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An analytical solution for dynamic behavior of a beam–column frame with a tip body
Applied Mathematical Modelling 37 (2013) 9086–9100
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An analytical solution for dynamic behavior of a beam–column frame with a tip body Kyung Taek Lee Department of Technology Education, Korea National University of Education, 250 Taeseongtabyeon-ro, Gangnae-myeon, Cheongwon-gun, Chungbuk 363-791, South Korea
a r t i c l e
i n f o
Article history: Received 20 March 2012 Received in revised form 21 March 2013 Accepted 8 April 2013 Available online 3 May 2013 Keywords: Beam Characteristic equation Mode shape Modal frequency Pure axial (PA) vibration Coupled axial-bending (CAB) vibration
a b s t r a c t This paper analyzes the vibration characteristics of a beam-column frame, typical examples of which are often found in optical pickup actuators of optical disc drives (ODDs) and many architectural structures. The dynamic behaviour of this beam structure is predicted by solving mathematically its vibration characteristics governed by beam configurations. For practical applications and simplicity in the analysis, the vibration analysis for the structure is limited to lateral and longitudinal directions of the beams. As a result, mode and modal frequencies are obtained from mathematical expressions. The accuracy of vibration characteristics, which is mathematically induced, is demonstrated by a finite element (FE) analysis. Finally, it is shown that mode shapes are modified by using design values with the mathematical expressions. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Many authors contributed to the studies which treated the vibration characteristics of a beam with a tip body. Laura et al. showed an analytical solution for the lateral vibration of single cross-sectional cantilever with a tip body [1]. Bhat et al. introduced the analytical solution for the lateral vibration of a uniform cantilevered beam with a tip body slender in the axial direction [2]. Rossi et al. obtained natural frequencies in a non-uniform cantilevered beam with a tip mass [3]. Auciello obtained the solution of a vibration problem in a linearly tapered cantilever beam with tip mass of rotary inertia and eccentricity [4]. Gokdag et al. presented an exact procedure to obtain natural frequencies and mode shapes of a system consisting of a beam with monosymmetric open cross section carrying a tip body and springs at one end [5]. Dokumaci presented natural frequencies and modes affected by the coupling of bending and torsional vibrations [6]. Farghaly presented the vibration of an axially loaded cantilever beam with an elastically mounted end mass [7]. Banerjee derived exact frequency equation and mode shape expressions for a coupled bending-torsional beam with cantilever end condition [8]. Nowadays, more complicated systems such as multi-beam structures are used in various mechanical equipments. Nevertheless, researches on vibration analysis of such structures are not plentiful. Lee mathematically analyzed the vibration characteristics of four parallel and uniform beams joined by a tip body at their free ends and derived modal frequencies and shapes of an optical pickup actuator [9]. That paper showed the vibration characteristics of uniform four beams with a tip body, which were related to a pure-axial vibration, a coupled bending-torsional vibration, and two coupled axial-bending vibrations. Based on the same approach of [9] which shows analytical derivation of the equations of motions and also boundary conditions in a flexible system, this paper aims to mathematically analyze and improve the vibration characteristics of a multi-beam structure joined by a tip body, which are related to a coupled axial-bending vibration of a E-mail addresses: [email protected], [email protected] 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.04.017
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
Nomenclature
Beam s l As Is
qs Es
number of beam length of beam cross-sectional area of sth beam areal moment of inertia of sth beam density of sth beam Young’s modulus of elasticity of sth beam
Tip body m mass of tip body a,c distances between the mass center of tip body and the free end of the 1st beam along the x1 and x2 axes, respectively J = mk2 mass moment of inertia of tip body about the axis through the mass center, perpendicular to x1–x2 plane k radius of gyration about the axis through the mass center, perpendicular to x1–x2 plane bs1 distance between mid-lines of 1st and sth beams Motion x1, x2, x3 Cartesian coordinates t time us(x1, t) axial deformation of sth beam in x1 direction vs(x1, t) lateral deflection of sth beam in x2 direction u0s ðx1 ; tÞ ¼ @us ðx1 ; tÞ=@x1 extensional strain of sth beam v 0s ðx1 ; tÞ ¼ @ v s ðx1 ; tÞ=@x1 lateral deflection angle of sth beam in x2 direction, derivative of u_ s ðx1 ; tÞ ¼ @us ðx1 ; tÞ=@t derivative of us(x1, t) with respect to t v_ s ðx1 ; tÞ ¼ @ v s ðx1 ; tÞ=@t derivative of vs(x1, t) with respect to t x radian natural frequency
vs(x1, t) with respect to x1
Expression normalised by time constant r and beam length l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ q1 l2 =E1 time-normalising factor x1 ¼ x1 =l non-dimensional coordinate t ¼ t=r non-dimensional time a ¼ a=l; c ¼ c=l normalised distances between the mass center of tip body and the free end of the 1st beam along the x1 and x2 axes, respectively k ¼ k=l normalised radius of gyration of tip body about the axis through the mass center of the tip body, perpendicular to x1–x2 plane bs1 ¼ bs1 =l normalised distance between middle lines of 1st and sth beams u0s ðx1 ; tÞ ¼ @us ðx1 ; tÞ=@x1 normalised extensional strain of sth beam v 0s ðx1 ; tÞ ¼ @ v s ðx1 ; tÞ=@x1 normalised lateral deflection angle of sth beam in the x2 direction, derivative of v s ðx1 ; tÞ with respect to x1 u_ s ðx1 ; tÞ ¼ @us ðx1 ; tÞ=@t normalised derivative of us ðx1 ; tÞ with respect to t v_ s ðx1 ; tÞ ¼ @ v s ðx1 ; tÞ=@t normalised derivative of v s ðx1 ; tÞ with respect to t x ¼ rx normalised radian natural frequency
beam-column frame with many beams and a tip body. For this purpose, the vibration behaviour of the beam-column frame, which is composed of n beams joined by a tip body, is mathematically investigated. Through this mathematical deployment, a combination of homogeneous and linear algebraic equations which can yield modal frequency and vibration mode is obtained. As a result, when whole beams are identical and symmetrically located at the upper and lower parts of the mass center of a tip body, two characteristic equations, associated with pure axial (PA) vibrations (all beams are equally and simultaneously compressed and expanded) and coupled axial-bending (CAB) vibrations (all beams are equally and simultaneously deflected with axial deformations), are obtained. On conditions of non-identical beams and non-symmetrical beam placement, a characteristic equation which explains coupled axial-bending (CAB) vibrations (all beams are deflected with axial deformations) is obtained. The result of such an analytical approach is compared with a simulation by a finite element method (FEM): it is demonstrated that this study expresses the vibration characteristics of this model well. Finally, it is shown that this study is effectively applied to the analysis of vibration characteristics for the beam configuration of optical pickup actuators.
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
2. Mathematical formulation 2.1. Energy expression A beam structure, shown in Fig. 1, includes a number of parallel beams and a tip body joining them at their free ends. The beams are assumed to be expandable, contractible along x1-axis and deflectable along x2-axis. Therefore, a lateral deflection vs(x1, t) along x2 direction as well as an axial deformation us(x1, t) along x1-axis in each beam should be considered to describe the vibration. In this model, the shear deformations of the beams are ignored because the beams are assumed to be sufficiently thin. Based on the energy conservation law, the sum of kinetic and potential (strain) energies is always constant. nþ1 nþ1 X @ X Ts þ Vs @t s¼1 s¼1
! ¼ 0;
ð1Þ
where s designates the beams (s ¼ 1 n) and the tip body (s = n + 1). The kinetic and potential energies of all beams are expressed as
n n Z l X X 1 Ts ¼ ðqAÞs u_ 2s ðx1 ; tÞ þ v_ 2s ðx1 ; tÞ dx1 ; 2 s¼1 s¼1 0 Z n n l X X 1 1 002 Vs ¼ ðEAÞs u02 s ðx1 ; tÞ þ ðEIÞs v s ðx1 ; tÞ dx1 : 2 2 s¼1 s¼1 0
ð2Þ ð3Þ
The kinetic energy equation of the tip body can be described by the motion of the free end of the 1st beam and the geometric conditions of the tip body.
T nþ1 ¼
2 1 2 1 1 m u_ 1 ðl; tÞ cv_ 01 ðl; tÞ þ m v_ 1 ðl; tÞ þ av_ 01 ðl; tÞ þ J v_ 02 ðl; tÞ; 2 2 2 1
ð4Þ
where a and c are the distances from the free end of the 1st beam (s = 1) to the mass center of the tip body along x1 and x2 axes, as shown in Fig. 2. J is the mass moment of inertia of the tip body about the axis through the mass center, perpendicular to x1–x2 plane and expressed: 2
J ¼ mk ;
ð5Þ
where k is the radius of gyration of the tip body about the axis through the mass center. Hence, Eq. (4) is
T nþ1 ¼
1 1 1 2 0 0 _ _ _ _ u mu_ 21 þ mv_ 21 þ mða2 þ c2 þ k Þv_ 02 mc v þ ma v v ; 1 1 1 1 1 2 2 2 x1 ¼l
where u1 = u1(x1, t) and deformable (Vn+1 = 0).
ð6Þ
v1 = v1(x1, t). The potential energy of the tip body is not considered because it is assumed not to be
Tip body
Beams
Fixed part
nth beam
vs us
sth beam
x2 x1
1st beam Fig. 1. Typical beam-column frame.
v′s
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
l0 b0
c e0
a
vs
b2
3
us
b1
2
x2
ρ s , l , d s , As , I s , Es
1
v′s
x1
Fig. 2. Simplified beam structure.
