Analytical Dynamics

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Institutional Subscription. Free Shipping Free global shipping No minimum order. Preface Chapter I. Lagrangian Systems Chapter II. Stability Chapter IV.

Invariant imbedding and nonvariational principles in analytical dynamics

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15. Introduction to Lagrange With Examples

Connect with:. Use your name:. Thank you for posting a review! We value your input. Share your review so everyone else can enjoy it too. The plate model is defined within the configuration space, the set of field variables, and the Lagrangian density. The field variables are determined by the coefficients of the biorthogonal expansion of the spatial displacement vector field with respect to the dimensionless normal coordinate.


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The dynamic equations are derived as Lagrange equations of the second kind of the two-dimensional continuum system. The dynamics of the plane elastic layer is considered as an example, the normal wave propagation is described on the basis of refined plate theories of various orders, and the convergence of approximate solutions to the exact solution of the three-dimensional elastodynamics problem is analyzed for different wavenumbers. Keywords: shells, plates, thin-walled waveguides, analytical dynamics, Lagrangian formalism, normal waves, phase and group velocities.

A plate is used nowadays as a mathematical model of many modern devices in machine industry.

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In general, the refinement of plate models consists in the accounting of supplementary degrees of freedom in addition to the translation and rotation of the middle surface point in the plate kinematics [1, ]. The refined plate and shell theories can be also useful in problems of interaction of acoustic waves and thin-walled structures based on approximate diffraction models e. Many methods of construction of refined plate and shell models can be used. The asymptotic integration approach [15] seems to be powerful and efficient method of the qualitative analysis of the plate and shell dynamics for instance, see [22].

On the other hand, the asymptotic method does not allow one to construct the full hierarchy of solutions [23] approximating the three-dimensional solution in various norms [24].

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At the same time the formal series expansion of the displacement vector, stress tensor etc. As well power series can be used [1, 4, 26, 27] as special function expansions [12, 13, 28].

Analytical Dynamics

One of the most powerful and universal approaches is based on generalized Fourier series [6, 9, 10, 24, 26, ]. Here the higher-order plate theory based on the Lagrangian formalism of analytical mechanics of continua combined with the dimensional reduction approach [9] is used.

The plate model interpreted as a two-dimensional continuum consists in the configuration space, the set of field variables being the biorthogonal expansion coefficients of the three-dimensional displacement vector field with respect to the thickness coordinate, and the Lagrangian density defined on the two-dimensional area corresponding to the plate middle surface for more details, see [11, 31, 32]. In particular, the boundary conditions shifted from the faces onto the middle surface become constraint equations, and the constrained variational problem is solved by the Lagrange multipliers method [33]; this approach allows one to obtain consistent low-order approximations [27] but seems to be a bit too complex when the order of the theory rises.

The well investigated problem of normal wave propagation in the plane elastic layer [34] can be used to analyse the properties of the constructed theories hierarchy and the convergence of the two-dimensional solutions [].

Analytical dynamics

Here the normal waveforms corresponding to some specific wavenumbers are analysed; some results unpublished in the cited articles are presented. The linear dynamics problem statement for a plate can be based on the Hamilton principle [11, 32]:. For linear systems such as the Eq. The density of Lagrangian can be now defined on S - as follows [32, 38]:.

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Here C i j k l are contravariant components of the elastic constants tensor C. The dynamic equations of the generalized plate theory of N th order can be obtained as Lagrange equations of the second kind [31] for the two-dimensional continuum system given by the Eq. The initial-boundary value problem statement, Eq.

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This theory allows one to obtain the simplest equations system and is preferable for higher orders. Let the normal waves be propagating along the axis O x 1. Finally, let us introduce the following dimensionless variables see also [11, ] :. Substituting Eq.


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The eigenvalues of the matrix A , Eq. The corresponding dispersion curves for the phase velocities are shown on Figs. The phase velocities computed on the basis of the spectral problem given by the Eqs. The same results for the phase frequencies are presented in [36].



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