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A note on the vibrations of a clamped-free beam with a mass at the free end
Journal of Sound and Vibration (1974) 37(2), 161-168
A NOTE
O N T H E VIBRATIONS OF A C L A M P E D - F R E E BEAM W I T H A MASS AT T H E FREE E N D P. A. A. LAURA, J. L. POMBOAND E. A. SUSEMIHL Department of Engineering, UniversidadNacional del Sur, Bahia Blanca, Argenthta
(Received4 December 1973, and in revisedform 18 April 1974) This note is concerned with the determination of natural frequencies and modal shapes of a clamped-free beam which carries a finite mass, M, at the free end. Accurate results are presented for different (concentrated mass)/(beam mass) ratios, M[Mo. It is also the purpose of this note to analyze the variation of the maximum dynamic stress as a function of the parameter M/M~ for the first mode of vibration.
1. INTRODUCTION
Characteristic functions and frequencies have been extensively tabulated for nearly all common types of beams: clamped-clamped, damped-free, clamped-supported, freesupported and supported-supported [1, 2]. This note deals with the determination of the first ten natural frequencies of a clamped-free beam which carries a finite mass, M, at the free end (Figure 1). Shear and rotatory inertia effects are disregarded.
:'1
L
1. / Figure 1. Beam-mass system studied in the present investigation. It is important to point out that the frequency equation was obtained by Prescott [3] but no extensive numerical information has been published in the open literature. Modal shapes and the effect of the mass M on the maximum dynamic stress when the structure vibrates in the first mode are also investigated.
2. MATHEMATICAL ANALYSIS The problem under consideration is described by the following differential system: 0* w
eIT
azw
+pA 161
=o,
(1)
162
P. A. A. LAURA, J. L. POMBO AND E. A. SUSEMIilL
aw
w(0, t) = ~
(2a, b)
(o, t) = 0,
ox
0 2 It'
Oxz (L, t) = 0,
-
(3)
] 02 w 0 3 w., (L, t) = M - - ~ ( L , - E l ax 3
t),
(4)
where E is the Young's modulus, I the constant moment of inertia, A the constant crosssectional area, p the mass density and M the concentrated mass at the free end. Using the standard method of separation of variables, one assumes "(5)
w(x,t)=X(x)T(t).
Substituting this form in equation (1) results in the equality X (~v) pA T" . . . . . . X EI T
k*.
(6)
The solution of the ordinary differential equation X ~tv~- k 4 X = 0 is simply X = C~ cos k x + C2 sin k x + Ca cosh k x + C4 sinh kx.
(7)
Substituting expression (7) in equation (2a, b) results in the equations C, + C3 = 0,
(8a)
c , + G = o.
(Sb)
From equations (7) and (3) one obtains - C~ cos k L - C2 sin k L + C3 cosh k L + C4 sinh k L = O.
(8c)
Substituting equation (7) in equation (4) results in the algebraic expression Ct sin k L - C2 cos k L + C3 sinh k L + C4 cosh k L = -1 - - - k M ( C t coskL + C 2 s i n k L + C 3 c o s h k L + C4sinhkL). pA
(8d)
The solution of equations (8) leads to the following determinantal equation in the eigenvalues, k: gll(k)
g12(k) = o,
(9)
CLAMPED-FREE BEAM WITIt A MASS
163
where gl l(k) = cos kL + cosh kL,
(10a)
g12(k) = sinkL + sinhkL,
(lOb)
g2~(k) = sin kL - sinh kL + (kL) M ( - cosh kL + cos kL),
(lOc)
(kL) M g22(k) = - cos kL - cosh kL + ~ ( - sinh kL + sin kL).
(lOd)
pAL
Equations (9) and (10) lead to the transcendental equation 1
l + cosycoshy
M
y sinycoshy-cosysinhy
(ll)
ML.'
where My is the beam mass (M~ = pAL) and y = kL. The frequencies, f~, are then given by
A
(12)
pA"
3. DETERMINATION OF THE EIGENVALUES In order to calculate the roots, yt, of the transcendental equation (11) it is convenient to express it in the form M
z(y) = - - y ( c o s y s i n h y - s i n y c o s h y ) + c o s y c o s h y + I = 0.