The boundary conditions are
us ð0; tÞ ¼ v s ð0; tÞ ¼ v 0s ð0; tÞ ¼ 0;
s ¼ 1 n:
ð7Þ
Considering the conditions of beam free ends joined by the tip body, the following joint conditions are obtained:
us ðl; tÞ ¼ u1 ðl; tÞ bs1 v 01 ðl; tÞ;
v
0 s ðl; tÞ
¼v
s ¼ 2 n;
ð8Þ
0 1 ðl; tÞ;
s ¼ 2 n;
ð9Þ
v s ðl; tÞ ¼ v 1 ðl; tÞ;
s ¼ 2 n;
ð10Þ
where bs1 is the distance between mid-lines of the 1st and sth beams. Employing these conditions, time derivatives of the kinetic and potential energies are derived from integrations by parts. nþ1 n Z l X @ X € s þ v_ s v€ s Þdx1 Ts ¼ ðqAÞs ðu_ s u @t s¼1 s¼1 0 h i € 1 þ mv_ 1 v€ 1 þ mða2 þ c2 þ k2 Þv_ 01 v€ 01 mcu € 1 v_ 01 mcu_ 1 v€ 01 þ mav€ 1 v_ 01 þ mav_ 1 v€ 01 þ mu_ 1 u
x1 ¼l
n n X @ X Vs ¼ @t s¼1 s¼1 "
Z
l
0
;
ð11Þ
ðEAÞs u00s u_ s þ ðEIÞs v sð4Þ v_ s dx1
n n n n X X X X _ þ ðEAÞs u0s u_ 1 ðEAÞs bs1 u0s v_ 01 þ ðEIÞs v 00s v_ 01 ðEIÞs v 000 s v1 s¼1
s¼2
s¼1
s¼1
# ;
ð12Þ
x1 ¼l
ð4Þ
where v s ¼ @ 4 v s ðx1 ; tÞ=@x41 . Insertion of Eqs. (11) and (12) into Eq. (1) yields motion Eqs. (13), (14) and continuity conditions (15)–(17) at x1 = l.
qs u€ s ðx1 ; tÞ Es u00s ðx1 ; tÞ ¼ 0; 0 < x1 < l; 0 < t; s ¼ 1 n;
ð13Þ
¼ 0; 0 < x1 < l; ðqAÞs v€ s ðx1 ; tÞ þ ðEIÞs v " # n X € 1 mcv€ 01 þ ðEAÞs u0s ¼ 0; mu
ð14Þ
ð4Þ s ðx1 ; tÞ
s¼1
" mv€ 1 þ
v
ma € 01
" 2
2
mða þ c þ k
2
v
Þ € 01
ð15Þ
#
¼ 0;
ð16Þ
x1 ¼l n n X X € 1 þ mav€ 1 mcu ðEAÞs bs1 u0s þ ðEIÞs v 00s s¼2
where us = us(x1, t) and
s ¼ 1 n;
x1 ¼l
n X ðEIÞs v 000 s s¼1
0 < t;
s¼1
# ¼ 0;
ð17Þ
x1 ¼l
vs = vs(x1, t).
2.2. Normalised expression Expressions in (18) and (19) are used to simplify the motion equations and all conditions and Eq. (20) to normalise them.
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
r¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q1 l2 =E1 ;
As ¼ As =A1 ; a ¼ a=l;
l ¼ m=q1 A1 l; a ¼
bs1 ¼ bs1 =l;
ð18Þ
qs ¼ qs =q1 ; Es ¼ Es =E1 ;
Is ¼ Is =I1 ;
c ¼ c=l;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I1 =A1 l ;
k ¼ k=l;
s ¼ 2 n;
ð19Þ
t ¼ t=r;
x1 ¼ x1 =l;
ð20Þ
where x1 and t are the normalised coordinate and time, respectively. Normalised forms for motion Eqs. (13), (14), boundary and joint conditions (7)–(10), and continuity conditions (15)–(17) are as follows.
u00s ðx1 ; tÞ
q € us ðx1 ; tÞ ¼ 0; 0 < x1 < 1; 0 < t; s ¼ 1 n; E
v sð4Þ ðx1 ; tÞ þ
s
1 qA v€ ðx ; tÞ ¼ 0; 0 < x1 < 1; 0 < t; s ¼ 1 n; a2 EI s s 1
us ð0; tÞ ¼ v s ð0; tÞ ¼ v 0s ð0; tÞ ¼ 0;
v ¼v v s ð1; tÞ ¼ v 1 ð1; tÞ; 0 1 ð1; tÞ;
ð22Þ
s ¼ 1 n;
u1 ð1; tÞ us ð1; tÞ bs1 v 01 ð1; tÞ ¼ 0; 0 s ð1; tÞ
ð21Þ
!
ð23Þ
s ¼ 2 n;
ð24Þ
s ¼ 2 n;
ð25Þ
s ¼ 2 n;
ð26Þ
n X lu€1 ð1; tÞ lcv€ 01 ð1; tÞ þ Es As u0s ð1; tÞ ¼ 0;
ð27Þ
s¼1
lv€ 1 ð1; tÞ þ lav€ 01 ð1; tÞ a2
n X Es Is v 000 s ð1; tÞ ¼ 0;
ð28Þ
s¼1
lða2 þ c2 þ k2 Þv€ 01 ð1; tÞ lcu€1 ð1; tÞ þ lav€ 1 ð1; tÞ
n n X X Es As bs1 u0s ð1; tÞ þ a2 Es Is v 00s ð1; tÞ ¼ 0; s¼2
ð29Þ
s¼1
2.3. Characteristic equation The solutions of normalised motion Eqs. (21) and (22) are assumed, to be
us ðx1 ; tÞ ¼ us ðx1 Þ cos xt;
s ¼ 1 n;
ð30Þ
v s ðx1 ; tÞ ¼ v s ðx1 Þ cos xt;
s ¼ 1 n;
ð31Þ
where x is a normalised radian natural frequency. Consequently, time functions are eliminated from Eqs. (21)–(29), and the following eigenvalue problems are obtained:
u00s ðx1 Þ þ X2s us ðx1 Þ ¼ 0;
0 < x1 < 1;
s ¼ 1 n;
v sð4Þ ðx1 Þ K4s v s ðx1 Þ ¼ 0; 0 < x1 < 1; us ð0Þ ¼ v s ð0Þ ¼ v 0s ð0Þ ¼ 0; s ¼ 1 n; u1 ð1Þ us ð1Þ bs1 v 01 ð1Þ ¼ 0;
v ¼v v s ð1Þ ¼ v 1 ð1Þ; 0 s ð1Þ
0 1 ð1Þ;
ð32Þ
s ¼ 1 n;
ð33Þ ð34Þ
s ¼ 2 n;
ð35Þ
s ¼ 2 n;
ð36Þ
s ¼ 2 n;
ð37Þ
lx2 u1 ð1Þ lc x2 v 01 ð1Þ
n X
Es As u0s ð1Þ ¼ 0;
ð38Þ
s¼1
lx2 v 1 ð1Þ þ la x2 v 01 ð1Þ þ a2
n X Es Is v 000 s ð1Þ ¼ 0;
ð39Þ
s¼1
lða2 þ c2 þ k2 Þx2 v 01 ð1Þ lc x2 u1 ð1Þ þ la x2 v 1 ð1Þ þ where X2s ¼ x2
q
E s
; K4s ¼
s¼2
x 2 qA
a
EI
n X
s
.
Es As bs1 u0s ð1Þ a2
n X Es Is v 00s ð1Þ ¼ 0; s¼1
ð40Þ
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
The solutions satisfying normalised boundary conditions (34) are
us ðx1 Þ ¼ Bs sin Xs x1 ;
s ¼ 1 n;
ð41Þ
v s ðx1 Þ ¼ C s ðcos Ks x1 cosh Ks x1 Þ þ Ds ðsin Ks x1 sinh Ks x1 Þ;
s ¼ 1 n;
ð42Þ
where Bs, Cs, and Ds are modal coefficients. Normalised boundary conditions (36) and (37) give us the following relations.
C s ¼ C 1 Ns þ D1 Ws ;
s ¼ 2 n;
ð43Þ
Ds ¼ C 1 Cs þ D1 Ps ;
s ¼ 2 n;
ð44Þ
where
fKs ðcos K1 cosh K1 Þðcos Ks cosh Ks Þ þ K1 ðsin K1 þ sinh K1 Þðsin Ks sinh Ks Þg ; 2Ks ð1 cos Ks cosh Ks Þ fK ðsin K1 sinh K1 Þðcos Ks cosh Ks Þ K1 ðcos K1 cosh K1 Þðsin Ks sinh Ks Þg Ws ¼ s ; 2Ks ð1 cos Ks cosh Ks Þ fðcos K1 cosh K1 Þ Ns ðcos Ks cosh Ks Þg fðsin K1 sinh K1 Þ Ws ðcos Ks cosh Ks Þg Cs ¼ ; Ps ¼ : ðsin Ks sinh Ks Þ ðsin Ks sinh Ks Þ
Ns ¼
The insertion of solutions (41) and (42) into condition (35), (38)–(40) yields the following matrix form:
½aij fg j g ¼ f0g;
i; j ¼ 1 ðn þ 2Þ:
ð45Þ
The form of matrix [aij] is
ð46Þ
The components of [aij] are
a1;1 ¼ sin X1 ;
a1;2 ¼ sin X2 ;
a1;nþ1 ¼ b1 K1 ðS þ SHÞ1 ;
a1;nþ2 ¼ b1 K1 ðC CHÞ1 ;
ð47-1; 2; 3; 4Þ
a2;1 ¼ sin X1 ;
a2;3 ¼ sin X3 ;
a2;nþ1 ¼ b2 K1 ðS þ SHÞ1 ;
a2;nþ2 ¼ b2 K1 ðC CHÞ1 ;
ð47-5; 6; 7; 8Þ
an2;1 ¼ sin X1 ;
an2;n1 ¼ sin Xn1 ;
an2;nþ1 ¼ bn2 K1 ðS þ SHÞ1 ;
an2;nþ2
¼ bn2 K1 ðC CHÞ1 ; an1;1 ¼ sin X1 ; an1;n ¼ sin Xn ; an;1 ¼ lx2 sin X1 X1 cos X1 ;
ð47-9; 10; 11; 12Þ an1;nþ1 ¼ bn1 K1 ðS þ SHÞ1 ; an;2 ¼ E2 A2 X2 cos X2 ;
an1;nþ2 ¼ bn1 K1 ðC CHÞ1 ;
an;3 ¼ E3 A3 X3 cos X3 ;
an;n1
¼ En1 An1 Xn1 cos Xn1 ; an;n ¼ En An Xn cos Xn ;
ð47-17; 18; 19; 20Þ
an;nþ1 ¼ lc x2 K1 ðS þ SHÞ1 ;
anþ1;nþ1 ¼ lx2 ðC CHÞ1 la x2 K1 ðS þ SHÞ1 þ a2
ð47-13; 14; 15; 16Þ
an;nþ2 ¼ lc x2 K1 ðC CHÞ1 ;
n X Es Is K3s ½Ns ðS SHÞs Cs ðC þ CHÞs ; s¼1
ð47-21; 22; 23Þ ð47-24Þ
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
anþ1;nþ2 ¼ lx2 ðS SHÞ1 þ la x2 K1 ðC CHÞ1 þ a2
n X Es Is K3s ½Ws ðS SHÞs Ps ðC þ CHÞs ;
ð47-25Þ
s¼1
anþ2;1 ¼ lc x2 sin X1 ;
anþ2;2 ¼ E2 A2 b1 X2 cos X2 ;
anþ2;n1 ¼ En1 An1 bn2 Xn1 cos Xn1 ;
anþ2;3 ¼ E3 A3 b2 X3 cos X3 ;
ð47-26; 27; 28Þ
anþ2;n ¼ En An bn1 Xn cos Xn ;
anþ2;nþ1 ¼ lða2 þ c2 þ k2 Þx2 K1 ðS þ SHÞ1 la x2 ðC CHÞ1 a2
n X
ð47-29; 30Þ
K2s Es Is ½Ns ðC þ CHÞs þ Cs ðS þ SHÞs ;
ð47-31Þ
s¼1
anþ2;nþ2 ¼ lða2 þ c2 þ k2 Þx2 K1 ðC CHÞ1 la x2 ðS SHÞ1 a2
n X K2s Es Is ½Ws ðC þ CHÞs þ Ps ðS þ SHÞs ;
ð47-32Þ
s¼1
where (C ± CH)s = cosKs ± cosh Ks and (S ± SH)s = sin Ks ± sinh Ks. The modal vector is
fg j g ¼ fB1 ; . . . Bn ; C 1 ; D1 gT :
ð48Þ
A specific case arises when whole beams are identical and symmetrically located at the upper and lower parts of the mass center of tip body. Because this case makes Xs ¼ X1 ¼ x and Ks = K1, it is induced that Ns = 1, Ws = 0, Cs = 0, and Ps = 1. Therefore
Cs ¼ C1; Ds ¼ D1 ;
s ¼ 2 n; s ¼ 2 n:
ð49Þ ð50Þ
Also
v s ðx1 Þ ¼ v 1 ðx1 Þ;
s ¼ 2 n:
ð52Þ
