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Wave control of vibrations in multi-story planar frame structures based on classical vibration theories
Journal of Sound and Vibration 330 (2011) 5530–5544
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Wave control of vibrations in multi-story planar frame structures based on classical vibration theories C. Mei n Department of Mechanical Engineering, The University of Michigan - Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA
a r t i c l e in f o
abstract
Article history: Received 29 November 2010 Received in revised form 17 April 2011 Accepted 27 June 2011 Handling Editor: M.P. Cartmell Available online 22 July 2011
An active control approach to vibrations in multi-story planar frame structures is presented in this paper. The controller is designed from a wave vibration standpoint, in which vibrations are described as waves that propagate along uniform waveguides, and are reflected and transmitted upon structural discontinuities. Regardless of the complexity of a structure, from the wave point of view it consists of only two basic types of structural components, namely, structural elements and joints. In this paper, vibrations in a multi-story planar frame structure are controlled through controlling its structural elements and its joints. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Multi-story planar frame structures can be found in many fields of engineering, ranging from big scale high-rise buildings and towers to micro-frames used in modern electronic equipment. These frame structures are often subject to harmful vibrations. Due to their complexity, vibrations in multi-story planar frame structures are often analyzed and controlled either based on approximated discrete models such as lumped mass/elasticity models, or using numerical approach such as the finite element analysis (FEA) approach. In this paper, vibrations in multi-story planar frame structures are analyzed from the wave vibration standpoint, in which vibrations are described as waves that propagate along a uniform waveguide (or structural element), and are reflected and transmitted at discontinuities (such as joints and boundaries) [1–3]. Controllers are designed from the wave standpoint to suppress vibrations in the frames. Active wave vibration control is most often feedforward, in which the disturbance is detected, and a control force applied somewhere downstream to produce a destructive signal to cancel the incoming wave or to absorb the energy associated with it. Feedforward wave control has been applied in controlling both single wave type [4–7] and multiple wave types [8–11] of structural waves. Wave control can also be feedback, and feedback control has been designed in controlling vibrations in beams [12–14]. In this paper, feedback wave control is designed to control vibrations in multi-story planar frame structures. In the next section, the equations for describing vibrations in a planar frame structure are presented. In Section 3, controllers are designed for controlling the reflection and transmission characteristics of incoming vibration waves in a multi-story planar frame. In Section 4, numerical examples are presented. Conclusions and acknowledgements are given in Section 5 and 6, respectively.
n
Tel.: þ 1 313 593 5369; fax: þ 1 313 593 3851. E-mail address: [email protected]
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.06.022
C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
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2. Wave analysis and equations of motion in a multi-story planar frame Fig. 1 shows a multi-story frame that is symmetrical about a vertical line through the centers of the spans, which is normally the case in practice. The vibration modes are either symmetrical or anti-symmetrical. It has been shown that vibrating in symmetrical modes, the mid-points of the cross-members behave as sliding ends (that is, the bending slope, the shear force, and the longitudinal deflection are zero); while vibrating in anti-symmetrical modes, the mid-points of the cross-members behave as rolling ends (that is, the transverse deflection, the bending moment, and the longitudinal force are zero). As a result, when the frame is vibrating in a symmetrical mode, each half of it will have the same modal form as the isolated half-frame shown in Fig. 2(a); and when it is vibrating in one of its anti-symmetrical modes, each half of it can be treated as if it is an isolated half-frame having the form shown in Fig. 2(b) [15,16]. Regardless of the complexity of a structure, from the wave standpoint it consists of only structural elements and structural joints. Vibrations propagate along a uniform structural element, and are reflected and/or transmitted at the joints and boundaries. The number of wave components involved depends on a specific application. In the multi-story planar frame under consideration, there exist bending and axial vibration components in the structure, which are denoted using wave vectors a, A, and c. The half-frame model consists of n horizontal and n vertical beam elements. The discontinuities in the half-frame model include one ‘‘L’’ joint, (n 1) ‘‘T’’ joints, one classical boundary, and n sliding/rolling boundaries. Assembling these
L LH Fig. 1. A plane n-story frame.
+
an
A +n−1
−
+
an
A −n−1
+ a n−1
− a n−1
A i+ ai+
A i− a i−
+
A2 + a2
+
A1 + a1
+
A0
+
c Ln − c Ln
c Rn − c Rn
+ c L (n−1) −
c L (n−1)
+ c R (n−1) c −R (n−1)
c +Li c −Li
c +Ri c −Ri
−
+
an
A+n−1
A −n−1
a+n−1 A i+ ai+
+
A2 − a2
+
+
c L2 c −L2
c R2 − c R2
+ c L1 c −L1
+ c R1 −
−
A1 − a1
−
A0
A2 + a2
+
c R1
−
an
A1 + a1
+
A0
−
a n −1 Ai− ai−
+
c Ln c −Ln +
+
c Rn − c Rn +
c L (n−1) c −L (n−1)
cR (n−1) − cR (n−1)
c +Li c −Li
c +Ri c −Ri
−
A2 − a2
+
c L2 − c L2
+
cR2 − c R2
−
A1 − a1
−
+
c L1 − c L1
A0
Fig. 2. Half-frames for symmetrical (a) and anti-symmetrical and (b) mode analysis.
+
c R1 − c R1
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propagation, reflection, and transmission matrices offers a concise and systematic approach for analyzing coupled bending and longitudinal vibrations in a multi-story frame structure. The n pairs of propagation relations along the uniform vertical beam elements are þ aiþ ¼ fðLÞAi1 , A i1 ¼ fðLÞai , where i ¼ 1,2,:::,n
(1)
The n pairs of propagation relations along the uniform horizontal beam elements are þ ¼ fðLH =2ÞcLiþ , c cRi Li ¼ fðLH =2ÞcRi , where i ¼ 1,2,:::,n
(2)
The reflection and transmission relations of the waves at the ‘‘T’’ joints are þ þ þ þ a i ¼ r11 ai þ t31 Ai þt21 cLi , Ai ¼ r33 Ai þt13 ai þ t23 cLi , cLi ¼ r22 cLi þt12 ai þ t32 Ai , where i ¼ 1,2,:::,n1
(3)
The reflection and transmission relations of the waves at the ‘‘L’’ joint are þ þ ¼ R22 c cLn Ln þ T12 an ,
þ a n ¼ R11 an þ T21 cLn
(4)
The reflection and transmission relations of the waves at the sliding or rolling boundaries are þ c Ri ¼ rsliding cRi ,
þ or c Ri ¼ rrolling cRi
(5)
The reflection at the classical boundary is A0þ ¼ r0 A 0 ai7 ,
Ai7 ,
CLi7 ,
(6)
7 CRi
and are wave vectors as shown in Fig. 2, f(x) is the wave propagation matrix for a distance x, R and where T denote the reflection and transmission matrices at the L joints, and r and t denote the reflection and transmission matrices at the T joints. Subscripts 1, 2, and 3 denote parameters related to the involved beam elements at a joint. Writing the above equations in matrix form gives Az ¼ 0,
(7)
where A is a (24n) by (24n) coefficient matrix, and z is a 24n wave component vector. Setting the determinant of the coefficient matrix A to zero, that is 9A9 ¼ 0,
(8)
gives the natural frequencies of the multi-story frame. The detailed propagation, reflection, and transmission matrices are derived from the equations of motion, and the continuity and equilibrium conditions at a specific structural discontinuity, such as the ‘‘L’’ and ‘‘T’’ joint involved. Both bending and longitudinal vibrations exist in a multi-story planar structure due to wave mode conversion at the ‘‘L’’ and ‘‘T’’ joints. This paper concerns small magnitude vibrations, where linear vibration theories apply. Based on classical vibration theories, the equations of motion are as follows [17]: EI
@4 yðx,tÞ @2 yðx,tÞ þ rA ¼ qðx,tÞ, 4 @x @t 2
(9)
@2 uðx,tÞ @2 uðx,tÞ EA ¼ pðx,tÞ, @t 2 @x2
(10)
rA
where x is the position along the beam axis, t is time, y(x,t) and u(x,t) are the transverse and longitudinal deflections of the centerline of the beam; q(x,t) and p(x,t) are the externally applied transverse and longitudinal forces; E and r are Young’s modulus and mass density, respectively. I is the area moment of inertia of cross section and A is the cross-sectional area. Waves propagate along uniform waveguide and are reflected and transmitted at structural discontinuities. Details on free wave propagation and wave reflections at classical boundaries can be found in the Appendix, which also describes how the shear force V(x,t), the bending moment M(x,t), and the longitudinal force F(x,t) at any section of the beam are related to the transverse deflection y(x,t), the bending slope c(x,t), and the longitudinal deflection u(x,t) according to classical vibration theory [18].
3. Wave vibration control of classical multi-story planar frame structures Regardless of the complexity of a structure, from the wave point of view it consists of only two basic types of structural components, namely, joints/supports and structural elements, as shown in Fig. 3. As a result, vibrations can be controlled through controlling the joints/supports or controlling the structural elements.
C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
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Joint a Input
Element i
Element j
Input
Element k
Joint b Element l Input
Fig. 3. Joints and elements—the basic components of a complex structure.
Incident waves a+
b+ Transmitted waves
a− Reflected waves
w
KT2 Fc
Hw KR KT1 Fig. 4. (a) Feedback control system. (b) Wave scattering at a point support.
