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Dynamical analysis of machines
DYNAMICAL ANALYSIS OF MACHINES.* BY R. EKSERGIAN, M.S., M.E.E., Ph.D., Consultant, Engineering Department,
E. I. DuPont de Nemours & Company" Member of the Institute. OSCILLATIONS OF A LINKAGE MECHANISM.
We will consider a m e c h a n i s m similar to a governor linkage which is r o t a t e d a b o u t a vertical shaft with angular velocity ¢. T h e linkage consists of four equal bars forming a rhombus, each of length 2a and of mass m and uniformly distributed. T h e lower vertical joint is fixed to the shaft, while the u p p e r vertical joint can slide freely along the shaft. T h e horizontal joints are connected by a spring 1 with m o d u l u s of elasticity u. T h e initial unstrained length of spring is 2c. T h e angle of a n y rod with respect to the vertical is 0, t h a t is the total angle at the b o t t o m or u p p e r joints between the rods is 20. T h e co6rdinates of the s y s t e m are evidently 0 and ¢. T h e kinetic energy can be shown to be, T = ma2[S(sin
2 0 + ~)02 + ~ sin 2 0.62]
and the potential energy due to g r a v i t y and the spring is, V = 8mgacosO+2u
= 8toga
f
2as~nOy _ c
c
°d y
cos 0 + ~-(2a sin 0 -- c) 2. c
T h e m o m e n t u m corresponding to the co6rdinate 0 is OT
0--0- = 2ma2[-S(sin2 0 + ½)0] =
Po
* Extension of a portion of a dissertation for the degree of doctor of philosophy submitted to Clark University, 1928. 1 Constructively, two parallel springs on either side of shaft could be used, or simply a fixed guide shaft at the top, the rotating shaft terminating at lower hinge. C o n c l u d e d from p. 505, vol. 2I I, April, 1931. v O L 2xx, ~o. 1265--43 627
628
R. F~KSERGIAN.
[J. F. I.
and the m o m e n t u m corresponding to the co6rdinate ¢ is 16
OT -
-
O4;
-
3
m
a
2sin 206 = p,,
which is obviously also equal to the angular m o m e n t u m , A.M., about the shaft axis. T h e equations of motion are, 16ma2[-(sin
16
2 0 + ½) 0 + sin 0. cos 0.02-] - - - m a 2 sin 0 cos 0q~2 3 = 8mga sin 0 -- 4au (2a sin 0 -- c) cos 0 c
and I6ma~[-sin2 0.~ + 2 sin 0 cos 0067 = 4. 3 T h e latter equation is obviously the rate of change of A.M. about the shaft axis equal to' the shaft torque 4. If we assume an oscillation about the kinetic equilibrium position, let o0 = the angular velocity corresponding to this position, and 00 the corresponding angle of the rods with respect to the vertical. Then, 0=00+~,
0=~,
~=~0+~.
T h e equilibrium position is given b y the expression, 16 4au (2a sin 00 - c) cos 00 = 8mga sin 00 + - - m a 2 sin 00 cos 00. co2. c 3 Neglecting the displacements and velocities from the equilibrium position of higher order the equations of oscillation are, I6ma 2
(
sin 200 + 3 --
i) [(4o ~+
- - c (2a cos 200 + c sin 00)
8mga c°s O° ) -- I6 - - m a 2 cos 20o.w 2 ] ~ 3 _
(I)
16 rna2 sin 20o~O,~= 0, 3
16 - - m a 2 [ s i n ~ 00.;/ -t- 2 sin 00 cos 00.w~ = 4. 3
(2)
May, 1931.]
629
DYNAMICAL ANALYSIS OF M A C H I N E S .
These equations m a y be written in terms of the coefficients, P, Q, K, R, and M and N. If we suppose an impressed harmonic torque of some multiple of the shaft frequency, a v e r y possible condition in a p p a r a t u s of this type, we have, P~ + (Q -
K,02)~ + R0~o = o,
Mi/ + N~0~ = ~0 cos m~ot. If we let, e = A sin mo~t, ~1 = B cos mo2t, then, A -
-
m°~2R~b° D '
B = - - (Q -
( m 2 P + K)°~2) D
and Imw2N, D =
_
-
( m ~ p + K ) w ~ + Q,
m2o)2M mco~R
•
If the shaft torque • is due to the reaction of a torsional spring coupling, t h e n ¢ = k ( ¢ - o ~ t ) = - k~ and the critical speeds are d e t e r m i n e d from the determinate, m~o2N, Q _
(k -
(rn2 P + K)0~2,
m2w2)M
_ m~o2R
= o.
Finally, if the a p p a r a t u s is connected t h r o u g h an elastic shaft to a mass of m o m e n t of interia I an additional degree of freedom 6 is introduced, then ¢---
--k(¢-¢)
=
-
k[-(4,-wt)
-
(¢,-wt)]
=
-k(n-q)
and the equations of oscillation become P~ + (Q - K~02)~ + R~0// = o, Mi/ + N~& = - k(n - q), = - q ) + ¢'. THE DYNAMICAL EQUATIONS FOR A SYSTEM OF VIBRATING DISKS.
We will neglect the gyroscopic forces. T h e polar m o m e n t of inertia t h r o u g h the c. of g. is m k 2. Assume the c. of g. offset from the center of shaft b y h. Let ~ be the torsion angle, i.e., the angle t u r n e d by vector h in the plane of the disk with respect to the y axis. T h e n for a single disk if Yu and zo are the co6rdinates of the c. of g., yl and zl the co6rdinates of the center of the disk,
630
R. ]~KSERGIAN.
[J. F. I.
the kinetic e n e r g y is, T = ½m(k2C + yg2 + ig2)
or, a l t e r n a t e l y , T = ½m((k 2 + h2)6 2 -+- ?)12 -~- il 2 -Jr- 2h6(i1 cos ¢ - #1 sin 4))).
T h e w o r k of the r e a c t i v e forces of the s h a f t a n d w e i g h t of the disk, are -
~V
=
-
Y~yt
-
Z~zl
-
m@zg
+
M~4)
where Yl = Y g - h c o s ¢ , ~Yl = ~Yg + h sin ¢ . 5 ¢ , The equations
of m o t i o n
zl = z ~ - h s i n ¢ , ~Zl = ~zg - h cos 4). 5¢. are, therefore,
with co6rdinates
(y z0¢) ] m?)o= - Y, rn~o= - Z - m g [ ( i ) mk2~ M - Yh sin 4) + Z h cos 4)
or a l t e r n a t e l y , w i t h c o 6 r d i n a t e s (ylz14)) m(91 - h sin 4).~; - h$ ~ cos 4)) = - Y 4) rn(~l + h cos 4).$ - h C sin 4)) = - Z - mg m [ ( k 2 + h2)6 + h(#l cos ¢ - #1 sin ¢) = M - mgh cos
(II)
T h e s y s t e m II is e x p r e s s e d in t e r m s of the c o S r d i n a t e s of t h e shaft. In s y s t e m I, t h e m o m e n t e q u a t i o n can be simplified for a single disk, since - Y 1 = Cyl = C(yg - h cos 4)) a n d - Z1 = - Czl = - C(zg - h sin 4)), so t h a t rnk2~ + rn(~gYg - ~gZg)
=
M - mg.yg,
w h i c h is the e q u a t i o n of a n g u l a r m o m e n t u m a b o u t t h e s h a f t .bearing axis. T h e e q u a t i o n s II s h o w t h a t the lateral a n d torsional v i b r a t i o n s are not i n d e p e n d e n t , e x c e p t for t h e special case w h e n t h e e c c e n t r i c i t y of the disk h is nil. If we neglect t h e w e i g h t of t h e disk, n o t i n g 4) = o~t + w h e r e ~0 = t h e m e a n a n g u l a r v e l o c i t y of t h e shaft, a n d ass u m i n g cos (o~t + ,) = cos 00t (approx.), we h a v e for t h e
May, 1931.]
DYNAMICAL ANALYSIS OF MACHINES.
631
loadings on t h e shaft,
-
Y = - - m ( # l - - ~h sin o~t - h(o~2 + 2o~) cos o~t), Z = - m@l + ~h cos ~t - h(~ 2 + 200~) sin o~t), M = - m ( k 2 + h~)'~ - m h ( ~ l cos o~t - #1 sin wt).
T h e s e e q u a t i o n s offer an i n t e r e s t i n g verification. If we a s s u m e r o t a t i n g axis yl a n d z ~, along, a n d n o r m a l to, h w i t h m o v i n g origin h a v i n g c o 6 r d i n a t e s (ylZl) e v i d e n t l y , t h e accelera t i o n along y~ or h for t h e c. of g. is #1 cos ~0t + ~1 sin o~t - h~ ~ - 2wh~ a n d t h e acceleration n o r m a l to h is ~1 cos o~t -
~1 sin ~0t + hi.
R e s o l v i n g along t h e fixed axis y a n d z t h e acceleration c o m p o n e n t s in t h e a b o v e e q u a t i o n s are o b t a i n e d . E v i d e n t l y t h e m o m e n t e q u a t i o n is t a k e n a b o u t the m o v i n g origin (ylzl) where t h e r e a c t i o n c o m p o n e n t s - Y a n d - Z of t h e s h a f t h a v e no m o m e n t s . F o r a single l o a d e d s h a f t , t h e p o t e n t i a l e n e r g y is, d u e to bending, V = ½all(y1 ~ + Zl 2) = ½ b n ( Y 2 + Z2),
then Y = ally1,
Z
= a11z1
where I
a11 = ~
OV Yx -- O Y -
-
-
b~iY,
,
Zl
=
bllZ.
If we a s s u m e a torsion m o m e n t , M = - c(o~t - ~ ) = - c~ t h e e q u a t i o n s become m i j l -4- a l l Y 1 = m h ( c o 2 -[- 2w~) cos ~ t + m'~h s i n o~t,
m~l + anzl
= rnh(o~ 2 + 2o~) sin o~t -
m ( k 2 + h2)[ + c~ = m h . ( # l
sin wt -
mh'i cos c0t, ~1 cos ~t).
F o r a s y s t e m of v i b r a t i n g disks a t t a c h e d to a m u l t i b e a r i n g s h a f t , t h e kinetic e n e r g y in t e r m s of t h e c 0 6 r d i n a t e s of t h e
632
R. EKSERGIAN.
[J. F. I.
shaft, is T = ½ ~ mm['(km 2 + hmS)~,m2 + Oms + ~,2 + 2hckm 1
× (~m cos Cm - O~ sin ¢~), and the potential energy has the form V = ½Jan(y1 s + zl s) + ass(ys' + Zss) + aa3(ya~ + zas) + 2am(yly2 + zlzs) + 2asa(ysya -]- zsza) + 2aal(yayl + ZaZl) + .. "2 + ½Ecls(~s - ¢1) ~ + cs3. (¢3 - ¢s) s . . . 1.
In general, applied torques act on various disks of the n a t u r e • = M s i n (nwt + a). On s u b s t i t u t i n g ¢ = cot + + ¢0 we have finally the system of equations, m101 + allY1 + alsys . . . = m l h l ( w 2 -1- 2W~l) cos ((2t -~- ~bOl) 2f_ ml[lhl sin (wt + ¢o1),
m20s + a21yl + assys . . . = msh~(oa ~ q - 20a~s) cos (cot --}- dp02) -~- m2"4sh2
sin (cot + Cos),
mlZl + allZl + a12z2 • • • = ~q,lhl(¢aO2 +
2¢2~1) sin
(~ot + (/)Ol) - ~V/l~lhl c o s (cot -t- (j~Ol),
mszs-+- aUlZl -I- Cts2Z= ' ' ' = m2hs(oa2 -1- 20a~S) sin (wt + ¢02) - ms[shs cos (oat + .Cos), m~(k, ~ + hlS)~l + c1~% - ~s) = mlhl(O~ sin (wt + ¢0~) -
~1 cos (wt + ¢01)) + ~ ,
ms(ks ~ + h=')'is + cl:(,s - ,1) + cs~(~s - ,~) = m=h=(i3=sin (wt + ¢0=) -- ~s cos (wt + q~0s) + ~s.
