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F LUTTER flight testing is a necessary section for every type of new aircraft in the process of designing and finalizing, and it continues to be a challenging research area because of the concerns with cost, time, and safety in expanding the envelope of new or modified aircraft. There is a variety of means to make flutter flight testing. Natural atmospheric turbulence was first used for most testing, and stick raps were frequently used with it. However, natural atmospheric turbulence is often difficult to find and seldom excites all of the aircraft’s structural modes [1], and stick raps typically do not excite structural modes above 5 Hz. Other means of excitation include sinusoidal control surface excitation, oscillating aerodynamic vanes, rotary inertia exciters, and pyrotechnic bonkers. However, implementing the method of sinusoidal control surface excitation requires modifications to the control system software, which can be very costly and time consuming. The oscillating aerodynamic vane system requires a relatively large amount of power and must be run from the aircraft’s hydraulic system, resulting in costly and time-consuming installations procedures [2]. For the rotary inertia exciter method, adding mass would increase the weight of the exciter, and a risk is taken because additional weight might affect the vibration characteristics of the aircraft [3]. The pyrotechnic bonkers method’s disadvantages include questionable reliability under arduous environmental conditions and the fact that the number of impulses per flight is limited [4]. The disadvantages of the aforementioned flutter excitation methods promote the motivation to develop a low-cost effective inflight structural excitation system. A novel excitation system that addresses these needs is the configuration that consists of a fixed vane and a rotating slotted cylinder (RSC) behind its trailing edge (as shown in Fig. 1a), which perfectly combines mechanical simplicity, controllability, and reliability [5]. This excitation system can produce high-frequency high-force aerodynamic excitation with minimal power and torque input. The two slots cut in the cylinder generate periodic lifting forces that excite the aircraft. As the cylinder rotates during flight, the flow is alternately deflected upward and downward through the slots, resulting in a periodic lift force at twice the cylinder’s rotation frequency. Figure 1b illustrates the working mechanism. Points A, B, C, D, and E, respectively, correspond to the five typical positions of RSCs, namely, point A zero lift position, point B maximum positive lift position, point C second zero lift position, point D maximum negative lift position, and point E third zero lift position. Note that the cylinder only rotates 180 deg for one full sinusoidal force cycle. The amplitude of the exciting force depends upon the dynamic pressure and the amount of slot opening. Studying the performance property of the excitation system is a necessary procedure before designing a satisfactory airfoil/RSC excitation system for a particular flutter testing. Compared to flight testing [6–9] and wind-tunnel testing [10], the numerical simulation method is much more low cost and effective. The numerical simulation research of the excitation system is scarce in published literature. Cizmas et al. [11] conducted a numerical simulation to the experimental airfoil/RSC system tested in a wind tunnel with a coming flow speed of 20 m∕s. In the simulation, the flowwas simplified to be steady, two–dimensional, and incompressible. However, strictly speaking, the steady result was not able to reflect the real characteristic of the unsteady flowfield caused by the rotating cylinder. In the present Note, the complicated unsteady flow around the excitation system is simulated by solving unsteady Navier– Stokes equations using an in-house code, and the performance property of the system is investigated in detail.

Application of the multisegment numerical integration technique is extended to the free-vibration analysis of macroscopically anisotropic filament-wound branched shells of revolution with ring stiffeners, considering the variation of the thickness and winding angle. The solution procedure is based on a modified-frequency trial method, which processes on the numerically integrated transformed fundamental shell equations that are obtained in terms of finite exponential Fourier transform of the fundamental shell variables. The full macroscopically anisotropic form of the constitutive relations, including first-order transverse shear deformation and all components of translatory and rotary inertia, are included in the analysis. To handle branched shells of revolution, modifications that are necessary to incorporate junctions are added to the solution procedure. Inclusion of asymmetric circumferential stiffeners, with respect to the middle surface of the shell, into the semi-analytical solution method is demonstrated by presenting two alternative methods of analysis. The present solution methodology also incorporates the variation of the thickness and winding angle along the meridian of filament-wound shells of revolution, with general meridional curvature, by assuming placement of filaments along the geodesic fiber path on the surface of the shell of revolution.

