Mech060513.qxd The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 __________________________________________ *Corresponding author e-mail: asiri.net@gmail.com Broadband Vibration Attenuation Using Hybrid Periodic Rods S. Asiri Department of Mechanical Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia Received 13 May 2006; accepted 17 December 2006 Abstract: This paper presents both theoretically and experimentally a new kind of a broadband vibration isolator. It is a table-like system formed by four parallel hybrid periodic rods connected between two plates. The rods consist of an assem- bly of periodic cells, each cell being composed of a short rod and piezoelectric inserts. By actively controlling the piezoelec- tric elements, it is shown that the periodic rods can efficiently attenuate the propagation of vibration from the upper plate to the lower one within critical frequency bands and consequently minimize the effects of transmission of undesirable vibra- tion and sound radiation. In such a system, longitudinal waves can propagate from the vibration source in the upper plate to the lower one along the rods only within specific frequency bands called the "Pass Bands" and wave propagation is efficient- ly attenuated within other frequency bands called the "Stop Bands". The spectral width of these bands can be tuned accord- ing to the nature of the external excitation. The theory governing the operation of this class of vibration isolator is present- ed and their tunable filtering characteristics are demonstrated experimentally as functions of their design parameters. This concept can be employed in many applications to control the wave propagation and the force transmission of longitudinal vibrations both in the spectral and spatial domains in an attempt to stop/attenuate the propagation of undesirable distur- bances. Keywords: Pass Band, Stop band, Broadband vibration isolator, Piezoelectricity, Vibration attenua- tion, Wave propagation, Tunable filtering Notation A Cross-section area b Width of piezo-insert CD Elastic modulus Dp Electrical displacement of the piezo-insert E Young's modulus Ep Electrical field intensity of the piezo-insert F Force FI Total interface force fN Excitation harmonic force at the end of strut Fp piezo-force hp Piezo-coupling constant ájQh~dG Úé¡dG ¿ÉÑ° b ∫ɪ©à°SÉH ¢ jôY ¥É£f ≈∏Y äGRGõàg’G ∞«ØîJ …Ò°ùY .¢S áá°°UUÓÓÿÿGGá©HQG øe πµ°ûàj ƒ¡a ¬dhÉ£dÉH ¬«Ñ°ûJ øµÁ ΩɶædG Gòg .¢ jôY ¥É£f ≈∏Y äGRGõàg’G ∫RGƒY øe G~j~L ÉYƒf á«ÑjôéàdG h ájô¶ædG Úà«MÉædG øe Ω~≤J ¬bQƒdG √òg : hõ«ÑdG ôKG §«°ûæàHh .»FÉHô¡c QõH ≥ë∏eh Ò°üb Ê~©e AõL øe á«∏N πc ¿ƒµàJ h á∏Kɪàe ÉjÓN áYƒª› øe ¿ƒµàJ ájQh~dG ¿ÉÑ° ≤dG .ÚMƒd ÚH π°üJ ájRGƒàe áæé¡e ¿ÉÑ° b É¡©æeh RGõàg’G øe ∞«ØîàdG É¡æµÁ ájQh~dG ¿ÉÑ° ≤dG ¿G åëÑdG Gòg í°VhCG.É©e á«dÉ©ŸG h ¢ ØîæŸG OOÎdG äGP äGRGõàg’G ∫õY ‘ ôKDƒe πµ°ûH ΩɶædG πª©j ¿G øµÁ »FÉHô¡µdG ∫ÓN πØ°SC’G ¤G ≈∏YC’G ìƒ∏dG ‘ RGõàg’G Q~°üe øe …ô°ùJ ¿CG øµÁ á«dƒ£dG äÉLƒŸG ΩɶædG Gòg ‘ .áHƒZôŸG ÒZ QÉKB’G π«∏≤J ‹ÉàdÉHh OOÎdG äÉbÉ£f øª°V ¿Éjô°ùdG øe Ò«¨J ™e É«∏ª©e ájô¶ædG áægôH ” ~bh .á«LQÉÿG ájRGõàg’G äGôKDƒŸG á©«Ñ£d É≤ah ¬ª«ª°üJ øµÁ äÉbÉ£ædG √òg ¢VôY .'∞bƒàdG ¥É£f ' äGOOÎdG äÉbÉ£f øª°V §≤a ¿ÉÑ° ≤dG ‘ ÊɵŸGh »Ø«£dG ÚdÉÛG ‘ äGRGõàg’G QÉ°ûàfG ∫õ©d IO~©àe äÉ≤«Ñ£J ‘ Ω~îà°ùj ¿G øµÁ Ωƒ¡ØŸG Gòg .≥«Ñ£àdGh ájô¶ædG ÚH ≥aGƒJ ∑Éæg ¿CG ~Lhh IôKDƒŸG πeGƒ©dG .áHƒZôŸG ÒZ äÉHGô£°V’G QÉ°ûàfG ∞«ØîJ hCG ±É≤j’G ádhÉfi ääGGOOôôØØŸŸGGáá««MMÉÉààØØŸŸGG .IôjÉ©ª∏d πHÉ≤dG í«°TÎdG ,äÉLƒŸG ¿Éjô°S ,äGRGõàg’G ¢ ØN ,á«FÉHô¡c hõ«ÑdG ,i~ŸG ¢ jôY RGõàg’G ∫RÉY ,∞bƒàdG ¥É£f ,QhôŸG ¥É£f : 8 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 1. Introduction The periodic rods in the proposed mechanical filter act as the transmission paths of the vibration from the upper plate where the source of vibrations is located to the lower plate. Therefore, a proper design of these rods is essential to the attenuation of the vibratory energy and the noise radiated into any structure connected to the lower plate. A periodic rod consists of an assembly of identical cells connected in a repeating array which together form a 1-D periodic structure. The study of periodic structures has a long history. Wave propagation in periodic systems relat- ed to crystals and optics has been investigated for approx- imately 300 years. Brillouin (1953) developed the theory of periodic structures for solid state applications and then, in the early seventies, the theory was extended to the design of mechanical structures (Mead, 1970; Cremer, et al. 1973). Since then, the theory has been extensively applied