Loudspeaker Rocking Modes (Part 1: Modeling)

Transcription

1 Loudspeaer Rocing Modes (Part 1: Modeling) Willia Cardenas, Wolfgang Klippel; Klippel GbH, Dresden 139, Gerany he rocing of the loudspeaer diaphrag is a severe proble in headphones, icro-speaers and other inds of loudspeaers causing voice rubbing which liits the aiu acoustical output at low frequencies. he root causes of this proble are sall irregularities in the circuferential distribution of the stiffness, ass and agnetic field in the gap. A dynaic odel describing the echanis governing rocing odes is presented and a suitable structure for the separation and quantification of the three root causes eciting the rocing odes is developed. he odel is validated eperientally for the three root causes and the responses are discussed conforing a basic diagnostics analysis. 1. INRODUCION Rocing odes are natural vibration patterns of the loudspeaer, producing undesired rotational vibrations. Loudspeaers drivers with substantial irregularities produced in the anufacturing/assebling process and drivers subjected to asyetric acoustics loads ehibit this phenoena. his rocing behaviour becoes ore critical in sall drivers such as headphones or icrospeaers, where sall irregularities in the stiffness, ass and agnetic field distributions, can affect draatically the dynaic behaviour of these tiny structures. Even when high precision in the production process and low tolerances in the assebly are reached, soe of these drivers present substantial ecitation of rocing odes. One of the ost iportant proble associated is the rub&buzz effect which liits the acoustic output before the noral echanical or theral echanis do. Since soe of the causes of the rocing odes are nonlinear lie: Bl() and K (), this proble can appear only at high aplitudes, for this reason, drivers that are very stable at low levels can becoe unstable at high operation levels. he detection and identification of this rocing behaviour is crucial in the loudspeaer developent process, since it will reveal the root cause of these particular sypto allowing the engineer to tae the necessary actions to eliinate or itigate instability proble. he ain idea is to develop an optial odel that can eplain the generation of rocing odes and that allows to quantify the agnitude and the direction of the entioned proble. his inforation needs to be easy to interpret and should be close to the physical phenoena eperienced by engineers and designers woring daily on loudspeaer developent and anufacturing. Since the natural odes are the siplest way to epress how a structure vibrates, epressing the aount of ecitation of those odes is a very intuitive approach that provides valuable inforation for understanding proble. his paper is distributed as follows: First the physical echanis and the consequences of each root cause are described. After that, the syste of equations are presented, the separation of the different causes as ecitation echanis are developed and the siplified odel is derived. In chapter four the odal analysis and the projection of the root-causes oents is described. Finally a set of validation eperients is discussed.. PHYSICAL MODEL.1. Rigid body otion assuption he rocing ode behavior of a loudspeaer can be considered as a low frequency echanis totally described by the equations of otion derived for the luped paraeter odel [1] coupled to another set of equations describing the tilting angle around a deterined rotation ais. In this contet, only the suspension parts are defored under the effect of applied forces and oents. his assuption is fulfilled by all ind of electro-dynaical transducers coposed by a rigid diaphrag, voice and forer attached to slightly softer suspension parts. In the case of icrospeaers and headphones, the diaphrag and the surround are ade of the sae aterial and the diaphrag needs to be defored to allow the rotational degree of freedo. In this special cases the diaphrag will be taen as the larger surface vibrating with soe negligible deforation. he frequency range for the rocing ode analysis is valid only below the first breaup ode of the diaphrag.

