`
`SCHAEFFLER EXHIBIT 2001, pg. 1
`
`SCHAEFFLER EXHIBIT 2001, pg. 1
`
`

`

`PREFACE TO THE FC
`
`This book grew from a course of 1
`Design School of the Westinghouse Cc
`period from 1926 to 1932, when the sul
`into the curriculum of our technical
`beginning of the war, it became a regula
`ing School, and the book was written f
`course, being first published in 1934.
`I:
`entirely by the author’s industrial expe
`editions have brought modifications a1
`problems published in the literature, b:
`by service during the war in the Burea
`The book aims to be as simple as
`complete treatment of the subject. M
`but in all cases the mathematical ap
`available.
`
`In the fourth edition the number of 1:
`substantially, rising from 81 in the fit
`second and third, and to 230 in this pr
`have been made in every chapter to brii
`to keep the size of the volume within l
`deletions as well as additions.
`During the life of this book, from
`engineering has grown at an astonishing
`has expanded with it. While in 1934
`covered more or less what was know:
`such claim can be made for this fourth
`our subject has become the parent of ti
`each of which now stands on its own fl
`body of literature. They are (1) elect
`the theory and practice of instrumentati
`trol or systems engineering, (3) aircraft f
`No attempt has been made to cover
`a superficial treatment would have mac'
`However, all three subjects are olishooV
`
`Copyright 0 1934, 1940, 1947, 1956, 1985 by J. P. Den Hartog.
`Copyright 9 renewed 1962, 1968, 1975, 1984 by J. P. Den Hartog.
`All rights reserved.
`
`This Dover edition, first published in 1985, is an unabridged,
`slightly corrected republication ofthe fourth edition (1956) ofthe work
`first published by the McGraw-Hill Book Company, Inc, New York,
`in 1934. A brief Preface has been added to this edition.
`
`Library of Congress Cataloging-in—Publication Data
`Den Harteg, J. P. (Jacob Pieter), 1901-
`Mechanical vibrations.
`
`Reprint. Originally published : 4th ed. New York : McGraw-Hill,
`1956. With a new preface.
`Includes index.
`I. Title.
`1. Vibration- 2. Mechanics, Applied.
`TA355.D4 1985
`620.3
`84-18806
`ISBN-13: 978—0-486—64785-2 (pbk.)
`ISBN-10: 0-486—64785-4 (pbk.)
`
`Manufactured in the United States by Courier Corporation
`64785418
`2015
`www.doverpublications.com
`
`
`
`SCHAEFFLER EXHIBIT 2001, pg. 2
`
`SCHAEFFLER EXHIBIT 2001, pg. 2
`
`

