Recent Technological Progress in High Speed Annealing* Continuous

16 Pages·2006·2.52 MB·English

Recent Technological Progress in High Speed Continuous
Annealing*
By Takeo FUKUSHIMA**
I. Preface the line speed of the horizontal furnace must be kept
Continuous annealing furnaces perform heat treat- relatively low. In the catenary type furnace, for ex-
ment with a certain heat cycle to process a cold rolled ample, the line speed remains 2O'-. 100 m/min.2 The
steel strip which is running through the processing maximum speed of the roller hearth type falls in the
line. range of 15O-2O0 m/min3~ as typically operated in
Specification of the equipment is basically deter- the continuous galvanizing line.
mined by kinds of steel to the procured, required heat Then, the vertical furnace used for stainless steell
cycles and required strip surface condition. processing, that is BA furnace, is also subject to a
There are several types of furnaces, any one of limited line speed because of the high annealing tem-
which is selected suitably complying with the kinds of perature resulting in diminished heat flux. The line
steel to be processed. In annealing such special steel speed of this furnace type is further restricted by a
strips as stainless steel and electrical steel, horizontal requirement to complete heating and cooling in a
type of furnaces are predominantly used to meet high single pass. In the vertical furnace for carbon steel;
annealing temperature and other requirements in however, much higher line speed is attainable in that
multi-strand pass provides a long heating section even
processing these groups of steel. The horizontal type
furnaces are subclassified, according to furnace atmo- with the small heat flux, and that the strip turns
sphere, into a catenary type for open air atmosphere around the supporting rolls to cause a large strip
and a roller hearth type for non-oxidizing atmo- holding force of rolls, thereby ensuring a good track-
sphere. In some cases, however, vertical type furnaces ing. A recently constructed continuous annealing
are used for the special steel as well. The stainless line achieves the line speed as high as 600 m/min.1~
steel is better processed by a vertical furnace when Bearing a general view as above in mind, I will
bright finishing is required for final products. This discuss in this paper about recent technological trend,
type of furnace is usually called a vertical bright an- focusing on the vertical type continuous annealing
nealing furnaces (BA Furnace). Electrical steel can furnace for carbon steel. The discussion is mainly
also be annealed in the ordinary vertical furnace at related to the technology for high speed processing
a high speed as a result of recently developed process particularly in light of strip conveying methods and
technology. 1) heat transfer, after reviewing briefly construction of
In the meantime, almost all of ordinary carbon the equipment and history of the annealing tech-
steel are annealed in vertical furnaces, except when nology.
processed by a continuous galvanizing line which
II. Construction of Continuous Annealing Equip-
normally incorporates a vertical furnace in the line.
These furnaces are filled with reducing gas to protect ment
the strip surface finished through cold rolling from A continuous annealing line consists of annealing
oxidization during heat treatment. The number of furnaces, and entry and delivery equipment. Figure
strand passes varies according to the production ca- 1 shows schematically a layout of a multi-purpose
continuous annealing line as a typical example.
pacity required. The capacity of the furnaces is prin-
cipally characterized in terms of the line speed and The equipment construction and function thereof
annealing temperature. are described hereunder based on this figure.
The horizontal furnace, as described above, pri-
marily processes the special steel which requires a 1. Entry Equipment
heating temperature ranging from 500 to 1 000 °C. Constituent equipment of the entry section en-
Since the heat flux from heat sources tends to dimin- compasses pay-off reels through No. 1 looper. The
ish as the temperature rises, a high strip temperature strip fed from the pay-off reel is cropped by the shear,
must be secured either by extending a heating section welded with a preceding strip, and led to a cleaning
or by reducing the line speed. Further, the strip is section where the rolling oil on the strip surface is
conveyed on supporting rolls without sufficient fric- removed. The cleaning section usually consists of a
tion between the rolls and strip (holding force) to dunk tank, No. 1 brush scrubber, an electrolytic clean-
ensure better tracking of the strip. Consequently, ing tank, No. 2 brush scrubber, a rinse tank and a
* Presented to the Nishiyama Memorial Seminars: the 88th Seminar at Nokyo Hall , Tokyo in February, 1983, and the 89th Seminar
at Kagaku Gijutsu Center, Osaka in March, 1983. Originally published in The 88-89th Nishiyama Memorial Seminars, ed. by ISIJ,
ISIJ, Tokyo, (1983), 137, in Japanese. English version received on September 13, 1984. © 1985 ISIJ
* * Hiroshima Shipyard & Engine Works
, Mitsubishi Heavy Industries, Ltd., Kan-non-shinmachi, Nishi-ku, Hiroshima 733.