2 .1.1. Syste of coordinates he selection of the syste of coordinates to be used in the derivation of the equations of otion will deterine the usefulness of the odel for the separation of the root causes of rocing odes. In the odel presented in [], the syste of equations is derived assuing that the pivot point of the rotations along and y coincides eactly with the center of gravity of the driver. Based on this coordinate syste, all the oents associated to a shifting of the center of gravity of the diaphrag around the pivot point are zero, in other words the equations are dynaically decoupled [3], as a consequence the causes due to ass proble cannot be independently etracted. Since the equations will describe eactly the sae behaviour independent of the syste of coordinates or variables used, by using the entioned coordinates, all the physical causes will be attributed to a static coupling ter produced in the suspension parts. For perforing accurate root-cause analysis of rocing odes including the quantification of proble associated to Mass, Stiffness and B-field distributions, the odel presented in this paper is etended to include the dynaic and static couplings as well as the oents ter associated to the ecitation force. Starting fro the contributions ade by MacLachlan in [4] and later by Bright in [] and [5] the rocing ode odel can be developed by using the equations of rigid body echanics. translational describing the position of the center of the along the z ais and two rotational τ and τ y describing the tilting angles around the y and ais respectively, see Figure 1. he echanical ass M is assued to be luped at the center of gravity of the diaphrag C.G and the total stiffness K including viscoelastic effects and R echanical resistance are associated to the suspension parts..1.. otal equivalent oents he iperfections produced in the production process, such as inhoogeneous distribution of the aterial or thicness over the diaphrag, geoetrical or aterial changes along the suspension parts and differences of the agnetic field in the gap usually are located in different angles and at different distances fro the center of the driver. As they can be seen as a oents acting around an arbitrary rotation angle (coincident with the direction of the rocing odes), all its contributions can be collapsed to a one equivalent oent producing the sae effect. he tilting angle produced by the actions of the oents around the rotation ais of the rocing ode is called τ α. he graphical representation of this concept is depicted in Figure. he force f sy corresponds to a part of the total applied echanical force coing fro the otor that produces only displaceent of the diaphrag in the z-direction (pure ecitation of fundaental ode), is called syetric force because it does not produces any tilting of the oving ass. he other oents are results of soe asyetrical forces acting on the syste. he total oent produced by proble in the agnetic field µ Blα is located at soe arbitrary direction at the distance where the voice forer and the diaphrag are in contact. In the other hand, the total oent µ α generated due to irregular stiffness distribution will appear at the contact points between the suspension parts and the oving ass. Figure 1 State variables describing rocing ode of a loudspeaer. he coplete behavior of a loudspeaer ehibiting rocing odes can be described by three states; one

3 Figure Separated total oents eciting a rocing ode he total oent contributing to the rocing ode due to a ass proble µ α can be seen as the effect of a shifting of the center of gravity C.G of the oving ass with respect to the geoetrical center. his oent can be generated as well by soe ecess of glue in the junctions with the suspension parts and is taen into account only if the origin of the syste of coordinates is located at the geoetrical center of the driver... Root causes and ecitation ter In the previous sections, it has been stated that the rocing odes in loudspeaers will be activated only if soe oents around the ais crossing the center of the driver eists. In other words, any ind of nonsyetrical force acting on the diaphrag will cause a tilting vibration nown as rocing ode. hey can be generated by irregular ass and stiffness distribution on the oving parts (passive ecitation) or fro the iperfections in the Bl distribution in the gap (active ecitation)...1. Mass Unbalance his proble is related with the shifting of the center of gravity of the oving parts out of the geoetrical center. It can happen due to non-unifor aterial distributions on the driver coponents, thicness variations, ecess of gluing in a certain direction, etc. he ass difference between two opposite regions of the loudspeaer, will produce a shifting of the center of gravity C.G which consequently generates a oent µ at a distance ε around the ais crossing the geoetrical center which will produce a titling angle τ as shown in Figure 3. Figure 3 ilting generation due to a ass unbalance he equilibriu of forces and oents acting on a infinitesial part of the diaphrag ass d located at C.G are: f d (1) M c d () I Assuing that the distributed ass of the loudspeaer M =ʃd is luped at center of gravity C.G the paraeter =εm quantifies the difference of ass between two etrees of the loudspeaer in the direction of the distance ε fro the geoetrical center of the oving assebly. he oent of inertia is defined as I c= ε M [3]. wo effects are produced here; a force f accelerating the diaphrag up and down d /dt and a oent µ acting around the center of the driver producing a tilting angle τ. Note that the two coordinates and τ are coupled through the paraeter.. If there is no shifting of the center of gravity ε=, the oent eciting the tilting angle is zero and the only reaining force will produce a perfect transversal otion. he aplitude of this tilting angle will depend on the aount of shifting of the center of gravity produced by the irregular ass distribution. he ass proble is a dynaic process that becoes doinant above the fundaental resonance frequency of the loudspeaer where the voice acceleration is high.... Stiffness asyetry Any ind of defect or irregularity which odifies locally the restoring force of the suspension along the peripheral lines where the diaphrag is attached, can be seen as a difference of the stiffness between two opposite contact points. Note that the total stiffness distribution acting on should be equivalent to K.

