New paradigms to control coupled powertrain and frame

Transcription

1 New paradigms to control coupled powertrain and frame motions using concurrent passive and active mounting schemes DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosoph in the Graduate School of The Ohio State Universit B Jared Liette, B.S., M.S. Graduate Program in Mechanical Engineering The Ohio State Universit 04 Dissertation Comttee: Prof. Rajendra Singh, Advisor Prof. Manoj Srinivasan Prof. Vishnu Baba Sundaresan Prof. Vadim Utkin Dr. Jason T. Dreer

2 Copright b Jared Liette 04

3 Abstract The topic of this scholarl research is motivated b the need for superior control of vehicle powertrain vibration, commonl accomplished using 3 or 4 passive mounts. However, emerging design trends (such as higher power densit powertrains and lightweight structures) necessitate a hbrid approach utiliing active and passive methods to meet more stringent sstem performance targets. The chief research objective is to acquire fundamental understanding of dnac interactions among multiple active and passive paths in a powertrain mounting sstem for improved control of multi-dimensional motion, in the presence of a rigid frame placed on four bushings. All hbrid paths are assumed to be an actuator in series with an elastomeric mount; and discrete linear timeinvariant deternistic sstems are assumed with small motions, harmonic ecitations, stead state behavior, and no kinematic nonlinear effects. Also, passive elements are assumed massless while active elements possess mass. Additionall, passive torque roll ais motion decoupling concepts are eplored to enhance active control capabilities given certain practical constraints. Analtical, computational, and eperimental methods are utilied though no real-time control is done. First, the torque roll ais motion decoupling concept is studied in a degree of freedom model of a realistic powertrain and coupled frame. Deficienc of prior literature neglecting the need for a phsicall realiable sstem is overcome b deriving improved ii

4 mount compatibilit conditions, implemented in new decoupling paradigms to ensure more realistic mount positions. It is mathematicall shown that full decoupling is not possible for a practical sstem, and partial decoupling paradigms are pursued to ensure that onl the powertrain roll motion is donant. This constitutes as a major contribution. The interaction between hbrid paths is studied net as part of a resonating two path source-path-receiver sstem with 6 degrees of freedom, simplified from a realistic powertrain and frame sstem. The main contribution of this work is derivation of a performance inde (dictated b passive sstem dnacs) that characteries this interaction for source mass motion control; two passive sstem parameters (hbrid path damping and disturbance force location) emerge that drasticall change the performance inde. Design paradigms are developed for desirable path interactions, and lited eperimental validation demonstrates motion control concept at 400 H. Finall, a new 4 degree of freedom mathematical model for a coupled powertrain and frame is developed with versatilit to select passive onl, active onl, or hbrid powertrain paths. Additionall, new torque roll ais decoupling paradigms are derived for non-identical powertrain mount properties and orientations, and the new model allows for arbitrar or torque roll ais designed mounts. Improved vibration control is achieved with hbrid mounts over active or passive onl, and it is found that a nimum of two actuators should be used for a realistic powertrain mounting sstem. Future work should include construction of a powertrain and frame eperiment to eane hbrid path effectiveness and torque roll ais mounting schemes, eanation of active motion control for transient powertrain events, and application of real-time control. iii

5 Dedication To Brandi, m fal, and peanut. iv

6 Acknowledgements First, I would like to thank m advisor, Prof. Rajendra Singh, for his guidance throughout m undergraduate, masters, and doctoral stud. His tremendous knowledge, eperience, and man conversations have been invaluable throughout m acadec career. I also would like to epress appreciation to Dr. Jason Dreer for his stead support and man technical discussions to enhance this work. Man thanks as well to m comttee members Prof. Manoj Srinivason, Prof. Vishnu Baba Sundaresan, and Prof. Vadim Utkin for their time to criticall review m work; also thanks to Caterina Runon- Spears for her careful reviews of this work and corresponding publications. Lastl, I would like to thank all the members of the Acoustics and Dnacs Lab for their patience and thoughtful feedback on the numerous research presentations over the ears. I would like to thank the member organiations such as Hundai Motor Compan (R&D Division) of the Smart Vehicle Concepts Center ( and the National Science Foundation Industr/Universit Cooperative Research Centers program ( ) for supporting this work. Most importantl, I am grateful to m wife and fal for their endless support and encouragement over m ten ear journe from undergraduate to PhD. v

7 Vita November Born in Coldwater, Ohio December B.S. Mechanical Engineering, The Ohio State Universit, Columbus, Ohio August 0...M.S. Mechanical Engineering, The Ohio State Universit, Columbus, Ohio March 009 to August 0...BS/MS Program, Mechanical Engineering, The Ohio State Universit, Columbus, Ohio Januar 00 to present...graduate Research Associate, SVC Fellowship, The Ohio State Universit, Columbus, Ohio Publications. J. Liette, J.T. Dreer, R. Singh, Dnac characteriation of the rectangular piston seal in a disc-caliper braking sstem using analtical and eperimental methods. Proc. IMechE Part D: Automobile Engineering 6 (0) J. Liette, J.T. Dreer, R. Singh, Interaction between two active structural paths for source mass motion control over d-frequenc range. Journal of Sound and Vibration 333 (04) Fields of Stud Major Field: Mechanical Engineering Main Stud Areas: Mechanical Vibrations, Sstem Dnacs, Active and Passive Vibration Control, Automotive Powertrain Mounting Schemes vi

8 Table of Contents Abstract... ii Acknowledgements... v Vita... vi Table of Contents... vii List of Tables... i List of Figures... ii List of Smbols... vii Chapter : Introduction.... Motivation.... Literature review Problem formulation... 5 References for Chapter... Chapter : Critical eanation of isolation paradigms for a coupled powertrain and frame: Partial torque roll ais decoupling methods given practical constraints Introduction Problem formulation... 7 vii

9 .3. Analtical model Powertrain sub-sstem Frame sub-sstem and coupled sstem Conditions for full decoupling Compatibilit conditions for a realiable sstem Analsis of decoupling conditions Design paradigms for partial decoupling Coupled sstem with arbitrar mount placement (Eample I) Coupled sstem mount design using onl the powertrain sub-sstem (Eample II) Miniation of decoupling conditions for coupled sstem design (Eample III) Design of frame mounts to improve partial decoupling Alignment of powertrain and frame TRA aes Full sstem frame mount design using onl the frame sub-sstem (Eample IV) Alternate design paradigms and practical constraints Conclusion References for Chapter viii

10 Chapter 3: Interaction between two hbrid structural paths for active source mass motion control over d-frequenc range Introduction Problem formulation Analtical model Design of a feasibilit eperiment Quantification of path interaction Calculation of ke design parameters Investigation of the second peak in L Investigation of the first peak in L Investigation of the valles in L and transitional parameter summar Eperimental validation Conclusion... 9 References for Chapter Chapter 4: Enhancement of vibration control for powertrain mounting scheme paradigms: Combination of active and passive methods Introduction Problem formulation Modeling the sstem configurations... 0 i

11 4.3.. Model schematics Mathematical formulations Active powertrain motion control Comparative stud of sstem configurations Analsis of hbrid path effectiveness Alternate activated path combinations Mechanisms controlling hbrid effectiveness Active control with arbitrar mount design Conclusion References for Chapter Chapter 5: Conclusion Summar Contributions Future work References for Chapter Bibliograph... 48

12 List of Tables Table Page. Powertrain and frame parameters that (a) remain the same for all eamples and (b) are specific to Eample I Mount locations and TRA Ω for (a) powertrain of Eamples II and III and (b) frame TRA TRA of Eample IV with and without q g = q g Comparison of natural frequencies ( π ) Ω Identified sstem parameters Predicted and measured insertion losses for a 400 H disturbance force Mounting sstem configurations and corresponding model designations Sstem parameters for (a) powertrain and frame and (b) path models for i = j =,, 3, Activated path (mount) permutations to target powertrain motions ( g5, g5) ( g5, g5) n θ ε or θ ε...4 i

13 List of Figures Figure Page. Conceptual source-path-receiver model to represent a realistic powertrain mounting scheme with active and passive structural path elements...5. Eample schematic for a coupled powertrain and frame mounting sstem Resonating two hbrid path source-path-receiver sstem with active and passive elements Schematic of a coupled powertrain and frame sstem with passive onl, active onl, and hbrid powertrain paths...0. Eample case with schematics and coordinate sstems for (a) coupled powertrain and frame sstem, (b) powertrain sub-sstem, and (c) frame sub-sstem. Ke:, inertial coordinates ( gj ) Γ ; - - -, mount coordinates ( Γ ) ; and, TRA TRA coordinates ( Γ )...8. Impractical mount locations for an eample in Hu and Singh [.] referenced to (a) powertrain sub-sstem and (b) frame sub-sstem. Ke:, mount #;, mount #; *, mount #3; and, mount # Rotation of mounts from the mount coordinates ( ) Γ to the TRA coordinates TRA ( Γ ) in the mounting plane...9 ii

14 .4 Powertrain displacement magnitude spectra for Eamples I and II in the TRA Γ coordinate sstem: (a) ε g, (b) ε g, (c) ε g, (d) θ g, (e) θ g, and (f) θ g. Ke:, Eample I and - - -, Eample II Powertrain displacement magnitude spectra for Eamples II and III in the TRA Γ coordinate sstem: (a) ε g, (b) ε g, (c) ε g, (d) θ g, (e) θ g, and (f) θ g. Ke: - - -, Eample II and, Eample III Comparison of powertrain mount locations relative to the powertrain center of gravit for Eamples I, II, and III. Ke:, Eample I (baseline);, Eample II; and, Eample III Comparison of frame mount locations relative to the frame center of gravit for TRA TRA Eamples II and IV. Ke:, Eample II and, Eample IV with q g q g Powertrain displacement magnitude spectra for Eamples II and IV in the TRA Γ coordinate sstem: (a) ε g, (b) ε g, (c) ε g, (d) θ g, (e) θ g, and (f) θ g. TRA TRA Ke: - - -, Eample II and - - -, Eample IV with q g q g Powertrain displacement magnitude spectra for Eample IV in the TRA Γ coordinate sstem: (a) ε g, (b) ε g, (c) ε g, (d) θ g, (e) θ g, and (f) θ g. Ke: - - -, TRA TRA TRA TRA Eample IV with q g q g and, Eample IV with q g = q g Powertrain displacement magnitude spectra for Eamples I and IV in the TRA Γ coordinate sstem: (a) ε g, (b) ε g, (c) ε g, (d) θ g, (e) θ g, and (f) θ g. TRA TRA Ke:, Eample I and, Eample IV with q g = q g Powertrain displacement magnitude spectra for Eamples II in the TRA Γ coordinate sstem with damping and stiffness modifications: (a) ε g, (b) ε g, (c) ε g, (d) iii

15 θ g, (e) g θ, and (f) θ g. Ke: - - -, Eample II;, Eample II with 4C g ; and *, Eample II with 0 K bi Incorporation of active path elements to nie source motion or sound radiation Schematic of the analtical model: (a) inertial coordinates and (b) eample case with new coordinates at the attachments of active or passive mounts Schematic of the eperiment Comparison of eperiment and model for a 400 H disturbance force. Ke:, measured and - - -, predicted Insertion loss L up to 500 H. Ke:, L and - - -, ero db Magnitude spectra of (a) H ξ 3 and (b) ( ξ det ) ξ κ H Effect of η m3 on (a) det ξ ξ L and (b) ( ) 3 κ H. Ke:, η 3 = 0.30 ; - - -, η 3 = 0.50 ;, η 3 = 0.70 ; and - - -, ero aes...77 m m 3.8 Phase β before and after transitional point of the first peak in L for (a) η m3 and (b) moment arm d. Ke: (a), η 3 =.467 ;, η 3 =.469 and (b), m d = 0.4;, d = Calculated control force amplitudes. Ke:, F 3 ; - - -, F 4 ; and - - -, ero ais Insertion loss L before and after transitional point for (a) the first and second valles for ( η m3 ), (b) the first peak for t ( m3 ) t the second peak for ( m3 ) m t m η, (c) the first peak for d t, and (d) η. Ke: (a), η 3 = ;, η 3 = ; (b), iv m m

16 η 3 =.467 ;, η 3 =.469; (c), d = 0.4;, d = ; and (d), m m η 3 = ;, η 3 = m m 3. Motion control of a 400 H disturbance force for (a) model and (b) eperiment Liting cases for powertrain and frame mounting sstem configurations: (a) passive onl and (b) active onl Schematics for three powertrain and frame mounting sstem models: (a) Model A, (b) Model B, and (c) Model C. Ke:, inertial coordinates ( Γ gj ) coordinates ( ) TRA Γ ; and, TRA coordinates ( ) ;, mount Γ Kinematic relationship between path configurations: (a) with passive mounts onl and (b) with active and passive mounts Rotation of active force vectors into TRA coordinates TRA Γ Miniation of selected motions in time domain for Model C with activated paths ( j, j ) = ( 3, 4) : (a) θ ( t) and (b) ( t) g5 ε Mount (path) locations (marked b X) relative to the powertrain (rigid bod) for (a) "arbitrar" design and (b) TRA design Powertrain motion (magnitude) spectra for (i) ε g5, (ii) ε g5, (iii) ε g5, (iv) θ g5, (v) θ g5, and (vi) θ g5 for alternate models: (a) Model A and (b) Model B. Ke: (a), Model A;, Model A with low damping;, Model A and (b), Model B;, Model B Powertrain motion (magnitude) spectra for (i) ε g5, (ii) ε g5, (iii) ε g5, (iv) θ g5, (v) θ g5, and (vi) θ g5 for Model C with alternate path # and # orientations: v g5

17 (a) ϕ = 30 and (b) ϕ = 0. Ke: (a), no activated paths;, two activated paths (, ) (, ) j j = targeting motions ( g5, g5) θ ε and (b), no activated paths;, one activated path j = targeting motion θ g5 ;, two activated paths (, ) (, ) j j = targeting motions ( g5, g5) θ ε Effect of active control on non-targeted motions as deterned b Model C metrics. Ke:, ε Φ ;, θ Φ ; and, F Φ Magnitude spectra of motion denonator dnacs with two activated paths (, ) (, ) j j = targeting motions ( g5, g5) θ ε for (a) individual effects and (b) combined effect. Ke: (a), det ( κ g ) ;, det ( G ) and (b), det ( κ g ) ;, det ( κ ) det ( G)...30 g 4. Powertrain eigenvector components for the first three modes: (a) Mode, (b) Mode, and (c) Mode 3. Ke:, ( j, j ) = (, ) and, (, ) (, 3) ( g5, g5) j j = targeting θ ε Magnitude spectra of powertrain motions (i) ε g5, (ii) ε g5, (iii) ε g5, (iv) θ g5, (v) θ g5, and (vi) g5 targeting motions ( g5, g5) θ for Model C with two activated paths ( j, j ) = (, ) θ ε. Ke:, Model C and, Model C...34 vi

18 Nomenclature a translational acceleration A constant coefficient for Λ component of A A sstem matri b normaliing constant B constant coefficient for Λ c viscous damping coefficient C constant coefficient for Λ = cosϕ C ϕ i List of Smbols matri cofactor C viscous damping matri matri of cofactors d disturbance moment arm D constant coefficient for Λ E constant coefficient for Λ E skew smmetric rotation matri f(t) time varing control force f control force vector F control force amplitude comple control force amplitude F control force amplitude vector vector of comple control force amplitudes g gravitational constant G sstem matri for control force calculation H dnac compliance H dnac compliance matri i square root of - I mass moment of inertia I identit matri J objective function k stiffness K component of K vii

19 K stiffness matri l length L insertion loss m mass matri nor M inertia matri N some integer p number of structural paths P constant for TRA decoupling condition (i) P mechanical power P matri of P constants O matri of eros q generalied displacement vector Q generalied displacement amplitude Q generalied displacement amplitude vector r coordinates of position vector r position vector R reaction force vector R rotational transformation matri S pieo stack sensitivit S ϕ i = sinϕ t time T torque ecitation amplitude T(t) time varing torque ecitation T torque ecitation amplitude vector U partitioning for disturbance force vector w(t) time varing disturbance force w disturbance force vector W disturbance force amplitude W disturbance force amplitude vector,, Cartesian coordinates Y component of Y Y output vector α constant value β phase corresponding to Ξ γ dimensionless mount position γ dimensionless mount position vector Γ Cartesian coordinate sstem ε translational displacement at center of mass ε translational displacement vector at center of mass translational displacement amplitude at center of mass η loss factor viii

20 θ θ Θ κ κ λ Λ μ ν ξ Ξ Π ρ ρ nk, n k σ m = km rg, m0 nk, n k σ b = kb rg, b0 rotational displacement rotational displacement vector rotational displacement amplitude dnac stiffness dnac stiffness matri = ω, eigenvalue compliance numerator dnacs proportional damping coefficients eigenvector components translational displacement at mounts translational displacement amplitude at mounts transformation matri component of ρ θ = M τ scaling constant ς constant coefficient for J γ n υ directional cosine φ Euler angle φ vector of Euler angles ϕ phase corresponding to f Φ metric for active powertrain motion control effectiveness χ effectiveness of the control forces ψ residual ψ vector of residuals Ψ acceleration root mean square value ω circular frequenc (rad sec - ) ω vector of circular ecitation frequencies Ω resonance frequenc (rad sec - ) Ω vector of resonance frequencies Subscripts 0,,,k,n general indices a active powertrain mount b passive base (frame) mount g referenced to discrete inertia element coordinates i mount inde j inertia inde m passive powertrain mount n n matri of dimension (n, n) r receiver i

21 s source t transitional value,, referenced to Cartesian coordinates γ referenced to equations for solving γ η referenced to quadratic for transitional loss factor λ referenced to quartic for transitional frequenc active path element modified to represent a passive element Superscripts 0,,,k,n general indices a alternating B referenced to constant B λ C referenced to constant C λ D referenced to constant D λ E referenced to constant E λ F control force G referenced to matri G I imaginar component m mean num numerator dnacs R real component T transpose,, referenced to Cartesian coordinates ε translational direction θ rotational direction κ referenced to matri κ ξ relating to equations of motion formulation using mount displacements d( )/dt d ( )/dt refers to sstem before active forces are applied comple valued normalied ʹ relating to inertial coordinates relating to elastic coordinates * reference parameter Operators ( ) magnitude of comple number ( ) phase of comple number t time average ( ) ( ) cross product

22 det( ) diag( ) dim{ } Im{ } Re{ } deternant diagonal matri dimension of matri or vector imaginar part of comple number real part of comple number Abbreviations TRA torque roll ais c.g. center of gravit i

23 Chapter : Introduction. Motivation Emerging design trends such as higher power densit powertrains, lighter weight structures, and hbrid electric powertrains impose noise and vibration issue concerns. Such eamples are higher transtted dnac forces, increased global motions, and new d-frequenc ecitation frequencies (sa from 00 to 000 H) which degrade acoustic comfort. This scholarl research is therefore formulated b the need for superior control of a vehicle powertrain dnac behavior given these new challenges. Conventional prime movers such as combustion engine powertrains generall have low frequenc vibration ecitations and are supported b 3 or 4 elastomeric mounts; a primar mount function is to reduce the transssion of dnac forces. Motion control techniques are also done at low frequencies, isolating the rigid bod kinetic energ in one or two degrees of freedom. Additionall, source mass motion control is needed for hbrid electric powertrains, as these produce significant d-frequenc ecitations that amplif the source regime and create significant structure-borne and radiated noise. Such control of vehicle powertrain dnac behavior is commonl accomplished using passive methods [.-0] with adequate success.

