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Research Papers

Static and Dynamic Response of Beams on Nonlinear Viscoelastic Unilateral Foundations: A Multimode Approach

[+] Author and Article Information
Udbhau Bhattiprolu

Ray W. Herrick Laboratories,
School of Mechanical Engineering,
Purdue University,
140 South Martin Jischke Drive,
West Lafayette, IN 47907
e-mail: udbhau@purdue.edu

Patricia Davies, Anil K. Bajaj

Ray W. Herrick Laboratories,
School of Mechanical Engineering,
Purdue University,
140 South Martin Jischke Drive,
West Lafayette, IN 47907

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 17, 2013; final manuscript received December 18, 2013; published online February 21, 2014. Assoc. Editor: Marco Amabili.

J. Vib. Acoust 136(3), 031002 (Feb 21, 2014) (9 pages) Paper No: VIB-13-1165; doi: 10.1115/1.4026435 History: Received May 17, 2013; Revised December 18, 2013; Accepted December 26, 2013

Nonlinear viscoelastic behavior is a characteristic of many engineering materials including flexible polyurethane foam, yet it is difficult to develop dynamic models of systems that include these materials and are able to predict system behavior over a wide range of excitations. This research is focused on a specific example system in the form of a pinned-pinned beam interacting with a viscoelastic foundation. Two cases are considered: (1) the beam and the foundation are glued so that they are always in contact and the foundation can undergo both tension and compression, and (2) the beam is not glued to the foundation and the foundation reacts only in compression so that the contact region changes with beam motion. Static as well as dynamic transverse and axial forces act on the beam, and the Galerkin method is used to derive modal amplitude equations for the beam-foundation system. In the second case of the beam on tensionless foundation, loss of contact between the beam and the foundation can arise and determination of the loss-of-contact points is integrated into the solution procedure through a constraint equation. The static responses for both cases are examined as a function of the foundation nonlinearity and loading conditions. The steady-state response of the system subject to static and harmonic loads is studied by using numerical direct time-integration. Numerical challenges and the accuracy of this approach are discussed, and predictions of solutions by the three-mode and five-mode approximate models are compared to establish convergence of solutions. Frequency responses are studied for a range of foam nonlinearities and loading conditions.

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References

Figures

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Fig. 1

Schematic of a pinned-pinned beam on a nonlinear tensionless viscoelastic foundation illustrating the parameters and terminology. F10, F20, F30, F2t are applied forces; K1, K3, and Cf are foundation parameters; Co, EI, L, and ρA are beam parameters.

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Fig. 2

Effect of number of modes on static solution. The results shown are for: three-mode approximation and bilateral foundation ( ), five-mode approximation and bilateral foundation ( ), three-mode approximation and unilateral foundation (–  –  –), five-mode approximation and unilateral foundation (——), and zero load equilibrium (———). k1 = 500, k3 = 0, (f10,ξ10) = (-40,-0.5), (f20,ξ20) = (10,0.5), p = 0.

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Fig. 3

Effect of number of modes on the time response to harmonic excitation of the beam on a bilateral foundation. Three-mode approximation ( ), five-mode (——); k1 = 500, k3 = 0, co = 1, cf = 1, static loads (load, position): (f10,ξ10) = (-40,-0.5), (f20,ξ20) = (10,0.5), p = 0; harmonic load (load, position): (f2τ,ξ2τ) = (-5,0).

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Fig. 4

Effect of time step on the steady-state time response of a beam with bilateral foundation to a harmonic excitation. Exact solution (——), Δτ = 0.0005 ( ), and Δτ = 0.0001 ( ) k1 = 500, k3 = 0, co = 1, cf = 1; static loads (load, position): (f10,ξ10), = (-40,-0.5), (f20,ξ20) = (10,0.5), p = 0; harmonic load (load, position): (f2τ,ξ2τ) = (-5,0).

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Fig. 5

Static deflection shapes of the beam on a linear foundation, subject to asymmetric loads. The foundation cases are as follows: unilateral, k1 = 1000 ( ); unilateral, k1 = 800 ( ); unilateral, k1 = 500 ( ); bilateral, k1 = 1000 ( ); bilateral, k1 = 800 ( ); bilateral, k1 = 500 ( ); zero load equilibrium (———). k3 = 0, (f10,ξ10) = (-40,-0.5), (f20,ξ20) = (10,0.5), p = 0.

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Fig. 6

Effect of nonlinearity on static deflection shapes of beam resting on unilateral and bilateral foundations and subject to antisymmetric loads. The foundation cases are as follows: k3 = 1500, unilateral ( ), bilateral ( ); k3 = 1000, unilateral ( ), bilateral ( ); k3 = 500, unilateral ( ), bilateral ( ); zero load equilibrium (———); k1 = 10, (f10,ξ10) = (-10,-0.5), (f20,ξ20) = (10,0.5), p=0.

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Fig. 7

Static deflection shapes of a beam in the case of symmetric loading. The foundation cases are as follows: k1 = 500, k3 = 0 unilateral ( ), bilateral ( ); k1 = 700, k3 = 0, unilateral ( ), bilateral ( ); k1 = 500, k3 = 1000, unilateral ( ), bilateral ( ); zero load equilibrium (———); (f10,ξ10) = (30,-0.5), (f20,ξ20) = (-100,0), (f30,ξ30) = (30,0.5), p = 0.

