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Vibrational power-flow analysis of a MIMO system using the transmission matrix approach
ARTICLE IN PRESS
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 21 (2007) 365–388 www.elsevier.com/locate/jnlabr/ymssp
Vibrational power-flow analysis of a MIMO system using the transmission matrix approach L. Suna,b, A.Y.T. Leunga, Y.Y. Leea,, K. Songb a
Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong b School of Mechanical Engineering, Shandong University, 73 Jingshi Road, Jinan 250061, PR China Received 11 March 2005; accepted 10 July 2005 Available online 24 August 2005
Abstract The transmission matrix approach is employed in this paper for the vibrational power-flow analysis of a multiple-input/multiple-output (MIMO) system. The substructures in each subsystem are modelled as a group of general MIMO transmission elements, which are described in terms of discrete or distributed properties. The global transmission matrix of the system is formed by assembling the transmission matrices together. Using the proposed approach, the output variables of any subsystem can be expressed in terms of the input state variables of that subsystem. Hence, the proposed approach is very systematic in analysing the dynamic behaviour of a complex system with a group of elements. In the numerical example, the power flow of a multi-mount system is modelled analytically. The transmission of vibratory power flow from the vibrating rigid body into a simply supported plate through elastic mounts is studied in detail. The effects on the power flows of various factors, such as the plate thickness, loading type, mounting locations, and mass and damping properties of the mounts, are addressed. r 2005 Elsevier Ltd. All rights reserved. Keywords: Vibration isolation; Transmission matrix; Power flow
Corresponding author.
E-mail address: [email protected] (Y.Y. Lee). 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.07.001
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1. Introduction In practice, a vibrating machine is always installed on a non-rigid foundation and causes unwanted noise and vibration. Isolators, which are used to reduce vibration transmissions, lead to more complicated systems as the number of degrees of freedom (dof) increases. Sometimes, a significant residual vibration still transmits through the isolators, and induces forces and moments at the connection points. Furthermore, the flexural and longitudinal resonances of the isolators mean that the isolation performance can be adversely affected. Hence, the investigation of the vibration transmission of a vibrating system that is mounted with isolators has received much attention in recent years. Much research has been published on the transmission of vibration from a machine to its supporting structure (e.g. [1–3]). The concept of mobility and impedance in vibration transmission and isolation has been widely adopted for decades. According to the mobility and impedance approaches [4–7], a system is divided into three constituent elements (i.e. a vibration source, a mounting system, and a receiving structure), which are characterised by their driving point or transfer motilities and impedances. Based on the continuity and equilibrium principles at the interfaces between the substructures, these three elements were assembled together to form the complete system. Thus, the velocity transmissibility and power flow of the system can be obtained and used as objective functions for evaluating the vibration control performance. Some researchers (e.g. [8,9]) focused on studying the driving point mobility of flexible structures. Force and moment mobility functions were obtained experimentally and they agreed well with the theoretical predictions. Moreover, a multiple degree of freedom (mdof) suspension system, which consisted of the three main components (i.e. elastic mount, supporting panel, and rigid body), was investigated in [10], where the masses of the elastic mounts were not considered. The effects of the loading pattern, resonance, and coupling between the three components on the power flows into the individual elements were studied and addressed. The masses of the elastic mounts of a multimount system were considered in [11–13], but their effects on the power flows were not investigated. In works that also considered the masses of the mounts [14,15], it was found that the isolation performance seriously deteriorated at the frequency range around the first two quasilongitudinal wave resonances of the mounts. However, when the mobility and impedance methods dealt with systems with two or more resilient mounts and multiple interconnections, the modelling procedures were tedious as the input and output variables were coupled in the mobility and impedance matrix equations. To solve problems of mechanical systems with multi-interconnections, the four-pole network theory was developed based on the network and transmission matrix approaches used for electrical systems. In the four-pole network theory, the input variables of a transmission component are represented by the product of the four-pole matrix and corresponding output variables. Hence, the transmission characteristics of the assembled system can be derived directly by synthesising the fourpole matrices. The definitions of four-pole mechanical elements and four-pole matrices of lumped elements and some distributed elements were given in [16]. Another early dynamic analysis of a mechanical system using the four-pole parameter theory can be found in [17]. The transfer mobility and impedance were derived in [18] for flexible structures, such as uniform rod, beam, and shaft, etc., and the four-pole matrices of the longitudinal, torsional, and flexural vibrations could be obtained accordingly. The four-pole network theory was extended to a multiple terminal approach
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for multiple-input/multiple-output (MIMO) mechanical systems, where the input and output variables were related by a transmission matrix [19]. Other studies described the one-dimensional wave effect and demonstrated its importance both theoretically and experimentally [20–22]. However, most of the studies that used the multiple terminal approaches did not derive a generalised formulation of the dynamics of a distributed mount. The mass property of the mounts/ isolators was neglected, so that the effects of the resonances and standing waves on the power flows vanished. As viscoelastic mounts/isolators are extensively used in engineering, the importance of the wave effects must be emphasised, especially for high-frequency isolation. The objectives of this paper are to develop a systematic approach for the formulation of the vibration transmission and power flow of a MIMO system with parallel and series substructures, and to study the effects of various factors on the power flows, such as the plate thickness, loading type, mounting locations, mass and damping properties of the mounts, etc. A general transmission matrix model for the multi-dimensional vibration analysis of assembled systems consisting of MIMO substructures is developed. To explore the effects of the various factors, the power-flow spectra of isolation systems with multiple mounts are investigated numerically.
2. Transmission matrix model Fig. 1 shows a system that consists of k subsystems S 1 ; S 2 ; . . . ; S k . Each subsystem represents a rigid/flexible substructure or a substructure that is assembled by inter-connected elements. Fig. 2 schematically illustrates the transmission characteristic of the system in Fig. 1. Each multi-terminal block represents a MIMO subsystem that consists of substructures. Each of the MIMO lumped or distributed substructures is characterised by transmission matrices, so that the input variables can be expressed in terms of the output variables. Thus, the transmission matrix equation of the system can be obtained from the synthesis of the transmission matrices of the subsystems. 2.1. Transmission matrix of a MIMO substructure A MIMO substructure (or transmission element) ti with n input terminals and p output terminals is shown in Fig. 3. Each transmission element contains 2 6(n+p) input and output parameters, where 6 is the maximum number of dof at each connecting point, and 2 takes into account the kinematic and dynamic parameters. All quantities are p inffiffiffiffiffiffi the ffi complex form with the time-dependent function of ejot (o is the excitation frequency, j ¼ 1, and t is time), which is omitted for brevity. The generalised forces and corresponding velocity responses at the input terminals of the substructure ti are given by 8 1 9 8 1 9 * * > > > > > > > F > ti > > Vti > > > > > > > > > > > > > > 2 2 > > > > * * > > > > < < = * * F ti Vti = F ti ¼ ; Vti ¼ , (1a,b) > > .. > .. > > > > > > > > > > > . > . > > > > > > > > > > > *nti > *nti > > > > :F ; :V > ; ti ti
ARTICLE IN PRESS L. Sun et al. / Mechanical Systems and Signal Processing 21 (2007) 365–388
S1
x
………
y
z
……
………
……
o
……….…
368
……… Sk-1
Sk supporting structure Note : k = 3 for a system consisting of an elastic mount, supporting panel, and rigid body
Fig. 1. A complex system with k subsystems in series connection. Note: k ¼ 3 for a system consisting of an elastic mount, supporting panel, and rigid body.
*1
*1
where the force and velocity vectors F ti , Vti are given by 8 1 9 8 1 9 vti x > > F > > > > ti x > > > > > > > > > > > > 1 > > > v 1 > > > > ti y > F > > > > ti y > > > > > > > > 1 > > > > > > > > v 1 = t z < < i = F ti z *1 *1 F ti ¼ , ; Vti ¼ _ 1 yti x > > > T 1ti x > > > > > > > > 1 > > 1 > > > > > > > > > > _ > T > > > > yti y > > ti y > > > > > > > > > > > > 1 > > : T 1t z > ; : y_ > ; i
(2a,b)
ti z
where F 1ti x , F 1ti y , F 1ti z , v1ti x , v1ti y , v1ti z are the forces and velocities of the first input terminal in the x-, y1 1 1 , and z- directions, and T 1 , T 1 , T 1 , y_ , y_ , and y_ are the moments and angular velocities ti x
ti y
about the x-, y-, and z-directions.
ti z
ti x
ti y
ti z
ARTICLE IN PRESS L. Sun et al. / Mechanical Systems and Signal Processing 21 (2007) 365–388 FS1
FS1
FS2
FS2
TS1 VS1
VS2
FSk
……
TS2
VS1
369
TSk VSk
VS2
Fig. 2. Representation of the transmission characteristics of a complex system.
Fti, Vti
Ft1i, Vt1i
Ft1i, Vt1i
Ft2i, Vt2i
Ft2i, Vt2i
n
---
Transmission element ti p
n
Fti, Vti
p
Fti,ti Vti ti
Fti,ti Vti ti
Tti
Fig. 3. Representation of the transmission characteristics of a general MIMO substructure.
Similarly, the generalised forces and corresponding velocity responses at the output terminals of the transmission element ti are given by 8 8 9 9 1 1 > > > F V > > > ti > ti > > > > > > > > > > > > 2 2 > > > > < Fti = < Vti > = ¼ Fti ¼ ; V . (3a,b) t i > ... > > ... > > > > > > > > > > > > pti > > pti > > > > > : Ft > : Vt > ; ; i i According to the multi-terminal transmission matrix method, the input vector of substructure ti is expressed in terms of the output vector as follows: 8* 9 " #( ) < Ft = Fti Tt11i Tt12i i ¼ , (4) * Vti Tt21i Tt22i :V ; ti i where Ttmn (m; n ¼ 1; 2) is the transmission submatrix of substructure ti. Each of these submatrices contains 6 n rows and 6 p columns, the elements of which depend on the ratios of the corresponding forces and velocities at the input and output terminals. In Eq. (4), each of the input forces or velocities is a linear combination of the output force and velocity components.
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Consider a subsystem Si consisting of nSi rigid or flexible substructures in parallel connection. The transmission equation of the subsystem can be concisely formulated from the submatrices of each substructure in the following way: 8 1 9 * > > > 8 9 > F Si > > > > 3> F1 > 2 1 > > 1 > > Si > > > > T T 2 * > > > > 11 12 > > > > > 2 > F Si > 7> 6 2 2 > > > > F > > > > T T S i 7> > > > > 6 11 12 > > > 7 6 > > > > . > > > > . 7 6 > > > . . . > .. > > . > > > 6 . . 7 > > > . . > > > > > > 7> 6 *nSi > > > > p S n n >F i > >F = > > 6 si si 7< < = T T 7 6 Si Si 11 12 7 6 (5) ¼6 1 7> V1 >, *1 > T122 > > > 7> 6 T21 Si > > > > > V > 2 > > Si > > > 6 7> > > > 6 7> T221 T222 VSi > > > > 2 > * > > > > 7 6 > > > > 6 > > > 7> . . > > > > V S . > > > > 7 6 . . i > > > . . . > . > > > > 4 5> > > > > > > . nsi nsi > p > > > > . > > > > S i T T > . > > 21 22 : VS ; > > > i n * Si > > > :V > ; Si where Tkm;n ðk ¼ 1; 2; . . . ; nSi ; m; n ¼ 1; 2Þ is the transmission submatrix of the kth substructure, *k
*k
F Si ; VSi ðk ¼ 1; 2; . . . ; nSi Þ are the input force and response velocity vectors of the kth substructure, and FkSi ; VkSi are the output force and response velocity vectors. In Eq. (5), the substructures within the overall subsystem are modelled separately and independently. The key point of this approach is that the input variables are expressed in terms of the output state variables, unlike those in the mobility and impedance methods. This relation between the input and output variables is conducive for systematically modelling a system that consists of a number of substructures both in parallel and in series connection.