(I 3)
Mr The function z(y) shows rapid oscillations, attaining very large values between successive roots. The slope of the function at each root is, therefore, very close to vertical. This characteristic of the function suggests the use of the method of false position to ensure convergence in the determination of the roots [4]. In order to initiate the iterative process, the roots were first bracketed by means of a straight search process. The algorithmic procedure was implemented in a Hewlett-Packard 9810 A. TABLE 1
Values ofthefirst ten roots of equation (1 l) for M/My = 0, 0-20, 0 . 4 0 , . . . , 10 M/My y~ yl Yz Ya Y4 Y5 Y6 Y7 Y8 .)9 Ylo
0 1.87510407 4.69409113 7.85475744 10-99554073 14.13716839 17.27875953 20.42035225 23.56194490 26.70353756 29-84513021
0.20 1.61639966 4.26706157 7.31837267 10.40156263 13.50670225 16.62335441 19.74686001 22.87475293 26.00561758 29.13858564
0.40 1.47240849 4.14443036 7.21548589 10.31780693 13-43667566 16.56341840 19.69456473 22.82841390 25.96403931 29.10089373
0.60 1-37566854 4.08665324 7.17252465 10-28498044 13.41020846 16.54128737 19.67556452 22.81177553 25.94924411 29-08757605
0.80 1.30408675 4.05307815 7.14898484 10.26748665 13-39631447 16.52977831 19.66574657 22.80321764 25-94166078 29.08076863
1"00 1-24791741 4.03113944 7.13413224 10.25662107 13.38775633 16.52272548 19.65975089 22.79800451 25-93704999 29.07663567
164
P. A. A. LAURA, J. L. POMBO AND E. A. SUSEMIIIL TABLE
Yi y, Y2 Y3 Y4 Y5 Y6 y7 Ys Y9 Ylo
Yt Yl
Yz Ya y, ):5 Y6 y~ Ys Y9 Y~o
1"20 1"20206578 4"01568357 7"12390909 10"24921773 13"38195599 16.51796088 19.65570932 22.79449594 25.93395048 29.07385994
2.20 1"05340227 3"97781398 7"09966406 10"23190000 13"36848374 16"50694191 19"64638946 22"78642196 25"92682893 29.06749002
l--conibmed M/Mo
1"40 1"16355780 4'00420785 7"11644347 10'24384957 13"37776560 16.51452647 19.65280046 22.79197344 25.93172388 29'07186719
1"60 1"13051695
3"99535061 7"11075258 10"23977894 13'37459663 16.51193347
19.65060668 22.79007255 25.93004697 29.07036709
2.40 1"03283409 3"97380051 7"09715917 10"23012968 13"36711397 16"50582527 19"64544708 22"78560685 25"92611082 29.06684831
2.~ 1"01412844 3"97037033 7"09502787 10"22862612 13"36595169 16"50487829 19"64464820 22"78491604 25"92550235 29.06630464
1"80 1-10168865 3"98830760 7"10627093 10"23658619 13"37211624 16.50990644 19.64889318 22.78858870 25.92873856 29.06919706
2.80 0"99700223 3"96740490 7"09319239 10"22733327 13"36495308 16"50406505 19"64396235 22"78432310 25"92498017 29.06583814
2"00 1"07619566 3"98257329 7"10265019 10"23401501 13-37012199 16.50827832 19.64751781 22-78739823 25.92768922 29.06825895
3 0'98123061 3"96481577 7"09159514 10"22620973 13"36408583 16"50335908 19"64336713 22"78380862 25"92452716 29.06543348
M/3r Yt
3"2
Yl
0"96663220
Y2
3"96253560
Ya Y4 y~ Y6 y~ Ya Y9 Ylo
7"09019258 10"22522429 13"36332563 16"50274047 19"64284570 22"78335800 25"92413041 29"06507912
3"4 0"95305873 3"96051219 7"08895114 10"22435297 13"36265381
16"50219396 19'64238512 22"78296003 25"92378007 29"06476623
3"6 0"94038758 3"95870450 7"08784459 10"22357702 13"36205581 16"50170762 19'64197535 22"78260599 25'92346844 29"06448793
3"8 0"92851621 3"95707978 7"08685207 10"22288160 13"36152009 16"50127205 19"64160840 22"78228900 25"92318944 29'06423879
4 0"91735814 3"95561158 7"08595681 10"22225479 13"36103740 16"50087969 19"64127790 22"78200353 25"92293820 29-06401445
MIMo Yl Yl Yz Ya y, Y5 Y6 Y7 Ys Y9 Ylo
4.2 0"90683976 3"95427833 7"08514519 10"22168692 13"36060024 16"50052441 19"64097868 22"78174510 25"92271077 29'06381139
4.4 0"89689794 3"95306224 7"08440601 10"22117005 13"36020247 16"50020119 19"64070650 22"78151004 25"92250393 29"06362671
4"6 0"88747820 3"95194852 7"08373000 10'22069760 13"35983898 16"49990588 19"64045784 22"78129531 25"92231499 29"06345804