(278) Review
Transactions ISIJ, Vol. 25, 1985 (279)
Fig. 2. Structure of the furnace and function of each section.
dryer. Then, the strip is sent to No. 1 looper. The
looper stores a sufficient length of strip to feed it con-
tinuously to the furnace at a constant speed while the
strip running in the entry section stops in order to
weld with a following strip.
2. Continuous Annealing Furnace
In continuous annealing, the heat cycle varies ac-
cording to kinds of steel to be treated.
Figure 2 indicates typical heat cycles applicable to
different kinds of steel, and the function of each fur-
nace section to realize an intended heat cycle in the
multi-purpose continuous annealing furnace. Photo-
graph 1 shows the multi-purpose furnace viewed from
the entry section.
1. Heating Section
The strip passes through a roll seal installed at the
Photo. 1. KM-CAL.
furnace entrance and is conveyed into the furnace
filled with reducing gas. In the furnace, a number the strip to a present annealing temperature. These
of rolls are arranged at the top and bottom of the tubes are arranged in several zones (6 zones in case
furnace chambers, through which the strip runs alter- of Fig. 1), and the furnace temperature is indepen-
nately between them. The distance between the top dently controlled for each zone enabling the furnace
and bottom rolls is more than 10 m, and a contact to realize any kind of heat cycle.
angle of the strip around the rolls is 180 deg. In the 2. Soaking Section
heating section, radiant tubes are installed to heat up The strip coming out of the heating section is then
(280) Transactions ISIJ, Vol. 25, 1985
sent to a soaking section where the strip temperature III. History of Continuous Annealing
is kept uniform to the width and length directions at
The first vertical type continuous annealing line
a fixed level.
was put in operation in 1936 at Baltimore Works of
The heat input in this section only compensates the
Crown Cork and Seal Co., Ltd. It was the continu-
radiation loss from the furnace rolls. However, the ous annealing line for black plates with a speed of 25
soaking system must be able to quickly change the fur-
m/min. (75 fpm) and employed electric resistance
nace temperature in response to a change in the heat heating.4~
cycle. The equipment shown in Fig. 1 employs a
Several years later, another line was built in Ohio
radiant tube heating system for this purpose. as a pilot plant, the purpose of which is not known
3. First Cooling Section
today though. But, the line was pulled down, since
This section meets the requirements of three differ- the product obtained was so hard and of so high
ent modes mentioned below. strength in comparison with batch annealed products
1) No cooling required for high-temper tinplates and that metallurgists did not admit the product. Metal-
electrical steel.
lurgical research on continuously annealed products
2) Rapid cooling down to an overaging temperature was under way thereafter in spite of this failure, and
for soft-temper tinplates and sheet products which
its result brought forth a continuous annealing line
need overaging treatment. built at Dominion Foundries & Steel Co., Ltd. in
3) Rapid cooling to 300 °C for dual phase steel. Canada during World War II4) which is practically
The strip is cooled down in the first cooling section comparable to a batch furnace. The abovemen-
by gas jet from nozzles arranged on both sides of the tioned pilot line reportedly employed radiant tubes
strip pass line. Cooling rate can be freely adjusted for strip heating, which was a remarkably innovative
with the maximum cooling rate of 50 °C/sec. technology in those days and provided a basis of cur-
4. Second Cooling Section
rent radiant tube design.
As shown in the heat cycles of Fig. 2, this section
The line at Dominion Foundries employed an elec-
employs two operation modes.
tric resistance heating method with a line speed of
1) Slow cooling to a predetermined temperature for 100 mf min (300 fpm). The completion of this line
high temper tinplates and electrical steel sheets.