2.2. Transmission characteristics of complex system Fig. 2 shows a system that consists of the subsystems S1 ; S 2 ; . . . ; Sk connected in series. The dynamic characteristics of the subsystems can be written in the form of a transmission matrix. 8* 9 ( ) < FS = FS 1 1 , (6a) ¼ TS1 * VS 1 :V ; S1 8* 9 < FS = 2
* :V ; S2
( ¼ TS2
FS 2 VS 2
) ,
(6b)
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8* < FS
k1
9 =
S k1
;
*
:V
.. .
(
¼ TSk1
FSk1 VSk1
371
)
(6c)
:
For the last subsystem Sk, the relation between the velocity and force vectors is given by *
VSk ¼ MSk F Sk ,
(7)
where TSi (i ¼ 1; 2; . . . ; k 1) is the * transmission * * matrix of subsystem, MS k is the mobility matrix of the receiver structure Sk, and F Sk ; VSk , F Si ; VSi , and FSi ; VSi (i ¼ 1; 2; . . . k 1) are the input and output state vectors of the subsystem. Considering the continuity condition of the force and velocity vectors at the interface between any two subsystems, the following relation is obtained: *
*
FSi ¼ F Siþ1 ; VSi ¼ VSiþ1 ,
(8a,b)
where i ¼ 1; 2; . . . ; k 1. Then, the transmission matrix equation of the complex system can be obtained by sequential multiplications of the transmission matrices of subsystems S 1 ; S2 ; . . . ; S k , 8* 9 8 9 < FS = <* F Sk = 1 ¼ K , (9) * :V ; : VSk ; S1
" where K ¼
K11
K12
K21
K22
# ¼ TS1 TS2 . . . TSk1 is the overall transmission matrix of the complex
system. The submatrices can be obtained by Kij ¼
2 X n¼1
2 X 2 X
TS1 ;il TS2 ;lm . . . TSk1 ;nj ;
i; j ¼ 1; 2.
m¼1 l¼1
Therefore, the force and velocity vectors of subsystem Si can be expressed in terms of 8* 9 ( ) FS i 1 < F S1 = ¼ TS 1 TS 2 TS i . * VS i :V ;
(10)
S1
By substituting Eq. (7) into Eq. (9), the input and output variables of the overall system can be obtained. *
*
VS1 ¼ ðK21 þ K22 MSk ÞðK11 þ K12 MSk Þ1 F S1 ,
(11a)
* F Sk
(11b)
*
¼ ðK11 þ K12 MSk Þ1 F S1 , *
VSk ¼ MSk ðK11 þ K12 MSk Þ1 F S1 .
(11c)
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The time average power PS1 of subsystem S1 that is input into the overall system and the power transmitted into the supporting structure PSk are given by H H * * * 1 1 PS1 ¼ Re F S1 VS1 ; PSk ¼ Re F S1 VSk , (12a,b) 2 2 where the symbol H represents the transpose and conjugate of a matrix. 2.3. Transmission matrix of a resilient mount For a complex coupled system in which all substructures are different, a general formulation of a lumped or distributed substructure is required to characterise the dynamic behaviour of the assembled system. In Sections 2.3–2.5, the transmission matrix formulations for MIMO source, transmission, and support substructures, which are the basic components of many vibration problems, are derived. The formulations are used to systemically describe the transmission characteristics of an assembled system with multiple connections. The vibration of a supporting structure is induced by the excitations that are transmitted through the isolators. The isolators can affect the radiated sound power from the receiving structure. Typically, linear isolator theory, which neglects the mass effects, can represent the stiffness and damping properties only. However, the transmissibility curve of an isolator system that is modelled by this linear isolator theory will overestimate the isolation performance at high frequencies, especially for cases of low damping. This occurs because the so-called wave effects in practical resilient mounts are neglected in typical isolator theory. In practice, an isolator has mass, and thus its resonant responses can affect its performance. The resonances of isolators may occur at high frequencies, when their dimensions are comparable with multiples of the half-wavelength of elastic waves travelling through the materials. In Fig. 4, the rubber isolator is modelled as a
Fig. 4. Schematic of a general elastic mount.
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continuous elastic circular rod. Forces and moments that act on the mount induce motions in the corresponding translational and rotational dof. This model can predict the longitudinal, flexural, shearing, and torsional wave effects in a mount. The top and bottom of the resilient mount are defined as the input and output terminals, respectively, where the subscripts 1 and 2 represent the input and output variables. The force and velocity vectors at the input and output terminals are denoted by
Fm1
8 9 F m1x > > > > > > > > > > F > m1y > > > > > > < F m1z > = ¼ ; > T m1x > > > > > > > > > T m1y > > > > > > > > :T ;
Vm1
m1z
Fm2
8 9 F m2x > > > > > > > > > > F > > m2y > > > > > > < F m2z = ¼ ; > T m2x > > > > > > > > > > T m2y > > > > > > > :T ;
Vm2
m2z
8 9 vm1x > > > > > > > > > v > > m1y > > > > > > < vm1z > = ¼ _ , > ym1x > > > > > > > > y_ m1y > > > > > > > > > : y_ ; m1z
(13a,b)
8 9 vm2x > > > > > > >v > > > > > m2y > > > > > > < vm2z = ¼ _ . ym2x > > > > > > >_ > > ym2y > > > > > > > >_ > : ym2z ;
(14a,b)
According to the transmission matrix of a substructure in Eq. (4), the transmission matrix equation of the mount can be given by (
Fm1 Vm1
)
" ¼
Tm11
Tm12
Tm21
Tm22
#(
Fm2 Vm2
) ,
(15)
where Tmij (i; j ¼ 1; 2) are the transmission submatrices, which can be derived from the frequency response functions of the distributed mount. The detailed derivation of the transmission submatrices for the mount is shown in the Appendix. 2.4. Transmission matrix of a rigid-body source Fig. 1 illustrates a general three-dimensional rigid-body source structure with the coordinate system O-xyz originated at the centre of gravity. The subsystem S1 schematically represents a series of source * substructures. It is assumed that the source structure is excited on its centre of gravity. The response velocity vector at by a general force vector F e that acts * the centre of gravity is denoted as Ve . The output force and velocity vectors at the n connection
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points are given by 8 19 8 19 Fa > Va > > > > > > > > > > > > > > > 2 > > < Fa > < V2a > = = Fa ¼ .. >; Va ¼ > .. >. > > > . > . > > > > > > > > > > > : Fn > : Vn > ; ; a a
(16a,b)
According to the theory of rigid-body dynamics, the equations of translational and rotational motion of the source structure can be expressed in the mobility form as (* ) " #( * ) M11 M12 Fe Ve ¼ , (17) M M 21 22 Va Fa where M11 ¼ ð1=joÞJ1 , M12 ¼ M11 TF , M21 ¼ Tv M11 , M22 ¼ Tv M11 TF , and J is the diagonal matrix of inertia relative to the center of gravity. TF and Tv are the matrices used to correlate the *
*
force and velocity vectors Fa and F e , Va and Ve by considering the rigid-body constraint * Fe
*
¼ TF Fa ; Va ¼ Tv Ve .
Rearranging the input and output terms in Eq. (17) gives 8* 9 " #( )
(18a,b)
(19)
e
2.5. Mobility matrix of a supporting structure In Fig. 2, a block diagram represents the supporting structure of the assembled system. As shown in Eq. (7), a mobility matrix represents the input–output characteristics of the receiving structure, which consider the force and moment mobilities at the connection points on the supporting structure. For some simple structures, the force and moment mobilities can be calculated analytically. In typical composite structures, such as buildings and ships, the supporting structure of a machine is usually constructed of beams and plates. Refs. [1,2] presented the formulas for the driving point mobility for beams, plates, and beam-stiffened plates of finite or infinite extent. The results showed that the first modes of the receiver structure greatly influence the transmission of vibration in a finite structure. The general modal formulation of the longitudinal, shear and flexural mobilities of plates and shells was presented in [23]. Other works (e.g. [10,12,24–29]) showed the generalized mobility and impedance equations for supporting beams, plates, and flexible cylinders subject to various boundary conditions. In this study, the support structure is modelled as a rectangular plate. At each connection point, the supporting plate is driven by concentrated forces (in the x-, y-, and z-directions) and moments (around the x-, y-, and z-axes). It is assumed that in-plane forces act in the middle of the plate cross-section. Only in-plane longitudinal waves, in-plane shear waves induced by the in-plane forces, and out-of-plane flexural waves induced by out-of-plane force and the flexural moment are
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taken into account. The twisting moment Tz on the plate surface does not generate any flexural waves in the plate. Thus, the driving point mobility and the transfer mobility of the supporting structure can be derived using wave equations [12]. Consider the out-of-plane motion and the rotations about the x- and y-axes as a coupling and superposing of the contributions from the connection points. The mobility submatrix MijSk (i; j ¼ 1; 2; . . . ; n; n is the number of points of connection on supporting structure) in matrix MSk is given by 3 2 ij mvx F x 0 0 0 0 0 7 6 ij 7 6 0 0 0 0 0 m 7 6 vy F y 7 6 ij ij ij 7 6 0 m m 0 0 m 7 6 vz F z vz T x vz T y (20) MijSk ¼ 6 7. 6 0 0 07 0 mijyx F z mijyx T x 7 6 7 6 6 0 0 mijyy F z 0 mijyy T y 0 7 5 4 0 0 0 0 0 0
3. Results and discussion In this section, the power transmission from a vibrating machine to a flexible foundation through multiple resilient mounts is investigated numerically. The aims of this investigation are (i) to illustrate the generalisation and effectiveness of the transmission matrix model proposed for complex systems; and (ii) to determine the power flow through each mount and explore the multi-dimensional wave effects in the mount and the multi-mount effects on the vibration transmission of isolation systems subject to various excitations. In the analysis, the machine is modelled as a 3dof system supported on a rectangular plate through multiple resilient mounts (see Fig. 5). The combined excitation that acts on the rigid body includes a vertical unit force Fey, a transverse unit force Fex, and a unit pitch moment Tex. The centre of gravity of the machine can vibrate in the vertical and transverse directions, and rotate about the x-axis. For a given external force vector and the corresponding velocity vector, the transmission matrices for the subsystems that represent the source, mount, and receiver can be obtained using Eqs. (15), (19), and (20). The overall transmission matrix in Eq. (9) is determined using sequential matrix multiplication. The input velocities and output forces and velocities at each connection point are then obtained by substituting the submatrices of the transmission overall matrix into Eqs. (11a–c). The power-flow spectra are calculated in decibel scale (dB, Pref ¼ 1012 W). In the numerical calculations, the following data are used.
Vibrating
machine: The mass and moment of inertia about the x-axis are m ¼ 105:3 kg and J ¼ 3:16 kg m2 , respectively. Cylindrical rubber isolator: The internal diameter, external diameter, and height are D d h ¼ 0:06 m 0:03 m 0:10 m; and the density, Young’s modulus, and loss factor are r ¼ 1000 kg=m3 , E ¼ 6:5 106 N=m2 , and Z ¼ 0:15. Supporting structure: The long side of the plate is parallel to the y-axis. The dimensions are a b s ¼ 1:0 m 1:5 m 0:005 m; and the density, Young’s modulus, shear modulus, loss
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Fey
1
…. .
2
n
x y z
Fig. 5. A multi-mount isolation system subject to combined excitation.
factor, and Poisson’s ratio are r ¼ 2700 kg=m3 , E ¼ 7 1010 N=m2 , G ¼ 2:4 1010 N=m2 , z ¼ 0:02, and m ¼ 0:33.
