1,721,053 research outputs found
Taussky' s theorem, symmetrizability and modal analysis revisited
No full textThis paper is concerned with symmetrization and diagonalization of real matrices and their implications for the dynamics of linear, second-order systems governed by equations of motion having asymmetric coefficient matrices. Results in the light of Taussky's theorem are presented. The connection of the symmetrizers with the eigenvalue problem is brought out. An alternative proof of Taussky's theorem for real matrices is presented. Diagonalization of two real symmetric (but not necessarily positive-definite) matrices is discussed in the context of undamped non-gyroscopic systems. A commutator of two matrices with respect to a given third matrix is defined; this commutator is found to play an interesting role in deciding simultaneous diagonalizability of two or three matrices. Errors in a few previously known results are brought out. Pseudo-conservative systems are studied and their connection with the so-called 'symmetrizable systems' is critically examined. Results for modal analysis of general non-conservative systems are presented. Illustrative examples are given
Waveguide modes in elastic rods
Full text linkThis paper is concerned with wave propagation in elastic rods of cross-sectional shapes that admit coupling between bending and torsional motions. Dispersion relations for progressive and evanescent waves are presented in closed form. The general case of triple coupling and the special case of double coupling are considered. The propagation characteristics are analytically presented in the limits of the long and the short waves. Waveguide modes in these limits are presented as asymptotic expansions. The polarization behaviour in various regimes of propagation is studied. Eigenvalue veering is observed for coupled modes. For the complete spectrum involving complex wavenumber, the problem of obtaining the dispersion relation and the propagation modes is cast as a standard eigenproblem. Illustrative examples are given. <br/
Elastic waves in Timoshenko beams: the 'lost and found' of an eigenmode
No full textThis paper considers propagating waves in elastic bars in the spirit of asymptotic analysis and shows that the inclusion of shear deformation amounts to singular perturbation in the Euler-Bernoulli (EB) field equation. We show that Timoshenko, in his classic work of 1921, incorrectly treated the problem as one of regular perturbation and missed out one physically meaningful 'branch' of the dispersion curve (spectrum), which is mainly shear-wise polarized. Singular perturbation leads to: (i) Timoshenko's solution ?(1)*??EB*[1+O(?2k*2)] and (ii) a singular solution ?(2)*?(1/2?2)+O(k*)2; ?, ?* and k* are the non-dimensional slenderness, frequency and wavenumber, respectively. Asymptotic formulae for dispersion, standing waves and the density of modes are given in terms of ?. The second spectrum—in the light of the debate on its existence, or not—is discussed. A previously proposed Lagrangian is shown to be inadmissible in the context. We point out that Lagrangian densities that lead to the same equation(s) of motion may not be equivalent for field problems: careful consideration to the kinetic boundary conditions is important. A Hamiltonian formulation is presented—the conclusions regarding the validity (or not) of Lagrangian densities are confirmed via the constants of motion
Group velocity as an eigensensitivity problem
No full textWe present the problem of computing the dispersion relation in the generic form ofa parameter dependent generalized eigenvalue problem–the wavenumber plays the role of thisparameter. By carrying out first order perturbation analysis in the spirit of Rayleigh, we obtainan exact expression for group velocity from this eigen-sensitivity. The connection of the eigensensitivityrelationship with the stationarity of the Rayleigh’s quotient is explored. Illustrativeexamples to demonstrate the idea are presented for a specific waveguide problem involving twoelastically coupled beams. Two numerical examples are given
Group velocity of elastic waves as eigensensitivity of a parameter dependent eigenvalue problem
No full textWe consider the problem of calculating group velocity of elastic waves along different branches of the dispersion curves without carrying out direct numerical differentiation along these curves. We obtain an expression for the rate of change of eigenvalues with respect to a parameter for a parameter-dependent ‘non-linear eigenvalue problem’. Eigenvalues have the meaning of frequency (or its square) and the role of the parameter is played by the wave number. We then define a quotient which shows Rayleigh-quotient-like behaviour in that it differs from unity by a quantity that scales as ??2, where ? is the magnitude of a parameter used to perturb an eigenvector. By carrying out first order perturbation analysis of a generalised eigenvalue problem involving two Hermitian matrices, at least one of which is positive definite, we obtain an exact expression for group velocity as a rate of change of an eigenvalue. This form is particularly suited for elastic waves. The connection of the eigensensitivity relationship with the stationarity of the Rayleigh's quotient is explored. Two specific examples of mechanical waveguides are presented: (i) the Timoshenko waveguide and (ii) the bending-torsion coupled elastic waveguide.<br/
