2003-01-1489 Vibration Characteristics of Cardboard Inserts in Shells Martin G. Foulkes and Jaes P. De Clerck General Motors Corporation Raendra Singh he Ohio State University Copyright 2003 SAE International ABSRAC A study has been conducted to deterine the noise and vibration effect of inserting a cardboard liner into a thin, circular cross-sectioned, cylindrical shell. he relevance of such a study is to iprove the understanding of the effects when a cardboard liner is used in a propeller shaft for noise and vibration control purposes. It is found fro the study that the liner adds significant odal stiffness, while an increase in odal ass is also observed for a particular shell type of ode. Further, the study has shown that the additional odal daping provided by the liner is not appropriately odeled by Coulob friction daping, a daping odel often intuitively associated with cardboard aterials. Rather, the daping is best odeled as proportional viscous daping. INRODUCION Propeller shaft resonant vibration can be a noise and vibration control issue in the operation of otor vehicles. While certain characteristics of noise and vibration originating fro the engine can be pleasing to an occupant, the drive line coponents are expected to be transparent. herefore, various treatents have been developed to reduce the resonant behavior in propeller shafts. hese treatents exist in fors ranging fro press-fit cardboard to foa inected into the shaft in a liquid state and expanded to fit during a curing process. While any copanies apply these treatents to the shaft, the physical phenoena behind the vibration attenuation are not fully known. Further knowledge of the attenuation echanis is required to design and optiize the treatents in a ore efficient anner during the product developent process. he paper presents the results of a study conducted to address the physical phenoenon behind a press fit cardboard liner used to attenuate resonant vibration in a circular cross-sectioned, thin-walled, cylindrical shell. Experientally deterined frequency response functions (FRFs) and curve-fitted odal paraeters of a thin cylinder are copared to deterine the ass, stiffness, and daping effects when the cardboard liner is applied to the interior wall of the shell. o aid in identifying the odal changes in the syste, a portion of thin shell theory is used to properly characterize the ode types in the shell syste. Furtherore, the appropriate daping odel for the cardboard liner is deterined by two different approaches. First, inspection of the acceleration response decay envelopes in the tie doain is used to evaluate the presence of Coulob friction daping; this daping odel is often intuitively associated with a cardboard treatent. Second, observations fro the daping atrices constructed using experientally curve-fitted coplex natural frequencies and ode shapes are presented in order to deterine if the appropriate daping odel is in fact viscous or soe other for of daping. MODAL EFFECS OF HE LINER A coon way of deterining the ass, stiffness, and daping changes in a syste is to copare FRFs when the change is ade to the syste. An increase in frequency for a particular ode usually iplies an increase in odal stiffness of the syste, while a decrease in frequency can be associated with an increase in odal ass. A reduction and widening of the resonant peak usually iplies an increase in odal daping. o be able to track the frequency changes of each ode, proper identification of the ode shape is required. o aid in this task, thin shell theory is eployed to characterize the spatial behavior of each ode in a circular cross-sectioned cylindrical shell. HIN SHELL HEORY A nuber of theories for the vibration behavior of circular cross-sectioned cylindrical shells are available in the literature. In particular, Ref [1] presents a coprehensive list of the theories and copares the published results for any different cases. he deterination of ode shapes for a free-free boundary condition is reproduced here. Certain liiting cases are assued when applying thin shell theory. hey are:
1. Constant wall thickness, sall deflections, and the iddle surface of the shell defors without stretching. 2. Effects dealing with initial stress, shear deforation, and rotary inertia are excluded. 