For small eccentricities, h, the lateral and torsional vibrations become i n d e p e n d e n t a n d the equations reduce to the a p p r o x i m a t e form. rnl01 + any1 + alsy2
.
.
.
msOs + aslyl + assys . . . . m ~ l + anz, + al2zs . . . .
.
mlhlW s cos (oat +
¢01),
m2h=ws cos (wt + ~b0s), rnlhloa2 sin (wt + ~01),
May, 1931.]
633
DYNAMICAL ANALYSIS OF ~.{ACHINES.
~rt2~2 -4- a21Zl "+- a22z2 . . . .
mxk12"~l +
c12(~1
-- ~2) =
m2k22"~2 + c12(~2 -
rn2h2co~- sin (cot + 4,o2), (I)1,
~1) + C2a(E2 - ~8) = ¢2, etc.
T h e condition for the whirling speeds is when the determinate all
--
mlco 2,
a21,
a12,
'''
a22 -- m2co2,
'''
= 0
for lateral vibrations, C..~12 --
"~$~Ikl2co2,
C12,
C12,
"'"
C12 + C23 + m~k22co2,
...
= 0
for torsional oscillations. Alternately, we m a y express the potential in terms of the applied forces and r e d u n d a n t reactions as for i n d e t e r m i n a t e bearing reactions. Since t h a t p a r t of the potential due to torsion is i n d e p e n d e n t of the lateral loadings, we will m a i n t a i n it in its previous form. We finally arrive at the lateral displacements as a linear function of the lateral loadings, t h a t is, yl = h l l Y 1 "~- h12Y2 -~- h13Y3 " . . , Y2 = h21Y1 + h22Y2 + h23Y3 " . , Zl = h l l Z 1 + h12Z2 + h13Z3 . . . , z2 = h21Z1 -+- h22Z2 + h2aZ3 . . . ,
where Y1 = - m1(#1 - hi(co2 + 2w[1) cos (cot + 4)01) [1hl sin (cot + ¢01)), -
Z1 = - m,(21 - hi(co2 + 2co~1) sin (cot + ¢01) - ~lhl cos (cot + ¢01), etc. T h e m o m e n t equations retain their previous form. W h e n h is small, in a first approximation and in view of neglecting gyroscopic reactions, we are quite justified in also neglecting the torsional effect on the lateral vibrations. T h e
634
R. EKSERGIAN.
[J, F. I.
e q u a t i o n s simplify t h e n to yl =
-
- ht~o2 cos (~ot + q~ox)) - m2h12(#2 - h~o 2 cos (~ot + q~02) m3h13(#3 - h3co2 cos (cot + 4~03)) " " ,
mlhn(fh
-
y2 =
-
rnlh21(#~ -
hxo~ 2
cos (cot + 4,01)) h20~ 2 cos (~ot + q~02)) m3h23(/)3 - h3~o2 cos (wt + cbo~)),
m2h22(f)2 -
-
-
z~ =
-
mthn(~t
sin (~ot + ~b01)) m2h~2(~2 - h~o ~ sin (~ot + q~02)) rn3h13(~3 - h3~o2 sin (o~t + 4,03)), -
-
h~w ~
-
z2 = - rnlh~l(~l - hl~o2 sin (~ot + 4~01) m ~ h 2 2 ( e 2 - h2~o2 sin (~ot + q~02)) m3h~3(~3 - h3~o2 sin (~ot + 4~03)). -
-
T h e s e e q u a t i o n s m a y be w r i t t e n in t h e f o r m m~hlO~ + m~hl~#~ + m3h~3~3 -
+ y~ = o ~ Q cos (o~t + ~'1),
-
m~h21?~l + m2h~2~)2 + m~h~3#~ - - - - + Y2 = o ~ R cos (o~t + ~,2),
w i t h similar e q u a t i o n s for zl. Q cos (wt + 7) =
mlhnhl
Evidently
cos (~ot + q~ol) + m~h12h2
mlhllhl
t a n "11
mxh~h~
sin q~01 + cos $01 +
m2hi2h2 m2h~2h2
cos (o~t + q~02),
sin ¢02 cos 402 etc.•
T h e condition for t h e whirling speeds is w h e n t h e determinate, ~mlhl~o~ 2 -
I
m2h12w 2
rrt2h21w 2
m2h22¢o2 -
m3h31oo 2
m3h32o~ 2
m3h~30~ 2
I
m3h23¢o 2 m3h33o~ 2 -- I
o.
It is to be n o t e d t h a t with elastic bearings of different e q u i v a l e n t spring c o n s t a n t s in t h e h o r i z o n t a l a n d vertical planes, t h e a's a n d h's are no longer t h e s a m e as coefficients of y a n d z. T h u s a d d i t i o n a l critical speeds will be o b t a i n e d d u e to the necessity of considering b o t h t h e y a n d z e q u a t i o n s .
May, i93i.]
DYNAMICAL ANALYSIS OF ~.IACHINES.
635
RELATIONS OF THE ELASTIC COEFFICIENTS.
The potential energy of the shaft due to the lateral loadings in terms of the displacements has the quadratic form, V = ~1 [ a n y l 2 nt- a~2y22 + a~3Y~2 + 2a12yly~. + 2a23y2y3 "l- 2a81y3y~ " " ] ,
where we note the reciprocal relations a12 = a21, a23 = a~2, etc., the loadings have the linear form, P1
OV Oyl'
a l l y , -k- a12Y2 -t- a233'3 . . . .
=
OV
P z = a2,yl -k- a22y2 Jr- a~3y~ . . . .
Oy2 ' OV
P3 = a31yl --}- a~y2 --k a~3y3 . . . .
Oy3 "
The potential energy, however, may also be expressed in terms of the loadings and the redundant reactions, which can be considered as " r e a c t i o n " co6rdinates. On Castigliano's theorem: If we express the elastic potential in terms of the loadings and the redundant reactions acting on the system, then for small displacements, if P~ . . . P~ are the applied forces and RI . . . R~ are the redundant reactions and moments, V = f ( P 1 " " P , , R1 . . . R,,,),
so that 3 V = O V 6P1 . ' . OPt
OV OV R ... + - ~ 6P , + --~110 ~
OV + -~-R-~ f Rm,
(i)
where otherwise, V = ½(Ply1 " " P~y~ + R , n , . . . Rmnm)
and ~ V = ½EPI~yl + yI~P1 " "
+ P~y~
+ Rl~n~ + nx~R1
...
+ y~P,, -}- R , ~ n m + nr~Rm-].
(2)
636
R. EKSERGIAN.
[J. F. I.
But also ~V = Play1 ' ' .
+ P ~ y n + RlSnl . . . q- Rm~nm.
(3)
Therefore, combining (2) and (3), V = yl~P1 ' ' '
(4)
-]- y,~P,~ --k nl~R1 . . . + n J R m .
On equating (I) and (4) we obtain OPt
Yl
~PI " " -k --
)
nl
OP,
~R1
Y'~
• • • -Jr-
~P'~
OR'--~m - n m
)
~Rm
=
o.
(5)
Since the loadings and redundant reactions are independent, then for any arbitrary variations of the loadings and redundant reactions their coefficients must be nil, so that, yl
OV OPI'
OV OPn'
,Yn
nl
OV OR,~'
nm
OV c)Rm
For constraint redundant reactions, V
=
yl~P1
• • • q- y.~P~
and ~V=
OV OV OV ~ P1 "'" + ~ n ~ P n -[~ ~R I ' ' ' OP~
OV ~R Jv OR----~ m,
so that (0~
Yl ) SPI ' ' ' + ( c) V
OP,~
Y'~ ) 6P" OV
-[-~-~1 ~gx
OV ...-t-0--R--£ ~Rm--o,
and OV Yl "= OPI'
OV Y'~ - OP,~'
OV OR1 = o,
OV ORm = o.
Slope Deflection Theorem
A very simple application is the slope deflection equation of a beam,--
May, 193I.]
637
D Y N A M I C A L ANALYSIS OF M A C H I N E S .
We have the statical relation, M.¢-
Me+S1
= o,
where Ma and M9 are the constraining moments at the ends and S is the shear reaction, 0a and 0~ are the angular deflections at A and B and y the relative deflection of B with respect to A. From the statical relation above, the reactions MA, Me, and S are not independent, so that V can be expressed as a function of any two of the reaction coSrdinates. OV
OV
. .
but ~V = - ~ S ' y -
~ M A ' O A -{- 6M~'OR
= - 6 S . y - ~MA'OA + = -- ~ S . y +
= k
( OMB OMs ) ~ ~MA -~- ~ .~S OB
" ~ A ' O B -- Oa
OS
~ M A + OMB •08" ~S
~
6MA.
013 - - OA
Therefore, 0V
OM~
0V
OS = OS "OB - y,
and
OMI~
O M A = OM,-----~"OR -
OA,
but since M~
= MA
"Jr- S l ,
OV O--S = lOR -- y,
OM~ OMa
OM~ - l; OS
I,
OV OM'-----~a"~-
OB
-
-
OA.
The bending moment, for any section is M a = M + S x so that OM OM = I, ~ - - = N. OMa OS
638
R. ]~KSERGIAN.
[J. F. I.
We have, then,
ov = f
OMdx=fMxdx__ • = lOn-y;
•OS
OS
EI
Mxdx • .
v = lO~ "
08 - OA =
f
-
f
EI
'
Mdx EI
The above shows the process of analysis required in the more complex problems in the application of Castigliano's theorem. In the procedure of a shafting problem, with applied loads P 1 P 2 P ~ ' " and r e d u n d a n t bearing reactions, R , , R , ~ . . . the potential energy has the quadratic form, V = ~(bnP1 1 2 + b22P22 + b33P32 .. . + Cm,"Rra2 -Jr- CnnR n"~ + 2bl~P1P2 + 262~P2P3 + 2b~lP3Pl " " + 2CI,"P1R," + 2C2,"P2R," + 2C3,"P3R," .." + 2CI,~P1R, + 2C2,P2R,~ + 2C3,P3R,~" .C,"nR,"R,...). The r e d u n d a n t reactions are obtained from the conditions, O---V-V= C,"mR," + C,~.R" + CamP~ + C2,"P2 + C3,.P3 = o, OR,. OV = C..R.~ + Cm.R," + C1.P1 + C2.P~ + C3.P~ = o, OR,,
or otherwise, in an actual calculation,
ov oR,"-
f M OM Z
E-I OR,. - °'
ov OR.-
f M oM Z
E I OR. - °"
Where the bending m o m e n t M is expressed in terms of the loadings and r e d u n d a n t reactions• F r o m these equations we obtain the r e d u n d a n t reactions as linear functions of PxP2 etc. T h u s for the case of two red u n d a n t reactions, R," = kmPa + k2,"P2 + ka,"Pa, R,~ = jl,Pa + je,P2 + j3nP3,
May, i93i. ]
639
DYNAMICAL ANALYSIS OF ~'tACHINES.
where
klm
ClmCnn -
C,.n 2 C,.C,.r.
j,n
Cmn2 -
ClnCmn
C..~C..
C2mCnn -- C2nCmn
'
k2m =
Cm. 2 -- C..,mC.. ' etc.
j2.
