Dynamic response of a viscously damped cantilever with a viscous end condition

ARTICLE IN PRESS JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 298 (2006) 132–153 www.elsevier.com/locate/jsvi

Dynamic response of a viscously damped cantilever with a viscous end condition M. Gu¨rgo¨ze, H. Erol Istanbul Technical University, Faculty of Mechanical Engineering, 34439, Istanbul, Turkey Received 17 January 2005; received in revised form 19 April 2006; accepted 24 April 2006 Available online 25 July 2006

Abstract This paper deals with the dynamical analysis of a cantilevered Bernoulli–Euler beam subjected to distributed external viscous damping in-span and with a viscous end condition by a single damper. In order to evaluate the vibration characteristics of the system, a procedure is presented where overdamped and underdamped ‘‘modes’’ are investigated simultaneously, via the dynamic stiffness matrix of the system. Further, the orthogonality conditions, which allow the decoupling of the equations of motion in terms of principal coordinates, are derived. Then, the complex frequency response function is obtained through a formula which was established for the receptance matrix previously. Finally, the dynamic impulse response function of the system is evaluated in the time domain. Comparison with the numerical results obtained via a boundary value formulation justifies the approaches used here. r 2006 Elsevier Ltd. All rights reserved.

1. Introduction The dynamic response of a cantilevered beam subjected to distributed external viscous damping in-span and with a viscous boundary condition by a single damper is important for structural engineers. Designers often use dampers in order to reduce force transmissibility and displacements of various structures. The need for reduced force transmissibility and displacements is evident since lower force levels permit simpler and lighter structural designs. Thus, dampers are currently critical in many systems, such as buildings, cars and airplanes. The dynamics of structures with fixed and free boundary conditions has been studied previously [1–3]. These analyses form the basis for many classical beam problems. Their self-adjoint operators are separated using mutually orthogonal modes to form models of structural dynamic response. However, these models admit only standing waves into the response and do not consider damping at the boundary. It is to be noted that damping is one of the most important, as well difficult and complicated problems in vibration theory of mechanical systems. Most serious problems occur in damped complex continuous systems. In systems having non-classical damping it should be taken into account that the dynamic analysis can not be carried out by the standard mode superposition method [4]. In fact, eigencharacteristics of the system are complex and the equations of motion have to be decoupled in the complex domain. Corresponding author. Tel.: +212 293 13 00; fax: +212 245 07 95.

E-mail address: [email protected] (M. Gu¨rgo¨ze). 0022-460X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2006.04.042

ARTICLE IN PRESS M. Gu¨rgo¨ze, H. Erol / Journal of Sound and Vibration 298 (2006) 132–153

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This paper represents an approach based on the dynamic stiffness method which allows an assembly technique that enables exact free vibration analysis of either a single structural element or a combination of structural elements with different orientations. Furthermore, in contrast to finite element methods, the results using the dynamic stiffness method are not only exact, but also independent of the number of the elements used in the analysis, and thus offer much better computational efficiency [5]. Recently, a number of papers have appeared investigating different effects in beams and bars [4–8]. For a fixed-free longitudinal bar with a viscous boundary condition at its free end, a series solution was formulated and the complex eigenfunctions were evaluated in closed form [5]. The study in Ref. [4] is concerned with the application of the complex mode superposition method to the dynamic analysis of a simply supported Bernoulli–Euler beam with two rotational viscous dampers attached at the boundaries. Ref. [6] deals with the determination of the frequency response function of a cantilevered Bernoulli–Euler beam which is viscously damped by a single damper. The beam is simply supported in-span and carries a tip mass. The frequency response function is obtained through a formula that was established for the receptance matrix of discrete linear systems subjected to linear constraint equations, by considering the simple support as a linear constraint imposed on generalized coordinates. Ref. [7] is concerned with the eigencharacteristics of a continuous multistep beam model with external damping in-span using the separation-of-variables approach. In that study, the beam considered has different physical properties in each of its parts. A cantilevered beam viscously damped by a damper with a concentrated mass at its free end has been examined in Ref. [8]. The study in Ref. [9] is concerned with the free and forced vibration analyses of a uniform cantilever beam carrying a number of spring–damper–mass systems with arbitrary magnitudes and locations were made by means of the analytical and numerical combined method. The problem to be handled in the present paper represents to some extent a mixture of the applications mentioned above. The aim of this paper is to perform the dynamical analysis of a cantilevered Bernoulli–Euler beam subjected to distributed external viscous damping in-span and with a viscous end condition created by a single damper, by using the dynamic stiffness method. In order to evaluate the vibration characteristics of the system, a procedure is presented where overdamped and underdamped ‘‘modes’’ are investigated simultaneously. Then, the orthogonality conditions which allow the decoupling of the equations of motion in terms of principal coordinates are derived. Further, the complex frequency response function is obtained through a formula which was established for the receptance matrix previously [10]. Then, the dynamic impulse response function of the system is evaluated in the time domain. A further aim is to determine the amplitude distribution of the beam due to a harmonically varying vertical force acting at a given point. Furthermore, a comparative study has been developed between the approaches used in the present study and through a boundary value problem approach of the system. Comparison with the numerical results obtained via the boundary value problem solution justifies the approach used here. 2. Theory The problem can best be stated referring to the cantilevered beam shown diagrammatically in Fig. 1. Let it be assumed that the laterally vibrating Bernoulli–Euler beam has length L, bending rigidity EI, mass per unit length m, respectively. The beam subjected to distributed external viscous damping in-span with damping coefficient s is assumed to be viscously damped by a damper of damping constant d at the free end. At the distance x ¼ gL from the fixed end, a harmonically varying force F(t) is acting on the beam. 2.1. Free vibration analysis of the system Due to the presence of viscous damping in the system it is more appropriate to work with complex variables. It will be assumed that the lateral displacements w(x,t) are the real parts of some complex quantities denoted as z(x,t). Keeping in mind that, actually we are interested only in the real parts of the expressions below, the equation of motion of the beam can be written as EIzIV ðx; tÞ þ m€zðx; tÞ þ s_zðx; tÞ ¼ 0,

(1)

ARTICLE IN PRESS M. Gu¨rgo¨ze, H. Erol / Journal of Sound and Vibration 298 (2006) 132–153

134

L F(t) L

B

w(x,t) A

x

m, EI,
d

Fig. 1. Cantilevered beam subjected to external viscous damping in-span with a viscous end condition by a single damper.

where x is the axial position coordinate along the beam. Dots and primes denote partial derivatives with respect to time t and position coordinate x, respectively. The corresponding boundary conditions are zð0; tÞ ¼ 0; z00 ðL; tÞ ¼ 0;

z0 ð0; tÞ ¼ 0,

EIz000 ðL; tÞ ¼ d z_ðL; tÞ.