Such a case makes the matrix simpler:
2
sin x
sin x
0
6 6 sin x 0 sin x 6 6 6 6 6 6 6 6 6 sin x 0 0 6 6 6 sin x 0 0 6 6 lx sin x 6 6 cos x cos x 6 cos x 6 6 6 6 6 6 0 0 0 6 6 6 6 6 6 6 6 6 6 lcx sin x b1 cos x b2 cos x 6 4
0
0
b1 K1 ðS þ SHÞ1
0
0
b2 K1 ðS þ SHÞ1
sin x
0
bn2 K1 ðS þ SHÞ1
0
sin x
bn1 K1 ðS þ SHÞ1
cos x
cos x
lcxK1 ðS þ SHÞ1
0
lK1 ðC CHÞ1 laK21 ðS þ SHÞ1
0
þnðS SHÞ1
lða2 þ c2 þ k2 Þ xK1 ðS þ SHÞ1 : : bn2 cos x bn1 cos x laxðC CHÞ1 naðC þ CHÞ1
b1 K1 ðC CHÞ1
3
7 b2 K1 ðC CHÞ1 7 7 7 7 7 7 7 7 7 bn2 K1 ðC CHÞ1 7 7 7 bn1 K1 ðC CHÞ1 7 7 7 7 lcxK1 ðC CHÞ1 7 7 7 7 7 lK1 ðS SHÞ1 7 7 2 þlaK1 ðC CHÞ1 7 7 7 7 nðC þ CHÞ1 7 7 lða2 þ c2 þ k2 Þ 7 7 xK1 ðC CHÞ1 7 7 7 laxðS SHÞ1 7 5 naðS þ SHÞ1 ð53Þ
Since Eq. (45) is a combination of homogeneous and linear algebraic equations from the components of modal vector (48), a non-trivial solution exists only if the determinant of the coefficients vanishes. That is,
det½aij ¼ 0:
ð54Þ
The determinant of the (n + 2) (n + 2) matrix can be calculated by a mathematical program [10]. Matrix (46) is too complicated to obtain its determinant yielding characteristic equations including parametric symbols. In this case, substituting real values into each component of the matrix gives us characteristic equations which are expressed as the function of x. Especially, calculating the determinant of matrix (53) composed by the condition of identical beams and symmetrical beam placement with respect to the mass center of the tip body yields the following characteristic equations:
n cos x lx sin x ¼ 0;
ð55Þ
Table 1 Analytic solutions and FEM results with respect to variation of n and d (CAB vibration). d
n 1
2
3
4
5
FEM
Eq. (56)
FEM
Eq. (56)
FEM
Eq. (56)
FEM
Eq. (56)
FEM
x1
0.2 0.4 0.6
3.0 11.9 25.7
3.0 11.9 25.7
8.7 32.4 65.6
9.3 31.4 65.8
10.6 38.0 74.2
10.6 38.0 74.0
12.0 42.1 80.0
12.1 42.1 79.8
13.3 45.3 84.2
13.4 45.3 84.1
x2
0.2 0.4 0.6
91.1 196.9 304.9
91.0 196.7 304.7
152.5 313.8 487.2
152.3 314.1 487.0
153.3 318.8 498.4
153.8 319.3 497.4
154.2 323.3 507.9
154.4 323.8 506.5
154.9 327.4 515.5
154.8 327.4 515.5
x3
0.2 0.4 0.6
248.9 601.8 944.1
248.7 601.4 943.4
417.2 842.0 1277.5
416.5 842.8 1276.9
417.9 846.5 1288.5
419.1 847.9 1285.9
418.7 851.3 1299.7
419.4 852.4 1295.8
419.5 855.8 1309.3
419.2 856.1 1309.4
x4
0.2 0.4 0.6
476.7 1166.9 1893.5
476.3 1165.9 1891.6
815.9 1635.5 2458.5
814.6 1637.3 2458.0
816.3 1637.9 2463.5
818.6 1640.2 2458.9
816.9 1641.5 2472.5
818.3 1643.6 2465.0
817.5 1645.1 2480.7
816.9 1647.3 2481.2
x5
0.2 0.4 0.6
850.2 1898.1 3110.1
849.7 1896.3 3106.3
1347.2 2691.2 4024.2
1344.8 2696.0 4016.3
1347.1 2689.5 4017.3
1350.8 2693.3 4005.8
1347.3 2690.9 4020.6
1349.6 2694.0 4004.5
1347.6 2692.5 4024.2
1346.7 2694.1 4019.7
x6
0.2 0.4 0.6
1371.1 2887.4 4619.2
1370.1 2884.4 4612.2
2010.8 4006.6 5964.6
2007.2 4004.3 5953.5
2009.9 3998.1 5937.0
2015.3 4014.6 5918.9
2009.6 3995.9 5930.8
2012.9 3998.7 5904.3
2009.4 3994.1 5926.6
2007.9 4016.8 5923.2
bs
c=0
b1 = 2c
b1 ¼ c b2 ¼ 2c
b1 ¼ 0:7c b2 ¼ 1:3c b3 ¼ 2c
b1 b2 b3 b4
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
Eq. (56)
¼ 0:6c ¼c ¼ 1:4c ¼ 2c
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
"
lK21 lk2 x sin x
( ) # n X ðbs1 cÞ2 þ c2 cos x ð1 cos K1 cosh K1 Þ s¼2
"
( ) # n X 2 2 nK1 lða þ k Þx sin x ðbs1 cÞ þ c cos x ðsin K1 cosh K1 þ cos K1 sinh K1 Þ 2
2
s¼2
þ n2 a sin xð1 þ cos K1 cosh K1 Þ nlaK1 sin xðsin K1 cosh K1 cos K1 sinh K1 Þ 2nlax sin x sin K1 sinh K1 ¼ 0;
ð56Þ
where c ¼ bn1 =2. Special cases (n = 1, 2) for Eq. (55) were reported in previous studies [11,12]. In light of these facts, one can say that Eq. (55) is the characteristic equation for the PA vibration of the multi-beam structure having n beams and a tip body on condition of identical beam and symmetrical beam placement. Using FEM by computer program [13] in the following section, it is investigated what kind of vibration motion is expressed by Eq. (56). 2.4. Vibration mode examination and accuracy verification In order to show what kind of vibration mode is expressed by characteristic Eq. (56) and verify the accuracy of that equation, the fundamental radian natural frequencies from Eq. (56) are compared with those from an FE model simulated in the CAB vibration. The following values for mechanical and dimensional parameters are used to obtain the radian natural frequencies from Eq. (56) and a simplified model generated by FE analysis, respectively. Fig. 2 shows the simulated model in which all beams have identical mechanical properties. Beam: l = 100 mm, q = 8250 kg/m3, E = 15 GPa, Tip body: q0 = 1400 kg/m3, b0 = 5 mm, e0 = 5 mm, l0 = 10 mm.
CAB mode ( x1 − x2 plane)
1st
2nd
3rd
4th
5th
6th
Fig. 3. Mode shapes.
Table 2 Analytic solutions and FEM results with respect to variation of n and d (CAB vibration). d
n 1
2
3
4
5 FEM
0.2 0.4 0.6
3.0 11.9 25.7
3.0 11.9 25.7
9.0 33.1 66.7
8.9 31.6 66.7
10.7 38.5 74.9
10.9 38.6 74.9
12.2 42.5 80.5
13.5 41.2 80.6
13.4 45.6 84.6
14.1 45.4 84.6
x2
0.2 0.4 0.6
81.1 191.2 301.0
81.0 191.0 300.8
152.6 314.4 488.7
152.3 313.7 489.0
153.4 319.4 499.7
153.1 318.6 500.0
154.3 324.0 509.5
153.9 323.1 509.5
155.0 328.0 516.8
154.7 327.2 517.1
x3
0.2 0.4 0.6
205.4 550.7 908.3
205.2 550.3 907.6
417.3 842.7 1279.4
416.4 840.6 1279.9
418.0 847.3 1290.5
417.1 845.2 1291.0
418.9 852.5 1303.0
417.9 850.2 1302.8
419.6 856.8 1312.1
418.7 854.7 1312.6
x4
0.2 0.4 0.6
441.3 1032.1 1756.2
441.0 1031.2 1754.5
816.0 1636.2 2460.6
814.2 1632.1 2460.9
816.5 1638.8 2466.2
814.6 1634.7 2466.8
817.2 1643.5 2478.4
815.2 1638.9 2477.3
817.7 1646.8 2488.7
815.8 1642.7 2486.1
x5
0.2 0.4 0.6
829.6 1754.4 2850.3
829.0 1752.8 2846.7
1347.4 2691.8 4026.6
1344.2 2684.7 4025.9
1347.3 2690.7 4020.9
1344.1 2683.7 4020.7
1347.7 2694.2 4030.6
1344.5 2686.2 4027.1
1347.9 2694.9 4032.4
1344.9 2688.3 4032.1
x6
0.2 0.4 0.6
1357.0 2775.7 4323.6
1356.0 2772.8 4316.9
2013.2 4025.0 6000.4
2006.2 3996.2 5964.2
2012.5 3999.4 5942.4
2005.3 3988.9 5940.3
2010.5 4000.7 5946.4
2005.3 3988.6 5939.5
2009.9 3996.7 5939.7
2005.3 3987.8 5937.5
Es ; bs
FEM
Mat. (46)
FEM
Mat. (46)
FEM
Mat. (46)
FEM
c=0
q2 ¼ 1:1q1 ;
E2 ¼ 1:1E1 ; b1 ¼ 2c
q2 ¼ q1 ;
E2 ¼ E1 ; q3 ¼ 1:1q1 ; E3 ¼ 1:1E1 ; b1 ¼ 1:1c b2 ¼ 2c
q2 ¼ q1 ;
E2 ¼ E1 ; q3 ¼ q1 ; E3 ¼ E1 ; q4 ¼ 1:1q1 ; E4 ¼ 1:1E1 ; b1 ¼ 0:7c b2 ¼ 1:5c b3 ¼ 2c
q2 ¼ q1 ; E2 ¼ E1 ;
q3 ¼ q1 ; E3 ¼ E1 ;
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
Mat. (46)
x1
qs ;
Mat. (46)
q4 ¼ q1 ; E4 ¼ E1 ;
q5 ¼ 1:1q1 ;
E5 b1 b2 b3 b4
¼ 1:1E1 ; ¼ 0:6c ¼c ¼ 1:5c ¼ 2c
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
c a
Fig. 4. Optical pickup actuator with six suspensions.
