The coupling effect of axial motion and joint mass on the lateral vibrations of a rigid-jointed triangular frame

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 15,447462 (1987)

THE COUPLING EFFECT OF AXIAL MOTION AND JOINT MASS

TRIANGULAR FRAME ON THE LATERAL VIBRATIONS OF A RIGID-JOINTED

A. N. KOUNADIS* AND K. MESKOURIS Lehrstuhl K I B I l l , Ruhr-Vniuersitat Bochum, Federal Republic of Germany

SUMMARY

In this investigation, the coupling effects of the axial motion and other parameters on the bending eigenfrequencies and eigenmodes of laterally vibrating frames are re-examined. To this end a n energy variational approach is performed on a rigid-jointed triangular frame, whose joint mass is eccentrically located with respect to its theoretical position. The governing partial differential equations subject to the appropriate boundary conditions are very conveniently formulated and successfully solved in a closed form by using generalized functions and Laplace transforms. Contrary to the usual assumptions of the standard dynamic analysis of continuous systems, herein the effect of axial contraction and extension is accounted for when establishing the translational kinematic boundary conditions. This may lead to considerable discrepancies that reveal the decisive role of the axial motion effect on the dynamic response of framed struciures. Such discrepancies are clearly confirmed through a thorough numerical discussion of the governing parameters: joint mass and its rotatory inertia, joint angle, positioning of the mass, slenderness ratios, stiffness and length ratios.

INTRODUCTION

When considering lateral vibrations of framed structures, the effect of axial motion on the bending eigenfrequencies is usually ignored, being considered of higher order of smallness. Such an omission, leading to a considerable simplification of the dynamic analysis, according to the pioneer work by Hohenemser and Prager,' published in 1933, is allowable for frames having bars with slenderness ratios greater than 40; for lower slenderness ratios it may lead to an error less than 10 per cent. Since then, the finding of these investigators has been widely accepted and used. Reviewing the present state-of-the-art the only pertinent work*-to the knowledge of the authors-refers to a specific two-storey single span frame, where this effect is taken into account but without any further discussion. Recently, Kounadis and his associate^,^-^ reconsidering the validity of the aforementioned finding, have shown that the omission of the axial motion under certain combinations of values of slenderness ratios, stiffness and length ratios may yield a serious error on the bending eigenfrequencies of rectangular frames; moreover, they have shown that this effect becomes considerable when the joint mass of the frame is taken into account, whereas the influence of transverse shear deformation and rotatory inertia: being practically negligible, can be ignored even for frames having bars with low slenderness ratios.

In this investigation, the previous works are extended to the study of bending eigenfrequencies and eigenmodes by using as model an arbitrary two-bar plane frame with various values of its joint angle as well as of its joint mass which is eccentrically located with respect to the point of intersection of the centre lines of the two bars. Thus, the influence of the axial motion is thoroughly discussed individually or in conjunction with the foregoing two parameters as well as with a variety of values of other parameters such as rotatory inertia of the joint mass, positioning of the joint mass, slenderness ratios, stiffness and length ratios.

* On leave from National Technical University of Athens, Greece.

OO98-8847/8 7/040447- 1 5$07.50 0 1987 by John Wiley & Sons, Ltd.

Received 12 May 1986 Revised 15 July 1986

448 A. N. KOUNADIS AND K. MESKOURIS

The mathematical problem under discussion leads to a system of four partial differential equations which is very conveniently formulated and readily manipulated using generalized functions.6 A closed form solution is successfully obtained with the aid of Laplace transforms. The use of generalized functions-besides their advantages in the derivation of the governing equations and solution technique-allows a very efficient study to be made of the effect of the positioning of the concentrated mass along the length of the bar. Thus, the present analysis may be extended to mechanical models in addition to the framed structures to which it is primarily related. A thorough discussion of the effect of the positioning of a concentrated mass on the bending eigenfrequencies of a cantilever is presented in Reference 6.

The field equations for this problem are those of the classical dynamic analysis with the unique exception that the shortening (or extension) of the bar axes due to axial compression (or tension) is accounted for. This fact leads to considerable discrepancies between the classical dynamic analysis and the more precise analysis proposed herein, as is clearly confirmed by a large variety of numerical results.

MATHEMATICAL FORMULATION

We consider the triangular, two-bar, plane frame shown in Figure 1 supported on two immovable hinges. Its joint mass M having a rotational inertia J is located at a distance e from the centre line of the column. Each bar has a length l i , mass per unit length mi, cross-sectional area Ai and second moment of area Z i ( i = 42).

The frame undergoes free transverse vibrations described by the lateral and axial displacement components W ( x i , t ) and Ui(x i , t) referring to an arbitrary point of the centre line of the ith bar with the sign convention shown also in Figure 1.