accelerated the metallurgical study on continuous an-
2) Temperature holding for overaging treatment of nealing of tinplate products, and the study resulted in
soft temper tinplates and sheet products. such superiorities of continuous annealing process over
This section is equipped with cooling tubes for slow batch annealing as product uniformity and improve-
cooling of strip and electric heaters for temperature
ment in mechanical properties, surface quality, flat-
holding which perform their respective functions ac- ness and corrosion resistance.4,5~
cording to any operation mode selected from the
From the 1940's to the first half of the 1950's, the
above. construction of continuous annealing lines gradually
5. Third Cooling Section
propagated, along with the line speed increased up to
The strip is then sent to the third cooling section
250 to 300 m/min. During the second half of the
where the cooled atmospheric gas blows directly on
1950's through the 1960's, the line speed jumped up
the strip surfaces to cool down to 7O'-'9O °C. to 450 N 600 m/min. Technological factors allowing
this achievement include improvement of strip cooling
3. Delivery Equipment methods in terms of heat transfer, in addition to vari-
The delivery equipment consists of No. 2 looper, ous improvements in component design. The con-
shears and tension reels. The strip annealed in the tinuous annealing lines with the line speed below 300
furnace is coiled by a tension reel. As the coil on m/min, built probably before 1955 used radiant cool-
the reel grows to a certain diameter, the strip is ing method to cool the strip by arranging cooling
sheared, and a new coiling operation starts alternately water boxes on both sides of the strip pass line.
using the other tension reel while the finished coil is Therefore, the heat flux from the strip to the cooling
unloaded from the first reel. water boxes diminishes as the strip temperature be-
This operation renders the line speed at the deliv- comes low, and a large heat transferring area is re-
ery section slow down to the shearing speed. Thus, quired to cool down to the target temperature. As
No. 2 looper stores the strip from the furnace to main- equipment design, a number of strand passes had to
tain the line speed in the furnace section during shear- be provided, and the strip was dipped into water to
ing. The shears used for this section is normally of cool the strip below 80 °C. However, the develop-
flying type, but the multi-purpose line shown in Fig. ment of gas jet cooling technique has significantly
1 is provided with a high speed light gauge shear and increased the strip cooling capacity of the plant, there-
a heavy gauge shear to meet a wide range of strip by reducing strand passes by half and enabling " dry
cooling " to be put to practical use.14) This new
gauge and line speed. The line to be processed only
sheet gauges is further provided with a skinpass mill technique has made a great contribution to a high
between the looper and the shears for temper rolling, speed continuous annealing.
although not expressly shown in Fig. 1. There are The continuous annealing lines constructed by 1959
two tension reels both of which are attached with belt are mostly located in the United States, where there
wrappers. are 18 plantsl4) as opposed of only 5 plants in other
parts of the world. The first plant in Japan was
Review
Transactions ISIJ, Vol. 25, 1985 (281)
built at Hirohata Works of Nippon Steel Corporation kind of steel can be produced only through the con-
in 1959 using the radiant tube heating technique with tinuous annealing process. As typically demonstrated
the line speed of 300 m/min.5) Since then, 10 con- by this example, the continuous annealing process has
tinuous annealing lines were installed for blackplates broadened its scope of steel grades to be processed,
processing with the line speed ranging 45O-6O0 m/ and a wide variety of heat cycles have been accord-
min in the 1960's through the 1970's. ingly developed.
Figure 3 shows a chronological trend of line speed Under such circumstances, " KM-CAL " (Kawa-
increase.2'5) saki Steel Multipurpose Continuous Annealing Line)
Looking at the application to cold rolled sheet prod- started its operation at Chiba Works of Kawasaki
ucts, research on continuous annealing technology for Steel Corporation in July, 1980.
sheet products set about in the 1930's, and it was re- So far, there are eight continuous annealing lines
ported that a pilot line (a horizontal furnace, though) for sheet products in operation and another line under
was operated at Ford Motor Co. in 1935 to 1936.6) construction in Japan. Meanwhile, nine lines are
Although the study on metallurgical process of currently in operation or under construction abroad.12>
sheet product annealing is almost as old as that of The most prevailing production capacity among these
blackplate annealing, it was not until the 1970's that plants are around 400 000 t/ y, followed by the second
the first continuous annealing line was put to use for prevailing capacity of 600 000 t/ y, and then 300 000
sheet products. This is primarily due to the lack of try and 500 000 try. The newest topic is the line with
means to sufficiently precipitate solute carbon in the the capacity exceeding 1 000 000 t/ y capacity that has
steel in a short time and to grow crystal grains of a been put into operation at Hirohata Works of Nippon
large size.7~ Steel Corporation in August, 1982.