4.8 0"87853317 3"95092476 7"08310937 10"22026408 13"35950553 16"49963501 19"64022979 22"78109840 25"92214173 29"06330336
5 0"87002146 3"94998049 7"08253760 10"21986487 13"35919854 16"49938568 19"64001989 22"78091716 25"92198228 29'06316102
165
CLAMPED-FREE BEAM WITH A MASS
TABLE l--continued
M/M~ y~
"5"2
5"4
5"6
5"8
6
Yl Yz 5'3 Y4 Y5 Y6 Y7 Ya Y9 YJo
0.86190669 3.94910680 7.08200914 10.21949605 13-35891498 16.49915540 19.63982605 22.78074981 25.92183504 29.06302958
0'85415675 3.94829605 7.08151923 10.21915429 13.35865227 16.49894209 19.63964650 22.78059480 25.92169867 29.06290785
0"84674314 3.94754169 7.08106382 10.21883670 13.35840819 16.49874392 19.63947972 22.78045082 25"92157201 29.06279479
0.83964046 3.94683803 7.08063939 10.21854081 13-35818083 16.49855934 19.63932438 22.78031673 25.92145405 29.06268950
0.83282600 3-94618012 7.08024287 10.21826447 13.35796852 16.49838700 19.63917935 22.78019154 25.92134393 29.06259121
M/Mo Y~
6-2
6.4
6.6
6"8
7
Yt Y2 Y3 Y4 Ys Y6 y7 Ya Y9 Ylo
0"82627938 3"94556364 7"07987160 10"21800581 13"35776982 16"49822573 19.63904364 22.78007440 25-92124089 29-06249925
0"81998222 3"94498480 7"07952324 10"21776317 13"35758347 16"49807448 19-63891637 22.77996455 25.92114428 29.06241301
0"81391791 3"94444026 7"07919574 10"21753513 13"35740833 16"49793235 19.63879679 22.77986134 25-92105349 29.06233199
0"80807141 3"94392705 7"07888727 10"21732039 13"35724344 16"49779855 19.63868421 22-77976418 25.92096804 29.06225572
0'80242903 3"94344254 7"07859623 10"21711783 13"35708792 16"49767235 19.63857804 22.77967255 25.92088745 29.06218380
M/M~ Yi
7"2
7"4
7"6
7"8
8
yl y2 y3 y4 y5 y6 y7 ys y9 ylo
0-79697828 3.94298439 7.07832118 I0.21692643 13.35694099 16.49755314 19-63847775 22.77958600 25-92081133 29.06211586
0-79170777 3.94255051 7.07806084 10.21674531 13.35680196 16.49744034 19.63838286 22.77950411 25.92073931 29.06205159
0.78660703 3.94213902 7.07781405 10.21657366 13.35667021 16.49733345 19.63829295 22.77942652 25-92067107 29.06199069
0.78166649 3.94174823 7.07757979 10.21641074 13-35654518 16.49723203 19.63820763 22.77935290 25.92060632 29.06193291
0.77687729 3.94137662 7.07735712 10.21625592 13.35642637 16.49713565 19.63812656 22.77928295 25.92054480 29.06187801
M/M~ y~
8.2
8.4
8.6
8.8
9
Yz Y2 Y3 Y4 Y5 Y6 Y7 Ys Y9 Ylo
0"77223130 3"94102281 7"07714521 10"21610860 13"35631332 16"49704396 19"63804944 22"77921639 25"92048628 29"06182578
0"76772098 3"94068554 7"07694329 10"21596825 13"35620564 16"49695661 19'63797597 22"77915300 25"92043053 29"06177604
0-76333937 3"94036369 7"07675068 10"21583439 13"35610293 16"49687332 19"6379059I 22"77909256 25"92037737 29"06172860
0"75908001 3"94005622 7"07656674 10"21570658 13"35600488 16"49679379 19"63783903 22-77903485 25"92032663 29-06168332
0"75493689 3"93976219 7.07639090 10"21558441 13"35591116 16"49671779 19"63777511 22"77897970 25"92027813 29'06164004
166
P. A. A. LAURA, J. L. POMBO AND E. A. TABLE
SUSEMIItL
l--continued
M/M. Yl yl y~ ya y4 )'5 Y6 y~ Ys Y9 Ylo
9"2
9"4
0-75090442 3.93948074 7.07622264 10.21546752 13-35582151 16"49664508 19'63771396 22"77892694 25.92023174 29"06159864
9"6
0.74697739 3.93921108 7.07606148 10.21535558 13.35573565 16"49657545 19"63765541 22.77887642 25.92018731 29-06155900
0.74315095 3.93895247 7.07590698 10.21524828 13.35565335 16"49650872 19"63759929 22-77882800 25-92014474 29'06152101
9"8 0.73942054 3.93870426 7-07575873 10.21514533 13.35557440 16"49644470 19"63754545 22.77878155 25"92010389 29-06148457
10 0.73578192 3.93846583 7.07561637 10.21504648 13.35549859 16"49638323 19'63749376 22.77873696 25.92006468 29'06144958
Table 1 shows the values of the first ten roots of equation (11) for M[Mv = 0, 0.20, 0.40, . . . , 10.