4 agnetization, non-constant gap width due to shifted positions of the iron parts, etc. he non-asyetrical transduction factor Bl can produce substantial changes of the echanical force generated at different angles on the voice producing a oent µ Bl around the rotation ais at a distance d, contact point of the with the oving parts. he action of this oent will produce a tilting angle τ Bl of the voice as seen in Figure 5. Figure 4 ilting generation due to stiffness asyetry Coputing the forces and oents for the siplified D case shown in Figure 4: f f f K K (3) L R K K (4) he forces f L and f R represent the distributed vertical coponent of the restoring force provided by the suspension parts. µ L and µ R the oents around the center of the driver produced by the stiffness. he paraeter K τ=k l is called the rotational stiffness around the rotational ais and K =K l, l is the distance fro the center to the contact point with the surround. his irregularity will generate a total oent µ around the ais crossing the center of the driver producing a tilting τ. Note that if the suspension parts ehibit soe ind of proble, its effect will be evidenced even if there is no oveent of the driver. For this reason this ind of tilting generation can be called static and is ainly active below the fundaental resonance frequency..3. B-field inhoogeneity he force factor Bl deterines the aount of echanical force that a loudspeaer otor can produce fro an applied current. his paraeter depends locally on the aount of agnetic field B crossing the voice with length l. Several reasons can cause variations of the agnetic field along the gap, such as non-hoogenous Figure 5 ilting generation due to B-field inhoogeneity he force and the oents produced by this proble are: f Bl Bli (5) id (6) Bl Bl his root cause is related to the input current of the loudspeaer driver and it will ecite the rocing resonators actively depending on its position over the voice circuference and the ode shape. his is active broadband but presents a dip at the resonance frequency due to the decreasing of current flowing through the voice. 3. ROCKING MODE MECHANISMS 3.1. Electroechanical syste By coputing the equilibriu of forces acting in the z- direction and the oents around the and y ais and coupling the electrical part of the driver through the current i(, the net set of otion equations can be derived: M y I c I y cy ( R ( y( R ( K ( R y y( y K y ( f ( K y y( sy Bl Bly Bl Bli( Bly (7) where M, R and K corresponds to the echanical paraeters of the driver, the product Bl is the force factor and they can be easured by using the conventional techniques [6] or [1]. he constant paraeters and y are the ter coupling the translational displaceent with the two rotational vibrations and τ represents the coupling