`

`
`MECHANICAL VIBRATIONB
`TWO DEGREES OF FREEDOM
`
`
`
`lgme’ans a rotary vibration of the bar about a point which lies at a distance of 2.161
`Eht pull. By changing the position of the nuts-
`, 18kt of the center of the bar for the first natural frequency and about a point
`_I is changed while the mass m remains 00W
`«35151 to the left of the center for the second natural frequency.
`.dJustment of the nuts the two natural frequencies r
`ly the same value. Then by pulling down and rel
`'n motion of the mass without twist is initiated. W
`rig occurs without vertical motion, and so on.
`ustrated in Fig. 3.3a, is the electrical analogue 0f ‘
`pages 27, 28). Two equal masses (inductancefl)
`main springs (condensers) C are coupled with 3-
`:ge coupling condenser 0; since to is equivalent to 1/ '
`In one mesh will after a time be completely transferfag
`and so on. Electrically minded readers may ’9’”
`ts flow in each of the two “natural modes” and Wm
`, and may also construct a figure similar to 3.4 Of 3‘
`
`
`
`
`
`553- The Undamped Dynamic Vibration Absorber. A machine or
`him Part on which a steady alternating force of constant frequency is
`“may take up obnoxious vibrations, especially when it is close to
`‘ 941cc.
`In order to improve such a situation, we
`i first attempt to eliminate the force. Quite often
`I not practical or even possible. Then we may
`the mass or the spring constant of the system in
`“tempt to get away from the resonance condition,
`‘_
`some cases this also is impractical. A third pos-
`lies in the application of the dynamic vibration
`invented by Frahm in 1909.
`Fm. 3.6. The ad-
`_, Fig. 3.6 let the combination K, M be the schematic
`dim" °f 3 ""511
`station of the machine under consideration, with
`k—m system to a
`.
`.
`.
`.
`.
`large machine
`0‘08 P0 sm wt acting on 1t. The Vibration absorber
`‘ is of a comparatively small vibratory system 10, m fig’iofiregnmfi
`L
`3,
`Wed to the main mass M. The natural frequency machine in spite
`1m oi the attached absorber is chosen to be equal to the
`gir:2;:§§:‘:;m¢
`4.
`w of the disturbing force.
`I6 will be shown that
`l
`the main mass M does not vibrate at all, and that the small system k, m
`in such a way that its spring force is at all instants equal and oppo-
`m ‘0 Po sin wt. Thus there is no net force acting on M and therefore
`.3 mass does not vibrate.
`0 prove this statement, write down the equations of motion. This
`a simple matter since Fig. 3.6 is a special case of Fig. 3.1 in which k2
`Qade zero. Moreover, there is the external force Po sin wt on the first
`lM. Equations (3.1) and (3.2) are thus modified to
`
`'
`
`,
`
`,
`
`m; .. 9,13 1‘.
`and
`«4 - 2.54-5-I
`in
`“i011: which can be written as
`action corresponding to these frequencies are found from
`
`r
`
`T318forced vibration of this system will be of the form
`x: 2 a: sin wt 1
`x = a Sin wt
`
`Mr. + (K + m, ~ km, 2 P0 sin cut}
`me. + km — x.) = 0
`
`(3‘10)
`
`(3-11)
`
`sfor
`
`1%
`3 k
`fl=_l7£wr+3
`d
`‘ ’u tf
`'
`w J 8 mm ,this becomes
`25-!)
`:3: +115
`fl) _ _o 15
`” ‘
`l?» 2
`'
`
`:32, but not the
`is evident since (3.10) contains only an, :81, and 2:2,
`/
`first derivatives at, and 232. A sine function remains a sine function after
`.
`.
`.
`.
`,WO dlfferentiations, and consequently, With the assumption (3.11), all
`in (3.10) will be proportional to sin wt. Division by sin mt trans-
`ms the differential equations into algebraic equations as was seen before
`
`SCHAEFFLER EXHIBIT 2001, pg. 3
`
`
`
`. bar of massm and length 21 is supported by two sprints 9”
`c). The springs are not equally stifi', their constants be!“
`espeotively. Find the two natural frequencies and the 8M”:
`iodes of vibration.
`I
`,
`e upward displacement of the center of the bar and o it! (01
`. Then the displacement of the left end is x + lg: and thaw?
`2
`The spring forces are k(x + lo) and 2k(z —— to), ma
`'
`I
`m+k(z +lw) +2k(x —l¢) ~0
`91ml!» + kl(z + ls) - 2kl(z ~ lo) - o
`
`ations. With the assumption of Eq. (3.3) we obtain
`
`(—mw' + 3km — 1on a 0
`40110 + (-}snuo’l‘ + 31:12)..." a. o
`
`a frequency equation
`
`mm2 + 3k)(—}~énm'l2 + 3k?) — k't' - 0
`"Nm
`m
`w‘ — 12"
`I+24(5)2 =0
`
`
`
`SCHAEFFLER EXHIBIT 2001, pg. 3
`
`

`

`For simplification we want to bring these into a dimensionless form an
`for that purpose we introduce the following symbols:
`x" = Po/K = static deflection of main system
`of, = k/m = natural frequency of absorber
`(2,2. = K/M = natural frequency of main system
`a = m/M = mass ratio = absorber mass/main mass
`
`‘
`
`L
`
`_
`The ratio
`
`88
`
`MECHANICAL vmazrrrous
`
`with Eqs. (3.1) to (3.4). The result is that
`
`- kaa = P0
`a1(—Mw’ + K +
`"160.1 + (120—471002 +
`= 0
`
`Then Eq. (3.12) becomes
`
`1
`
`“1
`
`k
`
`or
`
`k
`
`K'h‘z "hr—x"
`
`1 _ 2.2
`“’5
`9.1 =
`(1 + _k_ _ 9:) __ _k_
`(1 _
`9”"
`
`«:3
`K
`(I:
`K
`92 =
`1
`
`From the first of these equations the truth of our contention can be
`2
`"
`seen immediately. The amplitude a; of the main mass is zero when thé
`numerator 1 —- 12 is zero, and this occurs when the frequency of the for“
`wu
`.
`is the same as the natural frequency of the absorber.
`Let us now examine the second equation (3.14) for the case that a) =
`The first factor of the denominator is then zero, so that this equati':
`raduces to
`
`.
`
`2—
`
`ktt'“
`
`k
`
`With the main mass standing still and the damper mass having a
`motion ——Po/k ' sin cut the force in the damper spring varies as —Po sin
`;
`which is actually equal and opposite to the external force.
`It was seem
`These relations are true for any value of the ratio w/ 9".
`however. that the addition of an absorber has not much reason unless 1:11,
`
`
`
`
`SCHAEFFLER EXHIBIT 2001, pg. 4
`
`TWO DEGREES OF FREEDOM
`
`, wiginal system is in resonance or at least near it.
`m what follow, the case for which
`
`we = 9”
`
`or
`
`K
`k
`a = A7
`m
`1‘ = H
`
`or
`
`.
`.
`For this Spemal case’ (3’14) becomes
`m;
`1 _. _
`
`w:
`we
`
`war
`
`(l’w‘z><1+"”fi)"“
`1
`
`____
`
`5‘91
`3;“
`
`1:2
`
`(
`‘A striking peculiarity of this result and of Eq.
`womanators are equal. This is no coincidence b1
`.ieal reason. When multiplied out, it is seen that
`W a term proportional to (w2(w§)2, a term prl
`“fwd a term independent of this ratio. When equat
`‘iflgtor is a quadratic equation in (oz/w: which necr
`Thus for two values of the external frequency w
`1M5) become zero, and consequently :31 as well a:
`W, These two frequencies are the resonant or
`am system.
`If the two denominators of (3.15) n
`fighter, it could occur that one of them was zero at a (
`we not zero. This would mean that as; would be
`.
`.
`.
`at, But, if 321 IS infinite, the extensrons and comp:
`ring k become infinite and necessarily the fore.
`-
`gyms we have the impossible case that the amplit
`mmis finite while an infinite force k(x1 — x2) is:
`therefore, if one of the amplitudes becomes infinii
`and consequently the two denominators in (3.15) r
`r The natural frequencies are determined by sett
`equal to zero:
`
`0’2
`
`2
`
`
`
`SCHAEFFLER EXHIBIT 2001, pg. 4
`
`