3.1. Structural element control In wave control along a structural element, the reflection and transmission of waves are controlled using a collocated sensor and actuator, as shown in Fig. 4(a). The wave control force can be written in frequency domain as Fc ¼ HðoÞw
(11)
where H(o) is the frequency response of the controller. It can be seen from Eq. (11) that a feedback wave controller is similar to the case of an attached spring, as shown in Fig. 4(b), with a dynamic translational/rotational stiffness KT/R ¼H(o). The dynamic stiffness may be frequency dependent and complex under this circumstance. The propagating wave, when incident upon a discontinuity, will be partially transmitted and partially reflected. Denoting the incident propagating wave amplitudes as vector a þ , the transmitted and reflected wave amplitudes b þ and a can then be found in terms of the transmission and reflection matrices t and r as b
þ
¼ ta þ ,
a ¼ ra þ
(12)
The transmission and reflection matrices depend on the type of the discontinuity. For a given discontinuity, both the transmission and reflection matrices can be found through the particular equilibrium and continuity conditions. The size of the vectors depends on the wave types and the number of wave components that exists in the structure. A number of control strategies can be adopted in feedback wave control design [12,19,20]. For example a strategy may aim either to maximize the energy absorbed by the controller or isolate the vibrations (that is, stopping energy transmission from one part of the structure to another). Two examples will now be considered. The first involves adding optimal damping using velocity feedback, and the second involves maximizing the energy absorbed by the controller. In the first example, H(o) is constrained to be purely imaginary and is given in the frequency domain as H(o) ¼ioc. Such an arrangement is to have an analogy to a ‘‘D’’ (Derivative) controller. This is because ‘‘io’’ is a differentiator in the frequency domain. When the controller is confined in the forms of H(o) ¼ioc, the control gain c is equivalent to the control gain of a ‘‘D’’ controller. Here the control gain c is designed to maximize the absorbed incoming vibration energy, in other 2 2 words, to minimize 9r11 9 þ 9t11 9 , which is found to be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 HðoÞ ¼ io ðrAÞ3 EIo2 (13) In the second example, the controller is designed to maximize the vibration energy absorbed by the controller without the restriction for H(o) to be purely imaginary. In this case, the controller is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 HðoÞ ¼ 2 ðrAÞ3 EIð1 þ iÞo3=2 (14)
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An optimal damping controller that is designed to maximize the absorbed incoming longitudinal vibration energy, that 2 2 is, to minimize 9r33 9 þ 9t33 9 , is found by following a similar procedure: pffiffiffiffiffiffi HðoÞ ¼ i2o rEA (15) The causality of the controllers is discussed at the end of Section 4. A practical structure, of course, seldom consists of a single element. A multi-story planar frame consists of ‘‘L’’ and ‘‘T’’ joints that hold the structural elements together. Since vibration power flows from one element to another via the structural joints, active joints are designed to control such vibration flow. 3.2. L-joint control It should be pointed out that wave transmission and reflection at an angle joint in general introduce wave mode conversion. At an ‘‘L’’ joint, for example, an incident bending wave induces reflected and transmitted bending and axial waves in the members attached to the joint. This is evident from the coupled equilibrium and continuity relations below. Fig. 5 shows the free body diagram of an ‘‘L’’ joint in planar motion. The equilibrium conditions are F2 V1 KT yJ ¼ my€ J V2 F1 KT uJ ¼ mu€ J M2 M1 þV1
h2 h1 € þV2 KR cJ ¼ J c J 2 2
(16)
where F is the axial force in the beam and h the beam thickness. Subscripts 1 and 2 refer to beam 1 and beam 2, and uJ, yJ, and cJ are the displacements and rotation of the joint as indicated in the figure. The first two of these equations include the mass of the joint, while the third includes the moment of inertia of the joint and the moments induced by the off-set shear forces. The continuity equations at the joint are u1 ¼ uJ ,
u2 ¼ yJ
h2 h1 y1 ¼ yJ cJ , y2 ¼ uJ þ c 2 2 J c1 ¼ cJ , c2 ¼ cJ
(17)
þ
A set of positive going waves a incident upon the L-joint from one beam gives rise to transmitted and reflected waves b þ and a , which are related to the incident waves through the transmission and reflection matrices T and R by b
þ
¼ Ta þ ,
a ¼ Ra þ
(18)
Let us consider the general situation, that is, beam 1 and 2 are of different materials and/or dimensions, and denote beam 1 and beam 2 related wavenumbers as k1,2,3 and ka,b,c, respectively, and the rest of beam 1 and beam 2 related physical parameters using subscripts 1 and 2. With an incident wave from beam 1, from the continuity conditions, one has 2 3 2 3 2 3 ika h22 kb h22 1 1 1 0 1 1 0 6 7 6 6 1 þika h1 1 þk h1 0 7b þ 6 0 1 7 0 1 7 (19) 4 0 5a ¼ 4 0 5a þ b 2 4 5 2 0 0 ik1 k2 ik1 k2 kb 0 ika
Beam 2 h2 Transmitted waves
b+ V2 (y2) M2 (2)
Incident waves
V1 (y1)
a+
KT
a−
h1
Reflected waves
KR
M1 (1)
J
F1 (u1)
uJ V1 (y1)
KT
F2 (u2)
V2 (y2)
yJ KTyJ
Fig. 5. L- joint control.
KTuJ KRJ
C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
The equilibrium conditions give 2 3 0 0 þ ikc ðEAÞ2 mo2 þKT 6 7 þ 3 3 ika ðEIÞ2 kb ðEIÞ2 0 6 7b 4 5 h1 h1 2 3 2 2 3 2 0 ka ðEIÞ2 ika ðEIÞ2 2 ika J o þika KR kb ðEIÞ2 þkb ðEIÞ2 2 kb Jo þ kb KR 2 3 k32 ðEIÞ1 0 ik31 ðEIÞ1 6 7 0 0 ik3 ðEAÞ1 þmo2 KT 7a 6 4 5 k21 ðEIÞ1 ik31 ðEIÞ1 h22 k22 ðEIÞ1 þ k32 ðEIÞ1 h22 0 2 3 k32 ðEIÞ1 0 ik31 ðEIÞ1 6 7 0 0 ik3 ðEAÞ1 þ mo2 KT 7a þ ¼6 4 5 h2 h2 2 3 2 3 k1 ðEIÞ1 þ ik1 ðEIÞ1 2 k2 ðEIÞ1 k2 ðEIÞ1 2 0
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(20)
The transmission and reflection matrices T12 and R11 can be obtained from solving Eqs. (18)–(20). It is evident that the reflection and transmission coefficients are KT and KR dependent, hence parameters KT and KR can be designed to control the vibration flow through ‘‘L’’ joint. 3.3. T-joint control Similarly, wave transmission and reflection at a ‘‘T’’ joint introduce wave mode conversion. The transmission and reflection matrices are obtained from considering the continuity and equilibrium conditions at the joint. The free body diagram of a ‘‘T’’ joint in planar motion is shown in Fig. 6. The continuity equations at the joint are u1 ¼ uJ , u2 ¼ yJ , u3 ¼ uJ y1 ¼ yJ
h2 h h c , y2 ¼ uJ þ 1 cJ , y3 ¼ yJ þ 3 cJ 2 J 2 2 c1 ¼ cJ , c2 ¼ cJ , c3 ¼ cJ
(21)
And the equilibrium conditions are V3 þ F2 V1 KT yJ ¼ my€ J F3 V2 F1 KT uJ ¼ mu€ J M3 þ M2 M1 þ V3
h2 h2 h1 € þ V1 þV2 KR cJ ¼ Jc J 2 2 2
(22)
Again let us consider the general situation, that is, beam 1, 2, and 3 are of different materials and/or dimensions, and denote beam 1, 2, and 3 related wavenumbers as k1,2,3, ka,b,c, and kA,B,C, respectively, and the rest of beam 1, 2, and 3 related physical parameters using subscripts 1, 2, and 3.
Fig. 6. T- joint control.
