Effect of rotatory inertia and shear deformation on vibration of an inclined bar with an end constraint

Journal of Sound and Vibration (1985) 101(2), 171-180

EFFECT OF ROTATORY INERTIA AND SHEAR DEFORMATION ON VIBRATION OF AN INCLINED BAR WITH AN END CONSTRAINT C.

H.

CHANG AND

Y.

C. JUAN

Department of Engineering Mechanics, The University of Alabama, University, Alabama 35486 U.S.A.

(Received 18 June 1984, and in revised form 1 August 1984)

A set of equations for the free vibration of an inclined bar with an end constraint including the effect of rotatory inertia and shear deformation is derived by the variational method. The equations for the axial and transverse vibrations are coupled by the end constraint. Exact solutions for the linearized equations are obtained. The effect of rotatory inertia without and with shear deformation is examined. It is shown by the characteristic equation that when the shear deformation is included, the characteristic values may change from real values for low frequencies to imaginary values for high frequencies. Frequency spectra of the first six (some seven) modes for various end conditions are presented.

1. INTRODUCTION

An inclined bar with one end fixed in space and the other end moveable along a prescribed

path constitutes a variable end point problem. The moveable end condition is known as a transversality condition in the calculus of variations [1]. However, when the prescribed path is a vertical straight line, the approach may be simplified by incorporating the end constraint into the system by using a Lagrange multiplier. This has been illustrated for buckling [2] as well as vibration [3] of such inclined bars. The end constraint involves both the axial and transverse displacements. Thus in the vibration of such an inclined bar the two displacements are coupled. The coupling effect as shown in reference [3] on the frequencies of vibration depends on the angle ofinclination as well as on the slenderness ratio. The smaller the slenderness ratio, the larger the coupling effect will be. This is quite similar to what happens in beam vibrations in that the effect of the shear deformation and rotatory inertia will be larger if the slenderness ratio gets smaller. This leads to a question: How will the shear deformation and rotatory inertia influence the coupling effect in the vibration of inclined bars? This paper is concerned with this question. The variational method is used in the derivation of the equations of motion and boundary conditions. A brief discussion on the change of mode functions at high frequencies (the so-called "second spectrum") is made. The exact solutions for the linearized equations and basic frequencies of vibration for inclined bars with various end conditions are presented. The present work may be considered as an extension of the investigation described in reference [3]; therefore, the same nomenclature will be used herein, with some specified additions. 171 0022-460X/85/140171 + 10 $03.00/0

© 1985 Academic Press Inc. (London) Limited

172

C. H. CHANG AND Y. C. JUAN

2. FORMULATION

Let cjJ be the angle of rotation due to bending only. The kinetic energy and potential energy of an inclined bar undergoing free vibration may be given as, respectively, (La) (1b)

where (2)

in which K is the Timoshenko shear coefficient [4-8]. The upper end, moveable along a straight path, has a constrained condition I

V sin a + W cos a

=

°

at 1/

=

1.

(3)

By combining expressions (1) and (3), a functional equivalent to Lagrange's function may be formed:

L=T-V+A f

tl

(4)

[Vsina+Wcosa]1J=ldt.

to

Applying the following three Lagrange-Euler equations

dL)'+ (aL) - -0 (-au aV. 1J,1J - ,

. + (aL) -(-aL)' aw

aW,1J.1J

=0

'

aL) ' _ aL + (~) -0 ( a¢ acjJ a¢.1J,1J(Sa-c)

to functional (4) results in three equations of motion: (1/ /32) U- (v. 1J +4 w~1J).1J = 0, (1/ /32) W-[(

(6a)

v. 1J +4 w~1J) w,1JJ.1J -

S2("'.1J - cjJ)", = 0, 2 (1/ R2/32)~ - 0/ R ) cjJ.1JTJ - S2( w,1J - ¢) = 0.