The first vertical continuous annealing line for cold This may suggest the disappearance of batch an-
rolled sheets started the operation in a commercial nealing furnaces from cold rolling shops.
scale at Kimitsu Works of Nippon Steel Corporation Figure 4 illustrates chronological trend of produc-
in October, 1972. This line is called as " C.A.P.L." tion capacity achieved by continuous annealing lines
(Continuous Annealing & Processing Line) and has for sheet products including those under construction.
marked a new age of continuous annealing lines Figure 5 indicates the relationship between the
(CAL). production capacity and the line speed. The spead
However, the kinds of steel which can be processed of the lines with their capacities below 600 000 try
by a continuous annealing furnace for sheet products fall in the range of 200-250 m/ min regardless of these
are characterized in chemical compositions, and there- production capacities. This is attributable to the dif-
fore annealing processes for these steel grades differs ference in strandard strip size, which is set based on
depending who developed them. Nippon Kokan the product mix in planning the equipment specifica-
K.K, developed its own " NKK-CAL " process in tion. However, the equipment above 600 000 t/ y in
1971 and started commercial scale operation of a con-
production capacity must inevitably have a high level
tinuous annealing line for sheet products in 1976 at of line speed to achieve the expected production. The
Fukuyama Works.9~ 1 000 000 try line at Hirohata Works of Nippon Steel
Then, the oil crisis incurred a new trend of energy Corporation has the line speed of 450 zn/min, which
saving, and the automative industry, for instance, is as fast as those for tinplate annealing lines at an
pointed at light-weight vehicles resulting in develop- early stage. In conclusion, the age of tinplate con-
ment of various types of high tensile strength steel.
Among them, it was reported that the heat treatment
of dual phase steel required a cooling rate of 10-.
100 °C/sec10~ from the soaking temperature. This
Fig. 3. L me spee d variation per period on tin plate CAL. Fig. 4. Instalation of sheet CAL with production capacity.
(282) Transactions ISIJ, Vol. 25, 1985
Fig. 6. Model of strip on roll.
Fig. 5. Relation between production capacity and line
speed on sheet CAL.
tion of the strip. Such temperature differences easily
induce strip deformation due to different thermal ex-
tinuous annealing lasted until 1970 since an experi- pansion, which affects the contact condition of the
mental equipment had been built in 1936. strip with rolls and impairs the stability of strip run-
During this period, the development of annealing ning. Unsatisfactory tracking of the strip will often,
technology approached a marginal line speed of 600 as experienced by operators of continuous annealing
m/min and such technology established in the latter furnaces, lead to strip breakage.
half of the 1970's. Then, the 1970's was the age of On the other hand, even if the thermal condition
continuous annealing for sheet products, while the line is perfectly set up, inadequate design of the conveying
speed stayed in the same level as that of the 1970's. In system may cause such troubles on vibration and un-
the 1980's, however, continuous annealing lines with even distribution of strip tension which give an detri-
their capacities more than 600 000 try and incorpo- mental effect to the strip shape and tracking. They
rated multipurpose use into such continuous annealing will eventually spoil the product quality. In other
lines, that is, we are experiencing more diversified de- words, good tracking means the strip pass with the
velopment in high speed continuous annealing. As strip shape maintained properly, thereby securing
to the future development of annealing technology, smooth plant operation and high value added prod-
changes of requirements to continuous annealing proc- ucts.
ess, or changes in metallurgical process will play an While there are numerous issues involved in strip
leading role. tracking, this paper discusses a theoretical analysis of
The technology of the high speed operation will strip conveyance among others, particularly in view
also be much influenced by the direction of this metal- of a) the relationship between the strip and rolls, b)
lurgical development. vibration of components in the roll driving system,
and c) critical speed specified from the point of strip
Iv. Technology for High Speed Continuous An- vibration. The thermal effects on tracking will also
nealing be discussed later in connection with the heat transfer.
1. Outline 2. Relationships between Strip and Rolls
The technologies for continuous annealing process As explained in Chapter II " Structure of Continuous
are comprised of, Annealing Equipment ", the strip is conveyed by the
i) Strip conveying, rolls installed in the top and bottom of the annealing
ii) Heating, and furnace. Figure 6 illustrates schematically a model
iii) Cooling. of the strip running around a roll.
