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wheel system vibration under impulsive boundary excitation
Journal
ofSound
and Vibration
BAND/WHEEL
(1987)
115(2),
SYSTEM
203-216
VIBRATION
BOUNDARY
UNDER
IMPULSIVE
EXCITATION
K. W. WANG~ AND C. D. MOTE, JR. Department of Mechanical Engineering,
University of California, Berkeley, California, U.S.A.
(Received 9 July 1986) Measurements of band vibration undertaken here show that passage of the butt weld connecting the ends of continuous band over wheels excites vibration in the band/wheel system. A displacement impulse occurs each time the weld initially contacts and separates from the wheels. The excitation is periodic, and it can excite instability. The vibration and stability of the coupled band/wheel system under impulsive boundary displacements are analyzed in this paper. The theoretical and experimental findings show that resonance occurs in the system when the weld passage (impulse) period is an integer multiple of any system natural period. 1. INTRODUCTION AND MOTIVATION
The vibration of high speed, axially moving continuous bands is usually detrimental to their function. Examples of axially moving bands, driven by rotation wheels, include band saws, belts, chains, and magnetic tapes. Excitation of transverse vibration caused by the interactions of the band and the wheels is common and significant. Wheel eccentricities and band imperfections are major sources of these excitations. The contributions of wheel eccentricities have been discussed in the literature [ 1,2]. However, the excitation caused by thickness and stiffness imperfections in the continuous band running over rotating wheels has not received attention. An example of this excitation is the passage of the butt weld, connecting the ends of the metal belt or band saw blade, over the wheels. An impulsive input to the transverse vibration of the spans is observed as the weld enters and exits the free spans between the wheels and as it passes over bearing surfaces. Experiments, conducted on a table-top test stand, were carried out to examine the transverse vibration of the band under the excitations of weld passage. A schematic of the experimental set-up is shown in Figure 1. For purposes of discussion, one span is called the principal span, and the other is called the secondary span. The test stand consists of two, 200 mm diameter wheels and an axle to axle wheel separation of 860 mm. The band cross-section is rectangular, 0.5 mm x 19 mm, and the weld bump is 0.02 mm in height and l-5 mm in width. The times when the weld enters and exits the secondary span are detected with a photo-sensor. A displacement probe is positioned at the midsecondary span to monitor its transverse vibration. A spectral analyzer (HP5423A) is used for data acquisition and processing. The mid-span transverse vibration of the secondary span is shown in Figure 2. The upper trace shows the transverse vibration and the impulsive signals in the lower trace indicate times when the weld enters and exits the secondary span. The data shows the i Presently with Power Systems Michigan 48090-9055, U.S.A.
Research
Department,
General
Motors
Research
Laboratories,
Warren,
203 0022-460X/87/110203
+ 14 %03.00/O
0
1987 Academic
Press Inc. (London)
Limited
204
K. W.
WANG
AND
C.
I
D. MOTE,
JR
I
HP5423A Structure
onolyzer
Figure 1. Schematic of experimental set-up for observing the impulsive excitation caused by weld passage. 0, Weld; 0, reflector monitoring weld entering second span; I::, reflector monitoring weld leaving secondary span.
s
0.15
0
9
I
I
I
‘<
1
I
I
0.10
L
0.05
;E
0.00
gs g’ 0
-0.05
z Z
-0.10
8
-0.15
Weld
Weld
entering
E
leaving
span
spon
1.0.
/
I
.
.g
b
0.8 -
E-
?1f0
0.6
_
0.4
-
=> 0’
P
*
,o 0 p’
0.2 I 0.00
I 0.50
I 0.25
I 190
I 0.75 Time
I I .25
-:
I.50
($1
Figure 2. Measured band transverse response under weld path impulsive excitation.
transverse vibration from the impulsive excitation of the weld passage. These excitations occur periodically. The dynamic response of the band/wheel system is analyzed here by treating the weld disturbance as an impulsive displacement boundary excitation of the spans. 2. FORMULATION
The linearized, non-dimensional derived in reference [3], are ii+2SZ-@“-y(w*‘w’)‘=o,
equations of motion of a free band/wheel ; - 2& - &jv - y( \i*‘$‘)’ = 0,
system, (1,2)
IMPULSIVELY
EXCITED
BAND/WHEEL
205
SYSTEM
w + 2$’ - @‘I+ W’m- y(w*‘u’)‘_ r(u*‘w’)‘_$@( W*‘2Wfjf2w*‘w*“w’) = 0,
(3)
6 _ 2s$;’_ $$I+ $,U,_ y(@*I$)I_ r(3*‘~‘)‘_~(Y(~*~2~,,r+2~*f~t”~i(’) =O,
(4)
where C, u, G, and w are the non-dimensional longitudinal and transverse vibrations of the principal and secondary spans, respectively. The superscript * denotes the transverse and longitudinal displacements associated with the equilibrium configuration of the two spans. The superscript dot ( ’) and prime (‘) denote partial derivatives with respect to the non-dimensional time and space variables, respectively. The boundary conditions are (J,/2p:)(ii(o,
T)G(O,
T))_P(U’(O, 7)-LT’(O, 7))
- r(w*‘(o)w’(o, (M,/2)(ii(O, -
T)+i(o,
~(W*‘(o)W’(o,
7) - G*‘(o)G’(o, 7)) = 0,
T))+(KJ2)(u(O, 7)+6*‘(0)6’(0,
(J,/2&)@(1,
(5)
T)+C(o, 7))--p(u’(O,
T)+C’(o, 7))
T))=o,
T)-$1,
(6)
T))+p(U’(l,
T)-;‘(I,
7))
(7)
+y(W*‘(l)W’(l,T)-~*‘(l)+‘(l,T))=o,
(k&/2)(1(1,
T)+k(l,
T))+(&/2)(U(1,
T)+Zi(l,
T))+p(U’(l,
T)+G’(l,
7))
+y(W*‘(l)W’(l,T)+i*‘(l)i’(l,T))=o, W”(0,
W(0,
7)
fi(O,~)=;
=
-
3
n=I
(8) 7)
=
wo 6( 7 -
6”(0,
n&a),
T)
= 0,
W”(1,
w(~,T)=-?
=
+‘(I,
7)
=o,
“=I
T)=
3
(9,lO)
WO~(T-T,~-(PI-~)~,,,),
(11, 12)
w,~(T-T~~-(?I-~)~,).
(13,
n=, i(1,
W,6(~-~,~-(n-l)p,),
7)
14)
n=,
parameters and variables are illustrated in Figure 3 and defined in the nomenclature (Appendix 2). Equations (5)-(8) describe the rotation and translation of the wheels, and equations (9) and (10) state that the variation of the curvature of band of the end points, during oscillation about equilibrium, vanishes. Equations (1 l)-( 14) express the impulsive excitation of the band ends at the wheel contacts. The 6( ’) is the delta function, w. is the height of the weld, T@represents the time for the weld to travel from point i to point j in Figure 3, p,,, is the period of the weld circumnavigation cycle, and N,,, is the number of these cycles. The
I
L Weld path entering secondary span ot T=O
A-A Figure 3. Schematic
of the axially moving band and rotating
wheel system model.
