This invention relates to the processing of continuous webs such as paper, film, composites, or the like, in dynamic continuous processing operations. More particularly, the invention relates to controlling tension in such continuous webs during the processing operation, and to temporarily accumulating limited lengths of such continuous webs.
In the paper and plastic film industries, a dancer roll is widely used as a buffer between first and second sets of driving rolls in a line of processing machines. The first and second sets of driving rolls define respective first and second nips, which drive a continuous web. The dancer roll, which is positioned between the two sets of driving rolls, is also used in detecting the difference in speed between the first and second sets of driving rolls.
Typically, the basic purpose of a dancer roll is to maintain constant the tension on the continuous web which traverses the respective section of the processing line between the first and second sets of driving rolls, including traversing the dancer roll.
As the web traverses the section of the processing line, passing over the dancer roll, the dancer roll moves up and down in a track, serving two functions related to stabilizing the tension in the web. First, the dancer roll provides a tensioning force to the web. Second, the dancer roll temporarily absorbs the difference in drive speeds between the first and second sets of driving rolls, until such time as the drive speeds can be appropriately coordinated. However, the length of web which the dancer roll can absorb is limited to that length of web which traverses the upward path to the dancer roll and the downward path from the dancer roll.
A web extending between two drive rolls constitutes a web span. The first driving roll moves web mass into the span, and the second driving roll moves web mass out of the span. The quantity of web mass entering a span, per unit time, equals the web""s cross-sectional area before it entered the span, times its velocity at the first driving roll. The quantity of web mass exiting a span, per unit time, equals the web""s cross-sectional area in the span, times its velocity at the second driving roll. Mass conservation requires that over time, the web mass exiting the span must equal the mass entering the span. Web strain, which is proportional to tension, alters a web""s cross-sectional area.
Typically, the dancer roll is suspended on a support system, wherein a generally static force supplied by the support system supports the dancer roll against an opposing force applied by the tension in the web and the weight of the dancer roll. The web tensioning force, created by the dancer system, causes a particular level of strain which produces a particular cross-sectional area in the web. Therefore, the web mass flowing out of the span is established by the second driving roll""s velocity and the web tensioning force because the web tensioning force establishes web strain which in turn establishes the web""s cross-sectional area. If the mass of web exiting the span is different from the mass of web entering the span, the dancer roll moves to compensate for the mass flow imbalance.
A dancer roll generally operates in the center of its range of travel. A position detector connected to the dancer roll recognizes any changes in dancer roll position, which signals a control system to either speed up or slow down the first and/or second pairs of driving rolls to bring the dancer back to the center of its travel range and reestablish the mass flow balance.
When the dancer roll is stationary, the dancer support system force, the weight of the dancer roll, and the web tension forces are in static equilibrium, and the web tension forces are at their steady state values. Whenever the dancer moves, the web tension forces change from their steady state values. This change in web tension forces supplies the effort that overcomes friction, viscous drag, and inertia, and causes the dancer motion. When the dancer moves very slowly, viscous drag and inertia forces are low and therefore the change in web tension is slight. However, during abrupt changes in mass flow, as during a machine speed ramp-up or ramp-down, the viscous drag, and inertia forces may be several times the web""s steady state tension values.
The dancer roll""s advantages are that it provides a web storage buffer which allows time to coordinate the speed of machine drives, and the dancer provides a relatively constant web tension force during steady state operation, or periods of gradual change. A limitation of dancer rolls, as conventionally used, is that under more dynamic circumstances, the dancer""s ability to maintain constant web tension depends upon the dancer system""s mass, drag, and friction.
In processing apparatus for processing a such continuous web, it is common practice to employ both a dancer roll, for purposes of tension control, and a festoon, biased to accumulate and temporarily hold a limited length of the continuous web, but a length substantially greater than the capacity of a dancer roll. The accumulated limited length of web is then played out, or an additional length accumulated, when processing of the continuous web is temporarily interrupted. Such temporary interruption can be, for example and without limitation, change and splicing of a feed/supply roll, or change and splicing of a wind-up roll. Other temporary interruptions can also be accommodated by using the festoon as an accumulator while maintaining operation of various steps in the web manufacture without having to shut the line down.
