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ScienceDirect Procedia Engineering 79 (2014) 160 – 165
37th National Conference on Theoretical and Applied Mechanics (37th NCTAM 2013) & The 1st International Conference on Mechanics (1st ICM)
Effect of local Kelvin-Voigt damping on eigenfrequencies of cantilevered twisted Timoshenko beams Wei-Ren Chen* Department of Mechanical Engineering, Chinese Culture University, Taipei 11114, Taiwan
Abstract The present paper investigates the vibration behavior of a cantilevered twisted Timoshenko beam with partially distributed Kelvin-Voigt damping. A finite element method is used to derive the system equations of motion with a damping term. A quadratic eigenvalue problem of a damped system is formed to study the effects of the twist angle, the size and location of the damped segment on the eigenfrequencies of the twisted beams. © 2013 2014 The Elsevier Ltd. This is an open access Ltd. article under the CC BY-NC-ND license Authors. Published by Elsevier © (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the National Tsing Hua University, Department of Power Mechanical Selection and peer-review under responsibility of the National Tsing Hua University, Department of Power Mechanical Engineering. Engineering Keywords:vibration; twisted beam; Kelvin-Voigt damping; twist angle
1. Introduction Vibration analysis of the beam structure is important in the machine and structure design for better understanding the dynamic behavior of beam systems. Because damping has a profound influence on vibration and vibration control of beam structures, the effects of internal and/or external lumped or distributed damping had been extensively studied [1-5]. Recently, the vibration of elastic systems with Kelvin-Voigt damping had been analyzed by many investigators [6-10]. Due to their important applications in various engineering fields, the dynamic behavior of the twisted beam structures, such as turbine blades and fluted cutters, had been investigated in the past years by
* Wei-Ren Chen. Tel.: 886-2-28610511; fax: 886-2-28615241. E-mail address:
[email protected]
1877-7058 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/).
Selection and peer-review under responsibility of the National Tsing Hua University, Department of Power Mechanical Engineering doi:10.1016/j.proeng.2014.06.325
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Wei-Ren Chen / Procedia Engineering 79 (2014) 160 – 165
using Euler beam theory [11-13] or Timoshenko beam theory [14-18]. However, the vibration characterization of twisted Timoshenko beams with locally distributed Kelvin-Voigt damping has not yet been reported. The present work tends to study the vibration characterizations of a cantilevered twisted Timoshenko beam with partially distributed internal damping of Kelvin-Voigt type. Based on a finite element approach, the system equations of motion of the damped twisted beam are derived. Then, a quadratic eigenvalue problem is formulated to study the free vibration behavior of the cantilevered twisted beam having a local damped segment with Kelvin-Voigt damping. The effects of the twist angle, the size and location of the damped segment on the eigenfrequencies of the cantilevered twisted beams are investigated. Y
ly
lx X
b
lz"
h
Z1
lj
Z2
z""
j"
Z
L
Fig. 1 Beam configuration and coordinate systems
2. Problem formulation In the present study, a cantilevered twisted Timoshenko beam with locally distributed Kelvin-Voigt damping is considered as shown in Fig. 1. Z1 and Z2 are coordinates of the damped segment of the twisted beam, which are used to define the length Ld = Z2 - Z1 and the position Lg = (Z1 + Z2)/2 of the damped segment. Coordinate system XYZ denotes the inertial coordinate frame. The axes z and j of the twist coordinate frame zjZ are in the principal directions of the beam cross-section. The bending vibration equations of motion of such a twisted beam are readily derived by applying Hamilton’s principle to the Lagrangian (L) of the beam system.
f Ð Ldt ? Ð (fT / fV - fWd )dt ? 0 t1
t1
t0
t0
(1)
where T is the total kinetic energy; V is the potential energy; fWd is the virtual work due to the Kelvin-Voigt damping. They are represented by the following expressions. T ? V ?
1 L 2 2 2 2 2 Ð [m(u% X - u%Y ) - J XX l% X - 2 J XY l% X l% Y - J YY l% Y ]dZ 2 0
1 L {mGA[(u ¦X / l Y ) 2 - (uY¦ / l X ) 2 ] - EI XX (l ¦X ) 2 - 2 EI XY l ¦X l Y¦ - EI YY (l Y¦ ) 2 }dZ 2 Ð0
(2)
(3)
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fWd ? R?