206
K.
W.
WANG
AND
C’
3. SOLUTION
I).
MOTE,
JR.
METHODS
A formulation for discretization is obtained by transforming the non-homogeneous boundary conditions (11)-( 14) to a homogeneous form by a change of variable [4]. Let W(.&T)=U(&T)-(1-t)
%;
W,,fi(T-n&)-c
W,6(7-T,2-(n-l)&)
‘>
,I =,
(15)
n-l
and
+t
Upon substitution motion become
3 n=,
of expressions
=yW*”
3
w,~(T-T,~--(n--l)&).
(16)
(15) and (16) into equations
w~6(T-~jl,)-
3
( II=, ;
_
2&
-
4,“_
y(
(l)-( 14), the equations
wos(T-Tn-(n-l)!‘,)
(17)
,
n-l
of
)
G*‘;‘)’
N” =
+,*‘I
? ( fl=l
WO~(T-T,~-(~-~)P,)-
~+2s~‘-4v”+v”“-
1 n=,
y(w*‘u’)‘-
=(1-t)
c
,
w&T-Up,)+5
3
W&T-T,2-(n-l)&)
n=,
3
W,,~(T--np,)-
2
n=,
Woii(~-~,2-(n-l)p,)
“=,
+ yu*”
3 W,~(T-T,~-(~-~)P,)
w,~(T--np,)-
II=,
+3aw*'w*"
>
NM W,,@T--pm)-
c
W,6(T-T,z-(?I-I)&)
_
2&l
_
#g”
+
,-I,,, _y($*,qr_
(19)
=o,
n=l ;
(18)
y(U*‘U’)‘_~~(W*rzUI,+2W*IW*“V))
“=,
-2s
wos(T-TM-(n-l)&)
>
Y(~*I~f)I_5~(~*r2~ff+2~*‘~*“~‘)
N” =(,$-I)
c
W&T-T,.,-(n-l)&)+
3
n=,
-2s
5
Wob(~-~714-(n-l)p,
( n=l
-3
n=,
>
Wd(T--,j+-1)Pn.))
- y;*” ( _
Woii(~-~,,-(n--l)p,,,)
n=,
:
Wo6(7’-7,4-(n-l)p,)-
II=,
3&*‘$*”
2
W,6(T-T,3-(n-l)p,)
tl=,
> Y%
W,8(7-~,~-(n-l)p,,)-
1
~,8(~--,~-(n-l)p,)
=O,
(20)
IMPULSIVELY
EXCITED
BAND/WHEEL
207
SYSTEM
and the boundary conditions become
-
$*I
2 w,8(T-T*4-(n-l)p,)+~*'
F %~(~-~13--(~-lhL)
?I=1
“=I
(&/2PzR)(W,
T)-Q,
T))+p(u’(l,
T)-u”(1,
T))+ y(w*‘(l)v’(l,
)
T)-i*‘(l)u”(l,
3
(22) 7))
(23)
= y
(
-w*’
+;*+,
3 W,S(T-rInp,)+w*’ n=l
3 WoS(7-7,*-(n-l)&) “=I
wo6(T-T*4-(n-l)p,)-3*’
u”(0, T) = 3”(0, T) = 0, u(0, 7) = 0,
u(l,7)=0,
3
w,8(T-Tll-(n-l)pw)),
II=1
(24)
o”(l,T)=P(l,T)=o,
(25,26)
$0, 7) = 0,
(27-30)
??(1,7)=0.
Because an exact solution of equations (17)-(30) cannot be obtained in general, a Galerkin discretization of a mixed weak form is pursued [3]. U, 6, u, and u’are expanded as
u(59T, =
c”q(5)PjtT),
j=l
u(CT, 7) =
F y(if)qj(T),
(31-34)
j=l
wherepj(
T), qj( T),jj( T) and ij( T) are generalized co-ordinates. Uj(t), C&(S)are admissible functions. Vj(,$), F(t) are comparison functions which satisfy the boundary conditions (25)-(30). Here we choose y(.$)=_q(t)=sin (j~r[), which are the eigenfunctions of a simple-simple beam, and Uj([) = Uj(e) = cos (~TT[), which are the eigenfunctions of a free-free rod.
208
K. W. WANG AND C.
D. MOTE.
JR
discretized weak form is obtained from the inner product of the expansion functions with the domain and boundary residuals derived from equations (17)-(30): The
M~;(T)+G~~(T)+K~(T)=P(~).
(35)
The vector P(T) is obtained from the inner product of the expansion functions with the right-hand side of (17)-(30). The elements of M, G, K, and Y(T) are defined in Appendix 1.
4. STABILITY
ANALYSIS
In the state vector form, equation (35) becomes i(~)=Ar(r)+p,,(~),
(36)
r(r) = r!Jr)+r,(r),
(37)
where
The solution of equation (36) is
where rh( 7) = eA’r(0) is the free response under initial condition r(O), and T rr( 7) = eA7 eFAeP,( 6) de I0
(38)
(39)
is the forced response from PO(~). Assume the solution of i(r) = Ar(r) is periodic (stable), and the excitation PO(~) is periodic. Let the natural periods for each mode be T, , T2, . . . , Tk, and the impulse period be p,,,. The modes 1 through k include all vibration modes contributing to the band response under all excitations in the process. It is assumed that there exist integers m, . * = nkTk; then, after period T, rh( T) nk suchthat T=mpw=n,T,=n2T2=~ nl , n,,..., returns to the initial state r(0): i.e., rh( T) = r(O), or eAr = I. Thus, from equation (37), one obtains r(T) = r(0) + rf( T).
(40)
Therefore, when r,(T) has value other than zero, r(T) does not return to the initial state after period T. For r = kT, one can write r(kT) as the system solution after time T with the initial condition r(( k - 1) T): i.e., T r(kT)=eATr((k-l)T)+eAT e-A”Po(B)dB=r((k-l)T)+r~(T) I0 = r((k -2)T)
+2r,(T)
= r(O)+ kr,( T).
(41)
Therefore, the boundedness of r( kT) as T+ cc depends upon rf( T). From equation (41), if r,(T) has a value other than zero, r( kT) is unbounded as k + CO.
IMPULSIVELY
EXCITED
BAND/WHEEL
209
SYSTEM
5. SINGLE SPAN RESPONSE AND STABILITY For a high tension and low speed band system, the two spans nearly decouple, as discussed in reference [3]. To facilitate manipulations in closed form, an uncoupled, single span model is used to explicitly demonstrate the stability ideas presented above. The equation of motion for the transverse vibration of the simply supported decoupled secondary span is, as derived in reference [5], $ + 2&’ - *w”+ w””= 0.