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C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
There exist three sets of reflection and transmission relations, corresponding to incident waves from each of the three beam elements, respectively. First, let us consider waves incident from beam 1. The set of positive going incident waves a þ gives rise to reflected waves a , and transmitted waves b þ and e þ , which are related to the incident waves through the reflection and transmission matrices by a ¼ r11 a þ ,
b
þ
¼ t12 a þ ,
e þ ¼ t13 a þ
(23)
The displacements of the beam elements are as follows: ik1 x1 k2 x1 y1 ¼ a1þ eik1 x1 þa2þ ek2 x1 þ a þ a 1e 2e
u1 ¼ c þ eik3 x1 þc eik3 x1
c1 ¼
ik1 a1þ eik1 x1 k2 a2þ ek2 x1
ik1 x1 k2 x1 þ ik1 a þk2 a 1e 2e
y2 ¼ b1þ eika x2 þ b2þ ekb x2 u2 ¼ d þ eikc x2
c2 ¼ ika b1þ eika x2 kb b2þ ekb x2 y3 ¼ e1þ eikA x3 þe2þ ekB x3 u3 ¼ g þ eikC x3
c3 ¼ ikA e1þ eikA x3 kB e2þ ekB x3
(24)
From Eqs. (3–5), (22), and (24), one obtains the equilibrium conditions related equations þ
Et13 e þ þ Et12 b Er11 a ¼ E1 a þ
(25)
where 2 Et13
2 Et12
6 6 ¼6 4
6 ¼6 4
ðEIÞ3 ik3A
ðEIÞ3 k3B
0
ðEIÞ3 k2A h21 ik3A
0
ðEIÞ3 k2B þ h22 k3B
0
ðEIÞ2
6 Er11 ¼ 6 4
6 E1 ¼ 6 4
ðEIÞ2
ðEIÞ1 ik31
ik31
ðEIÞ1 ik31
0
ðEAÞ3 ikC 7 7 5 0 ðEAÞ2 ikc mo2 þ KT
ðEIÞ1 k21 þ h22 ik31
0
0
ðEIÞ1 k22 þ
h2 2
k32
ðEIÞ1 k32
3
ðEIÞ2 k3b h1 3 2 kb þ 2 kb J o2 kb þ kb KR
ðEIÞ1 k32
0
ðEIÞ1 k21 h22
2
0
ðEIÞ2 ik3a 2 h1 ka 2 ik3a J o2 ika þ ika KR 2
0
0
ðEIÞ1 k22 h22 k32
0 0
3 7 7 7 5
3
mo2 ðEAÞ1 ik3 KT 7 7 5 0 0
3
mo þðEAÞ1 ik3 KT 7 7 5 0 2
From Eqs. (23) and (25), one has Et13 t13 þ Et12 t12 Er11 r11 ¼ E1
(26)
The continuity conditions in Eq. (21) can be grouped into two sets, one relating beam 1 and beam 2, and the other relating beam 1 and beam 3 h2 c 2 J h1 y2 ¼ u1 þ c 2 J c1 ¼ c2 ¼ cJ
y1 ¼ u2
(27a)
h2 þ h3 cJ 2 u1 ¼ u3
y3 y1 ¼
c1 ¼ c3 ¼ cJ
(27b)
C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
From Eqs. (24) and (27a), one has 2 kb h22 ika h22 6 6 1 þ ika h1 1þ k h1 b 2 4 2 kb ika
3
2 1 7 þ 6 7 0 5b 4 0 ik1 0 1
and from Eqs. (24) and (27b), one has 2 1þ ikA h2 þ2 h3 1 þ kB h2 þ2 h3 6 0 0 4 kB
ikA
1
2 1 7 þ 6 0 1 5e 4 ik1 0 0
6 4
1 þ ikA h2 þ2 h3
1 þkB h2 þ2 h3
3
0
0
ikA
kB
0
3
2
3
2
1
1
6 1 7 5a ¼ 4 0
0 k2
From Eqs. (23) and (28a), and (23) and (28b), one has 2 3 2 kb h22 1 ika h22 1 6 7 6 1þ ika h1 1þ k h1 0 7t12 6 0 4 b 4 5 2 2 ik1 kb 0 ika 2
0 0
0
k2
0
0
3
1
6 1 7 5r11 ¼ 4 0
0
k2
0
1
1
6 7 1 5t13 4 0 ik1 0
ik1
0
0 k2
3
2
3 0 17 5a þ
k2
1
(28a)
0
1 0
ik1
2
1 7 5a þ
0 k2
ik1
3
0
3 2 0 1 7 1 5a ¼ 6 4 0
1 0
1
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k2
(28b)
0
0
3
1 7 5
(29a)
0
3
1
1
0
6 17 5r11 ¼ 4 0 0 ik1
0 k2
17 5 0
(29b)
The reflection and transmission matrices r11, t12, and t13 can be obtained from Eqs. (26), (29a), and (29b), respectively. It can be seen that the reflection and transmission coefficients are the point attachments dependent, hence parameters KT and KR can also be designed to control the vibration flow through the T-joint. It shall be noted that though the same control strategies applied to structural element control can be applied to joint control, the controllers are much more involved due to the increased complexity in the reflection and transmission relations at a structural joint compared to those along a structural element. 4. Numerical examples The two-story frame discussed by Petyt [21] is chosen as the example structure. The physical properties of the two-story frame are: lengths of vertical and horizontal beams are L1 ¼22.86 cm and L2 ¼45.72 cm, respectively, the cross section of the beam elements is 0.3175 cm by 1.27 cm, Young’s modulus E is 206.84 GN/m2, and mass density r is 7830 kg/m3. Boundary conditions are clamped–clamped. It shall be pointed out that though specific numerical examples are used to demonstrate the control theory, the generality of the theory holds true for general structures subject to bending and/or longitudinal vibrations. 4.1. Structural element control As shown in Fig. 7, an external excitation force G is applied at a point that is L11 distance away from the boundary. An element control force Fc is applied at a point L11 þL22 distance away from the boundary. The responses are measured at two locations: location 1 (x1) is in between forces G and F; location 2 (x2) is on the top horizontal bar. In the numerical example, the parameters are chosen as: L11 ¼0.38L, L11 þL22 ¼0.68L, x1 ¼0.52L, and x2 ¼0.35LH. These parameters are kept the same for L- and T-joint control as well. The analysis of forced responses follows a similar procedure as the free vibration analysis described in the introduction section. The difference is that there are two additional discontinuities at the external force applied point and the control force applied point, which modifies the propagation relations on that beam element. Below describe the details. (a) At the external force application point G þ þ g12 g11 ¼ q þmf, g (30) 12 g11 ¼ q þ m þf 2 3 1 0 i 6 7 Q 6 7 M 6 7 F where q ¼ 4 1 5 4EIk 3 , m ¼ 4 1 5 4EIk2 , f ¼ 4 0 5 2EAk , and Q , M, and F are the external point transverse force, bending 3 1 1 0 i 0 moment, and point longitudinal force, respectively. Detailed derivations can be found in Ref. [18]. (b) At the control force application point Fc
2
3
2
3
þ
f 22 ¼ rc f 22 þ tc f 23 ,
þ
þ
f 23 ¼ rc f 23 þtc f 22
(31)
where rc and tc are the control force related reflection and transmission matrices, which are functions of KT1, KT2, and KR.
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C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
a +n
a −n
+
A1
a1
+
a1
+ f 23
− f 23
+
f 22
−
A1
Fc
c +Ln c −Ln
−
+
c L1 − c L1
L33
−
f 22
L22 G
+ g12
− g12
g 11
+
g 11
+
A0
L
−
L11
−
A0
Fig. 7. Forced responses in a planar multi-story frame.
(c) The propagation relations on the beam element with the applied external disturbance and control force are modified to þ ¼ fðL11 ÞA0þ , g11
A 0 ¼ fðL11 Þg11
þ þ f 22 ¼ fðL22 Þg12 , þ þ a1 ¼ fðL33 Þf 23 ,
g 12 ¼ fðL22 Þf 22
f 23 ¼ fðL33 Þa 1
(32)
Assembling all the propagation, reflection, and transmission relations allows one to find the forced vibrations of a multi-story frame without difficulty. In matrix form one has Af zf ¼ F
(33)
where Af is a squre coefficient matrix, zf is a wave component vector, and F is a vector describing the externally applied forces and moments. The deflection of any point in the frame can then be found. For example, the deflection at x1 is given by y1 ¼ ½1
1
þ 1fðx1 L11 Þg12 þ ½1
1
1fððx1 L11 ÞÞg 12
(34)
Similarly, the deflection at x2 is y2 ¼ ½1
1
þ 1fðx2 ÞcLn þ½1
1
1fðx2 Þc Ln
(35)
Fig. 8 shows the receptance (displacement per unit force) frequency responses of the frame with (solid lines) and without (dotted lines) control, in which KR is set to zero, and KT1 and KT2 are designed to damp out optimally the flexural and longitudinal vibration energies, respectively. 4.2. L-joint control In the L-joint control shown in Fig. 9, KT is set to zero, and KR is designed to damp out optimally the reflected bending and longitudinal vibration energies. Fig. 10 shows the receptance frequency responses of the frame before and after control.
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-180 0
200
400 600 800 1000 1200 Frequency (Hz)
Fig. 8. Receptance frequency responses of the frame before (y) and after (___) structural element control. (a) Vertical leg, anti-symmetrical modes, (b) vertical leg, symmetrical modes, (c) horizontal bar, anti-symmetrical modes, and (d) horizontal bar, symmetrical modes.
KT KR KT
Fig. 9. L-joint control.
4.3. T-joint control In the T-joint control shown in Fig. 11, KT is set to zero, and KR is designed to damp out optimally the reflected bending and longitudinal vibration energies. Fig. 12 shows the receptance frequency responses of the frame with (solid lines) and without (dotted lines) control. From the above analysis, it can be seen that both the feedback wave element and joint control are able to suppress vibrations in a multi-story frame structure effectively. The vibration modes are damped in a broad frequency band, the resonant peaks are much less sharp as a result.
C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
-20
-40
-40
-60
-60
Magnitude (dB)
Magnitude (dB)
5540
-80 -100 -120 -140 -160
-120 -140 -160 -180 -200 -220
-180 0
200
400 600 800 1000 1200 Frequency (Hz)
-20
-40
-40
-60 Magnitude (dB)
Magnitude (dB)
-80 -100
-60 -80 -100 -120
0
200
400 600 800 1000 1200 Frequency (Hz)
0
200
400 600 800 1000 1200 Frequency (Hz)
-80 -100 -120 -140 -160
-140
-180
-160 0
200
400 600 800 1000 1200 Frequency (Hz)
Fig. 10. Receptance frequency responses of the frame before (y) and after (___) L-joint control. (a) Vertical leg, anti-symmetrical modes, (b) vertical leg, symmetrical modes, (c) horizontal bar, anti-symmetrical modes, and (d) horizontal bar, symmetrical modes.
KT
KR
KT KT
KR
KT
Fig. 11. T-joint control.
