Vibrations of inclined bars with end constraint

Journal of Sound and Vibration (1976) 44(3), 419-429

VIBRATIONS

OF INCLINED

BARS WITH

END CONSTRAINT

C. H. CHANC Department of Aerospace Engineering, Mechanical Engineering and Engineering Mechanics, The University of Alabama, University, Alabama 35486, U.S.A.

(Received 24 March 1975, and in revised form 15 July 1975)

Considered are vibrations of inclined bars with the bottom end fixed and the top end having a constraint that this end can move only in the vertical direction. During the vibrations of such bars, the axial and lateral motions are coupled. The frequencies of the coupled vibrations of small amplitudes of inclined bars with different end conditions are presented along with the normal functions. The application of the method of normalfunction expansion to the forced as well as transient vibrations of the inclined bars is also outlined.

1.

INTRODUCTION

The effect of axial force induced during beam vibration is appreciable only when the amplitudes of the lateral vibrations are large as compared with the least dimension of the beam as demonstrated in the works done by Woinowsky--Krieger [I] and recently by Rehfield [2]. In the latter reference [2], it was also shown that the amplitudes of axial vibrations are of higher orders as compared with that of lateral vibrations. Thus, in small vibrations, the axial and lateral vibrations are not coupled as treated in the conventional theories of longitudinal and beam vibrations [3,4]. The axisymmetrical vibrations of the frames shown in Figure 1 may be analyzed by a simplified model of an inclined bar with the bottom end fixed in space and the top end moving in a frictionless vertical slot as shown in Figure 2. Thus the top end has a constrained condition that the horizontal displacement resulted from axial displacement, 0, and lateral displacement, m, at that end must vanish (Figure 2(b)): i.e., Osinu+

Pcosu=O

at x = I,

(1)

in which CL (0 < a < ~12) is the angle of inclination as shown in Figures 1 and 2. Condition (1) indicates that in general, the axial displacement, u, and lateral displacement, IV, are of the same order of magnitude. Thus these two displacements are coupled even in small vibrations.

(0)

(b)

Figure 1. Frames with inclined bars. 419

420

C. H. CHANG

e

4 -u

a

w

(b)

(a)

Figure 2. An inclinedbar. In the present work the coupled vibrations of the inclined bars with different end conditions are studied. The frequencies of small vibrations of such bars are presented along with the application of the method of normal-function expansion to the forced and transient vibrations of such systems.?

2. FORMULATION Consider an elastic and prismatic inclined bar, AB, of length I as shown in Figure 2. Let x and y be the axial and lateral co-ordinates with the origin located at the centroid of the crosssection at the lower end, A, and let ?=

u = Q/1,

xl4

w=

w/r.

Pa-4

The kinetic energy of the system is

(3) in which A is the cross sectional area, E is the modulus of elasticity, a dot indicates the derivative with respect to time, I, and B” = Egly12,

(4)

where g is the gravitational acceleration and y is the weight of unit volume of the bar. The potential energy of the system for finite vibrations is

PJ,, +

+W,:)‘drl+ f

(5)

in which R is the slenderness ratio and a subscript preceded by a comma represents the appropriate derivative. Constrained condition (1) may be taken into consideration by use of the Lagrange multiplier, 1. Thus a functional equivalent to the Lagrange function is obtained as follows : I=T-

V-AEl~~[Usino+

Wcosa],,,dt.

f0

7 A study parallelto the present work of buckling of inclined bars is reported in reference (51.

(6)

421

END-CONSTRAINED INCLINED BAR VIBRATIONS

When Hamilton’s principle is applied to the functional, 1, and it is assumed that the virtual displacements 6U and 6 W vanish at the end points of the arbitrary time interval t,, < t < tl, the following two non-linear equations are obtained : (l/B’) U/D’) w+

V-J,, + WI)*,

0 - (V, + !ml>.,

= 0,

(74

w,, + (U, + 3%)

w,,, + O/R21 WJI,,, = 0.

(7’4

In view of condition (l), one assumes w=tw,+y*w,+***,

u=
@a, b)

in which r is the amplitude. For small vibrations, the following two equations for the first order functions, U, and WI, are obtained: (l/B’) U1 - U1,,, = 0,

(1 /#I’) w’1+ (1 lR2) w,,,,, = 0,

(9a, b)

which are identical to the equations used separately for axial and lateral vibrations of a bar [3,4]. The following boundary conditions are obtained from the variational processes for bar AB with end A fixed in space :

w,=o, WI,,=0

or

and

or

WI,,=0

Wl,vq=O,

atrj=

U,,,+Isina=O

at q = 0,

u1 = 0

and

1,

(lOa, b)

at n = 0,

WI,,,=0

at v = 1,

WI,,,, - R* I cos a = 0

(lla-c) (12a-c)

with constraint U,sina+

at?=

W,cosu=O

1.