(42)
The boundary conditions are w(l,7)=-3
w(0, T) = - 3 w, 8(T - npw), n=l
w”(1, 7)‘O.
w”(0, 7) = 0, Following the former development,
w,8(7-T,*-(n-l)p,),
n=,
(43,44) (45,46)
one substitutes
into equations (42)-(46). The equation of motion becomes N”
Y,
ii+2Sir’-$U”+V’“=(1--0
1
W&T-?I&)+,$
II=*
-2s
W&T-Try-(n-l)&)
c
“=*
3 W&T--&)+2S
5 II-1
II=1
W&T-T,2-(tl-l)p,)
and the boundary conditions are homogeneous, V(0, 7)
=
U( 1, 7)
= 0,
U”(c),
7)
=
U”( 1, 7) = 0.
and applying the Galerkin method, the ~(5, T) = c,?, y(5)qj(6), + of motion and boundary conditions are discretized into the form M&T) G,$T) + l&q(T) = P,. The M,, G,, and KS are N x N matrices, and P, and N x 1. The elements are Again,
by expanding
equation
G,(J, i) = 2s( I$, Vi),
W(j, i) = (%, V,), P.v(j)=(l--6,
V)
3
w~$(T-?I~,,,)~-(&
I#=1
The
k$)
3
i) = +
VI)+(Vj’,
VI’),
Woi(~-~,2-(n-l)~,,f)
?I=,
n=1
-2S(1, I$:.)3
K,(j,
W&T-P&)+2S(1,
%)
3 n=l
~~8(~-~,~-(n-l)p,).
(47)
discretized equations in the state vector form become (48)
i,(r)=A,r,(r)+P%(T), where rs(T)=($$.
As=(+&&-),
P,(T)=(&).
A transformation Q-'A& = A gives the diagonal 2 N x 2N matrix A, where A(j, j) = iwj N. The diagonal elements of A are the eigenand A(j+N, j+N)=-iwj forj=l,..., values of A,, and the column vectors of matrix Q are the eigenvectors of A,. Therefore, eAsr= Q e”‘Q-‘, where the diagonal 2N x 2N matrix en’ has elements e”‘(j, j) = eiwfr
210
K. W. WANG
AND
C. D. MOTE,
JR,
and e”‘(j+ N, j+ N) = ePiwlT, for j = 1,2,. . . , N. Because o,, w2, . . , wN are the imaginary parts of the eigenvalues of A,, they are the natural frequencies. For T = n, T, = n2Tz=n,T,=...=nNT,.,=mp then, T = n,(2r/w,) =. . . = nN(27r/wN) = mp,. From equation (40), r,(T) =u;,(O) +rrT( T), where rsl( T) = eAqJ
I
J
0
J
emA~‘P,(t9)de = Q en JQ-’
Q e-“‘Q-‘P,(
0) d0.
Then, substituting equation (47) into this equation gives rSf( T) = -4sA,wo[ QfiQ-’ - Q e-“‘MtQ-‘I +2A:w0[Q.0Q-‘(O/(l/jn))+ wherej=l,...,
Q e-A’12nQ-‘(Ol((-l)“‘/j~))l,
N. R is a diagonal 2N x 2N matrix with elements n(j,
j)
F
=
e-i(n,lm)*k~
and
L?(j+ N, j+ N)=
F ei(“~‘m)2k?r,
(49,50)
k=l
k=l
N and m = T/p,. wherej=l,..., According to the discussion presented above, a bounded response requires r,,(T) = 0. Therefore, summing the geometric series in equations (49) and (50), one concludes the following. If pw # nTk, Vn, ke Z, where Z is the set of positive integers (i.e., nk/m rl I), one has 0(j, j) =e N. That is, rSr( T) = 0, and, therefore, the system e i2qni)/(l-ei21m”,‘m)=0 for j=l,..., response is bounded. Ifp,=nTk,Vn,kEZ(nk/mEZ),onehasR(Sk)=n(k+N,k+N)=m;thenr,(T)f 0 in general, and the linear system response is unbounded as r + CO. -i2nnjlm(l
_e-i2mn,)/(1
_e-i2*nj/m)
=O
and
a(j+
N,
j+
N)
=
ei2vn,/m(l
0.5
;:
z
0
8 f
-0.5
0
50 T/T,
Figure 4. Calculated single span resonance
response under
impulsive boundary excitations pw = 16T,.
_
IMPULSIVELY
EXCITED
6. RESULTS
RAND/WHEEL
AND
211
SYSTEM
DISCUSSIONS
Numerical results have been obtained by direct integration of equations (36) and (48) for the coupled and uncoupled band/wheel system models, respectively. The initial states r(0)and r,(O)are zero. The weld is located at 5 = 0, on the secondary span at r = 0 (point 1 in Figure 3). The transverse responses of the single span model at .$=0*25 under resonance and non-resonance weld passage periods are shown in Figures 4 and 5, respectively. In Figure 4, the vibration amplitude becomes unbounded with time when the impulse period is
0.08
0.04 c. N”
0.00
0 t -0.04
-0.08
-0.121 0
I
I
20
I
40
60
80
r/T,
Figure 5. Calculated
single span non-resonance
response
under impulsive
boundary
I .
I
excitations,
I
p)<= 4.2T,
1
I Weld
entering
prlnclpal
t
span
0
G0 f
I
Weld leovlng secondary span
Non-dImensIonal
time,
T
Figure 6. Calculated coupled system model response under impulsive boundary excitations, strongly coupled case. (I = 12, p = 0.17007 x IO’, y = 0.14286 x 105, K, + cc, pR = pI = 428.6, R,, = 270. M, = M, = 111,s = 2.0, JR = JL = 1.02 x 10’.
212
K.
W.
WANG
Weld
AND
C.
D.
MOTE,
JR
lewng
Non-dunenslonol
time,
T
Weld entering prlnclpol spon Weld
’
leovlng
0.06, 5 + 0 f -0.06 0 Non-dlmenslonol
time,
I
Figure 7. Calculated coupled system model response under impulsive boundary conditions, weakly coupled s=2.0, JR=JL= case. (I = 12, p = 0.17007 x 108, y = 0.14286 x 105, K, +Q), pR=pL=428.6, M,=M,=lll, 1.02 x 10’. (a) Time range = 1.0, G scale = -00ll to O*OOl;before weld enters principal span; (b) time range = 1.2, i scale = -0.06 to O-06; after weld enters principal span.
pw = 16T In Figure 5, the span response is periodic with period T- 21 T, = Sp,(p, lb*2T,). p,,, is not equal to an integer multiple of any natural period. These numerical results are consistent with predictions of the stability theory in the previous sections. The undamped, coupled system model response is shown in Figures 6 and 7. The impulsive excitation in the coupled system occurs at the four boundary points in the two spans, and it drives the responses in both spans. The analysis incorporates the coupling of span and wheel responses and the impulsive displacement boundary excitation caused by weld passage. When the modulation wavelength is relatively short and the coupling of the spans is strong (see reference [3]), the impulsive input on either span excites significant transverse vibration in both spans. This is illustrated in Figure 6 with f, = 15.26, jz = 16.51, /ifi*= 1.25. When the modulated wavelength is long (f, = 19.66, fi = 19.77, A& = O-1l), the coupling of the excited span to the other span is not strong, as shown in Figure 7.