The annealing process progresses through close in- Representing the tension on the tight side by P1
teraction of temperature and time, and does not func- and that on the loose side by P2, then, from the condi-
tion unless constituent technologies controlling the tion of equilibrium in the radial direction in Fig. 6,
process are individually established and properly in-
terrelated among them. This inter-relationship regu-
Q.ds = P sin d8 2 +(P+dP) sin d~ 2 - Fd.c
lates common design criteria equally applicable to
constituent technologies in equipment design. Q . ds PdH-Fds
Holding a good strip tracking is particularly im-
since the centrifugal force Fds = m v2dO,
portant criterior in this respect. In the continuous g
annealing process, the strip undergoes such significant
temperature changes in a short time as heating, soak- Qds = P - m o2 de ........................ (1) g
ing and cooling while continuously conveyed, and
temperature differences may occur to the width direc- where, m=rbh (r : specific gravity of strip, b : strip
Transactions ISI1, Vol. 25, 1985 (283)
width, and h : strip thickness)
v : strip speed
g: gravitational acceleration
e : contact angle of strip.
From the condition of equilibrium in the circumferen-
tial direction.
P+dP=P+pQds
dP = pQ Qds .....................(2)
where, ~e : coefficient of friction between the strip and
roll.
From Eqs. (1) and (2)
dP=~ P-mv2 dD
g
P~ dp
= ~ °dB
P2 P _ m v2 n
g
Fig. 7.
1- m 2 Roll shape (example).
g =e°
... ; .....................(3)
m
P2 v2 direction of the arrow mark, point a comes to point
g
a'. Due to friction between the roller and strip, the
The maximum tensile stress, Amax of the strip between strip at point a moves also to point a' together with
rolls is given by the following equation, the roller, and the strip shifts to a position indicated
6max = Qp+Ug ..(4) by the dotted lines. As the roller keeps turning, the
strip continually shifts to the direction of the larger
where, ai, = P1: bh tensile stress due to tensile diameter and gets off as it goes over point b. The
roll profiles shown in Fig. 7, taking advantage of this
force P1 on the tight side
phenomenon, give to the strip the force to move to-
ward the center of the rolls, maintaining good track-
QF = bh F = r v2: tensile stress due to centrifugal
g ing.
force F. If excessive crown is given to the roll, however, a
For example, supposing the strip speed v=600 mf min, large stress is caused in the center part of the strip,
then ar=0.0784 N/mm2. This value only accounts and the high tensile force therefrom is liable for center
for O.8'-' 1.6 % of o which is 4.9'.'9.8 N/mm2. It buckle of the strip. Further, excessive self-centering
must be noted, however, that as the line speed in- force of the strip results in wrinkles due to buckling.
creases, OF multiplies in proportion to the square of Accordingly, optimal determination of roll profile and
the speed resulting in the increase of stress in the strip. dimensions as well as suitable control of strip tension
Also, as shown in Eq. (1), the bearing stress between matching various operation conditions are vitally im-
the strip and roll decreases. They give an negative
portant.
influence to strip running. Then, meandering of the strip is to be examined.
The rolls take a shape of mid-thick, of narrow body Assuming that the strip skews at an angle of cp from
or of crown as illustrated in Fig. 7, in order to give a the roll center line as shown in Fig. 10, and that the
self-centering movement to the strip. strip moves sideways at some instant by y, then the
Assuming that a strip turns around a conical roller strip displaces from the normal running position as
as shown in Fig. 9, the strip which is flexible winds indicated by displacement of contact points with the
tightly around the roller surface as indicated by solid roll from A and B to A' and B'. As a result, creep
lines. If the conical roller is turned 90 deg in the forces* Fx, Fy, and Mz around axis Z act on the
* Representing the strip running speed at any position by V
a, and the speed of strip
lateral distraction by Vs, slipping ratios of ra and ES are given by the following
equations.
Va-V VS
ea - V , r5= V ..............................( )
where, V: circumferential speed of roll
As shown in Fig. 8, the creep force P is almost proportional to the slipping ratio
when r is sufficiently small, but loses its proportionality as the ratio increases,
becoming asymptotic to the line of friction force. This is called the creep
phenomenon.
The strip running along the pathway around a cone tends to meander with a
geometrically constant amplitude. However, depending on the friction charac-
teristics between the strip and roll in the center part of the roll and other factors,
the strip does not necessarily meander. This phenomenon is yet to be clarified. Fig. 8. Creep.
(284) Transactions ISIJ, Vol. 25, 1985
Fig. 11. Example of roll drive system.