An eigenvalue approximation for parameter-dependent undamped gyroscopic systems
No full textParameter–dependent eigenvalue problem occurs in a host of engineering contexts. Structural design under dynamic loading is concerned with the evaluation of eigenvalues for a large number of structures, each evaluation corresponding to a combination of design parameters. Exact calculation of the natural frequencies of all the models considered is computationally expensive. Here structural problems that possess gyroscopy, typically encountered in the analysis of rotating elastic structures, are considered. Approximate but inexpensive calculations are sought for this class of problems. In the present work, an algorithm for approximating the natural frequencies of undamped gyroscopic systems is presented. Numerical examples show excellent accuracy, while affording significant computational economy compared to exact calculations. The computational gain is found to be relatively more notable when the size of the considered problem is large
Elastic stabilization of wrinkles in thin films by auxetic microstructure
No full textThin elastic sheets and membranes are known to wrinkle when they are stretched — the associated physics is highly non-linear. The unusual behavior exhibited by thin films upon stretching when they possess auxetic structure, i.e. when their apparent Poisson's ratio is negative, is reported here. Wrinkling is now suppressed within the bulk of auxetic films when tensioned, whereas localized creases confined to the clamps, which decay away exponentially, appear. These edge wrinkles are characterized for their amplitude and wavelength experimentally, theoretically, and computationally, which show excellent agreement with expected trends. The scaling for amplitude, wavelength and decay rate upon film properties and tension is obtained using simple analyses based on kinematic mismatch resulting from lateral Poisson's expansion.</p
A mode shape based approximation for parameter-dependent eigenvalue problem arising in structural vibration
No full textDynamics of two impacting beams with clearance nonlinearity
No full textAnalytical solutions describing the transient dynamics of two Euler-Bernoulli beams with tips separated by clearance, are obtained. The tips of the beams impact when one of the beams is harmonically excited. Expressions of transient dynamics are presented as a superposition of particular solutions that satisfy to inhomogeneous boundary conditions, and eigenfunctions series with time dependent coefficients and homogeneous boundary conditions. The transition from impact phase to out-of-contact phase and vice versa is implemented using conditions that switch, involving construction of expressions for shear forces and relative position of beam tips. After each transition from one phase to another, the functions describing the time dependent coefficients in the eigenfunctions series are updated. This update involves the solution of ordinary differential equations with initial conditions corresponding to the end of the previous phase. The system of impacting beams reveals complex dynamics, including chaotic behaviour. Transient dynamics surfaces, time histories of beams deflections, impact forces, coefficients of restitution and phase planes are presented
Elasticity of diametrically compressed microfabricated woodpile lattices
No full textModulus-porosity relationships are invaluable to rational material design of porous and structured solids. When struts in a lattice are compressed diametrically, the mechanics is rather complex. Herein, the problem of modulus-porosity in the spirit of scaling arguments and analyses based on simple ansatz followed by variational minimization of the elastic potential energy is addressed. Using scaling arguments, a simple power law where the apparent modulus of elasticity scales quadratically with the volume fraction for diametrically compressed elastic lattices is obtained. The modulus-porosity relationship is found to be consistent with computations and laboratory experiments on additively manufactured woodpile lattices with various cross-sectional shapes and lattice spacing. It is also shown that the persistence length of diametrically pinched elastic rods is small, so that the effect of compressive strain from neighboring sites can be ignored. The decay behavior is surprisingly accurately captured by the variational approach and is consistent with computations. Finally, the range of validity of the quadratic power law presented here, up to relative density similar to 80%, is identified. On the apparent modulus-porosity plane, the experimental data aligns well with the power law for modulus-porosity predicted from simple analyses and finite element calculations
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