The governing equations of motion will be established by using Hamilton’s variational principle given by

Jt I

where the functionals of the total kinetic energy K T and the total potential energy U, are given by the following expressions:

l 2 1 1 K T =- 2 i = l 1 j: mi(@+fi:)dxi +-M[k~(lz-e)+fi:(12-e)]+ZJk:(12-e) 2

where W l (12 - e), c:(12 - e), Fk$(lz - e) stand for { k2(12 - e)}*, { c2(Iz - e)}2, { k(lZ - e)}’, respectively. In the above expressions the dot denotes partial differentiation with respect to time t and the prime partial differentiation with respect to x i ; E is Young’s modulus of elasticity which is common for both bars made from the same material.

Figure 1 . Geometry and sign convention of a rigid-jointed triangular frame

COUPLING EFFECT OF AXIAL MOTION AND JOINT MASS 449

Using the relations with the Dirac generalized function 6 and its derivative

@ ; ( I 2 - e) = j: ik;(x,)S(x, -1, + e) dx,

G;(1, -e) = U,2(x2)8(x2 -I,+e)dx, Jd' *

-1, +e)dx2

formula (2a) becomes

KT = i l ' m l ( i k ? + i i : ) d x l 1 { [ m 2 + M 6 ( x 2 - 1 2 + e ) ] [ ~ ~ ( x 2 ) + ~ , Z ( x 2 ) ]

- J 6 ' ( x 2 - l 2 + e ) k i 2 (x,)} dx, (4)

Introducing expressions (2b) and (4) into equation (1) and performing its variation, we obtain the following

(5d

(5b)

(54

( 5 4

equations of motion:

EI,W;"' (xl , t )+m,i i ; (x , , t ) = 0

E Z ~ W ~ ( X , , t ) + [m2 + M ~ ( x , -1, +e)]ii;(x,, t ) - J ~ ' ( x , - 1 , +e)~;(x, , t ) = o EA,UI'(x,, ~ ) - m 1 U 1 ( X 1 , t ) = 0

EA2U; (x,. t ) - [m2 + M 6 (x2 - 1, + e ) ] c, (x,, t ) = 0

..

The geometric boundary conditions, a priori known, are

K(0, t ) = Ui(O, t ) = 0

Wi(/ , , t ) - w; (4, t ) = 0 ( i = 1,2)

U,(l,,t)+W,(I,,+ W,(I,,t)cota = 0 sin a

Wl(4 9 t ) sin a

U,(l , , t ) -7 - W2(4, t)cot a = 0

where a is the angle of the column axis with respect to the horizontal axis.

are given by By means of the last conditions the natural boundary conditions resulting from the variational equation (1)

y"(0,t) = 0 ( i = 1,2) (6h, i)

E I , W ~ ( l , , t)sina + E A , U ; ( I , , t ) -EA,U;(I , , t )cosa = 0 (6j) EI,W;"(I,, t ) sina + E A I U ; ( l , , t)cosa -EA,U;(I,, t ) = 0 (6k)

E I l W[(1,, t ) + EZ,W;'(l,, t ) = 0 (61)

For free vibrations, one can assume a solution of the system of equations (5a-d) of the form *

W(xi, t ) = w ( x i ) exp ( jot ) ( 7 4

Ui(xi, t ) = Ui(x i ) exp ( j o t ) (i = 42) (7b)

( i = 42) *

where j = J-1 is the imaginary unit and o the circular frequency of the free motion.


Introduction of the non-dimensionalized parameters

and substitution of relations (7a,b) into equations (5a-d) and (6a-1), yield

and

wi(0) = ui(0) = 0 W i ( l ) - W i ( l ) = 0

( i = 1,2)

u , ( l ) + p g + w l ( l ) c o t a = 0

u 2 ( l ) - W - w 2 ( l ) c o t a = 0 p sin a

wi” (0) = 0

wy (1) sin a + -A:ui(l) - Aiu; (1)cosa = 0

w y ( 1) sin a + (1)cosa - - Afu; (1) = 0

w;(l)+-w;’(l) = 0

(i = I, 2)

P2 P

P P2

P P

Taking the Laplace transforms of the system of equations (9a-d) and using conditions (1Oa-d) and (lOh, i), solving for the Laplace transforms of wi and ui (i = 1,2) and inverting the results, we obtain the following solution:

where ci ( i = 1, . . . , 6 ) are integration constants; H is the Heaviside generalized function, whereas F2 (t2 - 1 + a) and w2 ( 1 -2) as well as their derivatives are given by