to a wide variety of structures such as spring-mass systems Faulkne and Hong, 1985, periodic beams (Mead, 1970; Faulkner and Hong, 1985; Mead, 1971; Mead, 1971; Mead and Markus, 1983; Roy, 1986; Gupta, 1970; Mead, 1986; Richards and Pines, 2003; Mead, 1996), stiffened plates (Gupta, 1970; Mead, 1986; Mead and Yaman, 1991), ribbed shells (Mead, 1987) and space structures. Examples of such structures are found in many engineering applications. These include bulkheads, heli- copter drive shafts (Richads and Pines, 2003), airplane fuselages, vehicle engine mounting systems (Asiri, 2005), and helicopter gearbox supporting systems (Asiri, et al. 2002). Each such structure has a repeating set of stiffen- ers which are placed at regular intervals. Sackman, et al. Sackman, et al. (1999) presented a layered notch filter device that is limited only to high-frequency vibrations. Such a filter which was developed theoretically based on the Floquet theory is a periodically layered stack of two alternating materials with widely different densities and stiffnesses. The work presented here is an experimental implemen- tation of the work of Baz (2001) in which an active peri- odic spring mass system is employed theoretically, in a quasi-static manner, to control the wave propagation of longitudinal vibration. Periodic rods in passive mode of operation exhibit unique dynamic characteristics that make them act as mechanical filters for wave propagation. As a result, waves can propagate along the periodic rods only within specific frequency bands called the "Pass Bands" and wave propagation is attenuated within other frequency bands called the "Stop Bands". The spectral width and location of these bands are fixed for a 1-D pas- sive periodic structure, but are tunable in response to the structural vibration for active periodic structures (Baz, kpc Active piezo-stiffness due to the control gain K kps Structural piezo-stiffness Kd Dynamic stiffness matrix Kg Control gain Kij Appropriately partitioned matrices of the stiffness Kp Total stiffness of the piezo-insert L Element length La Length of cell a Lb Length of cell b Mij Appropriately partitioned matrices of the mass N Number of cells Qp Electrical charge Sp Strain of the piezo-insert tp Thickness of piezo-insert T Transfer matrix of the unit cell Tk Transfer matrix of the kth cell Tp Stress of the piezo-insert u Longitudinal deflection V Applied voltage Y State vector = {uL FL}T Yk Eigenvetor of transfer matrix of the unit cell Yk Eigenvetor of the transpose of transfer matrix of the unit cell α Logarithmic decay of amplitude of state vector β Phase difference between the adjacent cells εs Electrical permittivity λ Eigenvalue of transfer matrix of the unit cell ρ Density µ Propagation Constant ω Excitation frequency 9 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 2001; Asiri, et al. 2004). The shape memory alloy has been used as a source of irregularities to attenuate the wave propagation in periodic rods (Asiri, et al. 2004). The spectral finite element analysis and transfer matrix method (Baz, 2001; Asiri, et al. 2004; Ruzzene and Baz, 2000; Baz, 2000) will be used to analyze the hybrid peri- odic rod and determine the propagation parameter, which indicates the regions of stop bands and pass bands. This paper is organized in four sections. In Section 1, a brief introduction is given. Section 2 presents the theo- retical background of the hybrid periodic rod and Section 3 demonstrates the performance characteristics of the tun- able mechanical filter. Comparisons between the theoret- ical and experimental characteristics are also presented in Section 3. Section 4 summarizes the findings and the con- clusions of the present study and also outlines the direc- tion for future research. 2. System Modeling 2.1 Overview In this section, the emphasis is placed on studying the dynamics of the tunable mechanical filter in order to demonstrate its unique filtering capabilities. The dynam- ics of one-dimensional hybrid periodic rods in their active and passive modes of operation are determined using the transfer matrix method. The basic characteristics of the transfer matrices of periodic rods are presented and relat- ed to the physics of wave propagation along these rods. The methodologies for determining the pass and stop bands as well as the propagation parameters are presented. In this paper, the focus is placed on hybrid periodic rods consisting of a straight rod with periodically placed piezo- electric inserts as shown in Fig. 1. 