IMPULSIVELY
EXCITED
BAND/WHEEL
213
SYSTEM
7. EXPERIMENTS The response and stability predictions concluded above were experimentally verified on the test stand shown in Figure 1. In one band with easily excited natural periods ?, = O-0510 s (fr = 19425 Hz) and & = 04468 s ( jz = 21.375 Hz), resonance was observed by monitoring the transverse displacement of theAmid-point of the secondary span at the weld passage period jl,,, = O-255 s: that is, tiW= 5T,. The response shown in Figure 8(a) is dominated by this mode. (The unbounded response predicted at resonance in the analysis is a defect of the undamped linear system theory.) Following a small increase in wheel rotation speed so that cW= O-230 s, no single mode dominates the response; see
co
0
Time
Figure
i-t
8. Measured
secondary
(s I
span resonance
under
-1.5
weld passage
I
excitations.
I
-I
Time (sl Figure
9. Measured
secondary
span non-resonance
response
under
weld passage
excitations.
214
K.
W.
WANG
Figure_ 10. Measured spanAnon-resonance period. T = 1.23 s = 3p^, = 41 T, = 41 T,.
AN11
C.
D. MOTE.
JR.
steady state periodic response under weld passage excitations;
Figure 9(a). Furthermore, p(t) = ji w A2(0) de in Figures 8(b) and 9(b) shows that the rms response m/2 in Figure 9 is half that in Figure 8. The response in Figure 9 is not at resonance. Figure 10 shows the periodic response of the mid-secondary span recorded at 1000 sample/s. p*,,,is not an integer multiple of either natural period. In this case, $,,, = 0.41 s, f1 = f2 = 0.03 s, and the period T = 1.23 s = 3$,,, = 41 f, = 41 Tz: that is, f is the least common integer multiple of p*,, ?, and ?;. These observations show good agreement with the predictions of the analysis in the previous sections.
8. CONCLUSIONS
The analysis and experiments show that the impulsive excitation of the weld passage over the wheels can excite resonance even when the band translates at speeds much lower than the critical speed discussed in references [5,6-91. In many practical cases, it is important to consider this excitation. Adjusting the band speed, shifting the system natural frequencies and adding damping can be used to modulate the response at resonance.
ACKNOWLEDGMENTS
The authors would like to express their sincere thanks to the National Science Foundation, the University of California Forest Products Laboratory, and to the following corporations: AR1 AB (Sweden); California Cedar Products Co.; California Saw and Knife Works; CIRIS (France); Crown Zellerbach Co.: Hudson I.C.S.; P 8c M Cedar Products; Sun Studs, Inc.; Weyerhaeuser Co.; and the Wood Machining Institute for their interest and sponsorship. REFERENCES 1. W. L. MIRANKER 1960IBM Journal of Research and Development 4. The wave equations in a medium in motion. of guide 2. E. C. SCHROM 1984 University of California Berkeley, Technical Report. Contribution design to bandsaw vibration. Vibration 3. K. W. WANG and C. D. MOTE, JR. 1986Journal of Sound and Vibration 10!3,237-258. coupling analysis of band/ wheel mechanical systems. 4. R. D. MINDLIN and I. E. GOODMAN 1950Journal of Applied Mechanics 17, 377-380. Beam vibrations with time dependent boundary conditions. 5. C. D. MOTE, JR. 1965 Journal of the Franklin Znsritute 276, 430-444. A study of band saw vibrations. 6. D. W. ALSPAUGH 1967Journal of the Franklin Institute 283, 328-338. Torsional vibration of a moving band.
IMPULSIVELY
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BAND/WHEEL
215
SYSTEM
7. C. D. MOTE, JR. 1968Journal of the Franklin Institute 285, 329-346. Dynamic stability of an axially moving bank. 8. S. NAGULESWARAN and J. H. WILLIAMS 1968 International Journal of Mechanical Science 10, 239-250. Lateral vibration of band saw blades, pulley belts and the like. 9. A. G. ULSOY and C. D. MOTE, JR. 1982 Transactions of the American Society of Mechanical Engineers Journal of Engineering for Industry 104,71-78. Vibration of wide band saw blades.
APPENDIX The elements
of the matrices
and vectors
:;:i
i,l,
#;:
M;33
K=[:;;
p,t;q,A;e 3
1
in (35) are as follows:
G=[:f
:;
;;:
i2
iI],
Gi3
G;],
$1.
N d$lelnsional vectors, with elements P,(T), i;j( r), q,(T), ij( T), respectively. ,...) ) . . . ) K44 are N x N matrices. The elements of these matrices are Mll(j,i)=(q,
~ii)+((JL14PL)+f(ML/4))(~(0)ui(o))
+((JR14PR)+(MR14))(LS(1)U,(1)),
i) = ( w,, wl),
M33(j, Ml2(
i,i)=(-(JL14PL)+(ML/4))( uj(0)Q(“))
+(-(JR14rR)+(MR14))(~j(l)~i(l)),
Gll(j, Kll(j,
i)=2s(U,,
i)=4(U,!,
G33( j, i) = 2s( I+$, W:),
Vi),
U~)+S2(~(1)U~(1))+(KR/4)(~(1)Ui(l))
-S2(
uj(“)
ul(“))
+
(K.L/4)(
u,(o)
ui(o)),
K12(j,i)=(K,/4)(~,(0)~j(O))+(K,/4)(~iIi(1)~j(1)),
K13(j,
i)= y(w*‘WI,UJ,
K33(j,i)=$(W;,
W:)+(WT,
K31(j,
i) = y(w*‘Ui,
WJ,
W:l)+y(W~,u*‘W:)+$a(W~,
w*“W:).
Because U,(s) = fij(S) and Wj( 5) = kj( 0, and because of the symmetry of the equilibrium configurations w*(t) = -G*(t) and u*(t) = G*(t), the remaining matrices are M22(j,
i)= Mll(j, G22(j,
i),
M44( j, i) = M33( j, i ),
i) = -Gll(j,
i),
G44(j,
M21(j,
i) = -G33(j,
i) = M12(j, i),
i),
216
K. W.
WANG
K22(j, i)= Kll(j, K24( j, i) = K 13(j, i),
AND
i),
C.
b” L qx, t), E(x, t) %x9 t), ax, t) ‘,L, rlz y&&z
Jh. Jlp, FL,& -6
non-dimensional
JR.
K21(j, i) = K 12(j, i),
K42(j, i) = K31(j, i),
APPENDIX E A I R c
D. MOTE.
2
elastic modulus band cross-sectioned area band second moment of area band initial axial tension at band speed c band initial axial speed band mass per unit length band thickness span length longitudinal displacements of the spans transverse displacements of the spans radii of the wheels masses of the wheels rotational inertias of the wheels support stiffnesses the jth natural frequency parameters:
s=cLdqFI
ii(r$, T) = i/b
4 = L2((EA-mc)2/EI)
G(.$, r) = $jb
t,h= L2(( R - mc)2/ EI)
pL= rL/b
n =AL2/I
PR=rRlb
/3 = AL’/I
ML = i%?,/mL
y=ALb/i
MR = dfR/rnL
5=x/L
JL = jL/ mLb2
7=1/m
JR = _fRl mLb2
~(5, r) = i?/b
KL = RLL3/ EI
~(5, T) = O/b
KR = It,L’/ El
K44(j,
i) = K33(j,
i).
ofSound
and Vibration
BAND/WHEEL
(1987)
115(2),
SYSTEM
203-216
VIBRATION
BOUNDARY
UNDER
IMPULSIVE
EXCITATION
K. W. WANG~ AND C. D. MOTE, JR. Department of Mechanical Engineering,
University of California, Berkeley, California, U.S.A.