•R •R •R •R fu X fuY fl X fl Y •l%Y •l% X •u%Y •u% X
]
(4)
_
1 L }Ejb I XX (l%¦X )2 - 2Ejb I XY (l%¦X )(l%Y¦ ) - Ejb IYY (l%Y¦ )2 - GjsmA (u%¦X / l%Y )2 - (u%Y¦ / l% X )2 dZ 2 Ð0
(5)
Here R is the proposed dissipation function of the twisted beam at any time instant based on the model by Kocatürk and Şimşek [7]. Parameters jb and js denote the proportionality constants of the internal damping of the beam in bending and shearing, respectively. In this work, the partially distributed internal damping is considered so jb and js assume the subsequent forms.
j b ? j b ( H (Z / Z 1 ) / H (Z / Z 2 ))
(6a)
j s ? j s ( H (Z / Z 1 ) / H (Z / Z 2 ))
(6b)
The partial differential equations of motion of the investigated twisted Timoshenko beam in the twist frame can be directly obtained by substituting Eqs. (2)-(6) into Eq. (1) and using the transformation relationship [18] between the inertial and twist coordinates. Then a finite element method [18] is used to reduce the governing partial differential equations into a set of ordinary differential equations by using the Mindlin-type linear beam element with eight degrees of freedom. The displacement function of the beam element can be expressed as
Êu z Û Ç N1 Íu Í È0 Í jÍ d ( e ) ( z ) ? Ë Ü ? N( z )p ( e ) ? È È0 Íl z Í È ÍÌlj ÍÝ É0
0 N1 0 0
0 0 N1 0
0 0 0 N1
N2 0 0 0
0 N2 0 0
0 0 N2 0
Ê uz 1 Û Íu Í Í j1 Í 0 Íl z 1 Í Í Í 0 ÙÙ Ílj1 Í Ë Ü 0 Ù Íu z 2 Í ÙÍ Í N 2 Ú uj 2 Í Í Íl z 2 Í Ílj 2 Í Ì Ý
(7)
Here shape functions N1 = 1 – z/Le and N2 = z/Le, and Le is the beam element length. uz, uj, lz and lj are transverse displacements and angles of rotation of the twisted beam element. Substitute the displacement function (Eq. (7)) into the weak form of the governing partial differential equations and apply Galerkin’s criterion to yield the resulting beam element equations.
%% (e) - C(e) p% (e) - K (e) p (e) ? 0 M (e) p
(8)
M(e), C(e) and K(e) are the element mass matrix, element damping matrix and element stiffness matrix, respectively. Then directly assembling the element matrices and imposing the essential boundary conditions yield the global system of equations of the damped twisted Timoshenko beam as
%% - Cp% - Kp ? 0 Mp
(9)
where M, C and K are the corresponding global matrices. Introducing p = eytq into Eq. (9) leads to the quadratic eigenvalue problem. [ω2 M - ωC - K ]q ? 0
(10)
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Wei-Ren Chen / Procedia Engineering 79 (2014) 160 – 165 45
420 0 deg
0 deg
30 deg
30 deg 36
60 deg
416
60 deg
90 deg
27
Im(y) (Hz)
|Re(y)| (Hz)
90 deg
18
412
408
9
404
400
0 °
°♂″
°♂￥
°♂￠
°♂％
′
°
°♂″
°♂￥
°♂￠
°♂％
′
Ld/L
Ld/L
(a)
(b)
Fig. 2 Effect of damped segment size on mode 1 eigenfrequency of cantilevered twisted beams with different twist angles: (a) absolute real part and (b) imaginary part. 180
840
0 deg
0 deg 150
30 deg 820
30 deg
60 deg
60 deg
90 deg
90 deg
800
Im(y) (Hz)
|Re(y)| (Hz)
120
90
780
760
60
740
30
720
0 0
0.2
0.4
0.6 Ld/L
(a)
0.8
1
°
°♂″
°♂￥
°♂￠
°♂％
′
Ld/L
(b)
Fig. 3 Effect of damped segment size on mode 2 eigenfrequency of cantilevered twisted beams with different twist angles: (a) absolute real part and (b) imaginary part.