Cm,~2 -
-- Clr.Cm.
-- C
CmmCn,~ '
mC,..
CmmCnn , etc.
T h e deflections at the loadings P1P:P3 " " are, Yl
OV OPI'
Y2
OV OP2 '
Ya
OV OP3'
or
Yl = b~lP1 -k- bl~.P2 + bx3P3 -+- CI,.R.. -+- CI~Rn, Y2 = b21P1 -t- b22P~ + b23P3 + C2mR~ + C2.R,~, Y3 = b31P1 -Jr- b~2P~ + b33P~ + C3mRm -nt- C 3 . R . .
Eliminating the r e d u n d a n t reactions, we have yl = h n P l + hi:P2 + h13P3, Y2 = h21P1 + h22P2 + h23P3, y~ = h31P1 + ha2P2 + ha3P3,
where hll = bn + Cl,~k,.~ + CI,~jl,~, ht2 = b12 "-[- Clmk2m -~- C I . j : . = b21 @ C2mklm -~ C2njlm = h21; .'. h~2 = h21, in like manner, h:3 = ha2, etc.
T h e relations h12 = h21, h~a = h~,, etc. are also e v i d e n t from a direct application of the reciprocal theorem. In the case of a continuous system, for r e d u n d a n t reactions b e t w e e n a n y t w o parts, we note from the principle of action and reaction b e t w e e n the parts and the c o m m o n displacement of the interface dividing the parts that, OV1
OR
where V =
=~t,
0 V2 --OR
6,
VI-t- V2. 0 V, OR
0 V2 OR
0V --0 OR
for the r e d u n d a n t reaction R b e t w e e n the parts.
640
R.
EKSERGIAN.
[ J . F. I.
In the consideration of elastic bearing supports, we have additional terms for the potential in the form, R2n
"~ 1~ R2m ½C mYm'.-C, ,,
½C~yn2: ½- C~ etc. -,
Treating the shaft and springs of the supports as a continuous system, the r e d u n d a n t reactions are obtained from,
0 (I o (i
) )
OR,. --= --~-t- C,~,~ R,. + C,.,~R,~
OR,~-
-~-t- C ~
-t- ClmP1 -1"- C~,,,P2 + Ca,,,Pa = o,
R ~ - b C~mRm
"Jr- ClnPl "~- C2nP2 + Ca.Pa = o. The displacement at the loadings are
OV Y' - OPt'
OV Y2 = OPt'
OV Ya = OPa"
Substituting for the r e d u n d a n t reactions, we again have the displacements as a linear function of the loadings, i.e.,
yl = hnP1 nt- hl~P2 -t- hlaPa, y2 = h21P, -t- h~2P~ + h2aPa, ya = halP1 -Jr-ha~P2 + haaPa. By a simple transformation we m a y finally express the loadings as a linear function of the displacements, i.e., P1 = a11Yl -~ a12y2 -+- alaya, P2 = a21yl + a2=y2 q- a=aya,
Pa = aalyl -'k aa2y2 4- aaaya, where
I h2~h2a h32h3~ all
-
D
-
'
a22
-
h12h13 ha2haa a12
=
-
-
D
hllhl2 h21h~ I
I hllhla halh3a I D
'
aaa
=
]hnh12 halha21 ,
a2a =
D
,
aal
--
p
h21h22 h~h,2 D
May, I93I. ]
D Y N A M I C A L ANALYSIS OF M A C H I N E S .
641
where
hnh12h13 D
=
h21h22h23
h~,hs~h3~ This procedure can be extended to further degrees of freedom. ENERGY METHOD IN SHAFT ANALYSIS.
It is well known that the whirling speed of a shaft corresponds to the natural frequency of vibration for lateral vibrations. Good approximations for the fundamental frequency are obtained by assuming an arbitrary deflection curve of the shaft and equating the maximum strain energy in its deflected position to the maximum kinetic energy in its neutral position. This method was introduced by Rayleigh, and Morely applied it to the shaft problem, assuming the deflected curve to correspond to its static deflected position. When, however, this method is applied to disks subjected to angular vibrations sufficiently small so as not to affect the deflected curve corresponding to lateral vibrations alone, the natural frequency is found to be lowered whereas actually the whirling speed is raised. If the rotational speed is included in the expression of the kinetic energy for a deflected disk, Chree regarded the whirling speed as that speed for which the frequency of vibration becomes nil, that is, the critical speed at which the shaft ceases to vibrate. Lorenz, F6ppl, Lees, and others consider the critical speed with a slightly unbalanced disk simply as a condition of resonance, that is, when the forced vibration resulting from the centrifugal loading coincides with the natural frequency. We will regard the whirling speed as a condition of kinetic instability. The condition of stability approximating the critical speed means that the total input into the shaft and the work of the internal elastic reactions do not exceed the rotational kinetic energy at the maximum deflected position. If this work exceeds the rotational kinetic energy at the deflected configuration we have the condition of instability. Therefore, the whirling or critical speed corresponds to that
642
R. EKSERGIAN.
[J. F.
I.
speed at which the work done on the shaft from its neutral or undeflected position to its deflected position, including the corresponding change in the elastic potential, is just equal to the rotational kinetic energy alone in the deflected position. We may regard the work-from the neutral to the deflected position to occur approximately at the critical speed. At first, disregarding the angular turning of the planes of the disks, we have d(mo~y 2) ( I dt .o~.dt = d V + E d .I_moo2y2 + - m # 2
2
)
2
'
where the first term is the work of the impressed torque and V is the potential energy of the shaft. Then, o~.d(mo~y 2) = d V + ~ m(o~2ydy + foJ.do~ + f/dft), m(2~o2~dy + y20~do~) = d V + ~ m(o~2ydy + y%~do~ + / ) d / ) ) ; •.
~ mo~2ydy = d V + ~ rm)dft,
and the total work from the neutral to the deflected position, noting the lateral velocity is zero at the extreme positions, is
f0
E rno~2ydy = • . ~ m°~2Y2 -
/0
dV +
/0° ~
mydv;
V - ~ m g y (approx.).
2
2
T h a t is, the work of the centrifugal forces equals the elastic potential energy at its deflected position. If now we consider the angular deflections of the plane of the disks and assume the deflection to occur at synchronous precession, and if k0 = the polar radius of gyration of any disk and k its radius of gyration about a diameter, then for a deflection 0 for any disk, the angular momentum about the shaft aXis is approximately, A.M. = ~ mko%~ cos2 0 + ~ rnk~oo sin 2 0 + ~ mo~y2 = ~m(ko 2-
(ko2 -
k 2) sin 20)00+ ~mooy a.
The impressed torque on the shaft is, therefore, ev = - 2 ~ rnw(ko~ - k s) sin0. c o s 0 . 0 +
~
d ( m o~y 2) dt
May, I93I.]
DYNAMICALANALYSIS OF MACItlNES.
643
The change in kinetic energy from its neutral to its deflected position may be estimated as follows. The kinetic energy at the neutral position is simply 1 E mko%?. At the deflected position the kinetic energy is, 1 ~ mo~Zy2 + ½ ~ ink02002 c o s 2 0 4- ½ ~ mk%~ 2 sin s 0.
The change cf kinetic energy is ½ ~ mo?(y 2 -
(ko2 -
ks) sin" 0).
The equation of energy between the neutral and deflected positions is, therefore, f
¢oMt-
V =
!
E mo?(~-
2
(ko~ -
~
k 2) sin 2 0).
'
But since,
f
¢oMt = -- 2
5
~ m o ? (ko2 - k 2) sin 0 cos OdO+
£
~ d ( m o ? y ~ ) • o~
-- - ~ m o ? ( k o 2 - k 2) sin 2 0 + ~m~02y2; ..
~oPXm(y 2-(ko2-k
2) sin 2 0)= V.
For small displacements sin s 0 = 0s, and for circular disks k02 = 2k 2, and if V is measured by the potential energy corresponding to the static deflection curve, 1 S ~o~ Em(y
2
-
kSO2) = ½ E m g y .
Hence the whirling speed is given by ~°2 =
g ~ my ~ m ( y '2 -- k202)
On the other hand the natural frequency is p2 =
g~my E r n ( y 2 + k202) '
and neglecting the gyroscopic effect, p~ = o? = g E m y ]Emf VOL.
2II,
NO.
1265--44
644
R. F~KSERGIAN.
[J.
F. I.
ON INERTIA COEFFICIENTS AND VELOCITY RATIOS.
The kinetic energy of a dynamical system is a homogeneous quadratic function of the independent co6rdinates. Thus, for two degrees of freedom, T = ½(allq~l~ + a22622 q-2a12q~162), where the coefficients of inertias expressed in partial velocity ratios have the form, all = EM(K2~,I + K ~ , ) + EIK2~,,, a22 = E M ( K ~ , + K2v~,) + EIK2¢,,,, a12 = a21 = E M ( K , , , K ~ , + K,,,K~,,) + ~IKv,,IK~¢, '
where K,,,
Ox = OCj'
Ov Kv,~ - 0~1
and
Ox K~, = 0¢2'
Oy K,,, = 0¢2"
In a dynamical system, we haYe a variety of inertia distributions. The M's m a y refer to the total mass of a m e m b e r and I, its m o m e n t of inertia about its c. of g., so t h a t the velocity ratios are derived from the co6rdinates of the c. of g.'s and the angular displacements of the members. Or we m a y regard the M's as the weighted load's at the joints or pins and the m o m e n t of inertia I = M(ko 2 - z(l - z)), where z is the distance of the c. of g. from either pin and kg is the radius of gyration about the e. of g. For this case the velocity ratios are referred to the pin joints and the angular motion of the members. Let us assume generalized forces applied tO the system of the nature, OV
¢2 = ~ ( X K ~ ,
+ YKv,,) + ~LK,~,~
OV 0¢2'
where the L's are the applied torques or m o m e n t s and V is the potential function not otherwise included in the applied loading. It is to be noted reactions of constraints and internal reactions which have c o m m o n displacements as in rigid bodies, are excluded.
May, I93I. ]
645
DYNAMICAL ANALYSIS OF 5([ACHINES.
From the Lagrangian equations, noting the coefficients of inertias themselves are functions of the independent co6rdinates, we have for two degrees of freedom, the equations of motion expressed in terms of the inertia coefficients,
I Oai 1 ( Oal.2 +
o4~2
I Oa.22)
0(111
2 o¢,
4~~ + - g ~
614~ = ~'~,
I 0a22 -{--
( Oa,2
2I Oall Odp2 612
+ ~Oa~ l .... ~ 6162 ~- (I3'2•
The form of these equations are the basis of the Christoffel notation. We wish to show, however, that the equations in terms of partial velocity ratios previously derived from D'Alembert's principle, can also be derived from a consideration of the inertia coefficients in these equations. Since
an = EM(K2~, + K2,,,) -i- EIK2¢,~,, a22 = ~M(K2~,, + K~,,,) + F_.IK2¢.,,, aa~ = a2t = ZM(K,4,K~ + Kvc,K,,,) + ZIK#~K¢~,, we have
Oa~2 0 O 0¢2 - 0¢2 F.M(K~,,K~ + K,~K~,,) + ~ ~,IK¢,K~,~ = ~M
[(
K~,, 0¢3 + K~,,
)
+ (K,,, OK,,, r/
OK.,
+ x±//K.,-x~-~ + K,,, L\
I 0a22
-- 2I 0¢1 £
EM(
V~2
OK.,]] -ggT,, J' i 0
EM(K2~ + K2"*') + 2 ~ OK.,,
OK,,,,
EIK2~*= EIK~, O0_~,,.
646
R. EKSER(;IAN.
[J. F. I.