(6b) (6c)

Through the variational method, the following boundary conditions are obtained:

{ ucjJ ==o, 0,

W=O}

at1/=O;

(7a-d)

or ¢,1J = 0

( V,1J +4 W~TJ) w,1J + S2( W,TJ - cjJ) = -A cos a} V.TJ+!W:1J=A sin « { ¢ =0 or ¢.1J =0 V sin 0'+ W cos a

=0

at 1/ = 1;

(8a-d)

at 1/ = 1.

(3)

Eliminating the function ¢ from equations (6b) and (6c) converts the latter into

0/ /32" )W -

(s

2

.. 22 22" 2 1 + 1) w,TJ1J + /3 S W:TJTJ1J1J + s R W + /3 [( V. 1J +2: W,,,) W. 1J J,1J1J1J

- [( V,1J +4 W~1J) w,TJJ:~ - S2 R 2 /32[( V,1J +4 W~1J) w,1JJ,1J = O.

(9)

The cjJ's involved in the boundary conditions also may be expressed in terms of the functions Wand U. Probably no exact solutions of the non-linear equations (6a) and (9) may ever be obtained. For the purpose of the present study, these two equations may be linearized

173

INCLINED BAR WITH END CONSTRAINT

by expressing the two displacement functions as power series in the amplitudej, which is assumed to be small in comparison to the dimensions of the bar. Thus let

W=gW1+eW2 +· · ·,

U=gV I+g 2V2 + · · · ,

(lOa,b)

be substituted in equations (6a) and (9). For small vibrations, the equations associated with the first order terms in g are sufficient. They are (lla) VI - f32 VI."I'rJ = 0, 2

2"

24

222"

W 1 - f3 (l+s )W1,TlTl+S f3 WI.Tl1)"I"I+R f3 S W1=0.

(l lb)

The latter is known as the equation of the Timoshenko beam, of which the general solution has been discussed in references [9,10]. With (l2a, b)

WI = w( 7J) sin pt

VI = u( '17) sin pt, equations (11) become w",' + (el R

2)(1

u"+ (el R 2 )u

(I3a)

4

C13b)

= 0, + II S2)W" + k [(e l R 4 s 2 ) -1]w = 0,

with the following boundary conditions:

u =0,

w=O,

w' = 0 or w" =

u'+Asina=O,

w'=

°

or

Will -

°

2

R A cos

w"=o at

a

at

7)

= 0,

= 0,

'17 = 1.

(l4a-d) (ISa,b) (1Sc, d)

3, SOLUTIONS

The solution of equation (13a) satisfying conditions (l4a) and (l5a) is u( '17) = -CARl k

2

)

sin a sin

(el R)'I7/cos (el R).

(16)

Neglecting the effect of shear deformation and including the rotatory inertia alone simplifies equation (13b) to w" I + ( k 4 1R 2 )w" - e w =0.

(17)

The general solution of equations (17) and (13b) is w( TJ) =

CI

sin k, TJ + C2 cos k l 7) +c3 sinh k2 7) + C4 cosh k2 '17

(18)

in which k, and k2 are characteristic values obtained from,respectively, kl =

(e I R){±~+U+ (RI k)4]1/2}1/2,

(19)

2

k,. = (e I R){ ±~(I + II S2) + [;\(1 + II s2f + (RI k)4 - (II S2)]1/2}1/2,

(20)

2

The values of k, given by equation (19) are always real; hence the solution of equation 2

(18) is valid for all values of k. However, when the effect of shear deformation is taken into consideration, equation (18) is true onlyfor limited values of k. Equation (20) yields a real value of k 2 when k < RI';. For k> R.J S, k2 is an imaginary number; then the mode equation (18) needs to be changed accordingly, with the hyperbolic functions becoming sine and cosine functions.