3.1. Dynamic characteristics of the mdof isolation system Consider a case in which the source structure is isolated by two identical mounts that are symmetrically installed on the foundation plate. The three resonant frequencies that correspond to the vertical mode and the coupled transverse and pitch modes of the source vibration are 8.1, 4.7, and 10.8 Hz, respectively. The first two resonant frequencies of the longitudinal modes of the distributed mount clamped at both ends are 403.1 and 806.2 Hz, and the first resonant frequency of the flexural mode is 481.2 Hz. The six lowest resonant frequencies of the supporting plate are f 1;1 ¼ 17:7 Hz, f 1;2 ¼ 34:2 Hz, f 2;1 ¼ 54:7 Hz, f 1;3 ¼ 61:5 Hz, f 2;2 ¼ 71:7 Hz, and f 2;3 ¼ 98:5 Hz. In this subsection, the numerical results of the total power input to the source and power transmitted to the foundation plate through distributed mounts are presented. The power spectra of the isolation system are shown in Figs. 6a–d for the four excitation cases (i.e. combined excitation, vertical force, transverse force, and pitch moment). In Fig. 6a, the resonant peaks due to the three rigid-body modes are observed at about 4, 5, and 8 Hz, in the case of combined excitation (the peaks of 4 and 5 Hz almost coincide). Due to the coupling with the foundation plate, the values of these three resonance frequencies are different from those in the uncoupled case. In Fig. 6b, the source is excited by a unit vertical force. A resonant peak at about 5 Hz is induced by the vertical oscillation of the rigid-body machine in the case of vertical force excitation. In Figs. 6c–d, the source is excited by a unit transverse force/pitch moment. Two resonant peaks at about 4 and 7 Hz are observed. These two resonances correspond to the coupled modes of transverse and pitch oscillations of the machine. The resonances at about 22 and 40 Hz
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100
100 Power flow (dB)
120
Power flow (dB)
120
80 60 40 20
100 (a)
101 102 Frequency (Hz)
80 60 40 20
—— System ...... Foundation 103
100 (b)
100
100
80
80
40 20
100 (c)
Power flow (dB)
120
Power flow (dB)
120
60
101 102 Frequency (Hz)
103
101 102 Frequency (Hz)
103
40
100 (d)
System —— Po werinput into the ystem s ......Po wertrans mi tted tofo the undati on ate pl Foundation
60
20
—— System ...... Foundation
377
—— System ...... Foundation 101 102 Frequency (Hz)
103
Fig. 6. Power transmitted to the system and foundation plate, s ¼ 0:005 m: (a) combined excitation, (b) vertical force, (c) pitch moment and (d) transverse force.
are due to the first two flexural modes of the plate. Due to the coupling with the mounts, these two resonant frequencies are greater than those in the uncoupled case. In Figs. 6a–d, the vertical rigidbody mode and the coupled transverse and pitch modes of the system cause large power transmission at low frequencies. At frequencies above the resonances of rigid-body modes, many resonant peaks appear in the power-flow spectra. These resonances are caused by the flexural modes of the plate, which strongly affect the power transmission at higher frequencies. In particular, the resonances of the first two flexural modes induce much vibration power input into the system that is transmitted to the foundation. Moreover, the pitch moment causes more power flow than the vertical and transverse forces at the resonant frequencies of the flexural modes of the plate. This confirms that a pitch moment excitation greatly influences the power flow of a flexible isolation system. In Fig. 8, a foundation plate of 0.008 m thick is considered. The lowest four
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L. Sun et al. / Mechanical Systems and Signal Processing 21 (2007) 365–388 120
Power flow (dB)
100 80 60 40 20
—— System ...... Foundation
100
101
102
103
Frequency (Hz)
Fig. 7. Power transmitted to the system and thick foundation plate (s ¼ 0:008 m, vertical force).
120
Power flow (dB)
100 80 60 40 20
——
Symmetrically supported
...... Asymmetrically supported
100
101
102
103
Frequency (Hz)
Fig. 8. Power transmitted to the foundation plate through the two resilient mounts (s ¼ 0:005 m, vertical force).
resonant frequencies of the flexural modes are f 1;1 ¼ 28:5 Hz, f 1;2 ¼ 54:7 Hz, f 2;1 ¼ 87:6 Hz, and f 1;3 ¼ 98:5 Hz. A comparison of the thin and thick foundation plates in Figs. 6b and 7 show that the greater stiffness of the thick plate causes its resonant frequencies to be higher and thus the input of less power flow into the system. This observation indicates that the mobility of the supporting structure plays an important role in the vibration transmission of the whole system. In the above cases, the two identical mounts are installed symmetrically on the foundation plate. The first two resonances of the flexural modes of the plate cause a significant vibration power transmission. The case in which the mounts are installed asymmetrically on the plate is considered in Fig. 8. The resonant peaks of the vertical rigid-body mode shift to higher frequencies, and those of the first two flexural modes go to lower frequencies. Generally, there is more power flow in the case of asymmetrical mounts than in the case of symmetrical mounts.
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3.2. Resonances of the mounts Figs. 6 and 7 show that the two resonant peaks around 400 and 800 Hz can be clearly identified for the cases of combined excitation, vertical force excitation, and pitch moment excitation (see the peaks circled), which are due to the first two resonances of the longitudinal modes of the mounts. In the case of transverse force excitation, a peak is observed around 500 Hz, which is due to the first flexural wave resonance in the mounts. It can be seen that the resonances of the mounts can degrade the isolation performance. A comparison of Figs. 6b and 7 shows that the power input into the system around the resonant frequency of the mounts is higher in the case of the thick foundation plate. Fig. 9 shows the spectra of the two cases with and without isolator mass effects. The power transmission of the case with the mass effect is 10 dB higher than that without the mass effect at frequencies above 200 Hz. The peak values of the resonances of the rigid-body mode and the flexural modes are smaller in the case without the isolator mass effect. Fig. 10 shows the influence of the mount damping on the power that is transmitted to the foundation, in which the two cases of Z ¼ 0:15 and Z ¼ 0:3 are considered. The power transmission of the resonance of the rigid body mode and the flexural modes are reduced when the loss factor is increased to 0.3 (see the first three peaks). The power transmission due to the resonant responses of the mounts is also reduced, but not as much as those of the first three modes of the system. These observations show that the flexibility of the supporting structure is an important influence on the vibration transmission, and that the resonant effects of the mounts significantly increase the power transmission at higher frequencies. 3.3. Power flow of a system with multi-mounts In this section, the power transmission of an isolation system with three identical mounts is investigated. The three identical mounts are symmetrically installed on the foundation plate. The resonant frequencies of the vertical rigid-body mode and the coupled transverse and pitch modes 120
Power flow (dB)
100 80 60 40 20
100
With mass ...... Without mass 101
102
103
Frequency (Hz)
Fig. 9. Power transmitted to the foundation plate through different mounts (s ¼ 0:005 m, vertical force).
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Power flow (dB)
100 80 60 40 20
100
τ = 15.0
τ = 3.0 101
102
103
Frequency (Hz)
Fig. 10. Power flow transmitted to the foundation plate for different damping ratios (s ¼ 0:005 m, vertical force). 120
Power flow (dB)
100 80 60 40 20
100
—— system ...... Foundation 101
102
103
Frequency (Hz)
Fig. 11. Power transmitted to the system and foundation plate through three mounts (s ¼ 0:005 m, combined excitation).
are 8.4, 6.6, and 9.1 Hz, respectively, the first two resonant frequencies of the longitudinal modes of the mount are 353.6 and 707.1 Hz, and the first resonant frequency of the flexural mode is 422.3 Hz. Figs. 11 and 13 show the power transmission through the three mounts for the cases of vertical force or combined excitation (including a unit vertical force, unit transverse force, and unit pitch moment). When compared with those in the two-mount system in Figs. 6a–b, the power that is transmitted to the foundation plate through the three mounts decreases at low frequencies. Similar to the two-mount isolation system, two resonant peaks due to the first two longitudinal modes of the mounts are observed in Figs. 11 and 13. The power that is transmitted through mount 1 is slightly greater than the power that is transmitted through mount 3 in Fig. 12 (the case of combined excitation) but almost equal to that which is transmitted through mount 3 in Fig. 14 (the case of vertical force). Around the resonant frequency of the rigid-body modes, the power
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100
Power flow (dB)
50
0
-50 —— mount 1 —• mount 2 ...... mount 3
-100
-150 100
101
102 Frequency (Hz)
103
Fig. 12. Power transmitted to the foundation through a particular resilient mount (s ¼ 0:005 m, combined excitation).
120
Power flow (dB)
100 80 60 40 20
100
—— System ...... Foundation 101
102
103
Frequency (Hz)
Fig. 13. Power transmitted to the system and foundation plate through three mounts (s ¼ 0:005 m, vertical force).
that is transmitted through mount 2 can be negative (see the circled regions in Figs. 12 and 14). This implies that a net power flow is directed from the supporting structure towards the source. In other frequencies, the power transmitted through the mount 2 is null (Figs. 13 and 14)
4. Conclusions The transmission matrix approach is employed in this paper for the vibrational power-flow analysis of a multiple-input/multiple-output (MIMO) system. The substructures in each subsystem are modelled as a group of general MIMO transmission elements, which are independent of each other. The global transmission matrix of the system is formed by assembling the transmission matrices together. Using the proposed approach, the output variables of any
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L. Sun et al. / Mechanical Systems and Signal Processing 21 (2007) 365–388 100
Power flow (dB)
50
0
-50 —— —• ......
-100
-150 100
101
102
mount 1 mount 2 mount 3
103
Frequency (Hz)
Fig. 14. Power transmitted to the foundation through a particular resilient mount (s ¼ 0:005 m, vertical force).
subsystem can be expressed in terms of the input state variables of the subsystem subject to the external loads. Hence, the proposed approach is very systematic in analysing the dynamic behaviour of a complex system with a group of elements. In the numerical example, the power flow of a multi-mount system is modeled analytically. The transmission of vibratory power flow from the vibrating rigid body into a simply supported plate through elastic mounts is studied in detail. The following conclusions can be drawn. (1) The vertical rigid-body mode and the coupled transverse and pitch modes of the system are the dominant modes affecting the power that is transmitted to the system and supporting structure at low frequencies. (2) The resonances of the first few flexural modes of the plate are the dominant modes that affect much of the power that is transmitted to the system and supporting structure at middle frequencies. (3) The resonances of the mount are the dominant modes that affect much of the power that is transmitted to the system and supporting structure at high frequencies. (4) The mounts with mass effect can degrade the isolation performance 10 dB lower than those without mass effect. The power transmission due to the resonant responses of the mounts can be reduced slightly by increasing the damping of mounts. (5) The unit pitch moment excitation induces more power input into the system than the unit vertical and transverse forces at the resonant frequencies of the plate. (6) Generally, there is less power flow transmitted into the supporting structure in the case of symmetrical mounts and thick plates than in the asymmetrical and thin plate cases. (7) There is less power transmitted into the supporting structure at low frequencies in the case of three mounts than that of two mounts. Around the resonant frequency of the rigid-body modes, a net power flow is directed from the supporting structure to the source. At other frequencies, the power that is transmitted through the middle mount is null.
Acknowledgement This research was supported by the National Natural Science Foundation of China.