24 The emerging design trends, however, necessitate a hbrid approach of active and passive methods to meet more stringent sstem performance targets, utiliing active control schemes [.-33] and algorithms [.34-36] for improved motion and vibration control. Man practical litations hinder passive onl performance [.7], such as selecting either high damping for resonance control or low damping for high frequenc isolation. Active control alone also has several engineering litations in addition to the cost issue: actuator damping is generall low, effecting resonance control capabilities, and stiffness is generall high, increasing sstem resonances to the audible range which would degrade acoustic comfort. A hbrid approach of active and passive should overcome man litations of passive or active alone, and proper hbrid design could be driven b the sstem dnacs. In particular, the phase interaction (caused b passive sstem dnacs) between hbrid paths and the resulting sstem motion ma dictate the effectiveness of active control strategies. The primar research goal is therefore to acquire fundamental understanding of dnac interactions among multiple hbrid paths in a powertrain mounting sstem for improved control of multi-dimensional motion; both powertrain and frame rigid bod dnacs are considered for a realistic eample case [.37-40]. Passive motion decoupling ma improve active control capabilities, as the rigid bod kinetic energ is isolated and fewer actuators would be needed. Torque roll ais (TRA) motion decoupling [.8-0] is one such method that requires further investigation.

25 . Literature review The thrust of prior studies relevant to hbrid powertrain mounting schemes [.- 8] is usuall on control algorithms, where an actuator (active) and rubber or hdraulic element (passive) in series or parallel constitutes a hbrid path. Here, the passive element is onl utilied to provide static support and to ensure the control sstem stabilit, though a tuned reaction mass absorber [.] has been considered. For instance, decentralied velocit control is studied in sstems with multiple hbrid paths [.-], but the hbrid path interactions are not adequatel addressed. Additionall, such studies focus on vibration isolation, liting the dnac forces transtted through the mounts. Eample cases of source mass (powertrain) motion control include reduction of d-frequenc structure-borne and radiated noise via active [.9-33] and passive [.5,.6] patches, reducing the structure surface velocit to nie the radiated sound pressure or altering the radiating efficienc of the structure. Tpicall, the patches are bonded directl to the structure surface (thus no paths or interactions), and hbrid structural paths could more effectivel reduce the structure motion (and surface velocit) through appropriate design. Interactions among passive paths have been studied to some etent [.4,.4], but hbrid paths are fundamentall different due to the applied actuation force(s). Hbrid path interactions are not of interest in single mount [.3-8] or Stewart platform [.8- ] applications, but such configurations ma not be viable due to multi-aial forces or lited packaging space (geometric constraints). In general, literature is sparse on hbrid path interactions in the contet of active control, though such analsis is ke to maiing vibration control of multi-dimensional rigid bod motions. As such, studies 3

26 should first focus on simplified sstems (sa a two path sstem), and then etend to realistic sstems such as a coupled powertrain and frame. Another area where literature is sparse is the integration of active vibration control with passive TRA decoupling. One eample is Park and Singh [.3], where a single hbrid path is eaned in a four path powertrain mounting sstem with a rigid foundation (no frame dnacs). The focus is on modeling of realistic hdraulic and pieoelectric based actuators, as well as the powertrain comple eigensolution. Lited investigation is done on TRA decoupling capabilities, and it is concluded that decoupling is not achieved when a single hbrid path is used. Further eanation ma ield sufficient partial decoupling paradigms, as ght the use of two hbrid paths instead of one; frame dnacs should be included in such an analsis [.37-40]. problem Recentl, Hu and Singh [.8] proposed new aioms (leading to an eigenvalue TRA TRA TRA Kq = λ Mq ) to account for low frequenc dnac interactions between a powertrain mounting sstem and coupled compliant base sub-sstems, successfull decoupling all rotational and translation motions from the powertrain TRA. Here, the eigenvalue, TRA λ is TRA q is the displacement vector, superscript T indicates a transpose, M is the inertia matri of the powertrain and coupled sub-sstems, and K is the coupled stiffness matri. Through critical eanation, it is found that the derivation of K neglects the need for a phsicall realiable sstem. Namel, each powertrain mount is referenced to two different locations, and the sstem cannot be constructed. This issue must be addressed before integrating active vibration control with passive TRA decoupling as part of this scholarl research. 4

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28 deternistic and frequenc independent parameters. Vibrator motions are small and higher order terms are ignored, and thus the principle of superposition is valid. Additionall, each mass is assumed to be rigid with no fleural mode participation, and onl a known eternal harmonic disturbance force ecites the source mass. No real-time control is used, and closed form analtical control force epressions are instead derived to understand the stead state sstem behavior. In the hbrid structural paths, passive elements are assumed massless while active elements possess mass, and it is assumed that each actuator input can be represented b an applied force at a discrete mass with a constant actuator gain. Thus, actuators are assumed to be linear and well known with no hsteresis effects; rotational effects of the actuator inertia are also ignored. The actuator forces analticall applied at the corresponding discrete masses are transtted to both the source and receiver to account for mass participation effects. In general, a passive and active mount should be used in combination to provide vibration isolation, and all hbrid paths consist of a stiff active actuator (attached to source mass) in series with a passive rubber elastomer (attached to receiver mass), as suggested b Beard et al. [.43]. The TRA is defined as an ais around which rotation occurs when a torque pulse is eerted on a free rigid bod (engine and transssion combined) about an arbitrar direction [.9], often deviating from the crankshaft or inertial aes b up to 5 in man practical cases [.44]. The vector dictating the TRA orientation is deterned b the rigid bod inertia matri onl, though approimations such as connecting engine and transssion centers of gravit are suggested to estimate this ais [.7]. When constraints 6

29 (engine mounts) are added to the free rigid bod, an eigenvalue problem (silar to TRA TRA TRA Kq = λ Mq ) must be satisfied such that rotation about the TRA is also a natural mode of the sstem [.9]. Even if the powertrain foundation is assumed to be rigid, Kim [.45] has argued that full TRA decoupling is not possible for completel arbitrar mount locations. Thus, it is assumed that all powertrain mounts are located in the so called mounting plane relative to the TRA coordinates to improve decoupling capabilities [.9]. The specific objectives of this dissertation are outlined below along with corresponding sub-objectives, organied to parallel Chapters to 4. Objective : Etend TRA decoupling aioms recentl proposed b Hu and Singh [.8] for a degree of freedom powertrain and coupled frame; an eample case sstem schematic is shown in Fig... Through critical eanation, it is found that the aioms in [.8] neglect the need for a phsicall realiable sstem, and full passive motion decoupling is not possible with realistic mount locations (addressed in Chapter ). (a) Provide a mathematical proof that full TRA decoupling is not possible for a phsicall realiable powertrain and coupled frame sstem. (b) Propose powertrain and frame mount designs paradigms to enhance the partial decoupling of the powertrain TRA without imposing severe burden on the isolation sstem design. (c) Eane alternative isolation sstem design paradigms to further improve decoupling. 7

30 Engine Transssion Driveline Powertrain mounts Frame mounts Frame Rigid foundation Fig... Eample schematic for a coupled powertrain and frame mounting sstem. Objective : Eane the dnac interaction between hbrid paths (dictated b passive sstem dnacs) using a conceptual two path resonating source-path-receiver sstem, as shown in Fig..3. This is simplified from a realistic coupled powertrain and frame sstem, and the active control objective is source mass motion control of d-frequenc ecitations (addressed in Chapter 3). (a) Develop an analtical model of the sstem and define a performance inde to characterie the dnac hbrid path interaction for source mass motion control up to 000 H. (b) Identif ke passive design parameters and values dictating the defined performance inde; formulate appropriate design paradigms for effective active control. (c) Design a feasibilit eperiment and investigate source mass motion control at a representative d-frequenc ecitation of 400 H. 8

31 Source Mass Active Mount Passive Mount Receiver Mass Fig..3. Resonating two hbrid path source-path-receiver sstem with active and passive elements. Objective 3: Develop a new mathematical model for a coupled powertrain and frame with versatilit to allow passive onl, active onl, or hbrid powertrain paths to be selected with arbitrar or TRA designed mount locations; a representative sstem schematic is shown in Fig..4. Eane active powertrain motion control combined with TRA decoupling at low frequenc ecitations up to 70 H (addressed in Chapter 4). (3a) Formulate mathematical constructs and a closed form active powertrain motion control scheme for all powertrain path combinations. (3b) Quantitativel compare passive, active, and hbrid mounting paradigms for vibration control effectiveness. (3c) Identif and anale mechanisms controlling hbrid path behavior. Each chapter is organied to be self-sufficient including its own list of references; therefore, some items ma be duplicated within the dissertation for the sake of completeness. All chapters share a single list of smbols, however, given at the beginning 9

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33 References for Chapter [.] A. Inoue, R. Singh, G.A. Fernandes, Absolute and relative path measures in a discrete sstem b using two analtical methods. Journal of Sound and Vibration 33 (008) [.] S. Kim, R. Singh, Multi-dimensional characteriation of vibration isolators over a wide range of frequencies. Journal of Sound and Vibration 45(5) (00) [.3] R.A. Ibrahim, Recent advances in nonlinear passive vibration isolators. Journal of Sound and Vibration 34 (008) [.4] J.P. Den Hartog, Mechanical Vibrations. Dover Publications, New York (985). [.5] S.W. Kung, R. Singh, Vibration analsis of beams with multiple constrained laer damping patches. Journal of Sound and Vibration (5) (998) [.6] S.W. Kung, R. Singh, Development of approimate methods for the analsis of patch damping concepts. Journal of Sound and Vibration 9(5) (999) [.7] C.M. Harris, Shock and Vibration Handbook, McGraw-Hill, New York (995). [.8] J.-F. Hu, R. Singh, Improved torque roll ais decoupling aiom for a powertrain mounting sstem in the presence of a compliant base. Journal of Sound and Vibration 33(7) (0) [.9] T. Jeong, R. Singh, Analtical methods of decoupling the automotive engine torque roll ais. Journal of Sound and Vibration 34() (000) [.0] J.-Y. Park, R. Singh, Effect of non-proportional damping on the torque roll ais decoupling of an engine mounting sstem. Journal of Sound and Vibration 33 (008) [.] M. Serrand, S.J. Elliott, Multichannel feedback control of the isolation of baseecited vibration. Journal of Sound and Vibration 34(4) (000) [.] X. Huang, S.J. Elliott, M.J. Brennan, Active isolation of a fleible structure from base vibration. Journal of Sound and Vibration 63 (003) [.3] T.J. Yang, Z.J. Suai, Y. Sun, M.G. Shu, Y.H. Xiao, X.G. Liu, J.T. Du, G.Y. Jin, Z.G. Liu, Active vibration isolation sstem for a diesel engine. Noise Control Engineering Journal 60(3) (0) [.4] P. Gardonio, S.J. Elliott, Passive and active isolation of structural vibration transssion between two plates connected b a set of mounts. Journal of Sound and Vibration 37(3) (000) [.5] S.M. Kim, S.J. Elliott, M.J. Brennan, Decentralised control for multichannel active vibration isolation. IEEE Transactions on Control Sstems Technolog 9() (00)

34 [.6] P. Gardonio, S.J. Elliott, R.J. Pinnington, Active isolation of structural vibration on a multiple-degree-of-freedom sstem, part I: the dnacs of the sstem. Journal of Sound and Vibration 07() (997) [.7] P. Gardonio, S.J. Elliott, R.J. Pinnington, Active isolation of structural vibration on a multiple-degree-of-freedom sstem, part II: effectiveness of active control strategies. Journal of Sound and Vibration 07() (997) 95-. [.8] Z.J. Geng, L.S. Hanes, Si degree-of-freedom active vibration control using the Stewart platforms. IEEE Transactions on Control Sstems Technolog () (994) [.9] A. Preumont, M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, A. Abu Hanieh, A si-ais single-stage active vibration isolator based on Stewart platform. Journal of Sound and Vibration 300 (007) [.0] D. Stewart, A platform with si degrees of freedom. Proceedings of the Institute of Mechanical Engineers 80() (965) [.] A. Preumont, Vibration Control of Active Structures An Introduction. Kluwer Acadec Publishers, Massachusetts (00). [.] E. Garcia, S. Webb, J. Duke, Passive and active control of a comple fleible structure using reaction mass actuators. Journal of Vibration and Acoustics 7 (995) 6-. [.3] J.-Y. Park, R. Singh, Analsis of powertrain motions given a combination of active and passive isolators. Noise Control Engineering Journal 57(3) (009) [.4] B. Kim, G.N. Washington, R. Singh, Control of modulated vibration using and enhanced adaptive filtering algorithm based on model-based approach. Journal of Sound and Vibration 33 (0) [.5] B. Kim, G.N. Washington, R. Singh, Control of incommensurate sinusoids using enhanced adaptive filtering algorithm based on sliding mode approach. Journal of Vibration and Control 9(8) (0) [.6] G. Pinte, S. Devos, B. Stallaert, W. Smens, J. Swevers, P. Sas, A pieo-based bearing for the active structural acoustic control of rotating machiner. Journal of Sound and Vibration 39 (00) [.7] S.J. Elliott, Active control of structure-borne noise. Journal of Sound and Vibration 77(5) (994) [.8] V. Fakhari, A. Ohadi, Robust control of automotive engine using active engine mount. Journal of Vibration and Control 9(7) (0) [.9] E. Bianchini, Active vibration control of automotive like panels. SAE paper (008).

35 [.30] K. Wolff, H.-P. Lahe, C. Nussmann, J. Nehl, R. Wimmel, H Siebald, H. Fehren, M. Redaelli, A. Naake, Active noise cancellation at powertrain oil pan. SAE paper (007). [.3] J.P. Carneal, C.R. Fuller, An analtical and eperimental investigation of active structural acoustic control of noise transssion through double panel sstems. Journal of Sound and Vibration 7 (004) [.3] C.R. Fuller, Eperiments on active control of sound radiation from a panel using a pieocerac actuator. Journal of Sound and Vibration 50() (99) [.33] C.R. Fuller, Active control of sound radiation from a vibrating rectangular panel b sound sources and vibration inputs: an eperimental comparison. Journal of Sound and Vibration 45() (99) [.34] S.J. Elliott, A review of active noise and vibration control in road vehicles. ISVR Technical Memorandum 98 (008) -5. [.35] R. Alkhatib, M.F. Golnaraghi, Active structural vibration control: a review. The Shock and Vibration Digest 35(5) (003) [.36] C.R. Fuller, S.J. Elliott, P.A. Nelson, Active Control of Vibration. Acadec Press Inc., California (996). [.37] J.M. Lee, H.J. Yim, J.-H. Kim, Fleible chassis effects on dnac response of engine mounts sstems. SAE paper (995). [.38] H. Ashrafiuon, Design of optiation of aircraft engine mount sstems. ASME Journal of Vibration and Acoustics 5(4) (993) [.39] H. Ashrafiuon, C. Nataraj, Dnac analsis of engine-mount sstems. ASME Journal of Vibration and Acoustics 4() (99) [.40] M. Sirafi, M. Qatu, Accurate modeling for the powertrain and subframe modes. SAE paper (003). [.4] A. Inoue, R. Singh, G.A. Fernandes, Absolute and relative path measures in a discrete sstem b using two analtical methods. Journal of Sound and Vibration 33 (008) [.4] S. Kim, R. Singh, Multi-dimensional characteriation of vibration isolators over a wide range of frequencies. Journal of Sound and Vibration 45(5) (00) [.43] M.J. Beard, A.H. Von Flotow, D.W. Schubert, A practical product implementation of an active/passive vibration isolation sstem. Proceedings of IUTAM smposium on the Active Control of Vibration, Universit of Bath, UK, 994, pp [.44] R.M. Brach, Automotive powerplant isolation strategies. SAE paper 9794 (997). 3

36 [.45] B.J. Kim, Three dimensional vibration isolation using elastic aes. M.S. Thesis, Michigan State Universit, MI (99). [.46] L.L. Beranek, Noise and Vibration Control. Wile, INSE, New Jerse (988). 4

37 Chapter : Critical eanation of isolation paradigms for a coupled powertrain and frame: Partial torque roll ais decoupling methods given practical constraints.. Introduction Recentl, Hu and Singh [.] proposed new aioms to account for low frequenc dnac interactions between a powertrain mounting sstem and coupled compliant base sub-sstems, successfull decoupling all rotational and translation motions from the powertrain torque roll ais (TRA) using passive powertrain mount design. These aioms are defined for a discrete, proportionall damped sstem as an eigenvalue problem: TRA TRA TRA Kq = λ Mq. (.) Here, { } T TRA TRA TRA T λ is the eigenvalue, = ( g ) 6 TRA q q O, { } q T is g = the TRA vector for the powertrain, O nn is an n n null matri, superscript T indicates a transpose, M is the inertia matri of the powertrain and coupled sub-sstems, and K is the coupled stiffness matri,. While this criterion is strictl mathematicall correct, the derivation of K neglects the need for a phsicall realiable sstem. Namel, each mount coupling the powertrain to the compliant base is referenced to two different locations: from the powertrain to the mount and from the compliant base to the mount. If caution is not taken, the locations do not coincide, and a phsical isolation sstem cannot be 5