COUPLING EFFECT OF AXIAL MOTION AND JOINT MASS 45 1

1 F ' i ( C 2 - 1 -C) = 7 [COSh k2 ( 5 2 - 1 + U) -cosk2((2 - 1 + P)] (12b)

w,(l - P ) = c,sink,(l -P)+c4sinhk2(l -a) ( 124

w;(l -a)= c,k2cosk2(l-~)+c,k,coshk,(l-e) ( 1 2 4 u 2 ( l - P ) = c6sinu,(l -e) (124

2k2

Application of the remaining boundary conditions (lOe, f, g, j, k, 1) and the use of relations (lla-d) and (12a-e) lead to the following linear homogeneous system of equations with respect to the integration constants ci (i = 1, . . . . 6):

equation

In as much as due to relations (8) 1.4

where the analytical expressions of the elements aij (i , j = 1, . . . . 6) are given in the Appendix. For a non-trivial solution the determinant of the foregoing system must be zero, leading to the frequency

= O (14)

U: = - '% ( i = 1,2) and k2 = k, n:

the frequency equation (14) is a transcendental equation of the form

Note that when 2, = 2, and p = 1, owing to relation (15)k, = k, , regardless of the values of p and the other parameters.

Once the non-dimensionalized eigenfrequencies k i are determined through the frequency equation (14), the eigenmodes can be readily established by the system of equations (13), from which ci (i = 1,. . . ,5) are expressed in terms of c6( # 0).

Contrary to the analysis outlined above, in the classical (linear) dynamic analysis of Euler-Bernoulli the effect of axial extension (or contraction) is disregarded. Such an omission, implying that the joint of the frame is immovable, leads to a standard eigenvalue problem for determining the bending eigenfrequencies and eigenmodes.

NUMERICAL RESULTS AND DISCUSSION

The non-dimensionalized eigenfrequencies ki are established as functions of the parameters p, p , 11, I,, P, a and J b y solving numerically the frequency equation (14) with the aid of the Newton-Raphson technique; the corresponding eigenmodes can be subsequently determined by means of the system of homogeneous equations (13). A lot of numerical results (coveiing a large variety of geometrical configurations of the frame) which show the individual and coupling effect of the foregoing parameters on the eigenfrequencies and the corresponding eigenmodes are presented in both tabular (Tables I-V) and graphical (Figure 2-8) form. A comparison of the above results with those obtained by means of the classical Euler-Bernoulli theory (E-B) shows when the effect of longitudinal motion (L.M.) alone or in conjunction with other parameters is important and thus it should be taken into account.


The interest of this investigation is focused mainly on metal portal frames, whose response can be linear in accordance with the analysis presented herein. Such an analysis is also valid for frames with members of low slenderness ratios (i.e. I, = I, = 30), since, as has been shown, the influence of transverse shear deformation and rotatory inertia compared to longitudinal (L.M.) motion effect can be i g n ~ r e d . ~

Given that Reference 4 deals with rectangular frames, in this section particular emphasis is also given to numerical results corresponding to non-rectangular frames carrying a concentrated mass on their girders as well as to mechanical models; the latter can be considered either as extreme cases of frames with both members having solid cross-sections or usual frames with latticed members which appear frequently in practice.

In order to have comparable numerical results (for rectanguF frames) with those of Reference:, instead of the parameters M and J w e will use below the parameters M = M p I : / p I : ( = M/mlll) and !=Ip@/lt ( = J / m , l f ) ; the latter coincide with the corresponding parameters of Reference 4. Thus, if M (or J ) is a constant then M (or J ) is not necessarily constant.

In Table I, the fundamental eigenfreqFencies k, based on ESB theory and the analysis presented herein are given for a frame with I, = I, = 30, M = R pI: /PI: = 1, J = 7ppJ. i /A: = 0, various values of the mass offset Z?( = 0 and 0.15), joint angle a ( = 30°, 90" and 150" ), stiffness ratio j.i ( = 0~05,0~50,1~00,20~00) and many values of length ratios p . The numerical values between parentheses reflect the (L.M.) effect when k = J = 0.

The first two columns of Table I evaluated for k = 1 and .f = 0 give the fundamental eigenfrequencies k , obtained on the basis of the classical E-B theory according to which the joint of the frame is immovable to any direction (since the L.M. effect is neglected). Consequently, these results hold regardless of the magnitude of the joint angle. A comparison of the results of the first two columns corresponding to i? = 0 and i? = 0.15 shows that the effect of mass offset is important for low values ofp/p2 (= I : A 2 / l : A , ) ; thus for p / p 2 = 0 ~ 1 / 7 ~ = 0.002 the effect of mass offset implies an increase in the eigenfrequency k, by 267 per cent. Note also that the mass offset may provoke an increase in the eigenfrequency as in the case p = 0.05 and p = 1.50.