A bifilar (two-wire) pendulum is a torsional pendulum consisting of a test object suspended by two thin parallel wires. The pendulum oscillates about the vertical axis. The restoring torque of the bifilar pendulum is provided by the gravitational force as rotations from the rest state cause the test object to raise slightly. The mass moment of inertia is computed using dynamic modeling, measurements of the oscillation period, and the physical dimensions of the bifilar pendulum such as the length and separation displacement of the pendulum wires. A simulation technique is described that improves estimates of the mass moment of inertia by considering the nonlinear effects of damping and large angular displacements. An analysis of the error variance of mass moment of inertia measurements is also described. The resulting expression for the error variance is used to optimize the physical parameters of the bifilar pendulum to obtain the moment of inertia measurement with the minimum error variance. Monte Carlo simulations were used to validate the parameter optimization technique. Experimental results are presented for a uniform-density test object for which the moment of inertia is straightforward to compute from geometric considerations. Results are also presented for a small unmanned air vehicle, which was the intended application for this moment of inertia measurement technique.

isotropic/orthotropic/functionally graded material spherical shells suddenly exposed to thermal environment is rather meager in the literature and such studies are important to the structural designers. In the present work, the nonlinear dynamic thermal buckling of functionally graded spherical caps is investigated using a threenoded shear flexible axisymmetric curved shell element based on field-consistencyprinciple[6].Geometricnonlinearityisassumedin thepresentstudy,usingvonKarman’sstrain-displacement relations. In addition, the formulation includes in-plane and rotary inertia effects. The material properties are graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents of the material. The nonlinear governing equations derived are solved employing Newmark’s numerical integration method in conjunction with the modified Newton– Raphson iteration scheme. The critical dynamic buckling temperature difference is taken as the temperature difference between the shell surfaces corresponding to a sudden jump in the maximum average displacement in the time history [2,12]. Numerical results arepresented that considerdifferent values ofgeometrical parameter and power-law index.

A p-version, hierarchical finite element for moderately thick composite laminated plates is presented, where the effects of the rotary inertia, transverse shear, and geometrical nonlinearity are taken into account. The time-domain free-vibration equations of motion are obtained by applying the principle of virtual work. Those equations are mapped into the frequency domain by the harmonic balance method and solved by a predictor-corrector procedure. The linear natural frequencies of vibration of several plates are determined, and the convergence properties of the element are investigated. It is shown that the element is not prone to shear locking and that a moderate number of degrees of freedom is sufficient for accuracy. The influences of the plate's thickness, of the width to length ratio, and of the fiber orientation on nonlinear free vibrations are investigated.

Coupled thermal ‐structural analysis, which includes the interaction between structural deformations and incident heating, is performed for spinning thin-walled composite beam appendages. The beam model is assumed to have transverse shear deformation and rotary inertia, as well as primary and secondary warping effects. Additionally, the thermal e utter problem is studied by the stability analysis of the beam. For the thermal analysis, the two-dimensional heat transfer in the axial and circumferential directions of a thin-walled beam is considered; the changes of the heating surface should be involved due to the spinning motion of the beam. The thermal and the structural analyses are based on the principle of conservation of energy and Hamiltons principle, respectively. Numerical results are compared with those of the uncoupled analysis. A steady-state thermal response can be determined by the uncoupled analysis, whereas thermal e utter may be studied by the coupled thermal ‐structural analysis. Also, the coupled analysis provides data on the stability characteristics, as well as dynamic responses. Furthermore, the spinning speed plays an important role in the stability of the spinning beam under solar heat e ux. When the spinning speed is equal to the bending frequency of the beam, then the structure is unconditionally unstable.