2.2 Spectral Finite Element Method of the Rod The first development of the spectral Finite Element (SFE) method occurred in the early eighties (Patera, 1984). The main distinct between the SFE and Finite Element (FE) stems from the type of the shape functions used to approximate the model equations. In FE method, the shape function is only a function of a spatial variable, whereas in SFE the shape function has two independent variables: the spatial variable (x) and the spectral variable (ω). As a result, the SFE method is much more accurate than FE method. For example, one can show that the exact natural frequencies for a rod can be found by one element using SFE method, whereas a large number of elements needs to be used to get the same accuracy using the FE method. Consider the rod shown in Fig. 2 The equation of motion of the rod is given by: (1) where u is the longitudinal deflection, ρ is the density and E is Young's modulus. Then, assuming a solution u(x,t) = U (x) eiωt, reduces the equation of motion to: (2) Using the following spectral shape function: U(x) = Ae-ikx + Beiks, which is also a solution of Eq. (1), yields the spectral finite element description of the dynamics of the rod. This results in the following dynamic stiffness matrix of the rod: (3) The corresponding Transfer Matrix [T] takes the fol- lowing form: (4) where (5) 2.3 Dynamics of the Hybrid Rod Consider now the dynamics of the hybrid periodic rods which consists of a passive sub-cell and an active piezo- electric sub-cell as shown in Fig. 3. Piezoelectric inserts Cell Figure 1. Typical example of hybrid periodic rod u aL x L R Figure 2. Rod undergoing longitudinal vibrations ( / ) 0xx ttu E uρ− = 2( ) ( ) 0xxU x k U x+ = where k = wave number = / Eρ ω . [ ] 4ikL 3ikL 2ikLa a a a a 3ikL 2ikL 4ikL )ikLa a a a a a ( 1 e )ikL 2e ( 1 e )ikLEA K L 2e ( 1 e )ikL ( 1 e − − − − ⎡ − − − + ⎢= ⎢ − − + −⎢⎣ 1 1K K Kd LLd dLR LR [ T ] K 1d K K K KRR d d 1LL RLd d KLR RR dLR − −⎡ ⎤−⎢ ⎥ ⎢ ⎥= − − −⎢ ⎥− ⎢ ⎥⎣ ⎦ 4(1 ) EA ikLK e ikLd LL L −= − , ( )3 22 ( 1 ) ,EA ikL ikLK e e ikLd LR L −= − − + ( )3 22 ( 1 )−= − − +ikL ikLdRL EAK e e ikLL , and 4(1 )−= − ikLdRR EA K e ikL L . 10 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 2.3.1 Passive Sub-cell As shown in Eq. (4), the dynamic characteristics of the passive sub-cell (a) can be described as follows: (6) In a more compact form, Eq. (6) can be rewritten as: (7) where is the transfer matrix of the passive sub-cell a. One can show that the transfer matrix is a simplectic matrix (Asiri, et al. 2002). 2.3.2 Active Sub-cell The constitutive equations of the active piezoelectric insert are given by Baz (2001): (8) where Ep, Dp, Tp and Sp are the electrical field intensity, electrical displacement, stress and strain of the piezo- insert. Moreover, εS, hp and CD define the electrical per- mittivity, piezo-coupling constant and elastic modulus. Eq. (8) can be rewritten in terms of applied voltage Vp, Interface piezo-force FI, electrical charge Qp and net deflection (uR-uI) as follows: (9) where tp and b are the diameter and the length of the piezo insert. Eliminating the charge Qp from Eq. (9) gives: (10) Let the piezo-voltage Vp be generated according to the following control law: Vp = -Kg (uR-uI) (11) where Kg is the control gain which can be complex if phase shift is allowed. Appropriate phase shift means that the piezo-insert can act as a damper. Then, Eq (10) reduces to: (12) Eq. (12) can be used to generate the force vector {F1 FR}T acting on the piezo-insert rewritten as: (13) with kp = kpc+kps is the total stiffness of the piezo-insert which is complex if the gain is complex by virtue of the phase shift. Hence, the dynamic equation of the active sub-cell is given by: (14) Sub-cells rod a piezo-insert b rod a Piezo-insert b FL FI FI FR La Lb uL uI uI uR Left (L) Interface (I) Right (R) (a) Complete cell (b) passive sub-cell (c) – active sub-cell Figure 3. Unit cell of the hybrid periodic rod − − − ⎡ ⎤− ⎢ ⎥⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭− +⎢ ⎥ ⎣ ⎦ 1 1 dd dILIL ILaa aL I 1L Id d d dd dLL II LI LLIL ILa a a aa a K K K u u F FK K K K K K where u and F define the deflection and force vectors with subscripts L and I denoting the left and interface sides of the passive sub -cell and d ija K defines the elements o f dynamic stiffness matrix of the sub -cells a which can be calculated from Eq. (5). ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ L I a L I u u T F F S p p p Dp pp E 1/ - h D = T S-h C ε⎡ ⎤⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎣ ⎦ ( )2 / 4 Sp p p p b b D R I bI b p V /t 1/ - h Q /D L = (u - u )/LF / D -h C ε π ⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎪ ⎪⎪ ⎪ ⎩ ⎭⎣ ⎦⎩ ⎭ 2[( / 4)( ) ]( )S D 2 SI p b p b p b R IF = - h D V + D C - h /L u - uε π ε { [( ( ) ]}( )S D 2 SI p g b p p R IF = h b K + t b C - h /L u - uε ε ( ) { 1 1} Ipc ps R u k k u ⎧ ⎫ = + − ⎨ ⎬ ⎩ ⎭ where 2[( / 4)( ) ]S D 2 Sk = h D K and k = D C - h /Lpc p b g ps p bbε π ε with kpc and kps denoting the active piezo -stiffness due to the control gain Kg and the structural piezo -stiffness , respectively. ( ) { } ⎧ ⎫ ⎧ ⎫⎧ ⎫ +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎩ ⎭⎩ ⎭ ⎩ ⎭ I I pc ps R R F u-1 = k k -1 1 F 1 u ⎡ ⎤ ⎧ ⎫ = ⎢ ⎥ ⎨ ⎬ ⎢ ⎥ ⎩ ⎭⎣ ⎦ p p I p p R k - k u -k k u ⎡ ⎤ ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥ =⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎢ ⎥⎣ ⎦ d dII IRb b I I d d R RRI RRb b K K u F K K u F 11 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 Now, the state vectors at ends R and I of the passive sub-cell b are related through the transfer matrix derived from Eq. (14) as follows: (15) In a more compact form, Eq. (15) can be rewritten as: 2.3.3 Dynamics of Entire Cell The dynamics of the entire cell can be determined by the assembly of the dynamic equations of the passive and active sub-cells which are given by Eq. (3) and Eq. (15) respectively. This yields the following dynamic equa- tions: (16) In a more compact form, Eq. (16) can be rewritten as: (17) where Y and [Tk] denote the state vector = {uL FL}T and the transfer matrix of the k cells. For exactly periodic rods, [Tk] = [T] and the eigenval- ue problem of [T ] can be written as: (18) where λ is the eigenvalue of [T] Combining Eq. (17) and (18) gives: (19) Indicating that λ of the matrix [T] is the ratio between the elements of the state vectors at two consecutive cells. Hence, the magnitude of λ determine the nature of wave dynamics in the periodic rod as follows: A further explanation of the physical meaning of the eigenvalue λ can be extracted by rewriting it as: (20) where µ is defined as the "Propagation Factor" which is a complex number whose real part (α) represents the loga- rithmic decay of the state vector and its imaginary part (β) defines the phase difference between the adjacent cells. One can rewrite Eq. (19) as: (21) (22) From Eq. (21) and Eq.(22), we get: (23) Eq. (19) indicates that: Therefore, the equivalent conditions for the Pass and the Stop bands can be written in terms of the propagation constant parameters (α and β) as follows: The propagation factor, µ, can be calculated directly from the diagonal elements of overall transfer matrix as following: (24) In case of a rod element, the transfer function has two eigenvalues µ and 1/µ and according to the properties of where d ijb K defines the elements of dynamic stiffness matrix of the sub -cell b which can be calculated from Eq. (13). − − − − ⎡ ⎤− ⎧ ⎫⎧ ⎫ ⎢ ⎥⎪ ⎪ ⎪ ⎪= ⎢ ⎥⎨ ⎬ ⎨ ⎬ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ − + ⎩ ⎭ ⎢ ⎥⎣ ⎦ 1 1 dd dIIIR IRbb b IR 1 1 IR d d dd dRR II RRIR IRb b bb b K K K uu FF K K K K K ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ R I b R I u u T F F where bT is the transfer matrix of the active sub-cell b. + ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪= ⎡ ⎤⎨ ⎬ ⎨ ⎬⎣ ⎦ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ L L k L Lk 1 k u u T F F where ⎡ ⎤ ⎢ ⎥= = ⎢ ⎥⎣ ⎦ 11 12 k b a 21 22 t t T T . T t t + = ⎡ ⎤⎣ ⎦k 1 k kY T Y [ ] k kT y Yλ= k 1 kY Yλ+ = If 1λ = , the wave propagates along the rod without any change of amplitude, indicating a Pass Band and if not then the wave will be attenuated, indicating a Stop Band . ie eµ α βλ += = 1 L Li L Lk k u u e F F α β+ + ⎧ ⎫ ⎧ ⎫ =⎨ ⎬ ⎨ ⎬ ⎩ ⎭ ⎩ ⎭ and considering only the jth components L ju of the deflection vector uL, at cells k and k+1, which can be written as : i jk L L Ljk 1 jk 1 jk u U and u e φ + + = = where L jnU and jnφ denote the amplitude and phase shift of the jth component L ju at the n th cell. ( ) ( ) ( ) 1 1 1 L L L Ljk jk jk jk jk jk ln u / u ln U / U i iφ φ α β + + + = + = = + α = 1L Lj jk k ln(U / U ) + = Logarithmic decay of amplitude, and ( )1jk jkβ φ φ+= − = phase difference between the adjacent cells a. If α =0 (ie. µ is imaginary), then we have “Pass Band ” as there is no amplitude attenuation. b. If 0α ≠ (ie. µ is real or complex), then we have “Stop Band ” ” as there is amplitude attenuation defined by the value of α . 