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Frame Rigidity

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					Volume I, Number 1, June 1982

                                                   cooperative effort made possible. Gary Klein
                                                   and John Schubert collaborated on a design
IN THE LAB
                                                   over the telephone, an Klein built "Taran-
                                                   tula I" in his frame building shop. Bicycling

              Frame
                                                   magazine purchased the machine from Klein
                                                   and it now sits in our bike workshop. We
                                                   have done a few tests with the machine;
              Rigidity
                                                   this article, I'll present an analysis of how
                                                   and why the machine works and what ques-
                                                   tions it can answer for us.

        Crispin Mount Miller                                 Need to Isolate Results
  How rigid is a bicycle frame? How much
difference is there among different frames,           The fundamental purpose of the machine's
and what are the consequences? Few other           design is to apply loads to a frame in ways
questions about frames intrigue cyclists as        that resemble normal use, but to isolate the
much as these, but good answers have sel-          results: to sort out, from a bike's various de-
dom been available. Like many other aspects        flections under load, what portion is due to
of bicycle engineering, frame rigidity is a        deflection of the frame, and not of the
subject that has been buried in folklore and       wheels, drivetrain, or anything else.
mythology.                                            To perform such measurements reliably, a
  Some time ago we discovered that several         testing machine must provide well-
people had independently thought of the same       controlled, repeatable ways to do three
configuration for a frame rigidity testing jig.    things: support and hold the frame being
Among the people in this club were Gary            tested; apply force with a known magnitude,
Klein, Gary Fisher, John Schubert, and me.         location, and direction; and measure deflec-
We were scattered around the bicycle               tions of specific parts, in specific directions.
industry and didn't all know each other, but          The support must be designed to minimize
we'd all wished for the same kind of ma-           extraneous motion that would ;n . ate the
chine.                                             readings. This machine accomplishes this
                                                   principally by making all the supports many
                                                   times as rigid as the object being tested, so
                                                   that deflections of the supports are negligible;
                A Different                        and also by arranging the support
                                                   structure so that any deformations that
                                                   occur do not affect the measurement
             Kind of Machine
                                                      The principal support for the bicycle is a
   Our idea was to simulate riding stresses        rail representing the ground, consisting of a
by loading a bicycle frame as a rider would - by   steel pipe three inches in diameter and five
pressing on the pedal - and then measuring         feet long, three-dimensionally braced by
how the frame deflected under the load. The        several welded struts of smaller steel pipe.
frame would have to be supported in a realistic       The bicycle sits on this support rail on
fashion, with dropouts and head tube not           a pair of one by two inch steel pillars installed
clamped rigidly in place, but allowed to move      in place of wheels. These pillars are fastened
somewhat, as they do when a bike is ridden.        to the bicycle frame's dropouts, not to the
   In this respect, our machine would differ       testing machine; at the bottom end they are
from the few rigidity testing machines known       simply notched to sit on the support rail.
to exist inside the bicycle industry. Most         They simulate perfectly rigid wheels that
manufacturers test rigidity by clamping the        have perfect traction: (against lateral motion)
frame down and poking sideways on the bot-         on the ground. They are free, however, to
tom bracket shell. This is adequate for some       move limited ways (as rigid wheels would) so
kinds of comparisons, but it doesn't address       that they don't improperly restrain the motion
all the subtle questions that our pedaling load    o the frame itself.
simulator would answer.                               The front pillar is equipped with a freely
   Building such a machine was too time-           rotating "axle" at the top end and a "roller
consuming and costly for any of us, but a
skate" foot at the bottom, so that it can                                                              rear - as if a rider, standing up in a sprint,
move forward, as a real wheel would, when                    Loading and Measuring                     were tilting the frame a bit to the left while
the fork flexes forward under a load. The                                                              pushing straight down on the right pedal.
rear pillar simply stands still on the support                                                         The crank is positioned level, i.e., halfway
rail and is bolted solidly to the dropouts. This     The machine applies controlled loads to the       through the downstroke; and the force is ap-
pillar has two "rear sprockets" welded im-         frame with a pneumatic cylinder supplied with       plied to the pedal at a point five inches from
movably to it (with teeth staggered ¼-inch to      compressed air through a finely adjustable          the midplane of the frame. (On most bicycles,
allow fine adjustment of chain and crank           pressure regulator. In its present loca tion,       this point falls approximately in the middle of
position) and is braced against the sprockets'     the cylinder applies an oblique down ward           the foot-bearing part of the pedal.)
torque by a turnbuckle that reaches to the         force to the right pedal. Other ar-                   Having applied the force, one has to
bottom bracket from the lower end of the pil-      rangements we're considering are to apply a         choose which resulting deflection to mea-
lar. This rear pillar might seem to be immo-       vertical load to the seat, and to pull or push in   sure. When the machine pushes down on a
bile, but in fact it moves slightly with the       various directions on the handlebars.               pedal, the frame displays several different
frame, as a (locked) wheel would, when the           Once it applies the load, the testing ma-         motions at once: the fork tips flex forward
rear triangle tilts forward or swings side-        chine measures deflections with a simple and        and let the frame descend and tilt forward;
ways under a load.                                 direct instrument called a machinist's dial in      the middle of the frame bows away from the
                                                                                                       piston; and the bottom bracket rocks side-
                                                                                                       ways. All of these motions enable the pedal
                                                                                                       to descend somewhat without turning the
                                                                                                       rear wheel. Some of them have other impli-
                                                                                                       cations as well, such as shock absorption or
                                                                                                       effects on steering. Each is a worthwhile ef-
                                                                                                       fect to measure.
                                                                                                         A more interesting approach, though, is to
                                                                                                       take measurements that predict the actual
                                                                                                       amount of energy the frame absorbs from a
                                                                                                       pedal stroke.