`

`
`
`
`
`
`The ratio
`
`TWO DEGREES or FREEDOM
`.
`.
`.
`.
`-
`Orlgmal system is in resonance or at least near it. We therefore consider,
`in Whatf 11
`, th
`f
`-
`0 0W8
`6 case 01‘ Whlch
`wu=9n
`or %=
`p. =
`
`89
`
`01'
`
`%=
`
`:13
`
`SISEl»
`
`
`
`
`then defines the size of the damper as compared to the size of the main
`l3B’stcm. For this special case, (3.14) becomes
`
`
`
`(3.1511, b)
`
`
`
`JHANICAL VIBRATIONS
`‘
`.
`.he result is that
`2 + K + k) __ kaa = P0
`i + 02(‘7’1402 + k) = 0
`
`12)
`
`to bring these into a dimensionless form and
`ice the following symbols:
`
`tic deflection of main system
`ral frequency of absorber
`mral frequency of main system
`as ratio = absorber mass/main mass
`
`-——--— —a2—-=x.¢
`K
`93
`K
`
`(3.13)
`
`1 _ 23
`,
`WA“
`__
`(1 + “k— _ 9:) — E
`“3
`K
`9:
`K
`
`(3,14)
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`,A striking peculiarity of this result and of Eq. (3.14) is that the two
`
`fienOminators are equal. This is no coincidence but has a definite phys—
`real reason Wh
`‘
`'
`‘
`‘
`'
`gain
`-
`en multiplied out, it is seen that the denominator con—
`
`.was a term proportional to (ma/d)", a term proportional to (was)!
`_
`a term independent of this ratio. When equated to zero, the denom-
`
`gzzrfla a. quadratic equation in (oz/w: which necessarily has two roots.
`
`(3‘15) 3: two values of the external frequency or both denominators of
`‘zo‘:)(1+f“?22)’1?
`
`in”. TEE: :V:1")o,fr:n:ll consequently :61 as well as x; becomes infinitely
`en
`'
`_
`system.
`If the tam dgfisniiirifatto;fizozglylhfzvzzgxi 38133136113131? (if
`: equations the truth of our contention can be
`
`J“ 1‘, it could occur that one of them was zero at a certain w alhd the air;
`rmpfitude al of the main mass is zero when W
`
`’ not zero. This would mean that $1 would be infinite and 252 would
`and this occurs when the frequency of the fort8
`
`But, If an is infinite, the extensions and compressions of the damper
`393118
`1 frequency of the absorber.
`“
`k become infinite and necessarily the force in that spring also.
`
`second equation (3.14) for the case that w = f9!»
`3 W? hare the imposs1ble case that the amplitude as; of the damper
`.
`:nominator is then zero, so that this equatw!1
`“3338 m 18 finite while an infinite force k(xx —- x2) is acting on it. Clearly,
`
`,
`Ol'e, If one of the amplitudes becomes infinite, so must the other,
`
`consequently the two denominators in (3.15) must be the same.
`The natural frequencies are determined by setting the denominators
`
`Wltozero:
`2
`2
`
`
`
`n of an absorber has not much reason unless thG
`
`
`
`z =_£m -—_fl
`2
`[611— k
`
`itanding still and the damper mass having a
`force in the damper spring varies as ——Po Sin 0":
`1d opposite to the external force.
`a for any value of the ratio (0/ $2”.
`
`It was 6983‘
`
`
`
`(i)4‘<i)2(2+u)+1=0
`
`SCHAEFFLER EXHIBIT 2001, pg. 5
`
`SCHAEFFLER EXHIBIT 2001, pg. 5
`
`