An approach to detect and image multiple damages in a platelike structure is presented. A structural health monitoring system (SHMS) with a linear array of actuators/sensors is proposed. The integrated actuators/sensors are used to activate/receive lowest-order antisymmetric A0 mode Lamb waves. A migration technique used in geophysical exploration and seismic prospecting is adopted to interpret the backscattering wave e eld and to image thee awsinthestructure.Theproposed approachismodeledbya two-dimensional explicite nitedifferencemethod both in simulating the ree ection waves and in implementing the prestack migration. An analytical solution based on Mindlin plate theory (Mindlin, R. D., “ Ine uence of Rotary Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates,” Journal of Applied Mechanics , Vol. 18, No. 1, 1951, pp. 31 ‐38) is derived to verify the accuracy of the numerical algorithm. An excitation-time imaging condition specie cally for the migration of waves in a plate is introduced based on a ray tracing concept, and prestack reverse-time migration is proceeded to propagate the ree ection energy back to the damages. The plate is imaged in terms of the velocity of transverse deformation after migration,thus, thelocations, dimensions, and seriousness of thee awscan bevisually displayed. Numerical results show that multiple damages can be successfully detected and the image of the damages correlate well with the target damages. The conclusion is that prestack migration can be a prospective technique in SHMS applications.

The dynamic stability of a completely free isotropic circular cylindrical shell under a follower force is investigated. The first-order shear deformation and rotary inertia are included in the kinetic energy and the strain energy. A finite element model of the shell is formulated using the trigonometric ring element in the circumferential direction and the Lagrangian element in the longitudinal direction. Using the eigenvalue curves and the method of multiple scales, the dynamic stability is studied for the case of the follower force with a pulsating part as well as the constant follower force. The effect of the thickness ratio and the length ratio is presented in both cases. The numerical results for the shells are compared with those of a beam model having equivalent dimensions. The numerical results show that the shell under a constant follower force behaves like a beam in a certain range. The analysis of the shell under a pulsating follower force shows that the beamlike modes of the shell have larger unstable regions than the other modes. However, for the case of a pulsating load, the modes of higher circumferential wave number are of importance in that the combination resonance of difference type takes place frequently.

The coupled thermoelasticity of shells of revolution, based on second-order shell theory, is considered, and the governing equations including normal stress and strain as well as the transverse shear and rotary inertia are considered. The coupled energy equation based on the assumption of Lord and Shulman (Lord, H. W., and Shulman, Y., A Generalized Dynamical Theory of Thermoelasticity,' Journal of Mechanics and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309) is further considered, and the total system of equations is solved by means of Galerkin finite element method. It is concluded that the inclusion of normal stress in the coupled equation is significant and for thin shells can result in a noticeable difference in shell response compared to unassumed conditions.

This study presents the dynamic equations of a stiffened composite laminated conical thin shell under the influence of initial stresses. The governing equations of a truncated conical shell are based on the Donnell-Mushtari theory of thin shells including the transverse shear deformation and rotary inertia. The extension-bending coupling is considered in the derivation. The composite laminated conical shell is also reinforced at uniform intervals by elastic rings and/or stringers. The stiffening elements are relatively closely spaced, and therefore the stiffeners are smeared out along the conical shell. The inhomogeneity of material properties because of temperature, moisture, or manufacturing processes is taken into account in the constitutive equations. A generalized variational theorem is derived so as to describe the complete set of the fundamental equations of the conical shell. Next, the uniqueness is examined in solutions of the dynamic equations of the conical shell, and the boundary and initial conditions are shown to be sufficient for the uniqueness in solutions. The equations of the laminated composite conical shell are solved by the use of the finite difference method as an illustrative example. The accuracy of results is tested by certain earlier results, and a good agreement is found.