1 1 2 2aco sh 2 +⎛ ⎞= ⎜ ⎟ ⎝ ⎠ t t µ 12 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 the simplectic matrix (or transfer matrix), the trace is equal to the summation of the eigenvalues. So one can show that: (25) A better insight into the physical meaning of the eigen- values µ and µ-1 can be gained by considering the follow- ing transformation of the cell dynamics into the wave mode component domain: (26) Substituting Eq. (26) into Eq. (25), it reduces to: (27) (28) But, because of the particular nature of the eigenvalues of [T] as they appear in pairs (µ, µ-1), then the above equa- tion reduces to: (29) Eq. (29) can be expanded to give: (30) For the jth component of w, we have: (31) 3. System Performance 3.1 Overview In order to demonstrate the feasibility of the theoreti- cal concepts presented, experimental investigations are conducted. These investigations were carried out on two main steps. In the first step, the vibration attenuation char- acteristics of the hybrid periodic rod with two actuators on the shaker was studied and evaluated. In the second step, the tunable mechanical filter is used to evaluate its per- formance for attenuating the vibration transmission from the vibration source, the motor, to the lower plate. In the present study, the piezo actuators model 712A02 (see the index Table 1) from PCB piezotronics, Inc. (Depew, NY) are used. The actuator shown in Fig. 5 has 1 ac o s h 2 ⎛ ⎞+ ⎜ ⎟= ⎜ ⎟ ⎝ ⎠ λ λµ { } 1 1 1 1 L r k k L k L k r L L k k LL k L k u Y W w F u w and Y W F w Φ Φ Φ Φ+ + + + ⎧ ⎫⎪ ⎪= = =⎨ ⎬ ⎪ ⎪⎩ ⎭ ⎧ ⎫⎧ ⎫⎪ ⎪ ⎪ ⎪= = =⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ where Φ is the eigenvector matrix of the transfer matrix [ T]. Also, Wk is the wave mode component vector which has the right -going wave component wr and left -going wave component wL. [ ] ( ) [ ] 1 1 1 1 r r L L k k L L L Lk k r r L L L L L Lk k w w Y T Y T , w w w w or T w w Φ Φ Φ Φ + + − + ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ = = =⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪+⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ Note that the matrix ( )1 T−Φ Φ reduces to the matrix of eigenvalues of the matrix [ T], ie.: ( )1 2 2 1 2 1 r r L L n n L L L Lk k w w diag , ,..., , w w λ λ λ λ− + ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ 1 1 0 0 r r L L L L L Lk k w w w w Λ Λ− + ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ where [ ] 1 1 11 2 1 2diag , ,.. , and diag , ,.. .− − −⎡ ⎤Λ = λ λ Λ = λ λ⎣ ⎦ 1 1 1, and k k k k r r L L L L L Lw w w w+ + −= Λ = Λ It is clear from Eq. (31) that the eigenv alue jλ is the ratio between the amplitude of the right -going waves whereas 1jλ − defines the ratio between the amplitude of the left -going waves. Hence, if ( ) ( ) ( )1 11 1j j j j,then and the pair of ,λ λ λ λ− −< > and the pair of ( 1j j,λ λ − ) denote attenuation of the amplitude of wave propagation from cell k to cell k+1. This can be clearly understood by considering Fig. 5. Note that the right -going wave ( rL jk w ) is attenuated by jλ as it prop agates from cell k to k+1 and the resulting wave ( 1+ r L jk w ) has a lower amplitude as 1 r r jL Ljk jk w wλ + = from the first part of Eq. (31). The left-going wave ( 1+ L L jk w ) is amplified by 1jλ − as it propagate s from cell k+1 to cell k. Hence, the left - going wave ( 1+ L L jk w ), at cell k+1, has a lower amplitude than that at k. Accordingly, the amplitudes of both the right and left -going waves ( rL jk w and L L jk w ) at the left end of cell k are higher than the right and left -going waves ( 1+ r L jk w and 1j k L Lw + ) at left end of cell k+1. Therefore, if ( )j 1,λ < then the pair ( jλ , 1−jλ ) define attenuation (or stop bands) along the wave propagation direction from cell k to cell k+1. Also, if ( ) 1j ,λ = then the pair ( jλ , 1− jλ ) define pro pagation without attenuation ( ie. pass bands). 1 1 1, and j j j jk k k k r r L L L j L L j Lw w w wλ λ + + −= = and or drag 13 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 two built-in side bolts so that the short aluminum rods can be bolted from both sides. Figure 6 shows the frequency response characteristics of the actuator authority between 150-5000 Hz with a sensitivity of 0.015 pound/volt and a peak excitation voltage of 100 Volts. 