Since, however,
OK~, = OK~. _ _ _32x 0¢2 0¢1 0¢10¢2'
3Kv,, _ OKra,, _ 0¢2
0¢1
02y 0¢10¢2
and
0¢2
0¢~
0¢10¢2
we have
0a12 0¢2
OKu¢, I 0a22 [ OK~ 2 0¢~ - ~ M ~ K,,, - - ~ + K~,, 0¢2 ) OK ,,,
In like manner,
0a12 0¢1
I Oau 2 0¢2
-
OKv¢~ [ OK~,, + Kv~,~ 0¢1 ] EM~K~,~ - + y_.IK,;,~ OK¢,, 0¢1
Also, I Oau
2 o¢1 - E M \
(K OKz¢'I K OKv¢l~ *"--b-~ +
0¢----~= 2XM~K'*'-06-~2 +
0K¢¢'1
~I-~T1] + E I K ~ , aC---T' ~*'-~-(-2 ] +
0¢2
and
I 0a22 ( K OKx,, OK~,~ OK~,~ 2 0¢2 - E M \ ~,, a¢-----~+ K ~ * ' - ~ 2 / + EIK**, 0¢-----~' 0a22 ( K OK'v' OKv,,~ OK,,~ 0¢1 = 2 E M \ "*'~0¢1 + Kv,, -~--~// + 2 EIK~,, - -0¢1 • On substituting in the equations of motion in terms of the inertia coefficients, we obtain,
+ OK~,I
OK~,,
May, I931.] DYNAMICALANALYSISOF ~[ACHINES.
647
OKx~, 6162] + ~MK~,,[Kv¢,,¢I + Kv,~ga,,. + 2 0,#2 ,.,4;12 + OKv, . + 2 0Kv,,.0¢24;,4;2-,] + OKv, _ ~_. _ 2_~.4;2 + EIK~,, K,,161 + K~,2$., + - OK¢~ ~ , 4;2 +
K¢~, ]
OK #2~ 4;22 +200¢o'4;~4;2 = * ,
and
OK~,, .2
OK~,~ .,,
+ 2 04)1 4;,4;2 + ~MK~,~ K~,¢, + K~.~,, . ~ -~- 2--0,1 OK~,~ 4;t4;21 -1- OKu~ "--0-~1-14;12 -Jr-~OKv,, (~2 J
- ~ - ~ '4;22 -~- 2 0K¢~¢2 = . 2 ' 4;01,4;~2]1 where
OK,4,, _ OKx,, _ 02x , etc. 0¢2 0¢1 01#1002 In a similar way for three degrees of freedom, r = -}(a1~¢7 + a224;~ ~ + a ~ ¢ ~ 2
+ 2a~¢~¢~ + 2a~6~¢~ + 2a,~$~$~) for the kinetic energy in terms of the inertia coefficients, where as before,
a23 = a~2 = E M ( K , , , K ~ + Kv,=K~**) a33 = F.M(K2,,, + K2vh) + ~IK2,~, etc. From the Lagrangian equations for the qh co6rdinate, noting
648
R. EKSERGIAN.
[J. F. I.
the inertia coefficients are functions of the co6rdinates,
81161 "+- 81262 + 8]363 + I
081,
(08,2
I 08d] .
2
00,13 + \ 0¢3 (0812
I Oaa3~ 0all 2 0¢1! 632 + ~ 6162
Oaz~ + ~-g~ + 0813 --
,0811
= ~1
with similar expressions for ¢2 and ¢3 co6rdinates. To transform the coefficients to the form of velocity ratios, we note
0a12
I 0a22
0¢2
2 0¢1
+ EIK¢~,,
0a13
I 0(l,33
0¢3
2 0¢1
~M(K~,~ OK~"
+
K
of~[.
,
OKv*']
~,1 o¢--71 O¢a
and
0a12 o81. 0¢3 + - -
o8~. _ X~M/'K,
OA,,~
K
°K~*A
-~ ~-~.21K~¢~IOO~-~f2, etc. Therefore, the generalized equation of motion in terms of partial velocity ratios, for three degrees of freedom for the co6rdinate ¢1 is E M K~., K..,¢;1 + K~4;2 + K~,~43 + ~
6, ~
. 2 + OK~, . 2 + 2 OK~, 6142 + 20K~,, 6~6~ + OK~, - - ¢2 0¢2 ~ ¢3 0¢2 0¢3 0~. ] [
May, I93I. ]
649
DYNAMICAL ANALYSIS OF MACHINES.
OKy,1
OK~,,
04)3
OK ~ 3
_
0~1 OK~,~
OK~,~ 4)22 +
4)32
]}
+ 2 OK.%, 4)14,2 + 2 4)~4)3 + 2 4).~4), 04)2 04)3 04)i
= o~
with similar equations for ¢2 and 4)3. Evidently the terms in brackets are the generalized acceleration components and the multiplying partial velocity ratio reduces the generalized inertia loading to the particular cobrdinate under consideration. This latter is analogous to a mechanical varying gear ratio with respect to any particular co6rdinate, for any given inertia element of the mechanism. CONCLUSION.
The subject of dynamics of machinery offers a particular theoretical field for the application of general dynamical methods in a category quite comparable to gyrostatics and similar provinces in dynamics. Considering its practical importance it is fundamental in the field of technical engineering and offers a very unique field in applied physics. A primary object of these articles has been to maintain only the theoretical aspect of the subject as a branch of general dynamics. The development in this work of the dynamical equations directly in terms of velocity ratios and their derivatives for both one and several degrees of freedom is of particular use in the specialized subject of dynamics of machinery. It gives a connecting link between the now separate but well developed field of engineering kinematics and the dynamics of mechanisms. The preliminary analysis, however, offers considerable extension in both analytical and graphical procedures. Emphasis has also been made on the use of redundant co6rdinates with the equations of conditions or constrained velocity relations as a procedure of determining constraint reactions and internal reactions in a mechanism. Other branches of this subject that await considerable
650
R.F.KSERGIAN.
[J. F. I.
development are the application of gyroscopic theory in machinery, mechanical devices, and control apparatus. Also, an important phase is in connection with the oscillations of mechanical electromagnetic systems in control apparatus and with prime movers, synchronous motors, etc., in complex electric power systems. The field of vibration of machinery is under way, but it will always have a fertile field of m a n y applications and further extension in theory. Very closely connected with the subject and really an integral part pertains to the study of electromagnet reactions and fluid dynamics. These articles will quite fulfill their purpose if the presentation of the subject adds some further interest to the possibilities of the subject of dynamics of machinery as a branch of Applied Physics and Technical Engineering. ACKNOWLEDGMENT.
The presentation is an outcome of a proposed dissertation subject by the late Dr. A. G. Webster to w h o m t h e author is greatly indebted for much inspiration on the subject of dynamics and its theoretical applications. Particular acknowledgment is due Dr. R. H. Goddard and Dr. A. W. Duff and also Dr. E. B. Wilson and Dr. C. A. Adams for their criticisms and review of this work in its nucleus form as a portion of a Dissertation Subject. To the late Prof. G. Lanza and in particular to Dr. S. Timoshenko, to Dr. H. N. Davis, Dr. E. V. Huntington, Prof. C. E. Fuller, Dr. H. C. Richards and to Prof. J. C. Riley, the author is also indebted in connection with the discussion, advices, and suggestions on dynamical problems intimately connected with this work. Particular acknowledgment is also due Dr. Howard McClenahan for his interest and for his critical review of the manuscript as a subject for publication as well as for several fundamental suggestions in the presentation of the work. Thanks are also extended to Mr. A. Rigling in connection with the editorial work. Acknowledgment is due Mr. A. Madle for permission to consider the Relay Motor Oscillating axle for illustrating the use of velocity ratios in conjunction with a complex dynamical problem, and Mr. B. H. Slocum for suggestions as to an actual problem on the dynamics of a gear train on a rotating frame.
May, 1931. ]
DYNAMICAL ANALYSIS OF MACHINES.
651
BIBLIOGRAPHY.
I. 2. 3. 4. 5. 6. 7. 8. 9. Io.
1. 2. 3. 4.
Specifically Related to Dynamics of Machinery. Dynamics of Machinery, G. Lanza. John Wiley & Sons, I9I I. Mechanics of Machinery, Le Conte. Macmillan Company. Graphlsche Dynamik, F. Wittenbauer. Julius Springer, Berlin, 1923 Regelung der Kraftmaschinen, Max Tolle. Springer, 1921. M~eanique Appliqu~e aux Machines, J. Boulvin. L. Geisler, ImprimeurEditeur, Paris. The Balancing of Engines, W. E. Dalby. Longmans Green & Company. Theory of Machines, Angus. Kinematics and Kinetics of Machinery, Dent & Harper. J. Wiley & Sons. M~canique, Vol. 3., H. Resal. Cinematique appliquee et Mecanismes, L. Jacob. Octave Doin & Fils, Paris. On Vibrations in Machinery. Technische Schwingungslehre, W. Hort. J. Springer, Berlin. Vibration Problems in Engineering, S. Timoshenko. D. Van Nostrand Company. Die Bereehnung der Drehsehwingungen, H. Holzer. J. Springer. Mechanische Schwingungen, J. Geiger.
Closely Related Works, Containing Problems on the Subject. I. Vorlesungen fiber Technisehe Mechanik, A. Foppl. Vols. IV & VI, B. G. Teubner, Leipzig & Berlin. 2. The Dynamics of Particles and of Rigid, Elastic and Fluid Bodies, A. G. Webster. B.G. Teubner, Leipzig. 3. Technische Mechanik Starrer Gebilde, H. Lorenz. J. Springer, Berlin, 1924. 4. Einfahrung in die Theoretische Physik, C. Schaefer. Walter de Gruyter & Company. 5. Dynamique Appliquee, L. Lecornu. Gaston Doin, Editeur, Paris. 6. Higher Mechanics, H. Lamb. Cambridge Univ. Press. 7. Dynamics of a System of Rigid Bodies, E. J. Routh. Macmillan Company. Vols. I & II. 8. Handbueh der Physik, Vol. V. J. Springer. 9. Handbuch der Physikalischen und Technischen Mechanik, Auerbach & Hort. J. A. Barth, Leipzig. Vol. II. IO. Analytical Dynamics, E. T. Whittaker. Cambridge Univ. Press. I ~. Trait6 de M6canique Rationelle, P. Appell. Gauthier-Villars & Cie, Paris. I2. Theory and Design of Recoil Systems and Gun Carriages, R. Eksergian. U. S. A. Ordnance, Pub. 13. M6canique des Affuts, J. Challeat. O. Doin & Fils. I4. Gyrostatics and Rotational Motion, A. Gray. Macmillan Company• Paper and Articles. I. On the Whirling and Transverse Vibrations of Rotating Shafts. C. Chree. Philosophical Magazine, 19o4. 2. The Lateral Vibration of Loaded Shafts. H . H . Jeffcott. Phil. Mag. 1919.
652
R. EKSERGIAN.
[J. F. I.
3. Torsional Vibrations in the Diesel Engine, F. M. Lewis. Proc. Am. Soc. of Nay. Arch. I925. 4. Oscillations and Resonance in Systems of Parallel Connected Synchronous Machines, H. V. Putman. Journal Franklin Institute. 3. Power Application to Oscillating Axle, A. Madle. Proc. Applied Mechanics Div. A. S. M. E. 193o. 6. Stresses in Locomotive Frames, R. Eksergian. American Society of Mech. Engrs. 7. The Balancing and Dynamic Rail Pressure of Locomotives, R. Eksergian. Proc. R. R. Div. A. S. M. E. I928. 8. Torsional Vibration Dampers, J. P. Den Hartog & J. Ormondroyd. Proc. Applied Mechs. Div. 1929.
