Purdue University Purdue e-pubs Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering 7-2016 Response of a Shell Structure Subject to Distributed Harmonic Excitation Rui Cao Purdue University, cao101@purdue.edu J Stuart Bolton Purdue University, bolton@purdue.edu Follow this and additional works at: http://docs.lib.purdue.edu/herrick Cao, Rui and Bolton, J Stuart, "Response of a Shell Structure Subject to Distributed Harmonic Excitation" (2016). Publications of the Ray W. Herrick Laboratories. Paper 126. http://docs.lib.purdue.edu/herrick/126 This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information.
Response of a shell structure subject to distributed harmonic excitation Rui Cao, J. Stuart Bolton, Ray W. Herrick Laboratories School of Mechanical Engineering, Purdue University
I. Introduction Tire vibration problems can be modeled as ring/shell structure vibration problems x Tread vibration usually includes: In-plane vibration - y ( in the direction of travel) - x (axial direction) Out-of-plane vibration - z (radial direction) y z Ring model: S.C. Huang and W. Soedel. "Effects of coriolis acceleration on the free and forced in-plane vibrations of rotating rings on elastic foundation. Journal of Sound and Vibration 115.2 (1987): 253-274. MoViC&RASD 2
I. Introduction Experimental dispersion results of a tire with empty air cavity Tire surface mobility measurement MoViC&RASD 3
I. Introduction Experimental dispersion results of a tire with fibrous material filled air cavity A ring structure tire model can well capture the lower frequency wave types, but incapable of expressing the highlighted wave type in tire structures MoViC&RASD 4
I. Introduction 2D ring model 3D shell model Axial shearing motion in tires is experimentally found to be important in high frequency range Drawbacks: 1. No axial direction motion 2. Only allow radial and circumferential excitations 3. Not a good model for wide shell structures such as wide treadbands Advantage: 1. Axial motion is included 2. Allows 2D excitations 3. Suitable for modeling wide shell-like structures MoViC&RASD 5
II. Model description 1. Undeformed shell model 2. Pressurized from inside 3. Waves can travel in circumferential (θ) and axial (x) directions 4. Structure has motion in radial (r), circumferential (θ) and axial (x) directions 5. Discrete input forces are applied over an area to represent road surface input MoViC&RASD 6
III. Mathematical formulation Free vibration analysis Equations of motion Axial direction: L u, u, u h x x r 2 u t 2 x Tangential direction: Radial direction: L u, u, u h x r L u, u, u h r x r 2 u t 2 t 2 u 2 r L - linear differential operators, describing the stress strain relations in the shell MoViC&RASD Ref: W. Soedel. Vibrations of Shells and Plates. CRC Press, 2004. 7
III. Mathematical formulation Free vibration analysis Boundary conditions r Fixed BC in radial direction u (0, ) 0, u ( L, ) 0 r r θ x Fixed BC in circumferential direction u (0, ) 0, u ( L, ) 0 No constraint in axial direction N (0, ) 0, N ( L, ) 0 xx xx M (0, ) 0, M ( L, ) 0 xx xx MoViC&RASD 8
III. Mathematical formulation Free vibration analysis Solving for natural frequencies Assumed solutions satisfying the BCs k x u A x k L m cos cos n 2 h k k k A 11 12 13 2 k h k k B 21 22 23 2 k k h k C 31 32 33 0 k x u B k L m sin sin n k x u C r k L m sin cos n EOM Set determinant to zero k m - axial wavenumber k n - tangential wavenumber k xx - relating k m, k n and material/geometry constants Damping is incorporated through complex Young s Modulus MoViC&RASD 9
III. Mathematical formulation Forced vibration analysis N1 N2 2 h h u q mni mni mni mni i m1 n0 η mni modal participation factor, q i applied total force in i direction i = 1, axial wave motion i = 2, tangential wave motion i = 3, radial wave motion N 1 number of axial modes used N 2 number of tangential modes used The equations are multiplied by orthogonal modes and integrated over the structure domain to solve for all η mni MoViC&RASD 10
III. Mathematical formulation Forced vibration analysis Generic area excitations FN 1 FN 2 Frpq q x x R r p q p1 q1 FN 1 FN 2 F ij q x x R p1 q1 p q p index of drive point along axial direction q index of drive point along tangential direction FN 1 total number of axial drive points FN 2 total number of circumferential drive points FN FN 1 2 F F u ( x, ) F u ( x, ) mni r xmni p q mni p q p1 q1 x θ r F mni resultant modal force MoViC&RASD 11
III. Mathematical formulation Forced vibration analysis Because no initial phase was assumed, two orthogonal assumed solution need to be assumed to cover all possible solutions. Using derived modal participation factors, we can substitute to obtain the forced response. 3 N N 1 2 u u u x mni (1) xmni (1) mni ( 2 ) xmni ( 2 ) i 1 m 1 n 0 3 N N 1 2 u u u mni (1) mni (1) mni ( 2 ) mni ( 2 ) i 1 m 1 n 0 3 N N 1 2 u u u r mni (1) rmni (1) mni ( 2 ) rmni ( 2 ) i 1 m 1 n 0 MoViC&RASD 12
Free vibration - Dispersion relation m = 1, n = 1 V r V a V c n Flexural wave (radial disturbance) Low frequency range Slow wave speed Due to relatively low flexural stiffness MoViC&RASD 13