`

`90
`
`MECHANICAL VIBRATIONS
`
`with the solutions
`
`2
`
`‘1
`
`(3.16)
`u + "I
`i
`= (1 +
`This relation is shown graphically in Fig. 3.7, from which we find,
`for example, that an absorber of one-tenth the mass of the main system
`causes two natural frequencies of the combined system at 1.17 and 0.85
`times the natural frequency of the original system.
`The main result (3.15) is shown in Fig. 3.8a and b for u = )6, i.e.,
`for an absorber of one-fifth the mass of the main system.
`Follow the diagram 3.8a for an increasing frequency ratio (0/ S2,. = w/wa,
`It is seen that rel/21.; = 1 for w = 0, while for values somewhat larger
`than zero 1:; is necessarily positive, since both the numerator and the
`
`
`
`FIG. 3.7. The two natural or resonant frequencies of Fig. 3.6 as a function of the mass ratio
`m/M. expressed by Eq. 8.16.
`
`(3.15s) are positive. At the first resonance the
`denominator of Eq.
`denominator passes through zero from positive to negative, hence xl/x,‘
`becomes negative. Still later, at w -= 9,. = w“, the numerator becomes
`negative and x;/x.. becomes positive again, since both numerator and
`denominator are negative. At the second resonance the denominator _
`changes sign once more with negative x; as a result.
`The 232/27“ diagram passes through similar changes, only here the
`numerator remains positive throughout, so that changes in sign occur
`only at the resonance points.
`It was seen in the discussion of Fig. 2.13
`that such changes in sign merely mean a change of 180 deg. in the phase
`angle, which is of no particular importance to us. Therefore we draw
`the dotted lines in Fig. 3.8a and b and consider these lines as determining
`the amplitude, eliminating from further consideration the parts of the
`diagrams below the horizontal axes.
`The results obtained thus far may be interpreted in another manner,
`which is useful in certain applications.
`In Fig. 3.6 let the Frahm absorber
`
`TWO DEGREES 0F FREEDOM
`
`k, m be replaced by a mass mum. attached solidly
`and let this equivalent mass be so chosen that the
`as with the absorber. Since the absorber is more 1
`a, mass, it is clear that mm” cannot be constant but
`each disturbing frequency w. The downward forc
`absorber to the main system M is the spring force
`Eq. (3.10) , is equal to —mi2.
`If a mass mum, we
`M, its downward reaction force on M would be 1
`
`J:
`
`Fm. 3.8a and b.- Amplitudes of the main mass M and of the absc
`various disturbing frequencies w. The absorber mass is one—fit
`
`-meqma'él. For equivalence these two reactions In
`by Eq. (3.11) and the second Eq. (3.13), we have
`
`
`which is the well-known resonance relation, shown
`Thus it is seen that the Frahm dynamic-absorber s*
`by an equivalent mass attached to the main system,-
`mass is positive for slow disturbing frequencies,
`excitation at the absorber resonant frequency, an
`frequency excitation. This way of
`looking at 1
`absorber will be found useful on page 220.
`
`
`
`SCHAEFFLER EXHIBIT 2001, pg. 6
`
`SCHAEFFLER EXHIBIT 2001, pg. 6
`
`