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Appendix Consider the input and output variables at the yoz-plane Fm1y, Fm1z, Tm1x, Fm2y, Fm2z, Tm2z in the dynamic model of the resilient mount in Fig. 3. The driving point or transfer mobility frequency response functions of a uniform beam are given by [18] Rvm1 y F m1 y ¼ Rvm2 y F m2 y ¼ jo
cos lf h sinh lf h sin lf h cosh lf h , E s Il3f ðcos lf h cosh lf h 1Þ
Rvm1 y T m1 x ¼ Rvm1 y T m2 x ¼ Ry_ m x F m y ¼ Ry_ m x F m y ¼ jo 1
Rvm1 z F m1 z ¼ Rvm2 z F m2 z ¼ jo
1
2
2
cos ll h , AE s ll sin ll h
sin lf h sinh lf h 3 E s Ilf ðcos lf h cosh lf h
(A.5)
2
Rvm2 z F m1 z ¼ Rvm1 z F m2 z ¼ jo Ry_ m x T m x ¼ Ry_ m x T m x ¼ jo 1
1
2
(A.2)
(A.3)
1Þ
Rvm2 y F m1 x ¼ Rvm1 y T m2 x ¼ Ry_ m x F m y ¼ Ry_ m x F m y ¼ jo
2
sin lf h sinh lf h , lf h cosh lf h 1Þ
E s Il2f ðcos
(A.4)
2
Rvm2 y F m1 y ¼ Rvm1 y F m2 y ¼ jo
2
cos lf h sinh lf h þ sin lf h cosh lf h , E s Ilf ðcos lf h cosh lf h 1Þ
Ry_ m x T m x ¼ Ry_ m x T m x ¼ jo 1
1
(A.1)
AE s ll
1
1
2
, cos lf h cosh 2 E s Ilf ðcos lf h cosh
lf h , lf h 1Þ
1 , sin ll h
(A.6)
(A.7)
sin lf h þ sinh lf h , E s Ilf ðcos lf h cosh lf h 1Þ
(A.8)
where lf and ll are the complex wave numbers of the flexural and quasi-longitudinal waves in the rod, and h, A, and I are the height, area, and moment of inertia of the cross-section of the mount (with reference to the x-axis). E s ¼ E s ð1 þ jZÞ is the complex Young’s modulus, where Z is the structural loss factor (refer to [18] for more details). The equation of motion of the transmission structure can be described in terms of mobility ( ) ( ) ¯ m1 F¯ m1 U ¯ (A.9) ¯ m2 ¼ Rm F¯ m2 , U where ¯ m1 U
8 9 > < vm1y > = ¼ vm1z ; > > : y_ ; m1x
¯ m2 U
8 9 > < vm2y > = ¼ vm2z ; > > : y_ ; m2x
F¯ m1
8 9 > < F m1y > = ¼ F m1z ; > :T > ; m1x
F¯ m2
8 9 > < F m2y > = ¼ F m2z . > :T > ; m2x
(A.10213)
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¯ m are given by The submatrices in R 3 2 Rvm1 y F m1 y 0 Rvm1 y T m1 x 7 6 0 Rvm1 z F m1 z 0 7; ¯ m11 ¼ 6 R 5 4 Ry_ m x F m y 0 Ry_ m x T m x 1
1
1
2 6 ¯ m12 ¼ 6 R 4
1
Rvm1 y F m2 y
0
Rvm1 y T m2 x
0
Rvb1 z F b2 z
0
0
Ry_ m x T m x
Ry_ m x F m y 1
2
1
3 7 7, 5
2
(A.14,15) 2 6 ¯ m21 ¼ 6 R 4
Rvm2 y F m1 y
0
Rvm2 y T m1 x
0
Rvb2 z F b1 z
0
0
Ry_ m x F m x
Ry_ m x F m y 2
1
2
3
2
7 7; 5
6 ¯ m22 ¼ 6 R 4
1
Rvm2 y F m2 y
0
Rvm2 y T m2 x
0
Rvm2 z F m2 z
0
0
Ry_ m x T m x
Ry_ m x F m y 2
2
2
3 7 7. 5
2
(A.16,17) Then, by rearranging the input and output variables, Eq. (A.9) can be expressed in terms of a transmission matrix as follows: ( ) ( ) F¯ m1 F¯ m2 ¯ (A.18) ¯ m1 ¼ Tm U ¯ m2 , U ¯ m can be obtained using Eqs. (A.14)–(A.17) and are given by where the submatrices in T 3 2 G2 l f G1 0 6 2 2 7 7 6 1 6 0 7 cos ll h ¯ m22 ¼ 6 0 ¯ m21 R ¯ m11 ¼ R (A.19) T 7, 6 G3 G1 7 5 4 0 2lf 2 2 ¯ m12 ¼ R ¯ 1 T m21
G3 E s Il3f
6 2 1 6 6 0 ¼ 6 jo 6 4 G4 E s Il2f
0 E s All sin ll h 0
2
¯ m21 ¼ R ¯ 1 ¯ ¯ m12 R ¯ m11 R T m21 Rm22
3
7 7 7 7, 7 G2 E s Ilf 5 2
2
G2 6 2E s Il3f 6 6 6 0 ¼ jo6 6 6 6 G4 4 2E s Il2f
G4 E s Il2f 2 0
0
sin ll h E s All 0
G4 3 2 2E s Ilf 7 7 7 7 7, 0 7 7 G3 7 5 2E s Ilf
(A.20)
(A.21)
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2
¯ m22 ¼ T
¯ 1 ¯ m11 R R m21
G1 6 2 6 6 ¼6 0 6 4 G2 lf 2
G3 0 2lf cos ll h 0 G1 0 2
385
3 7 7 7 7, 7 5
(A.22)
where G1 ¼ cos lf h þ cosh lf h, G2 ¼ sin lf h sinh lf h, G3 ¼ sin lf h þ sinh lf h, and G4 ¼ cos lf h cosh lf hSimilarly, the transmission equation that is used to correlate the input and output variables at the xoz plane and the moment around the z-axis and the corresponding velocity is given by ( ) ( ) F^ m2 F^ m1 ¼ T^ m , (A.23) ^ m1 ^ m2 U U where F^ m1
8 9 > < F m1x > = ¼ T m1y ; > > ; :T m1z
^ m1 U
8 9 v > < m1x > = ¼ y_ m1y ; > > ; :_ ym1z
F^ m2
8 9 > < F m2x > = ¼ T m2y ; > > ; :T m2z
^ m are given by The submatrices in T 2 3 G1 G2 lf 0 6 2 7 2 6 7 ^ m11 ¼ 6 G2 lf 7, G1 T 6 7 0 4 2 5 2 0 0 cos ls h 2 ^ m12 T
6 6 2 1 6 2 6 ¼ 6 G4 E s Ilf jo 6 2 4 0 2
^ m21 T
G3 E s Il3f
6 6 6 6 ¼ jo6 6 6 6 4
G2 2E s Il3f G4 2E s Il2f
0
G4 2E s Il2f G3 2E s Ilf
0
(A.24227)
(A.28)
3
G4 E s Il2f
2 G2 E s Ilf 2 0
^ m2 U
8 9 v > < m2x > = ¼ y_ m2y . > > ; :_ ym2z
0 0 E t I p ls sin ls h
0
7 7 7 7, 7 7 5
(A.29)
3
7 7 7 7 7, 0 7 7 7 sin ls h 5 E t I p ls
(A.30)
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2
^ m22 T
G1 6 2 6 6 ¼ 6 G2 lf 6 4 2 0
G3 2lf G1 2 0
3 0 0 cos ls h
7 7 7 7, 7 5
(A.31)
where ls are the complex wave numbers of shearing wave, and E t is the complex shear modulus. Finally, using Eqs. (A.18) and (A.23), the transmission matrix of the mount in Eq. (14) Tm can be derived, with the submatrices given by
2
Tm11
G1 6 2 6 6 6 0 6 6 6 0 6 ¼6 6 0 6 6 6 6 G2 l f 6 4 2 0
2
Tm12
0 G1 2 0 G3 2lf
0
0
0
0
0
0
0
0
0
G2 lf 0 2 0 cos ll h G1 0 2
0
0
0
0
0
0
G3 E s Il3f
6 2 6 6 6 6 0 6 6 0 1 6 6 ¼ 6 jo 6 6 0 6 6 6 2 6 G4 E s Ilf 6 4 2 0
G2 lf 2
0
G3 E s Il3f 2 0
G4 E s Il2f 2 0 0
G1 2 0
0 0
(A.32)
cos ls h
0
0
0
7 7 7 7 7 7 7 7 7, 7 7 7 7 7 7 5
0
0
E s All
3
sin ll h
G4 E s Il2f 2 0 G2 E s Ilf 2 0 0
G4 E s Il2f 2
3 0
0
0
0
0
0
0
G2 E s Ilf 2 0
0 E t I p ls sin ls h
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
(A.33)
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Tm21
G2 6 2E Il3 s f 6 6 6 6 0 6 6 6 6 6 0 6 6 ¼ jo6 6 0 6 6 6 6 G4 6 6 6 2E s Il2f 6 6 4 0
0
0
G2 2E s Il3f
Tm22
0
sin ll h E s All
G4 2E s Il2f
0
G3 2E s Ilf
0
0
0
0
0
0
0
0
G1 2 0 G2 lf 2
cos ll h
0
0
0
0
0
0
0
0
G3 2lf 0 G1 2
0 0
G3 2E s Ilf 0
0
3
7 7 7 7 7 0 7 7 7 7 7 0 7 7 7, 7 0 7 7 7 7 7 0 7 7 7 sin ls h 7 5 E t I p ls
(A.34)
3
G3 2lf
0
0
0
0
0
0
0
G1 2 0
0
0
G4 2E s Il2f
2
G1 6 2 6 6 6 6 0 6 6 6 0 ¼6 6 6 0 6 6 6 6 G2 lf 6 4 2 0
0
0
0
G4 2 2E s Ilf
387
0 cos ls h
7 7 7 7 7 7 7 7 7. 7 7 7 7 7 7 7 5
(A.35)
For cases of vertical force excitation only, if the excitation frequency is much lower than the first resonant frequency of the longitudinal mode of the mount, then l1 h ! 0, cos l1 h ! 1, and sin l1 h ! l1 h. Hence, Eqs. (A.36)–(A.39) can be simplified to Tm11 ¼ 1;
Tm12 ¼ 0;
Tm21 ¼
jo ; K
Tm22 ¼ 1.
(A.36239)
References [1] H.G.D. Goyder, R.G. White, Vibrational power flow from machines into built-up structures, II. Wave propagation and power flow in beam-stiffened plates, Journal of Sound and Vibration 68 (1) (1980) 77–96. [2] R.J. Pinnington, R.G. White, Power flow through machine isolators to resonant and non-resonant beams, Journal of Sound and Vibration 75 (2) (1981) 179–197.