An analytical model for the impact response of laminated composite plates is presented. The governing equations, which apply to small deflection elastic response of specially orthotropic laminates, include the combined effects of shear deformation, rotary inertia, and the nonlinear Hertzian contact law. For simply supported boundary conditions, a Fourier series solution is presented that, in contrast to previously published work, retains the frequencies associated with rotary inertia effects throughout the analysis. Errors that are incurred in the analysis of impact events, where the contact force history is obtained as part of the solution process, are investigated and guidelines to achieve converged solutions are recommended. For a benchmark impact problem, the present solution converges more rapidly than other analytical solutions available in the literature. Present analytical predictions are also found to agree well with the experimental results for composite fiber-reinforced plastic plates impacted by instrumented projectiles launched from both gas-gun and drop-weight test setups. The efficiency and robustness of the model in handling the complexities of the impact response of composite plates is further demonstrated by comparing the analytical predictions of contact force and fiber strain histories with those generated using detailed finite element analyses. The good agreement obtained instills confidence in using the model as a foundation for predictions of impact damage and response to penetrating impact events.

The exact element method was applied to calculate the natural frequencies, the buckling loads, and the influence of the axial load on the natural frequencies and mode shapes of nonsymmetric laminated composite beams. The theoretical model is using a first-order shear deformation theory and includes the effects of rotary inertia, shear deformation, and coupling between the longitudinal and transverse displacements. A parametric study was performed to investigate the influence of boundary conditions, materials and layup sequence on the buckling loads and natural frequencies of rectangular, cross-ply laminated composite beams.

Consider a sandwich plate with anisotropic composite laminated faces and an ideally orthotropic honeycomb core. In this paper, a one-dimensional model considering the transverse shear effect and rotary inertia for the free vibration analysis of a sandwich plate with an across-the-width delamination located at the interface between the upper face and core is developed. With this model, the natural frequencies and mode shapes of the delaminated composite sandwich beams can be obtained by solving the eigenvalues and eigenvectors of 12 simultaneous homogeneous algebraic equations. Because there are no such general solutions presented in the literature, verification is done by some special cases such as delaminated composite beams (without core) and perfect composite sandwich beams (without delamination). Based upon this general solution, the effects of faces, core, and delamination on free vibration behavior of composite sandwiches are studied thoroughly.

The effects of transverse shear deformations and structural damping on the flutter phenomena of a composite rotor blade in hover have been investigated using the finite element method. First-order shear deformation theory with rotary inertia effects and a damped element model of composite laminates are employed for the structural formulation. A quasisteady aerodynamic theory with a dynamic inflow model is used. Torsion-related out-of-plane warping and noncirculatory aerodynamic components are also incorporated in the formulation. Using these structural and aerodynamic tools, several numerical studies are carried out, first to validate the current approach and second to show the effects of transverse shear deformations and structural damping on the aeroelastic stability of a composite rotor as a function of fiber orientation. The predictions derived by the frequency analysis of this model are found to be more accurate than those given by an alternative approach compared with experimental data. It is shown that the transverse shear flexibility tends to lower the frequency of the rotor, and generally has a destabilizing effect on the lag mode and a stabilizing effect on the flap mode. It is also presented that the magnitude of structural damping can be controlled by changing ply orientation angle. CT cd c, EA

This paper deals with the attitude motion of a dual-spin satellite with a finite sized rigid body attached to the end of a flexible beam. The equations of motion are derived using Lagrange's equations and are solved using the perturbation technique known as the Krylov-Bogoliubov-Mitropolsky method. The special case of torsional flexibility is presented in its entirety. The relationships between the satellite, beam and tip mass parameters, and pointing accuracy of the satellite are examined. KBM method instead of generalized averaging since the equa- tions of motion obtained for the satellite are not in periodic standard form. By deriving an approximate analytical solution of the non- linear equations of motion, the work of this paper allows examination of the effects of system parameters on the motion of the spacecraft in question. Although resonances are easily obtained from a simple examination of the fundamental fre- quencies of the problem, this work allows the examination of behavior away from resonance. This provides the opportunity to look for local minima of the perturbation from rigid-body behavior as a function of the system parameters and hence allows for a more optimally designed spacecraft. In addition, control of the spacecraft could be more efficiently imple- mented if a general idea of the form of the motion is available. With numerically computed results, it is not possible to exam- ine the effects of individual parameters without many repeated evaluations of the motion. The approach in this paper provides full-field parametric relationships for the examination of all system parameters. Also of note is the fact that the effects of finite size tip masses are investigated. Rotary inertia of the tip mass intro- duces a set of equations of motion for the tip mass that are coupled to the equations of motion for the system through the dynamics of the elastic beam. Very little has been done con- cerning analytical examination of such a tip mass. These re- sults will provide a new tool for the design and control of spacecraft of this type, as well as introducing the KBM ap- proach to the body of literature relating to this topic.

