Lateral displacements of a vibrating cantilever beam with a concentrated mass

Parnell, L. A.; Cobble, M.H.

doi:10.1016/0022-460x(76)90092-4

Cited by 41 publications

(13 citation statements)

References 4 publications

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“…The roots of this frequency equation for different end mass M‫ء‬ to cantilever mass m c ratios are shown [21][22][23] in Table II. Now we derive the equation of oscillation amplitude form X͑x͒ for the cantilever with a mass M‫ء‬ on the moving end.…”

Section: End Massmentioning

confidence: 99%

Atomic force microscope cantilever spring constant evaluation for higher mode oscillations: A kinetostatic method

Tseytlin

2008

Review of Scientific Instruments

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Our previous study of the particle mass sensor has shown a large ratio (up to thousands) between the spring constants of a rectangular cantilever in higher mode vibration and at the static bending or natural mode vibration. This has been proven by us through the derived nodal point position equation. That solution is good for a cantilever with the free end in noncontact regime and the probe shifted from the end to an effective section and contacting a soft object. Our further research shows that the same nodal position equation with the proper frequency equations may be used for the same spring constant ratio estimation if the vibrating at higher mode cantilever's free end has a significant additional mass clamped to it or that end is in permanent contact with an elastic or hard measurand object (reference cantilever). However, in the latter case, the spring constant ratio is much smaller (in tens) than in other mentioned cases at equal higher (up to fourth) vibration modes. We also present the spring constant ratio for a vibrating at higher eigenmode V-shaped cantilever, which is now in wide use for atomic force microscopy. The received results on the spring constant ratio are in good (within a few percent) agreement with the theoretical and experimental data published by other researchers. The knowledge of a possible spring constant transformation is important for the proper calibration and use of an atomic force microscope with vibrating cantilever in the higher eigenmodes for measurement and imaging with enlarged resolution.

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Section: End Massmentioning

confidence: 99%

Atomic force microscope cantilever spring constant evaluation for higher mode oscillations: A kinetostatic method

Tseytlin

2008

Review of Scientific Instruments

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“…Anderson [2] obtained the frequency equation for a cantilever with an asymmetrically attached tip mass. Parnell and Cobble [3] solved the displacement equation for a uniform cantilever beam with a concentrated mass at one end using the Laplace transform under generally distributed lateral load and arbitrary boundary and initial conditions. Laura et al [4] determined natural frequencies and modal shapes of a clamped-free beam which carried a "nite mass at the free end.…”

Section: Introductionmentioning

confidence: 99%

Natural Frequencies and Ope–loop Responses of an Elastic Beam Fixed on a Moving Cart and Carrying an Intermediate Lumped Mass

Park

Chung

Youm

et al. 2000

Journal of Sound and Vibration

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“…Moreover, m is the mass of the beam per unit length l; J ; J , and J are the principal mass moments of inertia; D 11 ; D 13 are the torsional and bending-twisting sti®nesses; D 22 ; D 33 are the sti®nesses for bending of w and v directions; EA is the longitudinal sti®ness; c and m a are the coe±cient of viscous damping and the lumped mass, respectively. Basic hypotheses are: (1) there is no transverse shear deformation, (2) there is no warping e®ect, (3) the Poisson's e®ect is negligible, (4) the EulerÀBernoulli beam theory applies to the beam, (5) the beam undergoes large deformations but small strains, meaning that the structure is nonlinear elastic, and (6) the order of dimensionless mass ðm a =mlÞ is "ð" ( 1Þ.…”

Section: Problem Formulationmentioning

confidence: 99%

Effect of Added Tip Mass on the Nonlinear Flapwise and Chordwise Vibration of Cantilever Composite Beam Under Base Excitation

Eftekhari

Mahzoon

Ziaei-Rad

2012

Int. J. Str. Stab. Dyn.

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In this paper, a comparative study is performed for a symmetrically laminated composite cantilever beam with and without a tip mass under harmonic base excitation. The base is subjected to both°apwise and chordwise excitations tuned to the primary resonances of the two directions and conditions of 2:1 autoparametric resonance. In the literature, the governing nonlinear equations of the same problem without tip mass have been derived using the extended Hamilton's principle. Extension is made in this study to include the e®ect of a tip mass on the response of the beam. The natural frequencies are obtained numerically using the diversity guided evolutionary algorithm (DGEA). Next, the multiple scales method is applied to determine the nonlinear response and stability of the system. A set of four¯rst-order di®erential equations describing the modulation of the amplitudes and phases of interacting modes are derived for the perturbation analysis. For veri¯cation, the above equations are reduced to the special case of the cantilever beam without tip mass for comparison with existing results. Finally, the e®ect of the tip mass on the stability of the¯xed points and on the amplitude of oscillation about the equilibrium points in both the frequency and force modulation responses is examined.

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Lateral displacements of a vibrating cantilever beam with a concentrated mass

Cited by 41 publications

References 4 publications

Atomic force microscope cantilever spring constant evaluation for higher mode oscillations: A kinetostatic method

Atomic force microscope cantilever spring constant evaluation for higher mode oscillations: A kinetostatic method

Natural Frequencies and Ope–loop Responses of an Elastic Beam Fixed on a Moving Cart and Carrying an Intermediate Lumped Mass

Effect of Added Tip Mass on the Nonlinear Flapwise and Chordwise Vibration of Cantilever Composite Beam Under Base Excitation

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