(2)

Let it be assumed that the solutions are of the form zðx; tÞ ¼ ZðxÞelt ,

(3)

where Z(x) is a complex function in general. l represents an eigenvalue of the system which is also complex in general. Substituting Eq. (3) into Eq. (1) gives Z IV ðxÞ
b4 ZðxÞ ¼ 0,

(4)

 m
s l þ l2 . EI m

(5)

with the abbreviation b4 ¼


The general solution of the differential Eq. (4) can be expressed as ZðxÞ ¼ C 1 ebx þ C 2 e
bx þ C 3 eibx þ C 4 e
ibx ,

(6)

where C i (i ¼ 1; . . . ; 4) denote complex constants to be determined. In terms of the Z(x), the boundary conditions in Eq. (2) can be formulated as Zð0Þ ¼ 0; Z00 ðLÞ ¼ 0;

Z0 ð0Þ ¼ 0,

EIZ000 ðLÞ ¼ dlZðLÞ.

(7)

The dynamic stiffness coefficients of a beam represent the end forces (FA and FB) and moments (MA and MB) resulting from the application of unit end displacements (ZA and ZB) and rotations (jA and jB) [11]. Thus, expressing the displacements and rotations at the two ends of the beam by means of the definitions given by Eq. (9) in matrix form lead to x ¼ U1 c¯ ,

(8)

ARTICLE IN PRESS M. Gu¨rgo¨ze, H. Erol / Journal of Sound and Vibration 298 (2006) 132–153

where

2

h x ¼ ZA

ZB

jA

jB

ZA ¼ ZðxA Þ;

iT

;

1 6 bL 6e U1 ¼ 6 6
b 4
bebL

h c¯ ¼ C 1

C2

C3

3

1 e
bL

1 eibL

1 e
ibL

b be
bL


ib
ibeibL

ib ibe
ibL

jA ¼ Z 0 ðxA Þ;

ZB ¼ ZðxB Þ;

135

C4

iT

7 7 7, 7 5

jB ¼ Z 0 ðxB Þ

.

(9)

Now expressing Eq. (6) and its first three derivatives in matrix form lead to f ¼ U2 c¯ ,

(10)

where  f ¼ FA 2

b3 6 6
b3 ð1
mÞebL U2 ¼ EI 6 6 b2 4
b2 ebL

FB


b3 b3 ð1 þ mÞe
bL 2

MA

MB


ib3 b3 ði þ mÞeibL 2

b


b


b2 e
bL

b2 eibL

T

,

3 ib3 7
b3 ði
mÞe
ibL 7 7; 7
b2 5 2
ibL b e

m¼

dl . EIb3

(11)

It is evident that the constants C¯ i can be evaluated via Eq. (8) and then substituting these into Eq. (10) leads to f ¼ U2 U
1 1 x.

(12)

The central matrix product in Eq. (12) which contains the dynamic stiffness information leads, by considering the corresponding boundary conditions [11], to " # " # zB FB ¼K , (13) jB MB where K denotes the dynamic stiffness matrix. It can be shown that it takes the form " # aF a EI , K¼ 1 þ e2ib
4eð1þiÞb þ e2b þ eð2þ2iÞb a aM

(14)

where b¯ ¼ bL, aF ¼ b3 ðð1
iÞ þ e2b ðð
1
iÞ þ mÞ þ eð2þ2iÞb ðð
1 þ iÞ þ mÞ þ m
4eð1þiÞb m þ e2ib ðð1 þ iÞ þ mÞÞ, a ¼ ib2 ð
1 þ e2ib Þð
1 þ e2b Þ, aM ¼ bð1 þ iÞðð
1 þ ie2ib
ie2b þ eð2þ2iÞb Þ.

(15)

The eigenvalues of the system are obtained by setting to zero the determinant of the dynamic stiffness matrix: ¯ mÞ ¼ 0. det Kðb;

(16)

The eigenfunctions of the system can be obtained after determination of the coefficients C i . Substitution of Eq. (6) into the boundary conditions in Eq. (7) yields the following set of homogeneous equations for the

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determination of the unknownsC i : C 1 þ C 2 þ C 3 þ C 4 ¼ 0, C 1
C 2 þ iC 3
iC 4 ¼ 0, C 1 eb þ C 2 e
b
C 3 eib
C 4 e
ib ¼ 0, C 1 ð
1
mÞeb þ C 2 ð1
mÞe
b þ C 3 ði
mÞeib þ C 4 ð
i
mÞe
ib ¼ 0.

(17)

It can be shown that the solution of this system of homogeneous equations yields the unknown complex 0 constants, C i s by the expressions 3 2 1 2 3 6 eb ðiþe2ib þð1þiÞeð1þiÞb Þ 7 C¯ 1 7 6 6 ¯ 7 6 ð
1
iÞeib
eb
ieð1þ2iÞb 7 7 6 C2 7 6 ib b ð2þiÞb 7 7 6 (18) c¯ ¼ 6 6 C¯ 3 7 ¼ 6 ie þð1þiÞeb þe Þ 7. 4 5 6 ð1
iÞeib
ie þeð1þ2iÞb 7 7 6 4 eib ð1þð1þiÞeð1þiÞb þie2b Þ 5 C¯ 4
b ib ð1þ2iÞb ð1
iÞe
ie þe

Having obtained the coefficients C i , the eigenfunctions Z(x) are at hand if these coefficients are substituted into Eq. (6). The differential equations which govern the mode shapes Zn and Zm can be written from Eq. (4) as 4 Z IV n ðxÞ
bn Z n ðxÞ ¼ 0,

(19)

4 Z IV m ðxÞ
bm Z m ðxÞ ¼ 0.

(20)

Multiplying Eq. (19) by Zm(x) and Eq. (20) by Zn(x), and integrating between 0 and L, leads to Z L Z L 4 Z IV Z dx
b Z n Zm dx ¼ 0, m n n 0

Z

L 0

(21)

0

4 Z IV m Z n dx
bm

Z

L

Z m Z n dx ¼ 0.

By integrating the Eqs. (21) and (22) by parts, and after some arrangements, Z L Z L 4 000 000 0 Zm ðLÞZ n ðLÞ
Z n Z m dx
bn Z n Z m dx ¼ 0, 0

Z n ðLÞZ 000 m ðLÞ


Z

L

0

4 0 Z000 m Z n dx
bm

0

Z

Z

L

Z m Z n dx ¼ 0

0

(24)

0

(25)

0

L

þ

(23)

0

are obtained. Integrating Eqs. (23) and (24) by parts once more, and performing some arrangements, yields Z L Z L 4 00 00 ðLÞ þ Z Z dx
b Z n Zm dx ¼ 0, Z m ðLÞZ000 n n n m

Z n ðLÞZ 000 m ðLÞ

(22)

0

Z00m Z 00

dx


b4m

Z

L

Z m Z n dx ¼ 0. 0

(26)

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137

By applying the boundary conditions and performing some rearrangements, Eqs. (25) and (26) may be written as Z L Z L don Zn ðLÞZm ðLÞ
b4n Z00n Z 00m dx þ Z n Z m dx ¼ 0, (27) EI 0 0 Z

L

0

Z 00m Z00n

dom Z m ðLÞZ n ðLÞ
b4m dx þ EI

Z

L

Z m Zn dx ¼ 0

(28)

0

with the abbreviation o2i ¼ ðbi LÞ4

EI . mL4

(29)

Subtraction of Eq. (27) from Eq. (28) yields


d ðom
on ÞZ m ðLÞZ n ðLÞ þ ðb4m
b4n Þ EI

Z

L

Z m Zn dx ¼ 0.