Free transverse vibration of a sandwich plate strip having a core of linearly varying thickness and symmetrical faces of constant thickness is considered. The materials of the core and the faces are assumed to be linearly elastic and isotropic. The effects of transverse shear deformation and rotary inertia are included in the core as well as in the faces. The equations of motion derived by Hamilton's energy principle are solved by the Chebyshev collocation method. Frequencies and mode shapes for the first four normal modes of vibration are plotted in figures by varying the taper angle, thickness of the faces, initial thickness of the core, and ratio of the Young's moduli of the faces and the core. Frequencies are also plotted by varying the taper angle and ratio of the densities of the faces to the core in such a way that the total weight of the plate remains constant. N recent years, sandwich constructions have gained a well- established position due to their applications in aerospace and marine industries. Sandwich constructions are lighter in weight and have very high rigidity. The structural elements of sandwich construction may be of varying thickness for further reduction in weight or due to some specific design require- ments. Sandwich plates of varying thickness are being used for trailing edges of airplane lifting surfaces. Habip,1 Plantema,2 and Bert3 have given reviews of the work on sandwich constructions available in literature until 1979. Almost all of the work available to date on sandwich structural elements is confined to elements of constant thick- ness. A few papers available after 1979 on sandwich beams and plates of varying thickness are as follows: Gupta and Jain4 and Paydar and Adams5 have considered the vibration of annular sandwich plates of linearly varying thickness by tak- ing faces as membranes. Taber and Viano6 have compared analytical and experimental results for free vibration of nonuniform composite beams. Paydar and Libove7'8 and Pay- dar9 have considered the static behavior of sandwich plates of varying thickness. Ko10 has analyzed free vibration of rotating sandwich tapered beams. Kar and Sujata11 have considered the dynamic stability of tapered symmetric sandwich beams. In the present paper, free transverse vibration of a sandwich plate strip of infinite length, finite breadth, and linearly vary- ing thickness along the breadth is considered. The faces of the plate are taken of constant thickness. They are taken identical in material and dimensions and equally inclined to the middle plane of the plate. The core of the plate is taken of linearly varying thickness. Apart from flexural rigidity, transverse shear and rotatory inertia of the core as well as of the faces are included in the analysis. No restriction is imposed on magni- tudes of the ratios between the thickness, material densities, and elastic constants of the core and the faces except that they are isotropic and linearly elastic. The equations of motion and edge conditions are derived by Hamilton's energy principle. They are solved by the Chebyshev collocation method. Fre- quencies for the first four normal modes of vibration are computed for clamped and cantilever plates for various values of the taper angle, thicknesses of the faces and the core, and the ratio of the Young's moduli of the faces and the core. Frequencies are also computed by varying the taper angle and ratio of the densities of the faces and the core in such a way that the total weight of the plate remains constant. Normal- ized transverse deflections are also computed for the first four normal modes of vibration. To check the validity of the theory and the numerical method, frequencies are compared with monocoque plates of linearly varying thickness and analytic solution of a sandwich plate of constant thickness. A close agreement is found. Formulation of the Problem A tapered sandwich plate of infinite length and finite breadth a is considered. The core of the plate is taken of thickness 2h\, varying linearly along the breadth. Each of the two faces of the plate are taken of constant thickness /?/. The plate is referred to rectangular coordinates by taking the long sides of the plate in the planes x = 0 and a, and extending to infinity in both of the directions of y axis. The middle plane, lower and upper interfaces, and bottom and top of the plate are taken as z = 0, =F h, and =F (hi + h2), respectively, where a is the taper angle of the core thickness and /*2 = /z/sec a. A section of the plate along the zx plane is shown in Fig. 1. Both of the faces of the plate are taken of the same material and dimensions but different from that of the core. The various quantities for the core, the lower face, and the upper face will be distinguished by the subscripts 1,2, and 3, respectively. The displacement components v/, w/, and w/, / = 1, 2, 3, in the directions of y, z, and x axes, respectively, are approxi- mated according to Yu 12 as