4. DETERMINATION OF MODAL SHAPES From equations (8a), (8b) and (8c) one obtains x
X(x/L) = Ct cosyi L
cosyl+coshyj . x x sinyi + sinhyl smyt-~ - c o s h y i ~ +
cosy~ + coshyi x] + ~ ~ sinhy,~].
(14)
The first five modes of vibration are shown in Figures 2 (a)-(f) for M[M~ = O, 0-20,'0.40, 0.60, 0.80 and 1.00. In the case where M[Mv = 0 the calculated co-ordinates of the nodes are in good agreement with those available in the technical literature. The only noticeable difference is in the case of the second node corresponding to the second mode since reference [5] indicates (x/L)2_2 = 0.774 while the result obtained in the present investigation is 0-783.
5. VARIATION OF THE MAXIMUM DYNAMIC STRESS WITH THE PARAMETER M/Mv For the first mode one has
W, = C1 X(x/L) V(t).
(15)
Taking Cx = l/X(x]L)lx_z and replacing this in equation (15) results in the expression
Wt =
1
X(xlL)I~_L
X(x/L) T(t).
(I 6)
In this form, the tip deflection is equal to unity. Using well-known results from the Bernoulli-Navier beam theory one can easily show
CLAMPED-FREE
BEAM
WlTII
167
A MASS
I
I
I
I
1
I
I
i
I
I
I
I
I
I
-I
I
-
#
0
~
0`1 0'2 03 0.4 0 5 0 . 6 0-7 08 0.9
I
o
0
o., ' o 2' o-3 ' c>, ' 0`~ ' o ' 6 o ! ~ o ,'.o . 9'
OI 0 " 2 0 " 3 0 4 0 5 0 6 0 " 7 0 8 0 " 9
x/L
(o) I
i
i
I
I
I
(b) I I
(c) l
l
ILl
I
I
I
I
I
I
I
I
I
-t~ ~
o
0.98?
I
-II
I
I
I
I
I
I
I
I
l
0 O'l 0 2 0"3 0"4 0 5 0 6 07 0 8 0 " 9
0
OI 0 " 2 0 3 0 4 0 " 5 0 6 0 " 7 0 8 0 " 9
I
0 0`1 0`20'30"4 0 5 0 6 07 0 8 0 9
x/L
(d)
(e)
(f)
Figure 2. Modes of vibration. (a) M/M,, = O; (b) M/M,, = 0.20; (c) MIM,, = 0.40; (d) M/M~= 0.60; {e) M/Mo = 0.S0; (f) M/M~ = 1.00.
I
168
P . A . A . LAURA, J. L. POMBO AND E. A. SUSEMilIL
that the amplitude of the maximum dynamic stress is given by [trxlm.x = E h / 2
x -(o)
(17)
X(x/Z.)lx.,
Evaluating X"(0) by means of equation (14), substituting this in equation (17) and conveniently defining a dimensionless parameter
I o'xl,.., L 2 Eh/2 one obtains (kL) 2 (18)
~, = - 2 -X ( x l L ) I . . L "
The numerical results are tabulated in Table 2 for several values of the ratio M[M,.. It is TABLE 2
Variation of the amplitude of the dynamic stress as a function of the parameter M/Mo y = I,,A~..v
M[Mo
(k~L)
Eh]2
0 0.2 0.4 0.6 0.8 I 2 3 4 5 6 7 8 9 10
1.87510407 1.61639966 1"47240849 1.37566854 1.30408675 1.24791741 1.07619566 0.98123061 0-91735814 0.87002146 0.83282600 0.80242903 0.77687729 0-75493689 0.73578192
3.52 3.28 3.19 3.14 3.11 3.10 3.05 3.04 3.03 3.02 3.02 3.02 3.01 3.01 3.01
observed that ), decreases as M]M~ increases, as is to be expected, in view of the fact that the frequency coefficient decreases.
REFERENCES 1. D. YOUNGand R. P. EELGAR,JR. ] 949 The University of Texas Publication, Austin, Texas, No. 4913. Tables of characteristic functions representing normal modes of vibration of a beam. 2. R. P. FELGAR,JR. 1950 The University of Texas, Austin, Texas, Bureau of Engineering Research, Circular No. 14. Formulas for integrals containing functions of a vibrating beam. 3. J. PRESCOrr 1924 Applied Elasticity. London: Constable and Company. See pp. 213-218. 4. S.D. CoN'rE 1965 Elementary NumericalAnalysis. New York: McGraw-Hill Book Company, Inc. See p. 40. 5. C. M. HARRISand C. E. CREDE (Editors) 1961 Shock and Vibration Handbook, Volume I. New York: McGraw-Hill Book Company, Inc. See pp. 1-14.
