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NOMENCLATURE Roman Letter Symbols a grinding cut depth, m ARMA output coefficient, dimensionless A state matrix, dimensionless A1 preexponential frequency factor in cure kinetics model, s

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Least Squares IfN consecutive measurements ofthe output y(i), 1 = 1 · · · N,

are made at successive sampling periods, these can be concatenated in a measured output vectorYas

Y= [y(n + 1)y(n + 2) · · ·y(n + N)]T (17.109)

These will be compared with their respective predictions Y by the ARMA process

model, in which the corresponding augmented vectorsΦ can be assembled in

ma-trixΦ:

Φ = [ΦT(n) ΦT(n + 1) · · · ΦT(n + N − 1)]T (17.110) where the elementary vectorsΦ also contain the previously measured outputsy (i),

i = 1 · · · n and inputs Q(i), i = 1 · · · n Thus the ARMA model ofthe thermal system

can be written as Y= ΦΘ The least squares method minimizes the quadratic index

ofthe output deviationsε =Y − Y (i.e., J = εTε) setting the parameters Θ to

Θ = (ΦTΦ)−1Y = Φ†Y (17.111) whereΦ† is the pseudoinverse of Φ.

NOMENCLATURE

Roman Letter Symbols

a grinding cut depth, m

ARMA output coefficient, dimensionless

A state matrix, dimensionless

A1 preexponential frequency factor in cure kinetics model,

s−1

Ac, Ao , A1 cross-sectional area, m2

b chip width, m

grinding width, m saturation point, dimensionless ARMA input coefficient, dimensionless

B input matrix, dimensionless

B1, B2, B3 parameters defined in eqs (17.26)–(17.28),

dimensionless

Bi Biot number, dimensionless

ˆc degree ofcrystallization, dimensionless

c specific heat capacity, kJ/kg· K

saturation level, dimensionless

C output matrix, dimensionless

C A instantaneous resin concentration in the resin–catalyst

mixture at any time,t, kg/m3

C A0 initial resin concentration in the resin–catalyst mixture,

kg/m3

d delay, dimensionless

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d s grinding wheel diameter, m

differential, dimensionless period, dimensionless

D direct matrix, dimensionless

D b degree ofbonding, dimensionless

D h degree ofhealing, dimensionless

Dic degree ofintimate contact, dimensionless

e error, dimensionless

E activation energy for viscosity in the chemorheological

model, kJ/kmol error (Laplace transform), dimensionless expectation, dimensionless

E1, E2 activation energies in the cure kinetics models, kJ/kmol

f transfer function, dimensionless

F, F forces associated with cutting, N

g controller transfer function, dimensionless

G transfer function, dimensionless

G discrete state matrix, dimensionless

h heat transfer coefficient, W/m2· K

depth ofcut, m thermoplastic tow thickness, m sensor transfer function, dimensionless discrete-time index, dimensionless

H plate thickness, m

H detectability matrix, dimensionless

discrete input matrix, dimensionless

HT total heat ofcrystallization, kJ/kg

Hu theoretical ultimate heat ofcrystallization, kJ/kg

∆H c heat ofcrystallization, kJ/kg

∆H R heat ofthe cure reaction, kJ/kg

i discrete time index, dimensionless

I identity matrix, dimensionless

J quadratic performance index, dimensionless

k thermal conductivity, W/m· K

discrete time index, dimensionless

k o Kozeny constant, dimensionless

K gain, dimensionless

K controller matrix, dimensionless

Ko modified Bessel function ofthe first kind, oforder zero,

dimensionless

K10, K20 preexponential frequency factors in the cure kinetics

models, s−1

l input number, dimensionless

length ofheated region, m

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l c frictional contact length, m

l s shear length, m

thickness, m characteristic length, m

L observer matrix, dimensionless

m, n exponents in the cure kinetics models, dimensionless

output number and state number, dimensionless

m1, m2



Pe±Pe2+ 4 Bi/2, dimensionless

N describing function, dimensionless

sample number, dimensionless

p stiffness of a fiber network, Pa

P perimeter, m

grinding power, W pressure, Pa

P pdrs matrix, dimensionless

Pe P´eclet number, dimensionless

q heat flux, W/m2

Q heat source strength, W

Q(t) heat release during cure per unit mass ofresin-catalyst

sample, kJ/kg

Q state penalty matrix, dimensionless

r, x, y, z spatial coordinates, m

r f fiber radius, m

R fraction of shear energy removed by the chip,

dimensionless void radius, m universal gas constant, kJ/kmol· K

R, X, Y coordinates, dimensionless

R resultant force, N

input penalty matrix, dimensionless

s Laplace variable, dimensionless

S radius ofa resin shell surrounding a void, m

S stabilizability matrix, dimensionless

T temperature, K

sampling period, s

u grinding energy, J/m3

integration variable, dimensionless

v velocity, m/s

void fraction, dimensionless output noise, dimensionless excitation, dimensionless

v a maximum fiber volume fraction, dimensionless

v f fiber volume fraction in the composite, dimensionless

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v I initial fiber volume fraction, dimensionless

V velocity, m/s

Lyapunov function, dimensionless

w state noise, dimensionless

W D deformation work per unit volume, J/m3

y parameter in the degradation kinetics model,

dimensionless output, dimensionless

y output measurement, dimensionless

Y measured output, dimensionless

z Z-transform variable, dimensionless

Greek Letter Symbols

α thermal diffusivity, m2/s

degree ofdegradation, dimensionless

β friction angle

shape constant, dimensionless

γ characteristic thickness, m [= A c /P ]

adaptation gain, dimensionless

γo rake angle, deg

ε fraction of grinding power entering the workpiece as

heat, dimensionless degree ofcure, dimensionless [= (C A0 − C A)/C AO] state deviation, dimensionless

θ temperature, dimensionless

Θ parameters, dimensionless

κ permeability, m2

λ constant in the chemorheological model, dimensionless

eigenvalue (observer), dimensionless

µ coefficient of friction, dimensionless

dynamic viscosity, Pa· s

eigenvalue (controller), dimensionless

ρ density, kg/m3

σ interfacial bond strength, Pa

τ time, dimensionless

time (Lagrangian), s

τs flow stress, N/m2

φ porosity, dimensionless

characteristic polynomial, dimensionless

φo shear plane angle, deg

ϕ resin volume fraction in the fiber–resin mixture,

dimensionless [= 1 − v f]

Φ augmented state, dimensionless

ω angular frequency, s−1

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Subscripts and Superscripts

∗ dimensionless

estimated

· time derivative

† pseudoinverse

atm atmospheric

controller controllability

cl closed loop

d desired reference

derivative

gas

ambient reference

proportional

refreference value

setting

x, y, z along the respective coordinate directions

∞ ultimate (maximum realizable) value

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