A modeling technique capable of determining the time response of a single body (rigid or flexible) that is, in general, undergoing large elastic deformations, coupled with large, nonsteady translational and rotational motions, is presented. The derivations of the governing equations of motion are based on Lagrange's form of d'Alembert's principle. The general dynamical equations of motion are expressed in terms of stress and strain tensors, kinematic variables, the velocity and angular velocity coefficients, and generalized forces. The formulation of these equations is discussed in detail. Numerical simulations that involve finite elastic deformations coupled with large, nonsteady rotational motions are presented for a beam attached to a rotating base. Effects such as centrifugal stiffening and softening, membrane strain effect, and vibrations induced by Coriolis forces are accommodated. The effects of rotary inertia as well as shear deformation are also included in the equations of motion. Although discussions here are restricted to a single body, the formulation allows the capability of a general dynamical formalism for handling multibody (rigid or flexible) dynamics.

IV. Discussion and Conclusions It is concluded that the theoretical model as described herein predicts frequencies and mode shapes that are in good agreement with experiment. It has been observed that accurate prediction of higher mode frequencies, for plates of the type discussed here, often requires inclusion of the effects of rotary inertia of attached masses in the analysis. This is not surprising as the masses will undergo harmonic rotation for some of the higher modes. Extension of the analysis to include these rotary inertia effects is currently underway. To the authors' knowledge, the work reported here represents the first comprehensive analytical and experimental study of this interesting and timely problem. The analysis provides the designer with a powerful tool for optimizing the distribution of rigid point supports and attached masses on the plate surface. The experimental data will provide other researchers with reference points against which they can compare their theoretical results.

Nomenclature A; B;C = deŽ ned in Eqs. (8), (9), and (10) a; b = undetermined coefŽ cients in Eq. (6) E = Young’s modulus, N/m2 I = area moment of inertia, m J = concentrated rotary inertia, kg ¢m2 k = stiffness of the rotational spring, N/rad L = length of the column, m M = concentratedmass, kg m = mass per unit length of the column, kg/m W = lateral displacement,m w = nondimensional lateral displacement ®1i®3 = undetermined coefŽ cients in Eq. (2) ° = nondimensional rotational spring stiffness parameter, kL =EI f = nondimensional frequency parameter,m!L=EI J = nondimensional concentrated rotary inertia parameter, J=mL M = nondimensional concentratedmass parameter,M=mL » = nondimensional axial coordinates, x=L ! = circular frequency, rad /s 0 = differentiationwith respect to »

A comprehensive computer algorithm used in the dynamic analysis of multibody systems is presented. The procedures developed combine Kane's equations, the strain energy, and modal analysis to describe the elastic bodies. The equations of motion are expressed in a form in which their coefficients are based on the partial velocity and partial angular velocity arrays and are easily coded for computers. The nonlinear geometric stiffness matrix is developed for three-dimensional beams that account for the couplings between the deformation components having significant effects in high-speed systems. The method presented in this paper is tailored for structures with variable cross-sectional beam elements such as spacecraft antennae, helicopter rotor blades, robot systems, and mechanisms. The effects of rotary inertia as well as shear deformation are automatically included into the equations of motion. A method for obtaining the shape function matrix consisting of assumed dynamic modes is also presented. In addition, the preceding formulations are used in a simulation of a space-based robotic manipulator, and the results are compared with those obtained by Kane et al. in Ref. 20.