174

C. H. CHANG AND Y. C. JUAN

This change in frequencies was first observed by Traill-Nash and Collar [11] in 1953 and was considered as a second spectrum of frequencies for the free vibration of Timoshenko's beam. Since then, this phenomenon has been discussed in a number of papers (see the references in reference [12]). Recently, however, it has been pointed out by Levinson and Cooke [12] that such a second spectrum is merely due to the fact, as observed in the last paragraph, that when the frequencies increase one of the characteristic values will change from real to imaginary. The frequencies, however, are still in sequence. Thus, there is no second spectrum. The discussions in references [11] and [12] were for simply supported beams. It is clear from the present equation (20) that such a change of frequencies from real to imaginary values and hence the change of normal mode functions may take place regardless of the end conditions; indeed the change is built into the characteristic equation. The present work is aimed at the comparison of the coupling effect on the frequencies of free vibration of an Euler-Bernoulli inclined bar, as discussed in reference [3], with that of a Timoshenko inclined bar, for frequencies up to those available in reference [3]. In the later numerical examples, R = 50 and S2 =~, which is for K' = i (for a rectangular section [4]),_E = 30 x 106 psi and G = 12 X 106 psi (for steel) will be used. The limiting k value is RJ s:= 37·99 which is much higher than those presented in reference [3]. Thus no change of the k 2 values from real to imaginary will take place and no further discussion about such changes is necessary for the present study. The four constants of integration, C\> C2, C3 and C4, in equation (18) are to be determined by the end conditions of the bars considered in what follows. 3.1. INCLINED BARS WITH HINGED-"HINGED" ENDS For an inclined bar with the bottom hinged and the top end free to rotate and moveable in the vertical direction, the solution of equation (18) satisfying conditions (14b, d) and (15b, d) assumes the form

( ) _ AR2 cos a k~ sin hk 2 sin k1'T/ + ki sin k 1 sin hk 2'T/ W 'T/ ki k~ k 2 sin k, cos hk2 - k, cos k. sin hk 2 •

(21)

Substitution of equations (16) and (21) into the constraint condition (3) for A¥-O yields the frequency equation tan 2 a+ k 2R cot (k 2I When a

= 0,

R)(2+2 1 1)

sin k] sin hk. 2 k 1 k 2 k, cos k. sin hk 2 - k2 sm k, cos hk;

0.

(22)

equation (22) calls for either cos

(el R) = 0, then k 21 R = 1'/2, 31'12, 57T/2, ... ,

or k =.j l'R12, J37TRI2, JS7TR/2, ... ,

(23)

or sin k, == 0, When a

then k] = 0,1', 27T, 37T, ....

(24)

= 7T12, either

sin (k 21 R)

= 0,

then k 21 R

= 1', 27T, 31T, ... ,

or k =J;R,.J);; R, & R, ... , (25)

or (26)

17,

i

,

I

I

I

I

I

I

17

I

'-'-'-'-'-'- -'-'-'

I

1

I

I

I

I

I

I

I

.:::4

J31TR~:~__

1

14 13

-

[;j[

121=

-'-'-'-'-'-'-'-

I

-

--"--'

.....j .rr;R

z

o

II 10

.---_.-.

hiR/zl 8

7

""-.

~

K

'""" .."..

~

I

l

J1TR/2 ""'-

8

71=- _._.-

6

-'-'--

c:Z

tIl

0

~.~

K

'-'-

""-.

-

""'-

I

- '- .- -

""'-

61-

I

r:D

>-

;>::J

~

:j

::r:

tn

Z 0

()

5

"-

51-

4

3F 2

0°

_

A a

Figure 1. Frequencies of hinged-"hinged" bars _. _ R' --,R&S. . "

I

0

4k-

I

.,z

31-

'\~

z.,

21-

A

en

~

0°

a Figure 2. Frequencies of hinged- "clamped" bars. - ' _ R' --,R&S. ' ,

......

......

v.