(13)

Thus the Lagrange multiplier, 1, physically is the non-dimensionalized horizontal reaction at the top end, B. This additional unknown, 1, involved in conditions (10) and (12), is compensated by the additional condition of (13).

3. NORMAL FUNCTIONS AND FREQUENCY EQUATIONS Let W,(q, t) = w(q) sinpt.

U&l, t) = U(V)sinpt,

(14a, b)

On substitution of these two functions into equations (9), one has a” + (k4/R2) u = 0,

w””- k4 w = 0,

(1%

b)

in which k = (pR/&‘l’.

(16)

Thusp or k is the circular frequency of vibration of this system. A prime indicates the ordinary derivative with respect to q. Conditions (10) to (13) become, respectively, u=o

at q = 0,

u’+~sinu=O

w=o

at?=

at q = 0,

1,

(17a, b)

w’= 0

or

w”= 0

at q = 0,

(18) (19a, b)

w’= 0

or

w” = 0

atq=

(2Oa, b)

WI- R*Icosa=O

at?=

usina+

atq=l.

wcosa=O

1, 1,

(21) (22)

422

C. H. CHANG

The solution of equation (15a) satisfying conditions (17) is u(q) = -AR sin u sin (k2/R)/k2 cos (k2/R).

(23)

The general solution of equation (15b) is w(q) = C, sin kq + C2 cos kq + C, sinh kq + Cd cash kq,

(24)

in which Cr, C,, C3 and C, are four constants of integration to be determined for different types of bars as follows. 3.1. INCLINED BARS WITH HINGED-“HINGED” ENDS

For this type of bar, with the satisfaction of conditions (18), (I 9b), (20b) and (21), solution (24) yields w(q) = -1 cos u R2(sinh k sin kq + sink sin kq)/k3 (sinh k cos k - cash k cos k).

(25)

On substitution of solutions (23) and (25) into condition (22), for I # 0, one obtains the frequency equation tan2 u + 2(R/k) cot (k2/R) (sin k sinh k)/(sinh k cos k - cash k sink) = 0.

(26)

With the slenderness ratio, R, as a parameter, the results of this frequency equation are presented in Figure 3 for frequency values, k, verms angle, a. The first seven normalized modes of vibrations of displacements U and W, for R = 100 and LY = 60”, are depicted in Figure 4.

21r -

54-

3.927

Tr_ 2a A

I-

1 0

I IO

I 20

I 30

I 40 o

I 50

I

1

I

60

70

80

lo 90

(degrees)

Figure 3. Frequencies of hinged-“hinged” bars. -,

R = 200; ----,

R = 100; -.-.-,

R = 50.

END-CONSTRAINED

INCLINED

-I

-I 0

7th

0

I

I

-I

-I 0

6th

0

I

I

-I

-I

0

5th

0

I

I -I

-I

4th

w

0

0

”

I

I -I

3rd

-I 0

0

I

I -I

-I

2nd

0

0 I

I -I

-I

1st

423

BAR VIBRATIONS

0

0 I

I 0

0.2

0.4

0.6

0.8

1.0

0

0.2

Figure 4. Normal modes of a hinged-“hinged”

0.4

0.6

0.0

I.0

bar. R = 100; a = 60”.

When CI= 0 equation (26) calls for either cos (k2/R) = 0,

or

(27)

sink = 0,

(28)

for k # 0. These two equations are satisfied by, respectively, k = a,

m,

v%&$,

.- .

and

k = a, 2n, 311,.. .,

(29) (30)

which are the frequencies of uncoupled axial and lateral vibrations, respectively, of a hinged“hinged” bar [3,4]. For CL= 7c/2,equation (26) is satisfied by either sin (k2/R) = 0 k=m,

and thus

d%%v%%,-..,

tanh k = tank

and hence

k = 0, 3.927, 7.069,. . -.

(31) or

(32) (33) (34)

424

C.

H. CHANG

These are the frequencies of uncoupled axial and lateral vibrations, respectively, of a hingedfree bar [3, 41. All these limiting k values are indicated in Figure 3. Equations (27) and (31), and hence the k values of equations (29) and (32) for CI= 0 and n/2, will also be obtained for the bars with other end conditions discussed below. Accordingly, these k values are indicated in Figures 5, 7, and 8. 3.2. INCLINED BARS WITH TOP “CLAMPED” AND BOTTOM HINGED With the fulfilment of conditions (18), (19b), (20a) and (21), solution (24) becomes w(q) = --AR’cos c1(cash k sin kq - cos k sinh kq)/(2k3 cos k cash k).

