diameter sizing

8/12/2019 Diameter Sizing

1/8

Diameter Sizing of Rolls in

Bridles that Utilize Powder

Clutches

BRIAN T. BOULTER

ROBERT F. LOCKHART


2/8

Diameter Sizing of Rolls in Bridles that Utilize Powder Clutches

Abstract - The use of powder clutches for regulation oftorque in entry and exit bridles of tension levelers musttake into consideration the constraints imposed by theslip requirements of the powder clutch. This paperprovides guidelines for roll diameter sizing as a functionof the maximum linear line speed and worst caseapplication tension profiles such that the powder clutchalways operates in a reasonable slip operating range.Key Words: Bridles, Powder-Clutches, TensionLevelers, Strip Tension.

1. INTRODUCTION

Scratching of the strip observed during commissioning ofthe tension leveler of an OEM tension leveling line wasattributed to slippage in the entry and exit bridles of the

tension leveler. Based on a careful analysis of thetension profiles, bridle roll static and dynamic frictioncoefficients, and motor power requirements of the entryand exit bridles of the tension leveler nothing was foundto explain the observed bridle slippage. It becameapparent that an additional unknown was responsible.To facilitate a better understanding of the problem a briefdescription of a powder clutch and the powertransmission system of the OEM tension leveler ispresented.

A powder clutch is a passive device, it is incapable ofsupplying power to a system. Briefly, it is a type ofelectromagnetic disk clutch in which the space between

the clutch members is filled with dry, finely dividedmagnetic particles; application of a magnetic fieldcoalesces these particles, creating friction forcesbetween the clutch members. Shaft torque can beregulated by controlling the strength of the magneticfield, and hence the friction between the clutchmembers. Powder clutches are designed to operate witha fairly linear relationship between torque and fieldcurrent within a given slip range (Fig. 1). The direction ofpower flow through the clutch is always from the shaftthat is rotating faster to the shaft that is rotating slower.

Fig. 2. is a schematic representation of the powertransmission system in the tension leveler. Beginningwith the exit bridle, a drive motor provides power to thesystem through a 1:1 transmission. The output shafts ofthe exit bridle transmission are all rotating at the samespeed. All four drive shafts feed power to the rollsthrough 96/16 gears. Power is supplied to the gear forroll #5 directly from the 1:1 transmission. For rolls #6, #7and #8 power is supplied through individual powderclutches. Strip tension in each section of the exit bridledecreases between higher roll numbers, with a minimumat the exit of the exit bridle. The bridle is said to bemotoring, therefore power must be supplied to theprocess from the drive motor as indicated by the

direction of the arrows representing the drive shafts. Theamount of torque on the individual shafts determines thedrop in strip tension across each roll and is controlledwith the powder clutch field current.

Another motor is connected to a differential gear box.The differential gearbox is also connected to the 1:1drive transmission to provide a path for powertransmission from the entry bridle to the exit bridle. Thedifferential configuration enables the average rotatingspeed of the entry bridle to be slightly less than the exitbridle thereby creating the necessary elongation fortension leveling. The output shafts of the entry bridle

differential transmission are, like the shafts on the exitbridle, all rotating at the same speed. Similarly all fourdrive shafts feed power to the rolls through 96/16 gears.Power is supplied to the gear for roll #4 directly from thedifferential transmission. For rolls #1, #2 and #3 power issupplied through individual powder clutches. Thetensions in each section of the entry bridle increasebetween higher roll numbers, with a maximum at the exitof the entry bridle. The bridle is said to be regeneratingtherefore power must be absorbed from the process tothe drive motor as indicated by the direction of the


3/8


arrows representing the drive shafts. Analogous to theexit bridle, the amount of torque on the individual shaftsdetermines the tension increase across each roll and iscontrolled with the powder clutch field current.

In the entry bridle, roll #4 is not connected to a powderclutch therefore the [rpm] of the shafts on the differentialtransmission side of the powder clutches for rolls #1, #2and #3 will rotate at the same speed as the drive shaftfor roll #4. To satisfy the requirement that the processside powder clutch shafts rotate faster than thetransmission side shafts (to absorb power from theprocess to the transmission) the roll diameters for rolls#1, #2 and #3 must be less than #4. A similar analogycan be drawn for the exit bridle showing that rolls #6, #7and #8 must have larger diameters than roll #5.