5 between the two rotational degree of freedo. hese paraeters are related to the position of center of gravity of the diaphrag and they are called dynaic couplings since they will be only active if there eists acceleration of the odel states. Note that any ass proble present in the real driver will be coded by those paraeters. he paraeters, y and τ play the sae role as in the ass atri, but for the stiffness. Since they are directly ultiplied by the states and not by soe of its derivatives, they are called static couplings. Any ind of non-unifor distribution of the stiffness of the driver will be described by these ter. he oents µ Bl and µ Bly eciting a tilting along the and y aes are depending on the paraeters Bl and Bly containing the inforation about the irregular B-field distribution in the gap. he diagonal ter I c,y, R τ,y and K τ,y are the ass oent of inertia, the echanical daping of the rotational vibration and the rotational stiffness of the diaphrag about the and y aes. Equation (7) can be transferred into Laplace doain as: s M sc K BI (8) where the state vector = [X (s) (s) y(s)] and the atrices M, C and K represent the ass, echanical resistance and stiffness, respectively. he vector B represents the coponents of the electro-dynaical transduction factor. For circular drivers, it has been shown in [] that the stiffness ter can be epressed analytically as Fourier series of sinusoidal functions around the angle in the circular coordinate called stiffness distribution functions. In the case of the odel presented here, this approach needs to be generalized since this odel is intended to be applied to loudspeaers with arbitrary shapes. he energy dissipation of the rotational (tilting) vibration is usually attributed to visco-theral effects present in the gap and the echanical resistance of the suspension. For siplicity, the daping atri is chosen to be diagonal, neglecting coupling ter due to the velocity of the odel states. 3.. Direct ecitation and feedbac sources he odel described by Equation (7) is a fully coupled syste of differential equations which is rather cople and difficult to interpret. A ore understandable for can be obtained by separating the inertia and elasticity ter producing oents around the rotational ais of the diaphrag as feedbac sources added to the direct ecitation fro the otor. Separating the diagonal ter M D and K D of the ass and stiffness atrices fro the non-diagonal (coupling) ter M and K gives s M sc K BI s M K (9) D D It can be clearly seen that the two new ter subtracted fro the direct source and ultiplied by the state vector can be considered as feedbac sources of the syste confored by the diagonal ter on the left side of (9). If the diagonal ter are grouped into a transfer function atri H τ and the ecitation ter associated to ass and stiffness are called F and F the full coupled differential equation taes the for: H F F F (1) Bl Equation (1) describes three different sources corresponding to the ass, stiffness and Bl root-causes, activating a set of independent (decoupled) second order syste. A scheatic representation of the derived Equation (1) is presented in Figure 6. U(s) Z e -1 I(s) s B M K Root-causes F F Bl - - F F Syste properties Figure 6 Mass and stiffness proble as feedbac sources and agnetic ecitation as a direct ter. he echanis governing rocing odes in loudspeaer diaphrag consist of a SIMO (single input ulti outpu syste with an internal feedbac structure (doinated by X ) through ter associated with ass and stiffness irregular distributions, activating a set of three second order resonators. he Z e -1 is the electrical ipedance converting the input voltage U(s) into the current I(s). he nd -order differentiator s transfor the state vector into the acceleration. he ter associated with Bl inhoogeneities are the direct forces and oents depending on the input current I(s). Fro the physical perspective, the only way to ecite the tilting angles of the diaphrag is by using a otor with soe inhoogeneities in the agnetic field, these introduce energy directly on the resonators. In the case of the ass and stiffness sources, they do not contribute actively with the total energy applied to the driver, but they redistribute the energy, that is supposed to ecite H τ

6 only the piston ode, into the other two tilting coponents. For this reason they appears as a feedbac sources in Figure 6. Note that the total ecitation vector F is coposed by the cople contributions of the translational and rotational states ultiplied with the different root-causes ter. A detailed analysis is required to deterine the relevant ter in practical diagnostics analysis Detailed oent generation and rootcauses separation echanis of the rocing odes in loudspeaers, aing transparent the relation between the physical causes and the sypto observed in real drivers. In the sae analysis it is possible to deterine the iportant ter and to neglect those which do not contribute significantly on the practical interpretation and diagnostics. he separation of the diagonal and non-diagonal ter leads to the following equation: he ai of this section is to describe the detailed derivation of a set of equations equivalent to equation (1) to provide a ore intuitive idea of the generation 1 H X Bl 1 H 1 H ( ) y s y Bl Bly I( s) s y yx ( s) y y y X ( s) y (11) he diagonal ter of the left side atri are the inverse of the transfer functions between the output states of the odel and the total input ecitation of each one. he transfer function: H 1 (1) s M sr K is the ratio between the echanical syetrical force f sy=bli(s) produced by the loudspeaer otor and the voice displaceent X (s). he other two transfer functions: 1 H i( s) s (13) Ici sr i K i where iϵ{,y} are the ratios between the tilting coponents i(s) and the total oents µ i(s) activating the. his resonators are introduced here as auiliary transfer functions which can describe the tilting oscillations of the diaphrag eactly coincident with the -y directions of the coordinate syste. hey should not be confused with the rocing resonators associated to the doinant rocing ode and its orthogonal counterpart, which can be generated in any direction. he coupling ter s i and i deterine the aount of interaction between the voice displaceent and the tilting angles. hese ter are doinant in practice. In the other hand the coupling ter s τ and τ deterine the interaction between the two rotational states. hese ter are usually negligible in practice. Solving equation (11) the transversal displaceent X (s) and the tilting angles (s) and y(s) can be described by second-order resonators and X H f (14) H (15) y H y (16) the total transversal ecitation force f f sy f f BlI s τ τ (17) and the total ecitation oents in and y directions: y Bl Bl Bly Bly I( s) s y y I( s) s y y y y (18) (19) Each ecitation force and oent coprise three coponents corresponding to root causes: ass, stiffness or Bl. he vectors τ, and y are called state sub-vectors and are defined as: τ y X y X he vectors, τ and yτ represent the ass paraeters: y y y and the vectors, τ and yτ contain the stiffness paraeters: y