In columns 3 8 are given t p fund$mental eigenfrequencies k, when the L.M. effect is taken into account for a frame with I1 = I, = 30, M = 1, J = 0 and three different values of joint angle a ( = 30", 90" and 150") for which the effect of mass offset B ( = 0 and 0 1 5 ) is also discussed. From columns 3-8 it is clear that the joint angle variation has, in general, a considerable effect upon the eigenfrequencies; in all cases considered the maximum value of the fundamental eigenfrequency k, occurs for a = 90" (rectangular frame); for joint angles different from a = 90" the eigenfrequencies decrease considerably, approximately with the same rate of change for joint angles smaller or greater than 90". The major effect occurs for p = 0.05 and p = 1.50, where for frames with a = 30" and a = 150" the eigenfrequency k, is about 40 per cent smaller than that corresponding to a = 90" and 2.4 times smaller than that obtained by E-B theory. Considering the values between parentheses corresponding to fi = 0, it is worth observing that for p = 0-05 and p = 1.5 the L.M. effect alone results in an analogous reduction of the eigenfrequencies k , for joint angles smaller or greater than a = 90". Even for the rectangular frame there is a considerable discrepancy between E-B theory and the present analysis, since the corresponding eigenfrequencies are 3.049366 and 2.349180, respectively (a reduction of almost 30 per cent). The last result agrees with that of Reference 4 (Table I, case 6) if one takes into account the small effect of shear deformation and rotatory inertia. The major effect of the mass offset appears for the case p = 1, p = 7 and amounts to 22 per cent. More specifically, in this case if a = 30", the presence of the mass offset causes a reduction of the eigenfrequency by about 22 per cent, whereas for a = 150" it causes an increase of the same rate in the eigenfrequency. Moreover, from all cases considered in Table I, it is apparent that the differences between E-B theory and the present analysis occur for small values of p and large values of p , or more precisely for small values of the quantity p / p 2 . Furthermore, it is deduced that the L.M. effect becomes more significant for non-rectangular frames of engineering importance than it is for rectangular frames. Note also that M varies from 0.035 to 30 for i6 = 1. Obviously this large value of M = 30 is due to the fact that the girder is very thin (case of latticed girder) compared to the column cross-section.

It is worthy of note that as the natural frequency o, is proportional to k:, the percentage discrepancies in natural frequency between E-B theory and the present analysis are larger.

From Table 11, evaluated for p = 005, p = 1, h = 1 (M = 20), 5 = 0, t? = 0 and l1 = A2 (= 30,80), one can study the effect of joint angle variation a on the eigenfrequency kl obtained on the basis of the present L.M.

* *

Tabl

e I.

Fund

amen

tal e

igen

freq

uenc

ies k

l bas

ed o

n E-

Bta

naly

sis

and

the

prop

osed

ana

lysi

s for

l1

= l

2 = 3

0, M =

1, J

= 0

, and

var

ious

val

ues o

f th

e pa

ram

eter

s p,

p a

nd a

.

* *

E-B

,M=

1,J

=O

L

.M?f

i =

1, j

= 0,1, =

1, =

30

p =

lJI

l p

= l2

/1,

ao

= a

rbit,

&,A, +

00

a =

30"

a

= 9

0"

a =

150

" -

-

e=

O

F=0.

15

F=

O

e=0.

15

i?=O

V

=O.1

5 e

=O

F =

0.1

5

0.05

0.50

1.00

20.0

0

1.50

1 .00

010

5.00

0.80

0.10

7.00

1 .00

0.15

i 15.0

0

8.00

I 0.70 3.04

9366

3.

2380

36

3.14

159

3.33

4129

3.32

2498

3.

3209

14

1.70

3306

1.

2792

90

3.24

1000

3.

1 116

05

3.72

8222

3.

7279

52

1.42

5710

1.

1251

91

3.14

157

2.98

5714

3.76

5589

3.

7651

70

0.87

4067

0.

8254

91

1.16

4234

1.

1248

16

3.67

7141

3.

6677

10

1.26

7931

1.

1988

53

(1.7

4208

3)

1.68

9049

1.

6308

71

(2.2

4765

4)

3.04

1 150

3.

0629

3 (3

.107

539)

0.

9585

05

0.81

4102

(1

.242

090)

2.82

671 1

2.

8037

54

(3.1

0807

6)

2.92

3756

2.

9448

12

(3.0

03 78

5)

0-90

4959

07

3937

7 (1

.104

972)

2.77

8335

2.