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[3] B.M. Gibbs, B.A.T. Petersson, S. Qiu, The characterization of structure-borne emission of building services machinery using the source descriptor concept, Noise Control Engineering Journal 37 (1991) 53–61. [4] P. Gardonio, M.J. Brennan, On the origins and development of mobility and impedance methods in structural dynamics, Journal of Sound and Vibration 249 (3) (2002) 557–573. [5] J.P. Den Hartog, Mechanical Vibrations, McGraw-Hill, New York, 1956. [6] J.C. Snowdon, Isolation of machinery vibration from non-rigid substructures using multiple anti-vibration mountings, Journal of the Acoustical Society of America (1973) 102–127. [7] A. Granhall, T. Kihlman, The use of mechanical impedance data in predicting vibration isolation efficiency, Journal of Noise Control Engineering 14 (2) (1980) 88–93. [8] Y.K. Koh, R.G. White, Analysis and control of vibrational power flow transmission to machinery supporting structures subjected to a multi-excitation system, Part I: driving point mobility matrix of beams and rectangular plates, Journal of Sound and Vibration 196 (1996) 469–493. [9] Y.K. Koh, R.G. White, Analysis and control of vibrational power flow transmission to machinery supporting structures subjected to a multi-excitation system, Part II: vibrational power analysis and control scheme, Journal of Sound and Vibration 196 (1996) 495–508. [10] J. Pan, J.Q. Pan, C.H. Hansen, Total power flow from a vibrating rigid body to a thin panel through multiple elastic mounts, Journal of the Acoustical Society of America 92 (1992) 895–907. [11] P. Gardonio, S.J. Elliott, R.J. Pinnington, Active isolation of structural vibration on a multiple-degree-of-freedom system, Part I: dynamics of the system, Journal of Sound and Vibration 207 (1997) 61–93. [12] P. Gardonio, S.J. Elliott, R.J. Pinnington, Active isolation of structural vibration on a multiple-degree-of-freedom system, Part II: effectiveness of the active control strategy, Journal of Sound and Vibration 207 (1997) 95–121. [13] Y. Du, R.A. Burdisso, E. Nikolaidis, D. Tiwari, Effects of isolators internal resonances on force transmissibility and radiated noise, Journal of Sound and Vibration 268 (2003) 751–778. [14] L.L. Sun, K.J. Song, Power flow analysis for complicated system, Proceedings of the ASME Design Engineering Technical Conference 5C (2003) 2103–2107. [15] L.L. Sun, Transmission matrix model of complex coupled system for passive and active vibration control 2004 (in Chinese). Ph.D. Dissertation, Shandong University. [16] C.T. Molloy, Use of four-pole parameters in vibration calculations, Journal of the Acoustical Society of America 29 (1957) 842–853. [17] J.C. Snowdon, Mechanical four-pole parameters and their application, Journal of Sound and Vibration 15 (1971) 307–323. [18] R.E.D. Bishop, D.C. Johnson, The Mechanics of Vibration, Cambridge University Press, Cambridge, 1960. [19] S. Rubin, Transmission matrices for vibration and their relation to admittance and impedance, Journal of Engineering for Industry 86 (1964) 9–21. [20] M. Harrison, A.O. Sykes, M. Martin, Wave effects in isolation mounts, Journal of the Acoustical Society of America 24 (1952) 62–71. [21] E.E. Ungar, C.W. Dietrich, High-frequency vibration isolation, Journal of Sound and Vibration 4 (1996) 224–241. [22] J.C. Snowdon, Vibration and Shock in Damped Mechanical Systems, Wiley, New York, 1968. [23] W. Soedel, Vibrations of Shells and Plates, second ed, Marcel Dekker, New York, 1993. [24] C.H. Hansen, S.D. Snyder, Active Control of Noise and Vibration, E & FN Spon, London, 1997. [25] D.J. Mead, R.G. White, X.M. Zhang, Power transmission in a periodically supported infinite beam excited at a single point, Journal of Sound and Vibration 169 (1994) 558–561. [26] J.M. Cuschieri, Vibration transmission through periodic structures using a mobility power flow approach, Journal of Sound and Vibration 143 (1990) 65–74. [27] N.H. Farag, J. Pan, Dynamic response and power flow in two-dimensional coupled beam structures under in-plane loading, Journal of Acoustical Society of America 95 (1996) 2930–2937. [28] P.E. Cho, R.J. Bernhard, Energy flow analysis of coupled beams, Journal of Sound and Vibration 211 (1998) 593–605. [29] R.S. Ming, J. Pan, M.P. Norton, The mobility functions and their application in calculating power flow in coupled cylindrical shells, Journal of the Acoustical Society of America 105 (1999) 1702–1713.
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 21 (2007) 365–388 www.elsevier.com/locate/jnlabr/ymssp
Vibrational power-flow analysis of a MIMO system using the transmission matrix approach L. Suna,b, A.Y.T. Leunga, Y.Y. Leea,, K. Songb a
Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong b School of Mechanical Engineering, Shandong University, 73 Jingshi Road, Jinan 250061, PR China Received 11 March 2005; accepted 10 July 2005 Available online 24 August 2005
Abstract The transmission matrix approach is employed in this paper for the vibrational power-flow analysis of a multiple-input/multiple-output (MIMO) system. The substructures in each subsystem are modelled as a group of general MIMO transmission elements, which are described in terms of discrete or distributed properties. The global transmission matrix of the system is formed by assembling the transmission matrices together. Using the proposed approach, the output variables of any subsystem can be expressed in terms of the input state variables of that subsystem. Hence, the proposed approach is very systematic in analysing the dynamic behaviour of a complex system with a group of elements. In the numerical example, the power flow of a multi-mount system is modelled analytically. The transmission of vibratory power flow from the vibrating rigid body into a simply supported plate through elastic mounts is studied in detail. The effects on the power flows of various factors, such as the plate thickness, loading type, mounting locations, and mass and damping properties of the mounts, are addressed. r 2005 Elsevier Ltd. All rights reserved. Keywords: Vibration isolation; Transmission matrix; Power flow
Corresponding author.
E-mail address: [email protected] (Y.Y. Lee). 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.07.001
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1. Introduction In practice, a vibrating machine is always installed on a non-rigid foundation and causes unwanted noise and vibration. Isolators, which are used to reduce vibration transmissions, lead to more complicated systems as the number of degrees of freedom (dof) increases. Sometimes, a significant residual vibration still transmits through the isolators, and induces forces and moments at the connection points. Furthermore, the flexural and longitudinal resonances of the isolators mean that the isolation performance can be adversely affected. Hence, the investigation of the vibration transmission of a vibrating system that is mounted with isolators has received much attention in recent years. Much research has been published on the transmission of vibration from a machine to its supporting structure (e.g. [1–3]). The concept of mobility and impedance in vibration transmission and isolation has been widely adopted for decades. According to the mobility and impedance approaches [4–7], a system is divided into three constituent elements (i.e. a vibration source, a mounting system, and a receiving structure), which are characterised by their driving point or transfer motilities and impedances. Based on the continuity and equilibrium principles at the interfaces between the substructures, these three elements were assembled together to form the complete system. Thus, the velocity transmissibility and power flow of the system can be obtained and used as objective functions for evaluating the vibration control performance. Some researchers (e.g. [8,9]) focused on studying the driving point mobility of flexible structures. Force and moment mobility functions were obtained experimentally and they agreed well with the theoretical predictions. Moreover, a multiple degree of freedom (mdof) suspension system, which consisted of the three main components (i.e. elastic mount, supporting panel, and rigid body), was investigated in [10], where the masses of the elastic mounts were not considered. The effects of the loading pattern, resonance, and coupling between the three components on the power flows into the individual elements were studied and addressed. The masses of the elastic mounts of a multimount system were considered in [11–13], but their effects on the power flows were not investigated. In works that also considered the masses of the mounts [14,15], it was found that the isolation performance seriously deteriorated at the frequency range around the first two quasilongitudinal wave resonances of the mounts. However, when the mobility and impedance methods dealt with systems with two or more resilient mounts and multiple interconnections, the modelling procedures were tedious as the input and output variables were coupled in the mobility and impedance matrix equations. To solve problems of mechanical systems with multi-interconnections, the four-pole network theory was developed based on the network and transmission matrix approaches used for electrical systems. In the four-pole network theory, the input variables of a transmission component are represented by the product of the four-pole matrix and corresponding output variables. Hence, the transmission characteristics of the assembled system can be derived directly by synthesising the fourpole matrices. The definitions of four-pole mechanical elements and four-pole matrices of lumped elements and some distributed elements were given in [16]. Another early dynamic analysis of a mechanical system using the four-pole parameter theory can be found in [17]. The transfer mobility and impedance were derived in [18] for flexible structures, such as uniform rod, beam, and shaft, etc., and the four-pole matrices of the longitudinal, torsional, and flexural vibrations could be obtained accordingly. The four-pole network theory was extended to a multiple terminal approach
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367
for multiple-input/multiple-output (MIMO) mechanical systems, where the input and output variables were related by a transmission matrix [19]. Other studies described the one-dimensional wave effect and demonstrated its importance both theoretically and experimentally [20–22]. However, most of the studies that used the multiple terminal approaches did not derive a generalised formulation of the dynamics of a distributed mount. The mass property of the mounts/ isolators was neglected, so that the effects of the resonances and standing waves on the power flows vanished. As viscoelastic mounts/isolators are extensively used in engineering, the importance of the wave effects must be emphasised, especially for high-frequency isolation. The objectives of this paper are to develop a systematic approach for the formulation of the vibration transmission and power flow of a MIMO system with parallel and series substructures, and to study the effects of various factors on the power flows, such as the plate thickness, loading type, mounting locations, mass and damping properties of the mounts, etc. A general transmission matrix model for the multi-dimensional vibration analysis of assembled systems consisting of MIMO substructures is developed. To explore the effects of the various factors, the power-flow spectra of isolation systems with multiple mounts are investigated numerically.
2. Transmission matrix model Fig. 1 shows a system that consists of k subsystems S 1 ; S 2 ; . . . ; S k . Each subsystem represents a rigid/flexible substructure or a substructure that is assembled by inter-connected elements. Fig. 2 schematically illustrates the transmission characteristic of the system in Fig. 1. Each multi-terminal block represents a MIMO subsystem that consists of substructures. Each of the MIMO lumped or distributed substructures is characterised by transmission matrices, so that the input variables can be expressed in terms of the output variables. Thus, the transmission matrix equation of the system can be obtained from the synthesis of the transmission matrices of the subsystems. 2.1. Transmission matrix of a MIMO substructure A MIMO substructure (or transmission element) ti with n input terminals and p output terminals is shown in Fig. 3. Each transmission element contains 2 6(n+p) input and output parameters, where 6 is the maximum number of dof at each connecting point, and 2 takes into account the kinematic and dynamic parameters. All quantities are p inffiffiffiffiffiffi the ffi complex form with the time-dependent function of ejot (o is the excitation frequency, j ¼ 1, and t is time), which is omitted for brevity. The generalised forces and corresponding velocity responses at the input terminals of the substructure ti are given by 8 1 9 8 1 9 * * > > > > > > > F > ti > > Vti > > > > > > > > > > > > > > 2 2 > > > > * * > > > > < < = * * F ti Vti = F ti ¼ ; Vti ¼ , (1a,b) > > .. > .. > > > > > > > > > > > . > . > > > > > > > > > > > *nti > *nti > > > > :F ; :V > ; ti ti
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S1
x
………
y
z
……
………
……
o
……….…
368
……… Sk-1
Sk supporting structure Note : k = 3 for a system consisting of an elastic mount, supporting panel, and rigid body
Fig. 1. A complex system with k subsystems in series connection. Note: k ¼ 3 for a system consisting of an elastic mount, supporting panel, and rigid body.
*1
*1
where the force and velocity vectors F ti , Vti are given by 8 1 9 8 1 9 vti x > > F > > > > ti x > > > > > > > > > > > > 1 > > > v 1 > > > > ti y > F > > > > ti y > > > > > > > > 1 > > > > > > > > v 1 = t z < < i = F ti z *1 *1 F ti ¼ , ; Vti ¼ _ 1 yti x > > > T 1ti x > > > > > > > > 1 > > 1 > > > > > > > > > > _ > T > > > > yti y > > ti y > > > > > > > > > > > > 1 > > : T 1t z > ; : y_ > ; i
(2a,b)
ti z
where F 1ti x , F 1ti y , F 1ti z , v1ti x , v1ti y , v1ti z are the forces and velocities of the first input terminal in the x-, y1 1 1 , and z- directions, and T 1 , T 1 , T 1 , y_ , y_ , and y_ are the moments and angular velocities ti x
ti y
about the x-, y-, and z-directions.
ti z
ti x
ti y
ti z
ARTICLE IN PRESS L. Sun et al. / Mechanical Systems and Signal Processing 21 (2007) 365–388 FS1
FS1
FS2
FS2
TS1 VS1
VS2
FSk
……
TS2
VS1
369
TSk VSk
VS2
Fig. 2. Representation of the transmission characteristics of a complex system.
Fti, Vti
Ft1i, Vt1i
Ft1i, Vt1i
Ft2i, Vt2i
Ft2i, Vt2i
n
---
Transmission element ti p
n
Fti, Vti
p
Fti,ti Vti ti
Fti,ti Vti ti
Tti
Fig. 3. Representation of the transmission characteristics of a general MIMO substructure.
Similarly, the generalised forces and corresponding velocity responses at the output terminals of the transmission element ti are given by 8 8 9 9 1 1 > > > F V > > > ti > ti > > > > > > > > > > > > 2 2 > > > > < Fti = < Vti > = ¼ Fti ¼ ; V . (3a,b) t i > ... > > ... > > > > > > > > > > > > pti > > pti > > > > > : Ft > : Vt > ; ; i i According to the multi-terminal transmission matrix method, the input vector of substructure ti is expressed in terms of the output vector as follows: 8* 9 " #( ) < Ft = Fti Tt11i Tt12i i ¼ , (4) * Vti Tt21i Tt22i :V ; ti i where Ttmn (m; n ¼ 1; 2) is the transmission submatrix of substructure ti. Each of these submatrices contains 6 n rows and 6 p columns, the elements of which depend on the ratios of the corresponding forces and velocities at the input and output terminals. In Eq. (4), each of the input forces or velocities is a linear combination of the output force and velocity components.