1st mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
a = −5 ~ 5
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2nd mode 1 0.8
0.2
a=5 a=4 a=3 a=2 a =1
0
a=0
0.6 0.4
-0.2
a = −1
-0.4
a = −5
a = −2
-0.6 -0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a = −3 a = −4
Fig. 5. Vibration modes with respect to a (at b1 = 1.1c, b2 = 2c) (a) 1st mode (b) 2nd mode.
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
In this model, the normalised radius of gyration is
k¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 b0 þ e20 =12=l:
ð57Þ
In addition, the normalised parameters and are established as follows.
a ¼ a=l ¼ e0 =2l;
c ¼ bn1 =2 ¼ ðb0 dÞ=2l:
ð58; 59Þ
According to the number of beams n, beam diameter d and the beam distances bs1 , the radian natural frequencies which are obtained from Eq. (56) and simulated by an FEM program are classified in Table 1. A radian natural frequency x from Eq. (56) is obtained as a normalised radian natural frequency x of Eq. (56) is divided by rðx ¼ x=rÞ. This table shows that the six fundamental radian natural frequencies obtained by the characteristic equations are in excellent agreement with those obtained by the FEM. Therefore, it is confirmed that Eq. (56) well expresses the vibration characteristics for this mechanical model and also recognised that the equation is the characteristic equation for CAB vibration. In order to help reader’s understanding, the mode shapes of CAB vibration simulated from FEM are provided in Fig. 3. In the same way, vibration problems, in which all beams with unsymmetrical beam placement do not have identical beam properties, are solved using the determinant of matrix (46). Contrary to the case of Eq. (56), qs and Es of each beam,
1st mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
b1 = 0.4c ~ 1.6c
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2nd mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
b1 = 0.4c ~ 1.6c
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 6. Vibration modes with respect to b1 (at a = 0, b2 = 2c) (a) 1st mode (b) 2nd mode.
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
which are expressed in Table 2, are slightly different. The radian natural frequencies, obtained from the determinant and from CAB vibration simulated by the FEM program, are classified in the table. The fundamental radian natural frequencies obtained from the determinant are accord with those obtained by the FEM. Therefore it is also confirmed that Eq. (45) defines CAB vibration of a multi-beam structure having non-identical beams or unsymmetrical beam placement condition. It should be noted that PA vibration motion can not be obtained from Eq. (45).
3. Mode shape control of optical pickup actuator 3.1. Structure of optical pickup actuator An optical pickup actuator is one of key components in ODDs. The actuator includes a moving part and suspensions to obtain focusing and tracking motions. In general, the optical pickup actuator for compact disc (CD) drives has four beams as electrical connectors as well as mechanical suspensions for two-axis motions [14]. However, optical disc systems such as digital versatile disc (DVD) or blue-ray disc (BD) drives require optical pickup actuators having six suspensions [15,16]. In particular, an actuator employing a liquid–crystal panel for securing multi-layer compatibility requires much more suspensions [17]. In a near field recording (NFR) system, Lee et al. used eight beams for four-axis control [18].
1st mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
b2 = 1.4c ~ 2c
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2nd mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
b2 = 1.4c ~ 2c
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 7. Vibration modes with respect to b2 (at a = 0, b1 = 1.1c) (a) 1st mode (b) 2nd mode.
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
9099
Nevertheless, studies related to the analysis of their dynamic behaviours are very rare. Lee analytically solved vibrations of a structure composed by identical four beams and also described vibration of optical pickup actuators to show the modification of mode shape by parametric analysis [9]. In the same manner, this paper treats the vibration motion of an actuator having multiple suspensions. 3.2. Mode shape control The dynamic behaviour of a simplified actuator model shown in Fig. 4, which has six suspensions in both sides of the moving part, is analyzed. Each mechanical property is listed as follows. Beams: d1 = d2 = 0.14 mm, d3 = 0.16 mm, l = 14 mm, q1 = q2 = q3 = 8250 kg/m3, E1 = E2 = E3 = 127 GPa, Tip body: m = 1 103 kg, k = 0.45 m2, Distance from 1st suspension to mass center: c = 2.4 mm. a, b1 and b2 are design parameters to improve vibration modes of this actuator. They are assumed to be varied in the following ranges, respectively.
5 6 a 6 5;
0:4c 6 b1 6 1:6c;
1:4c 6 b2 6 2c:
ð60; 61; 62Þ
Suspensions are located at both sides of the moving part and symmetrical in vertical direction. The six suspensions can be regarded as three ones on the vertical centreline as shown in Fig. 2. As a matter of course, the cross-sectional areas and areal moments of inertia of the suspension should be redefined as follows. 2
As ¼ 2 pds =4;
4
Is ¼ 2 pds =64:
ð63; 64Þ
For knowing the motion of the moving part, the mode shape of the 1st suspension is investigated. The mode shape is normalised by the difference of maximum and minimum values:
v 1 ðxÞ=fmaxðv 1 ðxÞÞ minðv 1 ðxÞÞg
ð63Þ
In this paper, two fundamental modes of the 1st suspension in low frequency range are observed on condition that several design parameters are varied. When the value of the parameter a is changed, the 2nd mode is remarkably changed as shown in Fig. 5. When a = 0, deflection of the end of suspension is suppressed but deflection angle is maximised. If jaj is increased, deflection angle is decreased while deflection is increased. The variation of parameter b1 scarcely affects the 2nd mode shape and the variation of parameter b2 slightly affects the 2nd mode shape as shown in Figs. 6 and 7, respectively. On the other hand, the 1st mode shape is never changed by these parameters. 4. Conclusions This paper mathematically analyzes the vibration characteristics of a beam-column frame having n beams joined by a tip body. As a result, modal frequencies and mode shapes for vibration motion of the beam-column frame in longitudinal and lateral directions are obtained. The accuracy of the mathematical deployment in this paper is demonstrated by comparing the natural frequencies obtained from FEM results. For proving practical use of this study, the result of this paper was used to a case study for the vibration analysis of an optical pickup actuator to improve its dynamic behaviour. Modal frequencies and shapes were calculated by applying the design parameters to the mathematical expression. Furthermore, it was shown that we could change the mode shapes by parametric analysis. References [1] P.A.A. Laura, J.L. Pombo, E.A. Susemihl, A note on the vibrations of a clamped-free beam with a mass at the free end, J. Sound Vib. 37 (2) (1974) 161– 168. [2] B. Bhat, H. Wagner, Natural frequencies of a uniform cantilever with a tip mass slender in the axial direction, J. Sound Vib. 45 (2) (1976) 304–307. [3] R.E. Rossi, P.A.A. Laura, R.H. Gutierrez, A note on traverse vibrations of a Timoshenko beam of non-uniform thickness clamped at one end and carrying a concentrated mass of at the other, J. Sound Vib. 143 (3) (1990) 491–502. [4] N.M. Auciello, Transverse vibrations of a linearly tapered cantilever beam with tip mass of rotary inertia and eccentricity, J. Sound Vib. 194 (1) (1996) 25–34. [5] H. Gokdag, O. Kopmaz, Coupled bending and torsional vibration of a beamwidth in-span and tip attachments, J. Sound Vib. 287 (2005) 591–610. [6] E. Dokumaci, An exact solution for coupled bending and torsion vibrations of uniform beams having single cross-sectional symmetry, J. Sound Vib. 119 (3) (1987) 443–449. [7] S.H. Farghaly, Bending vibration of an axially loaded cantilever beam with an elastically mounted end mass of finite length, J. Sound Vib. 156 (1992) 373–380. [8] J.R. Banerjee, Explicit frequency equation and mode shapes of a cantilever beam coupled in bending and torsion, J. Sound Vib. 224 (2) (1999) 267–281. [9] K.T. Lee, Analytical solutions for vibration characteristics of a multi-beam structure, Int. J. Mech. Sci. 52 (2010) 952–969. [10] Mathematica 6.5, Wolframe Research Inc., 2007. [11] R.R. Craig, A.J. Kurdila, Fundamentals of Structural Dynamics, John Wiley & Sons Inc., 2006. [12] G.L. Anderson, Natural frequencies of two cantilevers joined by a rigid connector at their free ends, J. Sound Vib. 57 (1978) 403–412. [13] Ansys 11, ANSYS Inc., 2006.