Free vibration - Dispersion relation m = 2, n = 1 V r V a V c n Flexural wave (radial disturbance) Low frequency range Slow wave speed Due to relatively low flexural stiffness MoViC&RASD 14
Free vibration - Dispersion relation m = 2, n = 2 V r V a V c n Flexural wave (radial disturbance) Low frequency range Slow wave speed Due to relatively low flexural stiffness MoViC&RASD 15
Free vibration - Dispersion relation m = 1, n = 1 V r V a V c n Longitudinal wave (Axial disturbance) Mid frequency range Mid wave speed Due to relatively high in-plane stiffness in the axial direction MoViC&RASD 16
Free vibration - Dispersion relation m = 2, n = 1 V r V a V c n Longitudinal wave (Axial disturbance) Mid frequency range Mid wave speed Due to relatively high in-plane stiffness in the axial direction MoViC&RASD 17
Free vibration - Dispersion relation m = 2, n = 2 V r V a V c n Longitudinal wave (Axial disturbance) Mid frequency range Mid wave speed Due to relatively high in-plane stiffness in the axial direction MoViC&RASD 18
Free vibration - Dispersion relation m = 1, n = 1 V r V a V c n Longitudinal wave (circumferential disturbance) High frequency range High wave speed Due to high in-plane stiffness in the circumferential direction MoViC&RASD 19
Free vibration - Dispersion relation m = 2, n = 1 V r V a V c n Longitudinal wave (circumferential disturbance) High frequency range High wave speed Due to high in-plane stiffness in the circumferential direction MoViC&RASD 20
Free vibration - Dispersion relation m = 2, n = 2 V r V a V c n Longitudinal wave (circumferential disturbance) High frequency range High wave speed Due to high in-plane stiffness in the circumferential direction MoViC&RASD 21
Forced vibration Transfer mobility x r θ Radial transfer mobility along the circumference, driven by radial harmonic point excitation MoViC&RASD 22
Forced vibration Dispersion (from wave decomposition) x r θ Dispersion due to radial harmonic point excitation Due to radial excitation, flexural waves are driven obviously Due to curvature coupling, axial in-plane waves are also driven, but at much lower amplitude MoViC&RASD 23
Forced vibration Dispersion (from wave decomposition) x r θ Dispersion due to radial harmonic point excitation Due to radial excitation, flexural waves are driven obviously Due to curvature coupling, axial in-plane waves are also driven, but at much lower amplitude MoViC&RASD 24
Forced vibration Transfer mobility x r θ Radial transfer mobility along the circumference, driven by radial harmonic line excitation MoViC&RASD 25
Forced vibration Dispersion (from wave decomposition) x r θ Dispersion due to radial harmonic line excitation The radial line input across the axial direction suppressed the axial in-plane and out-of-plane motion MoViC&RASD 26
Forced vibration Dispersion (from wave decomposition) x r θ Dispersion due to radial harmonic line excitation The radial line input across the axial direction suppressed the axial in-plane motion MoViC&RASD 27
Forced vibration Dispersion (from wave decomposition) x r θ Dispersion due to radial harmonic line excitation (1/10 normal damping) Axial and circumferential wave show up MoViC&RASD 28
Forced vibration Transfer mobility x r θ Radial transfer mobility along the circumference, driven by radial harmonic area excitation MoViC&RASD 29
Forced vibration Dispersion (from wave decomposition) x r θ Dispersion due to radial harmonic area excitation The area input not only suppressed the axial motion but also eliminated the response associated with certain wavenumbers MoViC&RASD 30
Forced vibration Dispersion (from wave decomposition) x r θ Dispersion due to radial harmonic area excitation The area input not only suppressed the axial motion but also eliminated the response associated with certain wavenumbers MoViC&RASD 31
Forced vibration Transfer mobility x r θ Tangential transfer mobility along the circumference, driven by 45 oblique harmonic area excitation The mobility along the circumference is no longer symmetric MoViC&RASD 32
Forced vibration Dispersion (from wave decomposition) x r θ Dispersion (from u r ) due to 45 oblique harmonic area excitation The oblique excitation drives the in-plane motion mostly at high frequency MoViC&RASD 33
Forced vibration Dispersion (from wave decomposition) x r θ Dispersion (from u θ ) due to 45 oblique harmonic area excitation The oblique excitation drives the in-plane motion mostly at high frequency MoViC&RASD 34
V. Conclusions In-plane shearing exist in both the circumferential and axial directions The circumferential shearing is associated with various mode shapes in the axial direction The axial shearing is due to the changing mode shapes in the circumferential direction Due to coupling, the shearing modes create out-of-plane motion which can radiate sound effectively due to high phase speeds Excitation area distribution creates a spatial domain windowing effect, which can suppress certain modes and wave types in the forced response MoViC&RASD 35