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Fig. 8

Steady-state response amplitude of beam on bilateral and unilateral foundations. The cases are as follows: midpoint deflection, wss(0) ( ) and contact length ( ); midpoint deflection, wss(0) on bilateral foundation (◣). Inset figures show the time response of the contact length at peak frequencies of the unilateral case. k1 = 500, k3 = 0, co = 4, cf = 2; static loads (load, position): (f10,ξ10) = (30,-0.5), (f20,ξ20) = (-100,0), (f30,ξ30) = (30,0), p = 0; harmonic load (load, position): (f2τ,ξ2τ) = (-22,0).

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Fig. 9

Steady-state response of the modal amplitudes, T1 (▪), T2(△), T3 (□), T4 (+), and T5 (•), for the unilateral case shown in Fig. 8; k1 = 500, co = 4, cf = 2; static loads (load, position): (f10,ξ10) = (30,-0.5), (f20,ξ20) = (-100,0), (f30,ξ30) = (30,0), p = 0; harmonic load (load, position): (f2τ,ξ2τ) = (-22,0)

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Fig. 10

Magnitude spectrum obtained from the DFT of the time response at the peak frequencies, (a) Ω = 2.25, and (b) Ω = 4.25; k1 = 500, k3 = 0, co = 4, cf = 2; static loads (load, position): (f10,ξ10) = (30,-0.5), (f20,ξ20) = (-100,0), (f30,ξ30) = (30,0), p = 0; harmonic load (load, position): (f2τ,ξ2τ) = (-22,0). Inset figures show the corresponding time response.

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Fig. 11

Poincaré section of the lift-off point response at the peak frequency, (a) Ω = 2.25, (b) Ω = 4.25; k1 = 500, k3 = 0, co = 4, cf = 2, (f10,ξ10) = (30,-0.5); static loads (load, position): (f20,ξ20) = (-100,0), (f30,ξ30) = (30,0), p = 0; harmonic load (load, position): (f2τ,ξ2τ) = (-22,0). Inset figures show a zoomed-in view. Numbers next to the data points in both the figures represent the number of cycles n.

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Fig. 12

Effect of damping on the steady-state response of a beam on unilateral foundation. co = 4, cf = 2: midpoint deflection, wss(0) ( ) and lift-off point coordinate ( ); co = 2, cf = 1.5: midpoint deflection, wss(0) (◣) and lift-off point coordinate (•). Inset figures show the time response at peak frequencies. k1 = 500, k3 = 0, (f10,ξ10) = (30,-0.5); static loads (load, position): (f20,ξ20) = (-100,0), (f30,ξ30) = (30,0), p = 0; harmonic load (load, position): (f2τ,ξ2τ) = (-22,0).

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Fig. 13

Effect of nonlinearity on the steady-state response of a beam on unilateral foundation. k3 = 0: midpoint deflection, wss(0) ( ) and lift-off point coordinate ( ); k3 = 1000: midpoint deflection, wss(0) (◣) and lift-off point coordinate (•). Inset figures show the time response at peak frequencies. k1 = 500, co = 4, cf = 2; static loads (load, position): (f10,ξ10) = (30,-0.5), (f20,ξ20) = (-100,0), (f30,ξ30) = (30,0), p = 0; harmonic load (load, position): (f2τ,ξ2τ) = (-22,0).

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Fig. 14

Steady-state response amplitude, wss(-0.5), of beam on a bilateral foundation for linear case, k1 = 100, k3 = 0, ( ) and nonlinear case, k1 = 100, k3 = 1500, (◣); co = 1, cf = 1; static loads (load, position): (f10,ξ10) = (-40,-0.5), (f2τ,ξ2τ) = (-25,0), (f30,ξ30) = (10,0.5), p = 0

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Fig. 15

Steady-state response amplitude of beam on a unilateral linear foundation and subject to asymmetric loading. (a) Lower damping: co = 1, cf = 1 and (b) higher damping: co = 2, cf = 3; deflection, wss(+0.5) ( ) and lift-off point ( ). Inset in Fig. 15 shows the time response at three frequencies, demonstrating the nonlinear nature of the response even for a linear foundation case. k1 = 500, k3 = 0; static loads (load, position) (f10,ξ10) = (-40,-0.5), (f30,ξ30) = (10,0.5), p = 0; harmonic load (load, position): (f2τ,ξ2τ) = (-5,0).

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Fig. 16

Midpoint response of the beam subject to swept-frequency cosine excitation: (a) bilateral foundation and (b) unilateral foundation. The frequencies are swept linearly between Ω = 0 to Ω = 20 in time τd = 100; parameters: k1 = 500, k3 = 0, co = 4, cf = 1.5; static loads (load, position): (f10,ξ10) = (30,-0.5), (f20,ξ20) = (-100,0), (f30,ξ30) = (30,0), p = 0. Amplitude of swept-frequency cosine signal (load, position): (f2τ,ξ2τ) = (-22,0). Inset figures show the zoomed in view of the time response around the resonant frequencies.

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