(30)

0

Eq. (30), for om aon , gives the first orthogonality condition as Z L ðom þ on Þ mZ m Z n dx
dZ m ðLÞZn ðLÞ ¼ 0,

(31)

0

Subtraction of Eq. (27) multiplied by om, from Eq. (28) multiplied by on, leads to Z L Z L don om
om Zn ðLÞZm ðLÞ þ b4n om Z 00n Z00m dx
Z n Z m dx EI 0 0 Z þon 0

L

Z 00m Z00n dx þ

dom on Zm ðLÞZ n ðLÞ
b4m on EI

Z

L

Z m Zn dx ¼ 0.

(32)

0

Eq. (32), for om aon and after some more arrangements, gives the second orthogonality condition as Z L Z L EIZ 00m Z 00n dx þ on om Z m Z n dx ¼ 0. (33) 0

0

Since to each eigenvalue ln, there is its conjugate ln , by taking lm ¼ ln ln þ ln ¼ 2xn jon j;

ln ln ¼ jon j2

(34)

are obtained, where xn are the modal damping factors. Thus, from Eq. (31), after some arrangements, djZ n ðLÞj2 s 2xn jon j ¼ R L þ 2 m 0 mjZ n j dx

(35)

and from the Eq. (33), after some rearrangements, 2

RL

jon j ¼ R0L 0

EIjZ00n j2 dx mjZ n j2 dx

(36)

are obtained. 2.2. Frequency response function of the system The equation of motion of the beam in Fig. 1 subjected to both distributed and discrete damping at the boundary and excited by the force F(t) can be written as [5,6] EIzIV ðx; tÞ þ m€zðx; tÞ þ s_zðx; tÞ ¼ F ðtÞdðx
gLÞ, where d(x) denotes the Dirac’s function.

(37)

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An approximate series solution of the differential equation (37) can be taken in the form zðx; tÞ 

n X

Zr ðxÞZr ðtÞ;

(38)

r¼1

where the Zr(x) are the orthogonal eigenfunctions of the bare clamped–free beam, normalized with respect to the mass density, and Zr(t) are the generalized coordinates. After substitution of expression (38) into the differential equation (37), the equation of motion becomes n X

½EIZ IV Zr ðtÞ þ sZ r ðxÞ_Zr ðtÞ ¼ F ðtÞdðx
gLÞ. r ðxÞZr ðtÞ þ mZ r ðxÞ€

(39)

r¼1

Then, according to Galerkin’s procedure, both sides of the equation are multiplied by the s’th eigenfunction Zs(x) and integrated over the beam length L, which, after some arrangements, yields i n h RL RL RL P EIZr ðtÞ 0 Z IV ðxÞZ ðxÞ dx þ m€ Z ðtÞ Z ðxÞZ ðxÞ dx þ s_ Z ðtÞ Z ðxÞZ ðxÞ dx s r s r s r r r 0 0 r¼1 (40) RL ¼ 0 F ðtÞdðx
gLÞZ s ðxÞ dx ðs ¼ 1; . . . ; nÞ: Integrating twice by parts and applying the boundary conditions (7), one obtains   Z L Z L Z n  X 000 00 00 EIZr ðtÞ Zr ðLÞZ s ðLÞ þ Z r ðxÞZ s ðxÞ dx þ m€Zr ðtÞ Z r ðxÞZ s ðxÞ dx þ s_Zr ðtÞ r¼1

Z

0

0



L

Z r ðxÞZ s ðxÞ dx 0

L

F ðtÞdðx
gLÞZ s ðxÞ dx

¼

ðs ¼ 1; . . . ; nÞ.

ð41Þ

0

Then, by using orthogonality conditions (31) and (33) of the eigenfunctions, after some algebraic manipulations, the system of the modal equations, i.e., the system of differential equations for the Zi(t), is obtained as [6] Z€ i ðtÞ þ

n n X Z i ðLÞZ j ðLÞ Z j ðgLÞ dX s Z_ i ðtÞ þ Z_ i ðtÞ þ b4i Zi ðtÞ ¼ F ðtÞ ; ði ¼ 1; . . . ; n; j ¼ 1; . . . ; nÞ. RL RL m j¼1 m Z i ðxÞZ j ðxÞ dx Z i ðxÞZ j ðxÞ dx j¼1 m 0

0

(42) Hence, upon making use of Eqs. (35) and (36), the system of differential equations in Eq. (42) becomes 2

Z€ i ðtÞ þ 2xi oi Z_ i ðtÞ þ oi Zi ðtÞ ¼ N i ðtÞ ði ¼ 1; . . . ; nÞ,

(43)

where, 2

oi ¼ ðbi LÞ4 EI=mL4 ; b1 L ¼ 1:875104068712; b2 L ¼ 4:694091132974; . . . ½6 N i ðtÞ ¼ F ðtÞ

n X j¼1

m

RL 0

Z j ðgLÞ Z i ðxÞZ j ðxÞ dx

.

(44)

At this point, it may be noted that Ni(t) is the ith generalized force. Assuming that the forcing function is of the form, F ðtÞ ¼ F 0 eiOt ,

(45)

the system of differential equations in Eq. (43) can be written in matrix notation as g€ ðtÞ þ D_gðtÞ þ x2 gðtÞ ¼ NðtÞ,

(46)

ARTICLE IN PRESS M. Gu¨rgo¨ze, H. Erol / Journal of Sound and Vibration 298 (2006) 132–153

where

 T gðtÞ ¼ Z1 ðtÞ . . . Zn ðtÞ ; NðtÞ ¼ F 0 Z ðgLÞeiOt ;

139


2 x2 ¼ diag oi ,

D ¼ sI þ dZðLÞZT ðLÞ,

ZðxÞ ¼ ½Z 1 ðxÞ . . . Z n ðxÞT .

ð47Þ

oi

(i ¼ 1, y, n) are the eigenfrequencies of the bare cantilevered beam. Substitution of gðtÞ ¼ geiOt

(48)

g ¼ HðOÞN,

(49)

into the matrix differential equation (46) yields

where N ¼ F 0 ZðgLÞ;

 T gðtÞ ¼ Z1 ðtÞ . . . Zn ðtÞ

(50)

Then, the receptance matrix is obtained in the form


1 . HðOÞ ¼
O2 I þ iOD þ x2

(51)

Considering Eq. (47), it can be arranged as 
1 HðOÞ ¼ K þ uvT ,

(52)

where K ¼ x2 þ Oðis
OÞI;

u ¼ iOdZðLÞ;

vT ¼ ZT ðLÞ.