38 constructed. Such is the case in Hu and Singh [.], and this issue must be addressed. Therefore, the chief goal of this chapter is to include compatibilit conditions in the derivation of K for Eq. (.) such that mounts alwas are referenced to a single location, allowing for a phsicall realiable mounting sstem. Also, it will be shown that conditions imposed b Eq. (.) result in a sstem that cannot be full decoupled. Accordingl, methods for partial decoupling of the TRA direction must be pursued without imposing severe burden on the isolation sstem design. In prior studies, Jeong and Singh [.] propose TRA decoupling aioms assung a rigid foundation (no frame dnacs) for a powertrain mounting sstem and a discrete, proportionall damped model. Arbitrar mount locations are assumed, and an eigenvalue problem silar to Eq. (.) is derived. These aioms are etended b Park and Singh [.3] to a non-proportionall damped model, where two eigenvalue problems must be concurrentl satisfied. It is suggested that sub-sstem dnacs cannot be neglected in a real-life vehicle, where the foundation dnacs (e.g. powertrain frame or cradle) ma have a significant effect on the dnac powertrain response [.4-7], especiall when frame natural frequencies and the ecitation frequencies lie in the same regime. For instance, Lee et al. [.4] calculate several modes in the to 0 H range for an uncoupled model of the frame connected to the vehicle bod and tires. Hu and Singh [.] calculate modes up to 5 H for the uncoupled frame onl, which are ecited b higher orders of the engine torque ecitation. Also considering that the weight of the frame is generall less than the powertrain itself [.8], there is a need to properl include the frame dnacs for TRA decoupling. Siraif and Qatu [.7] conclude as such b comparing 6

39 modal results of a powertrain model to that of full vehicle eperimental data, and inclusion of the frame dnacs drasticall improves results. Sub-structuring methods [.9-] and mode shifting [.] ma be used to lit the modal coupling, but these are outside the scope of this chapter... Problem formulation The TRA is defined as an ais around which rotation occurs when a torque pulse is eerted on a free rigid bod (engine and transssion combined) about an arbitrar direction [.], often deviating from the crankshaft or inertial aes b up to 5 in man practical cases [.3]. The vector dictating the TRA orientation is deterned b the rigid bod inertia matri onl, though approimations such as connecting engine and transssion centers of gravit are suggested to estimate this ais [.4]. When constraints (engine mounts) are added to the free rigid bod, conditions silar to Eq. (.) must be satisfied such that rotation about the TRA is also a natural mode of the sstem [.]. A conceptual model of the coupled powertrain and frame sstem is shown in Fig..(a) to illustrate a practical TRA orientation, where Γ gj are inertial Cartesian coordinates (,, ) at the center of gravit (c.g.) of the j th rigid bod, Γ are principal elastic Cartesian coordinates (,, ) of the i th mount (diagonal stiffness matrices), and TRA Γ are Cartesian coordinates (,, ) where the TRA is the -ais. It is assumed that all Γ gj are parallel with a vertical ais and an ais along the driveline. 7

40

41 Analsis of the sstem in Fig..(a) requires transformation of all ecitation and dnac reaction forces into a single coordinate sstem. Namel, the sstem should be analed in TRA Γ to better facilitate derivation of the decoupling conditions in Eq. (.). The scope of this chapter is lited to proportionall damped, discrete linear timeinvariant sstems with small motions (as also assumed b Hu and Singh [.]), and necessar transformations are easil implemented. Additionall, the scope is lited to TRA dnac decoupling methods of a powertrain and coupled frame mounting sstem with asmmetric inertia matrices. Even if the powertrain is assumed to have a rigid foundation, Kim [.5] has argued that full decoupling is not possible for completel arbitrar mount locations. Thus, it is assumed that all powertrain mounts are located in the so called mounting plane relative to TRA Γ to improve decoupling capabilities [.]. Specific objectives include the following: () provide a mathematical proof that full TRA decoupling is not possible for a phsicall realiable powertrain and coupled frame sstem, () propose powertrain and frame mount designs paradigms to enhance the partial decoupling of the powertrain TRA without imposing severe burden on the isolation sstem design, and (3) eane alternative isolation sstem design paradigms to further improve decoupling..3. Analtical model.3.. Powertrain sub-sstem Like Hu and Singh [.], the individual models for the decoupled powertrain and frame sub-sstems are formulated first; each sub-sstem is assumed to have si total 9

42 degrees of freedom with three translations { } T ε = ε ε ε and three rotations θ = { θ θ θ } T T T in a generalied displacement vector = { } T q ε θ. The mathematical model is formulated for N arbitraril located powertrain mounts, though it is assumed that four elastomeric mounts are used (like real-life cases in man vehicles). A discrete model of the decoupled powertrain sub-sstem is shown in Fig..(b) where ( t) T is a harmonic torque applied about the -ais (driveline), the eternal disturbance { } T ε θ w g t T t, g = diag { g g} force vector is ( ) = ( ) ( ) M M M is the powertrain inertia matri in Γ g with diag( ) as a diagonal matri operator, I I I ε θ M g = diag ({ m m m} ), M g = I I I (.a,b) I I I where m is mass and I is inertia, g, = { rg, rg, rg, } T r is the position vector from the powertrain c.g. to the i th mount elastic center in diag ({ k k k }) TRA Γ, and K = is the i th mount stiffness matri in Γ. For a tpical elastic mount, torsional stiffnesses are negligible [.4]. Thus, onl translational stiffnesses are included in K. Sstem matrices g w, and K should be transformed into M, ( t) g TRA Γ prior to deriving equations of motion. The TRA direction is defined b Jeong and Singh [.] in Γ g as 0

43 3 T { ρ ρ ρ } q = (.3) TRA g g g g where θ ρ g b M g and b is a normaliing constant for the first column of M θ g =. Net, ' ' υ υ υ ' ' ' R = υ υ υ (.4) ' ' ' υ υ υ TRA TRA is defined as an orthonormal rotation matri between q g and q g where υ are normalied directional cosines [.]. If M g is diagonal, θ ρ = ρ =, R = I where I 3 g g 0 TRA TRA is identit, and q g = q g ( M g is alread in TRA Γ ). Though this ma simplif the analsis, it is not realistic for a powertrain sstem. Finall, = { } Π R R is diag ( ) defined as the needed transformation matri, and the powertrain inertia matri in TRA Γ is T M g = ΠM gπ while g( t) = g( t) w Πw. A silar transformation K = R K R T is done where cosϕ cosϕ cosϕ sinϕ + sinϕ sinϕ cosϕ sinϕ sinϕ cosϕ sinϕ cosϕ R = cosϕ sinϕ cosϕ cosϕ sinϕ sinϕ sinϕ sinϕ cosϕ + cosϕ sinϕ sinϕ sinϕ sinϕ cosϕ cosϕ cos ϕ (.5) φ = ϕ ϕ ϕ are Euler angles from Γ to and { } T TRA Γ [.3]. In general, K = K is a full populated 33 matri. Both translational and rotational reaction T forces at each mount must be accounted for as ε Rg, = K I E g, q g, (.6) θ ε T ε T Rg, = rg, Rg, = Eg, Rg, = E g, K I Eg, q g, (.7)

44 respectivel, where 0 rg, r g, E g, = rg, 0 rg, (.8) rg, rg, 0 is a skew smmetric rotation matri [.]. Combining these reaction forces as ε θ Rg, = Rg, ; R g, results in the total resistance applied b each mount. Thus, K N N g, = K = K ( ) K E g g, T T i= i= KEg, Eg, KEg, (.9) T is the total stiffness contribution from all mounts on the powertrain with Kg = K g. Finall, ( t) + ( t) + ( t) = ( t) M q C q K q w (.0) g g g g g g g are the powertrain equations of motion in TRA Γ, where g C is a viscous (proportional) damping matri..3.. Frame sub-sstem and coupled sstem The decoupled frame sub-sstem discrete model is shown in Fig..(c). Parameters also appearing in Fig..(b) remain the same with additional parameters ε θ ({ }) Mg = diag Mg M g as the decoupled asmmetric frame inertia matri in diag ({ k k k }) bi bi bi bi Γ g, K = as the i th frame mount stiffness matri in Γ bi, { r r r } g, g, g, g, T r = as the position vector from the frame c.g. to the i th powertrain mount elastic center in TRA T Γ, and g, bi = { rg, bi rg, bi rg, bi} r as the position

45 vector from the frame c.g. to the i th frame mount elastic center in TRA Γ. The mathematical model is formulated in the same manner as the powertrain sub-sstem such that T M g = ΠMgΠ and K = R K R in T bi bi bi bi TRA Γ ; g( t ) = 6 w O with onl ( t) T as an ecitation. Also, K N N g, = K = ( ) g, m g, T T i= i= KEg, Eg, KEg, K K E (.) where r g, replace r g, in the skew smmetric rotation matri formulation of Eq. (.8), and K N N bi bi g, bi = K = ( ) g, b g, bi T T i= i= KbiEg, bi Eg, bikbieg, bi K K E. (.) The total stiffness matri is Kg = Kg, m + K g, b, and the equations of motion are in where C g is a viscous proportional damping matri: ( t) ( t) ( t) g g g g g g 6 TRA Γ M q + C q + K q = O. (.3) While K g and K g, m both arise due to the powertrain mounts, each involves different position vectors: rg, from the powertrain to the mount and r g, from the frame to the mount. Coupling between the sub-sstems comes through the powertrain mounts. R = R = K I E q is the translational reaction force on the Namel, ε ε g, g, g, g, g θ ε powertrain due to the frame motion. A rotational reaction Rg, g, = rg, R g, also occurs relative to the powertrain c.g., resulting in a coupling stiffness matri 3

46 N N g, K = K = T ( ) g, g g, g, T i= i= KEg, Eg, KEg, K K E. (.4) T Reciprocit eists, and Kg, g = K g, g is derived in the same manner. Equations of motion for the coupled sstem in ( t) + ( t) + ( t) = ( t) g g g g g g g TRA Γ are written compactl in matri form as Mq Cq Kq w (.5a) { 6} T T T T with w ( t) = w ( t) O, ( t) = ( t) ( t) g g { } T qg qg q g, g = µ g + µ g (proportional) damping matri with Raleigh coefficients µ and µ, C M K as a viscous M g M O = K K = ; (.5b,c) g 66 O66 M, g g, g K g g Kg, g Kg roughl 5% modal damping is assumed with µ = µ = Even though Eqs. (.5ac) are essentiall the same as Eqs. (7a-c) in [.], their derivation is included for the sake of completeness and clarit in later sections. The nomenclature is slightl different, with a focus on the different coordinate sstems and position vectors referencing the mounts..4. Conditions for full decoupling.4.. Compatibilit conditions for a realiable sstem The aioms derived b Hu and Singh [.] in Eq. (.) are epanded using the epressions in Eq. (.5), resulting in two sets of conditions (i) and (ii) that must be concurrentl satisfied: K q = λ M q, (.6a) TRA TRA TRA g g g g 4

47 K q = O. (.6b) TRA g, g g 6 Condition (i) is identical to that derived b Jeong and Singh [.] for a powertrain with a rigid foundation. These matrices are further epanded into si equations per condition. The first three equations for each condition are identical; and N N ( krg, krg, ) = 0, ( krg, krg, ) i= N i= ( krg, krg, ) = 0, (.7a,b) i= + = 0, (.7c) ( ( ) ( ) ) g g g g N TRA k r, + k r, k r, r, = λ I, (.7d) i= N ( g g g g g g ( g ) ) i= k r r + k r r k r r k r = λ I TRA,,,,,,, N ( g g g g g g ( g ) ) i= N i= k r r + k r r k r r k r = λ I TRA,,,,,,, ( krg, rg, krg, rg, krg, rg, krg, rg, ), (.7e), (.7f) + = 0, (.8a) N i= ( krg, rg, krg, rg, krg, rg, krg, rg, ) + = 0, (.8b) N i= ( krg, rg, krg, rg, krg, rg, krg, rg, ) + = 0 (.8c) are nine independent equations that must be satisfied for complete TRA decoupling. Eqs. (.7a-f) are from condition (i), and Eqs. (.8a-c) are from condition (ii). Using these equations, Hu and Singh [.] successfull designed a computational powertrain sstem with a compliant base (frame) that is full decoupled in the TRA 5

48 direction, denoted as Eample 4 in [.]. The locations of the mount elastic centers relative to both the powertrain and frame are given, reproduced graphicall in Fig.. (a) (b) Fig... Impractical mount locations for an eample in Hu and Singh [.] referenced to (a) powertrain sub-sstem and (b) frame sub-sstem. Ke:, mount #;, mount #; *, mount #3; and, mount #4. Both rigid bodies are assumed to be rectangular prisms with characteristic lengths l calculated from the provided inertias as 0 I j I T j 6 j j j = 0 I j I j m = j m j 0 I j I j {( ) ( ) ( ) }, (.9) and a position vector g, g = { rg, g rg, g rg, g} T r is assumed between the powertrain and frame centers of gravit (not considered in [.]). In order for the sstem to be realiable, the elastic center of each mount must eist in onl one phsical location. Fig.. shows mount # on the + side of both the 6

49 powertrain and frame. However, mount # is on the + side of the powertrain and the side of the frame. Therefore, there is no realistic r g, g which can fi the elastic center of both mount # and # to a single location, and the sstem cannot be phsicall constructed. Mathematicall, r = r r, (.0) g, g g, g, and different vectors r { } mm and r { }, T g g =, T g g = mm are needed for mounts # and #, respectivel. The same issue eists for mounts #3 and #4. To ensure that the sstem is realiable, the three compatibilit conditions in Eq. (.0) must be mathematicall incorporated into the TRA decoupling conditions..4.. Analsis of decoupling conditions Eq. (.0) is used in decoupling condition (ii) as rg, = rg, r g, g to implement the needed compatibilit conditions. After rearranging terms, N N ( ) + ( ) i= i= N = ( k ( rg, ) + k ( rg, ) krg, rg, ), r k r k r r k r k r g, g g, g, g, g g, g, i= N N ( ) + ( ) i= i= N = ( krg, rg, + krg, rg, krg, rg, k ( rg, ) ) r k r k r r k r k r g, g g, g, g, g g, g, i= N N ( ) + ( ) i= i= N = ( krg, rg, + krg, rg, krg, rg, k ( rg, ) ) r k r k r r k r k r g, g g, g, g, g g, g, i=, (.a) (.b) (.c) 7

50 are the resulting decoupling equations for condition (ii); and epressions from condition (i) appear in these new equations. Define N N ( g g ), P = ( krg, krg, ) P = k r k r,, i= N ( g g ) 3,, i=, (.a,b) i= P = k r + k r (.c) as summations from condition (i) which are nonall ero, used to re-write Eqs. (.ac) as 0 P P P, 3 = P3 0 P P P 0 r g, g I TRA P rg, g = I λ. (.3a,b) r g, g I If nonal P values are taken, P= O 33 and r g, g cannot be solved as rational values. Further, ( ) det P = 0 (where det( ) is a deternate operator) even if small values are taken for all P, as P is a n n skew-smmetric matri where n is odd, and Eq. (.3) still TRA has no solution. A trivial solution ma be achieved if λ = 0 ( r g, g is then arbitrar) or when r g, g. However, neither results in a phsicall realiable dnac sstem, and it is concluded that full decoupling of a powertrain and coupled frame in the TRA direction is not possible..5. Design paradigms for partial decoupling.5.. Coupled sstem with arbitrar mount placement (Eample I) In the mounting plane, ϕ = ϕ = 0 and each mount is onl rotated ϕ from Γ to TRA Γ, as illustrated in Fig..3. This simplifies 8 R and results in

51

52 Table. Powertrain and frame parameters that (a) remain the same for all eamples and (b) are specific to Eample I. (a) Common for all eamples Parameter Powertrain Frame mass (kg) m = 73. m = 3.9 inertia (kg m ) θ g = θ M M g = stiffness (N mm - ) = diag( { } ) position (mm), { } T g g = K K bi = K r -- (b) Parameters of Eample I (baseline design) Mount # i = i = i = 3 i = 4 Powertrain φ ( ) {30 0 0} T { } T {30 0 0} T { } T r (mm) { } T { } T { } T { } T g, Frame φ bi ( ) { } T { } T { } T { } T r (mm) { } T { } T { } T { } T g, bi Realistic stiffness and inertia parameters for the powertrain and frame in Eample I are listed in Table.(a) along with r g, g. These parameters remain the same throughout all eamples. Specific ϕ ϕ, r g,, and r g, bi are listed in Table.(b) for, bi Eample I, and these design parameters will be re-selected in later sections. Recall, 30

53 rg, = rg, r g, g and is thus not tabulated, and rn = rn R for all position vectors where n is a general inde..5.. Coupled sstem mount design using onl the powertrain sub-sstem (Eample II) Since full decoupling is not possible, alternate strategies must be pursued. Eample II designs the powertrain mount positions using onl decoupling condition (i), while intentionall neglecting the frame coupling entirel. This is analogous to assung the powertrain is attached to a rigid foundation, as analed b Jeong and Singh [.], though frame dnacs are included in the Eample II simulation. However, the solution strateg emploed is different than in [.], attempting here to create a solution more suited for design purposes and realiable sstems. Like Jeong and Singh [.], powertrain mounts are identical and assumed in the mounting plane with defined stiffness and orientation values. The same values used in Eample I are emploed for both the powertrain and frame mounts and inertias; the same r g, bi are also used. However, r g, and TRA λ are assumed arbitrar, resulting in thirteen total parameters that must be defined. Onl five can be solved from condition (i), while realistic values are selected for the rest. Jeong and Singh [.] obtain a closed form solution to calculate four locations and TRA λ b assung some smmetr in the mount locations. Conversel, the following procedure assumes no relationships and utilies both analtical and computational methods to obtain a reasonable solution. It is also relativel eas to interchange which locations are selected and which are solved. 3

54 n Define dimensionless locations γ > 0 to attempt to place mounts near the four corners of the powertrain and to ensure the powertrain mass is well distributed: γ r for i =, = for i = 3, 4 g, m0 g, γ rg, m0 r +, ( ) r, i g γ rg, m0 =, rg, = γ rg, m0. (.5a-c) Here, g, m0 = { rg, m0 rg, m0 rg, m0} T r are equal to the location of mount # in Eample I, referencing the improved design to the baseline. Eqs. (.7b-f) are epanded as k r ( γ γ γ γ ) k r ( γ γ γ γ ) =, (.6a) m g, m0 m m m3 m4 m g, m0 m m m3 m4 0 k r k ( γ γ γ γ ) k r ( γ γ γ γ ) =, (.6b) m g, m0 m m m3 m4 m g, m0 m m m3 m4 0 ( ) m( rg, m0) (( γm ) ( γm) ( γm3) ( γm4) ) ( r ) ( γ ) + ( γ ) + ( γ ) + ( γ ) m g, m0 m m m3 m4 + k ( ) k r r γ γ + γ γ + γ γ + γ γ = λ I, TRA m g, m0 g, m0 m m m m m3 m3 m4 m4 (.6c) k r ( γ γ γ γ γ γ + γ γ ) m g, m0 m m m m m3 m3 m4 m4 k r λ k r + = TRA m g, m0( γm γm γmγm γm3γm3 γm4γm4) rg, m0 ( γ γ + γ γ γ γ γ γ ) m g, m0 m m m m m3 m3 m4 m4 λ k r + = TRA m g, m0( γm γm γmγm γm3γm3 γm4γm4) rg, m0 I I, (.6d) (.6e) with all mounts identical. If γ are defined parameters, Eqs. (.6a-b,d-e) are linear with respect to γ and γ of equations in terms of. The solution strateg is to solve four γ and TRA λ and use these epressions in Eq. (.6c) to solve γ from a linear set TRA λ. 3