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[14] M. Nagasato, I. Hoshino, Development of a two-axis actuator with small tilt angles for one-piece optical heads, Jpn. J. Appl. Phys. 35 (1996) 392–397. [15] B. Zhang, J. Ma, L. Pan, X. Cheng, Y. Tang, High performance three-axis actuator in super-multi optical pickup actuator with low crosstalk force, IEEE Trans. Consum. Electron. 54 (4) (2008) 1743–1749. [16] S.J. Kim, T.Y. Heor, T.K. Kim, Y.M. Ahn, C.S. Chung, S.H. Park, High Response twin-objective actuator with radial tilt function for blu-ray disc recorder, Jpn. J. Appl. Phys. 44 (2005) 3393–3396. [17] T.K. Kim, Y.M. Ahn, S.J. Kim, T.Y. Heor, C.S. Chung, I.S. Park, Blu-ray disc pickup head for dual layer, Jpn. J. Appl. Phys. 44 (2005) 3397–3401. [18] K.T. Lee, J.E. Kim, J.K. Seo, S.N. Hong, I.H. Choi, E.S. Ko, B.H. Min, Dynamic tilt control of SIL with 4-axis actuator in NFR system, Proc. Opt. Data Storage (2007) WA7.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
An analytical solution for dynamic behavior of a beam–column frame with a tip body Kyung Taek Lee Department of Technology Education, Korea National University of Education, 250 Taeseongtabyeon-ro, Gangnae-myeon, Cheongwon-gun, Chungbuk 363-791, South Korea
a r t i c l e
i n f o
Article history: Received 20 March 2012 Received in revised form 21 March 2013 Accepted 8 April 2013 Available online 3 May 2013 Keywords: Beam Characteristic equation Mode shape Modal frequency Pure axial (PA) vibration Coupled axial-bending (CAB) vibration
a b s t r a c t This paper analyzes the vibration characteristics of a beam-column frame, typical examples of which are often found in optical pickup actuators of optical disc drives (ODDs) and many architectural structures. The dynamic behaviour of this beam structure is predicted by solving mathematically its vibration characteristics governed by beam configurations. For practical applications and simplicity in the analysis, the vibration analysis for the structure is limited to lateral and longitudinal directions of the beams. As a result, mode and modal frequencies are obtained from mathematical expressions. The accuracy of vibration characteristics, which is mathematically induced, is demonstrated by a finite element (FE) analysis. Finally, it is shown that mode shapes are modified by using design values with the mathematical expressions. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Many authors contributed to the studies which treated the vibration characteristics of a beam with a tip body. Laura et al. showed an analytical solution for the lateral vibration of single cross-sectional cantilever with a tip body [1]. Bhat et al. introduced the analytical solution for the lateral vibration of a uniform cantilevered beam with a tip body slender in the axial direction [2]. Rossi et al. obtained natural frequencies in a non-uniform cantilevered beam with a tip mass [3]. Auciello obtained the solution of a vibration problem in a linearly tapered cantilever beam with tip mass of rotary inertia and eccentricity [4]. Gokdag et al. presented an exact procedure to obtain natural frequencies and mode shapes of a system consisting of a beam with monosymmetric open cross section carrying a tip body and springs at one end [5]. Dokumaci presented natural frequencies and modes affected by the coupling of bending and torsional vibrations [6]. Farghaly presented the vibration of an axially loaded cantilever beam with an elastically mounted end mass [7]. Banerjee derived exact frequency equation and mode shape expressions for a coupled bending-torsional beam with cantilever end condition [8]. Nowadays, more complicated systems such as multi-beam structures are used in various mechanical equipments. Nevertheless, researches on vibration analysis of such structures are not plentiful. Lee mathematically analyzed the vibration characteristics of four parallel and uniform beams joined by a tip body at their free ends and derived modal frequencies and shapes of an optical pickup actuator [9]. That paper showed the vibration characteristics of uniform four beams with a tip body, which were related to a pure-axial vibration, a coupled bending-torsional vibration, and two coupled axial-bending vibrations. Based on the same approach of [9] which shows analytical derivation of the equations of motions and also boundary conditions in a flexible system, this paper aims to mathematically analyze and improve the vibration characteristics of a multi-beam structure joined by a tip body, which are related to a coupled axial-bending vibration of a E-mail addresses: [email protected], [email protected] 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.04.017
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
Nomenclature
Beam s l As Is
qs Es
number of beam length of beam cross-sectional area of sth beam areal moment of inertia of sth beam density of sth beam Young’s modulus of elasticity of sth beam
Tip body m mass of tip body a,c distances between the mass center of tip body and the free end of the 1st beam along the x1 and x2 axes, respectively J = mk2 mass moment of inertia of tip body about the axis through the mass center, perpendicular to x1–x2 plane k radius of gyration about the axis through the mass center, perpendicular to x1–x2 plane bs1 distance between mid-lines of 1st and sth beams Motion x1, x2, x3 Cartesian coordinates t time us(x1, t) axial deformation of sth beam in x1 direction vs(x1, t) lateral deflection of sth beam in x2 direction u0s ðx1 ; tÞ ¼ @us ðx1 ; tÞ=@x1 extensional strain of sth beam v 0s ðx1 ; tÞ ¼ @ v s ðx1 ; tÞ=@x1 lateral deflection angle of sth beam in x2 direction, derivative of u_ s ðx1 ; tÞ ¼ @us ðx1 ; tÞ=@t derivative of us(x1, t) with respect to t v_ s ðx1 ; tÞ ¼ @ v s ðx1 ; tÞ=@t derivative of vs(x1, t) with respect to t x radian natural frequency
vs(x1, t) with respect to x1
Expression normalised by time constant r and beam length l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ q1 l2 =E1 time-normalising factor x1 ¼ x1 =l non-dimensional coordinate t ¼ t=r non-dimensional time a ¼ a=l; c ¼ c=l normalised distances between the mass center of tip body and the free end of the 1st beam along the x1 and x2 axes, respectively k ¼ k=l normalised radius of gyration of tip body about the axis through the mass center of the tip body, perpendicular to x1–x2 plane bs1 ¼ bs1 =l normalised distance between middle lines of 1st and sth beams u0s ðx1 ; tÞ ¼ @us ðx1 ; tÞ=@x1 normalised extensional strain of sth beam v 0s ðx1 ; tÞ ¼ @ v s ðx1 ; tÞ=@x1 normalised lateral deflection angle of sth beam in the x2 direction, derivative of v s ðx1 ; tÞ with respect to x1 u_ s ðx1 ; tÞ ¼ @us ðx1 ; tÞ=@t normalised derivative of us ðx1 ; tÞ with respect to t v_ s ðx1 ; tÞ ¼ @ v s ðx1 ; tÞ=@t normalised derivative of v s ðx1 ; tÞ with respect to t x ¼ rx normalised radian natural frequency
beam-column frame with many beams and a tip body. For this purpose, the vibration behaviour of the beam-column frame, which is composed of n beams joined by a tip body, is mathematically investigated. Through this mathematical deployment, a combination of homogeneous and linear algebraic equations which can yield modal frequency and vibration mode is obtained. As a result, when whole beams are identical and symmetrically located at the upper and lower parts of the mass center of a tip body, two characteristic equations, associated with pure axial (PA) vibrations (all beams are equally and simultaneously compressed and expanded) and coupled axial-bending (CAB) vibrations (all beams are equally and simultaneously deflected with axial deformations), are obtained. On conditions of non-identical beams and non-symmetrical beam placement, a characteristic equation which explains coupled axial-bending (CAB) vibrations (all beams are deflected with axial deformations) is obtained. The result of such an analytical approach is compared with a simulation by a finite element method (FEM): it is demonstrated that this study expresses the vibration characteristics of this model well. Finally, it is shown that this study is effectively applied to the analysis of vibration characteristics for the beam configuration of optical pickup actuators.
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
2. Mathematical formulation 2.1. Energy expression A beam structure, shown in Fig. 1, includes a number of parallel beams and a tip body joining them at their free ends. The beams are assumed to be expandable, contractible along x1-axis and deflectable along x2-axis. Therefore, a lateral deflection vs(x1, t) along x2 direction as well as an axial deformation us(x1, t) along x1-axis in each beam should be considered to describe the vibration. In this model, the shear deformations of the beams are ignored because the beams are assumed to be sufficiently thin. Based on the energy conservation law, the sum of kinetic and potential (strain) energies is always constant. nþ1 nþ1 X @ X Ts þ Vs @t s¼1 s¼1
! ¼ 0;
ð1Þ
where s designates the beams (s ¼ 1 n) and the tip body (s = n + 1). The kinetic and potential energies of all beams are expressed as
n n Z l X X 1 Ts ¼ ðqAÞs u_ 2s ðx1 ; tÞ þ v_ 2s ðx1 ; tÞ dx1 ; 2 s¼1 s¼1 0 Z n n l X X 1 1 002 Vs ¼ ðEAÞs u02 s ðx1 ; tÞ þ ðEIÞs v s ðx1 ; tÞ dx1 : 2 2 s¼1 s¼1 0
ð2Þ ð3Þ
The kinetic energy equation of the tip body can be described by the motion of the free end of the 1st beam and the geometric conditions of the tip body.
T nþ1 ¼
2 1 2 1 1 m u_ 1 ðl; tÞ cv_ 01 ðl; tÞ þ m v_ 1 ðl; tÞ þ av_ 01 ðl; tÞ þ J v_ 02 ðl; tÞ; 2 2 2 1
ð4Þ
where a and c are the distances from the free end of the 1st beam (s = 1) to the mass center of the tip body along x1 and x2 axes, as shown in Fig. 2. J is the mass moment of inertia of the tip body about the axis through the mass center, perpendicular to x1–x2 plane and expressed: 2
J ¼ mk ;
ð5Þ
where k is the radius of gyration of the tip body about the axis through the mass center. Hence, Eq. (4) is
T nþ1 ¼
1 1 1 2 0 0 _ _ _ _ u mu_ 21 þ mv_ 21 þ mða2 þ c2 þ k Þv_ 02 mc v þ ma v v ; 1 1 1 1 1 2 2 2 x1 ¼l
where u1 = u1(x1, t) and deformable (Vn+1 = 0).
ð6Þ
v1 = v1(x1, t). The potential energy of the tip body is not considered because it is assumed not to be
Tip body
Beams
Fixed part
nth beam
vs us
sth beam
x2 x1
1st beam Fig. 1. Typical beam-column frame.
v′s
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K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
l0 b0
c e0
a
vs
b2
3
us
b1
2
x2
ρ s , l , d s , As , I s , Es
1
v′s
x1
Fig. 2. Simplified beam structure.