Using the Sherman-Morrison formula 
1 
1 ¼ K
1
K
1 u 1 þ vT K
1 u vT K
1 K þ uvT

(53)

(54)

which gives the inverse of the sum of a regular matrix and a dyadic product [6], the receptance matrix can be written as follows HðOÞ ¼ K
1
noting that K


1

K
1 iOdZðLÞZT ðLÞK
1 , 1 þ ZT ðLÞK
1 iOdZðLÞ

(55)

! 1 ¼ diag . o2i þ isO
O2

(56)

The receptance matrix can be expressed as

! 1 HðOÞ ¼ diag o2i þ isO
O2 2
 3 d 1 i mL OaðLÞaT ðLÞdiag o2 þisO
O 2 6 7
i  4I
5, d 1 T 1 þ i mL Oa ðLÞdiag o2 þisO
O2 aðLÞ

ð57Þ

i

where the following definitions are introduced: 1 1 ZT ðxÞ ¼ pffiffiffiffiffiffiffi aT ðxÞ ¼ pffiffiffiffiffiffiffi ½a1 ðxÞ . . . an ðxÞ, mL mL ai ðxÞ ¼ cosh bi


x x x x
cos bi
Zi sinh bi
sin bi ; L L L L

Zi ¼

cosh bi þ cos bi . sinh bi þ sin bi

(58)

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140

Finally, after some rearrangements, the receptance matrix can be expressed as 2
 3 ! T 1 a ð L Þa ð L Þdiag 1 o2 þisO
O2 6 7
i  HðOÞ ¼ diag I
5 2 4 2 mL oi þ isO
O þ aT ðLÞdiag 2 1 2 aðLÞ idO

(59)

oi þisO
O

I being the n  n unit matrix. Therefore, the displacements of the beam can be written as zðx; tÞ ¼ ZðxÞeiOt ,

(60)

where ZðxÞ ¼

n X

Z r ðxÞZr .

(61)

r¼1

It is easy to show that the above expression can be reformulated as  ZðxÞ ¼ ZT ðxÞHðOÞZðgLÞ F 0 .

(62)

As previously mentioned, the real part of z(x,t) represents the physical displacements. In the case of F 0 ¼ 1, the right-hand side of Eq. (62) represents the frequency response function of the beam in Fig. 1. 2.3. Modal impulse response function of the system In this section, the complex mode superposition method for the dynamical analysis of the system in time domain will be presented. At first, the modal impulse response functions will be derived. Then, these functions will be used to evaluate the response to a general input. Due to the impulsive nature of the excitation, the principal coordinates take the form [4] Zr ðtÞ ¼ C r eior t ,

(63)

which imply Z€ r ðtÞ ¼ ior Z_ r ðtÞ ¼
o2r Zr ðtÞ.

Z_ r ðtÞ ¼ ior Zr ðtÞ;

(64)

Through application of the boundary conditions (2) onto the differential equations (41), one obtains   Z L Z L n  X idor 00 00 Z r ðLÞZ s ðLÞ þ EIZr ðtÞ Z r ðxÞZ s ðxÞ dx þ m€Zr ðtÞ Z r ðxÞZs ðxÞ dx EI 0 0 r¼1  Z L Z L þs_Zr ðtÞ Zr ðxÞZ s ðxÞ dx ¼ F ðtÞdðx
gLÞZ s ðxÞ dx; ðs ¼ 1; . . . ; nÞ. ð65Þ 0

0

Then, by using orthogonality conditions (31) and (33) of the eigenfunctions and applying Eqs. (40) between the principal coordinates and their derivatives (64), after some algebraic manipulations, the following equations are found   Z L Z L n  X mZ r ðxÞZ s ðxÞ dx þ Z_ r ðtÞ dZr ðLÞZs ðLÞ þ s Z r ðxÞZs ðxÞ dx Z€ r ðtÞ2 r¼1

0

0

¼ F ðtÞZs ðgLÞ ðs ¼ 1; . . . ; nÞ.

ð66Þ

It may be noted that, by applying the orthogonality conditions (31) and (33), the previous expressions vanish for ras, this making it possible to decouple the equations of motion, which take the form   Z L Z L mZ 2r ðxÞ dx þ Z_ r ðtÞ dZ2r ðLÞ þ s Z 2r ðxÞ dx ¼ F ðtÞZ r ðgLÞ ðr ¼ 1; . . . ; nÞ. (67) Z€ r ðtÞ2 0

0

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141

At this stage, it is assumed that the forcing function is of the impulsive form: F ðtÞ ¼ IdðtÞ,

(68)

where I is the unit impulse and d(t) is the Dirac’s delta function. Integrating of Eqs. (67) in the time interval (0
,0+) leads to   Z L Z L þ þ 2 2 2 mZ r ðxÞ dx þ Zr ð0 Þ dZ r ðLÞ þ s Zr ðxÞ dx ¼ IZr ðgLÞ ðr ¼ 1; . . . ; nÞ Z_ r ð0 Þ2 0

(69)

0

from which the constants Cr in Eq. (63) are obtained as Cr ¼

RL

IZ r ðgLÞ

2ior 0 mZ 2r ðxÞ dx

þ dZ 2r ðLÞ þ s

RL 0

Z2r ðxÞ dx

¼ IBr

ðr ¼ 1; . . . ; nÞ,

(70)

Then, the complex modal impulse response functions can be written as hcr ðx; tÞ ¼ Br Z r ðxÞeior t

ðr ¼ 1; . . . ; nÞ.

(71)

The expression of the complex modal impulse response function is given in Appendix A due to space limitations. However, the dynamical response can be evaluated more conveniently by using real variables. This can be performed by summing each modal contribution and its conjugate one. Thus,   ior t hre ðr ¼ 1; . . . ; nÞ. (72) r ðx; tÞ ¼ 2Re Br Z r ðxÞe Additionally, the impulse response function and the frequency response function are related to each other. This relationship, by using Eq. (62), may be written as Z 1 þ1 T hðx; tÞ ¼ Z ðxÞHðoÞZðgLÞeiot do. (73) 2p
1 By using the impulse response function (73), the dynamical response to a general force F(t) may be obtained as 1 Z t X zðx; tÞ ¼ F ðtÞhre (74) r ðx; t
tÞ dt, r¼1

0

where the loading history has been divided into a sequence of impulses, as it is usual in the convolution integral method of dynamical analysis. When the forcing function is harmonic, the dynamical response may still be evaluated by means of Eq. (74). 2.4. Solution through the boundary value problem formulation In order to prove the validity of the above expressions, it is reasonable to compare the results with the results of a boundary value problem formulation. It will be assumed that the bending displacements wi(x,t), (i ¼ 1; 2) of the two parts of the beam are the real parts of some complex quantities denoted as zi(x,t). The bending vibrations of the two beam portions left and right to the external force F(t) shown in Fig. 1 are governed by the partial differential equations EIz{v zi ðx; tÞ þ s_zi ðx; tÞ ¼ 0 i ðx; tÞ þ m€

ði ¼ 1; 2Þ

(75)

which are formally the same as in Eq. (1). The corresponding boundary and matching conditions are as follows: z1 ð0; tÞ ¼ 0; z01 ð0; tÞ ¼ 0, z1 ðgL; tÞ ¼ z2 ðgL; tÞ; z01 ðgL; tÞ ¼ z02 ðgL; tÞ; 000 iOt EIz000 ¼ 0; 1 ðgL; tÞ
EIz2 ðgL; tÞ þ F 0 e

z001 ðgL; tÞ ¼ z002 ðgL; tÞ,

z002 ðL; tÞ ¼ 0;

EIz000 2 ðL; tÞ
d z_2 ðL; tÞ ¼ 0.