Fig. 9. Conical roll.
J,1 o: Moment of inertia of motor
(kg. m2)
JrO Moment of inertia of roll
K1 (kg•m2)
Kl : Tortional spring constant
J,o
1 KZ Jro K (Motor to reducer) (N • m/rad)
C1 Kc 2: Tortional spring constant
(Reducer to roll) (N. m/rad)
Kc : Spring constant of contact
CZ
point of reducer (N/m)
(a) C,, C2: Damping coefficient (N • m/
rad)
1/n: Reduction ratio r '
(a) Multi degree of freedom model
Fig. 10. Mode 1 of strip mistrack mg.
J„ 1: Moment of inertia concentrating
contact surface between the strip and roll, causing a to motor (kg.m2)=Jmo+4J2
constant vibration with a wave length virtually inde- K Jr : Moment of inertia concentrating J
r to roll (kg' m2) = (J
pendent of the speed. Jm ro/n2) +4J2
K: Tortional spring constant
y = Yo sin (cat+ jS), ~b = cj'o cos (cat+R) K Kl K2/n2
C = --
...........................(5) K1 +K2/n2
(b) C: Damping coefficient (N. m. s/rad)
1
where, c1'o = '~ . r (b) Single degree of freedom model.
1 rA ar
--- (Kr » K1, K2)
a Fig. 12. Model of mechanical system.
22r ; .....................(6)
W •V(= S v
ar ~ i 11 for simplicity. Figure 12(a) indicates a model de-
scribing torsional vibration characteristics in the driv-
ar S
1= 2~r ing system directly connected with the mechanical
system of Fig. 11. This model can be deemed as a
yo (or coo) and jS are constants determined by the ini- linear system since KG>>K1, K2 and ignoring the back-
tial conditions, and particularly S1 is called the geo- lash in gears and couplings. From this model, a sin-
metrical meandering wavelength.16 Roll arrange-
gle degree of freedom model is obtained as shown in
ment must be determined so that the distance between Fig. 12(b) by suitably dividing moment of inertia into
the rolls does not coincide with this wavelength S1. both sides.
This will suffice, in principle, for examination of
3. Roll Driving System17
responsiveness of the driving system, though not ade-
As already explained, strip speed control becomes
quate for analysing such high frequency range as re-
an important element to achieve stable running of the sponse in excitation due to intermeshing of gears.
strip as the line speed increases. The controllability From Fig. 12(b), the natural torsional vibration of
of the strip speed is virtually determined by design of the lowest degree w, (rad/s) in the driving system is
the roll driving system.
given by Eq. (7) hereunder,
Various technical problems stem from the fact that
the design of this system must treat a number of rolls (01= /K/J ...........................(7)
as an integrated system. Most of these problems lie
in the interface between manufacturing of rolls and J - JJm • Jr
m + Jr
electrical equipment including roll driving motors.
The discussion hereunder concentrates on the bridle where, K: torsional spring constant (N•m/rad).
roll driving system which plays a main role in the In the conventional speed control system, the me-
strip speed control. Although this system should be chanical system has mostly been designed assuming
treated as a system of multi degree of freedom con- the moment of inertia as Jo = Jm + Jr (kg • m2 ). The
sisting of several rolls in reality, the discussion here is block diagram of Fig. 13 shows the control system
based on a single roll driving system as shown in Fig. based on such assumption. Using the values in pa-
Transactions ISIJ, Vol. 25, 1985 (285)
rentheses in Fig. 13, the open loop transfer function Putting K=1.72 x 104 (N • m/rad), J~ =1.23 (kg. m2),
Ge0 (s) is expressed as Eq. (8) when Jo=10.1 (kg•m2). J=8.88 (kg. m2 ), and using the same values for other
terms as those for Eq. (8), the bode diagram Ge (jw)
Gep (s) = KsKc K,,F5 10.4 with C as a parameter is expressed by dotted lines in
J0S(1+TcS) SI 1-+-6 Fig. 14. Comparing them with the solid lines in Fig.