A NOTE
O N T H E VIBRATIONS OF A C L A M P E D - F R E E BEAM W I T H A MASS AT T H E FREE E N D P. A. A. LAURA, J. L. POMBOAND E. A. SUSEMIHL Department of Engineering, UniversidadNacional del Sur, Bahia Blanca, Argenthta
(Received4 December 1973, and in revisedform 18 April 1974) This note is concerned with the determination of natural frequencies and modal shapes of a clamped-free beam which carries a finite mass, M, at the free end. Accurate results are presented for different (concentrated mass)/(beam mass) ratios, M[Mo. It is also the purpose of this note to analyze the variation of the maximum dynamic stress as a function of the parameter M/M~ for the first mode of vibration.
1. INTRODUCTION
Characteristic functions and frequencies have been extensively tabulated for nearly all common types of beams: clamped-clamped, damped-free, clamped-supported, freesupported and supported-supported [1, 2]. This note deals with the determination of the first ten natural frequencies of a clamped-free beam which carries a finite mass, M, at the free end (Figure 1). Shear and rotatory inertia effects are disregarded.
:'1
L
1. / Figure 1. Beam-mass system studied in the present investigation. It is important to point out that the frequency equation was obtained by Prescott [3] but no extensive numerical information has been published in the open literature. Modal shapes and the effect of the mass M on the maximum dynamic stress when the structure vibrates in the first mode are also investigated.
2. MATHEMATICAL ANALYSIS The problem under consideration is described by the following differential system: 0* w
eIT
azw
+pA 161
=o,
(1)
162
P. A. A. LAURA, J. L. POMBO AND E. A. SUSEMIilL
aw
w(0, t) = ~
(2a, b)
(o, t) = 0,
ox
0 2 It'
Oxz (L, t) = 0,
-
(3)
] 02 w 0 3 w., (L, t) = M - - ~ ( L , - E l ax 3
t),
(4)
where E is the Young's modulus, I the constant moment of inertia, A the constant crosssectional area, p the mass density and M the concentrated mass at the free end. Using the standard method of separation of variables, one assumes "(5)
w(x,t)=X(x)T(t).
Substituting this form in equation (1) results in the equality X (~v) pA T" . . . . . . X EI T
k*.
(6)
The solution of the ordinary differential equation X ~tv~- k 4 X = 0 is simply X = C~ cos k x + C2 sin k x + Ca cosh k x + C4 sinh kx.
(7)
Substituting expression (7) in equation (2a, b) results in the equations C, + C3 = 0,
(8a)
c , + G = o.
(Sb)
From equations (7) and (3) one obtains - C~ cos k L - C2 sin k L + C3 cosh k L + C4 sinh k L = O.
(8c)
Substituting equation (7) in equation (4) results in the algebraic expression Ct sin k L - C2 cos k L + C3 sinh k L + C4 cosh k L = -1 - - - k M ( C t coskL + C 2 s i n k L + C 3 c o s h k L + C4sinhkL). pA
(8d)
The solution of equations (8) leads to the following determinantal equation in the eigenvalues, k: gll(k)
g12(k) = o,
(9)
CLAMPED-FREE BEAM WITIt A MASS
163
where gl l(k) = cos kL + cosh kL,
(10a)
g12(k) = sinkL + sinhkL,
(lOb)
g2~(k) = sin kL - sinh kL + (kL) M ( - cosh kL + cos kL),
(lOc)
(kL) M g22(k) = - cos kL - cosh kL + ~ ( - sinh kL + sin kL).
(lOd)
pAL
Equations (9) and (10) lead to the transcendental equation 1
l + cosycoshy
M
y sinycoshy-cosysinhy
(ll)
ML.'
where My is the beam mass (M~ = pAL) and y = kL. The frequencies, f~, are then given by
A
(12)
pA"
3. DETERMINATION OF THE EIGENVALUES In order to calculate the roots, yt, of the transcendental equation (11) it is convenient to express it in the form M
z(y) = - - y ( c o s y s i n h y - s i n y c o s h y ) + c o s y c o s h y + I = 0.