Such festoon is, by design, a low mass, low inertia device, and is typically biased so as to hold, at steady state operation, an accumulation of web material equivalent to approximately half its capacity for web accumulation. Thus, starting from steady state, the festoon can either accumulate more web if a downstream function is temporarily interrupted or can play out the accumulated length of web if an upstream function is temporarily interrupted. Critical to a festoon is its low mass, low inertia, design.
It is known to provide an active drive to the dancer roll, though such active drive is not known for a festoon, in order to improve performance over that of a static system, wherein the web is held under tension, but is not moving along the length of the web, whereby the dynamic disturbances, and the natural resonance frequencies of the dancer roll and the web are not accounted for, and whereby the resulting oscillations of the dancer roll can become unstable. Kuribayashi et al. xe2x80x9cAn Active Dancer Roller System for Tension Control of Wire and Sheet.xe2x80x9d University of Osaka Prefecture, Osaka, Japan, 1984.
More information about tension disturbances and response times is set forth in U.S. Pat. No. 5,659,229 issued Aug. 19, 1997, which is hereby incorporated by reference in its entirety. U.S. Pat. No. 5,659,229, however, controls the velocity of the dancer roll and does not directly control the acceleration of the dancer roll.
Thus, it is not known to provide an active dancer roll or an active festoon in a dynamic system wherein dynamic variations in operating parameters are used to calculate variable active drive force components for applying active and variable acceleration to the dancer roll or festoon, and wherein appropriate gain constants are used to affect response time without allowing the system to become unstable. Namely, it is not known to drive a dancer roll or festoon so as to nullify physical affects of actual mass and inertia of the dancer roll or festoon. Indeed, no variable drive parameter is known for a festoon.
This invention provides novel festoon apparatus and methods. Festoons of the invention control tension and tension disturbances in a continuous web during processing of the web. The festoons of the invention also hold accumulations of limited lengths of the web sufficient to enable continuity of the web processing operation while absorbing the affects of short-term interruptions of web processing, either upstream or downstream of the festoon. Festoons of the invention are controlled so as to nullify the affects of mass and inertia on the ability of the festoon to respond to speed and tension changes in the web traversing the given section of the processing line, or to respond to differences in web speed at the in-feed and take-away nips, or to respond to large scale changes in web speed at the in-feed or take-away nips.
The invention comprehends processing apparatus defining a processing line, for advancing a continuous web of material through a processing step along a given section of the processing line. The processing apparatus comprises first and second rolls defining a first nip; third and fourth rolls defining a second nip, the first and second nips collectively defining the given section of the web; a festoon, including upper and lower festoon rolls, operating on the web in the given section of the processing line, thereby to control tension in the web and to accumulate a limited length of the web sufficient to sustain operation of the process on the length of web during routine temporary stoppages of web feed to the given section of the processing line or taking the web away from the given section of the processing line; an actuator applying net translational force to the upper festoon rolls; and a controller driving the festoon, and computing and controlling net translational acceleration of the upper festoon rolls such that the festoon is effective to control tension, at a desired level of constancy, and to accumulate a limited length of the web, in the respective section of the processing line.
In some embodiments the actuator applies a first static force component to the festoon upper rolls, having a first value and direction, balances the festoon upper rolls against static forces and the average dynamic tension in the respective section of the web, the controller outputting a second variable force component, through the actuator, effective to control the net actuating force imparted to the upper festoon rolls by the actuator, and effective to periodically adjust the value and direction of the second variable force component, each such value and direction of the second variable force component replacing the previous such value and direction of the second variable force component, and acting in combination with the first static force component to impart the target net translational acceleration to the upper festoon rolls, the second variable force component having a second value and direction, modifying the first static force component, such that the net translational acceleration of the upper festoon rolls is controlled by the net actuating force enabling the festoon to control the web tension, and further comprising apparatus for computing acceleration (Ap) of the upper festoon rolls. The controller preferably comprises a computer controller providing control commands to the actuator based on the computed acceleration of the upper festoon rolls.