3. he aterial is considered isotropic and hoogeneous. he geoetric variables for the shell under consideration are shown in figure 1. he displaceent functions u, v, and w on the surface of the shell are deterined fro equations (1) (3): u = A X ( x) ( nθ ) cos( ω ) cos (1) t ( x) ( nθ ) cos( ω ) v = B X sin t (2) Figure 1: coordinate syste of a thin circular cylindrical shell (fro Ref [1]). ( x) ( nθ ) cos( ω ) w = C X cos t (3) Where, A, B, and C are aplitude coefficients, is the nuber of circuferential nodal circles, and n is the nuber of circuferential nodal waves. [] denotes the first derivative. he corresponding bea function, X, in (1) (3) for a free-free boundary condition is given in (4) as: X ( λ s) + cos( λ s) α [ sinh( λ s) sin( λ s) ] = cosh + for =1,2,3, (4a) X = 1 (4b) R x 1 X L = + (4c) l 2 Where s = x R, λ = R ε l, ε and α are tabulated in Ref [1]. Equations 4(b) and (4c) are special cases of in-extensional ode shapes: the Rayleigh type odes and the Love type odes, respectively. Figure 2 displays the ode shape for the first bending ode or an (,n) value of (1,1). he ode shapes for other odes are shown in Figure 9 in the appendix. Figure 2: first bending ode fro thin shell theory: (1,1). EXPERIMENAL APPROACH Once the ode shapes were deterined for the circular cylindrical shell, a series of experients were perfored to deterine the effects on odal stiffness, ass, and daping that the cardboard liner has on the shell syste. he effects were deterined by tracking the changes in frequency for each ode when the cardboard insert is introduced into the syste. An aluinu shell was used in the experients with the following geoetry: R/h=31.25, l/r=16.4, and h=2.032. o reain consistent with the ode shapes developed in the previous section, hanging each end of the shell by elastic cords siulated the free-free boundary condition. he low stiffness of the cords ensured that rigid body bounce odes well below the frequencies of interest in the shell. For the two test conditions, acceleroeters were placed in five equally spaced positions along the shell in the longitudinal (x) direction. In order to detect repeated roots arising fro the syetric geoetry of the shell, a second row of acceleroeters was placed orthogonal to the original row. A icrophone was also
placed in the vicinity of the shell to docuent the effects on acoustic radiation fro the addition of the cardboard insert. An ipact haer excited the shell. he ipact locations were at each end of the shell and in both horizontal and vertical directions, aligning with the directions of the acceleroeters. he ultiple excitation points were collected to ensure that the repeated roots of the syetric structure were detected through populating ultiple rows of the FRF atrix. he locations of the excitation and response locations are shown in figure 3. that the acoustic radiation of all of the odes was significantly reduced except for this (1,2) ode. herefore, it ay be stated that generally the liner adds stiffness and daping to the shell syste, and thus reduces the generation of sound into the field. However, there ay be special cases were soe odes are ore affected by the odal ass than stiffness, and the resonant behavior is not attenuated. his observed effect is not unique in propeller shaft liner applications. Fro the experient, accelerance, and acoustic sensitivity FRFs were calculated for each excitation and response location. Coercial odal analysis software was used to curve fit the FRFs to obtain the natural frequencies, daping ratios, and odal vectors. Due to the relatively light aount of daping in the syste and the ultiple excitation locations, the tie doain ultiple degree of freedo (MDOF) excitation odel was chosen for curve fitting. Figure 4: driving point FRFs at one end of the shell for the unlined and cardboard lined cases. op: accelerance. Botto: acoustic sensitivity. able 1: estiated natural frequencies and daping ratios for the unlined and cardboard lined shell. * Mode Unlined Shell Cardboard Lined Shell Shape f n (Hz) ζ (%) f n (Hz) ζ (%) Love ode, n=1 321 0.09 422 1.15 (1,1) 402 0.13 486 0.84 Figure 3: Excitation and response locations on the shell. RESULS Figure 4 displays a driving point accelerance FRF and an acoustic sensitivity FRF for the cases with and without the cardboard liner. able 1 suarizes the odal paraeters obtained by curve fitting the accelerance FRFs and coparing the ode shapes generated at each resonant frequency to the thin shell fors. It is seen fro figure 4 that the odal behavior is significantly altered when the liner is introduced. Inspection of the table reveals that generally the cardboard liner increases the odal stiffness and adds odal daping to the syste. Contrasting this observation, the (1,2) shell type ode experienced a decrease in the natural frequency and daping ratio, iplying that increase in the odal ass was the doinant effect. It ay also be noticed fro the figure (1,2) 690 0.32 618 0.22 (2,1) 704 0.25 718 0.55 (2,2) 721 0.03 1258 1.00 *Repeated roots not shown COULOMB FRICION DAMPING able 1, which suarized the odal behavior of the shell, reveals that generally the cardboard liner increases the odal stiffness and odal daping in the syste. he appropriate odel that provides this increase in odal daping is often assued to be Coulob friction daping, a priori. he purpose of this section is to present the results of a study conducted to deterine if the odal daping provided by the cardboard liner is indeed best odeled by Coulob friction daping.
