Civil Engineering Reference
In-Depth Information
ground surface position,
x
f
. Note that the compliance of strain gage force sensors has been shown to be
a significant part of the contact stiffness in applications where they are used (Book and Kwon, 1992).
Using Black's Law, the process dynamics in
Fig. 3.6
may be reduced to an algebraic equation. This
equation is simplified by assuming the effective damping of the servo stage and the structural damping
are negligible (
e.g.,
B
s
0). This assumption is made possible by the low speeds at which the grinding
wheel plunges into the part. Neglecting the effect of damping yields the following first order equation
→
x
f
()
s
K
P
K
S
V
/
A
------------
---------------------------------
.
(3.19)
x
e
()
s
s
K
P
K
S
V
/
A
3.4
Grinding Model Parameter Identification
Two types of experiments can be conducted to estimate the model parameters,
F
TH
and
K
P
, of Eq. (3.16).
These parameters are identified by conducting position controlled grinding experiments and force con-
trolled grinding experiments. The mathematical basis for identification in both types of experiments is
outlined in the following paragraphs. The purpose of these experiments is to validate the appropriateness
of the selected grinding model.
Parameter Identification via Position Control
The first type of experiment uses position control of the grinding system to provide position step inputs.
The response of the grinding wheel,
x
f
(
t
), to the position step input of magnitude (
X
e
), is given in Eq. (3.20).
K
P
K
S
V
/
A
X
e
s
1
()
---------------------------------
-----
x
f
t
(3.20)
s
K
P
K
S
V
/
A
Taking the inverse Laplace transform yields an exponential relationship
(
K
P
K
S
V
/
A
)
t
x
f
()
t
X
e
1
(
e
)
.
(3.21)
Similarly the transfer function from the controlled encoder position,
x
e
, to the predicted grinding normal
force,
F
N
, (between the part and the grinding wheel) can be formulated
F
N
s
()
X
e
s
s
.
-------------
K
S
---------------------------------
(3.22)
()
s
K
P
K
S
V
/
A
Again, the normal force response to a step position input of magnitude
(
x
e
)
is exponential in nature,
and is given by Eq. (3.23).
(
K
P
K
S
V
/
A
)
t
F
N
()
t
K
S
e
(
)
X
e
(3.23)
Based on Eqs. (3.22) and (3.23) for an ideal step function, the position response of the grinder (and ground
surface) is a first order system with an exponential rise from zero to a maximum value. The imposed
normal force will start at the maximum value and decay exponentially until it reaches the threshold force,
F
TH
. The exponent, (
K
P
K
s
V
/
A
), of Eq. (3.21) is estimated in the position control experiments off-line by
the use of an auto-regressive, least squares parameter estimation technique (ARX, AutoRegressive eXternal
input, Ljung, 1987).
After identifying the exponent, (
K
P
K
s
V
/
A
), of Eqs. (3.21) and (3.23),
K
P
is determined via substitution
of the known or calculated values of the constants. The residual value of normal grinding force when
there is no appreciable grinding is the estimate of the threshold force,
F
TH
.