(35)

Condition (22), for L # 0, yields the frequency equation tan2 c1+ (R/2k) cot (kZ/R) (tank - tanh k) = 0,

(36)

which fork # 0 also reduces to equation (33) for CI= 0. For c1= n/2, equation (36) yields tank=

and hence

cc)

(37a, b)

k = 7.~12,37112, 5n/2, . .,

which are the vibration frequencies of a guided-hinged beam [4]. The numerical results of equations (36) and (37) are depicted in Figure 5. The normalized displacements u and w for c1= 60” and R = 100 are shown in Figure 6.

J3aR/2--

‘-~---Y\~

14 -

&ZZ---._____ 12 II

-

____-~---’ ----___

IO.210

-..

/m

‘\\

‘----.-. 6

-.

-

i

‘\

J.

._J>

7.069

7.854

65-

4.712

3.927 '

23-

(I A

I.571

I-

0

II IO 20

I 30

I 40 a

Figure 5. Frequencies of hinged-“clamped”

I 50

I 60

I 70

I Bo

90

(degrees)

bars. -,

R =

200; ----,

R =

100; -.

-.

-,

R = 50.

425

END-CONSTRAINED INCLINED BAR VIBRATIONS

6th

5th

w

0

L

.

I

0

I.0

I

1

0.2

0.4

1 0.6

I

’ 0.8

I.0

?

Figure 6. Normal modes of a hinged-“clamped” bar. R = 100; a = 60”. 3.3. INCLINED BARS WITH TOP “HINGED” AND BOTTOM CLAMPED

Solution (24) satisfying equations (18), (19a), (20b) and (21) has the following form: w(q) = -[A cos aR2/2k3( 1 + cos k cash k)][(cos k + cash k) x (sin kq - sinh kq) - (sink + sinh k)(cos kq - cash kq)].

(38)

Condition (22), for A # 0, results in the following frequency equation : tan* a + (R/k) cot (k*/R) (sin k cash k - cos k sinh k)/( 1 + cos k cash k) = 0.

(39)

The results of this equation are presented in Figure 7. For one limiting case of a = 0, equation (39) also reduces to equation (33) for k # 0. For the other limiting case, a = x/2, one has 1 +coskcoshk=O,

from which

k = l-875,4*694, 7.855,. . a,

as given in references [3,4] for a clamped-free beam.

(40) (41)

426

C. H. CHANG

0

III1

I 10

20

30

40 a

Figure 7. Frequencies of clamped-“hinged”

3.4.

I 50

II 60

70

1 80

I so

(degrees)

bars. -,

R=

200; ----,

R = 100;-.--.--,R = 50.

INCLINED BARS WITH TOP CLAMPED IN ROTATION AND BOTTOM CLAMPED

With satisfaction

of conditions

w(q) = -[IR’cos

(1 S), (19a), (20a) and (21), solution

(24) becomes

cr/2(sin k cash k + cos k sinh k)][(sin k + sinh k) x

(sin kq - sinh kq) + (cos k - cash k)(cos kq - cash kq)]. The frequency

equation

obtained

from equation

(22) is

tan2 0: + (R/k) cot (k2/R) (1 - cos k cash k)/(sin k cash k + cos k sinh k) = 0. The results of this equation

are given in Figure 8. When c1= 0, equation 1 -coskcoshk=O,

[3,4] for clamped-clamped tank+tanhk=O,

(43)

(43) yields

from which

k = 4.730, 7.853, 10.996, . . ‘, as given in references

(42)

(44) (45)

beams. When a = n/2, one obtains and thus

k = 2.365, 5.498, 8.639,. . .,

(46) (47)

as given in reference [4] for clamped-guided beams. The normal modes of the last two types of bars are almost the same as those given in Figures 4 and 6, respectively, except the slopes at q = 0 vanish. It is noted that the slenderness ratio, R, which is not presented in the frequency equations of beam vibration such as equations (28), (33), (37a), (40), (44) and (46) for TV= 0 and 7r/2, plays an important role for 0 < tl < 7r/2, particularly for high modes or small R (Figures 3, 5,

427

END-CONSTRAINED INCLINED BAR VIBRATIONS

16 -

IO.996

-

IO .---.-___

bGmz= k

6-

4.730

. I

4-

8

32-

*

a

I -

0

I IO

I 20

I 30

I 40

I 50

I 60

I 70

I 80

_ 90

R =

200; ----, R = 100;-.-.--,

a (degrees)

Figure 8. Frequenciesof clamped-“clamped”bars. -,

R=

50.