3.2 Experimental Facilities Figure (7a) shows the dimensions of the hybrid period- ic rod used to build the tunable mechanical filter. It con- sists of three short aluminum rods shown (see for prop- erties Table 2 in the appendix) bolted together by the actu- ator as shown in Fig. (7b). The fundamental longitudinal natural frequency of the periodic rod is 640 HZ. The fre- quency response of such a rod is obtained both in its pas- sive (open-loop) and active (closed-loop) modes of oper- ation. Two experimental test rigs have been employed in this study to evaluate the performance of the tunable mechan- ical filter. The first test rig aims at monitoring the vibra- tion transmission characteristics of the hybrid rod alone as influenced by geometrical and material discontinuities. Figure 8 shows the details of the employed test facility. The objective is to measure the transfer function between the input excitation, the shaker, and the output response of the tip of the rod. First, for the passive plain rod (ie. no periodicity and open circuit), second, for the passive periodic rod (ie. with periodicity and open circuit), and third, for the hybrid periodic rod (ie. with periodicity and short circuit). The phase shifter has been used to introduce a feedback signal to the actuator to produce the destructive interference of waves. The experimental setup shown in Fig. 9 was used to demonstrate the feasibility of using the tunable mechani- cal filter to isolate the vibrations induced by a motor. The experiment has been done for the three cases described in first test rig. Two phase shifters have been used to pro- duce the negative feedback signals to the two actuators. Attenuation rL jk w jλ 1 r L jk w + Propagation L Cell k R L Cell k+1 R LL jk w 1jλ − 1 L L jk w + Direction Amplification High Amplitude Low Amplitude Figure 4. Attenuation of right and left-going waves in the propagation direction Figure 5. Piezoelectric actuator PCB Piezotronic Model 712A02 Frequency Range, Hz 150 to 5000 Dynamic Performance Broadband Force (min), Ib/volt 0.004 Clamped Force, Ib/volt 0.015 Static Performance Free Displacement, ì in/volt 2.56 Environmental Temperature, oF -10 to +150 Capacitance nF 65 Resistance ohm (min) 1 x 107 DC Excitation -125 to 500 AC Excitation (off resonance) ± 100 Electrical Input Voltage (max), Vpk AC Excitation (on resonance) ± 80 Weight, oz 1.27 Height 0.4 Size, in Diameter 2.0 Electrical connector BNC Plug Mounting Thread 10-32 Male Housing Material Titanium Mechanical Sealing Type Welded Hermetic Table 1. The main geometrical and performance parameters of the actuators Material Aluminum Density 2700 kg/m 2 Modulus of Elasticity 71 GPa Wave Speed 5128 m/s Table 2. Material properties of the aluminum rods 14 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 Figure 6. The frequency response characteristics of the actuator authority 3/4 in P iezo electric A ctuator s 1/2 in 1.6 in 1 9 in 6 in (a) (b) Figure 7. Geometrical parameters of the hybrid rod Signal conditioner Phase shifter Power amplifier FFT analyzer Hybrid rod Actuator Accelerometers Actuator Shaker Figure 8. Test setup for evaluating the vibration transmission characteristics of the rod Piezoelectric Actuators M ag ni tu de ( lb s/ V ) Ph as e (D eg .) Frequency (Hz.) Frequency (Hz.) 15 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 3.3 Experimental Results 3.3.1 Single Hybrid Rod In all the reported results, the hybrid periodic rod has provided a viable means for extending the width of the stop band between 0-4000 Hz. Also, attenuations of more than 20 dB are obtained with control voltages less than 5 volts as shown in Fig. 11 for the two control actuators. Note that frequency range higher than 4000 Hz was not considered because of the limitation imposed by the actu- ator bandwidth. Figure 12 displays the lateral vibration distribution over plain, passive periodic, and hybrid peri- odic rods at four different frequencies using the PSV-200 scanning laser vibrometer. Figure 12 shows that the later- al vibration transmission is attenuated also in addition to the longitudinal waves whenever the frequencies lie inside the stop band with the hybrid periodic rod out-performing the plain and the passive periodic rods. 3.3.2 The Table-like System Figure 13 shows the transfer function of the tunable mechanical filter with plain, passive periodic, and hybrid periodic rods when it is subjected to broad band excita- tions from vibration source, the motor. The excitation from the motor can be considered as a random excitation and the geometrical information of the configuration of rods is the same as shown in Fig. 7. The periodic rods in active mode are very efficient in the low frequency range while in the high