ð76Þ

ARTICLE IN PRESS M. Gu¨rgo¨ze, H. Erol / Journal of Sound and Vibration 298 (2006) 132–153

142

If harmonic solutions of the form zi ðx; tÞ ¼ Zi ðxÞeiOt

(77)

are substituted into Eq. (75), the previous ordinary differential equations in Eq. (4) are obtained for the amplitude functions Zi(x): Z {vi ðxÞ
b4 Z i ðxÞ ¼ 0

ði ¼ 1; 2Þ,

(78)

4

where b is now defined as mO2 þ sO . EI The general solutions of the differential equations (78) are similar in form to Eq. (6) b4 ¼


(79)

Z 1 ðxÞ ¼ c1 ebx þ c2 e
bx þ c3 eibx þ c4 e
ibx , Z 2 ðxÞ ¼ c5 ebx þ c6 e
bx þ c7 eibx þ c8 e
ibx .

ð80Þ

The corresponding boundary and matching conditions read now, as Z 1 ð0Þ ¼ 0;

Z 01 ð0Þ ¼ 0,

Z 01 ðgLÞ ¼ Z 02 ðgLÞ; Z 001 ðgLÞ ¼ Z002 ðgLÞ, F0 idO 000 Z 2 ðLÞ ¼ 0, ¼ 0; Z 002 ðLÞ ¼ 0; Z000 ð81Þ Z 000 1 ðgLÞ
Z 2 ðgLÞ þ 2 ðLÞ
EI EI where c1 to c8 are unknown integration constants to be determined, which can be complex in general. The substitution of expressions (80) into Eq. (81) yields the following set of eight inhomogeneous equations for the determination of the eight unknowns c1 to c8: 32 3 2 2 3 1 1 1 1 0 0 0 0 c1 0 76 7 6 6 1
1 i
i 0 0 0 0 7 6 c2 7 6 0 7 6 7 76 7 6 6 bg
bg ibg
ibg bg
bg ibg
ibg 7 6 c3 7 6 0 7 6e e e e
e
e
e
e 7 76 7 6 6 7 ibg
ibg bg
bg 76 7 6 6 ebg
e
bg ieibg
ie
ibg
e e
ie ie 7 6 c4 7 6 0 7 6 7 76 7 ¼ 6 6 bg , 7 6 c5 7 6 0 7 6e e
bg
eibg
e
ibg
ebg
e
bg eibg e
ibg 7 76 7 6 6 7 ibg
ibg ibg
ibg 76 7 6 F 0 7 6 ebg
e
bg
ie ie
ebg e
bg ie
ie 76 c6 7 6
EI 7 6 76 7 6 6 b
b ib
ib 7 6 c7 7 4 0 7 6 0 0 0 0 e e
e
e 5 54 5 4 b idO
b idO ib idO
ib 0 0 0 0 ð1
bidO Þe
ð1 þ Þe
ði þ Þe ði
Þe 3 3 3 3 c8 0 b EI b EI b EI EI Z 1 ðgLÞ ¼ Z 2 ðgLÞ;

(82) where b¯ ¼ bL. For the free vibration analysis, let the 8  8 matrix of coefficients in Eqs. (82) be denoted by A. For a nontrivial solution, the determinant of this matrix should be zero det A ¼ 0,

(83)

from which eigenvalues of the system can be obtained, which are complex numbers in general. If the eigenvalues of the system obtained from Eq. (83) are substituted into the coefficient matrix A in Eq. (82), the unknowns c1–c8 can be determined. Hence, Zi(x), i ¼ 1; 2 in Eqs. (80) are obtained. Thus, the bending displacements of the beam portions wi(x,t) are determined as wi ðx; tÞ ¼ Re½zi ðx; tÞ.

(84)

wi(x,t), i ¼ 1; 2 determine the bending displacement distribution over the length of the cantilever beam subjected to external viscous damping in-span and with a viscous end condition by a single damper, when it vibrates at an eigenvalue b. Absolute values of Zi(x) represent the amplitude distribution over the ith step of the beam.

From Eq. (15)

From Eq. (83) 725.84028i 7161.93829i 7453.43191i 7888.54556i 71468.82948i
31.28305715.00780i
26.976327149.34470i
28.677027446.16697i
29.107937883.31688i
29.2991371464.73711i
1.09566
5.328667113.62035i
18.163037369.79653i
37.131377774.62234i
61.5517071329.90543i

From Eq. (15)

725.84028i 7161.93829i 7453.43191i 7888.54556i 71468.82948i


31.28305715.00780i
26.976327149.34470i
28.677027446.16697i
29.107937883.31688i
29.2991371464.73711i


1.09566
5.328667113.62035i
18.163037369.79653i
37.131377774.62234i
61.5517071329.90543i

d¼0

d ¼ 10 kg=s

d ¼ 100 kg=s


1.07033
12.679717113.72146i
25.447657370.07296i
44.357437774.91183i
68.7228971330.16913i


10.18582
33.292377148.06339i
35.868767445.65466i
36.434927883.04899i
21.9303871464.86407i


7.40741724.75582i
7.407417161.76879i
7.407417453.37140i
7.407417888.51469i
7.4074171468.81080i

s ¼ 10 kg=ms

s¼0


1.07033
12.679717113.72146i
25.447657370.07296i
44.357437774.91183i
68.7228971330.16913i


10.18582
33.292377148.06339i
35.868767445.65466i
36.434927883.04899i 21.9303871464.86407i


7.40741724.75582i
7.407417161.76879i
7.407417453.37140i
7.407417888.51469i
7.4074171468.81080i

From Eq. (83)


0.88520
78.88160790.17755i
91.069317365.65266i
109.457817774.18387i
133.3348271330.58117i


3.26233
92.035127122.11581i
100.639707435.72825i
102.384997877.88844i
102.9889571461.42522i


4.65327
74.074077144.00362i
74.074077447.34050i
74.074077885.45257i
74.0740771466.96049i

From Eq. (15)

s ¼ 100 kg=ms


0.88520
78.88160790.17755i
91.069317365.65266i
109.457817774.18387i
133.3348271330.58117i


3.26233
92.035127122.11581i
100.639707435.72825i
102.384997877.88844i
102.9889571461.42522i


4.65327
74.074077144.00362i
74.074077447.34050i
74.074077885.45257i
74.0740771466.96049i

From Eq. (83)

Table 1 Eigenvalue parameters b of the system for three different viscous damping coefficients d at the free end and for three different external viscous damping coefficients s