E. I. DuPont de Nemours & Company" Member of the Institute. OSCILLATIONS OF A LINKAGE MECHANISM.
We will consider a m e c h a n i s m similar to a governor linkage which is r o t a t e d a b o u t a vertical shaft with angular velocity ¢. T h e linkage consists of four equal bars forming a rhombus, each of length 2a and of mass m and uniformly distributed. T h e lower vertical joint is fixed to the shaft, while the u p p e r vertical joint can slide freely along the shaft. T h e horizontal joints are connected by a spring 1 with m o d u l u s of elasticity u. T h e initial unstrained length of spring is 2c. T h e angle of a n y rod with respect to the vertical is 0, t h a t is the total angle at the b o t t o m or u p p e r joints between the rods is 20. T h e co6rdinates of the s y s t e m are evidently 0 and ¢. T h e kinetic energy can be shown to be, T = ma2[S(sin
2 0 + ~)02 + ~ sin 2 0.62]
and the potential energy due to g r a v i t y and the spring is, V = 8mgacosO+2u
= 8toga
f
2as~nOy _ c
c
°d y
cos 0 + ~-(2a sin 0 -- c) 2. c
T h e m o m e n t u m corresponding to the co6rdinate 0 is OT
0--0- = 2ma2[-S(sin2 0 + ½)0] =
Po
* Extension of a portion of a dissertation for the degree of doctor of philosophy submitted to Clark University, 1928. 1 Constructively, two parallel springs on either side of shaft could be used, or simply a fixed guide shaft at the top, the rotating shaft terminating at lower hinge. C o n c l u d e d from p. 505, vol. 2I I, April, 1931. v O L 2xx, ~o. 1265--43 627
628
R. F~KSERGIAN.
[J. F. I.
and the m o m e n t u m corresponding to the co6rdinate ¢ is 16
OT -
-
O4;
-
3
m
a
2sin 206 = p,,
which is obviously also equal to the angular m o m e n t u m , A.M., about the shaft axis. T h e equations of motion are, 16ma2[-(sin
16
2 0 + ½) 0 + sin 0. cos 0.02-] - - - m a 2 sin 0 cos 0q~2 3 = 8mga sin 0 -- 4au (2a sin 0 -- c) cos 0 c
and I6ma~[-sin2 0.~ + 2 sin 0 cos 0067 = 4. 3 T h e latter equation is obviously the rate of change of A.M. about the shaft axis equal to' the shaft torque 4. If we assume an oscillation about the kinetic equilibrium position, let o0 = the angular velocity corresponding to this position, and 00 the corresponding angle of the rods with respect to the vertical. Then, 0=00+~,
0=~,
~=~0+~.
T h e equilibrium position is given b y the expression, 16 4au (2a sin 00 - c) cos 00 = 8mga sin 00 + - - m a 2 sin 00 cos 00. co2. c 3 Neglecting the displacements and velocities from the equilibrium position of higher order the equations of oscillation are, I6ma 2
(
sin 200 + 3 --
i) [(4o ~+
- - c (2a cos 200 + c sin 00)
8mga c°s O° ) -- I6 - - m a 2 cos 20o.w 2 ] ~ 3 _
(I)
16 rna2 sin 20o~O,~= 0, 3
16 - - m a 2 [ s i n ~ 00.;/ -t- 2 sin 00 cos 00.w~ = 4. 3
(2)
May, 1931.]
629
DYNAMICAL ANALYSIS OF M A C H I N E S .
These equations m a y be written in terms of the coefficients, P, Q, K, R, and M and N. If we suppose an impressed harmonic torque of some multiple of the shaft frequency, a v e r y possible condition in a p p a r a t u s of this type, we have, P~ + (Q -
K,02)~ + R0~o = o,
Mi/ + N~0~ = ~0 cos m~ot. If we let, e = A sin mo~t, ~1 = B cos mo2t, then, A -
-
m°~2R~b° D '
B = - - (Q -
( m 2 P + K)°~2) D
and Imw2N, D =
_
-
( m ~ p + K ) w ~ + Q,
m2o)2M mco~R
•
If the shaft torque • is due to the reaction of a torsional spring coupling, t h e n ¢ = k ( ¢ - o ~ t ) = - k~ and the critical speeds are d e t e r m i n e d from the determinate, m~o2N, Q _
(k -
(rn2 P + K)0~2,
m2w2)M
_ m~o2R
= o.
Finally, if the a p p a r a t u s is connected t h r o u g h an elastic shaft to a mass of m o m e n t of interia I an additional degree of freedom 6 is introduced, then ¢---
--k(¢-¢)
=
-
k[-(4,-wt)
-
(¢,-wt)]
=
-k(n-q)
and the equations of oscillation become P~ + (Q - K~02)~ + R~0// = o, Mi/ + N~& = - k(n - q), = - q ) + ¢'. THE DYNAMICAL EQUATIONS FOR A SYSTEM OF VIBRATING DISKS.
We will neglect the gyroscopic forces. T h e polar m o m e n t of inertia t h r o u g h the c. of g. is m k 2. Assume the c. of g. offset from the center of shaft b y h. Let ~ be the torsion angle, i.e., the angle t u r n e d by vector h in the plane of the disk with respect to the y axis. T h e n for a single disk if Yu and zo are the co6rdinates of the c. of g., yl and zl the co6rdinates of the center of the disk,
630
R. ]~KSERGIAN.
[J. F. I.
the kinetic e n e r g y is, T = ½m(k2C + yg2 + ig2)
or, a l t e r n a t e l y , T = ½m((k 2 + h2)6 2 -+- ?)12 -~- il 2 -Jr- 2h6(i1 cos ¢ - #1 sin 4))).
T h e w o r k of the r e a c t i v e forces of the s h a f t a n d w e i g h t of the disk, are -
~V
=
-
Y~yt
-
Z~zl
-
m@zg
+
M~4)
where Yl = Y g - h c o s ¢ , ~Yl = ~Yg + h sin ¢ . 5 ¢ , The equations
of m o t i o n
zl = z ~ - h s i n ¢ , ~Zl = ~zg - h cos 4). 5¢. are, therefore,
with co6rdinates
(y z0¢) ] m?)o= - Y, rn~o= - Z - m g [ ( i ) mk2~ M - Yh sin 4) + Z h cos 4)
or a l t e r n a t e l y , w i t h c o 6 r d i n a t e s (ylz14)) m(91 - h sin 4).~; - h$ ~ cos 4)) = - Y 4) rn(~l + h cos 4).$ - h C sin 4)) = - Z - mg m [ ( k 2 + h2)6 + h(#l cos ¢ - #1 sin ¢) = M - mgh cos
(II)
T h e s y s t e m II is e x p r e s s e d in t e r m s of the c o S r d i n a t e s of t h e shaft. In s y s t e m I, t h e m o m e n t e q u a t i o n can be simplified for a single disk, since - Y 1 = Cyl = C(yg - h cos 4)) a n d - Z1 = - Czl = - C(zg - h sin 4)), so t h a t rnk2~ + rn(~gYg - ~gZg)
=
M - mg.yg,
w h i c h is the e q u a t i o n of a n g u l a r m o m e n t u m a b o u t t h e s h a f t .bearing axis. T h e e q u a t i o n s II s h o w t h a t the lateral a n d torsional v i b r a t i o n s are not i n d e p e n d e n t , e x c e p t for t h e special case w h e n t h e e c c e n t r i c i t y of the disk h is nil. If we neglect t h e w e i g h t of t h e disk, n o t i n g 4) = o~t + w h e r e ~0 = t h e m e a n a n g u l a r v e l o c i t y of t h e shaft, a n d ass u m i n g cos (o~t + ,) = cos 00t (approx.), we h a v e for t h e
May, 1931.]
DYNAMICAL ANALYSIS OF MACHINES.
631
loadings on t h e shaft,
-
Y = - - m ( # l - - ~h sin o~t - h(o~2 + 2o~) cos o~t), Z = - m@l + ~h cos ~t - h(~ 2 + 200~) sin o~t), M = - m ( k 2 + h~)'~ - m h ( ~ l cos o~t - #1 sin wt).
T h e s e e q u a t i o n s offer an i n t e r e s t i n g verification. If we a s s u m e r o t a t i n g axis yl a n d z ~, along, a n d n o r m a l to, h w i t h m o v i n g origin h a v i n g c o 6 r d i n a t e s (ylZl) e v i d e n t l y , t h e accelera t i o n along y~ or h for t h e c. of g. is #1 cos ~0t + ~1 sin o~t - h~ ~ - 2wh~ a n d t h e acceleration n o r m a l to h is ~1 cos o~t -
~1 sin ~0t + hi.
R e s o l v i n g along t h e fixed axis y a n d z t h e acceleration c o m p o n e n t s in t h e a b o v e e q u a t i o n s are o b t a i n e d . E v i d e n t l y t h e m o m e n t e q u a t i o n is t a k e n a b o u t the m o v i n g origin (ylzl) where t h e r e a c t i o n c o m p o n e n t s - Y a n d - Z of t h e s h a f t h a v e no m o m e n t s . F o r a single l o a d e d s h a f t , t h e p o t e n t i a l e n e r g y is, d u e to bending, V = ½all(y1 ~ + Zl 2) = ½ b n ( Y 2 + Z2),
then Y = ally1,
Z
= a11z1
where I
a11 = ~
OV Yx -- O Y -
-
-
b~iY,
,
Zl
=
bllZ.
If we a s s u m e a torsion m o m e n t , M = - c(o~t - ~ ) = - c~ t h e e q u a t i o n s become m i j l -4- a l l Y 1 = m h ( c o 2 -[- 2w~) cos ~ t + m'~h s i n o~t,
m~l + anzl
= rnh(o~ 2 + 2o~) sin o~t -
m ( k 2 + h2)[ + c~ = m h . ( # l
sin wt -
mh'i cos c0t, ~1 cos ~t).
F o r a s y s t e m of v i b r a t i n g disks a t t a c h e d to a m u l t i b e a r i n g s h a f t , t h e kinetic e n e r g y in t e r m s of t h e c 0 6 r d i n a t e s of t h e
632
R. EKSERGIAN.
[J. F. I.
shaft, is T = ½ ~ mm['(km 2 + hmS)~,m2 + Oms + ~,2 + 2hckm 1
× (~m cos Cm - O~ sin ¢~), and the potential energy has the form V = ½Jan(y1 s + zl s) + ass(ys' + Zss) + aa3(ya~ + zas) + 2am(yly2 + zlzs) + 2asa(ysya -]- zsza) + 2aal(yayl + ZaZl) + .. "2 + ½Ecls(~s - ¢1) ~ + cs3. (¢3 - ¢s) s . . . 1.
In general, applied torques act on various disks of the n a t u r e • = M s i n (nwt + a). On s u b s t i t u t i n g ¢ = cot + + ¢0 we have finally the system of equations, m101 + allY1 + alsys . . . = m l h l ( w 2 -1- 2W~l) cos ((2t -~- ~bOl) 2f_ ml[lhl sin (wt + ¢o1),
m20s + a21yl + assys . . . = msh~(oa ~ q - 20a~s) cos (cot --}- dp02) -~- m2"4sh2
sin (cot + Cos),
mlZl + allZl + a12z2 • • • = ~q,lhl(¢aO2 +
2¢2~1) sin
(~ot + (/)Ol) - ~V/l~lhl c o s (cot -t- (j~Ol),
mszs-+- aUlZl -I- Cts2Z= ' ' ' = m2hs(oa2 -1- 20a~S) sin (wt + ¢02) - ms[shs cos (oat + .Cos), m~(k, ~ + hlS)~l + c1~% - ~s) = mlhl(O~ sin (wt + ¢0~) -
~1 cos (wt + ¢01)) + ~ ,
ms(ks ~ + h=')'is + cl:(,s - ,1) + cs~(~s - ,~) = m=h=(i3=sin (wt + ¢0=) -- ~s cos (wt + q~0s) + ~s.