The boundary conditions are
us ð0; tÞ ¼ v s ð0; tÞ ¼ v 0s ð0; tÞ ¼ 0;
s ¼ 1 n:
ð7Þ
Considering the conditions of beam free ends joined by the tip body, the following joint conditions are obtained:
us ðl; tÞ ¼ u1 ðl; tÞ bs1 v 01 ðl; tÞ;
v
0 s ðl; tÞ
¼v
s ¼ 2 n;
ð8Þ
0 1 ðl; tÞ;
s ¼ 2 n;
ð9Þ
v s ðl; tÞ ¼ v 1 ðl; tÞ;
s ¼ 2 n;
ð10Þ
where bs1 is the distance between mid-lines of the 1st and sth beams. Employing these conditions, time derivatives of the kinetic and potential energies are derived from integrations by parts. nþ1 n Z l X @ X € s þ v_ s v€ s Þdx1 Ts ¼ ðqAÞs ðu_ s u @t s¼1 s¼1 0 h i € 1 þ mv_ 1 v€ 1 þ mða2 þ c2 þ k2 Þv_ 01 v€ 01 mcu € 1 v_ 01 mcu_ 1 v€ 01 þ mav€ 1 v_ 01 þ mav_ 1 v€ 01 þ mu_ 1 u
x1 ¼l
n n X @ X Vs ¼ @t s¼1 s¼1 "
Z
l
0
;
ð11Þ
ðEAÞs u00s u_ s þ ðEIÞs v sð4Þ v_ s dx1
n n n n X X X X _ þ ðEAÞs u0s u_ 1 ðEAÞs bs1 u0s v_ 01 þ ðEIÞs v 00s v_ 01 ðEIÞs v 000 s v1 s¼1
s¼2
s¼1
s¼1
# ;
ð12Þ
x1 ¼l
ð4Þ
where v s ¼ @ 4 v s ðx1 ; tÞ=@x41 . Insertion of Eqs. (11) and (12) into Eq. (1) yields motion Eqs. (13), (14) and continuity conditions (15)–(17) at x1 = l.
qs u€ s ðx1 ; tÞ Es u00s ðx1 ; tÞ ¼ 0; 0 < x1 < l; 0 < t; s ¼ 1 n;
ð13Þ
¼ 0; 0 < x1 < l; ðqAÞs v€ s ðx1 ; tÞ þ ðEIÞs v " # n X € 1 mcv€ 01 þ ðEAÞs u0s ¼ 0; mu
ð14Þ
ð4Þ s ðx1 ; tÞ
s¼1
" mv€ 1 þ
v
ma € 01
" 2
2
mða þ c þ k
2
v
Þ € 01
ð15Þ
#
¼ 0;
ð16Þ
x1 ¼l n n X X € 1 þ mav€ 1 mcu ðEAÞs bs1 u0s þ ðEIÞs v 00s s¼2
where us = us(x1, t) and
s ¼ 1 n;
x1 ¼l
n X ðEIÞs v 000 s s¼1
0 < t;
s¼1
# ¼ 0;
ð17Þ
x1 ¼l
vs = vs(x1, t).
2.2. Normalised expression Expressions in (18) and (19) are used to simplify the motion equations and all conditions and Eq. (20) to normalise them.
9090
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
r¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q1 l2 =E1 ;
As ¼ As =A1 ; a ¼ a=l;
l ¼ m=q1 A1 l; a ¼
bs1 ¼ bs1 =l;
ð18Þ
qs ¼ qs =q1 ; Es ¼ Es =E1 ;
Is ¼ Is =I1 ;
c ¼ c=l;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 I1 =A1 l ;
k ¼ k=l;
s ¼ 2 n;
ð19Þ
t ¼ t=r;
x1 ¼ x1 =l;
ð20Þ
where x1 and t are the normalised coordinate and time, respectively. Normalised forms for motion Eqs. (13), (14), boundary and joint conditions (7)–(10), and continuity conditions (15)–(17) are as follows.
u00s ðx1 ; tÞ
q € us ðx1 ; tÞ ¼ 0; 0 < x1 < 1; 0 < t; s ¼ 1 n; E
v sð4Þ ðx1 ; tÞ þ
s
1 qA v€ ðx ; tÞ ¼ 0; 0 < x1 < 1; 0 < t; s ¼ 1 n; a2 EI s s 1
us ð0; tÞ ¼ v s ð0; tÞ ¼ v 0s ð0; tÞ ¼ 0;
v ¼v v s ð1; tÞ ¼ v 1 ð1; tÞ; 0 1 ð1; tÞ;
ð22Þ
s ¼ 1 n;
u1 ð1; tÞ us ð1; tÞ bs1 v 01 ð1; tÞ ¼ 0; 0 s ð1; tÞ
ð21Þ
!
ð23Þ
s ¼ 2 n;
ð24Þ
s ¼ 2 n;
ð25Þ
s ¼ 2 n;
ð26Þ
n X lu€1 ð1; tÞ lcv€ 01 ð1; tÞ þ Es As u0s ð1; tÞ ¼ 0;
ð27Þ
s¼1
lv€ 1 ð1; tÞ þ lav€ 01 ð1; tÞ a2
n X Es Is v 000 s ð1; tÞ ¼ 0;
ð28Þ
s¼1
lða2 þ c2 þ k2 Þv€ 01 ð1; tÞ lcu€1 ð1; tÞ þ lav€ 1 ð1; tÞ
n n X X Es As bs1 u0s ð1; tÞ þ a2 Es Is v 00s ð1; tÞ ¼ 0; s¼2
ð29Þ
s¼1
2.3. Characteristic equation The solutions of normalised motion Eqs. (21) and (22) are assumed, to be
us ðx1 ; tÞ ¼ us ðx1 Þ cos xt;
s ¼ 1 n;
ð30Þ
v s ðx1 ; tÞ ¼ v s ðx1 Þ cos xt;
s ¼ 1 n;
ð31Þ
where x is a normalised radian natural frequency. Consequently, time functions are eliminated from Eqs. (21)–(29), and the following eigenvalue problems are obtained:
u00s ðx1 Þ þ X2s us ðx1 Þ ¼ 0;
0 < x1 < 1;
s ¼ 1 n;
v sð4Þ ðx1 Þ K4s v s ðx1 Þ ¼ 0; 0 < x1 < 1; us ð0Þ ¼ v s ð0Þ ¼ v 0s ð0Þ ¼ 0; s ¼ 1 n; u1 ð1Þ us ð1Þ bs1 v 01 ð1Þ ¼ 0;
v ¼v v s ð1Þ ¼ v 1 ð1Þ; 0 s ð1Þ
0 1 ð1Þ;
ð32Þ
s ¼ 1 n;
ð33Þ ð34Þ
s ¼ 2 n;
ð35Þ
s ¼ 2 n;
ð36Þ
s ¼ 2 n;
ð37Þ
lx2 u1 ð1Þ lc x2 v 01 ð1Þ
n X
Es As u0s ð1Þ ¼ 0;
ð38Þ
s¼1
lx2 v 1 ð1Þ þ la x2 v 01 ð1Þ þ a2
n X Es Is v 000 s ð1Þ ¼ 0;
ð39Þ
s¼1
lða2 þ c2 þ k2 Þx2 v 01 ð1Þ lc x2 u1 ð1Þ þ la x2 v 1 ð1Þ þ where X2s ¼ x2
q
E s
; K4s ¼
s¼2
x 2 qA
a
EI
n X
s
.
Es As bs1 u0s ð1Þ a2
n X Es Is v 00s ð1Þ ¼ 0; s¼1
ð40Þ
9091
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
The solutions satisfying normalised boundary conditions (34) are
us ðx1 Þ ¼ Bs sin Xs x1 ;
s ¼ 1 n;
ð41Þ
v s ðx1 Þ ¼ C s ðcos Ks x1 cosh Ks x1 Þ þ Ds ðsin Ks x1 sinh Ks x1 Þ;
s ¼ 1 n;
ð42Þ
where Bs, Cs, and Ds are modal coefficients. Normalised boundary conditions (36) and (37) give us the following relations.
C s ¼ C 1 Ns þ D1 Ws ;
s ¼ 2 n;
ð43Þ
Ds ¼ C 1 Cs þ D1 Ps ;
s ¼ 2 n;
ð44Þ
where
fKs ðcos K1 cosh K1 Þðcos Ks cosh Ks Þ þ K1 ðsin K1 þ sinh K1 Þðsin Ks sinh Ks Þg ; 2Ks ð1 cos Ks cosh Ks Þ fK ðsin K1 sinh K1 Þðcos Ks cosh Ks Þ K1 ðcos K1 cosh K1 Þðsin Ks sinh Ks Þg Ws ¼ s ; 2Ks ð1 cos Ks cosh Ks Þ fðcos K1 cosh K1 Þ Ns ðcos Ks cosh Ks Þg fðsin K1 sinh K1 Þ Ws ðcos Ks cosh Ks Þg Cs ¼ ; Ps ¼ : ðsin Ks sinh Ks Þ ðsin Ks sinh Ks Þ
Ns ¼
The insertion of solutions (41) and (42) into condition (35), (38)–(40) yields the following matrix form:
½aij fg j g ¼ f0g;
i; j ¼ 1 ðn þ 2Þ:
ð45Þ
The form of matrix [aij] is
ð46Þ
The components of [aij] are
a1;1 ¼ sin X1 ;
a1;2 ¼ sin X2 ;
a1;nþ1 ¼ b1 K1 ðS þ SHÞ1 ;
a1;nþ2 ¼ b1 K1 ðC CHÞ1 ;
ð47-1; 2; 3; 4Þ
a2;1 ¼ sin X1 ;
a2;3 ¼ sin X3 ;
a2;nþ1 ¼ b2 K1 ðS þ SHÞ1 ;
a2;nþ2 ¼ b2 K1 ðC CHÞ1 ;
ð47-5; 6; 7; 8Þ
an2;1 ¼ sin X1 ;
an2;n1 ¼ sin Xn1 ;
an2;nþ1 ¼ bn2 K1 ðS þ SHÞ1 ;
an2;nþ2
¼ bn2 K1 ðC CHÞ1 ; an1;1 ¼ sin X1 ; an1;n ¼ sin Xn ; an;1 ¼ lx2 sin X1 X1 cos X1 ;
ð47-9; 10; 11; 12Þ an1;nþ1 ¼ bn1 K1 ðS þ SHÞ1 ; an;2 ¼ E2 A2 X2 cos X2 ;
an1;nþ2 ¼ bn1 K1 ðC CHÞ1 ;
an;3 ¼ E3 A3 X3 cos X3 ;
an;n1
¼ En1 An1 Xn1 cos Xn1 ; an;n ¼ En An Xn cos Xn ;
ð47-17; 18; 19; 20Þ
an;nþ1 ¼ lc x2 K1 ðS þ SHÞ1 ;
anþ1;nþ1 ¼ lx2 ðC CHÞ1 la x2 K1 ðS þ SHÞ1 þ a2
ð47-13; 14; 15; 16Þ
an;nþ2 ¼ lc x2 K1 ðC CHÞ1 ;
n X Es Is K3s ½Ns ðS SHÞs Cs ðC þ CHÞs ; s¼1
ð47-21; 22; 23Þ ð47-24Þ
9092
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
anþ1;nþ2 ¼ lx2 ðS SHÞ1 þ la x2 K1 ðC CHÞ1 þ a2
n X Es Is K3s ½Ws ðS SHÞs Ps ðC þ CHÞs ;
ð47-25Þ
s¼1
anþ2;1 ¼ lc x2 sin X1 ;
anþ2;2 ¼ E2 A2 b1 X2 cos X2 ;
anþ2;n1 ¼ En1 An1 bn2 Xn1 cos Xn1 ;
anþ2;3 ¼ E3 A3 b2 X3 cos X3 ;
ð47-26; 27; 28Þ
anþ2;n ¼ En An bn1 Xn cos Xn ;
anþ2;nþ1 ¼ lða2 þ c2 þ k2 Þx2 K1 ðS þ SHÞ1 la x2 ðC CHÞ1 a2
n X
ð47-29; 30Þ
K2s Es Is ½Ns ðC þ CHÞs þ Cs ðS þ SHÞs ;
ð47-31Þ
s¼1
anþ2;nþ2 ¼ lða2 þ c2 þ k2 Þx2 K1 ðC CHÞ1 la x2 ðS SHÞ1 a2
n X K2s Es Is ½Ws ðC þ CHÞs þ Ps ðS þ SHÞs ;
ð47-32Þ
s¼1
where (C ± CH)s = cosKs ± cosh Ks and (S ± SH)s = sin Ks ± sinh Ks. The modal vector is
fg j g ¼ fB1 ; . . . Bn ; C 1 ; D1 gT :
ð48Þ
A specific case arises when whole beams are identical and symmetrically located at the upper and lower parts of the mass center of tip body. Because this case makes Xs ¼ X1 ¼ x and Ks = K1, it is induced that Ns = 1, Ws = 0, Cs = 0, and Ps = 1. Therefore
Cs ¼ C1; Ds ¼ D1 ;
s ¼ 2 n; s ¼ 2 n:
ð49Þ ð50Þ
Also
v s ðx1 Þ ¼ v 1 ðx1 Þ;
s ¼ 2 n:
ð52Þ