6736

5 (3

.140

303)

2.82

2496

2.

8503

45

(2.9

338 1

5)

0.78

8176

0.

7244

54

(079

6975

)

1.09

3327

1.

0424

41

(1.1

022 1

4)

2107

410

2.11

6166

(2

.1 50

755)

1.76

8655

1.

8115

05

(2.3

4918

0)

2.36

1 780

2.

4368

64

(2.8

1692

0)

3.29

8485

3.

3013

33

(3.3

0 14 1

7)

1.28

1873

1.

1645

709

(1.6

2972

7)

3.21

7505

3.

0765

46

(3.2

2045

1)

3.47

5264

3.

4882

2 (3

.511

380)

1.17

5795

1.

01 20

77

(1,3

6049

3)

3.12

1529

8 2.

9464

8 1

(3.1

2372

2)

3.56

6344

3.

5843

09

(3.6

1 328

0)

0.85

4328

0.

8038

28

(0.8

5583

8)

1.16

0390

1.

1205

549

(1.1

6057

5)

2.84

4973

2.

8447

69

(2.8

8502

7)

1.25

175

1.31

514

(1.7

3436

)

1.67

750

1.71

682

(2.2

4661

) 3.

3221

6 3.

3204

2 (3

.322

26)

0.94

446

1.10

168

(1.3

1 120

)

2.80

81 1

2.69

790

(3.0

2075

)

3.25

600

3.29

889

(3.4

0981

)

0907

55

1.10

575

(1.2

3489

)

2.72

966

2.60

954

(2.9

0408

) 3.

0454

3 3,

0986

7 (3

.286

81)

0.85

874

0.8 1

860

(0.8

6 19 1

0)

1.16

120

1.11

735

(1.1

61 81

)

2,09

783

2.10

709

(2.1

4267

)

E-B

= E

uler

-Ber

noul

li.

t L.M. = L

ongi

tudi

nal m

otio

n.

P

ul

w

454

3-

2-

1 -

A. N. KOUNADIS AND K. MESKOURIS

Table 11. Fundamental eigenfrequency k l for p = 1, p = 0.05, A = 1, J = 0, e = 0 and various values of joint angle a' combined with I, = 1,

= 30 and 1, = 1, = 80 *

L.M., M( = 1)

10 30 50 70 90

110 130 150 170

1.076840 1.689048 2.075018

For all 2.291882 cases 2.361780

3.14159 2.291824 2.073446 1.677501 0.9891 10

1.626896 2.65 1 147 2.999400 3.065396 3.076404 3.061545 2.990503 2.648457 1.61099 1

analysis as well as on the E-B theory. When the L.M. effect is ignored we obtain the same value of eigenfrequency k, = 3.14159-regardless of the joint angle variation-which corresponds to E-B theory. For both slenderness ratios the maximum eigenfrequency corresponds approximately to a = 90". The L.M. effect is more pronounced for I , = = 30 than it is for I, = A 2 = 80; however, the effect of joint angle variation on the change of the eigenfrequency k, is very pronounced in the case 1, = 1, = 80 for a = 170" and a = 10". The paramount importance of this effect on the fundamental eigenfrequency is also clearly shown in Figure 2.

From Table 111, evaluated for p = 005, I l = I , = 80, $f ( = 0, l), f = 0, t?( = 0,015) and a (= 90",30"), one can study th: effect of length ratio p on the eigenfrequency k l . Clearly, from the first two columns, one can see that when M = 0 the small discrepancy between E-B theory and L.M. analysis increases monotonically with increasing p. From the 3rd and 4th columns corresponding to M = 1 and a = 90" one can observe the considerable effect of the mass offset that increases also monotonically with increasingp; for the value p = 1.60 the mass offset causes a decrease in the eigenfrequency by about 42.5 per cent. Note that this effect becomes less pronounced for a = 30"; for instance for p = 1.60 the presence of a mass offset implies a reduction of the eigenfrequency by about 19 per cent. Comparing the 2nd and 5th columns we observe the considerable effect of

*

P E- B

04 c

0" 10' 30° 50' 70' 90' 110' 1300 1%" i7Oo a

* Figure 2. Fundamental eigenfrequency k , versus joint angle variation a for a frame with p = 1, p = 0.05, M = 1, B = 0, J = 0 and two

values of I , = I,( = 30 and 80)


