Page 570 - Mechatronics with Experiments

JWST499-Cetinkunt
JWST499-c07
556 MECHATRONICS Printer: Yet to Come October 9, 2014 8:41 254mm×178mm
The cylinder and load dynamics, including the fluid compressibility effect taken into
account,
(m + m ) ⋅ ̈ y(t) =−c ⋅ ̇ y(t) + p (t) ⋅ A − p (t) ⋅ A − F load (t) (7.559)
B
p
p
A
B
A
l
If up motion x ≥ 0.0 (7.560)
v
d
(p (t)) = (Q (t) − ̇ y(t) ⋅ A ) (7.561)
PA
A
A
dt V + y ⋅ A
hose,VA A
d
(p (t)) = (−Q BT (t) + ̇ y(t) ⋅ A ) (7.562)
B
B
dt V hose,VB + (l cyl − y) ⋅ A B
elseif down motion x < 0.0 (7.563)
v
d
(p (t)) = (−Q (t) − ̇ y(t) ⋅ A ) (7.564)
A
A
AT
dt V hose,VA + y ⋅ A A
d
(p (t)) = (Q (t) + ̇ y(t) ⋅ A ) (7.565)
PB
B
B
dt V hose,VB + (l cyl − y) ⋅ A B
end (7.566)
The position sensor is modeled as a constant gain device,
y (t) = K ⋅ y(t) (7.567)
fb
fb
The closed loop control algorithm is a PID control,
t
i cmd (t) = K (y (t) − y (t)) + K i (y ( ) − y ( ))d + K ( ̇ y (t) − ̇ y (t)) (7.568)
fb
d
d
d
p
fb
fb
d
∫
0
Now, let us consider the simulated conditions: the size related parameters of the
components, initial conditions of differential variables (i.e., pressures and spool position
and velocity), and input variables as a function of time.
3
D (t) = 20 ∗ 10 −6 m ∕rev pump displacement (7.569)
p
w shaft (t) = 3000 rev∕min input shaft speed of the pump (7.570)
2
3
p max = 3000 psi = 3000 psi ⋅ 6.895 ⋅ 10 (N∕m )∕psi (7.571)
2
6
= 20 685 kN∕m = 20 685 kPa = 20.685 ∗ 10 N∕m 2 (7.572)
relief valve max. presure setting (7.573)
3
Q = 1.0 ⋅ 10 −3 m ∕s at Δp = 1000 psi = 6895 kPa (7.574)
rv
r
rated flow rate of servo valve (7.575)
6
2
p = 6.895 ∗ 10 N∕m = 1000 psi (7.576)
r
x = 10 ⋅ 10 −3 m max. spool displacement of servo valve (7.577)
v,max
T = 1.0N max. actuation torque of servo valve (7.578)
v,max
= 0.6; damping ratio (7.579)
x v
w = 75 Hz = 2 ∗ 75 rad∕s; for ≤ 40% (7.580)
n
x
v,max
x v
= 25 Hz = 2 ∗ 25 rad∕s; for = 100% (7.581)
x v,max
natural frequency of valve spool motion (7.582)
= 0.010 s electrical time constant for current (7.583)
a
K = 1.0; current amplifier gain (7.584)
a
K = 1.0; current to torque gain (7.585)
t