A proposal for symbols and terminology in the C.I.R.P. Ma
Group
Citation for published version (APA):
Kals, H. J. J. (1971). A proposal for symbols and terminology in the C.I.R.P. Ma Group. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0279). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1971
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C.I.R.P. Ma Group".
Eindhoven University Press
This proposal has been made on request of the chairman of the C.l.R.P. Group Ma.
Its aim is only to make a first step to a uniform application of symbols and terminology within the group.
Remarks and additions should be made on the Group-meeting in Cracow, sept. 1971, or send to the address mentioned below.
Juni 1971. H.J.J. Kals Eindhoven University of Technology, H. G. 3.02, P.O. Box 513, EINDHOVEN. the Netherlands.
Symbol A
[AJ
B Significance Area Matrix Bandwith [B] Matrix ~] Matrix D Diameter Transverse force[0]
Decomposition matrix E s ~ Fj'
[F
c.F G I. ~ I p Im {oo} J. ~Young's modulus of elasticity
Cutting force
Nominal cutting force Coulomb friction force Feed force
Main cutting force
Shear force in the shear plane Force vector
Dynamic component of the cutting force
Projection of the dynamic component of the cutting force on the main direction of motion which is characterized by a
Shear modulus
Second axial moment of area (subscript indicates axis) Second polar moment of area Imaginary part of {o.}
Mass moment of inertia (subscript indicates axis)
Recommanded unit 2 m Hz m N N N N N N N N N 4 m 4 m 2 kg m Dimension -] T L -2 1M! -1 -2 L MT -2 1M! -2 1M! LMT-2 -2 1M! LMT-2 -2 1M! -2 1M! LMT-2 -1 -2 L MT
J
P Polar mass moment of inertia
2
kg m
[K]
Stiffness matrix[L] Lower triangular matrix
M.
J. Lumped mass (suffix denotes
mode number) kg [M] Mass matrix N P R R S. 1.
[s]
T T c Normal force PowerRatio of two natural frequencies for coupled systems
Plane
N
Nm/s (Watt)
Radius of workpiece m, rom
In-phase component miN, ~m/N
Upper triangular matrix Real part of {'1
Maximal negative in-phase
component miN, ~m/N
Maximal in-phase component of the structural transfer
function for a single
degree-of-freedom system miN, ~m/N
Maximal negative in-phase component of the structural transfer function for a single
degree-of-freedom system miN, ~m/N
Modal flexibility, i.e.
compliance at natural frequency (The suffix i denotes the mode
number) ~m/N, miN
Flexibility matrix
Period time
Transfer function of the cutting process (AF/Ah) Limit value for T on the
threshold of stabtlity s N/m, N/~m N/m, N/Ilm
-2-L~
M T -2 MT[T]
Direction-cosine matrixV Potential energy Nm L~-2
VB Width of the flank W'ear land
of the tool nun L
X Half of the peak-peak value of
displacement m, 11m L
x* Delayed X-value m, ].1m L
y Half of the peak-peak value
of displacement m, 11m L
y* Delayed Y-value m, ].1m L
a b g c c* c c c. ~ e f f o fd f. ~
Depth of cut m, rom
Admittance mlNs
Width of cut m, rom
Critical width of cut taking
into account the phase equation m, rom Limit value of width of cut m, rom
Damping coefficient Nslm
Optimal damping coefficient applying auxiliary systems for
damping Nslm
Damping coefficient related to
the instantaneous cut Nslm
Damping coefficient of the cutting process, related to the main
direction of motion Nslm
Damping coefficient related to the delayed path
Equivalent damping coefficient for structures
Specific process damping
coefficient related to the main direction of motion
(instantaneous cut)
Resultant specific process damping vector (instantaneous cut)
Overall damping coefficient of the machining system
Diameter of workpiece Deflection vector Eccentricity Width of gap Nslm Ns/m 2 Ns/m Nslm m, rom m, rom m, rom Frequency Hz
Undamped natural frequency Hz
Damped natural frequency Hz
Undamped natural frequenqy in a mUltiple degree-of-freedom system, where i
=
1,2,3,... Hz -4-L L L -1 MT -1 MT -1 -1 L MT -1 -1 Ii. MT L L L -1 T -1 T -1 T -1 Tfk Chatter frequency
Hz
T -1 f Frequency related to the limitn
value of the transfer function -1
of the cutting process
Hz
TfR+ Frequency for the maximal in-phase component of a
single-degree-of-freedom-system -1
transfer function
Hz
Tf R_ Frequency for the maximal negative part of a
single-degree-of-freedom-system -1
transfer function
Hz
TAcceleration of gravity 2 LT-2
g m/s
h Undeformed chip thickness m, rom L
h Nominal undeformed chip
0
thickness m, rom L
llh Chip thickness variation m, }.1m L
i Imaginary unity
k Structural stiffness N/m MT -2
Dynamic cutting coefficient N/m -2
kd MT
-2
k Equivalent stiffness N/m MT
e
N/m2 L- 1MT-2
k. Specific process stiffness
~
""'" Resultant specific process k.