55 m m g, m0 nk, n k Define known constants σ = k r where n and k are general indices and rewrite Eqs. (.6a-b,d-e) as Aγγ = Y γ in compact matri form. Here, γ is the vector of unknowns, A γ is the sstem matri, and Y γ is the output vector. This is epanded as ( + 3) ( + m3 ) ( + 3) ( + 3),, σm γm γm σm γm γ,,,, σm σm σm σ m γ,, σ m m γm γm σm γm γm,,,, σm σm σm σm γm4 TRA = λ I,,,,,, σm γm σm γm4 σm γm σm γ + σ m4 γ m γmγm γm γm + σm γmγm + γm γ m m rg, m0,,,, σm γm σm γm4 σm γm σm γ m4 γ m4 TRA λ I,, + σ m γmγm γm γm + σm γmγm + γm γm rg, m0 ( 3 3) ( 3 3) ( 3 3) ( 3 3) ; (.7) and the equations are easil manipulated to instead solve for γ m, γ m3, γ m, or γ m3 in γ. Analtical methods could be emploed to eplicitl solve for γ as linear functions of TRA n λ and define needed conditions for all 0 γ >. However, a derivation of rather length epressions would be tedious and still not ield tractable solutions. Instead, TRA computational methods and a smbolic toolbo [.6] are preferred to solve for ( λ ) incorporate them into Eq. (.6c), solve for obtain numerical values for γ using the calculated γ, TRA λ using the quadratic equation, and finall TRA λ. All parameters (both selected and solved for) are listed in Table.(a) for Eample II with TRA TRA Ω = λ as the TRA roll mode resonance frequenc. Note, not all γ > 0. This is not strictl required for a well distributed mass as it is the vertical coordinate; and γ > 0, γ > 0 are more pertinent. Further, selecting all γ helps 33

56 ensure that the mounts are located outside the powertrain bod. The resulting displacement magnitude spectra for q g are shown in Fig.4 for both Eamples I and II. Table. Mount locations and TRA TRA Eample IV with and without q g = q g. TRA Ω for (a) powertrain of Eamples II and III and (b) frame of Mount # (a) Eamples II and III i = i = i = 3 i = 4 Eample II TRA Ω π = 33. H γ (--) γ (--) γ (--) Eample III TRA Ω π = 3.4 H γ (--) γ (--) γ (--) (b) Eample IV TRA Eample IV: q g q g TRA Ω π = 55 H TRA γ bi (--) γ bi (--) γ bi (--) TRA TRA Eample IV: q g = q g TRA Ω π = 55 H γ bi (--) γ bi (--) γ bi (--)

57 db ref. mm (a) (b) (c) db ref. deg Freq [H] Fig..4. Powertrain displacement magnitude spectra for Eamples I and II in the TRA Γ coordinate sstem: (a) g Ke:, Eample I and - - -, Eample II Freq [H] (d) (e) (f) Freq [H] ε, (b) ε g, (c) ε g, (d) θ g, (e) θ g, and (f) θ g. Overall, Eample II design significantl reduces the magnitudes, particularl in the translational directions. The TRA roll motion in θ g is donant over most of the frequenc range, but there is still finite motion in the ε g and ε g directions due to the frame coupling. Selecting different change the magnitude spectra, though onl b to 3 db if n γ values or solving for alternate parameters will TRA λ is silar; Eample II is alwas an improved design. Akanda and Adulla [.7] have applied optiation 35

58 techniques for mount placement and design in a fied powertrain sstem, and such methods could also be used on the coupled sstem design of Eample II to further enhance the TRA decoupling Miniation of decoupling conditions for coupled sstem design (Eample III) Eample III attempts to improve on Eample II (results using onl condition (i)) b also including condition (ii). Since both conditions cannot be simultaneousl satisfied, niation of the decoupling equations is pursued using a total least squares method [.8]. Define parameters N ( ( ) ( ) ) g g g g P = k r + k r k r r, (.8a) 4,,,, i= N ( g g g g g g ( g ) ) P = k r r + k r r k r r k r, (.8b) 5,,,,,,, i= N ( g g g g g g ( g ) ) P = k r r + k r r k r r k r (.8c) 6,,,,,,, i= from Eqs. (.7d-f) in condition (i). All K are assumed in the mounting plane ( k TRA TRA TRA = k = 0 ); and P = 0, P 3 = 0, P4 = λ I, P5 λ I = 0, P6 λ I = 0, Pr 3 g, g P5 0 + =, Pr, + Pr 3, + P4 = 0, and Pr, + P6 = 0 are the eight necessar g g g g g g decoupling conditions. With r g, as defined parameters, onl P 4 is nonlinear with TRA respect to r g, and r g,. Thus, P4 = λ I is left as a constant; and the remaining TRA decoupling conditions are defined as residuals ψ = P, ψ = P3, ψ3 = P5 λ I, TRA ψ4 = P6 λ I, ψ 5 = Pr g, g + Pr 3 g, g + P4, ψ 6 = Pr 3 g, g + P5, and ψ 7 = Pr g, g + P6. The 36

59 T vector of residuals is = { ψ ψ ψ ψ ψ ψ ψ } ψ and equal weighting is used such that the objective function to nie is J N T = ψψ = ψ k ; here, k = J k = ψ k. The equations are again analed in terms of γ > 0 where ( γ γ γ γ ) ( γ γ γ γ ) P = k r k r + + +, (.9a) m g, m0 m m m3 m4 m g, m0 m m m3 m4 ( γ γ γ γ ) ( γ γ γ γ ) P = k r k r +, (.9b) 3 m g, m0 m m m3 m4 m g, m0 m m m3 m4 ( γ γ γ γ γ γ γ γ ) r ( γ γ γ γ γ γ γ γ ) P = k r r + 5 m g, m0 g, m0 m m m m m3 m3 m4 m4 + k r + m g, m0 g, m0 m m m m m3 m3 m4 m4, ( γ γ γ γ γ γ γ γ ) r ( γ γ γ γ γ γ γ γ ) P = k r r + 6 m g, m0 g, m0 m m m m m3 m3 m4 m4 + k r + m g, m0 g, m0 m m m m m3 m3 m4 m4 (.9c) (.9d) after epanding the summations; = { γm γm γm3 γm4 γm γm3 γm4} γ are defined as the unknowns with γ m and γ m selected. The function J is nied with T respect to γ such that N J γ = J γ = 0 for seven total equations. Considering n k n k = one J k γ n at a time, J γ P = P n γn J P 3, = P3 γn γn J γ TRA, = ( P5 λ I ) 3 5 n P γ n, (.30a-c) J 4 γ n P γ ( TRA 6 P6 λ I ) = n, (.30d) 37

60 J5 P P 3 = ( rg, gp + rg, gp3 + P4 ) rg, g + rg, g, γn γn γn (.30e) J6 P3 P 5 = ( rg, gp3 + P5 ) rg, g +, γn γn γn (.30f) J7 P P 6 = ( rg, gp + P6 ) rg, g + γn γn γn (.30g) are the resulting partial derivatives after appling the chain rule and simplifing epressions. Sumng over k and combining common terms gives the epression J γ = ς + Pς + Pς + Pς + Pς where n n0 n 3 n3 5 n5 6 n6 ς ς ς P P I P I P, (.3a) TRA n0 = λ I rg, g + rg, g γn γn I γn I γn ( ( ) ( ) ) P P = + r + r + r r + r 3 6 n g, g g, g g, g g, g g, g γn γn γn ( ( ) ( ) ) P P = r r + + r + r + r 3 6 n3 g, g g, g g, g g, g g, g γn γn γn P, (.3b) P, (.3c) ς P P 3 5 n5 = rg, g + γn γn P P 6, ς n6 = rg, g + γn γn ; (.3d,e) and partial derivatives in ς are known constants calculated from Eqs. (.9a-d):,,,, P γm P3 γm P5 γm P6 γ σ m m σm rg, mσm rg, mσ m,,,, P γm P3 γm P5 γm P6 γm σm σm rg, mσm rg, mσm,,,, P γm3 P3 γm3 P5 γm3 P6 γ σ m3 m σm rg, m3σm rg, m3σ m,,,, P γm4 P3 γm4 P5 γm4 P6 γm4 = σm σm rg, m4σm rg, m4σm. P γm P3 γm P5 γm P6 γ,,,, σ m m σm rg, mσm rg, mσ m,,,, P γm3 P3 γm3 P5 γm3 P6 γm3 σm σm rg, m3σm rg, m3σm P γ m4 P3 γm4 P5 γm4 P6 γ,,,, m4 σm σm rg, m4σm rg, m4σ m (.3) 38

61 The equation nγm+ nγm + n3γm3 + n4γm4 + n5γm + n6γm3 + n7γm4 = Yn is derived from J γ n = 0 b substituting in all P epressions and rearranging where = σ ς σ ς + σ r ς σ r ς, (.33a),,,, n m n m n3 m g, m n5 m g, m n6 = σ ς + σ ς σ r ς σ r ς, (.33b),,,, n m n m n3 m g, m n5 m g, m n6 = σ ς σ ς + σ r ς σ r ς, (.33c),,,, n3 m n m n3 m g, m3 n5 m g, m3 n6 = σ ς + σ ς σ r ς σ r ς, (.33d),,,, n4 m n m n3 m g, m4 n5 m g, m4 n6 = σ ς σ ς + σ r ς + σ r ς, (.33e),,,, n5 m n m n3 m g, m n5 m g, m n6 = σ ς + σ ς σ r ς + σ r ς, (.33f),,,, n6 m n m n3 m g, m3 n5 m g, m3 n6 = σ ς σ ς + σ r ς + σ r ς, (.33g),,,, n7 m n m n3 m g, m4 n5 m g, m4 n6,,,, ( ) Y = σ ς σ ς + σ r ς σ r ς γ ς. (.33h) n m n m n3 m g, m n5 m g, m n6 m n0 Finall, the seven equations for the nied cost function are constructed as m γ m γ m γ m γ m γ m γ m4 γ Y Y Y 3 = Y4, (.34) Y 5 Y6 Y 7 written compactl as Aγγ = Y γ. Computational methods and a smbolic toolbo [.6] are again utilied, following the same methods as in Eample II. The resulting displacement magnitude spectra for q g are shown in Fig..5 for Eamples II and III. 39

62 db ref. mm (a) -0 (b) -0 (c) db ref. deg (d) -0 (e) -0 (f) Freq [H] Freq [H] Freq [H] Fig..5. Powertrain displacement magnitude spectra for Eamples II and III in the θ g TRA Γ coordinate sstem: (a) g. Ke: - - -, Eample II and, Eample III. ε, (b) ε g, (c) ε g, (d) θ g, (e) θ g, and (f) TRA TRA Parameters for Eample III are selected such that ( λ ) ( λ ) II III, and the two produce a nearl identical roll motion θ g spectra which donates over most of the frequenc range. Overall, Eample II is the better design, achieving lower magnitude levels on all motions other than the desired θ g. This is somewhat unepected, as the frame dnacs are neglected entirel. However, decoupling conditions (i) and (ii) are highl contradictive of each other, and while niing the effect of both is mathematicall viable, the residuals are still significantl large in the nied state. It 40

63 is thus concluded that little can be done with the powertrain mounts to counteract the reactive motion from the frame coupling. However, adequate decoupling is still achieved with onl condition (i) considered for a realistic sstem (mm) 0-00 mount # mount # mount # 0 (mm) mount # (mm) 00 Fig..6. Comparison of powertrain mount locations relative to the powertrain center of gravit for Eamples I, II, and III. Ke:, Eample I (baseline);, Eample II; and, Eample III. All parameters (both selected and solved) are listed in Table.(a) for Eample III, and γ are again selected to locate the mounts outside the powertrain bod. As in Eample II, not all γ > 0 though this is still not required for a well distributed mass. However, not all γ > 0 either with γ m, γ m3, and γ m4 ver close to ero; the mass ma not be full supported. A graphical representation of the mount locations is shown in Fig..6 for Eamples I, II, and II; and little support for the mass is indeed provided in Eample III for the + portion of the mass. This further concludes that Eample II is the superior design. Interestingl, mounts #3 and #4 are now located in approimatel the 4

64 same location for Eample III, approaching a more common 3-point mounting sstem instead of a historical 4-point scheme..6. Design of frame mounts to improve partial decoupling.6.. Alignment of powertrain and frame TRA aes Further improvement in the partial TRA decoupling of Eample II is possible through an appropriate design of the frame mounts. Onl the powertrain mount properties are included in decoupling conditions (i) and (ii), and the frame mount properties have thus far been unaltered from Eample I (placed at corners of the frame with K bi = K bi ). One approach could be to lit the modal coupling between the frame and powertrain through the mount design [.9-], but this is outside the scope of the chapter. Instead, a TRA decoupling tpe approach is utilied. Consider a powertrain sstem where the roll motion θ g donates. This motion induces reaction forces and moments R g, from the powertrain mounts along some line of action, which are also transtted to the frame. Likewise, motions from the frame transt forces R g, g, to the powertrain. If the frame motion is also donated b θ g, the transtted forces align with R g, resulting force is along the same line of action as R g,, and the. Thus, the powertrain motion is still donated b θ g. Frame motions other than θ g alter the R g, g, line of action and induce other unwanted motions in the powertrain. It is therefore desired that the frame motion is donated b θ g, designing the mounts using TRA decoupling conditions. Ideall, the TRA of both the powertrain and 4

65 frame are parallel. This is not likel for a realistic sstem, though necessar modifications to the frame inertia could be practical. The TRA direction is defined in Eq. (.3), and conditions can be derived such that q TRA g and TRA q g align. The simplest solution θ θ is to design M g = τm g where τ is a dimensionless scaling constant, though this is also not realistic given the vastl different geometries of tpical powertrains and frames. Instead, define components ρ gj, θ ( ) I I ( I ) ρ gj, and 3 ρ gj in an analtical manner as ρ det M =, (.35a) gj gj gj gj gj θ ( ) ρ det M = I I + I I, (.35b) gj gj gj gj gj gj θ ( ) 3 ρ det M = I I + I I (.35c) gj gj gj gj gj gj from θ 3 ρ gj M gj. Modif normaliing constant b ( ρ gj ) ( ρ gj ) ( ρ gj ) = = + + as b = θ det ( M gj ) ( ( )) ( ) ( ) ( ( )) θ θ 3 θ gj det M gj + gj det M gj + gj det M gj ρ ρ ρ (.36) for θ gj = b gj ρ M, and normalied components ρ = n gj n θ ρ gj det ( M gj ) ( ( )) ( ) ( ) ( ( )) θ θ 3 θ gj det M gj + gj det M gj + gj det M gj ρ ρ ρ (.37) in q are full defined. Here, TRA gj 3 3 { ρ g ρ g ρ g} { ρ g ρ g ρ g} = (.38) 43

66 is the most general solution with highl nonlinear (and difficult to solve) equations. n θ n θ Instead, define ρ det ( g g) = τρ gdet ( g) M M which satisf Eq. (.38). Now, ( ) ( ) τ ( ) g g g g g g I I I = I I I, (.39a) ( ) I I + I I = τ I I + I I, (.39b) g g g g g g g g ( ) I I + I I = τ I I + I I (.39c) g g g g g g g g are the mathematical conditions to align q TRA g and TRA q g. Three parameters can be solved from the conditions of Eq. (.39), and two eample sets are selected: { τ Ig Ig } and { Ig τ Ig } with parameters solved from Eqs. (.39a), (.39b), and (.39c), respectivel. The former results in τ = 4.04, I = 0.04 kg m, and I = 0.56 kg m where g g g I significantl differs from the original frame. The latter results in I = 0.95 kg m, τ = 4.0, and I = kg m g g where both I g and g I significantl differ from the original frame. Altering frame geometr to match these inertias is a design specific procedure, though it ma not alwas be feasible. Also, different sets of parameters can be solved to produce more realistic results. For instance if I g, I g, and I g take nonal values, alternate sets { τ Ig Ig } = { } and { Ig Ig τ} { } = can be solved, where the latter is identical to the { I g τ I g } set. Possible sets are lited as Eqs. (.39a-c) are non-linear with respect to the unknown parameters and I g Ig 0 = = originall; a denonator is ero for some sets if either I g or g 44 I is

67 taken as a nonal value (no non-trivial solution eists). Additional sets are possible if I g, g I, or I g are considered as variables instead of I g, I g, I g, or τ..6.. Full sstem frame mount design using onl the frame sub-sstem (Eample IV) The frame is assumed to have the original inertia matri defined in Table.(a). TRA TRA Thus q g q g, and the effect of aligning these aes is eaned later in this section. Design of the coupled sstem is still not feasible due to the mount compatibilit conditions, and the decoupled frame sstem of Fig..(c) is instead considered. The TRA decoupling conditions (with the frame mounts in the mounting plane and ϕ =± 0 ) are N N 4 34 ( kbirg, bi kbi rg, bi ) = Kg, m, ( k r + k r ) = K i= ( ( ) ( ) ) bi g, bi bi g, bi g, m, (.40a,b) i= N TRA 44 kbi rg, bi + kbi rg, bi kbi rg, birg, bi = λ I Kg, m, (.40c) i= TRA 54 ( k r r k r r ) = λ I K N bi g, bi g, bi bi g, bi g, bi g, m, (.40d) i= TRA 64 ( k r r k r r ) = λ I K N bi g, bi g, bi bi g, bi g, bi g, m (.40e) i= n where stiffness matri components K g, m are known from Eample II. Define n dimensionless locations γ > 0 silar to Eq. (.5) for the frame mounts with bi bi r = γ r for i =, ; r, γ r, 0 for i 3, 4 g, bi bi g, b0 r g, bi bi g, b0 = = ; ( ) i + r g bi bi g b = γ r ; and g, bi bi g, b0 T = γ r. Here, r g, b0 = { rg, b0 rg, b0 rg, b0} are equal to the location of frame 45