(Received 9 July 1986) Measurements of band vibration undertaken here show that passage of the butt weld connecting the ends of continuous band over wheels excites vibration in the band/wheel system. A displacement impulse occurs each time the weld initially contacts and separates from the wheels. The excitation is periodic, and it can excite instability. The vibration and stability of the coupled band/wheel system under impulsive boundary displacements are analyzed in this paper. The theoretical and experimental findings show that resonance occurs in the system when the weld passage (impulse) period is an integer multiple of any system natural period. 1. INTRODUCTION AND MOTIVATION
The vibration of high speed, axially moving continuous bands is usually detrimental to their function. Examples of axially moving bands, driven by rotation wheels, include band saws, belts, chains, and magnetic tapes. Excitation of transverse vibration caused by the interactions of the band and the wheels is common and significant. Wheel eccentricities and band imperfections are major sources of these excitations. The contributions of wheel eccentricities have been discussed in the literature [ 1,2]. However, the excitation caused by thickness and stiffness imperfections in the continuous band running over rotating wheels has not received attention. An example of this excitation is the passage of the butt weld, connecting the ends of the metal belt or band saw blade, over the wheels. An impulsive input to the transverse vibration of the spans is observed as the weld enters and exits the free spans between the wheels and as it passes over bearing surfaces. Experiments, conducted on a table-top test stand, were carried out to examine the transverse vibration of the band under the excitations of weld passage. A schematic of the experimental set-up is shown in Figure 1. For purposes of discussion, one span is called the principal span, and the other is called the secondary span. The test stand consists of two, 200 mm diameter wheels and an axle to axle wheel separation of 860 mm. The band cross-section is rectangular, 0.5 mm x 19 mm, and the weld bump is 0.02 mm in height and l-5 mm in width. The times when the weld enters and exits the secondary span are detected with a photo-sensor. A displacement probe is positioned at the midsecondary span to monitor its transverse vibration. A spectral analyzer (HP5423A) is used for data acquisition and processing. The mid-span transverse vibration of the secondary span is shown in Figure 2. The upper trace shows the transverse vibration and the impulsive signals in the lower trace indicate times when the weld enters and exits the secondary span. The data shows the i Presently with Power Systems Michigan 48090-9055, U.S.A.
Research
Department,
General
Motors
Research
Laboratories,
Warren,
203 0022-460X/87/110203
+ 14 %03.00/O
0
1987 Academic
Press Inc. (London)
Limited
204
K. W.
WANG
AND
C.
I
D. MOTE,
JR
I
HP5423A Structure
onolyzer
Figure 1. Schematic of experimental set-up for observing the impulsive excitation caused by weld passage. 0, Weld; 0, reflector monitoring weld entering second span; I::, reflector monitoring weld leaving secondary span.
s
0.15
0
9
I
I
I
‘<
1
I
I
0.10
L
0.05
;E
0.00
gs g’ 0
-0.05
z Z
-0.10
8
-0.15
Weld
Weld
entering
E
leaving
span
spon
1.0.
/
I
.
.g
b
0.8 -
E-
?1f0
0.6
_
0.4
-
=> 0’
P
*
,o 0 p’
0.2 I 0.00
I 0.50
I 0.25
I 190
I 0.75 Time
I I .25
-:
I.50
($1
Figure 2. Measured band transverse response under weld path impulsive excitation.
transverse vibration from the impulsive excitation of the weld passage. These excitations occur periodically. The dynamic response of the band/wheel system is analyzed here by treating the weld disturbance as an impulsive displacement boundary excitation of the spans. 2. FORMULATION
The linearized, non-dimensional derived in reference [3], are ii+2SZ-@“-y(w*‘w’)‘=o,
equations of motion of a free band/wheel ; - 2& - &jv - y( \i*‘$‘)’ = 0,
system, (1,2)
IMPULSIVELY
EXCITED
BAND/WHEEL
205
SYSTEM
w + 2$’ - @‘I+ W’m- y(w*‘u’)‘_ r(u*‘w’)‘_$@( W*‘2Wfjf2w*‘w*“w’) = 0,
(3)
6 _ 2s$;’_ $$I+ $,U,_ y(@*I$)I_ r(3*‘~‘)‘_~(Y(~*~2~,,r+2~*f~t”~i(’) =O,
(4)
where C, u, G, and w are the non-dimensional longitudinal and transverse vibrations of the principal and secondary spans, respectively. The superscript * denotes the transverse and longitudinal displacements associated with the equilibrium configuration of the two spans. The superscript dot ( ’) and prime (‘) denote partial derivatives with respect to the non-dimensional time and space variables, respectively. The boundary conditions are (J,/2p:)(ii(o,
T)G(O,
T))_P(U’(O, 7)-LT’(O, 7))
- r(w*‘(o)w’(o, (M,/2)(ii(O, -
T)+i(o,
~(W*‘(o)W’(o,
7) - G*‘(o)G’(o, 7)) = 0,
T))+(KJ2)(u(O, 7)+6*‘(0)6’(0,
(J,/2&)@(1,
(5)
T)+C(o, 7))--p(u’(O,
T)+C’(o, 7))
T))=o,
T)-$1,
(6)
T))+p(U’(l,
T)-;‘(I,
7))
(7)
+y(W*‘(l)W’(l,T)-~*‘(l)+‘(l,T))=o,
(k&/2)(1(1,
T)+k(l,
T))+(&/2)(U(1,
T)+Zi(l,
T))+p(U’(l,
T)+G’(l,
7))
+y(W*‘(l)W’(l,T)+i*‘(l)i’(l,T))=o, W”(0,
W(0,
7)
fi(O,~)=;
=
-
3
n=I
(8) 7)
=
wo 6( 7 -
6”(0,
n&a),
T)
= 0,
W”(1,
w(~,T)=-?
=
+‘(I,
7)
=o,
“=I
T)=
3
(9,lO)
WO~(T-T,~-(PI-~)~,,,),
(11, 12)
w,~(T-T~~-(?I-~)~,).
(13,
n=, i(1,
W,6(~-~,~-(n-l)p,),
7)
14)
n=,
parameters and variables are illustrated in Figure 3 and defined in the nomenclature (Appendix 2). Equations (5)-(8) describe the rotation and translation of the wheels, and equations (9) and (10) state that the variation of the curvature of band of the end points, during oscillation about equilibrium, vanishes. Equations (1 l)-( 14) express the impulsive excitation of the band ends at the wheel contacts. The 6( ’) is the delta function, w. is the height of the weld, T@represents the time for the weld to travel from point i to point j in Figure 3, p,,, is the period of the weld circumnavigation cycle, and N,,, is the number of these cycles. The
I
L Weld path entering secondary span ot T=O
A-A Figure 3. Schematic
of the axially moving band and rotating
wheel system model.