Table 111. Fundamental eigenfrequency k , for p = 0.05, I, = I, = 80, d ( = 0 and I), .? = 0, C( = 0 and 0.15), a ( = 90" and 30") and various values of p

a=90" a = 30" f i = O A = 1 A=O f i = l

p = l , / l l E-B L.M. 2 = O C=0.15 L.M. C = O C=0.15

0.10 3.322498 3.319402 3.319345 3.318319 3.278234 3.275016 3.278028 0.50 3.178846 3.172239 3.17966 3.105267 3.143389 3.134433 3.107806 0.90 3.148641 3.1 13587 3.104105 2.676384 3.004470 2.802192 2.480533 1.00 3.1415927 3.094820 3.076404 2.526905 2.951771 2.651 147 2.301897 1.20 3.122875 3.048640 2990162 2.259699 2.822710 2.364722 2.015250 1.40 3.08521 1 2.988647 2.845929 2.042034 2.670028 2.127513 1.796355 1.50 3.049366 2.952812 2.755249 1.9491 120 2.58966 2.026407 1.705189

L.M. alone (i.e. k = 0) for large values of p when the joint angle reduces from 90" to 30". The foregoing results are also shown graphically in Figure 3. From Figure 4, one can also observe the effect of length ratio p and mass offset i? combined wit: stubby and relatively slender members corresponding to a rectangular frame with p = 0-05, 6 = 1 and J = 0. Obviously, this effect is very small for I, = A 2 = 30 and very pronounced for I, = I2 = 80. Note that M varies from 2 to 30 for 6 = 1.

From Table IV, evaluated for p = 0.05, p = 1.2, 1, = A2 = 30, a = 90°, 16 = 0.25 (@ = 6) one can study the effect of the positioning of the concentrated mass in conjunction with the influence of mass rotatory inertia j ( = 0,O.lO or J = 0,1*67) upon the fundamental eigenfrequency kl obtained on the basis of E-B theory and the present analysis. These results are also shown graphically in Figure 5. Clearly two important conclusions are drawn: the tremendou: effect on the eigenfrequency k , %fa small rotatory inertia of the concentrated mass and the coincidence for J = 0.10 (or small deviation for J = 0) of E-B theory and the present analysis.

a = 30°; MI= 0

a =30°; M*=l; 5=0.15

Figure 3. Fundamental eigenfrequency k , versus length ratio p for a frame with p = 0.05, A, = A2 = 80, .? = 0 and various values of fi ( = 0 and l), P( = 0 and 0.1 5) and a ( = 90" and 30")

456

1

A. N. KOUNADIS AND K. MESKOURIS

I I 1 I 1 1 V I I I I 1 I t ( 1 1

e = 0.15

Table IV. Fundamental eigenfrequency k, based on E-B theory and the* present analysis for a rectangular frame with p = 0.05, p = 1.2,1, = 1, = 30, M = 0.25

with J = 0 and J = 0.10, and the mass position C? * *

E-B L.M. k ( = 0.25) Mass position

e J = o 3=0 .10 3 = 0 3=0.10

0.0 0.1 0 2 0.3 0.4 0.5 0 6 0 7 0.8 0.9 1.0

3.122871 3.045795 2.595566 2.174391 1.942443 1.829175 1.804384 1.870732 2.07575 2.603442 3.122871

2.247764 1.5014 17 1.390517 1.378641 1.417714 1.495376 1.548855 1.462164 1.350347 1.2407 52 1.1 30742

2.453588 2.480525 2.501441 2.170414 1.9 393 1 1 1.826891 1.802697 1.869485 2.074956 2.591320 2.630492

2.239976 1.498273 1.389429 1.378208 1.417573 1.495 3 14 1.547941 1.460844 1.34931 1 1.240020 1.1 30266

From Table V, evaluated for p 7 0.05, p = 1.5, A1 = A 2 = 80 and various values ofjoint angle, one can study theeffectofaconcentratedmassM( = Oand 1)withJ = Ocombined withamassoffsetV( = OandO15)on the (square roots) of the three first eigenfrequencies k,, k z and k 3 obtained by means of E-B theory and the present analysis. The discrep2ncy between th:se analyses is obvious when comparing the corresponding three eigenfrequencies for M = 0, e = 0 and M = 1, V = 0.15. From this table, one can also observe the tremendous effect of the mass offset on the three first eigenfrequencies. The results of Table V can also be seen graphically in Figure 6. Some of these results are also presented in Reference 4.

*


't p

'1

] Jy =O E- B L M.