Preferred embodiments include a sensor for sensing tension in the web after the festoon, the controller being adapted to use the sensed tension in computing the value and direction of the second variable force component, and for imparting the computed value and direction through the actuator to the upper festoon rolls.
In some embodiments, the sensor is effective to sense tension at least 1 time per second, preferably at least 500 times per second, more preferably at least 1000 times per second, and the controller is effective to recompute the value and direction of the second variable force component, thereby to adjust the value and direction of the computed second variable force component a like number of times.
In preferred embodiments, the controller controls the actuating force imparted to the upper festoon rolls, and thus controls acceleration of the upper festoon rolls, including compensating for any inertia imbalance of the festoon not compensated for by the first static force component.
In some embodiments, the apparatus includes an observer for computing translational acceleration (Ap) of the upper festoon rolls, the observer comprising one of (i) a subroutine in the computer program or (ii) an electrical circuit, which computes an estimated translational acceleration and an estimated translational velocity of the upper festoon rolls.
The processing apparatus of the invention preferably includes first apparatus for measuring a first velocity of the web after the festoon; second apparatus for measuring a second velocity of the web at the festoon; third apparatus for measuring translational velocity of the upper festoon rolls; and fourth apparatus for sensing the position of the upper festoon rolls.
The invention can include fifth apparatus for measuring web tension before the festoon; and sixth apparatus for measuring web tension after the festoon.
One equation for calculating the servo force is
F*servo=F*d static+F*frictionSign(Vp)+ba(V*pxe2x88x92Vp)+ka(F*cxe2x88x92Fc)+Ma(A*pxe2x88x92Ap)
wherein the translational velocity set-point V*p of the upper festoon rolls reflects the equation:
Vp=[EA0/(EA0xe2x88x92Fc)][V2(1xe2x88x92Fb/EA0)-V3(1xe2x88x92Fc/EA0)],
to control the actuator based on the force so calculated, wherein:
F*d Static=static force component on the upper festoon rolls and is equal to Mg+2F*c.
Fc=tension in the web after the last movable festoon roller,
F*c=tension in the web, target set point, per process design parameters,
Fb=tension in the web ahead of the last movable festoon roller,
F*friction=Friction in either direction resisting movement of the upper festoon rolls,
F*servo=Force to be applied by the actuator,
ba=control gain constant regarding festoon translational velocity, in Newton seconds/meter,
ka=control gain constant regarding web tension,
Mg=mass of the upper festoon rolls times gravity,
MA=active mass,
Me=active mass and physical mass,
Vp=instantaneous translational velocity of the upper festoon rolls immediately prior to application of the second variable force component,
Sign(Vp)=positive or negative value depending on the direction of movement of the upper festoon rolls,
V2=velocity of the web at the last movable festoon roller,
V3=velocity of the web after the festoon,
V*p=reference translational velocity of the upper festoon rolls, set point,
r=radius of a respective pulley on the actuator,
E=Modulus of elasticity of the web,
Ao=cross-sectional area of the unstrained web,
A*p=target translational acceleration of the upper festoon rolls, set point, and
Ap=translational acceleration of the upper festoon rolls.
In some embodiments the target acceleration A*p is computed using the equation:
A*p=[V*pxe2x88x92Vp]/xcex94T
where xcex94T=scan time for the computer controller.
In preferred embodiments, the computer controller provides control commands to the actuator based on the sensed position of the upper festoon rolls, and the measured web tensions, acceleration and velocities, and thereby controls the actuating force imparted to the upper festoon rolls by the actuator thus either to maintain a substantially constant web tension or to provide a predetermined pattern of variations in the web tension.