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Consider a subsystem Si consisting of nSi rigid or flexible substructures in parallel connection. The transmission equation of the subsystem can be concisely formulated from the submatrices of each substructure in the following way: 8 1 9 * > > > 8 9 > F Si > > > > 3> F1 > 2 1 > > 1 > > Si > > > > T T 2 * > > > > 11 12 > > > > > 2 > F Si > 7> 6 2 2 > > > > F > > > > T T S i 7> > > > > 6 11 12 > > > 7 6 > > > > . > > > > . 7 6 > > > . . . > .. > > . > > > 6 . . 7 > > > . . > > > > > > 7> 6 *nSi > > > > p S n n >F i > >F = > > 6 si si 7< < = T T 7 6 Si Si 11 12 7 6 (5) ¼6 1 7> V1 >, *1 > T122 > > > 7> 6 T21 Si > > > > > V > 2 > > Si > > > 6 7> > > > 6 7> T221 T222 VSi > > > > 2 > * > > > > 7 6 > > > > 6 > > > 7> . . > > > > V S . > > > > 7 6 . . i > > > . . . > . > > > > 4 5> > > > > > > . nsi nsi > p > > > > . > > > > S i T T > . > > 21 22 : VS ; > > > i n * Si > > > :V > ; Si where Tkm;n ðk ¼ 1; 2; . . . ; nSi ; m; n ¼ 1; 2Þ is the transmission submatrix of the kth substructure, *k
*k
F Si ; VSi ðk ¼ 1; 2; . . . ; nSi Þ are the input force and response velocity vectors of the kth substructure, and FkSi ; VkSi are the output force and response velocity vectors. In Eq. (5), the substructures within the overall subsystem are modelled separately and independently. The key point of this approach is that the input variables are expressed in terms of the output state variables, unlike those in the mobility and impedance methods. This relation between the input and output variables is conducive for systematically modelling a system that consists of a number of substructures both in parallel and in series connection.
2.2. Transmission characteristics of complex system Fig. 2 shows a system that consists of the subsystems S1 ; S 2 ; . . . ; Sk connected in series. The dynamic characteristics of the subsystems can be written in the form of a transmission matrix. 8* 9 ( ) < FS = FS 1 1 , (6a) ¼ TS1 * VS 1 :V ; S1 8* 9 < FS = 2
* :V ; S2
( ¼ TS2
FS 2 VS 2
) ,
(6b)
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8* < FS
k1
9 =
S k1
;
*
:V
.. .
(
¼ TSk1
FSk1 VSk1
371
)
(6c)
:
For the last subsystem Sk, the relation between the velocity and force vectors is given by *
VSk ¼ MSk F Sk ,
(7)
where TSi (i ¼ 1; 2; . . . ; k 1) is the * transmission * * matrix of subsystem, MS k is the mobility matrix of the receiver structure Sk, and F Sk ; VSk , F Si ; VSi , and FSi ; VSi (i ¼ 1; 2; . . . k 1) are the input and output state vectors of the subsystem. Considering the continuity condition of the force and velocity vectors at the interface between any two subsystems, the following relation is obtained: *
*
FSi ¼ F Siþ1 ; VSi ¼ VSiþ1 ,
(8a,b)
where i ¼ 1; 2; . . . ; k 1. Then, the transmission matrix equation of the complex system can be obtained by sequential multiplications of the transmission matrices of subsystems S 1 ; S2 ; . . . ; S k , 8* 9 8 9 < FS = <* F Sk = 1 ¼ K , (9) * :V ; : VSk ; S1
" where K ¼
K11
K12
K21
K22
# ¼ TS1 TS2 . . . TSk1 is the overall transmission matrix of the complex
system. The submatrices can be obtained by Kij ¼
2 X n¼1
2 X 2 X
TS1 ;il TS2 ;lm . . . TSk1 ;nj ;
i; j ¼ 1; 2.
m¼1 l¼1
Therefore, the force and velocity vectors of subsystem Si can be expressed in terms of 8* 9 ( ) FS i 1 < F S1 = ¼ TS 1 TS 2 TS i . * VS i :V ;
(10)
S1
By substituting Eq. (7) into Eq. (9), the input and output variables of the overall system can be obtained. *
*
VS1 ¼ ðK21 þ K22 MSk ÞðK11 þ K12 MSk Þ1 F S1 ,
(11a)
* F Sk
(11b)
*
¼ ðK11 þ K12 MSk Þ1 F S1 , *
VSk ¼ MSk ðK11 þ K12 MSk Þ1 F S1 .
(11c)
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The time average power PS1 of subsystem S1 that is input into the overall system and the power transmitted into the supporting structure PSk are given by H H * * * 1 1 PS1 ¼ Re F S1 VS1 ; PSk ¼ Re F S1 VSk , (12a,b) 2 2 where the symbol H represents the transpose and conjugate of a matrix. 2.3. Transmission matrix of a resilient mount For a complex coupled system in which all substructures are different, a general formulation of a lumped or distributed substructure is required to characterise the dynamic behaviour of the assembled system. In Sections 2.3–2.5, the transmission matrix formulations for MIMO source, transmission, and support substructures, which are the basic components of many vibration problems, are derived. The formulations are used to systemically describe the transmission characteristics of an assembled system with multiple connections. The vibration of a supporting structure is induced by the excitations that are transmitted through the isolators. The isolators can affect the radiated sound power from the receiving structure. Typically, linear isolator theory, which neglects the mass effects, can represent the stiffness and damping properties only. However, the transmissibility curve of an isolator system that is modelled by this linear isolator theory will overestimate the isolation performance at high frequencies, especially for cases of low damping. This occurs because the so-called wave effects in practical resilient mounts are neglected in typical isolator theory. In practice, an isolator has mass, and thus its resonant responses can affect its performance. The resonances of isolators may occur at high frequencies, when their dimensions are comparable with multiples of the half-wavelength of elastic waves travelling through the materials. In Fig. 4, the rubber isolator is modelled as a
Fig. 4. Schematic of a general elastic mount.
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continuous elastic circular rod. Forces and moments that act on the mount induce motions in the corresponding translational and rotational dof. This model can predict the longitudinal, flexural, shearing, and torsional wave effects in a mount. The top and bottom of the resilient mount are defined as the input and output terminals, respectively, where the subscripts 1 and 2 represent the input and output variables. The force and velocity vectors at the input and output terminals are denoted by
Fm1
8 9 F m1x > > > > > > > > > > F > m1y > > > > > > < F m1z > = ¼ ; > T m1x > > > > > > > > > T m1y > > > > > > > > :T ;
Vm1
m1z
Fm2
8 9 F m2x > > > > > > > > > > F > > m2y > > > > > > < F m2z = ¼ ; > T m2x > > > > > > > > > > T m2y > > > > > > > :T ;
Vm2
m2z
8 9 vm1x > > > > > > > > > v > > m1y > > > > > > < vm1z > = ¼ _ , > ym1x > > > > > > > > y_ m1y > > > > > > > > > : y_ ; m1z
(13a,b)
8 9 vm2x > > > > > > >v > > > > > m2y > > > > > > < vm2z = ¼ _ . ym2x > > > > > > >_ > > ym2y > > > > > > > >_ > : ym2z ;
(14a,b)
According to the transmission matrix of a substructure in Eq. (4), the transmission matrix equation of the mount can be given by (
Fm1 Vm1
)
" ¼
Tm11
Tm12
Tm21
Tm22
#(
Fm2 Vm2
) ,
(15)
where Tmij (i; j ¼ 1; 2) are the transmission submatrices, which can be derived from the frequency response functions of the distributed mount. The detailed derivation of the transmission submatrices for the mount is shown in the Appendix. 2.4. Transmission matrix of a rigid-body source Fig. 1 illustrates a general three-dimensional rigid-body source structure with the coordinate system O-xyz originated at the centre of gravity. The subsystem S1 schematically represents a series of source * substructures. It is assumed that the source structure is excited on its centre of gravity. The response velocity vector at by a general force vector F e that acts * the centre of gravity is denoted as Ve . The output force and velocity vectors at the n connection
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points are given by 8 19 8 19 Fa > Va > > > > > > > > > > > > > > > 2 > > < Fa > < V2a > = = Fa ¼ .. >; Va ¼ > .. >. > > > . > . > > > > > > > > > > > : Fn > : Vn > ; ; a a
(16a,b)
According to the theory of rigid-body dynamics, the equations of translational and rotational motion of the source structure can be expressed in the mobility form as (* ) " #( * ) M11 M12 Fe Ve ¼ , (17) M M 21 22 Va Fa where M11 ¼ ð1=joÞJ1 , M12 ¼ M11 TF , M21 ¼ Tv M11 , M22 ¼ Tv M11 TF , and J is the diagonal matrix of inertia relative to the center of gravity. TF and Tv are the matrices used to correlate the *
*
force and velocity vectors Fa and F e , Va and Ve by considering the rigid-body constraint * Fe
*
¼ TF Fa ; Va ¼ Tv Ve .
Rearranging the input and output terms in Eq. (17) gives 8* 9 " #( )
(18a,b)
(19)
e
2.5. Mobility matrix of a supporting structure In Fig. 2, a block diagram represents the supporting structure of the assembled system. As shown in Eq. (7), a mobility matrix represents the input–output characteristics of the receiving structure, which consider the force and moment mobilities at the connection points on the supporting structure. For some simple structures, the force and moment mobilities can be calculated analytically. In typical composite structures, such as buildings and ships, the supporting structure of a machine is usually constructed of beams and plates. Refs. [1,2] presented the formulas for the driving point mobility for beams, plates, and beam-stiffened plates of finite or infinite extent. The results showed that the first modes of the receiver structure greatly influence the transmission of vibration in a finite structure. The general modal formulation of the longitudinal, shear and flexural mobilities of plates and shells was presented in [23]. Other works (e.g. [10,12,24–29]) showed the generalized mobility and impedance equations for supporting beams, plates, and flexible cylinders subject to various boundary conditions. In this study, the support structure is modelled as a rectangular plate. At each connection point, the supporting plate is driven by concentrated forces (in the x-, y-, and z-directions) and moments (around the x-, y-, and z-axes). It is assumed that in-plane forces act in the middle of the plate cross-section. Only in-plane longitudinal waves, in-plane shear waves induced by the in-plane forces, and out-of-plane flexural waves induced by out-of-plane force and the flexural moment are
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taken into account. The twisting moment Tz on the plate surface does not generate any flexural waves in the plate. Thus, the driving point mobility and the transfer mobility of the supporting structure can be derived using wave equations [12]. Consider the out-of-plane motion and the rotations about the x- and y-axes as a coupling and superposing of the contributions from the connection points. The mobility submatrix MijSk (i; j ¼ 1; 2; . . . ; n; n is the number of points of connection on supporting structure) in matrix MSk is given by 3 2 ij mvx F x 0 0 0 0 0 7 6 ij 7 6 0 0 0 0 0 m 7 6 vy F y 7 6 ij ij ij 7 6 0 m m 0 0 m 7 6 vz F z vz T x vz T y (20) MijSk ¼ 6 7. 6 0 0 07 0 mijyx F z mijyx T x 7 6 7 6 6 0 0 mijyy F z 0 mijyy T y 0 7 5 4 0 0 0 0 0 0
3. Results and discussion In this section, the power transmission from a vibrating machine to a flexible foundation through multiple resilient mounts is investigated numerically. The aims of this investigation are (i) to illustrate the generalisation and effectiveness of the transmission matrix model proposed for complex systems; and (ii) to determine the power flow through each mount and explore the multi-dimensional wave effects in the mount and the multi-mount effects on the vibration transmission of isolation systems subject to various excitations. In the analysis, the machine is modelled as a 3dof system supported on a rectangular plate through multiple resilient mounts (see Fig. 5). The combined excitation that acts on the rigid body includes a vertical unit force Fey, a transverse unit force Fex, and a unit pitch moment Tex. The centre of gravity of the machine can vibrate in the vertical and transverse directions, and rotate about the x-axis. For a given external force vector and the corresponding velocity vector, the transmission matrices for the subsystems that represent the source, mount, and receiver can be obtained using Eqs. (15), (19), and (20). The overall transmission matrix in Eq. (9) is determined using sequential matrix multiplication. The input velocities and output forces and velocities at each connection point are then obtained by substituting the submatrices of the transmission overall matrix into Eqs. (11a–c). The power-flow spectra are calculated in decibel scale (dB, Pref ¼ 1012 W). In the numerical calculations, the following data are used.