N/m2 -1 -2
~
stiffness vector L MT
~d
Delayed chip thickness modulationMT-2
coefficient N/m
~i
Instantaneous chip thickness-2 modulation coefficient N/m MT 1 Lenth m L m Mass kg M m Equivalent mass kg M e Rotational rls -1 n frequency T
q Amplification factor (resonance
factor, quality factor)
r e: 9 6,9 t u u., u .. J 1J v w x x* y z
Tool tip radius
Feed per revolution Arc length Compliance nnn m, lIlIll m
miN,
llm/N -6".. LNominal feed per revolution m, lIlIll L
Maxwell number of
cross-compliance. The suffix j denotes the station point where a unit force is applied., The suffix i
denotes the station point where
the deflection is obtained mIN or rad/Nm M-1T2 or M-1L-2T2
Feed variation m, llm L Time s T Thickness Displacement Directional factor m, lIlIll Cutting speed m/s Displacement m, llm Feed speed m/s
Variation of the feed speed m/s
Displacement m, llm
Displacement m, llm
Delayed displacement m, llm
Displacement, recommanded for the instantaneous d~flection of the tool, normal to the cut
surface m, llm
Delayed chip thickness modulation m, llm
Displacement
Number of teeth of the cutting tool m, llm L L L L L L L L
z
c Number of instantaneous active teeth of the cutting tool
y e: ~mt l;red ~s n
e
Angle between the main direction of motion and the direction of the chip thickness modulation Clearance angle of the cutting tool
Angular acceleration
Angle between the resultant specific process stiffness vector and the main direction of motion
Angle between the resultant
specific process stiffness vector and the direction of the chip thickness modulation o rad, o 2 radls o rad, o rad,
Angle between the resultant specific process damping vector
and the main direction of motion rad, 0
Rake angle of the cutting tool Shear strain
Angle between the resultant specific process damping vector and the direction of the chip thickness modulation
Slenderness ratio of the cut logarithmic strain
Variation, increment
Elastic strain
Damping ratio
Damping ratio of the structure in working conditions
Reduced damping ratio
Damping ratio of the system during cutting Dynamic viscosity Temperature Phase angle o rad o rad, 2 Nslm o rad, -8-1 -2 T -1 -1 L MT
A v p Q g w w o w c
Cutting edge inclination Wavelength
Eigenvalue
Logarithmic decrement
Coefficient of friction Overlap coefficient
Dimensionless angular frequency Kinematic viscosity Mass density Normal stress Shear stress Time constant Time delay
Phase angle between force and tool displacement Shear angle o m 2 s 2 m /s s s o rad, o rad,
Phase angle between force and
o
delayed chip thickness modulation rad,
Angular frequency of workpiece
or tool rad/s
Angular frequency of workpiece or tool, related to the limit value of the transfer function
of the cutting process rad/s
Angular frequency rad/s
Undamped natural angular
frequency rad/s
Angular frequency of system
pulse response during cutting rad/s Angular frequency of system pulse
response without cutting, but
with moving carriage rad/s
-3 ML -1 T -1 T -1 T -1 T -1 T -1 T
-10-Wd Damped natural angular -1
frequency rad/s T
w. Undamped natural angular
l.
frequency in a multiple
degree-of-freedom system, -1
where i
=
1,2,3, ••• rad/s T_1
Angular chatter frequency rad/s T •
w
k
W Angular frequency related to
n
the limit value of the transfer -1
function of the cutting process rad/s T
wR Angular frequency corresponding -1
+ with fR+ rad/s T
wR_ Angular frequency corresponding -I