(I 3)
Mr The function z(y) shows rapid oscillations, attaining very large values between successive roots. The slope of the function at each root is, therefore, very close to vertical. This characteristic of the function suggests the use of the method of false position to ensure convergence in the determination of the roots [4]. In order to initiate the iterative process, the roots were first bracketed by means of a straight search process. The algorithmic procedure was implemented in a Hewlett-Packard 9810 A. TABLE 1
Values ofthefirst ten roots of equation (1 l) for M/My = 0, 0-20, 0 . 4 0 , . . . , 10 M/My y~ yl Yz Ya Y4 Y5 Y6 Y7 Y8 .)9 Ylo
0 1.87510407 4.69409113 7.85475744 10-99554073 14.13716839 17.27875953 20.42035225 23.56194490 26.70353756 29-84513021
0.20 1.61639966 4.26706157 7.31837267 10.40156263 13.50670225 16.62335441 19.74686001 22.87475293 26.00561758 29.13858564
0.40 1.47240849 4.14443036 7.21548589 10.31780693 13-43667566 16.56341840 19.69456473 22.82841390 25.96403931 29.10089373
0.60 1-37566854 4.08665324 7.17252465 10-28498044 13.41020846 16.54128737 19.67556452 22.81177553 25.94924411 29-08757605
0.80 1.30408675 4.05307815 7.14898484 10.26748665 13-39631447 16.52977831 19.66574657 22.80321764 25-94166078 29.08076863
1"00 1-24791741 4.03113944 7.13413224 10.25662107 13.38775633 16.52272548 19.65975089 22.79800451 25-93704999 29.07663567
164
P. A. A. LAURA, J. L. POMBO AND E. A. SUSEMIIIL TABLE
Yi y, Y2 Y3 Y4 Y5 Y6 y7 Ys Y9 Ylo
Yt Yl
Yz Ya y, ):5 Y6 y~ Ys Y9 Y~o
1"20 1"20206578 4"01568357 7"12390909 10"24921773 13"38195599 16.51796088 19.65570932 22.79449594 25.93395048 29.07385994
2.20 1"05340227 3"97781398 7"09966406 10"23190000 13"36848374 16"50694191 19"64638946 22"78642196 25"92682893 29.06749002
l--conibmed M/Mo
1"40 1"16355780 4'00420785 7"11644347 10'24384957 13"37776560 16.51452647 19.65280046 22.79197344 25.93172388 29'07186719
1"60 1"13051695
3"99535061 7"11075258 10"23977894 13'37459663 16.51193347
19.65060668 22.79007255 25.93004697 29.07036709
2.40 1"03283409 3"97380051 7"09715917 10"23012968 13"36711397 16"50582527 19"64544708 22"78560685 25"92611082 29.06684831
2.~ 1"01412844 3"97037033 7"09502787 10"22862612 13"36595169 16"50487829 19"64464820 22"78491604 25"92550235 29.06630464
1"80 1-10168865 3"98830760 7"10627093 10"23658619 13"37211624 16.50990644 19.64889318 22.78858870 25.92873856 29.06919706
2.80 0"99700223 3"96740490 7"09319239 10"22733327 13"36495308 16"50406505 19"64396235 22"78432310 25"92498017 29.06583814
2"00 1"07619566 3"98257329 7"10265019 10"23401501 13-37012199 16.50827832 19.64751781 22-78739823 25.92768922 29.06825895
3 0'98123061 3"96481577 7"09159514 10"22620973 13"36408583 16"50335908 19"64336713 22"78380862 25"92452716 29.06543348
M/3r Yt
3"2
Yl
0"96663220
Y2
3"96253560
Ya Y4 y~ Y6 y~ Ya Y9 Ylo
7"09019258 10"22522429 13"36332563 16"50274047 19"64284570 22"78335800 25"92413041 29"06507912
3"4 0"95305873 3"96051219 7"08895114 10"22435297 13"36265381
16"50219396 19'64238512 22"78296003 25"92378007 29"06476623
3"6 0"94038758 3"95870450 7"08784459 10"22357702 13"36205581 16"50170762 19'64197535 22"78260599 25'92346844 29"06448793
3"8 0"92851621 3"95707978 7"08685207 10"22288160 13"36152009 16"50127205 19"64160840 22"78228900 25"92318944 29'06423879
4 0"91735814 3"95561158 7"08595681 10"22225479 13"36103740 16"50087969 19"64127790 22"78200353 25"92293820 29-06401445
MIMo Yl Yl Yz Ya y, Y5 Y6 Y7 Ys Y9 Ylo
4.2 0"90683976 3"95427833 7"08514519 10"22168692 13"36060024 16"50052441 19"64097868 22"78174510 25"92271077 29'06381139
4.4 0"89689794 3"95306224 7"08440601 10"22117005 13"36020247 16"50020119 19"64070650 22"78151004 25"92250393 29"06362671
4"6 0"88747820 3"95194852 7"08373000 10'22069760 13"35983898 16"49990588 19"64045784 22"78129531 25"92231499 29"06345804
