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Research Papers

Interplay Between Dissipation and Modal Truncation in Ball-Beam Impact

[+] Author and Article Information
Arindam Bhattacharjee, Anindya Chatterjee

Mechanical Engineering,
Indian Institute of Technology Kanpur,
Kanpur 208016, India

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 22, 2016; final manuscript received April 19, 2017; published online September 7, 2017. Assoc. Editor: Przemyslaw Perlikowski.

J. Comput. Nonlinear Dynam 12(6), 061018 (Sep 07, 2017) (8 pages) Paper No: CND-16-1634; doi: 10.1115/1.4036830 History: Received December 22, 2016; Revised April 19, 2017

We study a ball-beam impact in detail; and in particular, we study the interplay between dissipation and modal truncation. With Hertzian contact between a solid ball and an Euler–Bernoulli beam model, we find using detailed numerical simulations that many (well above 60) modes are needed before convergence occurs; that contact dissipation (either viscous or hysteretic) has only a slight effect; and that contact location plays a significant role. However, and more interestingly, we find that as little as 2% modal damping speeds up convergence of the net interaction so that only about 25 modes are needed. We offer a qualitative explanation for this effect in terms of the many subimpacts that occur in the overall single macroscopic impact. In particular, we find that in cases where the overall interaction time is long enough to damp out high modes yet short enough to leave lower modes undissipated, modal truncation at about 25 modes gives good results. In contrast, if modal damping is absent so that higher mode vibrations persist throughout the interaction, final outcomes are less regular and many more modes are needed. The regime of impact interactions studied here occurs for reasonable parameter ranges, e.g., for a 3–4 cm steel ball dropped at speeds of 0.1–1.0 m/s on a meter-long steel beam of net mass 1 kg. We are unaware of any prior similarly detailed numerical study which clearly offers the one summarizing idea that we obtain here.

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Figures

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Fig. 1

(a) Transverse impact of a Hertzian ball on a pinned-pinned Euler–Bernoulli beam at a distance b away from the end. The displacements of the beam and the ball are referred to as y(x, t) and zb(t), respectively. (b) The ball-beam contact. The unbroken lines show the actual configuration at contact. P is the notional contact point on the undeformed ball (i.e., the ball without localized contact deformation). Q is the notional contact point where the ball hits the beam. The distance from P to Q is the compression (positive in the sense shown); and a contact force exists when the compression is positive.

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Fig. 2

F-ξ hysteresis loops seen for input ξ=sin(2πt)+0.5 sin(8πt), see Eqs. (17) and (18). Parameters used here are Kh=1, K¯=4, θm=1.6,  β=1.4,  ε=1×10−4.

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Fig. 3

Impact with purely Hertzian contact with N = 25 modes and b = 0.4. The top figure shows the ball and beam motions during impact, using SI units but with displacements scaled up by a factor of 1000. In other words, the velocity scale is m/s and the displacement scale is mm. The middle figure shows the contact force F in Newtons. The bottom figure shows details of F on a magnified time scale (see color figure online).

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Fig. 4

Hertz contact, see Eq. (13) for F. Two impact locations are considered, left: b = 0.4, and right: b=(5−1)/2. Top: net restitution, and bottom: number of subimpacts, both against number of modes retained (N). For b = 0.4, mode numbers 5, 10, 15,…, are not excited at all. For this reason, it serves as a check to note that results for N = 4 and 5 are identical; results for 9 and 10 are identical; and likewise 14 and 15, 19 and 20, etc. For b=0.618034…, no such simple check is available.

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Fig. 5

Impact with N = 0. The viscous contact force F is plotted against ξ for ball impact velocities of 1.2 m/s, 0.9 m/s, and 0.6 m/s. The loops are rounded at the ends.

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Fig. 6

Impact with N = 27 modes. Left: F versus t, right: F versus ξ. The F versus ξ loops indicate dissipation of energy.

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Fig. 7

With and without viscous contact dissipation, see Eqs. (13) and (16). Two locations, left: b = 0.4, and right: b=(5−1)/2. Top: net restitution and bottom: number of subimpacts, both against N.

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Fig. 8

Impact with N = 0. Parameters used are θm=1.6,  β=1.4, K¯=0.4, ε=5×10−8, see Eqs. (17) and (18). The hysteretic force F is plotted against compression for ball impact velocities of 1.2 m/s, 0.9 m/s, and 0.6 m/s. The restitution values obtained are close to 0.95 in each case, decreasing very slightly with increasing impact velocity.

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Fig. 9

Impact with N = 27 modes. Left: the hysteretic contact force against time, right: hysteresis loops in the contact force, including minor loops.

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Fig. 10

See Sec. 3.4. Impact with rate-independent contact dissipation (red crosses) and without (blue circles). Left: b = 0.4 and right: b=(5−1)/2. Top: restitution and bottom: number of subimpacts, against N (see color figure online).

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Fig. 11

With modal damping (red crosses) and without (blue circles). Left: b = 0.4 and right: b=(5−1)/2. Top: restitution and bottom: number of subimpacts, against N. Convergence of a sort is seen for increasing N; certainly the variability is greatly reduced (see color figure online).

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Fig. 12

Ball-beam motion with N = 40 during impact at b=(5−1)/2 for damping ratios ζ = 0 and ζ = 0.02. In these two particular simulations, the final subimpacts differ dramatically; other pairs of simulations show differences in details, but the qualitative effect of light modal damping is the same.

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Fig. 13

Restitution against N for different impact velocities, with b=(5−1)/2, for ζ = 0.02 (red crosses) and ζ = 0 (blue circles) (see color figure online)

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Fig. 14

Impacts with ball diameters, of 4 cm and 2 cm at b=(5−1)/2, for ζ = 0.02 (red crosses) and ζ = 0 (blue circles). Top: restitution and bottom: number of subimpacts, against N (see color figure online).

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Fig. 15

See Sec. 4.2. Impacts with a ball diameter of 4 cm, for different contact locations b, with ζ = 0.02 and N = 40. Top: restitution and bottom: number of subimpacts, against b.

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