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144

Finally, in the following, the frequency response function of the beam is represented. From Eq. (82) the unknowns c1–c8 can be determined as 2 3 2 3 w1 c1 6 7 6w 7 6 c2 7 6 27 6 7 6 7 6 c3 7 6 w3 7 6 7 6 7 6 7 6 7 6 c4 7 6 w4 7 F 0 6 7¼
6 7, (85) 6 c5 7 6 w5 7 EIk 6 7 6 7 6 7 6 7 6 c6 7 6 w6 7 6 7 6 7 6 c7 7 6w 7 4 5 4 75 w8

c8

where k and w1–w8 are given in Appendix B. Having obtained Zi(x) (i ¼ 1; 2), it is possible to determine the steady-state amplitude at any point x of the beam, due to the harmonic force at a point x ¼ gL. Noting that according to Eq. (77) the real part of Z i ðxÞeiOt represents the physical displacements, absolute values of Zi(x) represent the amplitude distribution along  the cantilevered beam subjected to the harmonically varying vertical force at x ¼ gL. In the case of F 0 ¼ 1, Z i ðxÞ represents the frequency response function of the beam in Fig. 1. Then the impulse response function of the system, by using Eqs. (80), may be written as Z  1 þ1  hr ðx; tÞ ¼ Zr ðxÞeiot do; r ¼ 1; 2. (86) 2p
1 3. Numerical applications This section is devoted to the numerical evaluations of the formulae established in the preceding sections. In 3 the examples, the length L ¼ 1 ðmÞ, the bending rigidity EI ¼ ð7  1010 Þ 0:050:05 ðNm2 Þ, mass per unit length 12 m ¼ 0:675 ðkg=mÞ and the non-dimensional position of the harmonically varying vertical force g ¼ 1:0 are chosen, respectively. 14

200

12

150

10

100

8

50

6 4

-50

2

-100

0.2

0.4

0.6

0.8

1

-150 0.2

0.4

0.6

0.8

1

x [m]

x [m]

= ± 25.84028i

= ± 161.93829i

4000

100000

2x106

2000

50000

1x106

0.2

0.4

0.6

0.8

0.2

1

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

-1x106

-50000

-2000

-2x106 x [m]

x [m]

x [m]

= ± 453.43191i

= ± 888.54556i

= ± 1468.82948i

Fig. 2. Plots of eigenfunctions corresponding to the ‘‘first’’ five eigenvalues, for the external viscous damping coefficient s ¼ 0 and the viscous damping coefficient d ¼ 0 at the free end.

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145

Fig. 3. Three-dimensional plots of wi(x,t) and eigenfunctions corresponding to the ‘‘first’’ five eigenvalues, for the external viscous damping coefficient s ¼ 10 kg=ms and the viscous damping coefficient d ¼ 10 kg=s at the free end.

ARTICLE IN PRESS 146

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Fig. 4. Three-dimensional plots of wi(x,t) and eigenfunctions corresponding to the ‘‘first’’ five eigenvalues, for the external viscous damping coefficient s ¼ 100 kg=ms and the viscous damping coefficient d ¼ 100 kg=s at the free end.

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147

0.0025

0.006

0.001

0.005

0.002

0.004

0.0015

0.003

0.0008 0.0006

0.001

0.0004

0.0005

0.0002

0.002 0.001 200

400

600

200 400 600 800 1000 1200 1400

800 1000 1200 1400

200 400 600 800 1000 1200 1400







d=0,
=0

d = 0,
= 10 [kg/ms]

d = 0,
= 100 [kg/ms]

0.0008

0.0005

0.0006

0.0004

0.0008 0.0006

0.0003 0.0004

0.0004

0.0002 0.0002

0.0002 200 400 600 800 1000 1200 1400 

200 400 600 800 1000 1200 1400

200 400 600 800 1000 1200 1400

d = 10 [kg/s],
= 0 0.00012

0.0001





d = 10 [kg/s],
= 10 [kg/ms]

d = 10 [kg/s],
= 100 [kg/ms]

0.0001

0.00005

0.00008

0.00004

0.00006

0.00003

0.00004

0.00004

0.00002

0.00002

0.00002

0.00001

0.0001 0.00008 0.00006

200 400 600 800 1000 1200 1400

200 400 600 800 1000 1200 1400

200 400 600 800 1000 1200 1400







d = 100 [kg/s],
= 0

d = 100 [kg/s],
= 10 [kg/ms]

d = 100 [kg/s],
= 100 [kg/ms]

Fig. 5. Frequency responses of the beam for the three different viscous damping coefficients d at the free end and the three different external viscous damping coefficients s.

Prior to these numerical evaluations, eigenvalues of the beam subjected to distributed external viscous damping in-span and with a viscous end condition by a single damper are determined. Table 1 gives the ‘‘first’’ five eigenvalues of the system for three different viscous damping coefficients d at the free end and for three different external viscous damping coefficients s while the other parameters are kept constant. It is seen that the physical parameters in the damped case lead to both overdamped and underdamped ‘‘modes’’. The numerical values in the first columns represent the results of finding the roots of the determinantal equation in Eq. (16), whereas those of the second columns are results of finding the roots of the determinant in Eq. (83) based on the solution through the boundary-value problem formulation. Fig. 2 reflects the plots of eigenfunctions corresponding to the ‘‘first’’ five eigenvalues for the external viscous damping coefficient s ¼ 0 and the viscous damping coefficient d ¼ 0 at the free end. Fig. 3 shows both the three-dimensional plots of wi(x,t) and eigenfunctions corresponding to the ‘‘first’’ five eigenvalues for the external viscous damping coefficient s ¼ 10 ðkg=msÞ and the viscous damping coefficient d ¼ 10 ðkg=sÞ at the free end. The first pair of plots in Fig. 3, as the three-dimensional plots of wi(x,t) and eigenfunctions corresponding to the ‘‘first’’ eigenvalue, represents an overdamped mode, whereas the other modes are underdamped. It is seen from Fig. 3 that the amplitudes of the intersection curves corresponding to increasing t values decrease in accordance with the damped character of the system. Fig. 4 represents both the three-dimensional plots of wi(x,t) and eigenfunctions corresponding to the ‘‘first’’ five eigenvalues for the external viscous damping coefficient s ¼ 100 ðkg=msÞ and the viscous damping coefficient d ¼ 100 ðkg=sÞ at the free end. The first pair of plots in Fig. 4, as the three-dimensional plots of wi(x,t) and eigenfunctions corresponding to the ‘‘first’’ eigenvalue, represents an overdamped mode, whereas

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148

hr(x,t)

1 0.5 0 -0.5 -1 0

0.003 0.002 0.25 0.001

0.5

t [s]

0.75 x [m]

1 0 r=1

1 1 0.5 0 -0.5 -1 0

0.5 0.0008 0.0006 0.0004

0 -0.5 -1 0 0.25

0.25 0.5

0.0002

0.0001 0.5

0.0002

0.75

0.75 r=2

1 0

r=3

1

0

1

1 0.5 0 -0.5 -1 0

0.0002 0.00015 0.0001 0.25

0.5 0 -0.5 -1 0

0.0001 0.000075 0.00005

0.25 0.00005

0.5 0.75 r=4

0.5

0.000025 0.75

1 0

r=5

1 0

Fig. 6. Three-dimensional plots of the modal impulse response functions corresponding to the ‘‘first’’ five eigenvalues, for the external viscous damping coefficient s ¼ 0 and the viscous damping coefficient d ¼ 0 at the free end.