`

` MECHANICAL VIBRATIONB
`
`TWO DEGREES OF FREEDOM
`
`91
`
`M be replaced by a mass mm... attached solidly to the main mass M,
`let this equivalent mass be so chosen that the motion as, is the same
`3 With the absorber. Since the absorber is more complicated than Just
`3‘
`E
`\
`it is clear that mm“, cannot be constant but must be.different for
`as
`disturbing frequency w. The downward force transmitted by the
`ibfiorber to the main system M is the spring force k(x2 —_— 31), which, by
`(3.10), is equal to ~ma‘é2.
`If a mass mum were solidly. attached to
`M: its downward reaction force on M would be the pure inertia force
`
`45
`
`M
`
`20
`
`2.5
`
`0
`
`0.5
`
`2.0
`
`2.5
`
`1.5
`1.0
`(ll/n” ->,
`(bl
`he absorber mass 22 of Fig. 3.6 for
`‘
`'
`.8
`FIG.
`.
`b. Am litudes of the main mass 2:: and of't
`Var-leis ¢(miillslth‘ilrbing ffequenoies w. The absorber mass is one—fifth of the main mass.
`‘mmiyfiil. For equivalence these two reactions must be equal, so that,
`by Eq. (3.11) and the second Eq. (3.13), we have
`
`
`Mequav_§'§_g__£c}=g_2=
`12
`m
`:51
`231
`‘11 1_‘i
`w:
`
`3 shown graphically in Fig. 3.7, from which we find:
`. an absorber of one-ten
`‘
`th the mass of the main system
`5.1 frequencies of the combined system at 1.17 and 0.85
`frequency of the original system.
`It (3.15) is shown in
`Fig. 3.8a and b for u = }6, 110-,
`:' one-fifth the mass of the main system.
`ram 3.8a for an increas
`ing frequency ratio 0.3/9» = w/wa-
`/xu = 1 for
`'
`w = 0, while for values somewhat larger
`ecessarily positive, since both the numerator and the
`
`
`
`(3.15:1) are positive. At the first resonance the
`[.
`through zero from positi
`.
`ve to negative, hence xi/xu
`Still later, at w = 9n = can, the numerator becomes
`becomes positive again, since both numerator and
`:gative. At the second resonance the denominator
`Lore with negative an as a result.
`rm passes through similar changes, only here the
`positive throughout, so that changes in sign occur
`:epoints.
`It was seen in the discussion of Fig. 2.18
`Sign merely mean a change of 180 deg. in the phase
`9 particular importance to us. Therefore we draw
`.g. 3..8a and b and consider these lines as determining
`hating from further consideration the parts of the
`iorizontal axes.
`ed thus far may be interpreted in another manner,
`ain applications.
`In Fig. 3.6 let the Frahm absorber
`
`
`
`
`g!!£§‘-—..........
`-—Efi
`
`UI
`
`..
`
`.5
`w/nfl *
`
`which is the well—known resonance relation, shown in Fig. 2.18, page 44:
`Thus it is seen that the Frahm dynamic-absorber system can be replalcet
`by an equivalent mass attached to the main system, so that the eciuiva :21.
`mass is positive for slow disturbing frequencies, is infinite
`:rgeh. h
`excitation at the absorber resonant frequency, and is negatwe or
`1g
`This way of
`looking at the operation of the
`frequency excitation.
`absorber will be found useful on page 220.
`
`SCHAEFFLER EXHIBIT 2001, pg. 7
`
`SCHAEFFLER EXHIBIT 2001, pg. 7
`
`

`

`92
`
`MECHANICAL VIBRATIONS
`
`TWO DEGREES OF FREEDOD
`
`From an inspection of Fig. 3.8a, which represents the vibrations of the
`main mass, it is clear that the undamped dynamic absorber is useful only
`in cases where the frequency of the disturbing force is nearly constant.
`Then we can operate at (0/00., = «0/9,. = 1 with a very small (zem)
`amplitude. This is the case with all machinery directly coupled to
`synchronous electric motors or generators.
`In variable—speed machines,
`however, such as internal—combustion engines for automotive or aero-
`nautical applications, the device is entirely useless, since we merely
`replace the original system of one resonant speed (at w/Qn = 1) by
`another system with two resonant speeds. But even then the absorber
`can be made to work to advantage by the introductiongof a certain amount
`of damping in the absorber spring, as will be discussed in the next section.
`An interesting application of the absorber is made in an electric hair
`clipper which was recently put on the market.
`It is shown in Fig. 3.9
`and consists of a 60~cycle alternating-current magnet a, which exerts a
`‘\\
`
`\
`
`\
`\
`.w.
`I 60 cycles
`//
`
`small inertia force downward. The resultant of 1
`moving parts b,
`(1 therefore is an alternating fore
`the cutter d in Fig. 3.9.
`
`The effect of the absorber is to completely eliminate 1
`of the housing right under the absorber mass f, but it d0!
`from rotating about that motionless point. Complete e
`motion of the housing can be accomplished by mounting tv
`with a certain distance (perpendicular to the direction of 1
`their two masses. The two masses will then automatical
`as to cause two inertia forces which will counteract the for:
`the inertia action of the cutter assembly «1, b, or in differ
`will enforce two motionless points of the housing.
`
`
`
`For a torsional system, such as the crank shaft
`tion engine, the Frahm dynamic vibration absorl
`a, flyWheel A that can rotate freely
`on the shaft on bearings B and is
`held to it by mechanical springs
`k only (Fig. 3.10a). Since the
`torsional impulses on such an en—
`gine are harmonics of the firing
`frequency, $16., have a frequency
`proportional to the engine speed,
`(‘7’
`F
`the device will work for one engine
`absorlfedofa)Tvti
`Speed only, while there are two
`(b) With Cami“
`neighboring speeds at which the
`In order 4
`shaft goes to resonance (Fig. 3.8a).
`system is modified by replacing the mechanical
`by the “centrifugal spring” of Fig. 3.1%. The
`trifugal field of that figure acts in the same manner
`pendulum in which the field g is replaced by th(
`Since the frequency of a gravity pendulum is x/é
`centrifugal pendulum becomes to x/iT/Z, that is, 1
`speed. Thus a centrifugal pendulum will act
`2
`absorber that is tuned correctly at all engine speed
`this device are discussed on page 219.
`3.3. The Damped Vibration Absorber. Consider
`in which a dashpot is arranged parallel to the dam
`the masses M and m. The main spring K rem
`across itself. Newton’s law applied to the mass 114
`
`Mi! + K751 'i' 19(161 — x2) + 0(331 —- x2)
`
`and applied to the small mass m
`
`min + 10(352 —— x1) + c(:i:2 — (bl)
`
`=
`
`
`
`Fro. 3.9. Electric hair clipper with vibration absorber: a, magnet: b, armature tongue;
`c, pivot; d, cutter; e, guide for cutter; f, vibration absorber.
`
`120-cycle alternating force on a vibrating system 1). System b is tuned
`to a frequency near 120 cycles but sufficiently far removed from it (20
`per cent) to insure an amplitude of the cutter d, which is not dependent
`too much on damping. Thus the cutter blade d will vibrate at about the
`same amplitude independent of whether it is cutting much hair or no
`hair at all.
`
`The whole mechanism, being a free body in space without external
`forces, must have its center of gravity, as well as its principal axes of *
`inertia, at rest. Since the parts I), d are in motion, the housing must
`move in the opposite direction to satisfy these two conditions. The
`housing vibration is unpleasant for the barber’s hands and creates a new I
`kind of resistance, known as sales resistance. This is overcome to a great,
`_
`extent by the dynamic vibration absorber f, tuned exactly to 120 cycles
`per second, since it prevents all motion of the housing at the location of
`the mass f. With stroboscopie illumination the masses d and f are
`clearly seen to vibrate in phase opposition.
`The device as sketched is not perfect, for the mass f is not located
`correctly. At a certain instant during the vibration, the cutter 91 Wm ‘
`have a large inertia force upward, while the overhung and b will have 3
`
`
`
`SCHAEFFLER EXHIBIT 2001, pg. 8
`
`SCHAEFFLER EXHIBIT 2001, pg. 8
`
`