14, the stability increases significantly in the vicinity
(8) of anti-resonance point w2, but it is noted that no gain
margin exists near the resonance point of the mechani-
Solid lines in Fig. 14 describes the Bode diagram
cal system (01, and that there is a large rate of change
based on Eq. (8) assuming S= jw. The diagram im-
in the gain lag with the tendency of diminished phase
plies that the system is stable with the phase margin margin. In such system as this example, deteriora-
of 80 deg, and that sufficient gain margin can be ex- tion of response like Fig. 16(a) should be anticipated.
pected even if some dead time should exist in the sys- Further the stability could be immediately lost by the
tem.
existence of dead time at detecting elements. Figure
However, in actual mechanical system, the block
16(b) shows examples of deteriorated stability due to
diagram takes the form of Fig. 15 due to the vibration
excessive sensitivity of the system.
system formed as Fig. 12. Assuming the damping In the discussion so far, the mechanical system has
ratio
been assumed as a linear vibration system.
= c/(2 V'JK), and However, such components as reducers and cou-
WK/J, W2 =K/J., plings which inevitably include backlash have often to
be considered in the mechanical system. This sort of
the open loop transfer function for the system of Fig. system is non-linear, but can be treated as equivalent
15 becomes as follows.
Fig. 13. Example of block diagram for roll speed control.
(Mechanical system: concentrated inertia) Fig . 14. Bode dia gram of roll speed control.
Fig. 15. Example of block diagram for roll speed control.
(Mechanical system: Tortional vibration system
with single degree of freedom)
(286) Transactions ISIJ, Vol. 25, 1985
Fig. 17. Example of reduced tortional natural vibration
frequency of mechanical system. (Effect of back-
lash)
problem. Then, recent development of such elements
as inverters promotes the use of AC motors with
VVVF control system instead of DC motors. With
this system, there occurs a voltage ripple of about
30 % in the case of AC voltage type inverter (6 pulse
Fig. 16. Example of deteriorated response of roll drive sys- type) for example. This is substantially larger than
tem. 4.2 % of a conventional 3-phase all wave rectification,
and must be reflected on the mechanical system de-
sign as a torsional vibration factor with frequency six
linear system. Figure 17 indicates the relationship
times as high as that of power source for motors (ac-
between equivalent backlash and primary natural tor-
tual solution will be avoidance of resonance or in-
sional vibration using the aforementioned example.
creased damping).
As the backlash becomes large, the natural vibration
On the other hand, however, VVVF control sys-
frequency decreases substantially. In such situation,
tem offers advantages in maintenance work and linear
as described in Fig. 14, considerable increase of re-
control of motor torque and revolution. Also speed
sponse gain and dead time in the system due to back-
equalization can be secured through individual con-
lash will cause unstable vibration as shown in Fig.
trol of motors. It is readily expected that this system
16(c).
become a major specification for driving helper rolls,
Through the above discussion, using a single roll
etc.
driving system, it has been pointed out that assessing
Besides, there are some other issues relating to the
the characteristics of the mechanical system is very
roll driving system such as balancing and bending
important in the design of speed control system. This
vibration of fluid sealed rotors, which are left un-
is equally applicable to multi-roll and non-linear sys-
touched in this paper.
tem with which actual equipment design encounters,
although detailed study of the characteristics needs
4. Critical Speed18,19~
the aid of computers.
The distance between top and bottom rolls tends to
In practice, electric motors have recently been
extend as the line speed increases with the total length
much improved in response characteristics, while the
of continuous annealing lines delimited. At present
mechanical system tends to have lower natural vibra-
the roll span generally ranges from 15 to 20 m. In
tion frequency due to large sized machineries and use
this range, the critical speed of a line could be basical-
of high quality materials.
ly determined by the bending vibration of the strip
Consequently, it is essential for the design of me-
as explained hereunder.
chanical systems to determine dimensions in view of
Figure 18 shows a part of the strip running through
the aforementioned stability and improve the backlash
control and damping capacity. rolls at a line speed of v (m/sec). A symbol l (m) in
the figure is the span of rolls. A kinetic equation of
However, this does not suggest the roll system de-
the strip is expressed as follows along the ordinate X
sign on the basis of the driving system. To the con-
which moves at the same speed and to the same direc-
trary, constituent equipment and control systems must
tion as the strip.
be designed to organize well-balanced, integrated sys-
A symbol w (m) represents micro bending defor-
tem with the mechanical side as mainstay.