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149

0.1 hr(x,t)

0.05

1

0 -0.05 -0.1 0

0.75 0.5 0.25

t [s] 0.25

0.5 0.75

x [m]

1 0

r=1

0.2

0.5

0.1 0 -0.1 -0.2 0

0.001

0.25

0.25 0 -0.25 -0.5 0

0.0004 0.25

0.0005 0.5

0.0002 0.5 0.75

0.75 r=2

1 0

r=3

1

1

0.5 0 -0.5 -1 0

0.5 0 -0.5 -1 0

0.0004

0.0002

0.25

1 0

0.0004

0.0002

0.25 0.5

0.5

0.75

0.75 r=4

1 0

r=5

1 0

Fig. 7. Three-dimensional plots of the modal impulse response functions corresponding to the ‘‘first’’ five eigenvalues, for the external viscous damping coefficient s ¼ 10 kg=ms and the viscous damping coefficient d ¼ 10 kg=s at the free end.

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150

hr(x,t)

0.02 0 0.2 -0.02 0 0.1

0.25

t [s]

0.5 0.75 x [m]

1 0 r=1

20

10 5

0.04

0 -5 -10 0

0.03 0.02

10 0 0.002

-10 -20 0 0.001

0.25

0.25

0.5

0.01

0.5

0.75

0.75 r=2

1 0

r=3

1 0

10 5 0 -5 -10 0

0.0004 0.25

0.0002 0.5

4 2 0 -2 -4 0

0.0002 0.00015 0.0001 0.25 0.00005

0.5 0.75 r=4

0.75 1 0

r=5

1 0

Fig. 8. Three-dimensional plots of the modal impulse response functions corresponding to the ‘‘first’’ five eigenvalues, for the external viscous damping coefficient s ¼ 100 kg=ms and the viscous damping coefficient d ¼ 100 kg=s at the free end.

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151

the other modes are underdamped. Fig. 4 illustrates that the amplitudes of the intersection curves corresponding to increasing t values decrease in accordance with the damped character of the system. Fig. 5 gives the frequency responses of the beam for different viscous damping coefficients d at the free end and for different external viscous damping coefficients s while the other parameters are kept constant. Fig. 5 is obtained from formula (62), i.e., the present theory, where n ¼ 15 is taken in the series expansion (38), and b1 L to b15 L in Eq. (44) are taken from Ref. [6], which are correct up to 12 decimal places. This figure clarifies that increasing distributed external viscous damping in-span and the viscous damping coefficient at the boundary suppresses the frequency responses of the beam. Both of the distributed external viscous damping in-span and the viscous damping at the end are more effective at the resonance peaks in the system, as expected. Fig. 6 reflects the three-dimensional plots of the modal impulse response functions, normalized to 1, corresponding to the ‘‘first’’ five eigenvalues for the external viscous damping coefficient s ¼ 0 and the viscous damping coefficient d ¼ 0 at the free end. In this figure, for the undamped case, time history of the dynamic responses of the beam to an impulse are reported for the ‘‘first’’ five modes. Similarly, Fig. 7 represents the three-dimensional plots of the modal impulse response functions corresponding to the ‘‘first’’ five eigenvalues for the external viscous damping coefficient s ¼ 10 (kg/ms) and the viscous damping coefficient d ¼ 10 ðkg=sÞ at the free end. The modal impulse response functions are again normalized for clarity of the modes. Finally, Fig. 8 shows the three-dimensional plots of the modal impulse response functions corresponding to the ‘‘first’’ five eigenvalues for the external viscous damping coefficient s ¼ 100 ðkg=msÞ and the viscous damping coefficient d ¼ 100 ðkg=sÞ at the free end. For clarifying the modes, the modal impulse response functions are normalized too. Both distributed external viscous damping in-span and the single viscous damping at the boundary decrease magnitudes of the impulse response functions of the beam. However, it is seen that the single viscous damper at the free end is more effective on the suppression of the displacements of the free end. 4. Conclusions This study is concerned with the dynamical analysis of a cantilevered Bernoulli–Euler beam subjected to distributed external viscous damping in-span with a viscous end condition by a single damper. In order to evaluate the vibration characteristics of the system, a procedure is presented where overdamped and underdamped modes are investigated. Here, both of them will be handled simultaneously, by setting to zero the determinant of the dynamic stiffness matrix of the system. The orthogonality conditions, which allow the decoupling of the equation of motion in terms of principal coordinates, are derived. Then, the complex frequency response function is obtained through a formula which was established for the receptance matrix, previously. Finally, the impulse response function of the system is evaluated for the dynamical analysis in time domain. Comparison of the numerical results obtained via a boundary value problem formulation justifies the approaches used here. Appendix A hcr ðx; tÞ ¼ ðð1 þ iÞeð
1
iÞð2LþxÞbi þioi t ð
1 þ ie2iLbi
ie2Lbi þ eð2þ2iÞLbi ÞðeðLþixÞbi
ieðð1þ2iÞðLþixÞbi þð1
iÞeð2þiÞLþixÞbi þ eðiLþxÞbi þ ð1 þ iÞeðð1þ2iÞLþxÞbi þ ieðð2þiÞLþxÞbi
ieðiLþð1þ2iÞxÞbi
ð1 þ iÞeðLþð1þ2iÞxÞbi
eðð2þiÞLþð1þ2iÞxÞbi
ð1
iÞeðiLþð2þiÞxÞbi þ ieðLþð2þiÞxÞbi
eðð1þ2iÞLþð2þiÞxÞbi Þbi Þ =ð3ð4 sinðð1 þ iÞbi LÞ þ ð1
iÞ sinð2bi LÞ þ sinðð2 þ 2iÞbi LÞ
4 sinhðð1 þ iÞbi LÞ
ð1
iÞ sinhð2bi LÞ
sinhðð2 þ 2iÞbi LÞðs þ i2moi Þ þ 2bi ðð1
iÞsLð
i2 cosðð1 þ iÞbi LÞ þ cosð2bi LÞÞ þ dðð
1 þ iÞ cosðbi LÞ
i cosðð1 þ 2iÞbi LÞ þ cosðð2 þ iÞbi LÞ þð1 þ iÞsLð2 coshðð1 þ iÞbi LÞ þ i coshð2bi LÞÞ þ dðð
1 þ iÞ coshðbi LÞ
i coshðð1 þ 2iÞbi LÞ þ coshðð2 þ iÞbi LÞ þ ð1
iÞ sinðbi LÞ þ sinðð1 þ 2iÞbi LÞ þ sinðð2 þ iÞbi LÞ
ð1
iÞ sinhðbi LÞ
sinhðð1 þ 2iÞbi LÞ
sinhðð2 þ iÞbi LÞ
ð4 þ 4iÞmLðsinðbi LÞ þ sinhðbi LÞ2 oi ÞÞ