                                                                                                             How Much Energy is Wasted

                                                                                                          One of the most widely voiced complaints
                                                                                                       about frame flexibility is that it wastes en-
                                                                                                       ergy, because, when the rider depresses a
                                                                                                       pedal, some of the rider's work must be in-
                                                                                                       vested to "wind up" the frame, instead of
                                                                                                       turning the wheel.
                                                                                                          (At some later moment, the frame will
                                                                                                       "unwind" and release this energy; the ex-
                                                                                                       tent to which this energy is wasted, or in-
                                                                                                       stead ultimately helps to propel the bike, is a
                                                                                                       subtle question whose answer depends on
                                                                                                       the way muscles function, and also on each
                                                                                                       rider's pedaling technique. I plan to address
                                                                                                       this question in a future article, but for the
                                                                                                       moment, suffice it to say that some portion
                                                                                                       of this energy does get wasted.)
                                                                                                          Gary Klein, when he built the machine,
   (A vector diagram shows that the resulting      dicator: a dial face attached to a telescoping      equipped it to measure a frame's "wind-up"
pattern of forces due to the turnbuckle is         rod, with a hand on the dial that ,measures         energy directly.
equivalent to that for a bike with an "infi-       the telescoping motion in thousandths of an            One way to measure energy is as work
nitely high" gear, i.e., with a rear sprocket      inch. This instrument can be mounted                done on a moving object. "Work" in this
whose radius is zero or - except for the           against any of several parts of the frame to        sense is defined as the product of a force ap-
force on the seatstays - to that for a bike        measure specific deflections. By suitable lo-       plied in a given direction and the distance the
with its rear brake locked. While this isn't       cations of one or more cylinders and dial indi-     object moves in that direction while the force
completely realistic, the only difference for                                                          is exerted. For a constant force exerted par-
                                                   cators, the machine can be used to measure
the frame turns out to be that the net verti-                                                          allel to the object's motion, this can be
cal force on the bottom bracket is high by a       numerous aspects of rigidity under various          stated simply as
few percent - about 12 percent compared to         vertical, lateral, and pedaling loads.
                                                                                                                          W=FX
that for a bike in a 90-inch gear. This cor                                                                    where W = work; F = force;
rection can easily be deducted from the read-              Flexure under Heavy Load                                   and X = distance.
ings and is a small price for the simplification                                                          If the force or direction of motion varies
it allows in the structure and operation of the                                                        during the interval in question, the equation
machine.)                                            As the present location of the cylinder           becomes more complex, and often must be
   A spidery superstructure of additional          might imply, the first question we are looking      expressed as a calculus integral. However,
steel struts extends above and behind the          at is the classic one of frame flexure under a      for certain elastic objects like springs - and,
support rail. This upper structure provides        hard pedaling load. The piston is oriented to       to a good approximation, bicycle frames -
lateral support for the bike being tested, and     apply a force that simulates a hard push in a       there is a convenient special case. When
also offers a "place to push from" to apply        sprint: vertical when viewed from the side,         such objects are flexed, the force and dis
testing forces.                                    and slanted ten degrees toward the midplane
                                                   of the frame when viewed from the front or