In the meantime, a noteworthy development has mation of the strip.
been made recently in the driving system for helper
a2 a2w a4w
rolls and the like. Conventionally, DC motors are pA ate _P axe +EI ax4- = 0 ............(10)
most common means for variable speed roll driving,
which needs periodical replacement of parts (every where, p : strip density (kg/m3)
2 N 3 years) to remedy wear and spark damage of A : sectional area (m2)
brushes and commutors. Electrical noise is another t : time (sec)
Transactions ISIJ, Vol, 25, 1985 (287)
E : Young's modulus (N/m2 )
I: geometric moment of inertia (m4).
Therefore, pA (kg/m) represents the mass of unit
length of the strip, and El (Nm2) is the bending
rigidity of the strip.
P(N) is the tensile force acting on the strip and
generally expressed as the sum of following forces :
P = Po (Preset tensile force)+ (Strip weight)
+ (Centrifugal force of strip) ............... (11)
Using x=e-v.t, Eq. (10) can be transformed along
the static coordinates (, t) as follows :
Fig. 18. Strip pass.
a -V a 2 W_ 2 a 2 7~ +g2--- 4 _ 0
at ae ae2 ae4
equivalent compressive force becomes equal to the
........................(12)
limit load causing the strip buckling, 2rEI/l2. Name-
p2 = P/pA, q2= EI/pA ly,
Equation (12) is one dimensional wave equation vz = Pz+ 12 ~rEI A . ........................(16)
in the longitudinal direction of a strip running at v p
(m/sec), which is solved by seeking a synthesis of the Since the value of the second terms on the right side
forward wave p+ v, and the backward wave p-v.
is miniscule in case of the continuous annealing line,
Accordingly, defining the n-th degree natural vibra-
Eq. (16) concludes the solution of v= p, which corn-
tion cycle of the strip z (s) as the time that a wave
forms to the first definition of the critical speed.
once reciprocates between the distance l/n, the cycle
(Since the actual deformation of the strip is larger
z (s) can be computed by the following equation.
than the micro-displacement assumed here, the tensile
force increases accordingly resulting in the higher
L( -1--+ 1 2p1---
n p-v p+v n(p2-v2) critical speed.)
The critical speed Vti (n=1.......... corresponding
........................(13)
to each vibration mode) as defined above can be
(n=1,2,......) decided by solving Eq. (12) under the following
boundary conditions (providing a constant curvature
From Eq. (13), the n-th degree natural vibration fre-
at supporting point)
quency of the strip fn (Hz) is expressed as follows :
e=0; w=0, a2w/ae2 = coast.
.ffl = n(p2-v2) 2pl .....................(14) e=1; w=0, a2w/ae = const.
(n-1,2,......) The solution takes the following form.
This is a character important in analyzing resonance Vit = /p2+Rng2/c2 .....................(17)
caused by a forced external force. That is, as the
Rn = 1Cn, (n=1,2, ...... )
line speed v increases, f,~ gradually decreases and be-
comes zero at p = v. This state means that the wave w,(e) = A sin (p,~e/l)
suspends in the space and the strip is excited by static
Figure 19 shows, as an example, the safest critical
external forces such as differential pressure between
speed and corresponding natural frequencies f1 and
face and back sides, or own weight of the strip in
f2 of the strip which are determined under the condi-
horizontal position. Therefore, the line speed at this
tion that P = P0, and the bending rigidity of the strip
level should be considered as a kind of the critical
is ignored. The figure also indicates samples of the
speed.
present design speed which fall well within the safety
On the other hand, assuming a state in which the
range in terms of the strip vibration.
spatial movement of wave is small in the vicinity of
Then, as to factors to cause excitation for the strip
the critical speed, the terms of differential coefficient
vibration, besides the above mentioned spatially stand-
of time can be omitted from Eq. (12), resulting in
ing waves, torque pulsation in the driving system
the following equation.
(motors, couplings, bearings, gears, etc. ), forced vibra-
tion due to whirling of unbalanced rolls, roll eccen-
v2 , a2~ 2 a4~ ---- - 0 .(15)
tricity or unevenness of strip material, and vibration
due to coefficient excitation (the excitation caused by
Since v2 . pA-P can be regarded as compression force cyclical fluctuation of F in Eq. (12) can be the reasons
in the axial direction in the range of v> p, Eq. (15) therefore. The forced vibration is explained as reso-
is interpreted as representing the buckling of a long nance due to conformity with J in Eq. (14).
pillar. Consequently, the critical speed can also be The coefficient excitation includes the possibility of
defined in such way as the speed at which the above self excited vibration under certain conditions, par-