The roll diameters on the commissioned leveler bridleswere sized by the OEM such that the powder clutchesoperated with a very small slip (less than 0.1 [rpm]). Thiswas done without consideration of the differential inadjacent roll surface speeds that is required to maintaina given strip tension (see [1], [2], [3]). An analysis of theentry bridle that included this phenomena revealed thatat a given line speed the reduction in roll diameters ofrolls #1, #2, #3 was not sufficient enough to guaranteethat the process side powder clutch shafts would rotatefaster than the transmission side shafts. Bridle behaviorwould be completely unpredictable at or above this

speed.

Section 2 contains a derivation of a roll diameter sizingequation based on a commonly used strip tensiondynamic equation. Section 3 closes with someconclusions.

2. ANALYSIS

2.1 NOMENCLATURE

For the analysis the nomenclature is associated with theentry bridle model shown in Fig. 3

Ji The ith roll inertia [kg m^2]Vi The ith roll surface velocity [m/min]

i The i'th roll reflected rotational velocity [rpm]

Ri The ith roll radius [m]

Di The ith roll diameter [m]

GRi The ith roll gear ratio

Li The ith tension zone length [m] (includes 1/2

wrap length)

Ti The ith tension zone tension [kgf]

i The ith roll reflected shaft torque [kgf m]

E Modulus of elasticity [kgf/mm^2]A Cross sectional area [mm^2]AMIN Minimum strip cross sectional area

[mm^2]s The Laplace operatorLS Maximum line speed [m/min]

Si Slip at the ith roll powder clutch [rpm]

Si(MAX) Maximum powder clutch slip at the

i'th roll [rpm]

Ti(MAX) Maximum tension drop across the

i'th roll [kgf m]

2.2 STRIP TENSION VS SPEED DIFFERENTIAL

EQUATION:

The strip tension equation (1) is a commonly usedrepresentation ([1], [2], [3]) of the dynamics associatedwith the conveyance of strip through tension zones withthe use of bridles. It is based on the concept ofconservation of mass.

Assuming zero initial conditions, taking the Laplace

transform of (1) and solving for Ti yields:

For metals applications it is reasonable to assume the Viis approximately equal to Vi-1 [4], (2) can now beexpressed as:

Solving (2) for the condition where Ti-1, Viand Vi-1areconstant and assuming no disturbances, we can obtainan approximation of the speed differential required tomaintain a specific tension (5):


4/8


Therefore:

Solving for Vi-1we obtain:

Non-linear contributions such as strip viscous andsquare-law material damping have not been included in(6). Inclusion of these material properties would result ina speed differential other than that calculated with (6).

With this in mind the analysis that follows should beviewed as ideal. Some margin to accommodate theseexclusions should be left in the final design guidelines.

2.3 DESIGN RULES FOR ROLL DIAMETER

CALCULATIONS IN POWDER CLUTCH BRIDLE

APPLICATIONS

From (6) it is clear that the faster the line speed of agiven application the greater the speed differencebetween two driven rolls must be to maintain a given

tension differential (Ti). In the OEM bridle design the

values of GRiare equal to 96/19 on all the powder clutch

torque controlled rolls, therefore Tifor a range of linespeeds can only be achieved by either changing thespeed of rotation of the driven roll and/or by change inthe diameter of the roll.

As described in the introduction, for the entry bridle toregenerate power the powder clutch requires that theload side shaft be rotating faster than the motor sideshaft. To accomplish this the OEM increased thediameter of the driven rolls as the strip passes through

the bridle (i.e. D1< D2< D3< D4 in Fig. 3). If we ignorethe effect of stretch in the strip (Eq. 6) then with these

diameters 1> 2> 3> 4and the powder clutch slip

should theoretically be in the correct direction. Howeverwhen the effect of strip stretch (5) is considered therenow exists the possibility that the additional velocity

differential term may result in a condition where 1< 2< 3< 4to maintain tension in the strip. This is aphysically unrealizable operating condition for thepowder clutch. The entry bridle can no longer regeneratepower through the powder clutches and the tension inthe strip will not follow the desired tension profile. Thebridle behavior is no longer predictable. A similaranalogy exists for the exit bridle.