-...l

17,

I

J

I

I

- ' - '- - '-

16

i

I

j

j

- '-'- - ' - '- -

I

17/

15

14

- '- '- -- - '11TR

10' -- -- - - j~Rl2f-

-

- -- --

0\

I

1

_._._-_.- ._ .- - '- '- '- '-

l

o ;t

o

-=-

j~RI2F

8~ __ - '~-' - '

~ _ =.:..::::::..:.~ . _. _ . _ ._._ _"'='=-:--_;:,... _~

7 6

5

51

4r

__

~

__

4

~

3

2

00

00

10'

20' 30'

a

Figure 3. Frequencies of clamped -"hinged" bars. - . - , R; , R & S.

:t

» Z o » z o

< o

6

2

I

(0

K

3

I

--.-----

"r=---

- '-'-'-'--

8

71- -

I

/2

I

II

I

__-

13

12.-

K

I

14

'- '- '- '-' -

/3

I

J3~R;:r

I

J3.1TR/2k

,

Figure 4. Frequencies of c1amped- "c1amped" bars. - . - R· - -, R& S_ ••

.....

c z

»

177 2 Solutions of equations (23) and (25) for k j R are independent of Rand s. However, those of equations (24) and (26) are affected by these parameters. For a bar with R = 50 and S2 = the first seven sets of curves for the frequency k versus the angle a, as determined from equation (22), are presented in Figure l. In Figures 1-4, the dash-dot curves labeled "R" are obtained from equation (17) including rotatory inertia alone, and the solid lines, "R and S", are obtained from equation INCLINED BAR WITH END CONSTRAINT

to

1

TABLE

First six frequencies, R a

Eqs.

First

Second

Third

= 50,

s2_1 -3

Fourth

Fifth

Sixth

Hinged-"hinged" bars 0°

E-B (17) (13b)

3·142 3·136 3'127

6·283 6·257 6'188

8·862 8·862 8·862

9·425 9·336 9·116

12-566 12·373 11-901

15'350 15'350 14·510

90°

E-B (17) (13b)

0 0 0

3·927 3·919 3·908

7'069 7·036 6·951

10·210 10'115 9·866

12·533 12'533 12·533

13'352 13·134 12·610

Clamped-"clamped" bars 0°

E-B (17) (13b)

4·730 4·727 4·707

7'853 7-818 7·718

8·862 8'862 8·862

10·996 10·895 10'611

14·137 13·923 13·345

15·708 15·708 15·708

90°

E-B (17) (13b)

2-365 2·362 2·359

5-498 4·482 5·442

8·639 8'583 8·428

11-781 11-644 11-264

12·533 12-533 12-533

14·923 14-649 13'935

Fifth

Sixth

TABLE

2

First frequencies for R = 50, a

Eqs.

First

Second

Third

S2

== 1

Fourth

Bottom hinged-top "clamped" bars 0°

E-B (17) (l3b)

3-927 3·920 3-908

7·089 7-037 6·952

8·862 8·862 8'862

10·210 10'114 9·863

13·352 13'138 12'621

15·350 15'350 15·350

90°

E-B (17) (13b)

1-875 1·867 1-867

4·694 4·685 4·664

70855 7·817 7·717

10-996 10·828 10·612

12'533 12·533 12-533

14·137 13·904 13'339

Bottom clamped-top "hinged" bars 0°

E-B (17) (13b)

3·927 3-920 3·908

7·069 7·037 6·951

8'862 8·862 8·862

10·210 10·114 9·863

13'352 13·138 12·621

15'350 15·350 15·350

90°

E-B (17) (13b)

1·571 1·563 1·561

4·712 4·702 4·668

70854 70806 7-672

10·996 10·865 10·531

12·533 12-533 12·533

14'137 13·864 13·222

178

C. H. CHANG AND Y. C. JUAN

(13b) including both rotatory inertia and shear deformation. The labels "R" and "S" and the parameters of slenderness ratio, R, and shear coefficient, S, should not be confused. The first six frequencies for a = 0 and ni2 are listed in Table 1. For comparison, those obtained in reference [3] where the Euler-Bernoulli (E-B) equation was used are also listed on the first line in Tables 1 and 2 in each case. Equations (23) and (25) for the axial longitudinal vibration hold for all bars considered. 3.2. INCLINED BARS WITH BOTTOM HINGED-TOP "CLAMPED" When the boundary conditions of equations (l4b, d) and (15b, c) are satisfied, equation (18) yields