CHARACERISICS OF COULOMB FRICION DAMPING Coulob friction daping is characterized by a constant daping force opposing the haronic otion of the syste. his force is proportional to the coefficient of friction between the two surfaces. he iplication of this external force is that the decay envelope of free vibration is linear, rather than an exponential envelope found fro other odels such as viscous and hysteretic daping. Various sources in the literature address this phenoenon. In particular, Ref [2] shows that the decreent in aplitude per cycle is constant in the presence of friction and can be represented as: F x = xn+ 1 xn = 4 (5) k his result assues a constant daping force in a single degree of freedo syste. Here x n is the free vibration aplitude of the nth cycle, F is the external daping force, and k is the stiffness. Further, it is shown in Ref [2] for viscous daping that: x n+ δ x n+ 1 = 1 xne ln = δ (6) xn Here, δ is the logarithic decreent for sall daping. Note that the aplitude of vibration in (5) decreases in equal aounts as an arithetic series and produces a linear decay envelope. he result is obviously different for (6) where the aplitude is shown to decrease in equal percentage aounts as a geoetric series and produces an exponential decay envelope. EXPERIMENAL APPROACH In order to confir or contradict the presence of Coulob friction daping on a odal basis, the tie doain free response of the syste is needed at a particular resonant frequency. he free response is found after being subected to an initial excitation at that resonant frequency. herefore, acceleration responses of the shell were collected using the sae experiental approach for finding the FRFs described earlier in the report. In this case, propeller shaft attachent yokes were included in the experient to represent a realistic propeller shaft response. he yokes also added large end-asses to the syste and changed soe of the ode types to a fixed-fixed type of boundary condition. his boundary condition constrained the otion at the attachent points for the elastic cords on each end, and consequently, the external daping provided to the syste by these cords was iniized. FILER DESIGN Although an ipact haer excites a broad range of frequencies, the response of a single ode can be found by attenuating the tie doain results outside of a particular resonant frequency band through digital filtering. herefore, the acceleration responses in the tie doain were easured and a FF into the frequency doain was perfored on these responses. he filter was designed according to the identified resonant frequencies of the syste found fro the FF. he Elliptic (or Cauer) filter design was used to capture the acceleration response of a singe ode. his Infinite Ipulse Response (IIR) filter was chosen based on the characteristics given in Ref [4]: 1. he Elliptic filter yields a sharper cutoff frequency in coparison to other designs such as Chebyshev and Butterworth. 2. he equiripple in the pass band and stop band is the best that can be achieved in coparison to other filter designs for a given filter order. 3. he IIR filter design has closed for design forulas, and as a result is ore coputationally efficient than the Finite Ipulse Response (FIR) designs. However, a disadvantage of IIR filters is that they exhibit non-linear phase around the transition regions between the pass band and stop bands. o correct for the phase distortion, the filtered signal can be inverted and re-filtered, thus reversing the effects of the phase distortion on the signal. his inversion process also has the advantage of further attenuating the signal outside the pass band when the filter is applied a second tie. able 2 suarizes the lower and upper bounds chosen to isolate each resonant frequency for the fourth order Elliptic filter used in the study. An exaple of the filter frequency response is shown in figure 5, where sharp transition regions and iniized equiripple in both the pass and stop bands are present, as expected. able 2: lower and upper cutoff frequencies for the digital filter. f n (Hz) Lower Bound (Hz) Upper Bound (Hz) 394 320 400 424 405 435 447 435 460 528 480 625 708 650 850
Figure 5: frequency response of a fourth order Elliptic IIR filter with the pass band between 320 and 400 Hz. RESULS AND DISCUSSION Figure 6 displays the unfiltered and filtered acceleration aplitude, phase, and the filtered tie doain acceleration response. Inspection of c) shows that the response is doinated by an exponentially decaying envelope rather than a linearly decaying envelope. herefore, it ay be concluded that Coulob friction is not the appropriate daping odel for the cardboard insert-shell structure for this ode. It ay also be noted fro a) and b) in Figure 6 that signal outside the pass band of the filter is well attenuated with negligible phase distortion. Results for the other resonant odes are shown in the appendix. he sae conclusions ay be drawn in all cases. he conclusion that Coulob friction is not an appropriate daping odel is an interesting result. It has been found in any propeller shaft applications that the cardboard liner tends to disintegrate over tie; dust particles are often