In a practical implementation the causality of a controller must be guaranteed. A system is defined as causal if its impulse response h(t) ¼0 for t o0 [22,23]. The real part of the frequency response is the Hilbert transform of the imaginary part for a causal system. The optimal damping bending wave controller given by Eq. (13), for example, is purely imaginary and frequency dependent. It is therefore non-causal since, for a frequency dependent value, its Hilbert transform is nonzero. A causal controller that is an approximation to the ideal controller must be found for a practical implementation. One way of finding an approximate causal controller in a digital implementation is to truncate the non-causal part of the ideal controller. Another way is to tune the controller to be optimal at a certain frequency od, the controller given by Eq. (13),
-20
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-40
-60
-60
Magnitude (dB)
Magnitude (dB)
C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
-80 -100 -120 -140
-80 -100 -120 -140 -160 -180
-160
-200
-180
-220 0
200
400 600 800 1000 1200 Frequency (Hz)
-20
-40
-40
-60
-60
-80
Magnitude (dB)
Magnitude (dB)
5541
-80 -100 -120 -140
0
200
400 600 800 1000 1200 Frequency (Hz)
0
200
400 600 800 1000 1200 Frequency (Hz)
-100 -120 -140 -160
-160
-180 0
200
400 600 800 1000 1200 Frequency (Hz)
-20
-40
-40
-60
-60
-80
Magnitude (dB)
Magnitude (dB)
Fig. 12. Receptance frequency responses of the frame before (y) and after (___) T-joint control. (a) Vertical leg, anti-symmetrical modes, (b) vertical leg, symmetrical modes, (c) horizontal bar, anti-symmetrical modes, and (d) horizontal bar, symmetrical modes.
-80 -100 -120 -140 -160
-120 -140 -160 -180 -200
-180
-220 0
200
400 600 800 1000 1200 Frequency (Hz)
-20
-40
-40
-60 Magnitude (dB)
Magnitude (dB)
-100
-60 -80 -100 -120
0
200
400 600 800 1000 1200 Frequency (Hz)
0
200
400 600 800 1000 1200 Frequency (Hz)
-80 -100 -120 -140 -160
-140
-180
-160 0
200
400 600 800 1000 1200 Frequency (Hz)
Fig. 13. Receptance frequency responses of the frame before (y), after ideal (___), and after tuned (-.-.-.) structural element control. (a) Vertical leg, antisymmetrical modes, (b) vertical leg, symmetrical modes, (c) horizontal bar, anti-symmetrical modes, and (d) horizontal bar, symmetrical modes.
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for example, then becomes a constant c ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ðrAÞ3 EIo2d , which is definitely causal. Recall the receptance frequency
responses presented in Fig. 8, in which KR is set to zero, and KT1 and KT2 are designed to optimally damp out the flexural and longitudinal vibration energies, respectively. When tuning the non-causal controller KT1 to a causal ‘‘D’’ controller in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the form of io 4 ðrAÞ3 EIo2d , the controller can be easily implemented using a viscous damper with damping constant qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 4 ðrAÞ3 EIo2d in a practical application. Fig. 13 shows the receptance frequency responses of the frame before control, after ideal control, and after tuned causal control. In which the causal bending controller is tuned optimal at 200 Hz. It can be seen that the control performance of the tuned causal controller is the same as that of the ideal controller at the tuned optimal frequency od. The control performance at other frequencies is slightly compromised comparing to that of the ideal controller. However, the tuned controller still works very well in damping out the vibration resonances in the broad frequency band. 5. Conclusions In this paper, active wave control is designed based on classical vibration theory to suppress vibrations in multi-story planar frame structures. Regardless of the complexity of the structure, from the wave point of view it consists of only two basic types of structural components, namely, joints and structural elements. Consequently, controllers are designed both along structural elements and at structural joints. A number of control strategies have been adopted in feedback wave control design, such as adding optimal damping to the structure and maximizing the energy absorbed by the controller. Causality issues of the wave controllers are also addressed; two recommendations have been given for finding a causal approximation of a non-causal ideal controller. One way is to truncate the non-causal part of the ideal controller, the other is to tune the controller to be optimal at a certain frequency. Several numerical results are given, which include structural element control, L-joint control, and T-joint control. The numerical results show that after control, the resonances are much less sharp over the entire frequency range. It is evident that the wave controllers are very effective in suppressing vibrations in complex multi-story planar frame structures over a broad frequency band. It shall be noted that in-plane bending and longitudinal motions co-exist in in-plane vibrations of built-up planar frames due to wave-mode conversion at the joint(s). Neglecting either motion will result in inaccurate analysis. Should in-plane vibrations be not guaranteed, out-of-plane bending and torsional vibrations may also exist in the structure and will need to be addressed accordingly.
Acknowledgements The author gratefully acknowledges the support on this project from the Civil, Mechanical and Manufacturing Innovation Division of the National Science Foundation through Grant #0825761. Appendix A. Free wave propagation along a uniform waveguide and wave reflections at classical boundaries [18] A.1. Free wave propagation A.1.1. Bending vibrations First, consider the free bending vibration problem when no external force is applied to the beam. The differential equation of motion becomes EI
@4 yðx,tÞ @2 yðx,tÞ þ rA ¼0 @x4 @t 2
(A1)
Assuming time harmonic motion and using separation of variables, the solution to Eq. (A1) can be written in the form yðx,tÞ ¼ y0 eikx eiot , where o is the frequency and k is the wavenumber. Substituting this into Eq. (A1) gives a set of wavenumbers that are functions of the frequency o as well as the properties of the structure qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 k ¼ 7 rAo2 =EI (A2) The 7 sign outside the brackets indicates positive- and negative-going waves. With the time dependence eiot suppressed, the solutions to Eq. (A1) can be written as ik1 x k2 x yðxÞ ¼ a1þ eik1 x þ a2þ ek2 x þ a þa 1e 2e
(A3)
where the bending wavenumbers k1 ¼ k2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 rAo2 =EI
(A4)
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A.1.2. Longitudinal vibrations Now consider the free longitudinal vibration problem when no external force is applied to the beam. The differential equation for free longitudinal motion is
rA
@2 uðx,tÞ @2 uðx,tÞ EA ¼0 2 @t @x2
(A5)
Again assuming time harmonic motion and using separation of variables, the solution to Eq. (A5) can be written in the form uðx,tÞ ¼ u0 eikx eiot , where o is the frequency and k is the wavenumber. Substituting this into Eq. (10) gives the longitudinal wavenumber, which is a function of the frequency o as well rffiffiffiffi
r o
k¼ 7
(A6)
E
Similarly, the 7 sign indicates that longitudinal waves in the beam travel in both the positive and negative directions. With the time dependence eiot suppressed, the solutions to Eq. (A5) can be written as uðxÞ ¼ c þ eik3 x þ c eik3 x
(A7)
where the longitudinal wavenumber rffiffiffiffi k3 ¼
r o
(A8)
E
A.1.3. Propagation matrix Two points A and B on a uniform beam a distance x apart are considered. Waves propagate from one point to the other, with the propagation being determined by the appropriate wavenumber. Denoting the positive and negative going wave vectors at points A and B as a þ and a and b þ and b , respectively, they are related by
a ¼ fðxÞb ;
b
þ
¼ fðxÞa þ
(A9)
where 2 6 fðxÞ ¼ 4
0
0
0
ek2 x
0
0
0
eik3 x
is the propagation matrix for a distance x. Furthermore 2 þ3 2 3 a1 a1 6 7 6 7 a þ ¼ 4 a2þ 5, a ¼ 4 a 2 5, c
þ
3
eik1 x
c
2 b
þ
b1þ
7 5
(A10)
3
6 7 ¼ 4 b2þ 5, þ d
2
b 1
3
6 7 b ¼ 4 b 2 5 d
(A11)
A.2. Reflections at classical boundaries At a boundary, the incident waves a þ give rise to reflected waves a , which are related by a ¼ ra þ
(A12)
The reflection matrix r can be determined by considering equilibrium at the boundary, that is V ¼ EI ik1 x k2 x y ¼ a1þ eik1 x þ a2þ ek2 x þ a þa 1e 2e
where The reflection matrices are found as
@3 y @2 y @u , M ¼ EI 2 , F ¼ EA 3 @x @x @x
(A13)
and u ¼ c þ eik3 x1 þc eik3 x1 : 2
3 1 0 0 6 7 rs ¼ 4 0 1 0 5 0 0 1 2 3 i 1i 0 6 7 i 0 5 rc ¼ 4 1 þi 0 0 1 2 3 i 1 þ i 0 6 7 i 05 rf ¼ 4 1i 0 0 1 for simply supported, clamped, and free boundary conditions, respectively.