In some embodiments, the apparatus includes first apparatus for measuring translational velocity of the upper festoon rolls; second apparatus for measuring web tension force after the festoon; and third apparatus for sensing the current of the actuator, with the controller optionally comprising a computer controller computing a derivative of web tension force from the web tension force over the past sensing intervals, and including an observer computing the translational velocity of the upper festoon rolls, and the computer controller computing a derivative of the web tension force.
The controller can comprise a computer controller, and including a fuzzy logic subroutine stored in the computer controller for computing a derivative of web tension force from the web tension force and the translational velocity of the upper festoon rolls, the fuzzy logic subroutine inputting web tension force error, the derivative of web tension force error, and acceleration error, the fuzzy logic subroutine proceeding through the step of fuzzy inferencing of the above errors, and de-fuzzifying of inferences to generate a command output signal, the fuzzy logic subroutine being executed during each scan of the sensing apparatus.
The processing apparatus can further include first apparatus for measuring translational velocity of the upper festoon rolls; and second apparatus for sensing the current of the actuator.
In some embodiments, the controller computes the estimated translational acceleration of the upper festoon rolls from the equation:
Ape=[k1(Vpxe2x88x92Vpe)+kteIxe2x88x92F*d staticxe2x88x92F*frictionSign(Vp)]/M2e
where
Ape=estimated translational acceleration of the upper festoon rolls,
F*d static=static force component on the upper festoon rolls and is equal to Mg+2F*c.
F*friction=Friction in either direction resisting movement of the upper festoon rolls,
Sign(Vp)=positive or negative value depending on the direction of movement of the upper festoon rolls,
k1=Observer gain,
Vp=instantaneous translational velocity of the upper festoon rolls,
Vpe=estimated translational velocity,
kte=Servo motor (actuator) torque constant estimate,
I=actuator current, and
M2e=Estimated physical mass of the upper festoon rolls,
with the process optionally including a zero order hold for storing force values for application to the upper festoon rolls, and optionally actively compensating for coulomb and viscous friction, and acceleration, to actively cancel the effects of mass.
In some embodiments the invention further includes first apparatus for measuring translational position of the upper festoon rolls; second apparatus for measuring web tension force after the festoon; and third apparatus for sensing the motor current of the actuator, optionally including an observer for computing estimated translational velocity and estimated translational acceleration of the upper festoon rolls from the change in position of the upper festoon rolls.
In some embodiments, the invention further includes first apparatus for measuring translational position of the upper festoon rolls; and second apparatus for sensing the motor current of the actuator; and an observer for computing translational acceleration of the upper festoon rolls.
In some embodiments, the invention includes first apparatus for measuring web tension Fc after the festoon; and second apparatus for sensing the motor current of the actuator, optionally including an observer utilizing the motor current and force on the web, in combination with an estimate of system mass M2e, to compute an estimate of translational acceleration Ape of the upper festoon rolls, the observer optionally integrating the translational acceleration to compute an estimate of translational velocity Vpe and integrating the estimated translational velocity to compute an estimated web tension force Fce, and changing values until the estimated web tension force equals the actual web tension force.
In some embodiments, the controller provides the control commands to the actuator thereby controlling the actuating force imparted to the upper festoon rolls by the actuator, and thus controlling acceleration of the upper festoon rolls, such that the actuator maintains inertial compensation for the festoon system.
In some embodiments, the first nip comprises a wind-up roll downstream from the festoon and the second nip comprises driving rolls upstream from the festoon, the controller sending control signals to the wind-up roll and the driving rolls.
In some embodiments, the invention includes first velocity apparatus for measuring a first velocity of the web after the festoon, and second velocity apparatus for measuring a second velocity of the web at the festoon, the controller comprising a computer controller computing a velocity command V*p using the first and second sensed velocities and web tension before and after the festoon.