Such a case makes the matrix simpler:
2
sin x
sin x
0
6 6 sin x 0 sin x 6 6 6 6 6 6 6 6 6 sin x 0 0 6 6 6 sin x 0 0 6 6 lx sin x 6 6 cos x cos x 6 cos x 6 6 6 6 6 6 0 0 0 6 6 6 6 6 6 6 6 6 6 lcx sin x b1 cos x b2 cos x 6 4
0
0
b1 K1 ðS þ SHÞ1
0
0
b2 K1 ðS þ SHÞ1
sin x
0
bn2 K1 ðS þ SHÞ1
0
sin x
bn1 K1 ðS þ SHÞ1
cos x
cos x
lcxK1 ðS þ SHÞ1
0
lK1 ðC CHÞ1 laK21 ðS þ SHÞ1
0
þnðS SHÞ1
lða2 þ c2 þ k2 Þ xK1 ðS þ SHÞ1 : : bn2 cos x bn1 cos x laxðC CHÞ1 naðC þ CHÞ1
b1 K1 ðC CHÞ1
3
7 b2 K1 ðC CHÞ1 7 7 7 7 7 7 7 7 7 bn2 K1 ðC CHÞ1 7 7 7 bn1 K1 ðC CHÞ1 7 7 7 7 lcxK1 ðC CHÞ1 7 7 7 7 7 lK1 ðS SHÞ1 7 7 2 þlaK1 ðC CHÞ1 7 7 7 7 nðC þ CHÞ1 7 7 lða2 þ c2 þ k2 Þ 7 7 xK1 ðC CHÞ1 7 7 7 laxðS SHÞ1 7 5 naðS þ SHÞ1 ð53Þ
Since Eq. (45) is a combination of homogeneous and linear algebraic equations from the components of modal vector (48), a non-trivial solution exists only if the determinant of the coefficients vanishes. That is,
det½aij ¼ 0:
ð54Þ
The determinant of the (n + 2) (n + 2) matrix can be calculated by a mathematical program [10]. Matrix (46) is too complicated to obtain its determinant yielding characteristic equations including parametric symbols. In this case, substituting real values into each component of the matrix gives us characteristic equations which are expressed as the function of x. Especially, calculating the determinant of matrix (53) composed by the condition of identical beams and symmetrical beam placement with respect to the mass center of the tip body yields the following characteristic equations:
n cos x lx sin x ¼ 0;
ð55Þ
Table 1 Analytic solutions and FEM results with respect to variation of n and d (CAB vibration). d
n 1
2
3
4
5
FEM
Eq. (56)
FEM
Eq. (56)
FEM
Eq. (56)
FEM
Eq. (56)
FEM
x1
0.2 0.4 0.6
3.0 11.9 25.7
3.0 11.9 25.7
8.7 32.4 65.6
9.3 31.4 65.8
10.6 38.0 74.2
10.6 38.0 74.0
12.0 42.1 80.0
12.1 42.1 79.8
13.3 45.3 84.2
13.4 45.3 84.1
x2
0.2 0.4 0.6
91.1 196.9 304.9
91.0 196.7 304.7
152.5 313.8 487.2
152.3 314.1 487.0
153.3 318.8 498.4
153.8 319.3 497.4
154.2 323.3 507.9
154.4 323.8 506.5
154.9 327.4 515.5
154.8 327.4 515.5
x3
0.2 0.4 0.6
248.9 601.8 944.1
248.7 601.4 943.4
417.2 842.0 1277.5
416.5 842.8 1276.9
417.9 846.5 1288.5
419.1 847.9 1285.9
418.7 851.3 1299.7
419.4 852.4 1295.8
419.5 855.8 1309.3
419.2 856.1 1309.4
x4
0.2 0.4 0.6
476.7 1166.9 1893.5
476.3 1165.9 1891.6
815.9 1635.5 2458.5
814.6 1637.3 2458.0
816.3 1637.9 2463.5
818.6 1640.2 2458.9
816.9 1641.5 2472.5
818.3 1643.6 2465.0
817.5 1645.1 2480.7
816.9 1647.3 2481.2
x5
0.2 0.4 0.6
850.2 1898.1 3110.1
849.7 1896.3 3106.3
1347.2 2691.2 4024.2
1344.8 2696.0 4016.3
1347.1 2689.5 4017.3
1350.8 2693.3 4005.8
1347.3 2690.9 4020.6
1349.6 2694.0 4004.5
1347.6 2692.5 4024.2
1346.7 2694.1 4019.7
x6
0.2 0.4 0.6
1371.1 2887.4 4619.2
1370.1 2884.4 4612.2
2010.8 4006.6 5964.6
2007.2 4004.3 5953.5
2009.9 3998.1 5937.0
2015.3 4014.6 5918.9
2009.6 3995.9 5930.8
2012.9 3998.7 5904.3
2009.4 3994.1 5926.6
2007.9 4016.8 5923.2
bs
c=0
b1 = 2c
b1 ¼ c b2 ¼ 2c
b1 ¼ 0:7c b2 ¼ 1:3c b3 ¼ 2c
b1 b2 b3 b4
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
Eq. (56)
¼ 0:6c ¼c ¼ 1:4c ¼ 2c
9093
9094
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
"
lK21 lk2 x sin x
( ) # n X ðbs1 cÞ2 þ c2 cos x ð1 cos K1 cosh K1 Þ s¼2
"
( ) # n X 2 2 nK1 lða þ k Þx sin x ðbs1 cÞ þ c cos x ðsin K1 cosh K1 þ cos K1 sinh K1 Þ 2
2
s¼2
þ n2 a sin xð1 þ cos K1 cosh K1 Þ nlaK1 sin xðsin K1 cosh K1 cos K1 sinh K1 Þ 2nlax sin x sin K1 sinh K1 ¼ 0;
ð56Þ
where c ¼ bn1 =2. Special cases (n = 1, 2) for Eq. (55) were reported in previous studies [11,12]. In light of these facts, one can say that Eq. (55) is the characteristic equation for the PA vibration of the multi-beam structure having n beams and a tip body on condition of identical beam and symmetrical beam placement. Using FEM by computer program [13] in the following section, it is investigated what kind of vibration motion is expressed by Eq. (56). 2.4. Vibration mode examination and accuracy verification In order to show what kind of vibration mode is expressed by characteristic Eq. (56) and verify the accuracy of that equation, the fundamental radian natural frequencies from Eq. (56) are compared with those from an FE model simulated in the CAB vibration. The following values for mechanical and dimensional parameters are used to obtain the radian natural frequencies from Eq. (56) and a simplified model generated by FE analysis, respectively. Fig. 2 shows the simulated model in which all beams have identical mechanical properties. Beam: l = 100 mm, q = 8250 kg/m3, E = 15 GPa, Tip body: q0 = 1400 kg/m3, b0 = 5 mm, e0 = 5 mm, l0 = 10 mm.
CAB mode ( x1 − x2 plane)
1st
2nd
3rd
4th
5th
6th
Fig. 3. Mode shapes.
Table 2 Analytic solutions and FEM results with respect to variation of n and d (CAB vibration). d
n 1
2
3
4
5 FEM
0.2 0.4 0.6
3.0 11.9 25.7
3.0 11.9 25.7
9.0 33.1 66.7
8.9 31.6 66.7
10.7 38.5 74.9
10.9 38.6 74.9
12.2 42.5 80.5
13.5 41.2 80.6
13.4 45.6 84.6
14.1 45.4 84.6
x2
0.2 0.4 0.6
81.1 191.2 301.0
81.0 191.0 300.8
152.6 314.4 488.7
152.3 313.7 489.0
153.4 319.4 499.7
153.1 318.6 500.0
154.3 324.0 509.5
153.9 323.1 509.5
155.0 328.0 516.8
154.7 327.2 517.1
x3
0.2 0.4 0.6
205.4 550.7 908.3
205.2 550.3 907.6
417.3 842.7 1279.4
416.4 840.6 1279.9
418.0 847.3 1290.5
417.1 845.2 1291.0
418.9 852.5 1303.0
417.9 850.2 1302.8
419.6 856.8 1312.1
418.7 854.7 1312.6
x4
0.2 0.4 0.6
441.3 1032.1 1756.2
441.0 1031.2 1754.5
816.0 1636.2 2460.6
814.2 1632.1 2460.9
816.5 1638.8 2466.2
814.6 1634.7 2466.8
817.2 1643.5 2478.4
815.2 1638.9 2477.3
817.7 1646.8 2488.7
815.8 1642.7 2486.1
x5
0.2 0.4 0.6
829.6 1754.4 2850.3
829.0 1752.8 2846.7
1347.4 2691.8 4026.6
1344.2 2684.7 4025.9
1347.3 2690.7 4020.9
1344.1 2683.7 4020.7
1347.7 2694.2 4030.6
1344.5 2686.2 4027.1
1347.9 2694.9 4032.4
1344.9 2688.3 4032.1
x6
0.2 0.4 0.6
1357.0 2775.7 4323.6
1356.0 2772.8 4316.9
2013.2 4025.0 6000.4
2006.2 3996.2 5964.2
2012.5 3999.4 5942.4
2005.3 3988.9 5940.3
2010.5 4000.7 5946.4
2005.3 3988.6 5939.5
2009.9 3996.7 5939.7
2005.3 3987.8 5937.5
Es ; bs
FEM
Mat. (46)
FEM
Mat. (46)
FEM
Mat. (46)
FEM
c=0
q2 ¼ 1:1q1 ;
E2 ¼ 1:1E1 ; b1 ¼ 2c
q2 ¼ q1 ;
E2 ¼ E1 ; q3 ¼ 1:1q1 ; E3 ¼ 1:1E1 ; b1 ¼ 1:1c b2 ¼ 2c
q2 ¼ q1 ;
E2 ¼ E1 ; q3 ¼ q1 ; E3 ¼ E1 ; q4 ¼ 1:1q1 ; E4 ¼ 1:1E1 ; b1 ¼ 0:7c b2 ¼ 1:5c b3 ¼ 2c
q2 ¼ q1 ; E2 ¼ E1 ;
q3 ¼ q1 ; E3 ¼ E1 ;
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
Mat. (46)
x1
qs ;
Mat. (46)
q4 ¼ q1 ; E4 ¼ E1 ;
q5 ¼ 1:1q1 ;
E5 b1 b2 b3 b4
¼ 1:1E1 ; ¼ 0:6c ¼c ¼ 1:5c ¼ 2c
9095
9096
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
c a
Fig. 4. Optical pickup actuator with six suspensions.
