The Natural Frequencies and Mode Shapes of a Uniform Cantilever Beam With Multiple Two-Dof Spring–mass Systems

Wu, Jia‐Jang; Whittaker, A.R.

doi:10.1006/jsvi.1999.2324

Cited by 34 publications

(8 citation statements)

References 21 publications

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“…A spring-mass subsystem coupled with a uniform cantilever beam was investigated by Khalily et al [11], Siddiqui et al [12][13][14] and Wu [15][16][17][18]. In [15][16][17][18] the spring-mass subsystem was replaced by massless equivalent springs and the corresponding normal mode shape solution that satisfied spring constraint conditions was derived. Perturbation [12,13] and average acceleration [14] methods were used to analyse the distinctive oscillation features of the beam.…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis of a tapered cantilever beam under a travelling mass

Zhao

Heijden

2015

Meccanica

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We study the vibration of a tapered cantilever (Euler-Bernoulli) beam carrying a moving mass. The tapering is assumed to be parabolic. Using the Galerkin method we find approximate solutions in an energy formulation that takes into account dynamic mass-beam coupling due to inertial, Coriolis and centrifugal effects. The approximate solutions are expanded in terms of the mode shapes of the free tapered beam, which can be obtained analytically. We then study the effect the tapering as well as the magnitude and velocity of the mass have on the tip deflections of the beam. We consider two different initial conditions, one where the mass starts moving from a statically deformed beam and one where the beam is initially triggered to vibrate. We find that tip deflections are more irregular for strongly tapered beams. Our results are of interest for barreled launch systems where tip deflections may adversely affect projectile motion.

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Section: Introductionmentioning

confidence: 99%

Dynamic analysis of a tapered cantilever beam under a travelling mass

Zhao

Heijden

2015

Meccanica

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“…In 1999, Wu [39] proposed an alternative method to estimate natural frequencies and mode shapes. The approach consisted of applying an analyticnumerical combined method to the equivalent two degree of freedom spring-mass system.…”

Section: Thin-walled Beam Of Discontinuous Cross Sectionsmentioning

confidence: 99%

Probabilistic Analysis of a Thin-Walled Beam with a Crack

Kunaporn¹,

Singh²,

Patil³

et al. 2010

51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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Probabilistic Analysis of a Thin-walled Beam with a CrackChalitphan Kunaporn (ABSTRACT) It is reasonable to assume that an aircraft might experience some in-flight discrete source damage caused by various incidents. It is, thus, necessary to evaluate the impact of such damage on the performance of the aircraft. This study is focused on evaluating the effect of a simple discrete damage in an aircraft wing on its static and dynamic response.The damaged wing is modeled by a thin-walled beam with a longitudinal crack the response of which can be obtained analytically. As uncertainties are present in the location and size of the crack as well as in the applied loads, their effects are incorporated into the framework consisting of structural response, crack propagation and aeroelasticity.The first objective of this study is to examine the effect of damage represented by a crack on the wing flexibility that influences its deformation and aero-elastic divergence characteristics. To study this, the thin-walled beam is modeled by Benscoter thin-walled beam theory combined with Gunnlaugsson and Pedersen compatibility conditions to accurately account for the discontinuity at the interface of the cracked and uncracked beam segments. Instead of conducting a detailed finite element analysis, the solution is obtained in an exact sense for general distributed loads representing the wind pressure effects. This analytical approach is shown to provide very accurate values for the global beam response compared with the detailed finite element shell analysis. This analytical solution is, then, used to study the beam response probabilistically. The crack location and size are assumed to be uncertain and are, thus, characterized by random variable. For a specified limit state, the probability of failure can be conveniently calculated by the first order second moment analysis using the safety index approach. The same analytical solution is also used to study the aero-elastic divergence characteristics of a wing, the inner structure of which is represented by a thin-walled beam with a crack of uncertain size and position along the beam.iii The second objective of this study is to examine the time growth of a crack under dynamic gust type of loading to which a wing is likely to be exposed during flight.Damage propagating during operation further deteriorates the safety of the aircraft and it is necessary to study its time growth so that its impact on the performance can be evaluated before it reaches its unstable state. The proposed framework for the crack growth analysis is based on classical fracture mechanics where the remaining flight time is obtained by Monte Carlo simulation in which various uncertainties are taken into account. To obtain equivalent cyclic loading required for crack growth analysis, random vibration analysis of the thin-walled beam is conducted for stochastic wind load defined by a gust load spectral density function. The probability of failure represented by the crack size approaching the critical crack size wit...

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“…Wu and Chen [9] and Wu et al [10] have used the analytical-and-numerical-combined method to study the free vibration characteristics of a uniform cantilever beam carrying multiple one-dof spring-mass systems with dampers. Wu [11,12] and Wu and Whittaker [13] have performed the free vibration analyses of beams carrying multiple two-dof spring-mass systems. Chang and Chang [14] have investigated the free and forced vibration characteristics of beams carrying a two-dof spring-mass system by means of the Laplace transform with respect to the spatial variable.…”

Section: Introductionmentioning
confidence: 99%

Use of equivalent mass method for free vibration analyses of a beam carrying multiple two‐dof spring–mass systems with inertia effect of the helical springs considered
Wu
¹

2005
Numerical Meth Engineering
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SUMMARYThis paper investigates the free vibration characteristics of a beam carrying multiple two-degree-offreedom (two-dof) spring-mass systems (i.e. the loaded beam). Unlike the existing literature to neglect the inertia effect of the helical springs of each spring-mass system, this paper takes the last inertia effect into consideration. To this end, a technique to replace each two-dof spring-mass system by a set of rigidly attached equivalent masses is presented, so that the free vibration characteristics of a loaded beam can be predicted from those of the same beam carrying multiple rigidly attached equivalent masses. In which, the equation of motion of the loaded beam is derived analytically by means of the expansion theorem (or the mode superposition method) incorporated with the natural frequencies and the mode shapes of the bare beam (i.e. the beam carrying nothing). In addition, the mass and stiffness matrices including the inertia effect of the helical springs of a two-dof spring-mass system, required by the conventional finite element method (FEM), are also derived. All the numerical results obtained from the presented equivalent mass method (EMM) are compared with those obtained from FEM and satisfactory agreement is achieved. Because the equivalent masses of each two-dof spring-mass system are dependent on the magnitudes of its lumped mass, spring constant and spring mass, the presented EMM provides an effective technique for evaluating the overall inertia effect of the two-dof spring-mass systems attached to the beam. Furthermore, if the total number of two-dof spring-mass systems attached to the beam is large, then the order of the overall property matrices for the equation of motion of the loaded beam in EMM is much less than that in FEM and the computer storage memory required by the former is also much less than that required by the latter.

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The Natural Frequencies and Mode Shapes of a Uniform Cantilever Beam With Multiple Two-Dof Spring–mass Systems

Cited by 34 publications

References 21 publications

Dynamic analysis of a tapered cantilever beam under a travelling mass

Dynamic analysis of a tapered cantilever beam under a travelling mass

Probabilistic Analysis of a Thin-Walled Beam with a Crack

Use of equivalent mass method for free vibration analyses of a beam carrying multiple two‐dof spring–mass systems with inertia effect of the helical springs considered

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