4.8 0"87853317 3"95092476 7"08310937 10"22026408 13"35950553 16"49963501 19"64022979 22"78109840 25"92214173 29"06330336
5 0"87002146 3"94998049 7"08253760 10"21986487 13"35919854 16"49938568 19"64001989 22"78091716 25"92198228 29'06316102
165
CLAMPED-FREE BEAM WITH A MASS
TABLE l--continued
M/M~ y~
"5"2
5"4
5"6
5"8
6
Yl Yz 5'3 Y4 Y5 Y6 Y7 Ya Y9 YJo
0.86190669 3.94910680 7.08200914 10.21949605 13-35891498 16.49915540 19.63982605 22.78074981 25.92183504 29.06302958
0'85415675 3.94829605 7.08151923 10.21915429 13.35865227 16.49894209 19.63964650 22.78059480 25.92169867 29.06290785
0"84674314 3.94754169 7.08106382 10.21883670 13.35840819 16.49874392 19.63947972 22.78045082 25"92157201 29.06279479
0.83964046 3.94683803 7.08063939 10.21854081 13-35818083 16.49855934 19.63932438 22.78031673 25.92145405 29.06268950
0.83282600 3-94618012 7.08024287 10.21826447 13.35796852 16.49838700 19.63917935 22.78019154 25.92134393 29.06259121
M/Mo Y~
6-2
6.4
6.6
6"8
7
Yt Y2 Y3 Y4 Ys Y6 y7 Ya Y9 Ylo
0"82627938 3"94556364 7"07987160 10"21800581 13"35776982 16"49822573 19.63904364 22.78007440 25-92124089 29-06249925
0"81998222 3"94498480 7"07952324 10"21776317 13"35758347 16"49807448 19-63891637 22.77996455 25.92114428 29.06241301
0"81391791 3"94444026 7"07919574 10"21753513 13"35740833 16"49793235 19.63879679 22.77986134 25-92105349 29.06233199
0"80807141 3"94392705 7"07888727 10"21732039 13"35724344 16"49779855 19.63868421 22-77976418 25.92096804 29.06225572
0'80242903 3"94344254 7"07859623 10"21711783 13"35708792 16"49767235 19.63857804 22.77967255 25.92088745 29.06218380
M/M~ Yi
7"2
7"4
7"6
7"8
8
yl y2 y3 y4 y5 y6 y7 ys y9 ylo
0-79697828 3.94298439 7.07832118 I0.21692643 13.35694099 16.49755314 19-63847775 22.77958600 25-92081133 29.06211586
0-79170777 3.94255051 7.07806084 10.21674531 13.35680196 16.49744034 19.63838286 22.77950411 25.92073931 29.06205159
0.78660703 3.94213902 7.07781405 10.21657366 13.35667021 16.49733345 19.63829295 22.77942652 25-92067107 29.06199069
0.78166649 3.94174823 7.07757979 10.21641074 13-35654518 16.49723203 19.63820763 22.77935290 25.92060632 29.06193291
0.77687729 3.94137662 7.07735712 10.21625592 13.35642637 16.49713565 19.63812656 22.77928295 25.92054480 29.06187801
M/M~ y~
8.2
8.4
8.6
8.8
9
Yz Y2 Y3 Y4 Y5 Y6 Y7 Ys Y9 Ylo
0"77223130 3"94102281 7"07714521 10"21610860 13"35631332 16"49704396 19"63804944 22"77921639 25"92048628 29"06182578
0"76772098 3"94068554 7"07694329 10"21596825 13"35620564 16"49695661 19'63797597 22"77915300 25"92043053 29"06177604
0-76333937 3"94036369 7"07675068 10"21583439 13"35610293 16"49687332 19"6379059I 22"77909256 25"92037737 29"06172860
0"75908001 3"94005622 7"07656674 10"21570658 13"35600488 16"49679379 19"63783903 22-77903485 25"92032663 29-06168332
0"75493689 3"93976219 7.07639090 10"21558441 13"35591116 16"49671779 19"63777511 22"77897970 25"92027813 29'06164004
166
P. A. A. LAURA, J. L. POMBO AND E. A. TABLE
SUSEMIItL
l--continued
M/M. Yl yl y~ ya y4 )'5 Y6 y~ Ys Y9 Ylo
9"2
9"4
0-75090442 3.93948074 7.07622264 10.21546752 13-35582151 16"49664508 19'63771396 22"77892694 25.92023174 29"06159864
9"6
0.74697739 3.93921108 7.07606148 10.21535558 13.35573565 16"49657545 19"63765541 22.77887642 25.92018731 29-06155900
0.74315095 3.93895247 7.07590698 10.21524828 13.35565335 16"49650872 19"63759929 22-77882800 25-92014474 29'06152101
9"8 0.73942054 3.93870426 7-07575873 10.21514533 13.35557440 16"49644470 19"63754545 22.77878155 25"92010389 29-06148457
10 0.73578192 3.93846583 7.07561637 10.21504648 13.35549859 16"49638323 19'63749376 22.77873696 25.92006468 29'06144958
Table 1 shows the values of the first ten roots of equation (11) for M[Mv = 0, 0.20, 0.40, . . . , 10.
