Dynamics of a Flexible Beam Contacting a Linear Spring at Low Frequency Excitation: Experiment and Analysis

Fegelman, Karen J. L.; Grosh, Karl

doi:10.1115/1.1426073

Cited by 16 publications

(9 citation statements)

References 11 publications

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“…For t 0 40, the initial conditions z 0 and s 0 (29) are expressed in terms of the mode shapes and superpositions from the previous state. Defining i as the previous state, the state-to-state mapping follows from the orthogonality of the mode shapesĝ…”

Section: State-to-state Mappingmentioning

confidence: 99%

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Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints

Brake

Wickert

2010

Journal of Sound and Vibration

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To cite this version:M.R. Brake, J.A. Wickert. Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints. Journal of Sound and Vibration, Elsevier, 2010, 329 (7) A method for the modal analysis of continuous gyroscopic systems with nonlinear constraints is developed. This method assumes that the nonlinear constraint can be expressed as a piecewise linear force-deflection profile located at an arbitrary position within the domain. Using this assumption, the mode shapes and natural frequencies are first found for each state, then a mapping method based on the inner product of the mode shapes is developed to map the displacement of the system between the in-contact and out-of-contact states. To illustrate this method, a model for the vibration of a traveling string in contact with a piecewise-linear constraint is developed as an analog of the interaction between magnetic tape and a guide in data storage systems. Five design parameters of the guide are considered: flange clearance, flange stiffness, symmetry of the force-deflection profile in terms of flange stiffness and offset, and the guide's position along the length of the string. There are critical bifurcation thresholds, below which the system exhibits no chaotic behavior and is dominated by period one, symmetric behavior, and above which the system contains asymmetric, higher periodic motion with windows of chaotic behavior. These bifurcation thresholds are particularly pronounced for the transport speed, flange clearance, symmetry of the force deflection profile, and guide position. The stability of the system is sensitive to the system's velocity, and, compared to stationary systems, more mode shapes are needed to accurately model the dynamics of the system.

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Section: State-to-state Mappingmentioning

confidence: 99%

“…Unlike stationary strings or beams, the traveling string of Fig. 1(a) cannot be accurately approximated up to the first natural frequency by only three to five modes [28,29], as an unstable response is incorrectly predicted near o ¼ 0:68 for N ¼ 5. In Fig.…”

Section: Frequency Responsementioning

confidence: 99%

Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints

Brake

Wickert

2010

Journal of Sound and Vibration

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“…If the nonlinearity is expressed as a discontinuity in the domain, a non-smooth spatial and temporal transformation is shown to be effective in removing singular terms from the equation of motion [20], but the resulting equations still require numerical solution. When the nonlinearity is limited to being expressed as a piecewise-linear model, a number of modal mapping methods exist that calculate the mode shapes of the system in each regime of the piecewise-linear nonlinearity and then use the orthogonality of the mode shapes to map the displacement of the system across the non-smooth transitions of the nonlinearity [4,[21][22][23]. These methods, though, are often constrained in the order of the system [21,22] or the permissible parameter spaces [4,23].…”

Section: Introductionmentioning

confidence: 99%

Modelling localized nonlinearities in continuous systems via the method of augmentation by non-smooth basis functions

Brake

Segalman

2013

Proc. R. Soc. A.

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Existing solutions for continuous systems with localized, non-smooth nonlinearities (such as impacts) focus on exact methods for satisfying the nonlinear constitutive equations. Exact methods often require that the non-smooth nonlinearities be expressed as piecewise-linear functions, which results in a series of mapping equations between each linear regime of the nonlinearities. This necessitates exact transition times between each linear regime of the nonlinearities, significantly increasing computational time, and limits the analysis to only considering a small number of nonlinearities. A new method is proposed in which the exact, nonlinear constitutive equations are satisfied by augmenting the system's primary basis functions with a set of non-smooth basis functions. Two consequences are that precise contact times are not needed, enabling greater computational efficiency than exact methods, and localized nonlinearities are not limited to piecewiselinear functions. Since each nonlinearity requires only a few non-smooth basis functions, this method is easily expanded to handle large numbers of nonlinearities throughout the domain. To illustrate the application of this method, a pinned-pinned beam example is presented. Results demonstrate that this method requires significantly fewer basis functions to achieve convergence, compared with linear and exact methods, and that this method is orders of magnitude faster than exact methods.

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“…The dynamics of continuous systems like beams, strings, and rods subjected to impacts caused by motion-limiting constraints have been studied extensively. A representative set of studies can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The impact of these systems is modelled using two different methods.…”

Section: Introductionmentioning

confidence: 99%

“…To exactly satisfy the condition of sticking, as shown in Figure 2, when a chatter sequence is recognized the beam is brought to rest at the point of impact by applying an impulse using CoR = 0. The post-impact modal position and velocities can now be obtained from (17) and (28) using CoR = 0. The sticking Figure 2: Simulating a typical chatter sequence using CoR method with Lagrange multiplier.…”

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confidence: 99%

Nonsmooth Modeling of Vibro-Impacting Euler-Bernoulli Beam

Vyasarayani

Sandhu

McPhee

2012

Advances in Acoustics and Vibration

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A new technique to simulate nonsmooth motions occurring in vibro-impacting continuous systems is proposed. Sticking motions that are encountered during vibro-impact simulation are imposed exactly using a Lagrange multiplier, which represents the normal reaction force between the continuous system and the obstacle. The expression for the Lagrange multiplier is developed in closed form. The developed theory is demonstrated by numerically simulating the forced response of a pinned-pinned beam impacting a point-like rigid obstacle.

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Dynamics of a Flexible Beam Contacting a Linear Spring at Low Frequency Excitation: Experiment and Analysis

Cited by 16 publications

References 11 publications

Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints

Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints

Modelling localized nonlinearities in continuous systems via the method of augmentation by non-smooth basis functions

Nonsmooth Modeling of Vibro-Impacting Euler-Bernoulli Beam

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