For small eccentricities, h, the lateral and torsional vibrations become i n d e p e n d e n t a n d the equations reduce to the a p p r o x i m a t e form. rnl01 + any1 + alsy2
.
.
.
msOs + aslyl + assys . . . . m ~ l + anz, + al2zs . . . .
.
mlhlW s cos (oat +
¢01),
m2h=ws cos (wt + ~b0s), rnlhloa2 sin (wt + ~01),
May, 1931.]
633
DYNAMICAL ANALYSIS OF ~.{ACHINES.
~rt2~2 -4- a21Zl "+- a22z2 . . . .
mxk12"~l +
c12(~1
-- ~2) =
m2k22"~2 + c12(~2 -
rn2h2co~- sin (cot + 4,o2), (I)1,
~1) + C2a(E2 - ~8) = ¢2, etc.
T h e condition for the whirling speeds is when the determinate all
--
mlco 2,
a21,
a12,
'''
a22 -- m2co2,
'''
= 0
for lateral vibrations, C..~12 --
"~$~Ikl2co2,
C12,
C12,
"'"
C12 + C23 + m~k22co2,
...
= 0
for torsional oscillations. Alternately, we m a y express the potential in terms of the applied forces and r e d u n d a n t reactions as for i n d e t e r m i n a t e bearing reactions. Since t h a t p a r t of the potential due to torsion is i n d e p e n d e n t of the lateral loadings, we will m a i n t a i n it in its previous form. We finally arrive at the lateral displacements as a linear function of the lateral loadings, t h a t is, yl = h l l Y 1 "~- h12Y2 -~- h13Y3 " . . , Y2 = h21Y1 + h22Y2 + h23Y3 " . , Zl = h l l Z 1 + h12Z2 + h13Z3 . . . , z2 = h21Z1 -+- h22Z2 + h2aZ3 . . . ,
where Y1 = - m1(#1 - hi(co2 + 2w[1) cos (cot + 4)01) [1hl sin (cot + ¢01)), -
Z1 = - m,(21 - hi(co2 + 2co~1) sin (cot + ¢01) - ~lhl cos (cot + ¢01), etc. T h e m o m e n t equations retain their previous form. W h e n h is small, in a first approximation and in view of neglecting gyroscopic reactions, we are quite justified in also neglecting the torsional effect on the lateral vibrations. T h e
634
R. EKSERGIAN.
[J, F. I.
e q u a t i o n s simplify t h e n to yl =
-
- ht~o2 cos (~ot + q~ox)) - m2h12(#2 - h~o 2 cos (~ot + q~02) m3h13(#3 - h3co2 cos (cot + 4~03)) " " ,
mlhn(fh
-
y2 =
-
rnlh21(#~ -
hxo~ 2
cos (cot + 4,01)) h20~ 2 cos (~ot + q~02)) m3h23(/)3 - h3~o2 cos (wt + cbo~)),
m2h22(f)2 -
-
-
z~ =
-
mthn(~t
sin (~ot + ~b01)) m2h~2(~2 - h~o ~ sin (~ot + q~02)) rn3h13(~3 - h3~o2 sin (o~t + 4,03)), -
-
h~w ~
-
z2 = - rnlh~l(~l - hl~o2 sin (~ot + 4~01) m ~ h 2 2 ( e 2 - h2~o2 sin (~ot + q~02)) m3h~3(~3 - h3~o2 sin (~ot + 4~03)). -
-
T h e s e e q u a t i o n s m a y be w r i t t e n in t h e f o r m m~hlO~ + m~hl~#~ + m3h~3~3 -
+ y~ = o ~ Q cos (o~t + ~'1),
-
m~h21?~l + m2h~2~)2 + m~h~3#~ - - - - + Y2 = o ~ R cos (o~t + ~,2),
w i t h similar e q u a t i o n s for zl. Q cos (wt + 7) =
mlhnhl
Evidently
cos (~ot + q~ol) + m~h12h2
mlhllhl
t a n "11
mxh~h~
sin q~01 + cos $01 +
m2hi2h2 m2h~2h2
cos (o~t + q~02),
sin ¢02 cos 402 etc.•
T h e condition for t h e whirling speeds is w h e n t h e determinate, ~mlhl~o~ 2 -
I
m2h12w 2
rrt2h21w 2
m2h22¢o2 -
m3h31oo 2
m3h32o~ 2
m3h~30~ 2
I
m3h23¢o 2 m3h33o~ 2 -- I
o.
It is to be n o t e d t h a t with elastic bearings of different e q u i v a l e n t spring c o n s t a n t s in t h e h o r i z o n t a l a n d vertical planes, t h e a's a n d h's are no longer t h e s a m e as coefficients of y a n d z. T h u s a d d i t i o n a l critical speeds will be o b t a i n e d d u e to the necessity of considering b o t h t h e y a n d z e q u a t i o n s .
May, i93i.]
DYNAMICAL ANALYSIS OF ~.IACHINES.
635
RELATIONS OF THE ELASTIC COEFFICIENTS.
The potential energy of the shaft due to the lateral loadings in terms of the displacements has the quadratic form, V = ~1 [ a n y l 2 nt- a~2y22 + a~3Y~2 + 2a12yly~. + 2a23y2y3 "l- 2a81y3y~ " " ] ,
where we note the reciprocal relations a12 = a21, a23 = a~2, etc., the loadings have the linear form, P1
OV Oyl'
a l l y , -k- a12Y2 -t- a233'3 . . . .
=
OV
P z = a2,yl -k- a22y2 Jr- a~3y~ . . . .
Oy2 ' OV
P3 = a31yl --}- a~y2 --k a~3y3 . . . .
Oy3 "
The potential energy, however, may also be expressed in terms of the loadings and the redundant reactions, which can be considered as " r e a c t i o n " co6rdinates. On Castigliano's theorem: If we express the elastic potential in terms of the loadings and the redundant reactions acting on the system, then for small displacements, if P~ . . . P~ are the applied forces and RI . . . R~ are the redundant reactions and moments, V = f ( P 1 " " P , , R1 . . . R,,,),
so that 3 V = O V 6P1 . ' . OPt
OV OV R ... + - ~ 6P , + --~110 ~
OV + -~-R-~ f Rm,
(i)
where otherwise, V = ½(Ply1 " " P~y~ + R , n , . . . Rmnm)
and ~ V = ½EPI~yl + yI~P1 " "
+ P~y~
+ Rl~n~ + nx~R1
...
+ y~P,, -}- R , ~ n m + nr~Rm-].
(2)
636
R. EKSERGIAN.
[J. F. I.
But also ~V = Play1 ' ' .
+ P ~ y n + RlSnl . . . q- Rm~nm.
(3)
Therefore, combining (2) and (3), V = yl~P1 ' ' '
(4)
-]- y,~P,~ --k nl~R1 . . . + n J R m .
On equating (I) and (4) we obtain OPt
Yl
~PI " " -k --
)
nl
OP,
~R1
Y'~
• • • -Jr-
~P'~
OR'--~m - n m
)
~Rm
=
o.
(5)
Since the loadings and redundant reactions are independent, then for any arbitrary variations of the loadings and redundant reactions their coefficients must be nil, so that, yl
OV OPI'
OV OPn'
,Yn
nl
OV OR,~'
nm
OV c)Rm
For constraint redundant reactions, V
=
yl~P1
• • • q- y.~P~
and ~V=
OV OV OV ~ P1 "'" + ~ n ~ P n -[~ ~R I ' ' ' OP~
OV ~R Jv OR----~ m,
so that (0~
Yl ) SPI ' ' ' + ( c) V
OP,~
Y'~ ) 6P" OV
-[-~-~1 ~gx
OV ...-t-0--R--£ ~Rm--o,
and OV Yl "= OPI'
OV Y'~ - OP,~'
OV OR1 = o,
OV ORm = o.
Slope Deflection Theorem
A very simple application is the slope deflection equation of a beam,--
May, 193I.]
637
D Y N A M I C A L ANALYSIS OF M A C H I N E S .
We have the statical relation, M.¢-
Me+S1
= o,
where Ma and M9 are the constraining moments at the ends and S is the shear reaction, 0a and 0~ are the angular deflections at A and B and y the relative deflection of B with respect to A. From the statical relation above, the reactions MA, Me, and S are not independent, so that V can be expressed as a function of any two of the reaction coSrdinates. OV
OV
. .
but ~V = - ~ S ' y -
~ M A ' O A -{- 6M~'OR
= - 6 S . y - ~MA'OA + = -- ~ S . y +
= k
( OMB OMs ) ~ ~MA -~- ~ .~S OB
" ~ A ' O B -- Oa
OS
~ M A + OMB •08" ~S
~
6MA.
013 - - OA
Therefore, 0V
OM~
0V
OS = OS "OB - y,
and
OMI~
O M A = OM,-----~"OR -
OA,
but since M~
= MA
"Jr- S l ,
OV O--S = lOR -- y,
OM~ OMa
OM~ - l; OS
I,
OV OM'-----~a"~-
OB
-
-
OA.
The bending moment, for any section is M a = M + S x so that OM OM = I, ~ - - = N. OMa OS
638
R. ]~KSERGIAN.
[J. F. I.
We have, then,
ov = f
OMdx=fMxdx__ • = lOn-y;
•OS
OS
EI
Mxdx • .
v = lO~ "
08 - OA =
f
-
f
EI
'
Mdx EI
The above shows the process of analysis required in the more complex problems in the application of Castigliano's theorem. In the procedure of a shafting problem, with applied loads P 1 P 2 P ~ ' " and r e d u n d a n t bearing reactions, R , , R , ~ . . . the potential energy has the quadratic form, V = ~(bnP1 1 2 + b22P22 + b33P32 .. . + Cm,"Rra2 -Jr- CnnR n"~ + 2bl~P1P2 + 262~P2P3 + 2b~lP3Pl " " + 2CI,"P1R," + 2C2,"P2R," + 2C3,"P3R," .." + 2CI,~P1R, + 2C2,P2R,~ + 2C3,P3R,~" .C,"nR,"R,...). The r e d u n d a n t reactions are obtained from the conditions, O---V-V= C,"mR," + C,~.R" + CamP~ + C2,"P2 + C3,.P3 = o, OR,. OV = C..R.~ + Cm.R," + C1.P1 + C2.P~ + C3.P~ = o, OR,,
or otherwise, in an actual calculation,
ov oR,"-
f M OM Z
E-I OR,. - °'
ov OR.-
f M oM Z
E I OR. - °"
Where the bending m o m e n t M is expressed in terms of the loadings and r e d u n d a n t reactions• F r o m these equations we obtain the r e d u n d a n t reactions as linear functions of PxP2 etc. T h u s for the case of two red u n d a n t reactions, R," = kmPa + k2,"P2 + ka,"Pa, R,~ = jl,Pa + je,P2 + j3nP3,
May, i93i. ]
639
DYNAMICAL ANALYSIS OF ~'tACHINES.
where
klm
ClmCnn -
C,.n 2 C,.C,.r.
j,n
Cmn2 -
ClnCmn
C..~C..
C2mCnn -- C2nCmn
'
k2m =
Cm. 2 -- C..,mC.. ' etc.
j2.
Cm,~2 -
-- Clr.Cm.
-- C
CmmCn,~ '
mC,..