  tance bear a constant ratio to one another:                             (If one wishes to simulate a gear less than       four dial indicators against selected points on
                                                                        the "infinite" locked-wheel case repre-             its faces. These indicators will measure the
                      F=kX                                              sented by this machine, the use of this plate          otion and rotation of the bottom bracket in
where k is a constant called the "spring con-                           requires one correction factor: if the rear         ill the dimensions necessary to predict the
stant." For objects that exert varying forces                           wheel were stationary - i.e., did not rock          rests i ig motion of the pedal; given the ori-
in accord with this spring-constant equation,                           forward with the frame, but rather stayed still     entation and location of the piston shaft, we
the work required to produce a given deflec-                            in external coordinates, representing a true "      can then calculate the component of pedal
tion is                                                                 zero" reference for forward propulsion -            motion along the piston's line of action.
                  W= ½FX²                                               and the gear were finite, the crank would tilt         This technique may seem more complex,
                                                                        forward a bit less than the frame would,            but we find it attractive because it's more
          (often written as W = ½kX²)                                   because the chain would not move with the           versatile: in the course of determining the
where F is the force at full deflection and X is                        frame. To make this correction we place a
the magnitude of that deflection.                                       dial indicator under each end of the
   The work done by the piston depressing the                           lengthwise portion of the plate's mounting
pedal, then, is half the product of the final                           arm - this portion runs level and parallel to the
force and the final pedal displacement.                                 wheelbase - and measure its tilt by the
   But this figure, measured at the pedal, in-                          difference of the two vertical deflection read-
cludes work spent to wind up the drivetrain,                            ings. We then calculate the portion of the tilt
not just the frame; it includes crank flex, chain                       which would not occur in the simulated
                                                                        gear', and correct the plate-motion reading
stretch, and other factors. This figure may be                          by a corresponding amount. To be consistent
interesting as an evaluation of the whole                               in simulating the response to a finite gear, we
bicycle (and so would measurements of                                   also subtract the correction term that
wheel deflections) but it doesn't allow                                 corresponds to the downward force from the
comparison of frames by themselves.                                     turnbuckle mentioned earlier, since the net
                                                                        downward force on the bottom bracket de-
                     Where to Look                                      termines both its downward motion and its
                                                                        forward tilt, though not its lateral motions.)
   What's needed to determine the frame's
contribution is to measure how much pedal
motion is due specifically to frame deflection.                                  Predicting Pedal Motion
This requires measuring the motion of the
frame itself, not of the pedal, which is part of
the drivetrain. But while the measurement                                 A more abstract technique which we plan
                                                                        to try is to predict the pedal motion mathe-
must be made on the frame, the work equation                            matically by extrapolating from motions mea-
requires that the measurement must describe                             sured at the bottom bracket itself. Our
the motion at the point where the force is                              planned method of measuring this motion is
applied - the pedal - and there isn't any part                          to attach a rectangular metal block, an inch
of the frame there.
   This problem can be approached in various
                                                                        thick and three inches square, under the bot-
                                                                        tom bracket, and align it with the wheelbase          Detail of the floating mount
ways. The technique we have used so far                                 and midplane of the bicycle, and then place           for the front dropouts.
employs a very direct mechanical approach -
we measure the motion of a fixture on the
bottom bracket that presents an appendage
at the required location.
   This fixture is a small arm that is clamped to
the bottom bracket shell and supports a
small plate, roughly horizontal, just under-
neath the pedal. The plate is oriented per-
pendicular to the piston shaft, so that a dial
indicator placed under the plate, with its
plunger in line with the piston, will read mo-
tion only in the direction the piston pushes.
   Because this plate is attached rigidly to the
frame, it responds only to frame deflection and