In some embodiments, the controller comprises a computer controller intentionally periodically varying the variable force component to unbalance the system, and thus the tension on the web by periodically inputting command forces through the actuator causing sudden temporary alternating upward and downward movements of the upper festoon rolls such that the upper festoon rolls intermittently impose alternating higher and lower levels of tension on the web, the periodic input of force optionally causing the alternating movements of the upper festoon rolls to be repeated more than 200 times per minute.
The invention also comprehends, in a processing operation wherein a continuous web of material is advanced through a processing step defined by first and second spaced nips, each nip being defined by a pair of nip rolls, a method of controlling web tension, and of accumulating a limited length of the web, in the respective section of web. The method comprises providing a festoon, having upper and lower festoon rolls, operative on the respective section of web; applying a first generally static force component to the upper festoon rolls, the first generally static force component having a first value and direction; applying a second variable force component to the upper festoon rolls, the second variable force component having a second value and direction, modifying the first generally static force component, and thereby modifying (i) the effect of the first generally static force component on the upper festoon rolls and (ii) corresponding translational acceleration of the upper festoon rolls; and adjusting the value and direction of the second variable force component repeatedly, each such adjusted value and direction of the second variable force component (i) replacing the previous such value and direction of the second variable force component and (ii) acting in combination with the first static force component to provide a target net translational acceleration to the upper festoon rolls.
The method can include adjusting the value and direction of the second variable force component at least 500 times per second.
The method can include sensing tension in the web after the festoon, and using the sensed tension to compute the value and direction of the second variable force component.
The method can include sensing tension in the respective section of the web at least 1 time per second, recomputing the value and direction of the second variable force component and thereby adjusting the value and direction of the computed second variable force component at least 1 time per second, and applying the recomputed value and direction to the festoon at least 1 time per second.
The invention can include adjusting the force components and target net translational acceleration so as to maintain an average dynamic tension in the web throughout the processing operation while controlling translational acceleration such that system effective mass equals the polar inertia of the upper festoon rolls collectively, divided by outer radius of the rolls, squared.
The method can include periodically and intentionally varying the variable force component to unbalance the system, and thus the tension on the web by periodically inputting command forces through the actuator causing sudden temporary alternating upward and downward movements of the upper festoon rolls such that the upper festoon rolls intermittently impose alternating higher and lower levels of tension on the web, optionally the periodic input of force causing the upward movement of the upper festoon rolls to be repeated more than 200 times per minute.
In some embodiments, the method includes the first and second force components being applied simultaneously to the upper festoon rolls as a single force, by an actuator, and wherein the step of applying a force to the upper festoon rolls include measuring a first velocity of the web after the festoon; measuring a second velocity of the web at the festoon; measuring translational velocity of the upper festoon rolls; sensing the position of the upper festoon rolls; measuring web tension before the festoon; and measuring web tension after the festoon, and applying the force to the upper festoon rolls computed according to the equation:
F*servo=F*d static+F*frictionSign(Vp)+ba(V*pxe2x88x92Vp)+ka(F*cxe2x88x92Fc)+Ma(A*pxe2x88x92Ap)
wherein:
F*d static=static force component on the upper festoon rolls and is equal to Mg+2F*c.