`

` MECHANICAL VIBRATIONS
`
`b.
`nspection of Fig 3 8
`.
`a w
`.
`18 clear that the und’ampléiil Sewage-ms the Vibrations 0”
`re the frequency of the disturi?‘lamlc abs9rber ls useful 0”
`n Operate at m,“ = M," = mg .wae 15 nearly constan
`This is the case
`I
`.
`B ectnc motors or
`generators.
`,
`[.1 as Internal-combustion
`6
`Ications,
`the device is
`ngmal system of
`
`_
`v
`. In variable’speed machines,
`.
`ilfglnes for automotiva or gem.
`n irely useless, smce we merely
`
`)work to advant
`'
`the abSOTber s
`.
`pnng as w
`rig application of “’1
`
`-
`
`' But even the” the abs“
`8 8>1E?s:3—.221 noEcE»
`.
`-
`.
`111 be discussed
`
`93
`Two DEGREES on FREEDOM
`_
`m19:11 inertia force downward. The resultant of the inertia forces of the
`Wing parts b, d therefore is an alternating force located to the left of
`accuser d in Fig. 3.9.
`The efiect of the absorber is to completely eliminate 120-cycle motion of a point
`V
`the housing right under the absorber mass f, but it does not prevent the housing
`1m rotating about that motionless point. Complete elimination of all 120-cycle
`_
`.
`.
`.
`.
`.
`.
`“Minn of the housing can be accomplished by mounting two absorbersf in the device
`a certain dlstance (perpendicular to the direction of the cutter motion) between
`{hair two masses. The two masses will then automatically assume such amplitudes
`to cause two inertia forces which will counteract the force as well as the moment of
`'nertia action of the cutter assembly d, b, or in different words the two masses
`2 will enforce two motionless points of the housing.
`
`
`
`For a torsional system, such as the crank shaft of an internal-combus—
`’Tfiim engine, the Frahm dynamic vibration absorber takes the shape of
`EfiYWheel A that can rotate freely
`on the shaft on bearings B and is
`held to it by mechanical springs
`5: only (Fig.
`3.1011). Since the
`torsional impulses on such an en—
`81118 are harmonics of
`the firing
`frequency,
`i.e., have a frequency
`m
`(a)
`Proportional to the engine speed,
`Fro. 3.10. Torsional dynamic vibration
`the devme Will work for one engine
`absorber (a) with mechanical springs and
`Speed only, while there are We
`(b) with centrifugal springs.
`neighboring Speeds at which the
`the
`In order to overcome this,
`shaft goes to resonance (Fig. 3.8a).
`E‘ll’stcxn is modified by replacing the mechanical springs of Fig. 3.10a
`by the “centrifugal spring” of Fig. 3.1%. The pendulum in the cen-
`trifugal field of that figure acts in the same manner as an ordinary gravity
`pendulum in which the field 9 is replaced by the centrifugal field rw’.
`Since the frequency of a gravity pendulum is m, the frequency of a
`centrifugal pendulum becomes to m, that is, proportional to engine
`speed. Thus a centrifugal pendulum will act as a Frahm dynamic
`absorber that is tuned correctly at all engine speeds. Further details of
`this device are discussed on page 219.
`8.3. The Damped Vibration Absorber. Consider the system of Fig. 3.6
`in which a dashpot is arranged parallel to the damper spring k, between
`the masses M and m. The main spring K remains without dashpot
`across itself. Newton’s law applied to the mass M gives
`
`lug-current m
`
`It is shown in Fig. 3.9
`agnet a which exerts a
`
`
`
`~.
`5" {flipper with
`vibration b
`.
`,
`a scrber:
`and
`e for cutter; f, Vibration abmrbgj magnet, b, armature tongue;
`
`in
`
`rhanism b ‘
`_
`,
`emg a free bod
`pace without external
`‘its center of gravity,
`5
`y
`as well as its principal axes of
`since the parts 6, d ar
`e in motion, the housing must
`isite direction to satis
`I
`fy these two conditions. The
`arber s hands and creates a new
`known
`-
`as sales resrstan
`'
`~
`.
`-
`.
`<38. This ls
`’
`umc Vibration absorber f
`overcome to a great
`
`Md}; + K301 + M331 — $2) + C(dh - 9'32) = P0 Sin 0"
`
`(3‘17)
`
`and applied to the small mass m
`
`mafia + [C(xz ~ 531) + C(ic — 3'31) = 0
`
`(318)
`
`SCHAEFFLER EXHIBIT 2001, pg. 9
`
`SCHAEFFLER EXHIBIT 2001, pg. 9
`
`