7 y y y A scheatic representation of the generation of the transversal displaceent X (s) is shown as a signal flow chart in Figure 7. I(s) τ s Bl f scalar product f sy - - f f H X (s) Figure 7 Generation of forces producing voice displaceent due to coupling with tilting angles he echanis is a feedbac structure, which requires the prior coputation of the tilting angles τ. Physically the coupling forces f and f are dependent on the aount of tilting of the diaphrag. However, the forces f and f caused by the rocing odes are usually uch saller than the syetrical force f sy and can be neglected for the odeling of the voice displaceent X (s). I(s) i s Bl i i i µ Bli µ i - µ i - µ i Root-causes H τi Τ i (s) Figure 8 Generation of tilting angles due to oents produced by ass, stiffness and Bl proble he signal flow chart in Figure 8 shows the generation of the tilting angles i(s). he oents µ i and µ i representing ass and stiffness ecitations depends on the states i, doinated by X. he results obtained in Equation (18), and Equation (19) have a high diagnostic value since the root-causes of the rocing odes are clearly separated allowing an intuitive understanding of the proble presents in the loudspeaers and a clear interpretation of the paraeters. his brings benefits for the syste identification process which will be etended in the part of the present paper Siplified Model By applying the assuption that the feedbac sources due to ass and stiffness proble in Figure 8 are doinated by X on the vector i, and that the tilting angles τ are sufficiently sall in Figure 7, the odel reduces to the following feed forward structure: U(s) i H τ I(s) X s Bli i i µ Bli µ i - - µ i µ i H τi Τ i (s) Figure 9 Siplified feed forward odel generating the tilting angle i(s) of the rocing ode Note that the bloc H produces the voice displaceent and the current before distributing the in the networ in charge of producing the oents that are going to ecite the tilting resonators, it contains iplicitly H and Z e -1. Due to its feed forward structure, this odel is a uch ore powerful tool for analysis and diagnostics. Woring in the -y syste of coordinates siplifies soe steps for the odelling and for the syste identification but is not suitable for analysis and data etraction. For this reason, the oents and the tilting produced by the root-causes need to be epressed in a coordinate syste ore related to the physics of the loudspeaer vibration. A ore convenient alternative is the odal basis. 4. MODAL ANALYSIS At low frequencies the diaphrag vibration can be described by the superposition of three natural odes; the piston and the two orthogonal rocing odes. Fro the dynaic analysis the ode shapes and resonance frequencies of the structure depend only on the elasticity and inertial properties [8]. he odal analysis is a powerful tool which allows to analyse cople vibrating syste as a set of siple haronic oscillators with characteristics vibration patterns called Noral Modes [7]. In the analysis of rocing odes this is a convenient way to decouple the otion equation obtaining directly the rocing resonators and the ode shapes Rocing ode resonators Hα1, he first step to perfor a odal analysis is to find the set of orthonoral basis in which the otion equation will be projected. Soe drivers due to its suspension parts or particular acoustic loads present viscoelastic effects [9], [1] that liits slightly the perforance of the

8 odal analysis, since there is not siple way to decouple a syste of equations with a frequency dependent stiffness atri. In this contet the viscoelastic ter will be neglected. Assuing that there are no forces or oents acting on the loudspeaer and considering only the inertia and elasticity ter, the following eigenvalue proble is posed: 1 M K IΦ () n n Where the syste eigenvalues ω n corresponds with the resonance frequencies of the fundaental ode f s and the two orthogonal rocing odes f α1, in Hz. I states for the identity atri. Orthogonal Rocing Mode Φ α Figure 1 Mode shapes of a rectangular loudspeaer he eigenvectors Φ n are noralized with respect to the ass atri and correspond to the ode shapes of the fundaental ode Φ (piston ode) and the rocing odes Φ α1 and Φ α as depicted in Figure 1. he eigenvectors Φ n are orthogonal to each other and coprise three eleents: Φ Φ 1 1 y1 Φ y Piston Mode Φ he tilting coponents τ i and τ yi of the first and second rocing ode (i=1,) describe the direction of the arctan i i i 1, (1) yi aiu tilting in the y plane as shown in Figure 11. Doinant Rocing Mode Φ α1

