1,721,190 research outputs found
Second-order cone programming formulations for a class of problems in structural optimization
No full textThis paper provides efficient and easy to
implement formulations for two problems in structural
optimization as second-order cone programming
(SOCP) problems based on the minimum compliance
method and derived using the principle of complementary
energy. In truss optimization both single and
multiple loads (where we optimize the worst-case compliance)
are considered. By using a heuristic which is
based on the SOCP duality we can consider a simple
ground structure and add only the members which
improve the compliance of the structure. It is also
shown that thickness optimization is a problem similar
to truss optimization. Examples are given to illustrate
the method developed in this pape
A compliance based design problem of structures under multiple load cases
No full textThere are two popular methods concerning the optimal design of structures. The first is the minimization of the volume of the structure under stress constraints. The second is the minimization of the compliance for a given volume. For multiple load cases an arising issue is which energy quantity should be the objective function. Regarding the sizing optimization of trusses, Rozvany proved that the solution of the established compliance based problems leads to results which are awkward and not equivalent to the solutions of minimization of the volume under stress constraints, unlike under single loading 1. In this paper, we introduce the "envelope strain energy" problem where we minimize the volume integral of the worst case strain energy of each point of the structure. We also prove that in the case of sizing optimization of statically non-indeterminate2 trusses, this compliance method gives the same optimal design as the stress based design method.<br/
Mode shapes during asynchronous motion and non-proportionality indices
No full textWhen synchronous motion does not exist, it is not possible to draw the classical mode shapes. In this paper, a representative shape of motion during free vibration of a non-classically damped system is sought. It is noted that this shape provides an optimal representation of free motion. Interpretations of the optimality thus introduced are presented. Their connection with non-proportionality of damping and of gyroscopy is brought out. In the spirit of the optimality presented in this paper, two indices of non-proportionality are defined. Properties of these indices are discussed. Comparison with other indices of non-proportionality available in the literature is presented. Illustrative examples are given
Rayleigh's classical sensitivity analysis extended to damped and gyroscopic systems
No full textFirst order perturbation and rates of change of eigenvalues with respect to changes in system parameters are presented. Expressions for eigenvalue sensitivity for conservation (originally Rayleigh's) damped and gyroscopic system (discrete as well as continuous) are presented. Relevance to design is discussed
Gross modifications in structural dynamics via interpolated modes
No full textRepeated analysis of a structure for a range of parameter values is often encountered in engineering design and potimisation. In this paper, the problem of approximately predicting the natural frequencies of a system, when parameters undergo gross changes, is addressed. A method of 'interpolated modes' is developed. It is shown that reasonable estimates of the natural frequencies are obtained without a recourse to exact calculations for each value of the system parameter. Illustrative examples are given
Rayleigh quotient for non-conservative discrete systems
No full textRayleigh quotient is extended to non-conservative systems when the modes become complex due to the presence of damping. A critique of recently proposed quotients in this context is presented. Two new quotients are proposed here which provide approximation for the complex eigenvalues when the trial vectors approximate the actual latent vectors. The stationary values of the quotients proposed here coincide with the complex latent-roots of the system. In case of the first quotient, the complex trial vectors belong to the state-space. For the second, they belong to the configuration space. The expressions in the numerator and the denominator of these quotients involve bilinear forms as opposed to those in the original non-dissipative setting where they are quadratic forms. An example is given to illustrate the claimed stationarity of the proposed quotients
Criticality of damping in multi-degree-of-freedom systems
No full textThe concept of criticality in multi-degree-of-freedom systems is discussed. Sufficient conditions for overdamping, critical damping, and underdamping are derived in terms of the matrices appearing in the modal coordinates. It is noted that results available in the literature for the case of overdamping and mixed damping are erroneous. This has been pointed out by Bhaskar (1991,1992) for the cases of overdamping and mixed damping, and by Barkwell et al. (1992) for the case of overdamping. The error in the proof of the conditions for overdamping is brought out. A sufficient condition for overdamping is presented. Results obtained for the symmetric systems are then generalized to the symmetrizable systems. Theorems on eigenvalue bounds are applied to establish criticality
Dynamics of convecting elastic solids
No full textThe dynamics of a class of convecting elastic media is considered. On the basis of an appropriate variational principle, the general field equation governing small oscillations is derived. The variational formulation demands (i) conservations of mass, (ii) conservation of energy, and (iii) conservation of the identity of particles. Of these, conservation of mass needs to be satisfied explicitly as a constraint. This is achieved by constraining the classical mechanical Lagrangian using a Lagrange multiplier with the continuity equation. Hamilton's principle modified for a control volume in this way then leads to the equation of motion for small oscillations of convecting gyroelastic solids. The mathematical structure of the field equation thus derived is examined. The origins of the 'gyroscopic' and the 'centrifugal' effects are traced. These can be associated with various terms in the expression for the Lagrangian density. In particular, terms in the kinetic energy density that are independent the velocity field, those that are linear in the velocity field and those that are quadratic in the velocity field are associated with the centrifugal, gyroscopic, and inertia terms in the equation of motion respectively. A close mathematical analogy between the dynamics of this class of continua and the dynamics of discrete gyroscopic-centrifugal systems having fixed material particles is noted. The free vibration problem is posed in its generality. An appropriate Rayleigh quotient is defined. The stationarity associated with the quotient can potentially be used for computational work. Illustrative examples and applications are discussed
The effective Poisson ratio of random cellular matter having bending dominated architecture
No full textWe argue that the effective Poisson ratio of cellular and porous solids is independent
of the material of the solid phase, if the mechanism of the cell wall deformation is dominated
by beam bending —thus rendering it to be a purely kinematic quantity. Introducing a kinematic
simplification and requiring statistical isotropy, we prove a result of remarkable generality that the effective Poisson ratio of irregular planar structures equals 1 for all bending dominated random architectures. We then explore a deeper connection of this behavior with area-preserving deformation of planar closed elastic cells. We show that thin sheets and films made of such
microstructured material afford physical realizations of the two-dimensional analogue of incompressible matter.We term such non-stretchable sheet material as well as deformations as isoektasic
The use and limitations of continuum modes for response calculations of cellular structures
No full textIn this paper, a method for the approximate calculations of response of cellular structures is presented. The method is based on the idea that for low-frequency modes, the cellular structures behave in a way similar as a continuum does. The range of validity of the method on the mode number scale is also examined. It is shown that the response is estimated very accurately from a reduced order model based on continuum modes for low-frequency vibration. The accuracy starts to deteriorate with the increase in the mode number. Beyond certain mode number, the improvement in accuracy due to the inclusion of additional continuum modes in the approxijmation shows a point of diminishing returns. illustrative examples are given
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