Er:, 3 J* = 010

0 1 I 1 I I I 1 I I 1

1 I I I I I I I 1 c

0.0 0.2 0.L 0.6 0.8 1.0 P

Figure 5. Fundamental eigenfrequency k, versus mass position 2for a rectangular frame with p = 0.05, p = 1.2, A, = I, , = 30, fi = 0.25 combined with two values of .f(= 0 and 0.10)

Table V. F$st three eigen*frequencies k l , k2, k, for a frame with p = 0.05, v = 1.5. M (= 0 and 1). J = 0, P(= 0 and 0.15) and various joint angles tl

E-B L.M. * M = l 11 = 1 2 = 80

Joint angle regardless of 1 * a P = O P = 0 1 5 & = o , P = O M = l , P = 0 . 1 5

20"

40"

60"

80"

90"

loo"

120"

140"

160"

2252941 5 3.1 10889 4 1 598365 2775997 3.140675 4.5 12291 292021 5 3.18249 1 4,847746 2952453 3.211836 5,029266

For all For all cases: cases: 2952812 3.04937 1.951145 3,220257 3.30206 3.238036 5.054328 5.73006 3.701500

2.945410 3.225196 5.03 53 34 2,8993 18 3.226091 4.862460 2-753393 3.216420 4.514385 2255501 3.201680 4.1 10952

1.506090 2.528268 3.575241

1.814067 2.655815 3566160 1.910272 2.759051 3.551677 1.943239 2 8 19734 3.537390

1.9491 11 2836208 3533423

1.9498 72 2.844148 3533315 1.934261 2.832680 3.549492 1.872635 2773229 3.59063 1 1.628740 2.650610 3-625321

458

ki (i=1,2,3,) t A. N. KOUNADIS AND K. MESKOURIS

' 1 04 h

0" 20" 40° 60° 80°90" 100" 120" 140" 160" o

Figure 6. Three eigenfrequencies versus joint angle a for a frame with p = 0.05, p = 1.5, A, = I , = 80, M ( = 0 and I), J = 0 and P( = 0 and 0.15)

* *

Figure 7. First three bending eigenmodes obtained by E-B theory and this analysis for a rectangular frame with p = 0.05,1, = I2 = 80 p = 1,5 ,$f= 1 , f = O a n d i = 0 . 1 5

From Figure 7, one can compare the three first flexural eigenmodes obtained on the basis of E-B theory and the present analysis when the L.M. effect is taken into account. The serious discrepancies between these analyses show the importance of the L.M. effect, particularly when the mass of the joint must be accounted for. In Figure 8, one can see the first three eigenmodes obtained by the present analysis. It is worth noticing that in both cases of frames of Figures 7 and 8, the vertical joint displacement is nearly zero, since p = 0.05 and p = 1.5 yield A I = 45A2. This implies the coincidence of column eigenmodes between the E-B theory and the present analysis.


----__ 2nd --L-- -----

_c-

I i

t 8

Figure 8. First three bending eigenmodes for a rectangular frame with = 0.05, p = 2.5, 1, = 1, = 80, M = J = 0, F = 0

CONCLUSIONS

The following constitute the most important conclusions of this investigation. 1. A systematic analysis and solution methodology is comprehensively demonstrated through a simple

frame for establishing its bending eigenfrequencies and corresponding eigenmodes including the individual or coupling effect of longitudinal motion and other parameters.

2. Comparing the E-B theory with this dynamic analysis it is deduced that for frames of engineering importance the L.M. effect alone or particularly when combined with other parameters may reduce considerably the eigenfrequencies, as mentioned below.

3. The L.M. effect on the eigenfrequencies increases considerably with increase or decrease of the right angle deviation of the joint angle. The smallest value of this effect appears for rectangular frames. It should be noted that even for frames with relatively slender members the foregoing effect is tremendous for large or very small joint angles.

4. The L.M. effect when combined with a joint mass becomes significant, particularly for large values of p and small values of p, or more specifically for very small values of p / p z .

5. The mass offset may have an appreciable effect upon the eigenfrequencies under certain combinations, mainly of the parameters p and p.

6. As the joint mass is moved along the length of the girder the E-B theory gives practically the same results as those of this analysis provided that the rotatory inertia of the moving mass (regardless of its magnitude) is accounted for.

7. The L.M. effect alone or in conjunction with other parameters provokes also an appreciable reduction on the higher eigenfrequencies.

8. The discrepancies of this dynamicanalysis and E-B theory when computing the flexural eigenmodes may be of the same order of magnitude as that of the corresponding eigenfrequencies, even for frames with slender members.

ACKNOWLEDGEMENTS

This work was partially supported by the Deutsche Forschungsgemeinschaft and Ruhr-Universitat Bochum. The authors are grateful to these Institutions and Prof. Kratzig for providing Dr. Kounadis the opportunity to visit the Ruhr University for a short period, as visiting Professor.

The authors are also grateful to Professor G. Warburton for his kind criticism and valuable suggestions.