68 mount # in Eample II (same as I and III), referencing the improved decoupling design to the baseline. Eqs. (.40a-b,d-e) are re-written in matri form as, ( ) ( γb γb3) ( ) ( ) ( 3 3) ( 3 3) K + σ γ + γ σ +,,,, σb σb σb σ b γ b,,,, σb σb σb σb γb4 = λ,,,, σb γb σb γb4 σb γb σb γ b4 γb,,,, σb γb σb γb4 σb γb σb γ b4 γ b4 λ , g, m b b b3 b 34,, K g, m σb γb γb3 σb γb γb3 TRA 54 I Kg, m,, σb γbγb γb γb σb γbγb γ b γb rg, b0 TRA 64 I Kg, m,, σ b γb γb γb3γb3 σb γb γb γb3γb3 rg, b0 ( ) ( ) (.4) with σ = k r, and the same solution procedure for Eample II is repeated. All nk, n k b b g, b0 frame mount parameters (both selected and solved) as well as TRA Ω are listed in Table TRA TRA.(b) for Eample IV with q g q g ; frame mount locations for Eamples II and IV are shown in Fig (mm) mount # mount # mount # 0 (mm) mount # (mm) 000 Fig..7. Comparison of frame mount locations relative to the frame center of gravit TRA TRA for Eamples II and IV. Ke:, Eample II and, Eample IV with q g q g. 46

69 Here, all γ and γ > 0 for a well distributed mass with mounts outside the bi bi bod, and all γ > 0 ecept for γ m, which is a ver high negative number. This places mount # well above the frame and is not a realistic mounting position. The resulting displacement magnitude spectra for q g are shown in Fig..8 for both Eamples II and TRA TRA IV with q g q g. db ref. mm (a) -0 (b) -0 (c) db ref. deg (d) -0 (e) -0 (f) Freq [H] Freq [H] Freq [H] Fig..8. Powertrain displacement magnitude spectra for Eamples II and IV in the TRA Γ coordinate sstem: (a) g ε, (b) ε g, (c) ε g, (d) θ g, (e) θ g, and (f) q q. TRA TRA θ g. Ke: - - -, Eample II and - - -, Eample IV with g g 47

70 The TRA roll motion spectra θ g remains the same in both cases, and the frame mount design of Eample IV has successfull reduced the pronent coupling peak in ε g b 6 db. Now, g θ is the donant motion throughout the entire frequenc range considered. Coupling in the ε g, θ g, and θ g directions have increased due to q q, but not to significant magnitude levels. Results ma improve if TRA TRA g g TRA λ of the frame mount design matched that of the powertrain mount design. However, components in g = g, m + g, b K K K are much larger than those in Kg = K g, m due to stiff frame mounts and the combined effect of both powertrain ( K g, m) and frame ( K g, b) mounts. TRA The resulting Ω π = 55 H is thus high, and realistic frame mount locations cannot TRA be selected to lower it to sa Ω π = 33 H. Net, the TRA aes are aligned in Eample IV with τ = 4.04, I = 0.04 kg m, and I = 0.56 kg m TRA TRA. The resulting frame mount locations for q g = q g are listed in g Table.(b), and the q g displacement magnitude spectra for Eample IV with and TRA TRA without q g = q g are shown in Fig..9. Mount locations are silar for both cases TRA TRA with mount #4 now being wa above the frame bod in the q g = q g case (mount # g q q case). This ma cause a feasibilit issue, though a frame geometr could TRA TRA in g g TRA TRA be speciall designed on a case b case basis. The new design with q g = q g lowers the coupling peak in ε g b an additional 5 db for a total reduction of db from the 48

71 Eample II design. Additionall, couplings in the ε g, θ g, and θ g directions are decreased, with motion in the ε g direction being silar to Eample II. Note, solving n different parameters to satisf Eq. (.39) has a nimal effect on γ and q g. Eamples TRA TRA I and IV (with q g = q g ) are compared in Fig..0 to show the drastic improvement made over the original mounting sstem, and θ g is now the donant motion. db ref. mm (a) -0 (b) -0 (c) db ref. deg (d) -0 (e) -0 (f) Freq [H] Freq [H] Freq [H] Fig..9. Powertrain displacement magnitude spectra for Eample IV in the TRA Γ coordinate sstem: (a) ε g, (b) ε g, (c) ε g, (d) θ g, (e) θ g, and (f) θ g. TRA TRA TRA TRA Ke: - - -, Eample IV with q g q g and, Eample IV with q g = q g. 49

72 db ref. mm (a) 0 (b) 0 (c) db ref. deg (d) 0 (e) 0 (f) Freq [H] Freq [H] Freq [H] Fig..0. Powertrain displacement magnitude spectra for Eamples I and IV in the TRA Γ coordinate sstem: (a) g ε, (b) ε g, (c) ε g, (d) θ g, (e) θ g, TRA TRA and (f) θ g. Ke:, Eample I and, Eample IV with q g = q g..7. Alternate design paradigms and practical constraints Chapter has focused on one particular mounting sstem design paradigm for reduced noise and vibration. Other strategies ma be pursued (possibl in conjunction with TRA decoupling), and a few are discussed briefl in this section. One simple strateg is to add more damping (sa via mounts or structural damping treatments on the frame) to lower peak motions. This is demonstrated in Fig.. for Eample II with C g elements being four times as large. 50

73 db ref. mm (a) -0 (b) -0 (c) db ref. deg (d) -0 (e) -0 (f) Freq [H] Freq [H] Freq [H] Fig... Powertrain displacement magnitude spectra for Eamples II in the TRA Γ coordinate sstem with damping and stiffness modifications: (a) g ε, (b) ε g, (c) g ε, (d) θ g, (e) θ g, and (f) θ g. Ke: - - -, Eample II;, Eample II with 4C g ; and *, Eample II with 0 K. bi Though the roll mode peak TRA λ is no longer pronent, the g θ motion now donates over the entire frequenc range. This solution assumes adding evenl distributed damping over the entire sstem, which is costl and adds significant mass (often unattractive options in practice). Localied damping patches ma be effective if proper design procedures and mathematical analses are utilied. Yet another relativel straight forward solution is to stiffen the frame mounts, increasing frame rigidit and 5

74 shifting the coupled modes to higher frequencies. This is also demonstrated in Fig.. for Eample II where K bi elements are an order of magnitude higher. Here, the desired roll motion remains, while the coupled motions are greatl reduced. Such a solution ma not be satisfactor for vibration and structureborne isolation needs (for acoustic comfort) where compliant mounts are essential. Tuned hdraulic mounts with spectrall-varing properties are then a potential solution [.9]. Adjusting K bi or other stiffness components is a strateg congruent with sub-structuring [.9-] and mode shifting [.] methods, operating in the frequenc and modal domains, respectivel. Altering mass components is also an option, though financial and practical litations eist: more mass entails higher production cost and worse fuel econom while less mass generall creates higher vibration levels and becomes an issue for crash worthiness testing. A more design specific strateg could be to place powertrain mounts at rigid connection points on the frame, where little mode participation eists up to a prescribed frequenc. A simplifing assumption made in the frame model is that it behaves as a single rigid bod at low frequencies. Depending on the design, a more realistic frame model could either consider a continuous sstem [.0] or one with multiple discrete masses [.4]. In this case, hard points or anti-nodes could be optimal places to attach powertrain mounts, as nimal frame deflection occurs. Incorporating these desired locations into the TRA decoupling condition (i) would create a complication. Another design specific strateg could be ecitation shaping or input reduction (as observed b the powertrain inertial bod), focusing on control of the source as opposed to the paths (powertrain mounts) or receiver (frame). Methods such as designing torque balance shafts 5

75 [.8], increasing the number of engine clinders [.4], and controlled fuel injection [.,.] are viable options. Also, a multi-dimensional dnac vibration absorber could be attached to the powertrain or frame to reduce motions at selected resonance peaks; fundamental research is needed to eplore such an approach..8. Conclusion The aioms (leading to an eigenvalue problem TRA TRA TRA Kq = λ Mq ) recentl proposed b Hu and Singh [.] for a discrete, proportionall damped coupled powertrain and frame sstem are criticall eaned. It has been found that the derivation of K in [.] neglects the need for a phsicall realiable sstem. Namel, each powertrain mount is referenced to two different locations: from the powertrain to the mount and from the frame to the mount. Since these locations do not coincide in [.], the sstem cannot be constructed, and this deficienc in the decoupling analsis is overcome b deriving and implementing mount compatibilit conditions into the derivation of K such that mounts alwas are referenced to a single location. Further, it is mathematicall proven that a non-trivial solution does not eist for TRA TRA TRA Kq = λ Mq when compatibilit conditions are implemented, and full decoupling of the powertrain TRA is not possible for a phsicall realiable coupled powertrain and frame sstem. Thus, partial decoupling powertrain mount design paradigms which do not impose severe burden on the isolation sstem design are proposed via realistic eample cases. One paradigm nies the needed decoupling conditions for the coupled sstem using a total least squares method, resulting in a donant TRA roll motion θ g over 53

76 much of the frequenc range considered (up to 70 H) but with finite coupling motions in the ε g and ε g translational directions. This occurs because the decoupling conditions are highl contradictive of each other, and the residuals are still significantl large in the nied state. A second paradigm considers the decoupled powertrain onl, neglecting frame coupling. Through careful mount parameter selection, a nearl identical roll motion θ g as in the first paradigm occurs but with reduced coupling motions, thus resulting in an improved design; θ g now donates over nearl the entire frequenc range. Decoupling is improved further b implementing silar frame mount and inertia paradigms. Frame mounts are designed with the decoupled frame model onl, neglecting powertrain coupling, and the frame inertia is modified such that the powertrain and frame TRA aes are aligned. Coupled translational motions are further reduced, and θ g now donates over the entire frequenc range. Here, an important contribution immerges; design of the frame mounts and inertia for superior TRA partial decoupling appears to be an additive process to the powertrain mount design, not a simultaneous one. Such a result is not intuitive and onl concluded through the rigorous analsis of various paradigms conducted in this chapter. Alternative isolation sstem design paradigms to lit powertrain and frame coupling are also briefl discussed, and these ma be implemented independentl or simultaneousl with the TRA decoupling paradigms eaned. Such methods include adding more damping, increasing the frame mount stiffnesses, placing powertrain mounts at hard points or anti-nodes on the frame, ecitation shaping, multi-dimensional dnac 54

77 vibration absorbers, mode shifting, and sub-structuring. Litations of this article include the neglect of mount rotational stiffness effects, material frequenc dependence, driveline shaft influence, vehicle bod and tire dnacs, and fleural mode participation. Other future work could include a non-proportional damping formulation, silar to that of Park and Singh [.3], as well as active mounts in conjunction with TRA motion decoupling [.3]. Non-linearities could also be introduced into the sstem, though new decoupling paradigms other than a linear eigenvalue problem would need to be derived. Additionall, the paradigms investigated should be implemented in a controlled laborator eperimental setup. 55

78 References for Chapter [.] J.-F. Hu, R. Singh, Improved torque roll ais decoupling aiom for a powertrain mounting sstem in the presence of a compliant base. Journal of Sound and Vibration 33(7) (0) [.] T. Jeong, R. Singh, Analtical methods of decoupling the automotive engine torque roll ais. Journal of Sound and Vibration 34() (000) [.3] J.-Y. Park, R. Singh, Effect of non-proportional damping on the torque roll ais decoupling of an engine mounting sstem. Journal of Sound and Vibration 33 (008) [.4] J.M. Lee, H.J. Yim, J.-H. Kim, Fleible chassis effects on dnac response of engine mounts sstems. SAE paper (995). [.5] H. Ashrafiuon, Design of optiation of aircraft engine mount sstems. ASME Journal of Vibration and Acoustics 5(4) (993) [.6] H. Ashrafiuon, C. Nataraj, Dnac analsis of engine-mount sstems. ASME Journal of Vibration and Acoustics 4() (99) [.7] M. Sirafi, M. Qatu, Accurate modeling for the powertrain and subframe modes. SAE paper (003). [.8] J.P. Den Hartog, Mechanical Vibrations. Dover Publications, New York, 985. [.9] F. Bessac, L. Gagliardini, J.-L. Guader, Coupling eigenvalues and eigenvectors: a tool for investigating the vibroacoustic behavior of coupled vibrating sstems. Journal of Sound and Vibration 9(5) (996) [.0] E. Courteille, L. Léotoing, F. Mortier, E. Ragneau, New analtical method to evaluate the powerplant and chassis coupling in the improvement vehicle NVH. European Journal of Mechanics A/Solids 4(6) (005) [.] D.-H. Lee, W.-S. Hwang, C.-M. Kim, Design sensitivit analsis and optiation of an engine mount sstem using an FRF-based substructuring method. Journal of Sound and Vibration 55() (00) [.] D.D. Sivan, Y.M. Ram, Mass and stiffness modifications to achieve desired natural frequencies. Communications in Numerical Methods in Engineering (996) [.3] R.M. Brach, Automotive powerplant isolation strategies. SAE paper 9794 (997). [.4] C.M. Harris, Shock and Vibration Handbook. McGraw-Hill, NY (995). [.5] B.J. Kim, Three dimensional vibration isolation using elastic aes. M.S. Thesis, Michigan State Universit, MI (99). 56

79 [.6] MathWorks. Smbolic Math Toolbo, MathWorks, Natick, MA, USA, 04, ( (accessed.0.4). [.7] A. Akanda, C. Adulla, Application of evolutionar computation in automotive powertrain mount tuning. Shock and Vibration 3 (006) [.8] I. Markovsk, S.V. Huffel, Overview of total least-squares methods. Signal Processing 87 (007) [.9] J.-Y. Yoon, R. Singh, Dnac force transtted b hdraulic mount: Estimation in frequenc domain using motion and/or pressure measurements and quasi-linear models. Noise Control Engineering Journal 58(4) (00) [.0] I.M. Ibrahim, Non-linear simulation model for articulated vehicles with controllable dampers and fleible sub-structures. SAE paper (00). [.] A.J. Torregrosa, A. Broatch, R. Novella, L.F. Mónico, Suitabilit analsis of advanced diesel combustion concepts for essions and noise control. Energ 36 (0) [.] Z. Win, R.P. Gakkhar, S.C. Jain, M. Bhattachara, Investigation of diesel engine operating and injection sstem parameters for low noise, essions, and fuel consumption using Taguchi methods. Proc. IMechE Part D: Automobile Engineering 9(0) (005) [.3] J.-Y. Park, R. Singh, Analsis of powertrain motions given a combination of active and passive isolators. Noise Control Engineering Journal 57(3) (009)

80 Chapter 3: Interaction between two hbrid structural paths for active source mass motion control over d-frequenc range 3.. Introduction Active mounts have been suggested for several engineering sstems to primaril improve vibration isolation and resonance control, often with passive mounts used concurrentl [3.-3]. Conventional prime movers (e.g. automotive powertrains) generall have low frequenc vibration ecitations, and thus the primar mount function is to reduce the transssion of dnac forces. Some recent hbrid electric powertrains also produce significant d-frequenc ecitations (sa from 00 to 000 H) that amplif the source regime, creating significant structure-borne and radiated noise. Source mass motion control is thus needed to tigate such unwanted noise [3.4-8]. Active mounts are still a viable solution, though different design paradigms need to be considered. In particular, the phase interaction (caused b passive sstem dnacs) between active mounts and the resulting sstem motion becomes more important as the frequenc increases. This passive phase relationship ma dictate the effectiveness of active control strategies (even at low frequencies), and it is the main chapter thrust. Relevant literature on active vibration isolation focuses on control algorithms [3.-3]. For instance, decentralied velocit control is studied in sstems with multiple 58

81 active mounts in hbrid paths consisting of active and passive elements [3.-7], though the hbrid path interactions are not adequatel addressed. Interactions among passive paths have been studied to some etent [3.9, 3.0], but hbrid paths are fundamentall different due to the applied force. Hbrid path interactions are not of interest in single mount [3.8-] or Stewart platform [3., 3.3] applications, but such configurations ma not be viable due to multi-aial forces or lited packaging space. Reduction of dfrequenc structure-borne and radiated noise has been accomplished via active [3.4-8] and passive [3., 3.] patches, reducing the structure surface velocit to nie the radiated sound pressure or altering the radiating efficienc of the structure. Tpicall, the patches are bonded directl to the structure surface. Alternativel, hbrid structural paths (mounts) could more effectivel reduce the structure motion (and thus surface velocit). 3.. Problem formulation The scope of this chapter is lited to source mass motion control of a resonating source-path-receiver sstem that has two verticall oriented hbrid (active and passive) structural paths, addressing both hbrid path interactions and d-frequenc motion control. A simplified two path sstem is considered for conceptual analsis, and a schematic incorporating active path elements to nie source motion (or sound radiation) is shown in Fig. 3.. Relating this to sa a hbrid electric powertrain, the source mass would be the electric motor, the paths would be the motor mounts (e.g. rubber mount in series with an actuator), and the receiver mass would be the vehicle subframe. Specific objectives include the following: () develop an analtical model of the sstem and define a performance inde to characterie the hbrid path interaction for 59

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83 massless while active elements possess mass, and structural damping is assumed at each stiffness element with a constant loss factor. Passive mounts in this stud are made of rubber (elastomeric), and structural damping is a more valid model, especiall at higher frequencies [3.3, 3.4]. However, an equivalent Kelvin-Voigt configuration with viscous damping is used for lited time domain analsis. It is assumed that each actuator input can be represented b an applied force at a discrete mass with a constant actuator gain. Thus, actuators are assumed to be linear and well known with no hsteresis effects. Rotational effects of the actuator inertia are also ignored, and all active mounts considered are pieoelectric stack actuators (i.e. stacks). The stack force (generated from internal strain) is analticall applied at the corresponding discrete mass and is transtted to both the source and receiver masses. This accounts for the source and receiver mass participation effects observed at the stacks. A constant gain S of voltage over force (provided b manufacturer [3.5], though stack models have been discontinued) is used for each stack, where S accounts for the charge-strain relationship and the specific dielectric material constants. In general, a passive and active mount should be used in combination to provide vibration isolation, and all hbrid structural paths consist of a stiff active pieoelectric stack (attached to the source mass) in series with a passive rubber elastomer (attached to the receiver mass), as suggested b Beard et al. [3.6] Analtical model A schematic of the analtical model is shown in Fig. 3.(a) with inertial coordinates labeled. Motions are restricted to the - plane, and it is assumed that lateral 6