() W'Y]=-

AR2 cos O! k 2 cos hk 2sin kl'Y] - k 1cos k 1sin hk 2'Y] 2 2 • k 1k2(k 1 + k 2 ) cos k, cos hk;

(27)

The frequency equation is

2 k2R (k2/R) k 2 tan k 1 - k1 tan hk2 tan O!+ cot klk2(k~+k~) 0,

(28)

which for a =0 and '1T/2 reduces to, respectively, (29) tan k, =00,

k. = 71"/2,371"/2,571"/2, ....

(30)

The first seven frequencies are shown in Figure 2, and the first six frequencies when O! = 0 and a = 71"/2 listed in Table 2. 3.3. INCLINED BARS WITH BOTTOM CLAMPED-TOP "HINGED" When conditions (l4b, c) and (15b, d) are satisfied the general solution (18) becomes 2

w('Y]) = -

AR cos O! [ 2 2 ( • k, . ) B (k 1 cos k, + k 2 cos hk 2) sm k1'Y] - k S10 hk 2'Y] 2

(k~ sin k, + k 1k

2

sin hk2 )(cos k1'Y] - cos hk 2'Y])].

B = ki + klki+2k~k~ cos k, cos hk 2+ k~k2(k~ - k~) sin k, sin hk-:

(31) (32)

The frequency equation obtained from condition (3) for A ,e 0 is tan" a which, for a

+ Re cot (e R)(k~+ kD[k2 sin k, cos hk; - k, cos k, sin hk2] / k 2B = 0, =

(33)

0, reduces to equation (29), and for O! = '1T/2 for k1'rf 0 becomes

ki+ ki+2k~k; cos k, cos hk 2+ k 1k2(kf- kD sin k 1 sin hk2 = O.

(34)

The results are presented in Figure 3 and Table 2. 3.4. INCLINED BARS WITH BOTTOM CLAMPED-TOP CLAMPED IN ROTATION With conditions (14b, c) and (ISb, c) satisfied, the general solution (18) becomes

w('Y]) = -AR 2 cos 0![(k2 sin k1'Y] - k, sin hk, 'Y] )(k 1 sin k; + k 2 sin hk2)

+ k, k 2(cos k1'Y] - cos hk 2'Y] )(cos k, - cos hk2)]/ k 1k2 D,

(35)

179

INCLINED BAR WITH END CONSTRAINT

D

= (ki+ k~)(k] sin k, cos

h~+ k 2 cos k] sin hk2J.

(36)

The frequency equation then is

tan" a + Re cot (e/ R )[2 - 2 cos k] cos hk2 + (ki - ki) sin k, sin hk 2 / k, k 2J/ D = 0. (37) For a

= 0 and

a = 'TT'/2, equation (37) reduces to, respectively,

2 - 2 cos k, cos hk 2 + (k~ - ki) sin k, sin hk 2 / k, k 2 = 0,

(38)

k, tan k, + k2 tan hk2 = 0.

(39)

The basic frequencies are shown in Figure 4 and Table 1. In order to show the effect of slenderness ratio on the frequencies when both rotatory inertia and shear deformation are taken into consideration, frequency spectra for R = 200, 100, and 50 are presented in Figure 5 for the present clamped-"clamped" bars with the frequencies for a;:;:: and 'TT'/2 listed along the edges. The pattern is similar to the one given in reference [3]. After the second frequencies, the differences among the three groups of R = 200, 100, and 50 are substantial.

°

16 J37TR/2

.-

.-

......