The purpose of the following analysis is to identify

maximum bridle roll diameters such 1>

2>

3>

4for all strip cross sectional areas, all line speeds, and allbridle tension profiles. An analysis for the entry bridle rolldiameter calculations is presented, the exit bridle rolldiameters are calculated with a similar procedure. Thefollowing constraints need to be identified for theanalysis:

1. Identify the maximum slip desired in the powder

clutch: (S(MAX))2. Identify the maximum possible tension drop

requirement of the driven rolls. (Ti(MAX))3. Identify the minimum strip cross sectional area.

(AMIN

)

Let the rotational speed of the shaft on the motor side ofthe 96/16 gear-box for the speed regulated driven roll isgiven as:

Then the entry bridle analysis begins at the speedregulated roll (Roll #4) and moves out toward the entrybridle entry roll (Roll #1). The equations governing therotational speeds of the these rolls are obtained bycombining (6) and (7):

To meet the minimum slip requirement the analysisshould be performed for the worst case scenario. That is

for a minimum cross sectional area, maximum Ti's and

the desired S(MAX). This may be accomplished by solving(8a,b,c) for the roll radius and these operatingconditions. That is:


5/8


The worst case analysis above has a built in safetymargin based on the use of a minimum cross sectionalarea with an unrealistic tension differential through thebridle. However this approach provides foraccommodation of the unmodeled nonlinearitiesdescribed in section 2.2 above and leads to a morerobust design. A more general form of (8a,b,c) for n rolls(where the n'th roll is speed regulated and the other rollsare driven through powder clutches) can be expressedas:

Similar derivations can be made for the exit bridle where,in terms of flow, the first roll in the bridle is speedregulated:

Table 1 contains a calculated set of diameters based ona set of parameters for the entry bridle at the OEM's

user site. The author used a S i(MAX)of 3 [rpm]. This valuewas chosen based on the advice of several powderclutch vendors who recommend operating the powderclutch with as much slip as an application will allow. Thisis to avoid erratic torque behavior that occurs whenoperating with slips that approach zero. This erraticbehavior is attributed to a vacillation of the frictioncoefficient between the powder in the clutch and therotating surfaces. With extremely low slip values thefriction coefficient randomly varies from a static value(when the input and output speeds lock together) to adynamic value (when there is slight slip in the powderclutch). However it should be pointed out that theamount of maximum slip in the powder clutch must alsobe chosen based on a careful analysis of the thermalcharacteristics of the powder clutch and the desiredapplication torque for the clutch. This is to avoidoverheating during normal operation.

Ti(MAX)was calculated using 2100 [kg] entry tension

maximum, a coefficient of friction for the rolls equal to0.15 and a wrap angle of 210 [deg] on each roll asfollows:

For the exit bridle:

Ti(MAX)was calculated using 30000 [kg] entry tension

maximum, a coefficient of friction for the rolls equal to


6/8


0.15 and a wrap angle of 210 [deg] on each roll asfollows:

3. CONCLUSION

When sizing the roll diameters in applications withpowder clutches consideration must be made to providea reasonable minimum slip in the powder clutch. Inaddition the velocity differential attributable to stretch inmoving strip must also be accommodated.

References

1. Carter, W. C., "Reducing Transient Strains in ElasticProcesses,Control Engineering Mar. 1965. Pp. 84-87.

2. Boulter, B. T., "A Novel Approach for On-Line Self-Tuning Web Tension Regulation ", Proceedings ofthe 4th IEEE International Conference on ControlApplications, Pp. 91- 98, September 1995.

3. Fox, S. J., Lilley, D. G., "Computer Simulation OfWeb Dynamics ", Proceedings of the 1st IWHCInternational Web Handling Conference Tab 20.Oklahoma State University, March 1991.

4. [4] Hamby, R. C., "SER # 35, Tension Regulation,"Reliance Electric Engineering Reports 1968.


7/8


diameter sizing

Documents