(A14)
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
K.F. Graff, Wave Motion in Elastic Solids, Ohio State University Press, 1975. L. Cremer, M. Heckl, E.E. Ungar, Structure-Borne Sound, Springer-Verlag, Berlin, 1987. J.F. Doyle, Wave Propagation in Structures, Spring –Verlag, New York, 1989. B.R. Mace, Active control of flexural vibrations, Journal of Sound and Vibration 114 (2) (1987) 253–270. J. Lu, M.J. Crocker, P.K. Raju, Active vibration control using wave control concepts, Journal of Sound and Vibration 134 (2) (1989) 364–368. S.J. Elliott, L. Billet, Adaptive control of flexural waves propagating in a beam, Journal of Sound and Vibration 163 (2) (1993) 295–310. J.S. Vipperman, R.A. Burdisso, C.R. Fuller, Active control of broadband structural vibration using the LMS adaptive algorithm, Journal of Sound and Vibration 166 (2) (1993) 283–299. C.R. Fuller, G.P. Gibbs, R.J. Silcox, Simultaneous active control of flexural and extensional waves in beams, Journal of Intelligence, Material, System and Structure 1 (1990) 235–247. R.L. Clark, J. Pan, C.H. Hansen, An experimental study of the active control of multiple-wave types in an elastic beam, Journal of Acoustic Society of America 92 (2) (1992) 871–876 Pt. 1. A.H. Von Flotow, Traveling wave control for large spacecraft structures, Journal of Guidance 9 (4) (1985) 462–468. P. Gardonio, S.J. Elliott, Active control of multiple waves propagating on a one-dimensional system with a scattering termination, Active 95 (1995) 115–126. B.R. Mace, R.W. Jones, Feedback control of flexural waves in beams, Journal of Structural Control 3 (1–2) (1996) 89–98. S.J. Elliott, M.J. Brennan, R.J. Pinnington, Feedback control of flexural waves on a beam, Proceedings of Inter-Noise 93, Leuven, Belgium, 1993, pp. 843–846. M.J. Brennan, Active Control of Waves on One-dimensional Structure, PhD Thesis, University of Southampton, 1994. G.M.L. Gladwell, The vibration of frames, Journal of Sound and Vibration 4 (1964) 402–465. R.E.D. Bishop, D.C. Johnson, The Mechanics of Vibration, Cambridge University Press, Great Britain, 1960. D.J. Inman, Engineering Vibrations, Prentice-Hall, Inc., New Jersey, 1994. C. Mei, In-plane vibrations of classical planar frame structures—an exact wave-based analytical solution, Journal of Vibration and Control 16 (9) (2010) 1265–1285. C. Mei, B.R. Mace, R.W. Jones, Hybrid wave/mode active vibration control, Journal of Sound and Vibration 247 (5) (2001) 765–784. C. Mei, The analysis and control of longitudinal vibrations from wave viewpoint, ASME Journal of Vibration and Acoustics 124 (2002) 645–649. M. Petyt, Introduction to Finite Element Vibration Analysis, Cambridge University Press, Great Britain, 1990. ¨ and Kjæl, 1984. Technical Review: Hilbert Transform, ISSN 007-2621, No. 3, Bruel J.N. Pandey, The Hilbert Transform of Schwartz Distribution and Applications, John Wiley and Sons, Inc., New York, 1995.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Wave control of vibrations in multi-story planar frame structures based on classical vibration theories C. Mei n Department of Mechanical Engineering, The University of Michigan - Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA
a r t i c l e in f o
abstract
Article history: Received 29 November 2010 Received in revised form 17 April 2011 Accepted 27 June 2011 Handling Editor: M.P. Cartmell Available online 22 July 2011
An active control approach to vibrations in multi-story planar frame structures is presented in this paper. The controller is designed from a wave vibration standpoint, in which vibrations are described as waves that propagate along uniform waveguides, and are reflected and transmitted upon structural discontinuities. Regardless of the complexity of a structure, from the wave point of view it consists of only two basic types of structural components, namely, structural elements and joints. In this paper, vibrations in a multi-story planar frame structure are controlled through controlling its structural elements and its joints. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Multi-story planar frame structures can be found in many fields of engineering, ranging from big scale high-rise buildings and towers to micro-frames used in modern electronic equipment. These frame structures are often subject to harmful vibrations. Due to their complexity, vibrations in multi-story planar frame structures are often analyzed and controlled either based on approximated discrete models such as lumped mass/elasticity models, or using numerical approach such as the finite element analysis (FEA) approach. In this paper, vibrations in multi-story planar frame structures are analyzed from the wave vibration standpoint, in which vibrations are described as waves that propagate along a uniform waveguide (or structural element), and are reflected and transmitted at discontinuities (such as joints and boundaries) [1–3]. Controllers are designed from the wave standpoint to suppress vibrations in the frames. Active wave vibration control is most often feedforward, in which the disturbance is detected, and a control force applied somewhere downstream to produce a destructive signal to cancel the incoming wave or to absorb the energy associated with it. Feedforward wave control has been applied in controlling both single wave type [4–7] and multiple wave types [8–11] of structural waves. Wave control can also be feedback, and feedback control has been designed in controlling vibrations in beams [12–14]. In this paper, feedback wave control is designed to control vibrations in multi-story planar frame structures. In the next section, the equations for describing vibrations in a planar frame structure are presented. In Section 3, controllers are designed for controlling the reflection and transmission characteristics of incoming vibration waves in a multi-story planar frame. In Section 4, numerical examples are presented. Conclusions and acknowledgements are given in Section 5 and 6, respectively.
n
Tel.: þ 1 313 593 5369; fax: þ 1 313 593 3851. E-mail address: [email protected]
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.06.022
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2. Wave analysis and equations of motion in a multi-story planar frame Fig. 1 shows a multi-story frame that is symmetrical about a vertical line through the centers of the spans, which is normally the case in practice. The vibration modes are either symmetrical or anti-symmetrical. It has been shown that vibrating in symmetrical modes, the mid-points of the cross-members behave as sliding ends (that is, the bending slope, the shear force, and the longitudinal deflection are zero); while vibrating in anti-symmetrical modes, the mid-points of the cross-members behave as rolling ends (that is, the transverse deflection, the bending moment, and the longitudinal force are zero). As a result, when the frame is vibrating in a symmetrical mode, each half of it will have the same modal form as the isolated half-frame shown in Fig. 2(a); and when it is vibrating in one of its anti-symmetrical modes, each half of it can be treated as if it is an isolated half-frame having the form shown in Fig. 2(b) [15,16]. Regardless of the complexity of a structure, from the wave standpoint it consists of only structural elements and structural joints. Vibrations propagate along a uniform structural element, and are reflected and/or transmitted at the joints and boundaries. The number of wave components involved depends on a specific application. In the multi-story planar frame under consideration, there exist bending and axial vibration components in the structure, which are denoted using wave vectors a, A, and c. The half-frame model consists of n horizontal and n vertical beam elements. The discontinuities in the half-frame model include one ‘‘L’’ joint, (n 1) ‘‘T’’ joints, one classical boundary, and n sliding/rolling boundaries. Assembling these
L LH Fig. 1. A plane n-story frame.
+
an
A +n−1
−
+
an
A −n−1
+ a n−1
− a n−1
A i+ ai+
A i− a i−
+
A2 + a2
+
A1 + a1
+
A0
+
c Ln − c Ln
c Rn − c Rn
+ c L (n−1) −
c L (n−1)
+ c R (n−1) c −R (n−1)
c +Li c −Li
c +Ri c −Ri
−
+
an
A+n−1
A −n−1
a+n−1 A i+ ai+
+
A2 − a2
+
+
c L2 c −L2
c R2 − c R2
+ c L1 c −L1
+ c R1 −
−
A1 − a1
−
A0
A2 + a2
+
c R1
−
an
A1 + a1
+
A0
−
a n −1 Ai− ai−
+
c Ln c −Ln +
+
c Rn − c Rn +
c L (n−1) c −L (n−1)
cR (n−1) − cR (n−1)
c +Li c −Li
c +Ri c −Ri
−
A2 − a2
+
c L2 − c L2
+
cR2 − c R2
−
A1 − a1
−
+
c L1 − c L1
A0
Fig. 2. Half-frames for symmetrical (a) and anti-symmetrical and (b) mode analysis.
+
c R1 − c R1
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propagation, reflection, and transmission matrices offers a concise and systematic approach for analyzing coupled bending and longitudinal vibrations in a multi-story frame structure. The n pairs of propagation relations along the uniform vertical beam elements are þ aiþ ¼ fðLÞAi1 , A i1 ¼ fðLÞai , where i ¼ 1,2,:::,n
(1)
The n pairs of propagation relations along the uniform horizontal beam elements are þ ¼ fðLH =2ÞcLiþ , c cRi Li ¼ fðLH =2ÞcRi , where i ¼ 1,2,:::,n
(2)
The reflection and transmission relations of the waves at the ‘‘T’’ joints are þ þ þ þ a i ¼ r11 ai þ t31 Ai þt21 cLi , Ai ¼ r33 Ai þt13 ai þ t23 cLi , cLi ¼ r22 cLi þt12 ai þ t32 Ai , where i ¼ 1,2,:::,n1
(3)
The reflection and transmission relations of the waves at the ‘‘L’’ joint are þ þ ¼ R22 c cLn Ln þ T12 an ,
þ a n ¼ R11 an þ T21 cLn
(4)
The reflection and transmission relations of the waves at the sliding or rolling boundaries are þ c Ri ¼ rsliding cRi ,
þ or c Ri ¼ rrolling cRi
(5)
The reflection at the classical boundary is A0þ ¼ r0 A 0 ai7 ,
Ai7 ,
CLi7 ,
(6)
7 CRi
and are wave vectors as shown in Fig. 2, f(x) is the wave propagation matrix for a distance x, R and where T denote the reflection and transmission matrices at the L joints, and r and t denote the reflection and transmission matrices at the T joints. Subscripts 1, 2, and 3 denote parameters related to the involved beam elements at a joint. Writing the above equations in matrix form gives Az ¼ 0,
(7)
where A is a (24n) by (24n) coefficient matrix, and z is a 24n wave component vector. Setting the determinant of the coefficient matrix A to zero, that is 9A9 ¼ 0,
(8)
gives the natural frequencies of the multi-story frame. The detailed propagation, reflection, and transmission matrices are derived from the equations of motion, and the continuity and equilibrium conditions at a specific structural discontinuity, such as the ‘‘L’’ and ‘‘T’’ joint involved. Both bending and longitudinal vibrations exist in a multi-story planar structure due to wave mode conversion at the ‘‘L’’ and ‘‘T’’ joints. This paper concerns small magnitude vibrations, where linear vibration theories apply. Based on classical vibration theories, the equations of motion are as follows [17]: EI
@4 yðx,tÞ @2 yðx,tÞ þ rA ¼ qðx,tÞ, 4 @x @t 2
(9)
@2 uðx,tÞ @2 uðx,tÞ EA ¼ pðx,tÞ, @t 2 @x2
(10)
rA
where x is the position along the beam axis, t is time, y(x,t) and u(x,t) are the transverse and longitudinal deflections of the centerline of the beam; q(x,t) and p(x,t) are the externally applied transverse and longitudinal forces; E and r are Young’s modulus and mass density, respectively. I is the area moment of inertia of cross section and A is the cross-sectional area. Waves propagate along uniform waveguide and are reflected and transmitted at structural discontinuities. Details on free wave propagation and wave reflections at classical boundaries can be found in the Appendix, which also describes how the shear force V(x,t), the bending moment M(x,t), and the longitudinal force F(x,t) at any section of the beam are related to the transverse deflection y(x,t), the bending slope c(x,t), and the longitudinal deflection u(x,t) according to classical vibration theory [18].