APPENDIX

The values of the elements aij(i,j = 1,. . . ,6) of the matrix (13) are

all = k , cos k , , a,, = k , cosh k , 4

a, = -k, cosk, - A M k2 sink, (1 - P ) [ cosh k,P-cos k2Z] +- k2 - J cosk,(l -Z) [sinh k,e+ sin k,Z] 2 2

kf - k,4 - 2 2

a14 = -k2coshk2 ---Msinhk,(l - P ) [coshk,F-cosk,F] +-Jcoshk,(l-i?) [sinhk,E+sink,E]

a I s = a16 = 0

cosa . cos a a,, =--sink,, a,, = -sinhk,

P P

k 2 - . k 3 - 2 2

= sink,+-Msink,(l - e ) [sinhk,F-sink,F] --Jcosk,(l -a) [coshk,e-cosk,e]

3

= sinhk,+--M k2 - sinhk, (1 -a) [sinhk,F-sink,Z] --Jcoshk,(l k2 - -e) [ c o s h k 2 ~ - c o s k 2 ~ ] 2 2

sina . a25 =-mu, , = 0

P

sink, sinh k , a3, = -- , a32 = -__

p sin a p sin a 3

a33 = -cota{sink,+--M k 2 - . sink,(l -Z) [sinhk,Z-sink,F] --Jcosk,(l-2) k2 - [coshk2F-cosk,F]) 2 2

k k 3 - 2 2

a34 = - c o t u { s i n h k , + ~ ~ s i n h k , ( l -a) [sinhk,Z-sink,F] --Jcoshk,(l -F) [coshk,Z- C O S ~ , ~ ]

a35 = 0

a36 = sinu, -u,Msinu,(l -E)sinu,Z, a41 = a42 = 0, a43 = -k,3cosk2 -

4 6 +-A4 k 2 - . sink,(l - P ) [coshk,Z+ c o s k , ~ ] --Jcosk,(l k2 - -z) [sinhk,F-sink,Z]

2 2 6

a44 = k l c o s h k , + L R k4 sinhk,(l -Z) [coshk,P+ cosk,F] --Jcoshk,(l k2 - -Z) (sinhk2F-sink2F) 2 2

a4s =- P2A: ulcosul, a46 = -~I:cotau,[cosu, -u2ii?sinu2(1-2) c o s u , ~ ] p sin a

6

a44 = k l c o s h k , + L R k4 sinhk,(l -Z) [coshk,P+ cosk,F] --Jcoshk,(l k2 - -Z) (sinhk2F-sink2F) 2 2

a4s =- P2A: ulcosul, a46 = -~I:cotau,[cosu, -u2ii?sinu2(1-2) c o s u , ~ ] p sin a

a51 = -k:cosk, sina, a52 = k:coshk, sina, as3 = a54 = 0, a55 = A:u,cosu,cosa

a56 = -p2iI:u2[cosu2 P -u,R sinu2(1 -z) c o s u , ~ ]

a6, = --k:sink,, P a6, =Ek:sinhk, P P

3 5

o~~~ = - k t sink, +--Msink,(l k2 - . -z) [sinhk,F+ sink2F] --Jcosk,(l k2 - -F) [coshk2F+ cosk,F] 2 2

COUPLING EFFECT OF AXIAL MOTION AND JOINT MASS 46 1

3 5 k, - . k2 - a64 = k; sinhk,+--Msinhk,(l -i?) [sinhk,e+ sink,Z] ---Jcosk,(l -e) [coshk,e+ cosk,F] 2 2

a65 = a66 = 0

REFERENCES

1. K. Hohenemser and W. Prager, Dynamik der Stabwerke, Springer-Verlag. Berlin, (1933). 2. K. Meskouris, ‘Elektronische Ermittlung der Eigenfrequenzen und Eigenschwingungsformen der ebenen Rahmentragwerke’,

3. A. N. Kounadis, ‘Bending eigenfrequencies of a two-bar frame including the effect of axial inertia’, AlAA j . 23, 2000-2002 (1986). 4. A. N. Kounadis and D. Sophianopoulos, ‘The effect of axial inertia on the bending eigenfrequencies of a Timoshenko two-bar frame’,

5. A. Alexandropoulos, G. Michaltsos and A. Kounadis, ‘The effect of longitudinal motion and other parameters on the bending

6. A. N. Kounadis, ‘Dynamic response of cantilevers with attached masses’, J . eng. mech. diu. ASCE 101, 695-706 (1975).

Bautechnik No. 5 , 171-175 (1973).

Earthquake eng. struct. dyn., 14, 429-437 (1986).

eigenfrequencies of a simple frame’, J . sound uib. 106, 153-159 (1986).

Documents

The coupling effect of axial motion and joint mass on the lateral vibrations of a rigid-jointed triangular frame