where q is a constant vector and y is the eigenfrequency associated with the damping system. In the next, Eq. (10) is used to study the influence of the twist angle, the damped segment size and location on the eigenfrequencies of the cantilevered twisted Timoshenko beams with locally distributed Kelvin-Voigt damping. 3. Results and Discussions The material and geometric properties of the investigated twisted beam are E = 200 Gpa, p= 0.3, t = 7860 kg m-3, L = 0.2 m, b = 0.04 m, h = 0.02 m, m = 5/6 and"jb = js =10-4 s. Generally, the eigenfrequency y is a complex number for the damped beam. It has a real part and an imaginary part, which is denoted as Re( y) and Im( y) throughout the study. The real part is associated with the decay rate of free vibration of the damped system and the imaginary part is related to the damped oscillation frequency. Figs. 2 and 3 present the effect of the size of the damped segment on the mode 1 and mode 2 eigenfrequency of cantilevered twisted beams with various twist angles, respectively. The size Ld/L of the damped segment varies from 0.1 to 0.9 and its central position is kept at Lg/L = 0.5. The increasing twist angle decreases the absolute real (damping) part of mode 1 eigenfrequency and increases that of mode 2 eigenfrequency as 0 < Ld/L ａ 0.5, but it has an opposite tendency when Ld/L > 0.5. However, with the increase in twist angle, the imaginary (oscillating) part of mode 1 eigenfrequency increases but that of mode 2 eigenfrequency decreases regardless of Ld/L. The absolute real
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part of mode 1 and 2 frequency increases continuously when the length of the damped segment increases. With the increase in Ld/L, the imaginary part for both modes increases firstly and then starts to decrease when Ld/L Ω 0.6. As can be seen, the damped segment length has a significant impact on the absolute real part but has a minor effect on the imaginary part. The eigenfrequency of the twisted beam with a larger size of damped segment always has a greater absolute real part. Hence, the system response of such a beam system will decay toward zero with time more quickly. Figs. 4 and 5 present the effects of the location of the damped segment on the mode 1 and mode 2 eigenfrequency of cantilevered twisted beams with different twist angles, respectively. The damped segment size Ld is taken to be 0.2L and its central position Lg/L varies from 0.1 to 0.9. When the twist angle increases, the absolute real part of mode 1 eigenfrequency increases and that of mode 2 eigenfrequency decreases when 0 < Lg/L ａ 0.4. However, the increasing twist angle shows an opposite effect on the absolute real part of mode 1 and 2 eigenfrequency as Lg/L > 0.4. The absolute real part of mode 1 and 2 frequency decreases with the increasing Lg/L. When Lg/L increases, the imaginary part of both modes increases shortly and begins to reduce as Lg/L Ω 0.2. However, the increasing Lg/L has a minor impact on the imaginary part of eigenfrequencies. Similarly, the absolute real part of eigenfrequencies is significantly affected by the position of the damped segment. The twisted beam with damped segment positioned near to the fixed end has the higher absolute real part. When the damped segment is located near the free end of the beam, the absolute real part diminishes abruptly and approaches zero. Thus, to assure a better damping effectiveness, the damped segment should be positioned close to the fixed end of the cantilevered twisted beam.
420
35
0 deg
0 deg
30 deg
30 deg
30
60 deg
60 deg
416
90 deg
90 deg
Im(y) (Hz)
|Re(y)| (Hz)
25
20
15
412
408
10
404 5
400
0 °
°♂″
°♂￥
Lg/L
°♂￠
°♂％
°
′
°♂″
°♂￥
°♂￠
°♂％
′
Lg/L
(a)
(b)
Fig. 4 Effect of damped segment location on mode 1 eigenfrequency of cantilevered twisted beams with different twist angles: (a) absolute real part and (b) imaginary part. 840
120 0 deg
0 deg
30 deg 100
30 deg
820
60 deg
60 deg
90 deg
90 deg 800
Im(y) (Hz)
|Re(y)| (Hz)
80
60
780
40
760
20
740
720
0 °
°♂″
°♂￥
°♂￠
Lg/L
(a)
°♂％
′
°
°♂″
°♂￥
°♂￠
°♂％
′
Lg/L
(b)
Fig. 5 Effect of damped segment location on mode 2 eigenfrequency of cantilevered twisted beams with different twist angles: (a) absolute real part and (b) imaginary part.
Wei-Ren Chen / Procedia Engineering 79 (2014) 160 – 165
4. Conclusion Based on the results discussed earlier, some conclusions are addressed as follows. The increasing twist angle always increases the imaginary part of mode 1 eigenfrequency and decreases that of mode 2 eigenfrequency. Depending on the size and location of the damped segment, the increasing twist angle might decrease or increase the absolute real part of eigenfrequencies. The size and location of the damped segment significantly affects the absolute real part of eigenfrequencies of the cantilevered twisted beam. With the increase in damped segment length, the absolute real part increased. The absolute real part increases when the damped segment is located near the fixed end. The imaginary part of eigenfrequencies is slightly affected by the length and position of the damped segment.
Acknowledgement This study was funded by the National Science Council of the Republic of China under Grant NSC 101-2221-E034-009. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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