4. DETERMINATION OF MODAL SHAPES From equations (8a), (8b) and (8c) one obtains x
X(x/L) = Ct cosyi L
cosyl+coshyj . x x sinyi + sinhyl smyt-~ - c o s h y i ~ +
cosy~ + coshyi x] + ~ ~ sinhy,~].
(14)
The first five modes of vibration are shown in Figures 2 (a)-(f) for M[M~ = O, 0-20,'0.40, 0.60, 0.80 and 1.00. In the case where M[Mv = 0 the calculated co-ordinates of the nodes are in good agreement with those available in the technical literature. The only noticeable difference is in the case of the second node corresponding to the second mode since reference [5] indicates (x/L)2_2 = 0.774 while the result obtained in the present investigation is 0-783.
5. VARIATION OF THE MAXIMUM DYNAMIC STRESS WITH THE PARAMETER M/Mv For the first mode one has
W, = C1 X(x/L) V(t).
(15)
Taking Cx = l/X(x]L)lx_z and replacing this in equation (15) results in the expression
Wt =
1
X(xlL)I~_L
X(x/L) T(t).
(I 6)
In this form, the tip deflection is equal to unity. Using well-known results from the Bernoulli-Navier beam theory one can easily show
CLAMPED-FREE
BEAM
WlTII
167
A MASS
I
I
I
I
1
I
I
i
I
I
I
I
I
I
-I
I
-
#
0
~
0`1 0'2 03 0.4 0 5 0 . 6 0-7 08 0.9
I
o
0
o., ' o 2' o-3 ' c>, ' 0`~ ' o ' 6 o ! ~ o ,'.o . 9'
OI 0 " 2 0 " 3 0 4 0 5 0 6 0 " 7 0 8 0 " 9
x/L
(o) I
i
i
I
I
I
(b) I I
(c) l
l
ILl
I
I
I
I
I
I
I
I
I
-t~ ~
o
0.98?
I
-II
I
I
I
I
I
I
I
I
l
0 O'l 0 2 0"3 0"4 0 5 0 6 07 0 8 0 " 9
0
OI 0 " 2 0 3 0 4 0 " 5 0 6 0 " 7 0 8 0 " 9
I
0 0`1 0`20'30"4 0 5 0 6 07 0 8 0 9
x/L
(d)
(e)
(f)
Figure 2. Modes of vibration. (a) M/M,, = O; (b) M/M,, = 0.20; (c) MIM,, = 0.40; (d) M/M~= 0.60; {e) M/Mo = 0.S0; (f) M/M~ = 1.00.
I
168
P . A . A . LAURA, J. L. POMBO AND E. A. SUSEMilIL
that the amplitude of the maximum dynamic stress is given by [trxlm.x = E h / 2
x -(o)
(17)
X(x/Z.)lx.,
Evaluating X"(0) by means of equation (14), substituting this in equation (17) and conveniently defining a dimensionless parameter
I o'xl,.., L 2 Eh/2 one obtains (kL) 2 (18)
~, = - 2 -X ( x l L ) I . . L "
The numerical results are tabulated in Table 2 for several values of the ratio M[M,.. It is TABLE 2
Variation of the amplitude of the dynamic stress as a function of the parameter M/Mo y = I,,A~..v
M[Mo
(k~L)
Eh]2
0 0.2 0.4 0.6 0.8 I 2 3 4 5 6 7 8 9 10
1.87510407 1.61639966 1"47240849 1.37566854 1.30408675 1.24791741 1.07619566 0.98123061 0-91735814 0.87002146 0.83282600 0.80242903 0.77687729 0-75493689 0.73578192
3.52 3.28 3.19 3.14 3.11 3.10 3.05 3.04 3.03 3.02 3.02 3.02 3.01 3.01 3.01
observed that ), decreases as M]M~ increases, as is to be expected, in view of the fact that the frequency coefficient decreases.
REFERENCES 1. D. YOUNGand R. P. EELGAR,JR. ] 949 The University of Texas Publication, Austin, Texas, No. 4913. Tables of characteristic functions representing normal modes of vibration of a beam. 2. R. P. FELGAR,JR. 1950 The University of Texas, Austin, Texas, Bureau of Engineering Research, Circular No. 14. Formulas for integrals containing functions of a vibrating beam. 3. J. PRESCOrr 1924 Applied Elasticity. London: Constable and Company. See pp. 213-218. 4. S.D. CoN'rE 1965 Elementary NumericalAnalysis. New York: McGraw-Hill Book Company, Inc. See p. 40. 5. C. M. HARRISand C. E. CREDE (Editors) 1961 Shock and Vibration Handbook, Volume I. New York: McGraw-Hill Book Company, Inc. See pp. 1-14.