9 Figure 11 Rotation aes and oent projections Eploiting the orthogonality between the Eigenvectors the state vector in Eq. (8) can be written as a odal epansion with the odal atri ΦH(s) () 1 Φ Φ Φ Φ (3) coprising the orthogonal Eigenvectors Φ n (ode shapes) and the transfer function atri: H s ) H H H (4) ( 1 coprising the echanical transfer functions: 1 H (5) s s corresponding to the piston ode of the diaphrag and the transfer function of the two rocing odes: 1 H n n 1, s s (6) n n shown in Figure 1. Since the ode shapes are noralized with respect to the ass atri, the rocing resonators are totally described by only two paraeters, the resonance frequency and the daping factor [11]. n Figure 1 he aplitude and phase response of the odal transfer functions H (s), H α1(s) and H α(s) he resonance frequency ω =πf is the fundaental resonance of the driver and ω αn with n=1, are the resonance frequencies of the rocing odes. he odal daping factors η and η αn are the echanical losses of the odes ecluding the electrical daping. heoretically there are two rocing odes but in practice often only one of the is highly ecited asing the effect of the other or they degenerate and can be seen as only one with one resonance frequency [1]. 4.. Moents eciting rocing odes he total oent µ eciting the rocing odes μ μ Bl y μ Bl Bly μ y y (7) and the coponents µ Bl, µ and µ corresponding to the root causes ass, stiffness and force factor in Eq. (18) and (19) in Cartesian coordinates are transfored into the odal space to siplify the interpretation: μ 1 μ Bl Rμ Bl1 Bl R μ μ Bl μ 1 μ μ 1 (8) where the total oent µ α and its coponents are aligned with the direction of the odes by using the projection atri:

10 sin 1 cos1 R (9) sin cos based on the angles defined in Eq. (1) and shown in Figure 11. he signal flow chart in Figure 13 illustrates the odeling of the rocing odes in the odel space. Modal Analysis K M Eigenvalue proble ω n Φ Rocing Directions α 1 α U Rootcauses µ Bl µ µ Projection µ Blα1 µ α1 µ α1 µ Blα µ α µ α µα1 µ α H α1 H α α1 α and y coponents of oents Physical ecitation oents of rocing odes Figure 13 Generation of total oents activating rocing odes 5. DISCUSSION he proposed odel is validated by conducting several eperients related with the three root causes that need to be identified and separated in practical diagnostics. he idea is to intentionally ecite the rocing odes of the loudspeaer by odifying its echanical and agnetic properties and to quantify its effects by eans of the identification of the root-causes oents. he syste identification schee and the interpretation of the estiated paraeters will be discussed in greater detail in part. For instance, to produce a substantial tilting angle of the diaphrag due to a ass unbalances, the center of gravity of the oving parts is shifted by placing a sall ass on the surface. In the other hand, a stiffness asyetry was created by perforating the surround along a deterined area without changing the aount of ass displaced. he B-field inhoogeneity was generated by gluing two arrays of neodyiu agnets with inverse polarity in opposite directions to the bac plate. More details on the eperiental setup will be given in the second part of this paper. Figure 14 Aplitude response of the fundaental and two rocing odes easured (dotted) and odeled (solid) on a transducer perturbed by an additional ass shifting the center of gravity.