7 and 8). The ‘frequency curves have two alternative groups divided by frequencies of l/lrR, A&?@, . . . as given by equations (29) and (32). For instance, for R = 100, and 0 < k < V% = 12.533, k values decrease as a increases. The axial displacements are in compression and the lateral displacements at the top end move in the positive direction (downward) as they should be (Figures 4 and 6). After passing k = 12.533, k increases as a increases, and u becomes tension, while w at the top end moves in the negative direction until k = a. Thus for smaller R, the pattern changes alternatively more often. For large values of R, the basic modes belong to the first group and essentially are of vibrations of a bar with a = 0. However, when o!approaches n/2, the frequencies drop rather rapidly. k = m,

4. FORCED AND TRANSIENT VIBRATIONS For completeness, the general solutions of forced as well as transient vibrations of inclined bars undergoing small vibrations are outlined by the method of normal-function expansion as follows. The mth normal functions satisfy R*u;+k$u,,,=O,

w:-k:w,=O.

(4% b)

Following the Clebsch theorem [6] (k,4 - k:) 1 (w. w,,, + u, u,) dq = / (w; w, - R* u:, u,,,)drj - i(w: w, - R* u; u,) dtl 0

0

0

=[w.“w,,,-R*u~u,-~~w,+R*~~~&-[w~w~-~~w$,=~. (49)

428

C. H. CHANG

In the last step, boundary conditions (17) to (21) and constraint condition (22) were used. Thus the orthogonality or conjugate property of the normal functions reads 1 I

for n #

(W,,,W,+U,z4,,,)drl=O

m

(50)

0

and one may define I, = /(w.’ + u;) dq.

(51)

0

If the bar is considered to be subjected to an axial force &r])sinSZt and a lateral force $(q j sin I&, the equations of motion are U1 = B” Ui,,, + 4(q) sin Qr,

Wa, b)

W1 + (P2/R2) WI,,,,, = II/(q)sin %

with the following initial conditions : ~lh

0) = X(59,

ir,
0) = m>,

Wa, b)

W% 0) = f-m.

(5% b)

Wlh

One can expand the two force functions, 4 and $, as series of normal functions,

(5% W where (56) The complete solutions of equations (52) are U, = 2 u,(q)[B, cosp, t + C, sinp, t + D, sin at], n=l

(57a)

W,,(q)[B, cosp, t + C, sinp, t + D, sin Qtt],

(57b)

WI = 2 It=1

where in = k.2 BIR

D, = MP;

- Q’>,

1

B, = f s

(XU, + yWn)dtl,

(Gu,, + Hw,) dr] - Q D,

nO

5. CLOSING

(58a, b)

1

(58~. d)

REMARKS

The solutions as obtained in this study of inclined bar vibrations are exact if the amplitudes are small. For moderate amplitudes, the second order solutions U, and W, may be desirable. These solutions may be obtained by substitution of the series of equations (8) into equation (7), which will result in a pair of equations similar to equations (9) for U, and W,. Such equations could be solved, with lengthy computations but with little difficulty. The use of the Lagrange multiplier in dealing with the constraint may also be applied to other structures with inclined members. For instance, Figure 1(b) may simulate a meridional section of a conical frustum. Thus this approach may be followed for the analysis of vibrations of conical shells.

END-CONSTRAINEDINCLINED BAR VIBRATIONS

429

REFERENCES 1. S. WOINOWSKY-KRIEGER1950 Journal of Applied Mechanics, American Society of Mechanical Engineers 17,35-36. The effect of an axial force on the vibration of hinged bars. 2. L. W. REHPIELD1973 International Journal of Solids and Structures 9, 581-590. Non-linear free vibrations of elastic structures. 3. S. P. TIMOSHENKO1955 Vibration Problems in Engineering. New York: D. Van Nostrand Company, Inc. 4. W. C. HURTY and M. F. RUBINSTEIN 1964 Dynamics of Structures. Englewood Cliffs, N.J.: Prentice Hall, Inc. 5. C. H. CHANG 1974 Journal of the Engineering Mechanics Division, American Society of Civil Engineers 100,737-756. Buckling of slanted columns. 6. A. E. H. LOVE 1944 A Treatise on the Mathematical Theory of Elasticity. New York: Dover Publishers. See p. 180.