206
K.
W.
WANG
AND
C’
3. SOLUTION
I).
MOTE,
JR.
METHODS
A formulation for discretization is obtained by transforming the non-homogeneous boundary conditions (11)-( 14) to a homogeneous form by a change of variable [4]. Let W(.&T)=U(&T)-(1-t)
%;
W,,fi(T-n&)-c
W,6(7-T,2-(n-l)&)
‘>
,I =,
(15)
n-l
and
+t
Upon substitution motion become
3 n=,
of expressions
=yW*”
3
w,~(T-T,~--(n--l)&).
(16)
(15) and (16) into equations
w~6(T-~jl,)-
3
( II=, ;
_
2&
-
4,“_
y(
(l)-( 14), the equations
wos(T-Tn-(n-l)!‘,)
(17)
,
n-l
of
)
G*‘;‘)’
N” =
+,*‘I
? ( fl=l
WO~(T-T,~-(~-~)P,)-
~+2s~‘-4v”+v”“-
1 n=,
y(w*‘u’)‘-
=(1-t)
c
,
w&T-Up,)+5
3
W&T-T,2-(n-l)&)
n=,
3
W,,~(T--np,)-
2
n=,
Woii(~-~,2-(n-l)p,)
“=,
+ yu*”
3 W,~(T-T,~-(~-~)P,)
w,~(T--np,)-
II=,
+3aw*'w*"
>
NM W,,@T--pm)-
c
W,6(T-T,z-(?I-I)&)
_
2&l
_
#g”
+
,-I,,, _y($*,qr_
(19)
=o,
n=l ;
(18)
y(U*‘U’)‘_~~(W*rzUI,+2W*IW*“V))
“=,
-2s
wos(T-TM-(n-l)&)
>
Y(~*I~f)I_5~(~*r2~ff+2~*‘~*“~‘)
N” =(,$-I)
c
W&T-T,.,-(n-l)&)+
3
n=,
-2s
5
Wob(~-~714-(n-l)p,
( n=l
-3
n=,
>
Wd(T--,j+-1)Pn.))
- y;*” ( _
Woii(~-~,,-(n--l)p,,,)
n=,
:
Wo6(7’-7,4-(n-l)p,)-
II=,
3&*‘$*”
2
W,6(T-T,3-(n-l)p,)
tl=,
> Y%
W,8(7-~,~-(n-l)p,,)-
1
~,8(~--,~-(n-l)p,)
=O,
(20)
IMPULSIVELY
EXCITED
BAND/WHEEL
207
SYSTEM
and the boundary conditions become
-
$*I
2 w,8(T-T*4-(n-l)p,)+~*'
F %~(~-~13--(~-lhL)
?I=1
“=I
(&/2PzR)(W,
T)-Q,
T))+p(u’(l,
T)-u”(1,
T))+ y(w*‘(l)v’(l,
)
T)-i*‘(l)u”(l,
3
(22) 7))
(23)
= y
(
-w*’
+;*+,
3 W,S(T-rInp,)+w*’ n=l
3 WoS(7-7,*-(n-l)&) “=I
wo6(T-T*4-(n-l)p,)-3*’
u”(0, T) = 3”(0, T) = 0, u(0, 7) = 0,
u(l,7)=0,
3
w,8(T-Tll-(n-l)pw)),
II=1
(24)
o”(l,T)=P(l,T)=o,
(25,26)
$0, 7) = 0,
(27-30)
??(1,7)=0.
Because an exact solution of equations (17)-(30) cannot be obtained in general, a Galerkin discretization of a mixed weak form is pursued [3]. U, 6, u, and u’are expanded as
u(59T, =
c”q(5)PjtT),
j=l
u(CT, 7) =
F y(if)qj(T),
(31-34)
j=l
wherepj(
T), qj( T),jj( T) and ij( T) are generalized co-ordinates. Uj(t), C&(S)are admissible functions. Vj(,$), F(t) are comparison functions which satisfy the boundary conditions (25)-(30). Here we choose y(.$)=_q(t)=sin (j~r[), which are the eigenfunctions of a simple-simple beam, and Uj([) = Uj(e) = cos (~TT[), which are the eigenfunctions of a free-free rod.
208
K. W. WANG AND C.
D. MOTE.
JR
discretized weak form is obtained from the inner product of the expansion functions with the domain and boundary residuals derived from equations (17)-(30): The
M~;(T)+G~~(T)+K~(T)=P(~).
(35)
The vector P(T) is obtained from the inner product of the expansion functions with the right-hand side of (17)-(30). The elements of M, G, K, and Y(T) are defined in Appendix 1.
4. STABILITY
ANALYSIS
In the state vector form, equation (35) becomes i(~)=Ar(r)+p,,(~),
(36)
r(r) = r!Jr)+r,(r),
(37)
where
The solution of equation (36) is
where rh( 7) = eA’r(0) is the free response under initial condition r(O), and T rr( 7) = eA7 eFAeP,( 6) de I0
(38)
(39)
is the forced response from PO(~). Assume the solution of i(r) = Ar(r) is periodic (stable), and the excitation PO(~) is periodic. Let the natural periods for each mode be T, , T2, . . . , Tk, and the impulse period be p,,,. The modes 1 through k include all vibration modes contributing to the band response under all excitations in the process. It is assumed that there exist integers m, . * = nkTk; then, after period T, rh( T) nk suchthat T=mpw=n,T,=n2T2=~ nl , n,,..., returns to the initial state r(0): i.e., rh( T) = r(O), or eAr = I. Thus, from equation (37), one obtains r(T) = r(0) + rf( T).
(40)
Therefore, when r,(T) has value other than zero, r(T) does not return to the initial state after period T. For r = kT, one can write r(kT) as the system solution after time T with the initial condition r(( k - 1) T): i.e., T r(kT)=eATr((k-l)T)+eAT e-A”Po(B)dB=r((k-l)T)+r~(T) I0 = r((k -2)T)
+2r,(T)
= r(O)+ kr,( T).
(41)
Therefore, the boundedness of r( kT) as T+ cc depends upon rf( T). From equation (41), if r,(T) has a value other than zero, r( kT) is unbounded as k + CO.
IMPULSIVELY
EXCITED
BAND/WHEEL
209
SYSTEM
5. SINGLE SPAN RESPONSE AND STABILITY For a high tension and low speed band system, the two spans nearly decouple, as discussed in reference [3]. To facilitate manipulations in closed form, an uncoupled, single span model is used to explicitly demonstrate the stability ideas presented above. The equation of motion for the transverse vibration of the simply supported decoupled secondary span is, as derived in reference [5], $ + 2&’ - *w”+ w””= 0.