frequency range they have the same effi- Figure 9. Experimental setup used to demonstrate the feasibility of using the table-like to isolate the vibrations induced by the motor Figure 10a shows the magnitude of the transfer functions of plain, passive periodic, and hybrid periodic rods when they are subjected to random excitation in the axial direction. These transfer functions quantify the response of the free end of the ro ds to input excitations at the other end of the rods. Figure 10a indicates clearly that the passive periodic rod exhibits experimentally a broad stop band between 650 -4000 Hz whereby the vibration transmission through the rod is almo st eliminated. Further more, Fig. 10a indicates that the hybrid rod provides an effective means for attenuating the vibration transmission over a very broad frequency range. It is particularly effective for stopping the low frequency vibration in the range below 600 Hz where the passive periodic rod has been ineffective. This result is particularly important in using a hybrid rod that combines both the passive and active strategies to stop high as well as low frequency wave propagation. Figure 10b shows that the widths of the st op bands can be clearly predicted theoretically by plotting the real part α of the propagation parameter µ . For values of α ≠ 0, the stop bands can be cle arly identified and match closely the experimental results for both the passive and hybrid rods. 16 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 ciency of the passive mode. That is why the blue and green lines in Fig. 13 have the same trend above 1500 Hz. The corresponding control voltage of the upper control actuators is shown in Fig. 14. 4. Conclusions This paper has presented the tunable mechanical filter as a new tool for isolating vibrations. It consists of four Figure 10. The transfer functions of the plain, passive periodic, and hybrid periodic rods with the propagation factor in both passive and active modes 1 1 10-2 10-4 10-1 10-2 10-3 10-4 Figure 11. The control voltage of (a) the lower actuator, and (b) the upper actuator (a) (b) 17 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 (a) (b) (c) Figure 12. The vibration of plain, passive periodic and hybrid periodic rods at: (a) 160 Hz (in pass band), (b) 800 Hz (in stop band), and (c) 1520 Hz (in stop band) At 160 Hz At 800 Hz At 1520 Hz 18 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 hybrid periodic rods installed between two plates. The theory governing the operation of this class of rods has been presented. The factors governing the design of effec- tive periodic rods have been identified. The performance characteristics of passive and hybrid periodic rods alone have been measured experimentally and compared with the theoretical predictions of the real part of the propaga- tion factor. Close agreement between theoretical predic- tions and experimental results has been achieved. The performance of the tunable mechanical filter with a motor assembly has also been monitored experimentally. The predictions of the stop bands have also been found to be in close agreement with the experimental results. The periodic rods in active mode are very efficient in the low frequency range while in the high frequency range they have the same efficiency of the passive mode. As a result, the mechanical filter with hybrid periodic rods is found to be more effective in attenuating the transmission of vibra- tions from the source of vibrations to the lower plate over a broader frequency range extending from 150-4000 Hz with control voltages not exceeding 5 volts. This concept of the tunable mechanical filter can be implemented efficiently in many applications. For instant, the gearbox support system of helicopter and automotive vehicle engine mounting systems. With such unique filter- ing characteristics, it would be possible to control the wave propagation both in the spectral/spatial domains in an attempt to stop/confine the propagation of undesirable disturbances. Figure 13. The transfer function in case of collocated sensor/actuator arrangement with broadband excitation from the motor Excitation Frequency, Hz Tr an sf er F un ct io n, d B 0 500 1000 1500 2000 2500 3000 Excitation Frequency, Hz Figure 14. The control voltage applied to the upper control actuator 101 10o 10-1 10-2 10-3 19 The Journal of Engineering Research Vol. 5, No.1 (2008) 7-19 References Asiri, S., 2005, "Vibration Isolation of Automotive Vehicle Engine Using Periodic Mounting