F*friction=Friction in either direction resisting movement of the upper festoon rolls,
Fc=tension in the web after the upper festoon rolls,
F*c=tension in the web, target set point, per process design parameters,
F*servo=Force generated by the actuator,
ba=control gain constant regarding translational velocity of the upper festoon rolls, in Newton seconds/meter,
ka=control gain constant regarding web tension,
Mg=mass of the upper festoon rolls times gravity,
MA=active mass,
Me=active mass and physical mass,
Vp=instantaneous translational velocity of the upper festoon rolls immediately prior to application of the second variable force component,
Sign(Vp)=positive or negative value depending on the direction of movement of the upper festoon rolls,
A*p=reference translational acceleration of the upper festoon rolls, set point,
Ap=translational acceleration of the upper festoon rolls, and
wherein the translational velocity set-point V*p of the upper festoon rolls reflects the equation:
V*p=[EAo/(EAoxe2x88x92Fc)][V2(1xe2x88x92Fb/EAo)xe2x88x92V3(1xe2x88x92Fc/EAo)],
to control the actuator based on the force so computed, wherein:
Fb=tension in the web ahead of the last movable festoon roller,
V2=velocity of the web at the last movable festoon roller,
V3=velocity of the web after the festoon,
V*p=reference translational velocity of the upper festoon rolls, set point,
r=radius of a respective pulley on the actuator,
E=Modulus of elasticity of the web, and
Ao=cross-sectional area of the unstrained web, and optionally the target acceleration A*p being computed using the equation:
A*p=[V*p=Vp]/xcex94T
where xcex94T=scan time, the computations being repeated and the force adjusted at least 1 time per second.
In other embodiments, the first and second force components are applied simultaneously to the upper festoon rolls as a single force, and wherein applying a force to the upper festoon rolls includes measuring translational velocity of the upper festoon rolls; measuring web tension force after the festoon; and sensing the current of the actuator, such measuring and sensing occurring during periodic sensing intervals. and computing a derivative of web tension force from the web tension force based on present and past sensing intervals; computing the translational velocity of the upper festoon rolls; and computing a derivative of the web tension force, the applying of a force to the upper festoon rolls optionally including executing a fuzzy logic subroutine by inputting web tension force error, the derivative of web tension force error, and acceleration error, the fuzzy logic subroutine proceeding through the step of fuzzy inferencing of the above errors, and de-fuzzifying inferences to generate a command output signal, the fuzzy logic subroutine being executed during each of the measuring and sensing intervals.
In some embodiments, the first and second force components are applied simultaneously to the upper festoon rolls as a single force, and wherein applying a force to the upper festoon rolls includes measuring the translational velocity of the upper festoon rolls; sensing the current of an actuator; and computing the estimated translational acceleration of the upper festoon rolls from the equation
Apexe2x80x9d[F*d static+F*frictionSign(Vp)+k1(Vpxe2x88x92Vpe)+kteI]/M2e
where:
Ape=estimated translational acceleration of the upper festoon rolls,
F*d static=static force component on the upper festoon rolls and is equal to Mg+2F*c.
F*fraction=Friction in either direction resisting movement of the upper festoon rolls,
Sign(Vp)=positive or negative value depending on the direction of movement of the upper festoon rolls,
k1=Observer gain,
Vp=instantaneous translational velocity of the upper festoon rolls,
Vpe=estimated translational velocity,
kte=Servo motor (actuator) torque constant estimate,
I=actuator current, and
M2e=Estimated physical mass of the upper festoon rolls.
In some embodiments, the first and second force components are applied simultaneously to the upper festoon rolls as a single force, and applying a force to the upper festoon rolls includes measuring the translational position of the upper festoon rolls; measuring web tension force after the festoon; and sensing the motor current of an actuator applying the force to the upper festoon rolls, the above measuring and sensing occurring at each sensing interval , the method further including computing a derivative of web tension from the present measured web tension and the web tension measured in the previous sensing interval, optionally including computing estimated translational velocity and estimated translational acceleration of upper festoon rolls from the change in position of the upper festoon rolls.
In some embodiments, the first and second force components are applied simultaneously to the upper festoon rolls as a single force, and applying a force to the upper festoon rolls includes measuring the translational position of the upper festoon rolls; and sensing the motor current of an actuator applying the force to the upper festoon rolls; computing an estimated translational velocity of the festoon upper rolls by subtracting the previous sensed value for translational position from the present sensed value of translational position and then dividing by the time interval between sensing of the values; and computing a new force command for application to the actuator in response to the earlier computed values.