(42)
The boundary conditions are w(l,7)=-3
w(0, T) = - 3 w, 8(T - npw), n=l
w”(1, 7)‘O.
w”(0, 7) = 0, Following the former development,
w,8(7-T,*-(n-l)p,),
n=,
(43,44) (45,46)
one substitutes
into equations (42)-(46). The equation of motion becomes N”
Y,
ii+2Sir’-$U”+V’“=(1--0
1
W&T-?I&)+,$
II=*
-2s
W&T-Try-(n-l)&)
c
“=*
3 W&T--&)+2S
5 II-1
II=1
W&T-T,2-(tl-l)p,)
and the boundary conditions are homogeneous, V(0, 7)
=
U( 1, 7)
= 0,
U”(c),
7)
=
U”( 1, 7) = 0.
and applying the Galerkin method, the ~(5, T) = c,?, y(5)qj(6), + of motion and boundary conditions are discretized into the form M&T) G,$T) + l&q(T) = P,. The M,, G,, and KS are N x N matrices, and P, and N x 1. The elements are Again,
by expanding
equation
G,(J, i) = 2s( I$, Vi),
W(j, i) = (%, V,), P.v(j)=(l--6,
V)
3
w~$(T-?I~,,,)~-(&
I#=1
The
k$)
3
i) = +
VI)+(Vj’,
VI’),
Woi(~-~,2-(n-l)~,,f)
?I=,
n=1
-2S(1, I$:.)3
K,(j,
W&T-P&)+2S(1,
%)
3 n=l
~~8(~-~,~-(n-l)p,).
(47)
discretized equations in the state vector form become (48)
i,(r)=A,r,(r)+P%(T), where rs(T)=($$.
As=(+&&-),
P,(T)=(&).
A transformation Q-'A& = A gives the diagonal 2 N x 2N matrix A, where A(j, j) = iwj N. The diagonal elements of A are the eigenand A(j+N, j+N)=-iwj forj=l,..., values of A,, and the column vectors of matrix Q are the eigenvectors of A,. Therefore, eAsr= Q e”‘Q-‘, where the diagonal 2N x 2N matrix en’ has elements e”‘(j, j) = eiwfr
210
K. W. WANG
AND
C. D. MOTE,
JR,
and e”‘(j+ N, j+ N) = ePiwlT, for j = 1,2,. . . , N. Because o,, w2, . . , wN are the imaginary parts of the eigenvalues of A,, they are the natural frequencies. For T = n, T, = n2Tz=n,T,=...=nNT,.,=mp then, T = n,(2r/w,) =. . . = nN(27r/wN) = mp,. From equation (40), r,(T) =u;,(O) +rrT( T), where rsl( T) = eAqJ
I
J
0
J
emA~‘P,(t9)de = Q en JQ-’
Q e-“‘Q-‘P,(
0) d0.
Then, substituting equation (47) into this equation gives rSf( T) = -4sA,wo[ QfiQ-’ - Q e-“‘MtQ-‘I +2A:w0[Q.0Q-‘(O/(l/jn))+ wherej=l,...,
Q e-A’12nQ-‘(Ol((-l)“‘/j~))l,
N. R is a diagonal 2N x 2N matrix with elements n(j,
j)
F
=
e-i(n,lm)*k~
and
L?(j+ N, j+ N)=
F ei(“~‘m)2k?r,
(49,50)
k=l
k=l
N and m = T/p,. wherej=l,..., According to the discussion presented above, a bounded response requires r,,(T) = 0. Therefore, summing the geometric series in equations (49) and (50), one concludes the following. If pw # nTk, Vn, ke Z, where Z is the set of positive integers (i.e., nk/m rl I), one has 0(j, j) =e N. That is, rSr( T) = 0, and, therefore, the system e i2qni)/(l-ei21m”,‘m)=0 for j=l,..., response is bounded. Ifp,=nTk,Vn,kEZ(nk/mEZ),onehasR(Sk)=n(k+N,k+N)=m;thenr,(T)f 0 in general, and the linear system response is unbounded as r + CO. -i2nnjlm(l
_e-i2mn,)/(1
_e-i2*nj/m)
=O
and
a(j+
N,
j+
N)
=
ei2vn,/m(l
0.5
;:
z
0
8 f
-0.5
0
50 T/T,
Figure 4. Calculated single span resonance
response under
impulsive boundary excitations pw = 16T,.
_
IMPULSIVELY
EXCITED
6. RESULTS
RAND/WHEEL
AND
211
SYSTEM
DISCUSSIONS
Numerical results have been obtained by direct integration of equations (36) and (48) for the coupled and uncoupled band/wheel system models, respectively. The initial states r(0)and r,(O)are zero. The weld is located at 5 = 0, on the secondary span at r = 0 (point 1 in Figure 3). The transverse responses of the single span model at .$=0*25 under resonance and non-resonance weld passage periods are shown in Figures 4 and 5, respectively. In Figure 4, the vibration amplitude becomes unbounded with time when the impulse period is
0.08
0.04 c. N”
0.00
0 t -0.04
-0.08
-0.121 0
I
I
20
I
40
60
80
r/T,
Figure 5. Calculated
single span non-resonance
response
under impulsive
boundary
I .
I
excitations,
I
p)<= 4.2T,
1
I Weld
entering
prlnclpal
t
span
0
G0 f
I
Weld leovlng secondary span
Non-dImensIonal
time,
T
Figure 6. Calculated coupled system model response under impulsive boundary excitations, strongly coupled case. (I = 12, p = 0.17007 x IO’, y = 0.14286 x 105, K, + cc, pR = pI = 428.6, R,, = 270. M, = M, = 111,s = 2.0, JR = JL = 1.02 x 10’.
212
K.
W.
WANG
Weld
AND
C.
D.
MOTE,
JR
lewng
Non-dunenslonol
time,
T
Weld entering prlnclpol spon Weld
’
leovlng
0.06, 5 + 0 f -0.06 0 Non-dlmenslonol
time,
I
Figure 7. Calculated coupled system model response under impulsive boundary conditions, weakly coupled s=2.0, JR=JL= case. (I = 12, p = 0.17007 x 108, y = 0.14286 x 105, K, +Q), pR=pL=428.6, M,=M,=lll, 1.02 x 10’. (a) Time range = 1.0, G scale = -00ll to O*OOl;before weld enters principal span; (b) time range = 1.2, i scale = -0.06 to O-06; after weld enters principal span.
pw = 16T In Figure 5, the span response is periodic with period T- 21 T, = Sp,(p, lb*2T,). p,,, is not equal to an integer multiple of any natural period. These numerical results are consistent with predictions of the stability theory in the previous sections. The undamped, coupled system model response is shown in Figures 6 and 7. The impulsive excitation in the coupled system occurs at the four boundary points in the two spans, and it drives the responses in both spans. The analysis incorporates the coupling of span and wheel responses and the impulsive displacement boundary excitation caused by weld passage. When the modulation wavelength is relatively short and the coupling of the spans is strong (see reference [3]), the impulsive input on either span excites significant transverse vibration in both spans. This is illustrated in Figure 6 with f, = 15.26, jz = 16.51, /ifi*= 1.25. When the modulated wavelength is long (f, = 19.66, fi = 19.77, A& = O-1l), the coupling of the excited span to the other span is not strong, as shown in Figure 7.