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152

Appendix B k ¼ 16 EIðEI b3 þ coshðbLÞðEIb3 cosðbLÞ þ idO sinðbLÞ
idO cosðbLÞ sinhðbLÞÞ, w1 ¼ eð
1
iÞð1þgÞbL EIð2eðð1þiÞþigÞbL EIb3 þ ð1 þ iÞeðð1þiÞþð1þ2iÞgÞbL EIb3 þ eð1þ2iÞgbL ðð1 þ iÞEIb3
2dOÞ þ eð2þigÞbL ðEIb3
ð1 þ iÞdOÞ þ eðð2
2iÞþigÞbL ðEIb3 þ ð1
iÞdOÞ þ eð2þiÞgbL ð
iEIb3 þ ð1 þ iÞdOÞ þ ieð2iþð2þiÞgÞbL ðEIb3 þ ð1 þ iÞdOÞ þ eð2iþgÞbL ðð1
iÞEIb3 þ 2dOÞÞ w2 ¼
eð
1
iÞð1þgÞbL EIðð1 þ iÞeðð1þiÞþgÞbL EIb3 þ ð1
iÞeðð1þiÞþð1þ2iÞgÞbL EIb3 þ 2eðð1þiÞþð2þiÞgÞbL EIb3 þ eð2þð1þ2iÞgÞbL ðð1
iÞEIb3
2dOÞ þ eð2þiÞgbL ðEIb3
ð1
iÞdOÞ þ eðð2þigÞbL ðiEIb3 þ ð1
iÞdOÞ
ieðð2þ2iÞþigÞbL ðEIb3 þ ð1 þ iÞdOÞ þ eð2iþð2þiÞgÞbL ðEIb3 þ ð1 þ iÞdOÞ þ eðð2þ2iÞþgÞbL ðð1 þ iÞEIb3 þ 2dOÞÞ w3 ¼ eð
1
iÞð1þgÞbL EIðð
1 þ iÞeðð1þiÞþgÞbL EIb3 þ 2ieðð1þiÞþgÞbL EIb3 þ ð1 þ iÞeðð1þiÞþð2þiÞgÞbL EIb3 þ eð2þiÞgbL ðð1 þ iÞEIb3
2dOÞ þ eð2þð1þ2iÞgÞbL ðEIb3
ð1 þ iÞdOÞ
eð1þ2iÞgbL ðEIb3
ð1
iÞdOÞ þ eðð2þ2iÞþgÞbL ðiEIb3 þ ð1 þ iÞdOÞ þ ieð2iþgÞbL ðEIb3 þ ð1 þ iÞdOÞ þ eð2þigÞbL ðð
1 þ iÞEIb3 þ 2dOÞÞ, w4 ¼
eð
1
iÞð1þgÞbL EIðð1 þ iÞeðð1þiÞþigÞbL EIb3 þ 2ieðð1þiÞþð1þ2iÞgÞbL EIb3
ð1
iÞeðð1þiÞþð2þiÞgÞbL EIb3 þ eð2iþð2þiÞgÞbL ðð
1 þ iÞEIb3
2dOÞ þ ieð1þ2iÞgbL ðEIb3
ð1
iÞdOÞ þ eð2þð1þ2iÞgÞbL ðiEIb3 þ ð1
iÞdOÞ
eðð2þ2iÞþgÞbL ðEIb3 þ ð1
iÞdOÞ þ ieð2iþgÞbL ðEIb3 þ ð1 þ iÞdOÞ þ eðð2þ2iÞþigÞbL ðð1 þ iÞEIb3 þ 2dOÞÞ, w5 ¼ eð
1
iÞð1þgÞbL EIð
2eðð1þiÞþigÞbL EIb3 þ ð1
iÞeðð1þiÞþgÞbL EIb3 þ ð1 þ iÞeðð1þiÞþð2þiÞgÞbL EIb3 þ eð1þ2iÞgbL ðð1 þ iÞEIb3
2dOÞ
eigbL ðEIb3
ð1
iÞdOÞ þ eð2þiÞgbL ð
iEIb3 þ ð1 þ iÞdOÞ
eð2þgÞbL ðEIb3 þ ð1 þ iÞdOÞ þ ieð2iþð2þiÞgÞbL ðEIb3 þ ð1 þ iÞdOÞ þ eð2iþigÞbL ðð1
iÞEIb3 þ 2dOÞÞ, w6 ¼ e
ðiþð1þiÞgÞbL EIðð
1
iÞeðiþgÞbL EIb3
ð1
iÞeðiþð1þ2iÞgÞbL EIb3 þ eðð1þ2iÞþgÞbL ðð
1
iÞEIb3
2dOÞ þ eð1þð2þiÞgÞbL ðEIb3
ð1 þ iÞdOÞ þ eð1þigÞbL ð
iEIb3
ð1
iÞdOÞ þ eðð1þ2iÞþð2þiÞgÞbL ðEIb3 þ ð1
iÞdOÞ þ eðð1þ2iÞþigÞbL ðiEIb3 þ ð1 þ iÞdOÞ þ eð1þð1þ2iÞgÞbL ðð
1 þ iÞEIb3 þ 2dOÞÞ, w7 ¼ eð
1
iÞð1þgÞbL EIðð
1 þ iÞeðð1þiÞþigÞbL EIb3
2ieðð1þiÞþgÞbL EIb3 þ ð1 þ iÞeðð1þiÞþð2þiÞgÞbL EIb3 þ eð2þiÞgbL ðð1 þ iÞEIb3
2dOÞ þ eð2þð1þ2iÞgÞbL ðEIb3
ð1 þ iÞdOÞ þ eð2þgÞbL ð
iEIb3
ð1
iÞdOÞ
eð1þ2iÞgbL ðEIb3
ð1
iÞdOÞ þ egbL ð
iEIb3 þ ð1 þ iÞdOÞ þ eð2þigÞbL ðð
1 þ iÞEIb3 þ 2dOÞÞ, w8 ¼
e
ð1þð1þiÞgÞbL EIðð1 þ iÞeð1þigÞbL EIb3
2ieð1þð1þ2iÞgÞbL EIb3
ð1
iÞeð1þð2þiÞgÞbL EIb3 þ eðiþð2þiÞgÞbL ðð
1 þ iÞEIb3
2dOÞ
eðð2þiÞþgÞbL ðEIb3
ð1
iÞdOÞ
ieðð2þiÞþð1þ2iÞgÞbL ðEIb3 þ ð1
iÞdOÞ þ eðiþgÞbL ðEIb3 þ ð1 þ iÞdOÞ
ieðiþð1þ2iÞgÞbL ðEIb3 þ ð1 þ iÞdOÞ þ eðð2þiÞþigÞbL ðð1 þ iÞEIb3 þ 2dOÞÞ.

References [1] P.M. Morse, Vibration and Sound, Macmillan, New York, 1948. [2] L. Meirovitch, Analytical Methods in Vibrations, Macmillan, New York, 1967. [3] W.T. Thomson, Theory of Vibration with Applications, Englewood Cliffs, NJ, Prentice-Hall, 1981.

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