Vibrating
machine: The mass and moment of inertia about the x-axis are m ¼ 105:3 kg and J ¼ 3:16 kg m2 , respectively. Cylindrical rubber isolator: The internal diameter, external diameter, and height are D d h ¼ 0:06 m 0:03 m 0:10 m; and the density, Young’s modulus, and loss factor are r ¼ 1000 kg=m3 , E ¼ 6:5 106 N=m2 , and Z ¼ 0:15. Supporting structure: The long side of the plate is parallel to the y-axis. The dimensions are a b s ¼ 1:0 m 1:5 m 0:005 m; and the density, Young’s modulus, shear modulus, loss
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L. Sun et al. / Mechanical Systems and Signal Processing 21 (2007) 365–388 Fez Tex
Fey
1
…. .
2
n
x y z
Fig. 5. A multi-mount isolation system subject to combined excitation.
factor, and Poisson’s ratio are r ¼ 2700 kg=m3 , E ¼ 7 1010 N=m2 , G ¼ 2:4 1010 N=m2 , z ¼ 0:02, and m ¼ 0:33.
3.1. Dynamic characteristics of the mdof isolation system Consider a case in which the source structure is isolated by two identical mounts that are symmetrically installed on the foundation plate. The three resonant frequencies that correspond to the vertical mode and the coupled transverse and pitch modes of the source vibration are 8.1, 4.7, and 10.8 Hz, respectively. The first two resonant frequencies of the longitudinal modes of the distributed mount clamped at both ends are 403.1 and 806.2 Hz, and the first resonant frequency of the flexural mode is 481.2 Hz. The six lowest resonant frequencies of the supporting plate are f 1;1 ¼ 17:7 Hz, f 1;2 ¼ 34:2 Hz, f 2;1 ¼ 54:7 Hz, f 1;3 ¼ 61:5 Hz, f 2;2 ¼ 71:7 Hz, and f 2;3 ¼ 98:5 Hz. In this subsection, the numerical results of the total power input to the source and power transmitted to the foundation plate through distributed mounts are presented. The power spectra of the isolation system are shown in Figs. 6a–d for the four excitation cases (i.e. combined excitation, vertical force, transverse force, and pitch moment). In Fig. 6a, the resonant peaks due to the three rigid-body modes are observed at about 4, 5, and 8 Hz, in the case of combined excitation (the peaks of 4 and 5 Hz almost coincide). Due to the coupling with the foundation plate, the values of these three resonance frequencies are different from those in the uncoupled case. In Fig. 6b, the source is excited by a unit vertical force. A resonant peak at about 5 Hz is induced by the vertical oscillation of the rigid-body machine in the case of vertical force excitation. In Figs. 6c–d, the source is excited by a unit transverse force/pitch moment. Two resonant peaks at about 4 and 7 Hz are observed. These two resonances correspond to the coupled modes of transverse and pitch oscillations of the machine. The resonances at about 22 and 40 Hz
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100
100 Power flow (dB)
120
Power flow (dB)
120
80 60 40 20
100 (a)
101 102 Frequency (Hz)
80 60 40 20
—— System ...... Foundation 103
100 (b)
100
100
80
80
40 20
100 (c)
Power flow (dB)
120
Power flow (dB)
120
60
101 102 Frequency (Hz)
103
101 102 Frequency (Hz)
103
40
100 (d)
System —— Po werinput into the ystem s ......Po wertrans mi tted tofo the undati on ate pl Foundation
60
20
—— System ...... Foundation
377
—— System ...... Foundation 101 102 Frequency (Hz)
103
Fig. 6. Power transmitted to the system and foundation plate, s ¼ 0:005 m: (a) combined excitation, (b) vertical force, (c) pitch moment and (d) transverse force.
are due to the first two flexural modes of the plate. Due to the coupling with the mounts, these two resonant frequencies are greater than those in the uncoupled case. In Figs. 6a–d, the vertical rigidbody mode and the coupled transverse and pitch modes of the system cause large power transmission at low frequencies. At frequencies above the resonances of rigid-body modes, many resonant peaks appear in the power-flow spectra. These resonances are caused by the flexural modes of the plate, which strongly affect the power transmission at higher frequencies. In particular, the resonances of the first two flexural modes induce much vibration power input into the system that is transmitted to the foundation. Moreover, the pitch moment causes more power flow than the vertical and transverse forces at the resonant frequencies of the flexural modes of the plate. This confirms that a pitch moment excitation greatly influences the power flow of a flexible isolation system. In Fig. 8, a foundation plate of 0.008 m thick is considered. The lowest four
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L. Sun et al. / Mechanical Systems and Signal Processing 21 (2007) 365–388 120
Power flow (dB)
100 80 60 40 20
—— System ...... Foundation
100
101
102
103
Frequency (Hz)
Fig. 7. Power transmitted to the system and thick foundation plate (s ¼ 0:008 m, vertical force).
120
Power flow (dB)
100 80 60 40 20
——
Symmetrically supported
...... Asymmetrically supported
100
101
102
103
Frequency (Hz)
Fig. 8. Power transmitted to the foundation plate through the two resilient mounts (s ¼ 0:005 m, vertical force).
resonant frequencies of the flexural modes are f 1;1 ¼ 28:5 Hz, f 1;2 ¼ 54:7 Hz, f 2;1 ¼ 87:6 Hz, and f 1;3 ¼ 98:5 Hz. A comparison of the thin and thick foundation plates in Figs. 6b and 7 show that the greater stiffness of the thick plate causes its resonant frequencies to be higher and thus the input of less power flow into the system. This observation indicates that the mobility of the supporting structure plays an important role in the vibration transmission of the whole system. In the above cases, the two identical mounts are installed symmetrically on the foundation plate. The first two resonances of the flexural modes of the plate cause a significant vibration power transmission. The case in which the mounts are installed asymmetrically on the plate is considered in Fig. 8. The resonant peaks of the vertical rigid-body mode shift to higher frequencies, and those of the first two flexural modes go to lower frequencies. Generally, there is more power flow in the case of asymmetrical mounts than in the case of symmetrical mounts.
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3.2. Resonances of the mounts Figs. 6 and 7 show that the two resonant peaks around 400 and 800 Hz can be clearly identified for the cases of combined excitation, vertical force excitation, and pitch moment excitation (see the peaks circled), which are due to the first two resonances of the longitudinal modes of the mounts. In the case of transverse force excitation, a peak is observed around 500 Hz, which is due to the first flexural wave resonance in the mounts. It can be seen that the resonances of the mounts can degrade the isolation performance. A comparison of Figs. 6b and 7 shows that the power input into the system around the resonant frequency of the mounts is higher in the case of the thick foundation plate. Fig. 9 shows the spectra of the two cases with and without isolator mass effects. The power transmission of the case with the mass effect is 10 dB higher than that without the mass effect at frequencies above 200 Hz. The peak values of the resonances of the rigid-body mode and the flexural modes are smaller in the case without the isolator mass effect. Fig. 10 shows the influence of the mount damping on the power that is transmitted to the foundation, in which the two cases of Z ¼ 0:15 and Z ¼ 0:3 are considered. The power transmission of the resonance of the rigid body mode and the flexural modes are reduced when the loss factor is increased to 0.3 (see the first three peaks). The power transmission due to the resonant responses of the mounts is also reduced, but not as much as those of the first three modes of the system. These observations show that the flexibility of the supporting structure is an important influence on the vibration transmission, and that the resonant effects of the mounts significantly increase the power transmission at higher frequencies. 3.3. Power flow of a system with multi-mounts In this section, the power transmission of an isolation system with three identical mounts is investigated. The three identical mounts are symmetrically installed on the foundation plate. The resonant frequencies of the vertical rigid-body mode and the coupled transverse and pitch modes 120
Power flow (dB)
100 80 60 40 20
100
With mass ...... Without mass 101
102
103
Frequency (Hz)
Fig. 9. Power transmitted to the foundation plate through different mounts (s ¼ 0:005 m, vertical force).
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Power flow (dB)
100 80 60 40 20
100
τ = 15.0
τ = 3.0 101
102
103
Frequency (Hz)
Fig. 10. Power flow transmitted to the foundation plate for different damping ratios (s ¼ 0:005 m, vertical force). 120
Power flow (dB)
100 80 60 40 20
100
—— system ...... Foundation 101
102
103
Frequency (Hz)
Fig. 11. Power transmitted to the system and foundation plate through three mounts (s ¼ 0:005 m, combined excitation).
are 8.4, 6.6, and 9.1 Hz, respectively, the first two resonant frequencies of the longitudinal modes of the mount are 353.6 and 707.1 Hz, and the first resonant frequency of the flexural mode is 422.3 Hz. Figs. 11 and 13 show the power transmission through the three mounts for the cases of vertical force or combined excitation (including a unit vertical force, unit transverse force, and unit pitch moment). When compared with those in the two-mount system in Figs. 6a–b, the power that is transmitted to the foundation plate through the three mounts decreases at low frequencies. Similar to the two-mount isolation system, two resonant peaks due to the first two longitudinal modes of the mounts are observed in Figs. 11 and 13. The power that is transmitted through mount 1 is slightly greater than the power that is transmitted through mount 3 in Fig. 12 (the case of combined excitation) but almost equal to that which is transmitted through mount 3 in Fig. 14 (the case of vertical force). Around the resonant frequency of the rigid-body modes, the power
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100
Power flow (dB)
50
0
-50 —— mount 1 —• mount 2 ...... mount 3
-100
-150 100
101
102 Frequency (Hz)
103
Fig. 12. Power transmitted to the foundation through a particular resilient mount (s ¼ 0:005 m, combined excitation).
120
Power flow (dB)
100 80 60 40 20
100
—— System ...... Foundation 101
102
103
Frequency (Hz)
Fig. 13. Power transmitted to the system and foundation plate through three mounts (s ¼ 0:005 m, vertical force).
that is transmitted through mount 2 can be negative (see the circled regions in Figs. 12 and 14). This implies that a net power flow is directed from the supporting structure towards the source. In other frequencies, the power transmitted through the mount 2 is null (Figs. 13 and 14)
4. Conclusions The transmission matrix approach is employed in this paper for the vibrational power-flow analysis of a multiple-input/multiple-output (MIMO) system. The substructures in each subsystem are modelled as a group of general MIMO transmission elements, which are independent of each other. The global transmission matrix of the system is formed by assembling the transmission matrices together. Using the proposed approach, the output variables of any
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L. Sun et al. / Mechanical Systems and Signal Processing 21 (2007) 365–388 100
Power flow (dB)
50
0
-50 —— —• ......