14·076 ~= 13'907 13'345 .-._ .12

._

14·844 14·630 13·935

..=:;~_

_._._'_._.-

--------- --- - - - -

.firR 11·741 11·635 11 ·2.64

10'967 &::=-==.....=====-:-::::"""'_..;:;::;.,.." 10·890 10·6 11

K

17TR/2

- -'-'-'-'-- ._ ._.-

-

8·62.3 8·583 8·428

7' 843 t;".,==,..,..."==",===-="'=-==-::::---,, 7·817 7' 718

5-494 5·483 5 ·442

6 4·728 4 ·7'23 ~------_....-:==:-:::::-=::::::-... ....... -....,": ....... 4·707

'-'-

,

......... . -, '\ \

\.

3

2

O·

10 ·

2 ·362 2 ·361 2'359

20· 30· 40· 50· 60· 70· 80· 90·

a Figure 5. Frequencies of clamped-"clamped" bars. - - , R = 200; - - - . R = 100; -' -' -, R = 50.

180

C. H. CHANG AND Y. C. JUAN 4. CLOSING REMARKS

In Figures 1-4, for the first frequencies of the two equations (17) and (l3b) no difference could be shown. Thus one curve represents both cases. As the frequencies get higher, the differences are more apparent, especially after the frequency of axial vibration k == (7rR/2) 1/2. It is also seen from Figure 5 that after the frequency of axial vibration the coupling effect makes the frequency spectra much more complicated than the simple beam vibration. R, were deterIt was mentioned earlier that when a = 0 and 7r/2, the frequencies, mined from equations (23) and (25), which are frequency equations for axial vibration and are independent of the slenderness ratio, R, and the shear coefficient, s. However, for 0 < Q < 'IT /2, these curves as shown in Figures 1 to 4 are affected by these parameters but converge to the values given by equations (23) and (25) for a =0 and 7r/2. This phenomenon clearly indicates the coupling effect between the axial and transverse vibrations for inclined bars. The effect of rotatory inertia and shear deformation on the normal mode curves was also examined for the case of hinged-c'hinged" and clamped-vclamped" bars. It was seen that, except for the slight changes of the amplitudes, up to the sixth mode these effects were not much different from those presented in reference [3].

e/

REFERENCES 1. J. M. GELFAND and S. V. FORMIN 1963 Calculus of Variations. Englewood Cliffs, New Jersey:

Prentice-Hall. 2. C. H. CHANG 1974 Journal of the Engineering Mechanics Division, American Society of Civil Engineers 100, 737-756. Buckling of slanted columns. 3. C. H. CHANG 1976 Journal of Sound and Vibration 44, 419-429. Vibration of inclined bars with end constraint. 4. S. TrMOSHENKO, D. H. YOUNG and W. WEAVER, JR 1974 Vibration Problems in Engineering. New York: D. Van Nostrand Company. 5. R. D. MINDLIN and H. DERESIEWICZ 1955 Proceedings of the 2nd U.S. Congress of Applied Mechanics 175-178. Timoshenko's shear coefficient for flexural vibrations of beams. 6. G. R. COWPER 1966 Journal of Applied Mechanics 33, 335-340. The shear coefficient in Timoshenko's beam theory. 7. Y. W. Hsu 1975 Journal of Applied Mechanics 42, 226-228. The shear coefficient of beams of circular cross section. 8. J. J. JENSEN 1983 Journal of Sound and Vibration 87, 621-635. On the shear coefficient in Timoshenko's beam theory. 9. R. A. ANDERSON 1953 Journal ofApplied Mechanics 20, 504-540. Flexural vibrations in uniform beams according to the Timoshenko theory. 10. T. C. HUANG 1961 Journal of Applied Mechanics 28, 579-584. The effect of rotatory inertia and of shear deformation on the frequency and normal mode equation of uniform beams with simple end conditions. 11. R. W. TRAIL-NASH and A. R. COLLAR 1953 Quarterly Journal of Mechanics and Applied Mathematics 6, 186-222. The effect of shear flexibility and rotatory inertia on the bending vibrations of beams. 12. M. LEVINSON and D. N. COOKE 1982 Journal of Sound and Vibration 84,319-326. On the two frequency spectra of Timoshenko beams.