not to drivetrain deflection. But since its
location is the same as a pedal's, it responds
to angular deflections of the bottom bracket
in the same way a pedal would. The motion
of the plate, therefore, represents the de-
sired measurement: the portion of the pedal's
motion due to frame deflection only.
    'This portion is equal to the tilt of the frame multiplied by the
ratio of rear cog size to chainwheel size: since a stationary rear
cog would be rotating backward, relative to the frame, it would
drive the crank backward (relative to the frame) a proportional
amount. For example, a crank geared 52:13 would rotate in the
                                                                                                                              (Right) photo taken during
frame ¼ as much as the rear cog, so that its net tilt relative to the                                                         application of 200-pound load.
ground would be 25 percent less than the tilt of the frame.
pedal motion, it will also measure several            a pedal stroke bears no fixed ratio to the
motions of the bottom bracket, which are in-          maximum downward pedal force. Against                               Bottom Bracket Tilt
teresting in their own right; and it is more          the estimate in Table 1, though, which I con-
                                                      sider reasonable, the steel frames' deflec-                This mostly horizontal character of the
quickly adaptable to modifications in the location
                                                      tion energy is about one percent of the total           plate's motion raised an interesting ques-
and direction of force applied to the pedal, since    work in a stroke; and the aluminum frame's
the only adjustment required are changes of                                                                   tion. Tilting of the bottom bracket is often
                                                      is about 1½ percent. Contrary to                        thought of as a rotation relative to the imme-
numbers in the calculation formulas, instead of       traditional expectations, in this case the              diately adjacent parts of the frame, due to
mechanical adjustments of the measuring               larger of the two steel frames is the more              flexibility in the bottom bracket shell and the
device itself.                                        rigid one; but it weighs a good bit more, too.          down and seat tubes. This local flexing is of-
                                                         But I also discovered various things about           ten suspected by people with narrow front
               Initial Test Results                   the geometry of frame deflection.                       derailleurs who find that the chain may rub
                                                         The first thing that struck me was the di-           when they pedal hard and not rub when
                                                      rection of the indicator plate's motion.                they don't. (Deflection in the crank and
   Our testing so far has been limited to three       Though the force applied in these tests was             spindle could also cause this effect, of
bikes, with about a dozen load cycles run on          nearly vertical, the indicator plate's deflec-          course.)
each, and with eight data points taken for each       tion is mostly horizontal.                                 The large lateral motion of the bottom
                                                         The piston must swing sideways, following
cycle. I can't make far-reaching claims about                                                                 bracket suggested a second source for the
                                                      the pedal, more than it extends
the validity of the test data until more cycles       downward. (The energy calculations are                  rotation: perhaps the whole rear triangle ro-
are run, but so far the results have been             corrected for this swing.) While there is               tates rigidly (since it is tetrahedrally braced
quite consistent and repeatable. (There's a           still a moderate vertical motion at the                 to a large extent) and the bottom bracket
small chance that the relatively few cycles I'        indicator plate (directly below the pedal),             just happens to be the part whose rotation is
ve taken will eventually turn out to be on one        angular extrapolations indicate that the                measured.
side of the scatter that a larger data base           vertical motion at the midplane of the                     This possibility seemed fascinating, but I
would display.)                                       bicycle is very small - about a tenth of the            lacked the measurements to check on it. All
                                                      horizontal motion. Most of the plate's vertical         the measurements I had taken were of mo-
   Some interesting findings emerged when I
                                                      motion is due to tilting of the bottom                  tions near the bottom bracket. Time did not
finished and sat down with my calculator. (See
                                                      bracket; and not to downward motion of                  permit me to do all the additional measure-
Table 1.) I found numbers for the de-
                                                      the frame.                                              ments I would have liked (a run of numbers
flection energy ½F X. Putting them in context
is difficult because the total work done in                                                                   takes almost a full day, and the press dead