3. Wave vibration control of classical multi-story planar frame structures Regardless of the complexity of a structure, from the wave point of view it consists of only two basic types of structural components, namely, joints/supports and structural elements, as shown in Fig. 3. As a result, vibrations can be controlled through controlling the joints/supports or controlling the structural elements.
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Joint a Input
Element i
Element j
Input
Element k
Joint b Element l Input
Fig. 3. Joints and elements—the basic components of a complex structure.
Incident waves a+
b+ Transmitted waves
a− Reflected waves
w
KT2 Fc
Hw KR KT1 Fig. 4. (a) Feedback control system. (b) Wave scattering at a point support.
3.1. Structural element control In wave control along a structural element, the reflection and transmission of waves are controlled using a collocated sensor and actuator, as shown in Fig. 4(a). The wave control force can be written in frequency domain as Fc ¼ HðoÞw
(11)
where H(o) is the frequency response of the controller. It can be seen from Eq. (11) that a feedback wave controller is similar to the case of an attached spring, as shown in Fig. 4(b), with a dynamic translational/rotational stiffness KT/R ¼H(o). The dynamic stiffness may be frequency dependent and complex under this circumstance. The propagating wave, when incident upon a discontinuity, will be partially transmitted and partially reflected. Denoting the incident propagating wave amplitudes as vector a þ , the transmitted and reflected wave amplitudes b þ and a can then be found in terms of the transmission and reflection matrices t and r as b
þ
¼ ta þ ,
a ¼ ra þ
(12)
The transmission and reflection matrices depend on the type of the discontinuity. For a given discontinuity, both the transmission and reflection matrices can be found through the particular equilibrium and continuity conditions. The size of the vectors depends on the wave types and the number of wave components that exists in the structure. A number of control strategies can be adopted in feedback wave control design [12,19,20]. For example a strategy may aim either to maximize the energy absorbed by the controller or isolate the vibrations (that is, stopping energy transmission from one part of the structure to another). Two examples will now be considered. The first involves adding optimal damping using velocity feedback, and the second involves maximizing the energy absorbed by the controller. In the first example, H(o) is constrained to be purely imaginary and is given in the frequency domain as H(o) ¼ioc. Such an arrangement is to have an analogy to a ‘‘D’’ (Derivative) controller. This is because ‘‘io’’ is a differentiator in the frequency domain. When the controller is confined in the forms of H(o) ¼ioc, the control gain c is equivalent to the control gain of a ‘‘D’’ controller. Here the control gain c is designed to maximize the absorbed incoming vibration energy, in other 2 2 words, to minimize 9r11 9 þ 9t11 9 , which is found to be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 HðoÞ ¼ io ðrAÞ3 EIo2 (13) In the second example, the controller is designed to maximize the vibration energy absorbed by the controller without the restriction for H(o) to be purely imaginary. In this case, the controller is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 HðoÞ ¼ 2 ðrAÞ3 EIð1 þ iÞo3=2 (14)
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An optimal damping controller that is designed to maximize the absorbed incoming longitudinal vibration energy, that 2 2 is, to minimize 9r33 9 þ 9t33 9 , is found by following a similar procedure: pffiffiffiffiffiffi HðoÞ ¼ i2o rEA (15) The causality of the controllers is discussed at the end of Section 4. A practical structure, of course, seldom consists of a single element. A multi-story planar frame consists of ‘‘L’’ and ‘‘T’’ joints that hold the structural elements together. Since vibration power flows from one element to another via the structural joints, active joints are designed to control such vibration flow. 3.2. L-joint control It should be pointed out that wave transmission and reflection at an angle joint in general introduce wave mode conversion. At an ‘‘L’’ joint, for example, an incident bending wave induces reflected and transmitted bending and axial waves in the members attached to the joint. This is evident from the coupled equilibrium and continuity relations below. Fig. 5 shows the free body diagram of an ‘‘L’’ joint in planar motion. The equilibrium conditions are F2 V1 KT yJ ¼ my€ J V2 F1 KT uJ ¼ mu€ J M2 M1 þV1
h2 h1 € þV2 KR cJ ¼ J c J 2 2
(16)
where F is the axial force in the beam and h the beam thickness. Subscripts 1 and 2 refer to beam 1 and beam 2, and uJ, yJ, and cJ are the displacements and rotation of the joint as indicated in the figure. The first two of these equations include the mass of the joint, while the third includes the moment of inertia of the joint and the moments induced by the off-set shear forces. The continuity equations at the joint are u1 ¼ uJ ,
u2 ¼ yJ
h2 h1 y1 ¼ yJ cJ , y2 ¼ uJ þ c 2 2 J c1 ¼ cJ , c2 ¼ cJ
(17)
þ
A set of positive going waves a incident upon the L-joint from one beam gives rise to transmitted and reflected waves b þ and a , which are related to the incident waves through the transmission and reflection matrices T and R by b
þ
¼ Ta þ ,
a ¼ Ra þ
(18)
Let us consider the general situation, that is, beam 1 and 2 are of different materials and/or dimensions, and denote beam 1 and beam 2 related wavenumbers as k1,2,3 and ka,b,c, respectively, and the rest of beam 1 and beam 2 related physical parameters using subscripts 1 and 2. With an incident wave from beam 1, from the continuity conditions, one has 2 3 2 3 2 3 ika h22 kb h22 1 1 1 0 1 1 0 6 7 6 6 1 þika h1 1 þk h1 0 7b þ 6 0 1 7 0 1 7 (19) 4 0 5a ¼ 4 0 5a þ b 2 4 5 2 0 0 ik1 k2 ik1 k2 kb 0 ika
Beam 2 h2 Transmitted waves
b+ V2 (y2) M2 (2)
Incident waves
V1 (y1)
a+
KT
a−
h1
Reflected waves
KR
M1 (1)
J
F1 (u1)
uJ V1 (y1)
KT
F2 (u2)
V2 (y2)
yJ KTyJ
Fig. 5. L- joint control.
KTuJ KRJ
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The equilibrium conditions give 2 3 0 0 þ ikc ðEAÞ2 mo2 þKT 6 7 þ 3 3 ika ðEIÞ2 kb ðEIÞ2 0 6 7b 4 5 h1 h1 2 3 2 2 3 2 0 ka ðEIÞ2 ika ðEIÞ2 2 ika J o þika KR kb ðEIÞ2 þkb ðEIÞ2 2 kb Jo þ kb KR 2 3 k32 ðEIÞ1 0 ik31 ðEIÞ1 6 7 0 0 ik3 ðEAÞ1 þmo2 KT 7a 6 4 5 k21 ðEIÞ1 ik31 ðEIÞ1 h22 k22 ðEIÞ1 þ k32 ðEIÞ1 h22 0 2 3 k32 ðEIÞ1 0 ik31 ðEIÞ1 6 7 0 0 ik3 ðEAÞ1 þ mo2 KT 7a þ ¼6 4 5 h2 h2 2 3 2 3 k1 ðEIÞ1 þ ik1 ðEIÞ1 2 k2 ðEIÞ1 k2 ðEIÞ1 2 0
5535
(20)
The transmission and reflection matrices T12 and R11 can be obtained from solving Eqs. (18)–(20). It is evident that the reflection and transmission coefficients are KT and KR dependent, hence parameters KT and KR can be designed to control the vibration flow through ‘‘L’’ joint. 3.3. T-joint control Similarly, wave transmission and reflection at a ‘‘T’’ joint introduce wave mode conversion. The transmission and reflection matrices are obtained from considering the continuity and equilibrium conditions at the joint. The free body diagram of a ‘‘T’’ joint in planar motion is shown in Fig. 6. The continuity equations at the joint are u1 ¼ uJ , u2 ¼ yJ , u3 ¼ uJ y1 ¼ yJ
h2 h h c , y2 ¼ uJ þ 1 cJ , y3 ¼ yJ þ 3 cJ 2 J 2 2 c1 ¼ cJ , c2 ¼ cJ , c3 ¼ cJ
(21)
And the equilibrium conditions are V3 þ F2 V1 KT yJ ¼ my€ J F3 V2 F1 KT uJ ¼ mu€ J M3 þ M2 M1 þ V3
h2 h2 h1 € þ V1 þV2 KR cJ ¼ Jc J 2 2 2
(22)
Again let us consider the general situation, that is, beam 1, 2, and 3 are of different materials and/or dimensions, and denote beam 1, 2, and 3 related wavenumbers as k1,2,3, ka,b,c, and kA,B,C, respectively, and the rest of beam 1, 2, and 3 related physical parameters using subscripts 1, 2, and 3.
Fig. 6. T- joint control.