found in the inside of the shaft after any cycles. Such an observation would lead one to conclude that friction is present in the linerpropeller shaft syste. However, fro this analysis it has been shown that the friction is not significant as a vibration daping echanis. A ore appropriate daping odel is investigated further in the next section. Figure 6: a) frequency doain acceleration aplitude response, b) frequency doain acceleration phase response, c) corresponding filtered tie doain between 320 and 400 Hz. VISCOUS DAMPING It was shown in the previous section that Coulob friction daping is not an appropriate odel for a cardboard liner in a thin shell structure. herefore, an analysis to deterine the appropriate odel by a second approach is presented in this section. Specifically, the hypothesis regarding viscous daping as the appropriate daping odel is analyzed. CHARACERISICS OF VISCOUS DAMPING Ref [4] presents a ethod for deterining the daping echanis in a syste by using estiated coplex natural frequencies and ode shapes fro a set of experientally deterined transfer functions. It is shown in the reference that the iaginary parts of the natural frequencies are responsible for the diagonal ters of the daping atrix (in physical coordinates) and the iaginary parts of the ode vectors are responsible for the off-diagonal ters. he real parts of the coplex natural frequencies and odes are the sae as the undaped natural frequencies and odes, respectively. o obtain these results, it is assued that the daping in the syste is light so that a first order perturbation ethod for deterining coplex natural frequencies and ode shapes is a sufficiently accurate procedure. By constructing the daping atrix in this anner, the reference shows by exaple that if a non-syetric daping atrix is obtained then it ay be deduced that the physical law behind the daping echanis in the structure is not viscous. he procedure outlined in Ref [4] is restated in the appendix for convenience.
RESULS By using the transfer functions obtained earlier in the study, the daping atrix of the unlined shell as a function of the easured degrees of freedo is shown in figure 7. Figure 8 displays the atrix for the cardboard lined shell. In both cases, the atrices are syetric iplying that viscous daping is an appropriate daping odel to represent the daping in the syste. Indeed, when coparing the unlined atrix to the lined atrix, the diagonal ters of the lined atrix becoe ore pronounced than the unlined atrix. One ay deduce fro this increased doinance of the diagonal ters that there is an increase in proportional viscous daping when the cardboard liner is present. Further inspection of the atrices reveals that the aplitudes of viscous daping are obviously incorrect. he errors are thought to arise fro the difficulty in noralizing coplex odal vectors. Consequently, the atrix ay not directly be used in luped paraeter and finite-eleent ethods for calculation of transfer functions. Nevertheless, the syetric nature of the atrices, and the increased doinance of the diagonal ters in the cardboard lined atrix, are still interesting results. he iplication of such results is that the daping ratios obtained fro an experient can be used in finite-eleent-calculations as proportional viscous daping on a ode-by-ode basis with greater confidence for the daping odel. he causes of the aplitude errors obtained in the daping atrices are the subect of further study. Figure 8: daping atrix for the cardboard lined shell case. CONCLUSION It has been shown in the study that the application of a cardboard liner to a thin circular-cylindrical shell significantly changes the odal behavior of the shell. he liner adds odal stiffness to ost of the odes in the syste, while an increase odal ass is also an observed effect for a certain shell type ode. It has also been shown that the liner adds daping to the sae odes where an increase in stiffness was the observed effect. hrough inspection of the free vibration decay envelope, it was shown that Coulob friction is not an appropriate daping odel for a cardboard liner. Further, by construction and inspection of the daping atrix fro experientally deterined coplex natural frequencies and odal vectors, it was found that proportional viscous daping appears to be an appropriate daping odel. Because proportional viscous daping was found to be an appropriate odel for the cardboard liner, experientally deterined odal daping ratios ay be used in luped paraeter and finite-eleent calculations with a greater degree of confidence. Because of the results obtained in this study, the analytical tools ay be ore effectively used to analyze thin shell structures such as propeller shafts and the need for costly trial-and-error hardware testing is reduced. ACKNOWLEDGMENS Figure 7: daping atrix for the unlined shell case. he first author would like to thank Mark Gehringer for suggesting work on the proect. In addition, thanks to Pa Nielsen for assistance with the proect. he first author would also like to recognize Dr. Ji De Clerck and Dr. Raendra Singh for technical guidance. Finally, thanks to Dr. Ji Woodhouse for help in understanding his paper on the daping atrix.