`

`TWO DEGREES 01“ FREEDOM
`
`dauper spring (92'; — :52). Hence the current is through lmi
`ence of i; and (i1 —— £2). The equivalence of the electrical Cl!
`system is thus established.
`We are interested in the main current ii. The impedance
`condenser is l/ij’, that of a resistance simply R.
`Imp
`expressed in complex, add directly, and impedances in p
`Thus the impedance of the c, r branch is r +
`and that of
`two branches in parallel have an impedance
`
`fl.._1___
`1
`1
`ifuse + in
`To this has to be added the impedance of the other elements
`
`1
`'_1_
`_ .
`Z —]wL +1100 +———~1——-~:l =
`T'i'l/jwc
`jail
`
`By performing some algebra on this expression and translati
`the result (3.20) follows.
`
`The complex expression (3.20) can be reduced to
`
`$1 = P0(A1 +131)
`
`where A; and 31 are real and do not contain j.
`has to be attached to (3.20) is then that in vector re
`placement at; consists of two components, one in ph:
`and another a quarter turn ahead of it (compare F
`Adding these two vectors geometrically, the magnitl
`by
`
`1 = P0 V A? + 13%
`But (3.20) is not yet in the form (3.21); it is rather
`
`A + jB
`C + jD
`which can be transformed as follows:
`
`$1=Po
`
`$1=P
`
`
`_(A +jB)(C ~jD) * P _(AC+BD;
`° (0+jD)(0—jD) " °
`0
`Hence the length of the x1 vector is
`
`94
`
`MECHANICAL vrsas'rrons
`
`The reader should derive these equations and be perfectly clear on the
`various algebraic signs. The argument followed is analogous to that of
`page 25 and of page 80. The four terms on the left-hand side of (3.17)
`signify the “inertia force” of M, the main~spring force, the damper.
`spring force, and the dashpot force. We are interested in a solution for
`the forced vibrations only and do not consider the transient free vibration.
`Then both an and 702 are harmonic motions of the frequency w and can
`be represented by vectors. Any term in either (3.17) or (3.18) is rep-
`resentable by such a vector rotating with velocity to. The easiest manner
`of solving these equations is by writing the vectors as complex numbers.
`The equations then are
`
`—-Mw’:c1 + K31 + k(x1 —" m2) +jw0<$1 ~' x2) = P0
`——mw2x2 + Mm: — 331) +ij(£B2 "* (E1) = 0
`
`where an and x2 are (unknown) complex numbers, the other quantities
`being real.
`Bringing the terms with x1 and as together:
`
`[—sz + K + k +jwc]x1 — [k +jwc1x2 = P0]
`—ik +jwcixl + [‘"W2 + ’0 +jwci132 = 0
`These can be solved for x; and 902. We are primarily interested in the
`motion of the main mass 211, and, in order to solve for it, we express an in
`terms of 9:1 by means of the second equation of (3.19) and then substitute
`in the first one. This gives
`131 =
`
`(3.19)
`
`
`
`P
`
`(k ~— mm“) +jwc
`° “~sz + K)(—mw2 + k) — mw’k} +jwc{ ~—Mw2 + K ~ W}
`(3.20)
`For readers somewhat familiar with alternating electric currents this result will
`also be derived by means of the equivalent electric
`circuit shown in Fig. 3.11. The equivalence can be
`established by setting up the voltage equations and
`comparing them with (3.17) and (3.18) or directly
`by inspection as follows. The extension (or value.
`ity) of the spring K, the displacement (or velocity)
`of M, and the displacement (or velocity) of the
`force Po are all equal to 1:1 (or 23;). Consequently
`the corresponding electrical elements 1 /C, L, and
`En must carry the same current (111) and thus must
`be connected in series. The velocities across It or
`across the dashpot (x1 —— in) are also equal among
`themselves, so that 1/c and r electrically must be in
`3',11‘11;8Eq:r;:fienlt flffltffgigig
`series but must carry a different current from that
`'n
`- —
`in the main elements L, C, and E0. The velocity of
`tmp corresponds to the absorber.
`m is :32, equal to the difierenoe of the velocity of M(an) and the velocity across the
`
`[142+
`
`9.,
`
`F. a
`
`(A0 + 131))2 + (BO — AD)2
`
`02 +"D‘'2‘
`
`02 + D2
`
`(A2
`A202 + B2D2 + B202 + AZDS —.
`(02 + D2)2
`“ W
`
`—
`'" 0"
`Jame
`= m7
`
`
`
`SCHAEFFLER EXHIBIT 2001, pg. 10
`
`SCHAEFFLER EXHIBIT 2001, pg. 10
`
`