In some embodiments, the first and second force components are applied simultaneously to the upper festoon rolls as a single force, and applying a force to the upper festoon rolls includes measuring web tension Fc after the festoon;
(b) sensing motor current of an actuator; and utilizing the motor current and force on the web, in combination with an estimate of system mass M2e, to compute an estimate of translational acceleration Ape, optionally including integrating the translational acceleration to compute an estimate of translational velocity Vpe and integrating the estimated translational velocity to compute an estimated web tension force Fce.
Some embodiments of the invention include, in a processing operation wherein a continuous web of material is advanced through a processing step, a method of controlling the tension in the respective section of the web. The method comprises providing a festoon, having upper and lower festoon rolls, operative for controlling tension on the respective section of web; providing an actuator to apply an actuating force to the upper festoon rolls; measuring a first velocity of the web after the festoon; measuring a second velocity of the web at the festoon; measuring motor current of the actuator; measuring web tension before the festoon; measuring web tension after the festoon; measuring translational velocity of the upper festoon rolls; sensing the position of the upper festoon rolls; measuring acceleration of the upper festoon rolls; providing force control commands to the actuator based on the above measured values, including computed acceleration A*p of the upper festoon rolls, to thereby control the actuating force imparted to the upper festoon rolls by the actuator to control the web tension, optionally including providing force control commands to the actuator based on the equation
F*servo=F*d static+F*frictionSign(Vp)+ba(V*p=Vp)+ka(F*cxe2x88x92Fc)+Ma(A*pxe2x88x92Ap),
wherein the translational velocity set-point V*p of the upper festoon rolls reflects the equation
xe2x80x83Vp=[EAo/(EAoxe2x88x92Fc)][V2(1xe2x88x92Fb/EAo)xe2x88x92V3(1xe2x88x92Fc/EAo)],
to control the actuator based on the force so calculated wherein:
F*d static=static force component on the upper festoon rolls and is equal to Mg+2F*c,
F*friction=Friction in either direction resisting movement of the upper festoon rolls,
F*servo=Target force to be applied by the actuator,
Fc=tension in the web after the festoon,
F*c=target tension in the web, set point,
Fb=tension in the web ahead of the last movable festoon roller,
ba=control gain constant re translational velocity of the upper festoon rolls, in Newton seconds/meter,
ka=control gain constant re web tension,
Mg=mass of the upper festoon rolls times gravity,
MA=active mass,
Me=active mass and physical mass,
Vp=instantaneous translational velocity of the upper festoon rolls,
Sign(Vp)=positive or negative value depending on the direction of movement of the upper festoon rolls,
V2=velocity of the web at the last movable festoon roller,
V3=velocity of the web after the festoon,
V*p=target translational velocity of the upper festoon rolls, set point,
r=radius of a respective pulley on the actuator,
E=Modulus of elasticity of the web,
Ao=cross-sectional area of the unstrained web,
A*p=target translational acceleration of the upper festoon rolls, set point, and
Ap=translational acceleration of the upper festoon rolls, optionally including computing the target acceleration A*p using the equation:
A*p=[Vp=Vp]/xcex94T
where xcex94T=scan time or interval between sensing of translational velocity.
Some embodiments include applying the actuator and thereby controlling acceleration of the upper festoon rolls, such that the actuator maintains inertial compensation for the upper festoon rolls.
Some embodiments comprehend processing apparatus defining a processing line, for advancing a continuous web of material through a processing step along a given section of the processing line. The processing apparatus comprises a first and second rolls defining a first nip; third and fourth rolls defining a second nip, the first and second nips collectively defining the given section of the web; a web storage buffer operating on the web in the given section of the processing line, thereby to control tension in the web and to accumulate a limited length of the web sufficient to sustain operation of the process on the length of web during routine temporary stoppages of web feed to the given section of the processing line or taking the web away from the given section of the processing line; an actuator applying net translational force to the web storage buffer; and a controller driving the web storage buffer, and computing and controlling net translational acceleration of the web storage buffer such that the web storage buffer is effective to control tension, at a desired level of constancy, and to accumulate a limited length of the we, in the respective section of the processing line.