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There exist three sets of reflection and transmission relations, corresponding to incident waves from each of the three beam elements, respectively. First, let us consider waves incident from beam 1. The set of positive going incident waves a þ gives rise to reflected waves a , and transmitted waves b þ and e þ , which are related to the incident waves through the reflection and transmission matrices by a ¼ r11 a þ ,
b
þ
¼ t12 a þ ,
e þ ¼ t13 a þ
(23)
The displacements of the beam elements are as follows: ik1 x1 k2 x1 y1 ¼ a1þ eik1 x1 þa2þ ek2 x1 þ a þ a 1e 2e
u1 ¼ c þ eik3 x1 þc eik3 x1
c1 ¼
ik1 a1þ eik1 x1 k2 a2þ ek2 x1
ik1 x1 k2 x1 þ ik1 a þk2 a 1e 2e
y2 ¼ b1þ eika x2 þ b2þ ekb x2 u2 ¼ d þ eikc x2
c2 ¼ ika b1þ eika x2 kb b2þ ekb x2 y3 ¼ e1þ eikA x3 þe2þ ekB x3 u3 ¼ g þ eikC x3
c3 ¼ ikA e1þ eikA x3 kB e2þ ekB x3
(24)
From Eqs. (3–5), (22), and (24), one obtains the equilibrium conditions related equations þ
Et13 e þ þ Et12 b Er11 a ¼ E1 a þ
(25)
where 2 Et13
2 Et12
6 6 ¼6 4
6 ¼6 4
ðEIÞ3 ik3A
ðEIÞ3 k3B
0
ðEIÞ3 k2A h21 ik3A
0
ðEIÞ3 k2B þ h22 k3B
0
ðEIÞ2
6 Er11 ¼ 6 4
6 E1 ¼ 6 4
ðEIÞ2
ðEIÞ1 ik31
ik31
ðEIÞ1 ik31
0
ðEAÞ3 ikC 7 7 5 0 ðEAÞ2 ikc mo2 þ KT
ðEIÞ1 k21 þ h22 ik31
0
0
ðEIÞ1 k22 þ
h2 2
k32
ðEIÞ1 k32
3
ðEIÞ2 k3b h1 3 2 kb þ 2 kb J o2 kb þ kb KR
ðEIÞ1 k32
0
ðEIÞ1 k21 h22
2
0
ðEIÞ2 ik3a 2 h1 ka 2 ik3a J o2 ika þ ika KR 2
0
0
ðEIÞ1 k22 h22 k32
0 0
3 7 7 7 5
3
mo2 ðEAÞ1 ik3 KT 7 7 5 0 0
3
mo þðEAÞ1 ik3 KT 7 7 5 0 2
From Eqs. (23) and (25), one has Et13 t13 þ Et12 t12 Er11 r11 ¼ E1
(26)
The continuity conditions in Eq. (21) can be grouped into two sets, one relating beam 1 and beam 2, and the other relating beam 1 and beam 3 h2 c 2 J h1 y2 ¼ u1 þ c 2 J c1 ¼ c2 ¼ cJ
y1 ¼ u2
(27a)
h2 þ h3 cJ 2 u1 ¼ u3
y3 y1 ¼
c1 ¼ c3 ¼ cJ
(27b)
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From Eqs. (24) and (27a), one has 2 kb h22 ika h22 6 6 1 þ ika h1 1þ k h1 b 2 4 2 kb ika
3
2 1 7 þ 6 7 0 5b 4 0 ik1 0 1
and from Eqs. (24) and (27b), one has 2 1þ ikA h2 þ2 h3 1 þ kB h2 þ2 h3 6 0 0 4 kB
ikA
1
2 1 7 þ 6 0 1 5e 4 ik1 0 0
6 4
1 þ ikA h2 þ2 h3
1 þkB h2 þ2 h3
3
0
0
ikA
kB
0
3
2
3
2
1
1
6 1 7 5a ¼ 4 0
0 k2
From Eqs. (23) and (28a), and (23) and (28b), one has 2 3 2 kb h22 1 ika h22 1 6 7 6 1þ ika h1 1þ k h1 0 7t12 6 0 4 b 4 5 2 2 ik1 kb 0 ika 2
0 0
0
k2
0
0
3
1
6 1 7 5r11 ¼ 4 0
0
k2
0
1
1
6 7 1 5t13 4 0 ik1 0
ik1
0
0 k2
3
2
3 0 17 5a þ
k2
1
(28a)
0
1 0
ik1
2
1 7 5a þ
0 k2
ik1
3
0
3 2 0 1 7 1 5a ¼ 6 4 0
1 0
1
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k2
(28b)
0
0
3
1 7 5
(29a)
0
3
1
1
0
6 17 5r11 ¼ 4 0 0 ik1
0 k2
17 5 0
(29b)
The reflection and transmission matrices r11, t12, and t13 can be obtained from Eqs. (26), (29a), and (29b), respectively. It can be seen that the reflection and transmission coefficients are the point attachments dependent, hence parameters KT and KR can also be designed to control the vibration flow through the T-joint. It shall be noted that though the same control strategies applied to structural element control can be applied to joint control, the controllers are much more involved due to the increased complexity in the reflection and transmission relations at a structural joint compared to those along a structural element. 4. Numerical examples The two-story frame discussed by Petyt [21] is chosen as the example structure. The physical properties of the two-story frame are: lengths of vertical and horizontal beams are L1 ¼22.86 cm and L2 ¼45.72 cm, respectively, the cross section of the beam elements is 0.3175 cm by 1.27 cm, Young’s modulus E is 206.84 GN/m2, and mass density r is 7830 kg/m3. Boundary conditions are clamped–clamped. It shall be pointed out that though specific numerical examples are used to demonstrate the control theory, the generality of the theory holds true for general structures subject to bending and/or longitudinal vibrations. 4.1. Structural element control As shown in Fig. 7, an external excitation force G is applied at a point that is L11 distance away from the boundary. An element control force Fc is applied at a point L11 þL22 distance away from the boundary. The responses are measured at two locations: location 1 (x1) is in between forces G and F; location 2 (x2) is on the top horizontal bar. In the numerical example, the parameters are chosen as: L11 ¼0.38L, L11 þL22 ¼0.68L, x1 ¼0.52L, and x2 ¼0.35LH. These parameters are kept the same for L- and T-joint control as well. The analysis of forced responses follows a similar procedure as the free vibration analysis described in the introduction section. The difference is that there are two additional discontinuities at the external force applied point and the control force applied point, which modifies the propagation relations on that beam element. Below describe the details. (a) At the external force application point G þ þ g12 g11 ¼ q þmf, g (30) 12 g11 ¼ q þ m þf 2 3 1 0 i 6 7 Q 6 7 M 6 7 F where q ¼ 4 1 5 4EIk 3 , m ¼ 4 1 5 4EIk2 , f ¼ 4 0 5 2EAk , and Q , M, and F are the external point transverse force, bending 3 1 1 0 i 0 moment, and point longitudinal force, respectively. Detailed derivations can be found in Ref. [18]. (b) At the control force application point Fc
2
3
2
3
þ
f 22 ¼ rc f 22 þ tc f 23 ,
þ
þ
f 23 ¼ rc f 23 þtc f 22
(31)
where rc and tc are the control force related reflection and transmission matrices, which are functions of KT1, KT2, and KR.
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a +n
a −n
+
A1
a1
+
a1
+ f 23
− f 23
+
f 22
−
A1
Fc
c +Ln c −Ln
−
+
c L1 − c L1
L33
−
f 22
L22 G
+ g12
− g12
g 11
+
g 11
+
A0
L
−
L11
−
A0
Fig. 7. Forced responses in a planar multi-story frame.
(c) The propagation relations on the beam element with the applied external disturbance and control force are modified to þ ¼ fðL11 ÞA0þ , g11
A 0 ¼ fðL11 Þg11
þ þ f 22 ¼ fðL22 Þg12 , þ þ a1 ¼ fðL33 Þf 23 ,
g 12 ¼ fðL22 Þf 22
f 23 ¼ fðL33 Þa 1
(32)
Assembling all the propagation, reflection, and transmission relations allows one to find the forced vibrations of a multi-story frame without difficulty. In matrix form one has Af zf ¼ F
(33)
where Af is a squre coefficient matrix, zf is a wave component vector, and F is a vector describing the externally applied forces and moments. The deflection of any point in the frame can then be found. For example, the deflection at x1 is given by y1 ¼ ½1
1
þ 1fðx1 L11 Þg12 þ ½1
1
1fððx1 L11 ÞÞg 12
(34)
Similarly, the deflection at x2 is y2 ¼ ½1
1
þ 1fðx2 ÞcLn þ½1
1
1fðx2 Þc Ln
(35)
Fig. 8 shows the receptance (displacement per unit force) frequency responses of the frame with (solid lines) and without (dotted lines) control, in which KR is set to zero, and KT1 and KT2 are designed to damp out optimally the flexural and longitudinal vibration energies, respectively. 4.2. L-joint control In the L-joint control shown in Fig. 9, KT is set to zero, and KR is designed to damp out optimally the reflected bending and longitudinal vibration energies. Fig. 10 shows the receptance frequency responses of the frame before and after control.
-20
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-40
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-60
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Magnitude (dB)
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-80 -100 -120 -140 -160 -180 -200 -220
-180 0
200
400 600 800 1000 1200 Frequency (Hz)
-20
-40
-40
-60
-60
Magnitude (dB)
Magnitude (dB)
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-80 -100 -120 -140
0
200
400 600 800 1000 1200 Frequency (Hz)
0
200
400 600 800 1000 1200 Frequency (Hz)
-80 -100 -120 -140 -160
-160
-180 0
200
400 600 800 1000 1200 Frequency (Hz)
Fig. 8. Receptance frequency responses of the frame before (y) and after (___) structural element control. (a) Vertical leg, anti-symmetrical modes, (b) vertical leg, symmetrical modes, (c) horizontal bar, anti-symmetrical modes, and (d) horizontal bar, symmetrical modes.
KT KR KT
Fig. 9. L-joint control.
4.3. T-joint control In the T-joint control shown in Fig. 11, KT is set to zero, and KR is designed to damp out optimally the reflected bending and longitudinal vibration energies. Fig. 12 shows the receptance frequency responses of the frame with (solid lines) and without (dotted lines) control. From the above analysis, it can be seen that both the feedback wave element and joint control are able to suppress vibrations in a multi-story frame structure effectively. The vibration modes are damped in a broad frequency band, the resonant peaks are much less sharp as a result.