`

`
`
`MECHANICAL VIBRATIONS
`
`.d derive these equations and be perfectly clear on the
`signs. The argument followed is analogous to that of
`age 80. The four terms on the left—hand side of (3.17)
`'tia force” of M, the main—spring force, the damper-
`the dashpot force. We are interested in a solution for
`ms only and do not consider the transient free vibration.
`l 502 are harmonic motions of the frequency w and can
`i vectors. Any term in either (3.17) or (3.18) is rep-
`1 a vector rotating with velocity to. The easiest manner
`quations is by writing the vectors as complex numbers.
`an are
`
`81 + K901 + k(a;1 — x2) +jwc(x1 ~— :52) = Po
`~mao2xz + Mar; -- x1) +jwc(x2 —— x1) = 0
`
`ire (unknown) complex numbers, the other quantities
`
`ms with x, and x; together:
`
`)2 + K + k +jw01x1 —- [k +jwc]a:3 = Po
`Us + jwchn + [~—mw2 + k +jwc]x2 = 0
`'ed for x; and :32. We are primarily interested in the
`[1 mass 2:1, and, in order to solve for it, we express 132 in
`he of the second equation of (3.19) and then substitute
`This gives
`
`(3.19)
`
`'o-sin tut
`
`(k - mm“) +jwc
`[-~mw2 + k) -— mwzk} +jwc(—-Mw2 + K ~— mw2}
`(3.20)
`hat familiar with alternating electric currents this result will
`also be derived by means of the equivalent electric
`circuit shown in Fig. 3.11. The equivalence can be
`established by setting up the voltage equations and
`comparing them with (3.17) and (3.18) or directly
`by inspection as follows. The extension (or veloc-
`ity) of the spring K, the displacement (or velocity)
`of M, and the displacement (or velocity) of the
`force Po are all equal to z; (or 33;). Consequently
`the corresponding electrical elements 1/C, L, and
`E0 must carry the same current (1.") and thus must
`be connected in series. The velocities across k 01'
`across the dashpot (at; —— in) are also equal among
`.
`'
`fetifisx themselves, so that 1/0 and r electrically must be in
`absorber‘
`series but must carry a different current from that
`in the main elements L, C, and E9. The velocity of
`ifierence of the velocity of M(23;) and the velocity across the
`
`'rwo nnennns or rannnou
`
`,.
`90
`
`MP9! Spring (:él — $1). Hence the current in through I must be equal to the differ-
`ence 0‘ 131 and (1'1 - £2). The equivalence of the electrical circuit and the mechanical
`ainstem is thus established.
`_
`‘
`'
`We are interested in the main current 11. The impedance of a cell ls JwL, that of a
`condenser is l/ij, that of a resistance simply It.
`lmpedances in series, when
`expressed in complex, add directly, and impedances in parallel add recrprocally.
`Thus the impedance of the c, r branch is r +
`and that of the lbranch is jwl. The
`two branches in parallel have an impedance
`__.._1_.._.
`1
`l
`r' + 1 mm + jwl
`
`To this has to be added the impedance of the other