CmmCnn , etc.
T h e deflections at the loadings P1P:P3 " " are, Yl
OV OPI'
Y2
OV OP2 '
Ya
OV OP3'
or
Yl = b~lP1 -k- bl~.P2 + bx3P3 -+- CI,.R.. -+- CI~Rn, Y2 = b21P1 -t- b22P~ + b23P3 + C2mR~ + C2.R,~, Y3 = b31P1 -Jr- b~2P~ + b33P~ + C3mRm -nt- C 3 . R . .
Eliminating the r e d u n d a n t reactions, we have yl = h n P l + hi:P2 + h13P3, Y2 = h21P1 + h22P2 + h23P3, y~ = h31P1 + ha2P2 + ha3P3,
where hll = bn + Cl,~k,.~ + CI,~jl,~, ht2 = b12 "-[- Clmk2m -~- C I . j : . = b21 @ C2mklm -~ C2njlm = h21; .'. h~2 = h21, in like manner, h:3 = ha2, etc.
T h e relations h12 = h21, h~a = h~,, etc. are also e v i d e n t from a direct application of the reciprocal theorem. In the case of a continuous system, for r e d u n d a n t reactions b e t w e e n a n y t w o parts, we note from the principle of action and reaction b e t w e e n the parts and the c o m m o n displacement of the interface dividing the parts that, OV1
OR
where V =
=~t,
0 V2 --OR
6,
VI-t- V2. 0 V, OR
0 V2 OR
0V --0 OR
for the r e d u n d a n t reaction R b e t w e e n the parts.
640
R.
EKSERGIAN.
[ J . F. I.
In the consideration of elastic bearing supports, we have additional terms for the potential in the form, R2n
"~ 1~ R2m ½C mYm'.-C, ,,
½C~yn2: ½- C~ etc. -,
Treating the shaft and springs of the supports as a continuous system, the r e d u n d a n t reactions are obtained from,
0 (I o (i
) )
OR,. --= --~-t- C,~,~ R,. + C,.,~R,~
OR,~-
-~-t- C ~
-t- ClmP1 -1"- C~,,,P2 + Ca,,,Pa = o,
R ~ - b C~mRm
"Jr- ClnPl "~- C2nP2 + Ca.Pa = o. The displacement at the loadings are
OV Y' - OPt'
OV Y2 = OPt'
OV Ya = OPa"
Substituting for the r e d u n d a n t reactions, we again have the displacements as a linear function of the loadings, i.e.,
yl = hnP1 nt- hl~P2 -t- hlaPa, y2 = h21P, -t- h~2P~ + h2aPa, ya = halP1 -Jr-ha~P2 + haaPa. By a simple transformation we m a y finally express the loadings as a linear function of the displacements, i.e., P1 = a11Yl -~ a12y2 -+- alaya, P2 = a21yl + a2=y2 q- a=aya,
Pa = aalyl -'k aa2y2 4- aaaya, where
I h2~h2a h32h3~ all
-
D
-
'
a22
-
h12h13 ha2haa a12
=
-
-
D
hllhl2 h21h~ I
I hllhla halh3a I D
'
aaa
=
]hnh12 halha21 ,
a2a =
D
,
aal
--
p
h21h22 h~h,2 D
May, I93I. ]
D Y N A M I C A L ANALYSIS OF M A C H I N E S .
641
where
hnh12h13 D
=
h21h22h23
h~,hs~h3~ This procedure can be extended to further degrees of freedom. ENERGY METHOD IN SHAFT ANALYSIS.
It is well known that the whirling speed of a shaft corresponds to the natural frequency of vibration for lateral vibrations. Good approximations for the fundamental frequency are obtained by assuming an arbitrary deflection curve of the shaft and equating the maximum strain energy in its deflected position to the maximum kinetic energy in its neutral position. This method was introduced by Rayleigh, and Morely applied it to the shaft problem, assuming the deflected curve to correspond to its static deflected position. When, however, this method is applied to disks subjected to angular vibrations sufficiently small so as not to affect the deflected curve corresponding to lateral vibrations alone, the natural frequency is found to be lowered whereas actually the whirling speed is raised. If the rotational speed is included in the expression of the kinetic energy for a deflected disk, Chree regarded the whirling speed as that speed for which the frequency of vibration becomes nil, that is, the critical speed at which the shaft ceases to vibrate. Lorenz, F6ppl, Lees, and others consider the critical speed with a slightly unbalanced disk simply as a condition of resonance, that is, when the forced vibration resulting from the centrifugal loading coincides with the natural frequency. We will regard the whirling speed as a condition of kinetic instability. The condition of stability approximating the critical speed means that the total input into the shaft and the work of the internal elastic reactions do not exceed the rotational kinetic energy at the maximum deflected position. If this work exceeds the rotational kinetic energy at the deflected configuration we have the condition of instability. Therefore, the whirling or critical speed corresponds to that
642
R. EKSERGIAN.
[J. F.
I.
speed at which the work done on the shaft from its neutral or undeflected position to its deflected position, including the corresponding change in the elastic potential, is just equal to the rotational kinetic energy alone in the deflected position. We may regard the work-from the neutral to the deflected position to occur approximately at the critical speed. At first, disregarding the angular turning of the planes of the disks, we have d(mo~y 2) ( I dt .o~.dt = d V + E d .I_moo2y2 + - m # 2
2
)
2
'
where the first term is the work of the impressed torque and V is the potential energy of the shaft. Then, o~.d(mo~y 2) = d V + ~ m(o~2ydy + foJ.do~ + f/dft), m(2~o2~dy + y20~do~) = d V + ~ m(o~2ydy + y%~do~ + / ) d / ) ) ; •.
~ mo~2ydy = d V + ~ rm)dft,
and the total work from the neutral to the deflected position, noting the lateral velocity is zero at the extreme positions, is
f0
E rno~2ydy = • . ~ m°~2Y2 -
/0
dV +
/0° ~
mydv;
V - ~ m g y (approx.).
2
2
T h a t is, the work of the centrifugal forces equals the elastic potential energy at its deflected position. If now we consider the angular deflections of the plane of the disks and assume the deflection to occur at synchronous precession, and if k0 = the polar radius of gyration of any disk and k its radius of gyration about a diameter, then for a deflection 0 for any disk, the angular momentum about the shaft aXis is approximately, A.M. = ~ mko%~ cos2 0 + ~ rnk~oo sin 2 0 + ~ mo~y2 = ~m(ko 2-
(ko2 -
k 2) sin 20)00+ ~mooy a.
The impressed torque on the shaft is, therefore, ev = - 2 ~ rnw(ko~ - k s) sin0. c o s 0 . 0 +
~
d ( m o~y 2) dt
May, I93I.]
DYNAMICALANALYSIS OF MACItlNES.
643
The change in kinetic energy from its neutral to its deflected position may be estimated as follows. The kinetic energy at the neutral position is simply 1 E mko%?. At the deflected position the kinetic energy is, 1 ~ mo~Zy2 + ½ ~ ink02002 c o s 2 0 4- ½ ~ mk%~ 2 sin s 0.
The change cf kinetic energy is ½ ~ mo?(y 2 -
(ko2 -
ks) sin" 0).
The equation of energy between the neutral and deflected positions is, therefore, f
¢oMt-
V =
!
E mo?(~-
2
(ko~ -
~
k 2) sin 2 0).
'
But since,
f
¢oMt = -- 2
5
~ m o ? (ko2 - k 2) sin 0 cos OdO+
£
~ d ( m o ? y ~ ) • o~
-- - ~ m o ? ( k o 2 - k 2) sin 2 0 + ~m~02y2; ..
~oPXm(y 2-(ko2-k
2) sin 2 0)= V.
For small displacements sin s 0 = 0s, and for circular disks k02 = 2k 2, and if V is measured by the potential energy corresponding to the static deflection curve, 1 S ~o~ Em(y
2
-
kSO2) = ½ E m g y .
Hence the whirling speed is given by ~°2 =
g ~ my ~ m ( y '2 -- k202)
On the other hand the natural frequency is p2 =
g~my E r n ( y 2 + k202) '
and neglecting the gyroscopic effect, p~ = o? = g E m y ]Emf VOL.
2II,
NO.
1265--44
644
R. F~KSERGIAN.
[J.
F. I.
ON INERTIA COEFFICIENTS AND VELOCITY RATIOS.
The kinetic energy of a dynamical system is a homogeneous quadratic function of the independent co6rdinates. Thus, for two degrees of freedom, T = ½(allq~l~ + a22622 q-2a12q~162), where the coefficients of inertias expressed in partial velocity ratios have the form, all = EM(K2~,I + K ~ , ) + EIK2~,,, a22 = E M ( K ~ , + K2v~,) + EIK2¢,,,, a12 = a21 = E M ( K , , , K ~ , + K,,,K~,,) + ~IKv,,IK~¢, '
where K,,,
Ox = OCj'
Ov Kv,~ - 0~1
and
Ox K~, = 0¢2'
Oy K,,, = 0¢2"
In a dynamical system, we haYe a variety of inertia distributions. The M's m a y refer to the total mass of a m e m b e r and I, its m o m e n t of inertia about its c. of g., so t h a t the velocity ratios are derived from the co6rdinates of the c. of g.'s and the angular displacements of the members. Or we m a y regard the M's as the weighted load's at the joints or pins and the m o m e n t of inertia I = M(ko 2 - z(l - z)), where z is the distance of the c. of g. from either pin and kg is the radius of gyration about the e. of g. For this case the velocity ratios are referred to the pin joints and the angular motion of the members. Let us assume generalized forces applied tO the system of the nature, OV
¢2 = ~ ( X K ~ ,
+ YKv,,) + ~LK,~,~
OV 0¢2'
where the L's are the applied torques or m o m e n t s and V is the potential function not otherwise included in the applied loading. It is to be noted reactions of constraints and internal reactions which have c o m m o n displacements as in rigid bodies, are excluded.
May, I93I. ]
645
DYNAMICAL ANALYSIS OF 5([ACHINES.
From the Lagrangian equations, noting the coefficients of inertias themselves are functions of the independent co6rdinates, we have for two degrees of freedom, the equations of motion expressed in terms of the inertia coefficients,
I Oai 1 ( Oal.2 +
o4~2
I Oa.22)
0(111
2 o¢,
4~~ + - g ~
614~ = ~'~,
I 0a22 -{--
( Oa,2
2I Oall Odp2 612
+ ~Oa~ l .... ~ 6162 ~- (I3'2•
The form of these equations are the basis of the Christoffel notation. We wish to show, however, that the equations in terms of partial velocity ratios previously derived from D'Alembert's principle, can also be derived from a consideration of the inertia coefficients in these equations. Since
an = EM(K2~, + K2,,,) -i- EIK2¢,~,, a22 = ~M(K2~,, + K~,,,) + F_.IK2¢.,,, aa~ = a2t = ZM(K,4,K~ + Kvc,K,,,) + ZIK#~K¢~,, we have
Oa~2 0 O 0¢2 - 0¢2 F.M(K~,,K~ + K,~K~,,) + ~ ~,IK¢,K~,~ = ~M
[(
K~,, 0¢3 + K~,,
)
+ (K,,, OK,,, r/
OK.,
+ x±//K.,-x~-~ + K,,, L\
I 0a22
-- 2I 0¢1 £
EM(
V~2
OK.,]] -ggT,, J' i 0
EM(K2~ + K2"*') + 2 ~ OK.,,
OK,,,,
EIK2~*= EIK~, O0_~,,.
646
R. EKSER(;IAN.
[J. F. I.