C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
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-40
-40
-60
-60
Magnitude (dB)
Magnitude (dB)
5540
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-120 -140 -160 -180 -200 -220
-180 0
200
400 600 800 1000 1200 Frequency (Hz)
-20
-40
-40
-60 Magnitude (dB)
Magnitude (dB)
-80 -100
-60 -80 -100 -120
0
200
400 600 800 1000 1200 Frequency (Hz)
0
200
400 600 800 1000 1200 Frequency (Hz)
-80 -100 -120 -140 -160
-140
-180
-160 0
200
400 600 800 1000 1200 Frequency (Hz)
Fig. 10. Receptance frequency responses of the frame before (y) and after (___) L-joint control. (a) Vertical leg, anti-symmetrical modes, (b) vertical leg, symmetrical modes, (c) horizontal bar, anti-symmetrical modes, and (d) horizontal bar, symmetrical modes.
KT
KR
KT KT
KR
KT
Fig. 11. T-joint control.
In a practical implementation the causality of a controller must be guaranteed. A system is defined as causal if its impulse response h(t) ¼0 for t o0 [22,23]. The real part of the frequency response is the Hilbert transform of the imaginary part for a causal system. The optimal damping bending wave controller given by Eq. (13), for example, is purely imaginary and frequency dependent. It is therefore non-causal since, for a frequency dependent value, its Hilbert transform is nonzero. A causal controller that is an approximation to the ideal controller must be found for a practical implementation. One way of finding an approximate causal controller in a digital implementation is to truncate the non-causal part of the ideal controller. Another way is to tune the controller to be optimal at a certain frequency od, the controller given by Eq. (13),
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-80 -100 -120 -140 -160 -180
-160
-200
-180
-220 0
200
400 600 800 1000 1200 Frequency (Hz)
-20
-40
-40
-60
-60
-80
Magnitude (dB)
Magnitude (dB)
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-80 -100 -120 -140
0
200
400 600 800 1000 1200 Frequency (Hz)
0
200
400 600 800 1000 1200 Frequency (Hz)
-100 -120 -140 -160
-160
-180 0
200
400 600 800 1000 1200 Frequency (Hz)
-20
-40
-40
-60
-60
-80
Magnitude (dB)
Magnitude (dB)
Fig. 12. Receptance frequency responses of the frame before (y) and after (___) T-joint control. (a) Vertical leg, anti-symmetrical modes, (b) vertical leg, symmetrical modes, (c) horizontal bar, anti-symmetrical modes, and (d) horizontal bar, symmetrical modes.
-80 -100 -120 -140 -160
-120 -140 -160 -180 -200
-180
-220 0
200
400 600 800 1000 1200 Frequency (Hz)
-20
-40
-40
-60 Magnitude (dB)
Magnitude (dB)
-100
-60 -80 -100 -120
0
200
400 600 800 1000 1200 Frequency (Hz)
0
200
400 600 800 1000 1200 Frequency (Hz)
-80 -100 -120 -140 -160
-140
-180
-160 0
200
400 600 800 1000 1200 Frequency (Hz)
Fig. 13. Receptance frequency responses of the frame before (y), after ideal (___), and after tuned (-.-.-.) structural element control. (a) Vertical leg, antisymmetrical modes, (b) vertical leg, symmetrical modes, (c) horizontal bar, anti-symmetrical modes, and (d) horizontal bar, symmetrical modes.
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for example, then becomes a constant c ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ðrAÞ3 EIo2d , which is definitely causal. Recall the receptance frequency
responses presented in Fig. 8, in which KR is set to zero, and KT1 and KT2 are designed to optimally damp out the flexural and longitudinal vibration energies, respectively. When tuning the non-causal controller KT1 to a causal ‘‘D’’ controller in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the form of io 4 ðrAÞ3 EIo2d , the controller can be easily implemented using a viscous damper with damping constant qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 4 ðrAÞ3 EIo2d in a practical application. Fig. 13 shows the receptance frequency responses of the frame before control, after ideal control, and after tuned causal control. In which the causal bending controller is tuned optimal at 200 Hz. It can be seen that the control performance of the tuned causal controller is the same as that of the ideal controller at the tuned optimal frequency od. The control performance at other frequencies is slightly compromised comparing to that of the ideal controller. However, the tuned controller still works very well in damping out the vibration resonances in the broad frequency band. 5. Conclusions In this paper, active wave control is designed based on classical vibration theory to suppress vibrations in multi-story planar frame structures. Regardless of the complexity of the structure, from the wave point of view it consists of only two basic types of structural components, namely, joints and structural elements. Consequently, controllers are designed both along structural elements and at structural joints. A number of control strategies have been adopted in feedback wave control design, such as adding optimal damping to the structure and maximizing the energy absorbed by the controller. Causality issues of the wave controllers are also addressed; two recommendations have been given for finding a causal approximation of a non-causal ideal controller. One way is to truncate the non-causal part of the ideal controller, the other is to tune the controller to be optimal at a certain frequency. Several numerical results are given, which include structural element control, L-joint control, and T-joint control. The numerical results show that after control, the resonances are much less sharp over the entire frequency range. It is evident that the wave controllers are very effective in suppressing vibrations in complex multi-story planar frame structures over a broad frequency band. It shall be noted that in-plane bending and longitudinal motions co-exist in in-plane vibrations of built-up planar frames due to wave-mode conversion at the joint(s). Neglecting either motion will result in inaccurate analysis. Should in-plane vibrations be not guaranteed, out-of-plane bending and torsional vibrations may also exist in the structure and will need to be addressed accordingly.
Acknowledgements The author gratefully acknowledges the support on this project from the Civil, Mechanical and Manufacturing Innovation Division of the National Science Foundation through Grant #0825761. Appendix A. Free wave propagation along a uniform waveguide and wave reflections at classical boundaries [18] A.1. Free wave propagation A.1.1. Bending vibrations First, consider the free bending vibration problem when no external force is applied to the beam. The differential equation of motion becomes EI
@4 yðx,tÞ @2 yðx,tÞ þ rA ¼0 @x4 @t 2
(A1)
Assuming time harmonic motion and using separation of variables, the solution to Eq. (A1) can be written in the form yðx,tÞ ¼ y0 eikx eiot , where o is the frequency and k is the wavenumber. Substituting this into Eq. (A1) gives a set of wavenumbers that are functions of the frequency o as well as the properties of the structure qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 k ¼ 7 rAo2 =EI (A2) The 7 sign outside the brackets indicates positive- and negative-going waves. With the time dependence eiot suppressed, the solutions to Eq. (A1) can be written as ik1 x k2 x yðxÞ ¼ a1þ eik1 x þ a2þ ek2 x þ a þa 1e 2e
(A3)
where the bending wavenumbers k1 ¼ k2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 rAo2 =EI
(A4)
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A.1.2. Longitudinal vibrations Now consider the free longitudinal vibration problem when no external force is applied to the beam. The differential equation for free longitudinal motion is
rA
@2 uðx,tÞ @2 uðx,tÞ EA ¼0 2 @t @x2
(A5)
Again assuming time harmonic motion and using separation of variables, the solution to Eq. (A5) can be written in the form uðx,tÞ ¼ u0 eikx eiot , where o is the frequency and k is the wavenumber. Substituting this into Eq. (10) gives the longitudinal wavenumber, which is a function of the frequency o as well rffiffiffiffi
r o
k¼ 7
(A6)
E
Similarly, the 7 sign indicates that longitudinal waves in the beam travel in both the positive and negative directions. With the time dependence eiot suppressed, the solutions to Eq. (A5) can be written as uðxÞ ¼ c þ eik3 x þ c eik3 x
(A7)
where the longitudinal wavenumber rffiffiffiffi k3 ¼
r o
(A8)
E
A.1.3. Propagation matrix Two points A and B on a uniform beam a distance x apart are considered. Waves propagate from one point to the other, with the propagation being determined by the appropriate wavenumber. Denoting the positive and negative going wave vectors at points A and B as a þ and a and b þ and b , respectively, they are related by
a ¼ fðxÞb ;
b
þ
¼ fðxÞa þ
(A9)
where 2 6 fðxÞ ¼ 4
0
0
0
ek2 x
0
0
0
eik3 x
is the propagation matrix for a distance x. Furthermore 2 þ3 2 3 a1 a1 6 7 6 7 a þ ¼ 4 a2þ 5, a ¼ 4 a 2 5, c
þ
3
eik1 x
c
2 b
þ
b1þ
7 5
(A10)
3
6 7 ¼ 4 b2þ 5, þ d
2
b 1
3
6 7 b ¼ 4 b 2 5 d
(A11)
A.2. Reflections at classical boundaries At a boundary, the incident waves a þ give rise to reflected waves a , which are related by a ¼ ra þ
(A12)
The reflection matrix r can be determined by considering equilibrium at the boundary, that is V ¼ EI ik1 x k2 x y ¼ a1þ eik1 x þ a2þ ek2 x þ a þa 1e 2e
where The reflection matrices are found as
@3 y @2 y @u , M ¼ EI 2 , F ¼ EA 3 @x @x @x
(A13)
and u ¼ c þ eik3 x1 þc eik3 x1 : 2
3 1 0 0 6 7 rs ¼ 4 0 1 0 5 0 0 1 2 3 i 1i 0 6 7 i 0 5 rc ¼ 4 1 þi 0 0 1 2 3 i 1 þ i 0 6 7 i 05 rf ¼ 4 1i 0 0 1 for simply supported, clamped, and free boundary conditions, respectively.
(A14)
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C. Mei / Journal of Sound and Vibration 330 (2011) 5530–5544
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