-100
-150 100
101
102
mount 1 mount 2 mount 3
103
Frequency (Hz)
Fig. 14. Power transmitted to the foundation through a particular resilient mount (s ¼ 0:005 m, vertical force).
subsystem can be expressed in terms of the input state variables of the subsystem subject to the external loads. Hence, the proposed approach is very systematic in analysing the dynamic behaviour of a complex system with a group of elements. In the numerical example, the power flow of a multi-mount system is modeled analytically. The transmission of vibratory power flow from the vibrating rigid body into a simply supported plate through elastic mounts is studied in detail. The following conclusions can be drawn. (1) The vertical rigid-body mode and the coupled transverse and pitch modes of the system are the dominant modes affecting the power that is transmitted to the system and supporting structure at low frequencies. (2) The resonances of the first few flexural modes of the plate are the dominant modes that affect much of the power that is transmitted to the system and supporting structure at middle frequencies. (3) The resonances of the mount are the dominant modes that affect much of the power that is transmitted to the system and supporting structure at high frequencies. (4) The mounts with mass effect can degrade the isolation performance 10 dB lower than those without mass effect. The power transmission due to the resonant responses of the mounts can be reduced slightly by increasing the damping of mounts. (5) The unit pitch moment excitation induces more power input into the system than the unit vertical and transverse forces at the resonant frequencies of the plate. (6) Generally, there is less power flow transmitted into the supporting structure in the case of symmetrical mounts and thick plates than in the asymmetrical and thin plate cases. (7) There is less power transmitted into the supporting structure at low frequencies in the case of three mounts than that of two mounts. Around the resonant frequency of the rigid-body modes, a net power flow is directed from the supporting structure to the source. At other frequencies, the power that is transmitted through the middle mount is null.
Acknowledgement This research was supported by the National Natural Science Foundation of China.
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383
Appendix Consider the input and output variables at the yoz-plane Fm1y, Fm1z, Tm1x, Fm2y, Fm2z, Tm2z in the dynamic model of the resilient mount in Fig. 3. The driving point or transfer mobility frequency response functions of a uniform beam are given by [18] Rvm1 y F m1 y ¼ Rvm2 y F m2 y ¼ jo
cos lf h sinh lf h sin lf h cosh lf h , E s Il3f ðcos lf h cosh lf h 1Þ
Rvm1 y T m1 x ¼ Rvm1 y T m2 x ¼ Ry_ m x F m y ¼ Ry_ m x F m y ¼ jo 1
Rvm1 z F m1 z ¼ Rvm2 z F m2 z ¼ jo
1
2
2
cos ll h , AE s ll sin ll h
sin lf h sinh lf h 3 E s Ilf ðcos lf h cosh lf h
(A.5)
2
Rvm2 z F m1 z ¼ Rvm1 z F m2 z ¼ jo Ry_ m x T m x ¼ Ry_ m x T m x ¼ jo 1
1
2
(A.2)
(A.3)
1Þ
Rvm2 y F m1 x ¼ Rvm1 y T m2 x ¼ Ry_ m x F m y ¼ Ry_ m x F m y ¼ jo
2
sin lf h sinh lf h , lf h cosh lf h 1Þ
E s Il2f ðcos
(A.4)
2
Rvm2 y F m1 y ¼ Rvm1 y F m2 y ¼ jo
2
cos lf h sinh lf h þ sin lf h cosh lf h , E s Ilf ðcos lf h cosh lf h 1Þ
Ry_ m x T m x ¼ Ry_ m x T m x ¼ jo 1
1
(A.1)
AE s ll
1
1
2
, cos lf h cosh 2 E s Ilf ðcos lf h cosh
lf h , lf h 1Þ
1 , sin ll h
(A.6)
(A.7)
sin lf h þ sinh lf h , E s Ilf ðcos lf h cosh lf h 1Þ
(A.8)
where lf and ll are the complex wave numbers of the flexural and quasi-longitudinal waves in the rod, and h, A, and I are the height, area, and moment of inertia of the cross-section of the mount (with reference to the x-axis). E s ¼ E s ð1 þ jZÞ is the complex Young’s modulus, where Z is the structural loss factor (refer to [18] for more details). The equation of motion of the transmission structure can be described in terms of mobility ( ) ( ) ¯ m1 F¯ m1 U ¯ (A.9) ¯ m2 ¼ Rm F¯ m2 , U where ¯ m1 U
8 9 > < vm1y > = ¼ vm1z ; > > : y_ ; m1x
¯ m2 U
8 9 > < vm2y > = ¼ vm2z ; > > : y_ ; m2x
F¯ m1
8 9 > < F m1y > = ¼ F m1z ; > :T > ; m1x
F¯ m2
8 9 > < F m2y > = ¼ F m2z . > :T > ; m2x
(A.10213)
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¯ m are given by The submatrices in R 3 2 Rvm1 y F m1 y 0 Rvm1 y T m1 x 7 6 0 Rvm1 z F m1 z 0 7; ¯ m11 ¼ 6 R 5 4 Ry_ m x F m y 0 Ry_ m x T m x 1
1
1
2 6 ¯ m12 ¼ 6 R 4
1
Rvm1 y F m2 y
0
Rvm1 y T m2 x
0
Rvb1 z F b2 z
0
0
Ry_ m x T m x
Ry_ m x F m y 1
2
1
3 7 7, 5
2
(A.14,15) 2 6 ¯ m21 ¼ 6 R 4
Rvm2 y F m1 y
0
Rvm2 y T m1 x
0
Rvb2 z F b1 z
0
0
Ry_ m x F m x
Ry_ m x F m y 2
1
2
3
2
7 7; 5
6 ¯ m22 ¼ 6 R 4
1
Rvm2 y F m2 y
0
Rvm2 y T m2 x
0
Rvm2 z F m2 z
0
0
Ry_ m x T m x
Ry_ m x F m y 2
2
2
3 7 7. 5
2
(A.16,17) Then, by rearranging the input and output variables, Eq. (A.9) can be expressed in terms of a transmission matrix as follows: ( ) ( ) F¯ m1 F¯ m2 ¯ (A.18) ¯ m1 ¼ Tm U ¯ m2 , U ¯ m can be obtained using Eqs. (A.14)–(A.17) and are given by where the submatrices in T 3 2 G2 l f G1 0 6 2 2 7 7 6 1 6 0 7 cos ll h ¯ m22 ¼ 6 0 ¯ m21 R ¯ m11 ¼ R (A.19) T 7, 6 G3 G1 7 5 4 0 2lf 2 2 ¯ m12 ¼ R ¯ 1 T m21
G3 E s Il3f
6 2 1 6 6 0 ¼ 6 jo 6 4 G4 E s Il2f
0 E s All sin ll h 0
2
¯ m21 ¼ R ¯ 1 ¯ ¯ m12 R ¯ m11 R T m21 Rm22
3
7 7 7 7, 7 G2 E s Ilf 5 2
2
G2 6 2E s Il3f 6 6 6 0 ¼ jo6 6 6 6 G4 4 2E s Il2f
G4 E s Il2f 2 0
0
sin ll h E s All 0
G4 3 2 2E s Ilf 7 7 7 7 7, 0 7 7 G3 7 5 2E s Ilf
(A.20)
(A.21)
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2
¯ m22 ¼ T
¯ 1 ¯ m11 R R m21
G1 6 2 6 6 ¼6 0 6 4 G2 lf 2
G3 0 2lf cos ll h 0 G1 0 2
385
3 7 7 7 7, 7 5
(A.22)
where G1 ¼ cos lf h þ cosh lf h, G2 ¼ sin lf h sinh lf h, G3 ¼ sin lf h þ sinh lf h, and G4 ¼ cos lf h cosh lf hSimilarly, the transmission equation that is used to correlate the input and output variables at the xoz plane and the moment around the z-axis and the corresponding velocity is given by ( ) ( ) F^ m2 F^ m1 ¼ T^ m , (A.23) ^ m1 ^ m2 U U where F^ m1
8 9 > < F m1x > = ¼ T m1y ; > > ; :T m1z
^ m1 U
8 9 v > < m1x > = ¼ y_ m1y ; > > ; :_ ym1z
F^ m2
8 9 > < F m2x > = ¼ T m2y ; > > ; :T m2z
^ m are given by The submatrices in T 2 3 G1 G2 lf 0 6 2 7 2 6 7 ^ m11 ¼ 6 G2 lf 7, G1 T 6 7 0 4 2 5 2 0 0 cos ls h 2 ^ m12 T
6 6 2 1 6 2 6 ¼ 6 G4 E s Ilf jo 6 2 4 0 2
^ m21 T
G3 E s Il3f
6 6 6 6 ¼ jo6 6 6 6 4
G2 2E s Il3f G4 2E s Il2f
0
G4 2E s Il2f G3 2E s Ilf
0
(A.24227)
(A.28)
3
G4 E s Il2f
2 G2 E s Ilf 2 0
^ m2 U
8 9 v > < m2x > = ¼ y_ m2y . > > ; :_ ym2z
0 0 E t I p ls sin ls h
0
7 7 7 7, 7 7 5
(A.29)
3
7 7 7 7 7, 0 7 7 7 sin ls h 5 E t I p ls
(A.30)
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386
2
^ m22 T
G1 6 2 6 6 ¼ 6 G2 lf 6 4 2 0
G3 2lf G1 2 0
3 0 0 cos ls h
7 7 7 7, 7 5
(A.31)
where ls are the complex wave numbers of shearing wave, and E t is the complex shear modulus. Finally, using Eqs. (A.18) and (A.23), the transmission matrix of the mount in Eq. (14) Tm can be derived, with the submatrices given by
2
Tm11
G1 6 2 6 6 6 0 6 6 6 0 6 ¼6 6 0 6 6 6 6 G2 l f 6 4 2 0
2
Tm12
0 G1 2 0 G3 2lf
0
0
0
0
0
0
0
0
0
G2 lf 0 2 0 cos ll h G1 0 2
0
0
0
0
0
0
G3 E s Il3f
6 2 6 6 6 6 0 6 6 0 1 6 6 ¼ 6 jo 6 6 0 6 6 6 2 6 G4 E s Ilf 6 4 2 0
G2 lf 2
0
G3 E s Il3f 2 0
G4 E s Il2f 2 0 0
G1 2 0
0 0
(A.32)
cos ls h
0
0
0
7 7 7 7 7 7 7 7 7, 7 7 7 7 7 7 5
0
0
E s All
3
sin ll h
G4 E s Il2f 2 0 G2 E s Ilf 2 0 0
G4 E s Il2f 2
3 0
0
0
0
0
0
0
G2 E s Ilf 2 0
0 E t I p ls sin ls h
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
(A.33)
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2
Tm21
G2 6 2E Il3 s f 6 6 6 6 0 6 6 6 6 6 0 6 6 ¼ jo6 6 0 6 6 6 6 G4 6 6 6 2E s Il2f 6 6 4 0
0
0
G2 2E s Il3f
Tm22
0
sin ll h E s All
G4 2E s Il2f
0
G3 2E s Ilf
0
0
0
0
0
0
0
0
G1 2 0 G2 lf 2
cos ll h
0
0
0
0
0
0
0
0
G3 2lf 0 G1 2
0 0
G3 2E s Ilf 0
0
3
7 7 7 7 7 0 7 7 7 7 7 0 7 7 7, 7 0 7 7 7 7 7 0 7 7 7 sin ls h 7 5 E t I p ls
(A.34)
3
G3 2lf
0
0
0
0
0
0
0
G1 2 0
0
0
G4 2E s Il2f
2
G1 6 2 6 6 6 6 0 6 6 6 0 ¼6 6 6 0 6 6 6 6 G2 lf 6 4 2 0
0
0
0
G4 2 2E s Ilf
387
0 cos ls h
7 7 7 7 7 7 7 7 7. 7 7 7 7 7 7 7 5
(A.35)
For cases of vertical force excitation only, if the excitation frequency is much lower than the first resonant frequency of the longitudinal mode of the mount, then l1 h ! 0, cos l1 h ! 1, and sin l1 h ! l1 h. Hence, Eqs. (A.36)–(A.39) can be simplified to Tm11 ¼ 1;
Tm12 ¼ 0;
Tm21 ¼
jo ; K
Tm22 ¼ 1.
(A.36239)
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