Table 1: Pedal-Related
Deflections at 200-
Pound Load'




'Load applied as described in text. Each value at     3Based  on an assumed total work per stroke of          gests that this formula for total work per stroke
200 pounds is interpolated from -the best-fit line    maximum force X total vertical pedal travel, or         approximates the upper limit approached by
from readings at four points ranging from 61.6 to     2677 inch-pounds for 170-mm cranks and 2598             smoothly pedaling riders. For a rougher pedaling
246.4 pounds, usually averaged from two runs.         inch-pounds for 165-mm cranks. The actual forces        style (especially likely during a sprint) the work
                                                      during a pedal stroke are much more complex, but        per stroke would be a smaller multiple of the max-
2This figure is a rate and does not imply that this
                                       high           information published by P. Cavanagh and I. Faria (in   imum force, and the energy absorbed by the
a load was applied. Our maximum testing load          their book The Physiology and Biomechanics of           frame would be a larger percentage of the total
was 300 pounds.                                       Cycling, John Wiley and Sons, 1978) strongly sug        work.
line had arrived), but I was able to make some                                                          the rotation of the down tube to a small ex-
quick checks of the seat lug and rear                          Pattern of Deflection                    tent.
dropout motions on the Bianchi Limited. Be-                                                                So what does it all mean? We've learned
cause we do not yet have a fixture to hold                                                              some things, but the numbers don't tell the
the dial indicator in these positions, I took the                                                       whole story; they answer some questions,
                                                         To the extent that the rear triangle does
measurements with a measuring tape.                                                                     but they always raise new ones. Some ques-
                                                      rotate, though, what pattern of deflection lets
   From three runs' worth of tape measure-                                                              tions about where the flex takes place have
                                                      it do that? What part of the frame is being
ments (Table 2), my tentative conclusion is
                                                      flexed? This is a subject for numerous            been mentioned before. Others - particularly
that the effect exists, but it isn't the only
                                                      future projects, of course; at this point, our    the question of how much deflection energy is
source of bottom bracket tilt. Under this
                                                      analysis is largely conjecture. But the rear-     lost and how much is put back into the
type of load, the rear triangle does rotate,
                                                      triangle motion does imply certain things. I      drivetrain - keep popping up every time we
almost rigidly, around an axis that runs ap-
                                                      expect to find the following pattern:             sit down and talk about the machine. As we
proximately from the foot of the rear wheel
                                                         The only parts outside the rear triangle       run more tests and analyses, we'll let you
pillar to the midpoint of the seat tube. The
                                                    are the top tube, down tube, and fork. If the       know with future Bike Tech articles.
magnitude of the rotation is about 1.9 de-
                                                    rotation axis runs as the preliminary mea-
grees under a 200-pound load.
                                                    surements indicate, it's almost parallel to the
   This rotation can be expressed as the sum
                                                    down tube. The rear triangle rotation there-
of two component rotations: one around a
                                                    fore requires torsion in the down tube (and                          Table 2
vertical axis and the other around a front-to-
                                                    this torsion is made more severe by the addi-
rear horizontal axis. The vertical-axis com-
                                                    tional rotation of the bottom bracket beyond        Lateral Deflection of Points on Rear Triangle
ponent - 1.6 degrees - agrees closely with
                                                    that of the rear triangle). In addition, since      under 200-Pound Load on Right Pedal (Bian-
the lateral rotation of the bottom bracket it-
                                                    the rotation axis misses the head tube (which       chi Limited)
self. The component around the horizontal
                                                    is the more-or-less fixed restraint for the
axis, however - 1.0 degrees - is barely over                                                            bottom bracket                    +.294 ± .005
                                                    front of the frame) the top tube and down tube
half as great as the sideways-tilting rotation
                                                    must both bend sideways to stay with the            seat lug                              - . 3 1 ± .01
of the bottom bracket. Bottom bracket tilt,
                                                    head tube. (The top tube will bear some             rear dropout                          - . 2 2 ± .01
then, appears to be partly due to bodily
                                                    amount of torsion, too, but not much, because
rotation of the rear triangle, but almost                                                               (Average of three loading cycles. Deflection to left
                                                    it's largely crosswise to the axis of ro tation.
equally due to the traditional culprit of flex                                                          considered positive, to right, negative. Bottom
                                                    ) Finally, the fork will flex sideways
occurring near the bottom bracket.                                                                      bracket deflection measured with dial indicator,
                                                    somewhat, allowing the head tube to follow
                                                                                                        others with steel measuring tape; tolerances are
                                                                                                        maximum deviation of readings from mean.)

				
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