REFERENCES 1. Leissa, Arthur. Vibration of Shells. he Acoustical Society of Aerica, 1993. 2. Den Hartog, J. P. Mechanical Vibrations. New York: Dover Publications, Inc. 1985. 3. Oppenhei, Alan V. and Ronald W. Schafer. Digital Signal Processing. New Jersey: Prentice- Hall Inc., 1975. 4. Adhikari, S. and J. Woodhouse. Identification of Daping: Part 1, Viscous Daping. J. Sound and Vib. (2001) 243(1), 43-61. 5. Matlab, Student Version 5.3.0. Online help files. 6. Eail discussion with Dr. Jaes Woodhouse, May 2002. 7. Discussions with Dr. Raendra Singh, Septeber 2001 June 2002. 6. Finally, carry out the transforation [( Uˆ Uˆ ) Uˆ ' ] C ( Uˆ Uˆ ) 1 [ Uˆ ] C = to get the daping atrix in physical coordinates. ADDIIONAL RESULS 1 CONAC Martin G. Foulkes is a Noise and Vibration proect engineer at General Motors Corporation. He ay be contacted at: artin.g.foulkes@g.co. Jaes P. De Clerck is a proect leader for structural vibration and vibration ethods at General Motors Corporation. He ay be contacted at: aes.p.declerck@g.co. Figure 9: calculated ode shapes fro thin shell theory: a) Love Mode n=1, b) Shell ode (1,2), c) Bending Mode (2,1), d) Shell Mode (2,2). Raendra Singh is the Donald D. Glower Chair and Professor of Mechanical Engineering at the Ohio State University. He ay be contacted at: singh.3@osu.edu. APPENDIX A. DEERMINAION OF HE DAMPING MARIX he ethod for constructing the daping atrix fro coplex natural frequencies and odal vectors in Ref [5] is restated here for convenience. 1. Measure a set of transfer functions H i (ω). 2. Choose the nuber of odes to be retained in the study. Deterine the coplex natural frequencies λˆ and coplex ode shapes for the transfer functions, for all = 1. Obtain the coplex ode shape atrix N Zˆ = [ z1, z 2,..., z ] C. 3. Estiate the undaped natural frequencies as ω = R( ˆ λ ). 4. Set Uˆ = R[ Zˆ ] and Vˆ = I[ Zˆ ], and fro these obtain W = Uˆ Uˆ and S = Uˆ Vˆ. Now denote 1 B = W S. ' 2 2 5. Fro the B atrix get Ck = ( ˆ ω ˆ ωk ) B ˆ k ω ' for C = 2I λˆ. k and ( ) ẑ Figure 10: a) frequency doain acceleration aplitude response, b) frequency doain acceleration phase response, c) corresponding filtered tie doain between 405 and 435 Hz.
Figure 11: a) frequency doain acceleration aplitude response, b) frequency doain acceleration phase response, c) corresponding filtered tie doain between 435 and 460 Hz. Figure 13: a) frequency doain acceleration aplitude response, b) frequency doain acceleration phase response, c) corresponding filtered tie doain between 650 and 850 Hz. Figure 12: a) frequency doain acceleration aplitude response, b) frequency doain acceleration phase response, c) corresponding filtered tie doain between 480 and 625 Hz.