1st mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
a = −5 ~ 5
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2nd mode 1 0.8
0.2
a=5 a=4 a=3 a=2 a =1
0
a=0
0.6 0.4
-0.2
a = −1
-0.4
a = −5
a = −2
-0.6 -0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a = −3 a = −4
Fig. 5. Vibration modes with respect to a (at b1 = 1.1c, b2 = 2c) (a) 1st mode (b) 2nd mode.
9097
K.T. Lee / Applied Mathematical Modelling 37 (2013) 9086–9100
In this model, the normalised radius of gyration is
k¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 b0 þ e20 =12=l:
ð57Þ
In addition, the normalised parameters and are established as follows.
a ¼ a=l ¼ e0 =2l;
c ¼ bn1 =2 ¼ ðb0 dÞ=2l:
ð58; 59Þ
According to the number of beams n, beam diameter d and the beam distances bs1 , the radian natural frequencies which are obtained from Eq. (56) and simulated by an FEM program are classified in Table 1. A radian natural frequency x from Eq. (56) is obtained as a normalised radian natural frequency x of Eq. (56) is divided by rðx ¼ x=rÞ. This table shows that the six fundamental radian natural frequencies obtained by the characteristic equations are in excellent agreement with those obtained by the FEM. Therefore, it is confirmed that Eq. (56) well expresses the vibration characteristics for this mechanical model and also recognised that the equation is the characteristic equation for CAB vibration. In order to help reader’s understanding, the mode shapes of CAB vibration simulated from FEM are provided in Fig. 3. In the same way, vibration problems, in which all beams with unsymmetrical beam placement do not have identical beam properties, are solved using the determinant of matrix (46). Contrary to the case of Eq. (56), qs and Es of each beam,
1st mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
b1 = 0.4c ~ 1.6c
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2nd mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
b1 = 0.4c ~ 1.6c
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 6. Vibration modes with respect to b1 (at a = 0, b2 = 2c) (a) 1st mode (b) 2nd mode.
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which are expressed in Table 2, are slightly different. The radian natural frequencies, obtained from the determinant and from CAB vibration simulated by the FEM program, are classified in the table. The fundamental radian natural frequencies obtained from the determinant are accord with those obtained by the FEM. Therefore it is also confirmed that Eq. (45) defines CAB vibration of a multi-beam structure having non-identical beams or unsymmetrical beam placement condition. It should be noted that PA vibration motion can not be obtained from Eq. (45).
3. Mode shape control of optical pickup actuator 3.1. Structure of optical pickup actuator An optical pickup actuator is one of key components in ODDs. The actuator includes a moving part and suspensions to obtain focusing and tracking motions. In general, the optical pickup actuator for compact disc (CD) drives has four beams as electrical connectors as well as mechanical suspensions for two-axis motions [14]. However, optical disc systems such as digital versatile disc (DVD) or blue-ray disc (BD) drives require optical pickup actuators having six suspensions [15,16]. In particular, an actuator employing a liquid–crystal panel for securing multi-layer compatibility requires much more suspensions [17]. In a near field recording (NFR) system, Lee et al. used eight beams for four-axis control [18].
1st mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
b2 = 1.4c ~ 2c
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2nd mode 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
b2 = 1.4c ~ 2c
-0.8 -1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 7. Vibration modes with respect to b2 (at a = 0, b1 = 1.1c) (a) 1st mode (b) 2nd mode.
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Nevertheless, studies related to the analysis of their dynamic behaviours are very rare. Lee analytically solved vibrations of a structure composed by identical four beams and also described vibration of optical pickup actuators to show the modification of mode shape by parametric analysis [9]. In the same manner, this paper treats the vibration motion of an actuator having multiple suspensions. 3.2. Mode shape control The dynamic behaviour of a simplified actuator model shown in Fig. 4, which has six suspensions in both sides of the moving part, is analyzed. Each mechanical property is listed as follows. Beams: d1 = d2 = 0.14 mm, d3 = 0.16 mm, l = 14 mm, q1 = q2 = q3 = 8250 kg/m3, E1 = E2 = E3 = 127 GPa, Tip body: m = 1 103 kg, k = 0.45 m2, Distance from 1st suspension to mass center: c = 2.4 mm. a, b1 and b2 are design parameters to improve vibration modes of this actuator. They are assumed to be varied in the following ranges, respectively.
5 6 a 6 5;
0:4c 6 b1 6 1:6c;
1:4c 6 b2 6 2c:
ð60; 61; 62Þ
Suspensions are located at both sides of the moving part and symmetrical in vertical direction. The six suspensions can be regarded as three ones on the vertical centreline as shown in Fig. 2. As a matter of course, the cross-sectional areas and areal moments of inertia of the suspension should be redefined as follows. 2
As ¼ 2 pds =4;
4
Is ¼ 2 pds =64:
ð63; 64Þ
For knowing the motion of the moving part, the mode shape of the 1st suspension is investigated. The mode shape is normalised by the difference of maximum and minimum values:
v 1 ðxÞ=fmaxðv 1 ðxÞÞ minðv 1 ðxÞÞg
ð63Þ
In this paper, two fundamental modes of the 1st suspension in low frequency range are observed on condition that several design parameters are varied. When the value of the parameter a is changed, the 2nd mode is remarkably changed as shown in Fig. 5. When a = 0, deflection of the end of suspension is suppressed but deflection angle is maximised. If jaj is increased, deflection angle is decreased while deflection is increased. The variation of parameter b1 scarcely affects the 2nd mode shape and the variation of parameter b2 slightly affects the 2nd mode shape as shown in Figs. 6 and 7, respectively. On the other hand, the 1st mode shape is never changed by these parameters. 4. Conclusions This paper mathematically analyzes the vibration characteristics of a beam-column frame having n beams joined by a tip body. As a result, modal frequencies and mode shapes for vibration motion of the beam-column frame in longitudinal and lateral directions are obtained. The accuracy of the mathematical deployment in this paper is demonstrated by comparing the natural frequencies obtained from FEM results. For proving practical use of this study, the result of this paper was used to a case study for the vibration analysis of an optical pickup actuator to improve its dynamic behaviour. Modal frequencies and shapes were calculated by applying the design parameters to the mathematical expression. Furthermore, it was shown that we could change the mode shapes by parametric analysis. References [1] P.A.A. Laura, J.L. Pombo, E.A. Susemihl, A note on the vibrations of a clamped-free beam with a mass at the free end, J. Sound Vib. 37 (2) (1974) 161– 168. [2] B. Bhat, H. Wagner, Natural frequencies of a uniform cantilever with a tip mass slender in the axial direction, J. Sound Vib. 45 (2) (1976) 304–307. [3] R.E. Rossi, P.A.A. Laura, R.H. Gutierrez, A note on traverse vibrations of a Timoshenko beam of non-uniform thickness clamped at one end and carrying a concentrated mass of at the other, J. Sound Vib. 143 (3) (1990) 491–502. [4] N.M. Auciello, Transverse vibrations of a linearly tapered cantilever beam with tip mass of rotary inertia and eccentricity, J. Sound Vib. 194 (1) (1996) 25–34. [5] H. Gokdag, O. Kopmaz, Coupled bending and torsional vibration of a beamwidth in-span and tip attachments, J. Sound Vib. 287 (2005) 591–610. [6] E. Dokumaci, An exact solution for coupled bending and torsion vibrations of uniform beams having single cross-sectional symmetry, J. Sound Vib. 119 (3) (1987) 443–449. [7] S.H. Farghaly, Bending vibration of an axially loaded cantilever beam with an elastically mounted end mass of finite length, J. Sound Vib. 156 (1992) 373–380. [8] J.R. Banerjee, Explicit frequency equation and mode shapes of a cantilever beam coupled in bending and torsion, J. Sound Vib. 224 (2) (1999) 267–281. [9] K.T. Lee, Analytical solutions for vibration characteristics of a multi-beam structure, Int. J. Mech. Sci. 52 (2010) 952–969. [10] Mathematica 6.5, Wolframe Research Inc., 2007. [11] R.R. Craig, A.J. Kurdila, Fundamentals of Structural Dynamics, John Wiley & Sons Inc., 2006. [12] G.L. Anderson, Natural frequencies of two cantilevers joined by a rigid connector at their free ends, J. Sound Vib. 57 (1978) 403–412. [13] Ansys 11, ANSYS Inc., 2006.
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[14] M. Nagasato, I. Hoshino, Development of a two-axis actuator with small tilt angles for one-piece optical heads, Jpn. J. Appl. Phys. 35 (1996) 392–397. [15] B. Zhang, J. Ma, L. Pan, X. Cheng, Y. Tang, High performance three-axis actuator in super-multi optical pickup actuator with low crosstalk force, IEEE Trans. Consum. Electron. 54 (4) (2008) 1743–1749. [16] S.J. Kim, T.Y. Heor, T.K. Kim, Y.M. Ahn, C.S. Chung, S.H. Park, High Response twin-objective actuator with radial tilt function for blu-ray disc recorder, Jpn. J. Appl. Phys. 44 (2005) 3393–3396. [17] T.K. Kim, Y.M. Ahn, S.J. Kim, T.Y. Heor, C.S. Chung, I.S. Park, Blu-ray disc pickup head for dual layer, Jpn. J. Appl. Phys. 44 (2005) 3397–3401. [18] K.T. Lee, J.E. Kim, J.K. Seo, S.N. Hong, I.H. Choi, E.S. Ko, B.H. Min, Dynamic tilt control of SIL with 4-axis actuator in NFR system, Proc. Opt. Data Storage (2007) WA7.