IMPULSIVELY
EXCITED
BAND/WHEEL
213
SYSTEM
7. EXPERIMENTS The response and stability predictions concluded above were experimentally verified on the test stand shown in Figure 1. In one band with easily excited natural periods ?, = O-0510 s (fr = 19425 Hz) and & = 04468 s ( jz = 21.375 Hz), resonance was observed by monitoring the transverse displacement of theAmid-point of the secondary span at the weld passage period jl,,, = O-255 s: that is, tiW= 5T,. The response shown in Figure 8(a) is dominated by this mode. (The unbounded response predicted at resonance in the analysis is a defect of the undamped linear system theory.) Following a small increase in wheel rotation speed so that cW= O-230 s, no single mode dominates the response; see
co
0
Time
Figure
i-t
8. Measured
secondary
(s I
span resonance
under
-1.5
weld passage
I
excitations.
I
-I
Time (sl Figure
9. Measured
secondary
span non-resonance
response
under
weld passage
excitations.
214
K.
W.
WANG
Figure_ 10. Measured spanAnon-resonance period. T = 1.23 s = 3p^, = 41 T, = 41 T,.
AN11
C.
D. MOTE.
JR.
steady state periodic response under weld passage excitations;
Figure 9(a). Furthermore, p(t) = ji w A2(0) de in Figures 8(b) and 9(b) shows that the rms response m/2 in Figure 9 is half that in Figure 8. The response in Figure 9 is not at resonance. Figure 10 shows the periodic response of the mid-secondary span recorded at 1000 sample/s. p*,,,is not an integer multiple of either natural period. In this case, $,,, = 0.41 s, f1 = f2 = 0.03 s, and the period T = 1.23 s = 3$,,, = 41 f, = 41 Tz: that is, f is the least common integer multiple of p*,, ?, and ?;. These observations show good agreement with the predictions of the analysis in the previous sections.
8. CONCLUSIONS
The analysis and experiments show that the impulsive excitation of the weld passage over the wheels can excite resonance even when the band translates at speeds much lower than the critical speed discussed in references [5,6-91. In many practical cases, it is important to consider this excitation. Adjusting the band speed, shifting the system natural frequencies and adding damping can be used to modulate the response at resonance.
ACKNOWLEDGMENTS
The authors would like to express their sincere thanks to the National Science Foundation, the University of California Forest Products Laboratory, and to the following corporations: AR1 AB (Sweden); California Cedar Products Co.; California Saw and Knife Works; CIRIS (France); Crown Zellerbach Co.: Hudson I.C.S.; P 8c M Cedar Products; Sun Studs, Inc.; Weyerhaeuser Co.; and the Wood Machining Institute for their interest and sponsorship. REFERENCES 1. W. L. MIRANKER 1960IBM Journal of Research and Development 4. The wave equations in a medium in motion. of guide 2. E. C. SCHROM 1984 University of California Berkeley, Technical Report. Contribution design to bandsaw vibration. Vibration 3. K. W. WANG and C. D. MOTE, JR. 1986Journal of Sound and Vibration 10!3,237-258. coupling analysis of band/ wheel mechanical systems. 4. R. D. MINDLIN and I. E. GOODMAN 1950Journal of Applied Mechanics 17, 377-380. Beam vibrations with time dependent boundary conditions. 5. C. D. MOTE, JR. 1965 Journal of the Franklin Znsritute 276, 430-444. A study of band saw vibrations. 6. D. W. ALSPAUGH 1967Journal of the Franklin Institute 283, 328-338. Torsional vibration of a moving band.
IMPULSIVELY
EXCITED
BAND/WHEEL
215
SYSTEM
7. C. D. MOTE, JR. 1968Journal of the Franklin Institute 285, 329-346. Dynamic stability of an axially moving bank. 8. S. NAGULESWARAN and J. H. WILLIAMS 1968 International Journal of Mechanical Science 10, 239-250. Lateral vibration of band saw blades, pulley belts and the like. 9. A. G. ULSOY and C. D. MOTE, JR. 1982 Transactions of the American Society of Mechanical Engineers Journal of Engineering for Industry 104,71-78. Vibration of wide band saw blades.
APPENDIX The elements
of the matrices
and vectors
:;:i
i,l,
#;:
M;33
K=[:;;
p,t;q,A;e 3
1
in (35) are as follows:
G=[:f
:;
;;:
i2
iI],
Gi3
G;],
$1.
N d$lelnsional vectors, with elements P,(T), i;j( r), q,(T), ij( T), respectively. ,...) ) . . . ) K44 are N x N matrices. The elements of these matrices are Mll(j,i)=(q,
~ii)+((JL14PL)+f(ML/4))(~(0)ui(o))
+((JR14PR)+(MR14))(LS(1)U,(1)),
i) = ( w,, wl),
M33(j, Ml2(
i,i)=(-(JL14PL)+(ML/4))( uj(0)Q(“))
+(-(JR14rR)+(MR14))(~j(l)~i(l)),
Gll(j, Kll(j,
i)=2s(U,,
i)=4(U,!,
G33( j, i) = 2s( I+$, W:),
Vi),
U~)+S2(~(1)U~(1))+(KR/4)(~(1)Ui(l))
-S2(
uj(“)
ul(“))
+
(K.L/4)(
u,(o)
ui(o)),
K12(j,i)=(K,/4)(~,(0)~j(O))+(K,/4)(~iIi(1)~j(1)),
K13(j,
i)= y(w*‘WI,UJ,
K33(j,i)=$(W;,
W:)+(WT,
K31(j,
i) = y(w*‘Ui,
WJ,
W:l)+y(W~,u*‘W:)+$a(W~,
w*“W:).
Because U,(s) = fij(S) and Wj( 5) = kj( 0, and because of the symmetry of the equilibrium configurations w*(t) = -G*(t) and u*(t) = G*(t), the remaining matrices are M22(j,
i)= Mll(j, G22(j,
i),
M44( j, i) = M33( j, i ),
i) = -Gll(j,
i),
G44(j,
M21(j,
i) = -G33(j,
i) = M12(j, i),
i),
216
K. W.
WANG
K22(j, i)= Kll(j, K24( j, i) = K 13(j, i),
AND
i),
C.
b” L qx, t), E(x, t) %x9 t), ax, t) ‘,L, rlz y&&z
Jh. Jlp, FL,& -6
non-dimensional
JR.
K21(j, i) = K 12(j, i),
K42(j, i) = K31(j, i),
APPENDIX E A I R c
D. MOTE.
2
elastic modulus band cross-sectioned area band second moment of area band initial axial tension at band speed c band initial axial speed band mass per unit length band thickness span length longitudinal displacements of the spans transverse displacements of the spans radii of the wheels masses of the wheels rotational inertias of the wheels support stiffnesses the jth natural frequency parameters:
s=cLdqFI
ii(r$, T) = i/b
4 = L2((EA-mc)2/EI)
G(.$, r) = $jb
t,h= L2(( R - mc)2/ EI)
pL= rL/b
n =AL2/I
PR=rRlb
/3 = AL’/I
ML = i%?,/mL
y=ALb/i
MR = dfR/rnL
5=x/L
JL = jL/ mLb2
7=1/m
JR = _fRl mLb2
~(5, r) = i?/b
KL = RLL3/ EI
~(5, T) = O/b
KR = It,L’/ El
K44(j,
i) = K33(j,
i).