84 motions in the -direction are negligible. Here, m and I are the source mass and inertia ( refers to the Cartesian coordinate), m and I are the receiver mass and inertia, m 3 and m 4 are the masses of the stacks, i valued stiffnesses with are path mount comple =, k ( i = k + η ) k as the real stiffness and η as the loss factor (m refers to a path mount, i is a mount inde, and refers to the Cartesian coordinate), ( i ) k = k + η are compliant base (receiver mass) mount comple valued stiffnesses bi bi bi with k bi as the real stiffness and η bi as the loss factor (b refers to a base mount), sn are characteristic lengths of the source (s refers to source and n is a general inde), rn are characteristic lengths of the receiver (r refers to receiver), t is time, ( t) ε are vertical oscillator motions (j is an inertia inde and g refers to a center of mass motion), θ ( t) are rotational oscillator motions, f ( t ) are vertical control forces, w ( ) disturbance force, and d is the moment arm for w ( t ). Thus, w ( ) and moment on the source mass if d 0. n gj gj t is a vertical t imparts both a force The sstem of Fig. 3.(a) has si degrees of freedom: two rotational and four translational. The resulting equations of motion (derived using Newton s Second Law) are written in compact matri form in inertial coordinates as ( t) + ( t) = ( t) + ( t) Mq Kq f w (3.) { g g g3 g4 g g } where ( t) = ε ( t) ε ( t) ε ( t) ε ( t) θ ( t) θ ( t) q is the generalied displacement vector, superscript T indicates a transpose, M is the inertia matri, K is the 6 T

85 comple valued stiffness matri (assung harmonic ecitation and response), the control { 3 4 } force vector is f ( t) = f ( t) f ( t) ( ) = ( ) ( ) T, and the disturbance force vector is { } w t w t w t d 0 T. Here, ({ m m m3 m4 I I} ) M = diag, (3.) k m + k m 0 k m k m k ms k ms 0 k + k + k + k k k 0 k k + k k k m + k m3 0 k m s k m3 s K = k m + k m4 k ms k m4s smmetric k ms + k m s 0 k + k + k + k m3 m4 b b m3 m4 m4 s m3 s b r b r m3 s m4 s b r b r (3.3) where diag( ) is a diagonal matri operator. Instead of motions at the source and receiver center of masses, the model is analed at the mount attachment coordinates, labeled in Fig. 3.(b) where ( t) ξ is the translational motion at path mount i attached to inertia j., gj ξ A transformation ( t) = ( t) q Πq is needed; and q ξ s s 0 0 s+ s s + s O s s Π O I O 0 0 s+ s s + s O s s s s = s s s s { m, g m, g g3 g4 m, g m, g } ( t) = ξ ( t) ξ ( t) ε ( t) ε ( t) ξ ( t) ξ ( t), (3.4) T, O nn is an n n null matri, and I nn is an n n identit matri. Note that Π is time-invariant and 63

86

87

88 actuation. The disturbance force is supplied b an electrodnac shaker (Ling Electronics model 07, up to 4 lbf [3.8]) with a corresponding force transducer (PCB model 08C0,.4 mv/n sensitivit [3.5]) attached at the stinger. It is assumed that motions are relativel small (on the order of a cron), suitable for stacks and should not violate the small deflection assumption. Net, model parameter values are identified for the eperiment. Masses are weighed, moments of inertia are found from SolidWorks models [3.9], and lengths are measured. To deterne k values, two single degree of freedom sstems are eaned. The first is a stack attached directl to a ver large mass (assumed as a rigid foundation) used to find k m and k m. A chirp voltage signal ecites the stack mass, and an accelerometer measures the response, resulting in a single resonance peak. In general, k = Ω m where Ω n is the corresponding natural frequenc (rad sec - ) and m is the n corresponding stack mass ( m3 for k m and m 4 for m k ). Also, η ( ω ω ) = Ω where n n n ω n and ω n are the corresponding half power frequencies (rad sec - ). The second setup is a stack and rubber mount in series with the mount attached directl to a ver large mass, used to find k m3 and k m4. A chirp voltage signal again ecites the stack, and parameters are found in the same manner. Finall, k bi parameters are found b matching model and eperiment natural frequencies ( Ω ) and selecting reasonable rubber loss factors. Table n 3. summaries Ω n values, with the model and eperiment agreeing relativel well. The two highest Ω n correspond to modes donated b stack mass motions, but these are not 66

89 measured with the attached instrumentation and are above the frequenc range of interest. Table 3. summaries all identified model parameters. Table 3. Comparison of natural frequencies ( π ) Ω. Mode # Model [H] Eperiment [H] n * * *Correspond to stack mass resonances. Not identified since the are above the frequenc range of interest and not measured with the attached instrumentation. Table 3. Identified sstem parameters. Parameter Value Units m = m.08 kg m kg m kg I = I g m k 5.46( + i0.034) m kn mm - k.48( + i0.036) m kn mm - k 0.6( + i0.300) m3 kn mm - k 0.53( + i0.56) m4 kn mm - k b = k b 0.4 ( + i0.300 ) kn mm - s = s 00 mm r = r 36 mm d 50 mm 67

90 The entire mass of each stack is attributed to m 3 or m 4, respectivel. However, a portion of each mass could participate with the source mass, and a more representative model ma require lower m 3 and m 4 values. Since these masses are small and the associated Ω n are beond 000 H (well above the frequenc range of interest), errors in the mass values should have nimal effect. To test this, the model is perturbed ±0% of the nonal m 3 and m 4 values. The resulting first four the original values, and thus this source of error can be neglected. Ω n change b less than % from Net, the model and eperiment are compared at a ω π = 400 H ecitation frequenc, and the measured disturbance force time histor w ( t) ( ωt).6sin N is used as an input into the analtical model for a more realistic comparison. Thus, this comparison should be done in the time domain. The structural damping model using η is onl valid in the frequenc domain, so equivalent viscous damping c= ηk ω (3.6) is instead implemented at each stiffness element in a Kelvin-Voigt configuration. A ξ Simulink model [3.30] and the derived equations of motion are used to predict ( t) q, and displacements are twice differentiated with respect to time to calculate translational accelerations as a ( t) ξ ( t) n =. These can be directl compared to measured n accelerations. Four high resolution channels are available on the dspace board; and a ( t ), a ( t ), a ( t ), and w ( ) m, g m, g m, g t are measured with a 5 kh sampling rate. The predictions are compared to the eperiment in Fig. 3.4 (normalied b the gravitational 68

91 constant g), and a reasonable match is obtained, giving confidence in the proposed approach. Band-lited white noise is added to the predictions for realistic testing conditions, and the largest displacement is.8 μm, validating the small deflection assumption. a m,g [ g] a m,g [ g] a m,g [ g] t [sec] Fig Comparison of eperiment and model for a 400 H disturbance force. Ke:, measured and - - -, predicted Quantification of path interaction The relative phases between the harmonic forces and motions are of critical importance, and all analsis is done assung comple valued variables. The applied forces are defined as i ( ) = w t W ω, (3.7) e t 69

92 i( ωt φ3 ) f t = F +, (3.8) ( ) 3 3 e i( ωt φ4 ) f t = F + (3.9) ( ) 4 4 e where F n and φ are the amplitude and phase of f ( t) n and W is the amplitude of n ( t) w. Four control variables are thus available: F 3, F 4, φ 3, and φ 4. To simplif notation, define ξ ( t) = ξ ( t) and ( ) ξ t = ξ ( t ) m, g as the m, g motions to be reduced to virtuall ero (motion control). For effective vibration isolation, ξ ( ) and ( ) m, g t ξ could instead be nied. Superposition is valid since the m, g t sstem is assumed linear, and the source mass motions are defined as where n 3 4 ( t) ( ) ξ = Ξ +Ξ e +Ξ e e, (3.0) iφ iφ iωt ( t) ( ) ξ = Ξ +Ξ e +Ξ e e (3.) iφ iφ iωt 3 4 Ξ is the comple valued amplitude due to w ( t) amplitude due to f ( t), and n 4 3, Ξ n3 is the comple valued Ξ is the comple valued amplitude due to f ( t) 4. Each Ξ n has an associated magnitude and phase, and it is desired to reduce the total motion to ero. Ideall, this would be done b phase matching all Ξ n and setting the magnitude summation equal to ero. Define the dnac stiffness matri ξ ξ ξ in the mount coordinates, κ = ω M + K { m, g m, g g3 g4 m, g m, g} ξ T Q = Ξ Ξ Ξ Ξ as the comple valued displacement amplitude vector, as the comple valued amplitude of ε ( t), and gj 70 gj ξ ξ κ as the H =

93 ξ ω ξ iωt e with F and W i t comple valued compliance matri. In general Q e = H { F+ W} as the control and disturbance force amplitude vectors, respectivel, and each force is considered separatel: { W Wd } Q, (3.) ξ i i e ω t T t = H ξ e ω ξ i( 3 ) i( 3 ) e ω t+ φ ξ T t 0 0 F e ω + Q = H φ, (3.3) 3 3 { } ξ i( 4 ) i( 4 ) e ω t+ φ ξ T t F 0 0 e ω + Q = H φ. (3.4) { } 4 4 The general description of and the H ξ ξ H is ξ ξ ξ ξ ξ ξ H H H 3 H 4 H 5 H 6 H ξ H ξ H ξ 3 H ξ 4 H ξ 5 H ξ 6 H ξ 3 H ξ 3 H ξ 33 H ξ 34 H ξ 35 H ξ 36 = ξ ξ ξ ξ ξ ξ, (3.5) H 4 H 4 H 43 H 44 H 45 H 46 ξ ξ ξ ξ ξ ξ H 5 H 5 H 53 H 54 H 55 H 56 ξ ξ ξ ξ ξ ξ H 6 H 6 H 63 H 64 H 65 H 66 Ξ ξ ξ n of interest are thus Ξ = ( H + H 5d) W ξ ξ, Ξ = ( H + ) H d W, 5 Ξ 3 H ξ 3F, 3 Ξ H ξ F = =, Ξ = H ξ F, and Ξ = H ξ F. Eqs. (3.0) and (3.) are re written in terms of magnitude and phase as iβ i( 3 3 ) i( 4 4 ) β + φ β + φ iωt ξ ( t) = Ξ e +Ξ 3 e +Ξ4 e e, (3.6) iβ i( 3 3 ) i( 4 4 ) β + φ β + φ iωt ξ ( t) = Ξ e +Ξ 3 e +Ξ4 e e (3.7) where β = Ξ, ( ) is a phase operator, and ( ) is a magnitude operator. The n n resulting phases are ξ ξ β = ( H + H ξ ξ 5d), β = ( H + H5d) 7, β = H ξ, 3 3

94 β 3 = H ξ 3, 4 H ξ 4 β =, and β 4 = H ξ 4. As an eample case for some phsical meaning of these phases, β 3 is the phase dela between the applied force F 3 and the resulting motion at ξ. Likewise, β is the phase dela between W and the resulting motion at ξ. For motion control, Ξ of either ξ ( t) or ( t) n ξ can be phase matched, but not both. This is evident from Eqs. (3.0) and (3.), as each motion contains three Ξ n (si total), and onl two phase control variables are available. Here, optimal control and an appropriate cost function could be used to nie both motions. Alternativel, all Ξ of ( t) ξ are phase matched using 3 φ and φ 4, defined as n φ = β β, (3.8) 3 3 φ = β β. (3.9) 4 4 Eq. (3.8) aligns the resulting motions at ( t) aligns the resulting motions at ( t) create out-of-phase motions at ( t) ξ due to W and F 3 ; likewise, Eq. (3.9) ξ due to W and F 4. Thus, the control forces can ξ relative to the motion caused b W. It is then assumed that β3 + φ3 = β and β4 + φ4 = β, phase matching all Ξ of ( t) n ξ and enabling out-of-phase motions at ( t) ξ due to W and F n. The validit of these assumptions deternes the reduction amount of ξ ( t), which indicates the effect of the passive interaction between the hbrid paths on motion control performance. If both 7

95 motions are reduced, the performance is desirable. If onl ( t) ξ is reduced, the performance is undesirable, and optimal control would be needed to provide some reduction to ξ ( t) b not reducing ( t) nimum. Eqs. (3.6) and (3.7) are re-written as ( t) = ( Ξ +Ξ +Ξ ) ξ all the wa to ero, reaching an overall i( ωt+ β ) ξ, (3.0) 3 4 e i( ωt+ β ) ξ. (3.) ( t) = ( Ξ +Ξ +Ξ ) 3 4 e To reduce motions to ero, set Ξ +Ξ 3 +Ξ 4 = 0 and Ξ +Ξ 3 +Ξ 4 = 0. In matri form, ξ ξ ξ ξ H 3 H 4 F H + H 5d 3 W ξ ξ = H F ξ ξ 3 H 4 4 H + H 5d (3.) where derived epressions are plugged in for Ξ n. Finall, ξ ξ ξ ξ ξ ξ F W H H + H d H H + H d = ξ ξ ξ ξ F ξ ξ ξ ξ ξ ξ 4 H 3 H 4 H 4 H 3 H 3 H + H 5d H 3 H + H 5d (3.3) is derived, full defining the control force epressions of Eqs. (3.8) and (3.9) in terms of known parameters. ( t) Using the derived control forces, ( t) ξ ma increase or decrease. An effectiveness ξ is reduced to ero for an ω, while 73

96 χ = i 3 i 4 ( H ξ ξ ξ φ ξ φ + H 5d) W + H 3F3 e + H 4F4 e H + H dw ξ ξ 5 (3.4) is defined to quantif the reduction of ( t) divided b ( t) ξ. Here, χ is ( t) ξ with control forces ξ without control forces, derived from Eq. (3.). This can be epressed as an insertion loss on a db basis as L ( χ ) = 0 log, (3.5) 0 and L is now the performance inde b which the path interaction is characteried. Considering one ω at a time in 0. H increments, L is calculated from 0 to 000 H. It is plotted in Fig. 3.5 from 0 to 500 H, with no changes between 500 and 000 H. 80 Peak # Peak # 60 L [db] Valle # and # Frequenc [H] Fig Insertion loss L up to 500 H. Ke:, L and - - -, ero db. 74

97 A positive value indicates a decrease in ( t) ξ and thus a desirable performance. Here, the sstem is assumed smmetric (unlike in the eperiment) to simplif the analsis in the net section: s = s = s, r = r = r, k m = k, m k m = k, 4 m k 3 b = k, b m = m, m = m, and I = I ; and the natural frequencies are slightl altered to ield 4 3 { } Ω T n π = H. In general, L is positive below the first rigid bod mode (86 H) and negative beond the fourth mode (30 H). Several interesting features are labeled in Fig. 3.5: two peaks (# and #) occur, creating a ver desirable performance, and two valles (# and #) occur, creating a ver undesirable performance. these do not correspond to sstem resonances, the are investigated net Calculation of ke design parameters Investigation of the second peak in L The second peak frequenc in L is about 80 H, and Eq. (3.) is re-written as 3 4 β ( ) ξ ξ H H ( t) = ( H + H d) W + ( H F + H F ) ξ e e e e (3.6) ξ ξ ξ i ξ i i iωt where derived epressions for ξ ξ ξ H ξ H ( H3F3 + H4F4 ) Ξ n and φ n are used. The second term i 3 i 4 iβ is of interest, as it dictates how the control forces affect ( t) e e e ξ. Since the sstem is assumed smmetric, H ξ ξ 3 = H 4, and the H ξ 3 magnitude spectra is plotted in Fig. 3.6(a). Note, numerical methods are used to calculate ξ ξ from the analtical dnac stiffness H = κ in inertial coordinates: κ = ω M+ K ω ω. (3.7) 75 κ ξ ξ ξ = M + K = M + K Π = κπ

98 ξ Sstem resonances are present in Fig. 3.6(a) due to denonator dnacs from det ( ) κ, where det( ) is a deternant operator. To eane numerator dnacs onl, det ξ ξ ( κ ) H 3 magnitude spectra is plotted in Fig. 3.6(b). A ero is evident at 80 H, matching the L second peak frequenc. Thus, the second peak appears to be controlled b the passive H ξ ξ 3 = H 4 numerator dnacs (a) db ref..0 nm/n db ref..0 (N/µm) (b) Frequenc [H] Fig Magnitude spectra of (a) H ξ 3 76 and (b) ( ξ det ) ξ κ H 3.