Since, however,
OK~, = OK~. _ _ _32x 0¢2 0¢1 0¢10¢2'
3Kv,, _ OKra,, _ 0¢2
0¢1
02y 0¢10¢2
and
0¢2
0¢~
0¢10¢2
we have
0a12 0¢2
OKu¢, I 0a22 [ OK~ 2 0¢~ - ~ M ~ K,,, - - ~ + K~,, 0¢2 ) OK ,,,
In like manner,
0a12 0¢1
I Oau 2 0¢2
-
OKv¢~ [ OK~,, + Kv~,~ 0¢1 ] EM~K~,~ - + y_.IK,;,~ OK¢,, 0¢1
Also, I Oau
2 o¢1 - E M \
(K OKz¢'I K OKv¢l~ *"--b-~ +
0¢----~= 2XM~K'*'-06-~2 +
0K¢¢'1
~I-~T1] + E I K ~ , aC---T' ~*'-~-(-2 ] +
0¢2
and
I 0a22 ( K OKx,, OK~,~ OK~,~ 2 0¢2 - E M \ ~,, a¢-----~+ K ~ * ' - ~ 2 / + EIK**, 0¢-----~' 0a22 ( K OK'v' OKv,,~ OK,,~ 0¢1 = 2 E M \ "*'~0¢1 + Kv,, -~--~// + 2 EIK~,, - -0¢1 • On substituting in the equations of motion in terms of the inertia coefficients, we obtain,
+ OK~,I
OK~,,
May, I931.] DYNAMICALANALYSISOF ~[ACHINES.
647
OKx~, 6162] + ~MK~,,[Kv¢,,¢I + Kv,~ga,,. + 2 0,#2 ,.,4;12 + OKv, . + 2 0Kv,,.0¢24;,4;2-,] + OKv, _ ~_. _ 2_~.4;2 + EIK~,, K,,161 + K~,2$., + - OK¢~ ~ , 4;2 +
K¢~, ]
OK #2~ 4;22 +200¢o'4;~4;2 = * ,
and
OK~,, .2
OK~,~ .,,
+ 2 04)1 4;,4;2 + ~MK~,~ K~,¢, + K~.~,, . ~ -~- 2--0,1 OK~,~ 4;t4;21 -1- OKu~ "--0-~1-14;12 -Jr-~OKv,, (~2 J
- ~ - ~ '4;22 -~- 2 0K¢~¢2 = . 2 ' 4;01,4;~2]1 where
OK,4,, _ OKx,, _ 02x , etc. 0¢2 0¢1 01#1002 In a similar way for three degrees of freedom, r = -}(a1~¢7 + a224;~ ~ + a ~ ¢ ~ 2
+ 2a~¢~¢~ + 2a~6~¢~ + 2a,~$~$~) for the kinetic energy in terms of the inertia coefficients, where as before,
a23 = a~2 = E M ( K , , , K ~ + Kv,=K~**) a33 = F.M(K2,,, + K2vh) + ~IK2,~, etc. From the Lagrangian equations for the qh co6rdinate, noting
648
R. EKSERGIAN.
[J. F. I.
the inertia coefficients are functions of the co6rdinates,
81161 "+- 81262 + 8]363 + I
081,
(08,2
I 08d] .
2
00,13 + \ 0¢3 (0812
I Oaa3~ 0all 2 0¢1! 632 + ~ 6162
Oaz~ + ~-g~ + 0813 --
,0811
= ~1
with similar expressions for ¢2 and ¢3 co6rdinates. To transform the coefficients to the form of velocity ratios, we note
0a12
I 0a22
0¢2
2 0¢1
+ EIK¢~,,
0a13
I 0(l,33
0¢3
2 0¢1
~M(K~,~ OK~"
+
K
of~[.
,
OKv*']
~,1 o¢--71 O¢a
and
0a12 o81. 0¢3 + - -
o8~. _ X~M/'K,
OA,,~
K
°K~*A
-~ ~-~.21K~¢~IOO~-~f2, etc. Therefore, the generalized equation of motion in terms of partial velocity ratios, for three degrees of freedom for the co6rdinate ¢1 is E M K~., K..,¢;1 + K~4;2 + K~,~43 + ~
6, ~
. 2 + OK~, . 2 + 2 OK~, 6142 + 20K~,, 6~6~ + OK~, - - ¢2 0¢2 ~ ¢3 0¢2 0¢3 0~. ] [
May, I93I. ]
649
DYNAMICAL ANALYSIS OF MACHINES.
OKy,1
OK~,,
04)3
OK ~ 3
_
0~1 OK~,~
OK~,~ 4)22 +
4)32
]}
+ 2 OK.%, 4)14,2 + 2 4)~4)3 + 2 4).~4), 04)2 04)3 04)i
= o~
with similar equations for ¢2 and 4)3. Evidently the terms in brackets are the generalized acceleration components and the multiplying partial velocity ratio reduces the generalized inertia loading to the particular cobrdinate under consideration. This latter is analogous to a mechanical varying gear ratio with respect to any particular co6rdinate, for any given inertia element of the mechanism. CONCLUSION.
The subject of dynamics of machinery offers a particular theoretical field for the application of general dynamical methods in a category quite comparable to gyrostatics and similar provinces in dynamics. Considering its practical importance it is fundamental in the field of technical engineering and offers a very unique field in applied physics. A primary object of these articles has been to maintain only the theoretical aspect of the subject as a branch of general dynamics. The development in this work of the dynamical equations directly in terms of velocity ratios and their derivatives for both one and several degrees of freedom is of particular use in the specialized subject of dynamics of machinery. It gives a connecting link between the now separate but well developed field of engineering kinematics and the dynamics of mechanisms. The preliminary analysis, however, offers considerable extension in both analytical and graphical procedures. Emphasis has also been made on the use of redundant co6rdinates with the equations of conditions or constrained velocity relations as a procedure of determining constraint reactions and internal reactions in a mechanism. Other branches of this subject that await considerable
650
R.F.KSERGIAN.
[J. F. I.
development are the application of gyroscopic theory in machinery, mechanical devices, and control apparatus. Also, an important phase is in connection with the oscillations of mechanical electromagnetic systems in control apparatus and with prime movers, synchronous motors, etc., in complex electric power systems. The field of vibration of machinery is under way, but it will always have a fertile field of m a n y applications and further extension in theory. Very closely connected with the subject and really an integral part pertains to the study of electromagnet reactions and fluid dynamics. These articles will quite fulfill their purpose if the presentation of the subject adds some further interest to the possibilities of the subject of dynamics of machinery as a branch of Applied Physics and Technical Engineering. ACKNOWLEDGMENT.
The presentation is an outcome of a proposed dissertation subject by the late Dr. A. G. Webster to w h o m t h e author is greatly indebted for much inspiration on the subject of dynamics and its theoretical applications. Particular acknowledgment is due Dr. R. H. Goddard and Dr. A. W. Duff and also Dr. E. B. Wilson and Dr. C. A. Adams for their criticisms and review of this work in its nucleus form as a portion of a Dissertation Subject. To the late Prof. G. Lanza and in particular to Dr. S. Timoshenko, to Dr. H. N. Davis, Dr. E. V. Huntington, Prof. C. E. Fuller, Dr. H. C. Richards and to Prof. J. C. Riley, the author is also indebted in connection with the discussion, advices, and suggestions on dynamical problems intimately connected with this work. Particular acknowledgment is also due Dr. Howard McClenahan for his interest and for his critical review of the manuscript as a subject for publication as well as for several fundamental suggestions in the presentation of the work. Thanks are also extended to Mr. A. Rigling in connection with the editorial work. Acknowledgment is due Mr. A. Madle for permission to consider the Relay Motor Oscillating axle for illustrating the use of velocity ratios in conjunction with a complex dynamical problem, and Mr. B. H. Slocum for suggestions as to an actual problem on the dynamics of a gear train on a rotating frame.
May, 1931. ]
DYNAMICAL ANALYSIS OF MACHINES.
651
BIBLIOGRAPHY.
I. 2. 3. 4. 5. 6. 7. 8. 9. Io.
1. 2. 3. 4.
Specifically Related to Dynamics of Machinery. Dynamics of Machinery, G. Lanza. John Wiley & Sons, I9I I. Mechanics of Machinery, Le Conte. Macmillan Company. Graphlsche Dynamik, F. Wittenbauer. Julius Springer, Berlin, 1923 Regelung der Kraftmaschinen, Max Tolle. Springer, 1921. M~eanique Appliqu~e aux Machines, J. Boulvin. L. Geisler, ImprimeurEditeur, Paris. The Balancing of Engines, W. E. Dalby. Longmans Green & Company. Theory of Machines, Angus. Kinematics and Kinetics of Machinery, Dent & Harper. J. Wiley & Sons. M~canique, Vol. 3., H. Resal. Cinematique appliquee et Mecanismes, L. Jacob. Octave Doin & Fils, Paris. On Vibrations in Machinery. Technische Schwingungslehre, W. Hort. J. Springer, Berlin. Vibration Problems in Engineering, S. Timoshenko. D. Van Nostrand Company. Die Bereehnung der Drehsehwingungen, H. Holzer. J. Springer. Mechanische Schwingungen, J. Geiger.
Closely Related Works, Containing Problems on the Subject. I. Vorlesungen fiber Technisehe Mechanik, A. Foppl. Vols. IV & VI, B. G. Teubner, Leipzig & Berlin. 2. The Dynamics of Particles and of Rigid, Elastic and Fluid Bodies, A. G. Webster. B.G. Teubner, Leipzig. 3. Technische Mechanik Starrer Gebilde, H. Lorenz. J. Springer, Berlin, 1924. 4. Einfahrung in die Theoretische Physik, C. Schaefer. Walter de Gruyter & Company. 5. Dynamique Appliquee, L. Lecornu. Gaston Doin, Editeur, Paris. 6. Higher Mechanics, H. Lamb. Cambridge Univ. Press. 7. Dynamics of a System of Rigid Bodies, E. J. Routh. Macmillan Company. Vols. I & II. 8. Handbueh der Physik, Vol. V. J. Springer. 9. Handbuch der Physikalischen und Technischen Mechanik, Auerbach & Hort. J. A. Barth, Leipzig. Vol. II. IO. Analytical Dynamics, E. T. Whittaker. Cambridge Univ. Press. I ~. Trait6 de M6canique Rationelle, P. Appell. Gauthier-Villars & Cie, Paris. I2. Theory and Design of Recoil Systems and Gun Carriages, R. Eksergian. U. S. A. Ordnance, Pub. 13. M6canique des Affuts, J. Challeat. O. Doin & Fils. I4. Gyrostatics and Rotational Motion, A. Gray. Macmillan Company• Paper and Articles. I. On the Whirling and Transverse Vibrations of Rotating Shafts. C. Chree. Philosophical Magazine, 19o4. 2. The Lateral Vibration of Loaded Shafts. H . H . Jeffcott. Phil. Mag. 1919.
652
R. EKSERGIAN.
[J. F. I.
3. Torsional Vibrations in the Diesel Engine, F. M. Lewis. Proc. Am. Soc. of Nay. Arch. I925. 4. Oscillations and Resonance in Systems of Parallel Connected Synchronous Machines, H. V. Putman. Journal Franklin Institute. 3. Power Application to Oscillating Axle, A. Madle. Proc. Applied Mechanics Div. A. S. M. E. 193o. 6. Stresses in Locomotive Frames, R. Eksergian. American Society of Mech. Engrs. 7. The Balancing and Dynamic Rail Pressure of Locomotives, R. Eksergian. Proc. R. R. Div. A. S. M. E. I928. 8. Torsional Vibration Dampers, J. P. Den Hartog & J. Ormondroyd. Proc. Applied Mechs. Div. 1929.