Systems," SPIE 2005, San Diego, California, USA. Asiri, S., Baz, A. and Pines, D., 2004, "Active Periodic Struts for Gearbox support System," Proc. of SPIE, Smart Structures and Materials 2004: Damping and Isolation, Vol. 5386, pp. 347-358. Asiri, S., Baz, A. and Pines, D., 2002, "Periodic Struts for Gearbox Support System," Proc. Of the 2002 International Congress and Exposition on Noise Control Engineering, Inter Noise 2002, Paper number IN02-644, Dearborn, MI, USA. Baz, A., 2001, "Active Control of Periodic Structures," ASME Journal of Vibration and Acoustics, Vol. 123, pp. 472-479. Baz, A., 2002, "Vibration Damping," Class Notes, University of Maryland at College Park. Baz, A., 2000, "Spectral Finite Element Modeling of Longitudinal Wave Propagation in Rods with Active Constrained Layer Damping," Smart Materials and Structures, Vol. 9(3), pp. 372-377. Brillouin, L., 1953, "Wave Propagation in Periodic Structures," 2nd ed. Dover. Cremer, L., Heckel, M. and Ungar, E., 1973, "Structure- Borne Sound," Springer-Verlag, New York. Doyle, J., 1997, "Wave Propagation in Structures," 2nd ed., Springer-Verlag, New York. Faulkner, M. and Hong, D., 1985, "Free Vibration of a Mono-Coupled Periodic System," J. of Sound and Vibrations, Vol. 99, pp. 29-42. Gupta, S., 1970, "Natural Flexural Waves and the Normal Modes of Periodically-Supported Beams and Plates," J. of Sound and Vibration, Vol. 13, pp. 89-111. Mead, D. J. , 1986, "A New Method of Analyzing Wave Propagation in Periodic Structures; Applications to Periodic Timoshenko Beams and Stiffened Plates," J. of Sound and Vibration, Vol. 114, pp. 9-27. Mead, D.J., 1970, "Free Wave Propagation in Periodically Supported, Infinite Beams," J. of Sound and Vibration, Vol. 11, pp. 181-197. Mead, D. J., 1971, "Vibration Response and Wave Propagation in Periodic Structures," J. of Engineering for Industry, Vol. 21, pp. 783-792. Mead, D. J., 1975, "Wave Propagation and Natural Modes in Periodic Systems: I. Mono-Coupled Systems," J. of Sound and Vibration, Vol. 40, pp. 1-18. Mead, D. J., 1996, "Wave Propagation in Continuous Periodic Structures: Research Contributions from Southampton," J. of Sound and Vibration, Vol. 190, pp. 495-524. Mead, D. J. and Bardell, N. S., 1987, "Free Vibration of a Thin Cylindrical Shell with Periodic Circumferential Stiffeners," J. of Sound and Vibration, Vol. 115, pp. 499-521. Mead, D. J. and Markus, S., 1983, "Coupled Flexural- Longitudinal Wave Motion in a Periodic Beam," J. of Sound and Vibration, Vol. 90, pp. 1-24. Mead, D. J. and Yaman, Y., 1991, "The Harmonic Response of Rectangular Sandwich Plates with Multiple Stiffening: A Flexural Wave Analysis," J. of Sound and Vibration, Vol. 145, pp. 409-428. Orris, R. and Petyt, M., 1974, "A Finite Element Study of Harmonic Wave Propagation in Periodic Structures," J. of Sound and Vibration, Vol. 33(2), pp. 223-237. Patera, A., 1984, "A Spectral Element Method for Fluid Dynamics: Laminar Flow in Channel Expansion," J. of Comput. Physics, Vol. 54, pp. 468-488. Pierre, C., 1988, "Mode Localization and Eigenvalue Loci Veering Phenomena in Disordered Structures," J. of Sound and Vibration, Vol. 126, pp. 485-502. Ravindra, B. and Mallik, K., 1992, "Harmonic Vibration Isolation Characteristics of Periodic Systems," J. of Sound and Vibration, Vol. 154(2), pp. 249-259. Richards, D. and Pines, D.J., 2003, "Passive Reduction of Gear Mesh Vibration Using a Periodic Drive Shaft," J. of Sound and Vibration, Vol. 264(2), pp. 317-342. Roy, A. and Plunkett, R., 1986, "Wave Attenuation in Periodic Structures," J. of Sound and Vibration, Vol. 114, pp. 395-411. Ruzzene, G. and Baz, A., 2000, "Control of Wave Propagation in Periodic Composite Rods using Shape Memory Inserts," J. of Vibration and Acoustics, Vol. 122, pp. 151-159. Sackman, S., Kelly, J. and Javid, A. A., 1999, "Layered Notch Filter for High-Frequency Dynamic Isolation," J. of Pressure Vessel Technology, Vol. 111, pp. 17-24. Singh, A., Pines, D. and Baz, A., 2004, "Active/Passive Reduction of Vibration of Periodic One-Dimensional Structures using Piezoelectric Actuators," Smart Materials and Structures, Vol. 13, pp. 698-711. Snowden, J.C., 1979, "Vibration Isolation: Use and Characterization," NBS Handbook, Vol. 128, US National Bureau of Standards. Szefi, J., Smith, E. and Lesieutre, G., 2004, "Design and Testing of a Compact Layered Isolator for High- Frequency Helicopter Gearbox Isolation," Prec. Of 45th AIAA Structures, Structural Dynamics and Materials Conference, Palm Spring, CA, USA.