11 5.1. Doinant ass proble he eperiental results and the fitted odel of the rocing behavior due to a ass perturbation are shown in Figure 14. he oent µ α1 eciting the first ode generated by the additional ass is proportional to the acceleration which is the nd derivative of the transversal displaceent X (see Figure 9) but the tilting angle α1 shown in Figure 14 reveals an additional resonance pea at 3 Hz and an additional decay 1 db per octave at higher frequency caused by the odal resonator with the transfer function H α1(s). characteristic of the tilting angle α1(s). he odal resonator with the transfer function H α1(s) generates the pea at 3 Hz and decay at higher frequencies. he aplitude response α(s) of the second ode reveals a ass unbalance as found in the original transducer without any perturbation. Note that the aplitude response of the tilting angle Τ α of the second ode reveals other root causes activating the second resonator H α(s) having a resonance frequency at 35 Hz. he output paraeters and oents coputed in the identified syste (solid line) will be eplained in the second part of the paper. Figure 16 Aplitude response of the fundaental and two rocing odes easured (dotted) and odeled (solid) on a transducer perturbed by an asyetrical force factor distribution in the agnetic gap Doinant Bl proble Figure 15 Aplitude response of the fundaental and two rocing odes easured (dotted) and odeled (solid) on a transducer perturbed by an asyetrical distribution of the circuferential stiffness. 5.. Doinant stiffness proble he easured and odeled aplitude responses of the transversal displaceent X and the tilting angles of the three odes of a transducer perturbed by an asyetrical stiffness distribution are shown in Figure 15. he oent µ α1 eciting the first ode generated by the stiffness is in accordance to Figure 9 proportional to the voice displaceent X generating the low pass Figure 16 shows the aplitude responses of the first three odes of a transducer with an asyetrical Bl distribution in the gap. Contrary to the root causes associated to ass and stiffness, the oent µ Blα1 generated by the force factor asyetry has a relative flat response corresponding to the spectru of the electrical current I(s) used in the ecitation ter in Figure 9. he ipedance aiu fundaental resonance frequency f = 1 Hz causes a dip in the aplitude response of the tilting angle α1(s). he following odal resonator with the transfer function H α1(s) generates the resonance pea and the decay above 3 Hz. he frequency response of the nd rocing ode α(s) reveals a sall stiffness asyetry found in the original transducer. 6. CONCLUSIONS A new odel has been presented describing the generation of the fundaental ode and the first two rocing odes. he quantification of the oents

12 eciting the odal resonators reveal the root causes of the rocing odes which are caused by asyetrical distributions of ass in the oving eleents, stiffness in the suspension and electrodynaical force factor. he particular characteristics of each proble has been eplained fro the physical and fro the syste oriented odelling approach clarifying the sypto associated to each proble. Creep, J. Audio Eng. Soc., vol. 41, pp. 3-18, (Jan./Feb. 1993) [11] A. Sondipon, Structural Dynaic Analysis with Generalized Daping Models: Analysis,Wiley- ISE, (January 14) [1] B. M. Deutsch, Non-degenerate noral-ode doublets in vibrating flat circular plates, Aerican Journal of Physics 7(),pp, (February 4) he transfer responses associated to ass, stiffness and Bl proble are discussed and the differences between the associated sypto have been clearly separated. he odel has been eperientally validated by the perturbation analysis of the three causes. Nuerical odeling based on the theory developed in this paper is the basis for siulating the echanical vibration but also for the root cause analysis of rocing odes and other inds of diagnostics. he identification of the free odel paraeters based on easured data provided by laser scanning are the basis for perforing practical diagnostics on a particular transducer. 7. REFERENCES [1] R. H. Sall, Direct Radiator Loudspeaer Syste Analysis, J. Audio Eng. Soc., Vol., No. 5, pp (197 June). [] A. Bright, Vibration Behaviour of Single- Suspension Electrodynaic Loudspeaers, presented at the AES19th Convention, Los Angeles, CA, USA, Septeber -5. [3] S. R. Singiresu Mechanical Vibrations, Fifth Edition, Printice Hall, pp (11) [4] N. W. McLachlan, Loud Speaers, heory, Perforance, esting, and Design. pp Oford University Press, London, U. K. (1934). [5] A. Bright, Active control of loudspeaers: An investigation of practical applications. Ph.D. thesis. [6] J. R. Ashley and M. D. Swan, Eperiental Deterination of Low-Frequency Loudspeaer Paraeters in Loudspeaers J. Audio Eng. Soc., Vol. 17, No. 5, pp (1969 October). [7] D.J Ewins, Modal esting: heory and Practice Research Studies Press, Herts, UK, (1986). [8] C. W. de Silva, Vibration Fundaentals and Practice, CRC Press, () [9] F. Agervist, A Study of the Creep Effect in Loudspeaer Suspension, presented at the AES15th Convention, San Francisco, CA, USA, (8 October -5). [1] M.H. Knudsen and J.G. Jensen, Low-Frequency Loudspeaer Models that Include Suspension