7.776–7.782). Then we need y(t )and ̇ y(t ), p (t ), p (t ), p (t ) initial conditions to be
A
0
0
0
P 0
0
B
specified. Numerical solution requires only the solution of a set of ordinary differential equations.
All of the algebraic equations (5-9) can be solved directly (without any iterative algorithm) by using
the available initial conditions on pressures.
We simulate this condition without the accumulator and with the accumulator in the circuit.
The accumulator is added to the circuit model by including the V and Q terms in Equation (4),
acc acc
and the equations describing the dynamics of the accumulator state, p (t)and V (t) (Equations
acc acc
(10), (11), and (12)). We need to specify the initial pressure and volume of the accumulator (p (t ),
acc
0
V (t )), as well as the discharge volume capacity (V disch ) and operating pressure range (p min , p max ).
acc
0