99 80 60 ω ω (a) L [db] ω Frequenc [H] 0. (b) Im [(N/µm) 5 ] ω ω ω 3 Fig Effect of η m3 on (a) det ξ ξ L and (b) ( ) 3 η 3 = 0.50 ;, η 3 = 0.70 ; and - - -, ero aes. m Re [(N/µm) 5 ] m κ H. Ke:, η 3 = 0.30 ; - - -, m One parameter that ma have fleibilit when designing a sstem is the rubber mount structural damping in the hbrid paths, ηm3 = ηm4. Its effect on the second peak of L is numericall investigated in Fig. 3.7(a) where values of η 3 = 0.30, 0.50, and m

100 are all plotted. As η m3 increases, the peak frequenc (labeled ω, ω, and ω 3 ) increases, and the width of the peak decreases. Both have interesting design implications, as one could use η m3 to shift the peak to a heavil ecited frequenc regime (e.g. a motor operating speed). A wider width is likel preferred, as a desirable performance occurs at more frequencies, suggesting that lower damping is better when considering the performance in a broadband sense. With enough damping, the second peak disappears entirel. It is of interest to analticall calculate the transitional value ( m3 ) should avoid η 3 ( η 3) det ξ ξ ( κ ) H 3 m m t η where this occurs, as practical designs t. The effect of η m3 is further investigated b plotting on the comple plane, shown in Fig. 3.7(b), with the peak frequencies from { H ξ 3} ξ Fig. 3.7(a) labeled. Here, ( κ ) Im det 0 when ω n occurs, where Im{ } takes the { } ξ ξ imaginar part of the argument; Im det ( κ ) H 3 0 should be ero when ( m3 ) and ω = ωt ( t { ξ } ξ ω is the transitional frequenc). Further, ( κ ) η is used Re det H 3 crosses the imaginar ais between 0.50 < 3 < 0.70, where Re{ } takes the real part of the η m argument. This causes the phase to go from 80 to 0, specificall before and after ξ ξ { ( κ ) H 3} Re det = 0. Such a phase shift completel switches the effect of H ξ ξ 3 = H 4 on the control forces, causing a desirable performance to become undesirable. The mathematical conditions for the transitional point are thus ξ ξ { ( κ ) H 3} Im det = 0, (3.8) 78 t

101 ξ ξ { ( κ ) H 3} Re det = 0. (3.9) To analticall calculate ( m3 ) η, t ξ κ must first be smbolicall defined. B substituting Eqs. (3.), (3.3), and (3.4) into Eq. (3.7), m m k ω ω m k m k m k m 0 0 ω m ω m 0 0 k m3 k m3 k m3+ k b k m3+ k b k m 0 k m+ k m3 ω m3 0 k m3 0 ξ κ = 0 k m 0 k m+ k m3 ω m3 0 k m3 ω I ω I k ms k ms k ms k m 0 0 s s s k b r ω I k b r ω I 0 0 k m3s k m3s k m3s + k m3 s + s s s s (3.30) is derived where s = s = s, r = r = r, k m = k, m k m = k, 4 m k 3 b = k, b m = m, m = m, and I = I have been applied due to smmetr in the sstem. This is defined 4 3 more compactl as κ ξ ξ ξ ξ ξ κ κ κ 3 κ3 0 0 ξ ξ ξ ξ 0 0 κ 3 κ 3 κ 5 κ5 ξ ξ ξ κ3 0 κ33 0 κ3 0 = ξ ξ ξ, (3.3) 0 κ3 0 κ33 0 κ3 ξ ξ ξ ξ κ 5 κ 5 κ53 κ ξ ξ ξ ξ 0 0 κ3 s κ3s κ 66 κ66 taking advantage of an identical matri components. In general, ( ) H = κ = det κ ξ ξ T ξ matri nor, ( ) 3, where is the matri of cofactors. Further with as a 3 = + 3 = 3. Thus, H ξ 3 = 3 det ( κ ξ ) = 3 det ( κ ξ ). Substituting in for 3 and removing the denonator dnacs results in 79

102 κ κ κ det κ κ κ 0 0 ξ ξ ξ 3 3 ξ ξ ξ ξ 0 κ 3 κ 3 κ 5 κ5 0 0 ξ ξ κ33 0 κ3 ξ ξ ξ ξ ξ ξ ξ κ3 s κ3s κ 66 κ66 = det ξ ( κ ) H ξ 3. (3.3) The left hand side of Eq. (3.3) is epanded, but the process is tedious and otted for the sake of brevit. Eventuall, a comple valued function Λ= det 4 ξ s H ξ 3 mmi 3 ( s m I ) ( κ ) (3.33) is derived, where several constant terms have been moved from the left hand side of Eq. (3.3) to the right hand side. Still, Λ retains the H ξ 3 numerator dnacs and is further defined as Λ= + + (3.34) 4 3 Aλλ Bλλ Cλλ Dλλ Eλ where λ = ω and A λ, Bλ, Cλ, Dλ, and Eλ are all comple valued constants in terms of s, r, I j, m j, k, and k bi. Referring to Eqs. (3.8) and (3.9), mathematical conditions for the transitional point are defined as 4 3 { } { A λ} λt { Bλ} λt { C λ} λt { Dλ} λt { Eλ} Im Λ = Im Im + Im Im + Im = 0, (3.35) { } ( ηm3) ( ηm3) Re Λ = A + B + C = 0 (3.36) η t η t η where Im{ A λ }, Im{ B λ }, Im{ C λ }, Im{ D λ }, Im{ E λ } constants; the last three depend on λ. To calculate ( η m3 ), A η, B η, and C η are real valued t, λ t is first found graphicall from Eq. (3.35) and then incorporated into Eq. (3.36) to solve for ( m3 ) 80 η using the t

103 quadratic equation. Since changing η m3 moves the peak frequenc, this is an iterative process. After four iterations, ( m ) { λ} η 3 = and ω π = H are calculated. t The real valued constant coefficients in Eq. (3.35) are defined as Im{ A λ} = k m η m, (3.37a) Im { λ} ( ηm ) α ( η η ) 3α3 ( η η 3) B km α s B = k + k + ( I sm) + k + B m b m b B m m m C ( km3) α ηm ηm3 ηm ( ηm3) t, (3.37b) ( ) ( ( ) ) α ( η η η η ) α ( η η ( η ) η ) + C + km kbα ηm + ηb ηm ηb s r C Im { C λ} = k m ( b) 3 m b m( b) + k + ( I sm) C + km km3 4 m+ m3 m m3 + k k α η + η + η η η η 4 m ( ) C m3 b 5 m m3 b m m3 b, (3.37c) D ( km3) kb α ( ηm + ηm3( ηm ηm3 ηm ηb ηm3ηb ) + ηb ) D + 4km ( km3) α ( ηm + ηm3( ( ηm ) ηm ηm3) ) ( ) α ( η η η ( η η η η η η )), ( kb ) ( ( ( ) )) D + km km3kbα3 ( ηm ( ηm ηm3 ηm ηb ηm3ηb ) + ηm3 + η b) (3.37d) 4 s r D Im D = k 3 m + km3 kb m + m3 + b m m3 m b m3 b ( I sm) m 3 D + km α ηm + ηb ηm ηm ηb 8

104 { λ} 8 ( I sm) m ( ) ( ( ) ) ( ηm ηb ( ηm ) ηb ) ( ) ( ) η η η η η m3 m b m b 6 s r E Im E = 4 ( km ) ( km3) kb α + ηm3 ηm ηm ηb m3 + + (3.37e) where B α n are constants in B λ, is a constant in E λ. These constants are C α n are constants in C λ, D α n are constants in D λ, and α E I m α = B sm3 m3, (3.38a) α I m B r r = + sm s I, (3.38b) α = I + I m m, (3.38c) B s 3 sm sm3 I m3 ( I ) m m α = + I I + + m m C s s s s s s 4 s rmm 3 rm rm3 r I r m3 I r r m3, (3.39a) ( I ) m I I m m m α = + + +, (3.39b) C s s s rmm 3 rm3 sm3 m3 I r m3 m3 C I sm α3 = +, (3.39c) sm I ( I ) m α = + m I m C s s s 4 s rmm 3 r m3 I rm3 r m3, (3.39d) 8

105 α I m m ( I ) C s s 5 = sm m3 r r m3 s rmm 3 sm s m s m s m +, I r m3 r m3 r m3 (3.39e) ( I ) I I D s s s s smm sm rm r r I α = + + ( I ) I m D s 3 3 s smm sm I α = + α ( I ) D s s 3 = r s smm sm I m, (3.40a), (3.40b) I m, (3.40c) α I I I m m. (3.4) E s s s s s = sm sm rm r r I I r The real valued constant coefficients in Eq. (3.36) are defined as ( k ( ) k ( ( ) )) C s m D D λα t + λ t bα ηm ηb + mα η m ( I sm) m3 s r Aη = k m( km3) 4, ( I ) 4 s m E sm + k mkbα (( ηm ) + ηm ηb ) ( I sm) m 3 (3.4a) B η λαη λ αη αη α η η ( I sm) ( k k k ( )) 3 B r C C C t 3 m t m3 m + m 4 m + b 5 m + b ( ) ( ) (( ) ) D D km3kb α ( ηm + ηb ) + ( kb ) α ηm + ηb ηm ( ηb 4 ) s s r m k mkm3 λ t ( I m) ( ) m s I m s 3 D D + 8k m k m 3αη m + k m k b α3 ηm + ηb ( ηm ) ηb = s r + ( I sm) m 3 k E m k m3 k bα ηm ηb ηm ηb m3, (3.4b) 83

106 C η = k m ( ) ( ) ( ( η ) ) ( k ) k k ( ) λ λ α η η λ C + km3kb α5 ( ηm ηb) D D ( km3) kb α ( ηm ηb ) + km ( km3) α (( ηm ) ) D 4 + km3( kb ) α ( ηm ηb+ ( ηb ) 4 ) s r m + λ 3 t ( I ) m 3 D sm + km ( kb ) α (( ηm ) + 4ηm ηb+ ( ηb ) ( ηm ) ( ηb ) ) D km km3kbα3 (( ηm ) ηm ηb + + ) 6 8 s r m E + 4 k m( k m3) k bα ( ( ηm ) ηm ηb) ( I sm) m3 C C m3 α + m bα ηm ηm ηb B km α (( η ) m ) C ( ) 3 ( ) 4 3 b m b b s B + k α η η η s r t + t b ( m b + k ) + t ( I ) sm ( I sm) C B k km km3α4 m m3α + 3. (3.4c) Investigation of the first peak in L i 3 i 4 iβ of Eq. ξ ξ ξ H ξ H Each component in the second term ( H3F3 + H4F4 ) e e e (3.6) is again eaned for the first peak, and β is the passive controlling component, epressed as ξ ξ ( H H d) β = + (3.43) 5 from the derivation in Section 3.5. Analtical epressions could be derived for H ξ and H ξ 5, but computational studies are instead done for the sake of brevit. As with the second peak, the effect of increasing η m3 is eaned. The peak frequenc again increases, and the peak width decreases. Here, ( m ) computationall found, and β is plotted before and after ( m3 ) η 3 =.468 and ωt π = 00.8 H are t η in Fig. 3.8(a). As with t 84

107 for the second peak, the phase of H ξ 3 β changes b 80 at ( η m3 ) t, causing a desirable performance to become undesirable (a) Angle [deg] (b) Angle [deg] 0-00 Fig Phase β before and after transitional point of the first peak in L for (a) η m3 and (b) moment arm d. Ke: (a), η m3 =.467 ;, η 3 =.469 and (b), d = 0.4;, d = m Frequenc [H] 85

108 From Eq. (3.3), it is evident that the first peak is not onl effected b passive sstem dnacs in H ξ and H ξ 5, but also b the eternal disturbance force moment arm d. Define d = d s as a normalied moment arm with d = 0.50 initiall. Through a numerical stud, d t = 0.40 and ω π = 73.9 H are computationall found, and β t is plotted before and after d t in Fig. 3.8(b). The phase at the transitional point again changes b 80, and this suggests that the path interaction is more desirable when the disturbance force is moved awa from the center of mass, as the first peak disappears when d is too low. Note, decreasing d onl increases the peak frequenc slightl but greatl reduces the peak width Investigation of the valles in L and transitional parameter summar While the other L features are due to passive sstem properties dictating the path interactions, the valles arise due to numerical issues. When calculating the control force amplitudes in Eq. (3.3), H ξ 3 H ξ ξ ξ 4 H 4 H 3 is in the denonator; it crosses the ero ais at the valle frequencies, causing F 3 and F 4 to go to ±. This is illustrated in Fig. 3.9, and F n vs. ω is a smooth curve elsewhere. In a realistic sstem, Fn ± would be addressed with defined lits on the control forces or an appropriate cost function in optimal control. Increasing η m3 can also fi this, as it shifts H ξ 3 H ξ ξ ξ 4 H 4 H 3 so that no ero crossings occur; ( m ) found. A plot of η 3 = and ωt π = 04. H are computationall L before and after ( m3 ) t η is shown in Fig. 3.0(a) for both valles. This t 86

109 is also shown for the first peak in Fig. 3.0(b) and the second peak in Fig. 3.0(d). Finall, L is plotted before and after d t for the first peak in Fig. 3.0(c) Force [N] Frequenc [H] Fig Calculated control force amplitudes. Ke:, F 3 ; - - -, F 4 ; and - - -, ero ais. Recall, it is noted in Section 3.4 that a portion of each stack mass could participate with the source mass, effectivel lowering m 3 and m 4 and creating a source of error. B perturbing the model around ±0% of both nonal mass values and observing less than % change in the first four Ω n, this source of error is concluded to be negligible. However, it ma affect the calculated transitional values. The model is again perturbed around ±0% of m3 = m4 (smmetric sstem assumed) to test this. Less than 87

110 ±0.5% change is observed in ( m3 ) observed in t η for the second peak, and less than ±% change is t d for the first peak. A significant change of up to ±5% is observed in ( η m3 ) for the first peak; however, this does not alter an conclusions drawn or the analsis procedure outlined. t L [db] (a) (b) (c) (d) Fig Insertion loss L before and after transitional point for (a) the first and second valles for ( η m3 ), (b) the first peak for t ( η m3 ) t for d t, and (d) the second peak for ( m3 ) t, (c) the first peak η. Ke: (a), η 3 = ;, η 3 = ; (b), η 3 =.467 ;, η 3 =.469; (c), d = 0.4;, m Frequenc [H] Frequenc [H] Frequenc [H] Frequenc [H] m d = ; and (d), η 3 = ;, η 3 = m 88 m m m

111 In general, the identified values of ( m3 ) η are unrealisticall high for a rubber mount, though the equivalent loss factor for a hdraulic mount ma be as high as 0.70 [3.3]. Also, ( m3 ) t t η ma be lower for a different phsical sstem, and it is important to repeat the analsis as part of a design process. Even if ( m3 ) η ( η ) m3 m3 t η are high, selection of some < can be done to target a specific frequenc regime or to increase the width of the peak b lowering η m3, keeping in nd other design needs (such as passive vibration t isolation). The d t value identified is realistic, and d > d should be selected if possible. t 3.7. Eperimental validation Lited eperimental validation for source mass motion control of a representative d-frequenc 400 H disturbance force is done using the pieoelectric stack actuators and the eperimental setup of Fig No real time control is used, instead calculating needed control forces from the derivations in Section 3.5 and using a human-in-the-loop tpe feedback to manuall make small control parameter adjustments. Specificall, a 400 H sine wave is first applied using the attached shaker. The amplitude of this force is measured in real time and used to calculate the control forces. Net, the forces are converted to voltage units with the stack sensitivit S and used to ecite the stacks. Finall, small adjustments are made to F 3, F 4, φ 3, and φ 4 b the user in real time until a reasonable reduction is achieved. Measured source mass accelerations are compared to the model in the time domain, where equivalent viscous damping is used; band-lited white noise is added to 89

112 simulate a nois environment. Predicted results are shown in Fig. 3.(a) for ( ) = ξ ( ) and a ( t) ξ ( t) a t t ( ) ( π ) =, and measured results are shown in Fig. 3.(b). Here, w t 7.sin 800 t N, and a sampling frequenc of 5 kh is used. The results appear silar and are quantified with root mean square accelerations ( ) where t Ψ= a, is a time average operator during stead state sstem behavior. The insertion loss of Eq. (3.5) is re-defined as L = 0log0 ( ˆ ) ΨΨ, where ˆΨ are before control forces are applied. All insertion losses are summaried in Table 3.3 for the model and eperiment, where L refers to a and L refers to a. Observe that the model is able to reduce a to the noise floor, resulting in a ver large L. The stacks used in the eperiment have some internal noise issues and thus do not generate perfect sine waves. As a result, the eperiment does not reduce a as much. Nevertheless, a significant L is achieved. The model predicts a small increase in a, and while the eperiment does not replicate this, nimal L is measured. Thus, the model and eperiment are in reasonable agreement. n t Table 3.3 Predicted and measured insertion losses for a 400 H disturbance force. Insertion Loss Predicted [db] Measured [db] L 40 L -6 90

113 Disturbance turned on Control forces turned on Shaker turned on Left stack turned on Right stack turned on a [g] 0 - a [g] a [g] 0 a [g] t [sec] t [sec] (a) Fig. 3.. Motion control of a 400 H disturbance force for (a) model and (b) eperiment. (b) 3.8. Conclusion Some recent engineering sstems, such as hbrid electric powertrains, produce d-frequenc ecitations which can amplif the source mass and generate noise. Source mass motion control is thus needed to tigate this noise, which can be achieved using hbrid structural paths consisting of active and passive elements. The phase interaction (caused b passive sstem dnacs) between active mounts and the resulting sstem motion becomes more important as the frequenc increases, and it ma dictate the effectiveness of active control strategies (even at low frequencies). The main contribution of this stud is the derivation of a performance inde L quantifing the hbrid path interaction due to passive sstem dnacs for a conceptual two path resonating source-path-receiver sstem. A closed form analtical solution is 9

114 found for L b phase matching the left source mass motion ( t) force phases and assung the right source mass motion ( t) This assumption leads to ( t) ξ with the two control ξ is also phase matched. ξ being reduced at some frequencies and increased at others, depending on the passive sstem dnacs. In general, L is positive (desirable performance) below the first rigid bod mode and negative (undesirable performance) beond the fourth mode. Several interesting features are also identified in L : two peaks occur, creating a ver desirable performance, and two valles occur, creating a ver undesirable performance. Passive parameters dictating these features are identified, and corresponding ke design values are calculated. The hbrid path (rubber) damping η m3 in particular is investigated, calculating transitional values beond which the peaks and valles are no longer observed and formulating appropriate design paradigms for effective active control. For eample, η m3 can be selected to shift a peak to a heavil ecited frequenc regime in the sstem (e.g. a motor operating speed) or alter a peak width to increase the bandwidth of desirable performance. For the latter, lower damping is better. The eternal disturbance force moment arm d affects the first peak identified, and a transitional value is calculated. The path interaction is more desirable when the disturbance force is awa from the center of mass, as the first peak is not observed when d is too low. A nor, though necessar, contribution is eperimentall demonstrating active source mass motion control of a representative d-frequenc 400 H disturbance in a source-path-receiver sstem using hbrid structural paths (with pieoelectric stack 9

115 actuators). Future work would require an application of real time control strategies for motion control over a wider range of frequencies. Also, disturbances that are not pure sinusoids could be studied. Additional future work is to eane the interaction between two hbrid structural paths for vibration isolation. Lited investigation shows that a silar performance inde (sa L 4 referring to ξ4 = ξ m, g ) with silar features eists. This stud could also be etended to a three or four path sstem with various combinations of hbrid and passive onl paths. Litations not in the scope of this stud include material frequenc dependence and nonlinearities, kinematic nonlinear effects, source and receiver fleural mode participation, rubber mount mass participation, and pieoelectric stack behavior compleities (e.g. hsteresis); these should be investigated in the future. 93

116 References for Chapter 3 [3.] M. Serrand, S.J. Elliott, Multichannel feedback control of the isolation of baseecited vibration. Journal of Sound and Vibration 34(4) (000) [3.] X. Huang, S.J. Elliott, M.J. Brennan, Active isolation of a fleible structure from base vibration. Journal of Sound and Vibration 63 (003) [3.3] T.J. Yang, Z.J. Suai, Y. Sun, M.G. Shu, Y.H. Xiao, X.G. Liu, J.T. Du, G.Y. Jin, Z.G. Liu, Active vibration isolation sstem for a diesel engine. Noise Control Engineering Journal 60(3) (0) [3.4] P. Gardonio, S.J. Elliott, Passive and active isolation of structural vibration transssion between two plates connected b a set of mounts. Journal of Sound and Vibration 37(3) (000) [3.5] S.M. Kim, S.J. Elliott, M.J. Brennan, Decentralised control for multichannel active vibration isolation. IEEE Transactions on Control Sstems Technolog 9() (00) [3.6] P. Gardonio, S.J. Elliott, R.J. Pinnington, Active isolation of structural vibration on a multiple-degree-of-freedom sstem, part I: the dnacs of the sstem. Journal of Sound and Vibration 07() (997) [3.7] P. Gardonio, S.J. Elliott, R.J. Pinnington, Active isolation of structural vibration on a multiple-degree-of-freedom sstem, part II: effectiveness of active control strategies. Journal of Sound and Vibration 07() (997) 95-. [3.8] B. Kim, G.N. Washington, R. Singh, Control of modulated vibration using and enhanced adaptive filtering algorithm based on model-based approach. Journal of Sound and Vibration 33 (0) [3.9] B. Kim, G.N. Washington, R. Singh, Control of incommensurate sinusoids using enhanced adaptive filtering algorithm based on sliding mode approach. Journal of Vibration and Control 9(8) (0) [3.0] G. Pinte, S. Devos, B. Stallaert, W. Smens, J. Swevers, P. Sas, A pieo-based bearing for the active structural acoustic control of rotating machiner. Journal of Sound and Vibration 39 (00) [3.] S.J. Elliott, Active control of structure-borne noise. Journal of Sound and Vibration 77(5) (994) [3.] Z.J. Geng, L.S. Hanes, Si degree-of-freedom active vibration control using the Stewart platforms. IEEE Transactions on Control Sstems Technolog () (994) [3.3] A. Preumont, M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, A. Abu Hanieh, A si-ais single-stage active vibration isolator based on Stewart platform. Journal of Sound and Vibration 300 (007)

117 [3.4] E. Bianchini, Active vibration control of automotive like panels. SAE paper (008). [3.5] K. Wolff, H.-P. Lahe, C. Nussmann, J. Nehl, R. Wimmel, H Siebald, H. Fehren, M. Redaelli, A. Naake, Active noise cancellation at powertrain oil pan. SAE paper (007). [3.6] J.P. Carneal, C.R. Fuller, An analtical and eperimental investigation of active structural acoustic control of noise transssion through double panel sstems. Journal of Sound and Vibration 7 (004) [3.7] C.R. Fuller, Eperiments on active control of sound radiation from a panel using a pieocerac actuator. Journal of Sound and Vibration 50() (99) [3.8] C.R. Fuller, Active control of sound radiation from a vibrating rectangular panel b sound sources and vibration inputs: an eperimental comparison. Journal of Sound and Vibration 45() (99) [3.9] A. Inoue, R. Singh, G.A. Fernandes, Absolute and relative path measures in a discrete sstem b using two analtical methods. Journal of Sound and Vibration 33 (008) [3.0] S. Kim, R. Singh, Multi-dimensional characteriation of vibration isolators over a wide range of frequencies. Journal of Sound and Vibration 45(5) (00) [3.] S.W. Kung, R. Singh, Vibration analsis of beams with multiple constrained laer damping patches. Journal of Sound and Vibration (5) (998) [3.] S.W. Kung, R. Singh, Development of approimate methods for the analsis of patch damping concepts. Journal of Sound and Vibration 9(5) (999) [3.3] E.E. Ungar, C.W. Dietrich, High-frequenc vibration isolation. Journal of Sound and Vibration 4() (966) 4-4. [3.4] J.C. Snowdon, The reduction of structure-borne noise. Acustica (Beiheft ) (956) 8. [3.5] PCB. PCB pieotronics. PCB Group, Inc., Depew, NY, USA, (03, accessed 9 Jul 03). [3.6] M.J. Beard, A.H. Von Flotow, D.W. Schubert, A practical product implementation of an active/passive vibration isolation sstem. Proceedings of IUTAM smposium on the Active Control of Vibration, Universit of Bath, UK, 994, pp [3.7] dspace. dspace sstems. dspace Inc., Wiom, MI, USA, pub/home/products/sstems.cfm (03, accessed 9 Jul 03). [3.8] Ling Electronics. Ling Electronics Acquired b Data Phsics, San Jose, CA, USA, (03, accessed 4 Jul 03). 95

118 [3.9] SolidWorks. Dassault Sstèmes SolidWorks Corporation, Waltham, MA, USA, (03, accessed 9 Jul 03). [3.30] MathWorks. Simulink, simulation and model-based design, MathWorks, Natick, MA, USA, (03, accessed 9 Jul 03). [3.3] J.-Y. Yoon, R. Singh, Dnac force transtted b hdraulic mount: Estimation in frequenc domain using motion and/or pressure measurements and quasi-linear models. Noise Control Engineering Journal 58(4) (00)

119 Chapter 4: Enhancement of vibration control for powertrain mounting scheme paradigms: Combination of active and passive methods 4. Introduction Control of rigid bod motions and reduction of transtted forces is commonl accomplished using passive methods [4.-0, Chapter ] with adequate success. However, growing consumer epectations (e.g. higher power densit) necessitate a hbrid approach of active and passive methods to meet more stringent sstem design targets, utiliing active control schemes [4.-9] and algorithms [4.30-3] for improved motion and vibration control. Proper hbrid design is application specific and driven b the passive sstem dnacs; thus a quantitative comparison of passive vs. hbrid schemes is better facilitated with a specific mounting scheme eample such as for an automotive powertrain. Passive methods such as path isolator designs [4.-4] and damping patches [4.5-6] are traditionall used for vibration isolation and global source or path motion targets, but there are several practical litations hindering performance [4.7]. Capabilities can be improved through passive phsical decoupling of the sstem motions; for eample, proper powertrain mounting sstem design decouples all rotational and translation motions from the powertrain torque roll ais (TRA) given a torque pulse ecitation [4.8-0, Chapter ]. Yet this introduces additional litations, such as 97

120 unrealistic mounting locations and packaging (geometric) space issues [Chapter ]. Active control method alone also has litations: actuator damping is generall low, effecting resonance control capabilities, and stiffness is generall high, increasing sstem resonances to the audible range which would degrade acoustic comfort. A hbrid approach of active and passive should overcome man litations of passive or active alone. Studies relevant to powertrain mounting schemes include multiple [4.-3] and single [4.4-8] hbrid path sstem configurations, where an actuator (active) and rubber or hdraulic element (passive) in series or parallel constitutes a hbrid path. The thrust of prior research is usuall on control algorithms, and the passive element is onl utilied to provide static support and to ensure the control sstem stabilit, though Stewart platforms [4.8-] and a tuned reaction mass absorber [4.] have been considered. Analsis of passive sstem dnacs is ke in maiing vibration control performance, and lited relevant studies include Liette et al. [4.3] quantifing the effect of passive sstem dnacs on active control capabilities as well as Park and Singh [4.4] eaning a passive TRA powertrain mounting scheme with one hbrid path. This chapter epands prior research [4.3, 4.4] b considering a realistic powertrain mounting scheme which includes frame dnacs, aing for comparative analsis leading to a better design concept. 4. Problem formulation The powertrain inertia is approimated as a si degree of freedom rigid bod b connecting engine and transssion at an equivalent center of gravit [4.7], and a four point mounting scheme is assumed. The frame dnacs are also represented b a si 98

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122 hbrid path behavior. Suitable configurations of Models A and A are developed in Sections.5. and.6. of Chapter as Eamples I and IV, respectivel; corresponding schematics and parameters are shown in Fig., Table., and Table.. Here, discrete linear time-invariant deternistic sstems are assumed with small motions, proportional viscous damping, no kinematic nonlinear effects, identical rubber (elastomeric) powertrain and frame mounts with constant properties, and harmonic ecitations up to 70 H. For low frequencies at stead state sstem behavior, the rubber mounts are assumed massless. Even if the powertrain is assumed to have a rigid foundation, Kim [4.33] has argued that full TRA decoupling is not possible for completel arbitrar mount locations. Thus, it is assumed that all powertrain mounts are located in the so called mounting plane relative to the TRA coordinates to improve decoupling capabilities [4.9]. Models B and C require new formulations, and additional assumptions include linear and well known actuators which possess mass. Silar to Liette et al. [4.3], each actuator input is represented b an applied force at a discrete actuator mass with a constant gain. Also, no real-time control is used. Several design issues will be investigated as part of the stated objectives. For eample, onl partial TRA decoupling is achievable when frame dnacs are considered [Chapter ], which ma lit the effectiveness of configuration. Also, engine mounts generall possess nimal damping for better high frequenc performance, degrading resonance control capabilities and liting effectiveness of the passive paths. The number of active mounts available is a practical concern due to cost and sstem compleit: one 00

123 or two is reasonable but all four active paths are not financiall viable. This will lit active control effectiveness. Table 4. Mounting sstem configurations and corresponding model designations. Paths (Mounts) Model Configurations Passive onl A Active onl B Active and Passive C Arbitrar mount design TRA mount design 4.3 Modeling the sstem configurations Model schematics To anale hbrid path effectiveness, mathematical constructs are first formulated for the models in Table 4.. Model A with passive onl powertrain paths is schematicall shown in Fig. 4.(a) with architecture identical to the coupled powertrain and frame model analed in Chapter.3. Each rigid bod is assumed to have si degrees of freedom with three translations { } T ε = ε ε ε and three rotations gj gj gj gj { } T T T θ gj = θgj θgj θgj in a generalied displacement vector { } T gj = gj gj coordinate sstems are considered where q ε θ. Multiple Γ gj are inertial Cartesian coordinates (,, ) 0

124 at the center of gravit (c.g.) of the j th rigid bod, Γ and Γ bi are principal elastic Cartesian coordinates (,, ) of the i th mount (diagonal stiffness matrices), and are Cartesian coordinates (,, ) where the TRA is the -ais. It is assumed that all are parallel with a vertical ais and an ais along the driveline. TRA Γ Inertia properties are M g 5 for the powertrain and M g 6 for the frame in Γ gj with ε θ diag ({ }) M = M M, diag( ) as a diagonal matri operator, gj gj gj I j I j I j ε θ M gj = diag ({ mj mj mj} ), M gj = I j I j I j, (4.a,b) I j I j I j m as mass, and I as inertia. Stiffness components include { = k k k} the i th powertrain mount and { bi = k bi k bi k bi } diag ( ) diag ( ) Γ gj K as K as the i th frame mount in Γ and Γ bi, respectivel. For a tpical elastic mount, torsional stiffnesses are negligible [4.7], and it is assumed that all mounts are connected to the powertrain and frame b rigid brackets. Position vectors connecting the j th center of gravit to the i th elastic center in TRA T Γ are gj, = { rgj, rgj, rgj, } r and gj, bi = { rgj, bi rgj, bi rgj, bi} r for the powertrain and frame mounts, respectivel, and position vector r g5, g6 = { rg5, g6 rg5, g6 rg5, g6} connects the powertrain and frame centers of gravit. The sstem ecitation is torque pulse ( t) T about the driveline. Ecept for the powertrain paths, all parameters for 5 Model A are summaried in Table 4.(a); these remain unchanged in Models B and C. T T 0

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126 Table 4. Sstem parameters for (a) powertrain and frame and (b) path models for i = j =,, 3, 4. (a) Powertrain & frame Parameter Powertrain Frame mass (kg) m 5 = 73. m 6 = inertia (kg m θ ) g5 = θ M M g = stiffness (kn mm - ) * K bi = diag( { } ) position (mm) r 5, 6 { } T g g = -- (b) Path models for i = j =,, 3, 4 Parameter Passive Active Active + Passive m (kg) 0.00 j K (kn mm - ) diag( { }) diag( { }) diag( { }) K ai (kn mm - ) diag( { }) diag( { }) diag( { }) µ (--) µ (--) (mm) gj, (mm) gj, ai *Powertrain mount stiffnesses depends on path models Model B with active onl powertrain paths, as shown in Fig. 4.(b), is modified from Model A b replacing passive powertrain elements with active ones. New smbols include actuator inertias M ε gj = M gj in Γ with corresponding displacement vectors qgj = ε gj for i = j =,, 3, 4 and actuator stiffness matrices { K ai = kai kai kai } diag ( ) 04

127 in Γ ; K ai accounts for internal actuator stiffness and attachment method to the powertrain. Additionall, K = K ai for Model B is roughl an order of magnitude higher than for Model A. Model C, as shown in Fig. 4.(c), has identical notation to Model B, but now, however, the powertrain paths are a hbrid of active and passive elements. Silar to Liette et al. [4.3], all paths consist of a stiff actuator (attached to the powertrain) in series with a passive rubber elastomer (attached to the frame), as suggested b Beard et al. [4.34]; a parallel configuration is also commonl done, though a series configuration allows path fleibilit for a negligible localied torsional stiffness [4.7]. Rubber mount stiffness K and actuator stiffness K ai from Models A and B, respectivel, are used in Model C, and frame inertia and mount properties in Model A are identical for Models B and C (Eamples I and IV [Chapter ] for configurations and, respectivel) Mathematical formulations A discrete mathematical model is first formulated for Model C from Fig. 4.(c). To facilitate configuration (TRA mount design), all stiffness and inertia matrices are transformed into TRA Γ. Detailed mathematical analsis in Chapter.3. results in T M gj = ΠMgjΠ for j = 5, 6 (powertrain and frame) where = { } Π R R, diag ( ) ' ' υ υ υ ' ' ' R = υ υ υ (4.) ' ' ' υ υ υ 05

128 is a rotational transformation matri, and υ are normalied directional cosines. Silarl, K = R K R where T cosϕ cosϕ cosϕ sinϕ + sinϕ sinϕ cosϕ sinϕ sinϕ cosϕ sinϕ cosϕ R = cosϕ sinϕ cosϕ cosϕ sinϕ sinϕ sinϕ sinϕ cosϕ + cosϕ sinϕ sinϕ sinϕ sinϕ cosϕ cosϕ cosϕ (4.3) and = { ϕ ϕ ϕ } T Γ to φ are Euler angles from TRA Γ. In general, K = K is T a full populated 33 matri, and mounts K = R K R and T bi bi bi bi K = R K R are T ai ai ai ai rotated in the same manner with R ai = R. Actuator inertia matrices are diagonal with onl translational motions considered; thus M gj = M gj for j =,, 3, 4 (actuators). { } Lastl, powertrain ecitation ( ) = ( ) w T 0 0 T is rotated as g5 t 5 t { } ( t) = ( t) = ( t) ( t) ( t) w Πw T T T. g5 g The full sstem inertia matri in T TRA Γ with 4 degrees of freedom is ({ } ) M = diag M M M M M M (4.4) g g g g g g g with corresponding displacement vector { } T T T T T T ( t) = ( t) ( t) ( t) ( t) ( t) ( t) q q q q q q q. (4.5) g g g g g g g For the actuator rigid bodies, Kgj = Kai + K for i = j =,, 3, 4. The powertrain and frame bodies must consider the total stiffness of all mounts as well as torsional effects: T 06

129 K K 4 4 K K E K K = =, (4.6a) ai ai g5, ai i= i= g5 g5 g5 4 4 T T g5 g5 ( ai g5, ai ) ( g5, ai ) K K K E E KaiEg5, ai i= i= 4 4 K K E K K = =. (4.6b) g 6, i= i= g6 g6 g T T g6 g6 ( g 6, ) ( g 6, ) K K K E E KEg 6, i= i= Derivation is included in Chapter.3. where E is a skew smmetric rotation matri consisting of position vector components. The full sstem stiffness matri is constructed as K g = K g K g ; K K g g where K g Kg O33 O33 O33 O33 Kg O33 O33 =, O33 O33 Kg3 O33 O3 3 O33 O33 Kg4 K g Kg5 Kg5 O33 O 33 Kg5 Kg5 O33 O33 =, (4.7a,b) O33 O33 Kg6 Kg6 O3 3 O33 Kg6 Kg6 K Ka Ka Eg5, a Km Km Eg6, m K K E K K E a a g5, a m m g6, m T g = = Kg Ka3 Ka3Eg5, a3 K m3 K m3eg6, m3 K K E K K E a4 a4 g5, a4 m4 m4 g6, m4, (4.7c) and O nn is an n n null matri. Finall, the equations of motion in ( t) + ( t) + ( t) = ( t) + ( t) g g g g g g g g TRA Γ are Mq Cq Kq w f (4.8) { } T where ( t) = ( t) w O O O O w O is the disturbance torque g g T T T T vector, the control force vector is ( t) = ( t) ( t) ( t) ( t) with general control forces ( t) gj T { } f f f f f O O g g g g g f applied at the actuators to be eaned in the net 07 T

130 section, and Cg = µ Mg + µ K g is a viscous (proportional) damping matri with Raleigh coefficients µ and µ. Roughl 5% modal damping is assumed for Model C with µ = µ = The formulation for Model C in Eq. (4.8) is emploed for Models A and B with appropriate path modifications to enhance its versatilit. From Tables. and. in Chapter, all r g5, are defined for both arbitrar (Eample I) and TRA (Eample IV) mount designs; Models A and A use these values. Assung powertrain and frame rigid bodies remain in the same spatial coordinates for all models, a kinematic relationship between r g5, ai and r g5, is illustrated in Fig. 4.3 as r = r + + (4.9) g5, ai g5, gj, gj, ai where rg5, ai = rg5,, rg5, ai = rg5,, rg 5, = R rg 5,, and rg 5, ai = R airg 5, ai. Thus, r g5, ai is defined for Model C b assigning reasonable values of, = 0 mm and, = 5 mm, assung the rubber mount is thicker than the actuator. Silarl,, =, = 5 mm for Model B assung smmetric actuator attachment points from the center of gravit. gj gj gj ai gj ai Modifications also must be made to the actuator properties gj M, K,and K ai. Transitioning from Model C to B onl needs K = K ai and µ = as the changes (roughl.5% modal damping with onl actuators). Transitioning from Model C to A requires the actuator dnacs to be made negligible. This is achieved with m j = mj 000 and K ai = 000K ai as active path elements modified to represent passive ones, resulting in K donating path dnacs at low frequencies. Parameters for the 08

131

132 i ( ) = e t i, f ω ( t) = e t i, ( ) = w t W ω n n n n q t Q ω (4.0a-c) n e t n where n is a general inde, ω is the ecitation frequenc in rad sec -, i=, W n is the disturbance force amplitude with ero reference phase for disturbance force w ( t) the comple valued control force amplitude for control force f ( t) comple valued displacement amplitude for displacement q ( t). Eq. (4.8) is then rewritten as where { } n, and n n, n is n Q is the iωt iωt ω Mg + iωcg + K g Q ge = Wg + F g e (4.) Q g, W g, and F g are vectors of displacement, disturbance force, and control force amplitudes, respectivel. Finall, { } g g g g Q g are solved as Q = H W + F (4.) where g = ω g + iω g + g is the comple valued compliance matri. H M C K Before calculating the control forces, the must be defined in the TRA Γ coordinates. It is assumed that all actuators onl appl aial forces in the "-direction such that F gj = { 0 0 j} T. Assung the path is oriented in the mounting plane with ϕ = ϕ = 0, a rotation of ϕ onl is needed to transform from Γ to TRA Γ [4.9]. This is illustrated in Fig. 4.4, and F = R F = 0 cosϕ sinϕ 0 0 sinϕ cosϕ j gj gj (4.3) 0

133