Equations of motion for a transversely isotropic plate in a general state of nonuniform initial stress where the effects of transverse shear and rotary inertia are included are derived by two methods. The first method is to perturb the nonlinear equations of elasticity by an incremental deformation. The resulting equations are linearized and integrated through the thickness of the plate to obtain the plate equations. The second method is to derive nonlinear equations of motion for a thick plate variationally by Hamilton's principle. These equations are then perturbed and suitably linearized to obtain the same equations as were obtained by the first method. A reduced set of equations for a thin plate are also given. Finally, the thick plate equations are solved for a simply supported rectangular plate in a state of uniform compressive stress plus a uniform bending stress both acting in the same direction.

In this paper, nonlinear equations of motion are derived for a transversely isotropic thick circular plate in a general state of nonuniform initial stress. The equations include the effects of transverse shear and rotary inertia. The large-amplitude flexural vibrations of simply supported and clamped thick circular plates subjected to initial stress are also investigated. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. Nonlinear equations of motion are used to solve the vibration problems by the Galerkin method. In addition, the effects of various parameters on the nonlinear vibration frequencies are studied.

Forced motions of laminated composite plates are investigated using a finite element that accounts for the transverse shear strains, rotary inertia, and large rotations (in the von Karman sense). The present results when specialized for isotropic plates are found to be in good agreement with those available in the literature. Numerical results of the nonlinear analysis of composite plates are presented showing the effects of plate thickness, lamination scheme, boundary conditions, and loading on the deflections and stresses. The new results for composite plates should serve as bench marks for future investigations. mation are assumed to remain straight and normal to the midsurface after deformation (i.e., transverse shear strains are zero), has been used to calculate frequencies, static response, and dynamic response under applied loads. Recent studies in the analysis of plates have shown that the effect of the transverse shear strains on the static and dynamic response of plates is significant. For example, the natural frequencies of vibration predicted by the classical plate theory are 25% higher, for plate side-to-thickness ratio of 10, than those predicted by a shear deformation theory (SDT). In transient analysis of plates the classical plate theory predicts unrealistically large phase velocities in the plate for shorter wavelengths. The Timoshenko beam theory,3 which includes transverse shear and rotary inertia effects, has been extended to isotropic plates by Reissner 4'5 and Mindlin,6 and to laminated anisotropic plates by Yang et al.7 A generalization of the von Karman nonlinear plate theory for isotropic plates to include the effects of transverse shear and rotary inertia in the theory of orthotropic plates is due to Medwadawski,8 and that for anisotropic plates is due to Ebcioglu.9 With the increased application of advanced fiber composite material to jet engine fan or compressor blades, and in high performance aircraft, studies involving transient response of plates made of such materials are needed to assess the capability of these materials to withstand the forces of impact due to foreign objects (e.g., the ingestion of stones, nuts and bolts, hailstones, or birds in jet engines). Previous in- vestigations into the linear transient analysis of composite plates include Moon's10'11 investigation of the response of infinite laminated plates subjected to transverse impact loads at the center of the plate; Chow's12 study of laminated plates (with transverse shear and rotary inertia) using the Laplace transform technique; the Wang et al. 13 investigation, by the method of characteristi cs, of unsymmetrical orthotropic laminated plates; and Sun and Whitney's14'15 study of plates under cylindrical bending. More recently, the present author16'17 investigated the linear transient response of layered anisotropic composite rectangular plates and presented extensive numerical results for center deflection and stresses.

The effects of mass redistribution on the rotational motion of a free body, initially rotating uniformly about its axis of maximum moment of inertia, are studied using a simple three-body model composed of two "control" points masses and a triaxial rigid body. An exact analytical solution for the motion of the original spin axis after redistribution of the masses is developed from the classical solution for the rotational motion of a free triaxial rigid body. Yhis motion in interpreted geometrically using the reciprocal inertia ellipsoid of the redistributed system and the angular momentum integral. Application of simple mass redistribution to the problem of spacecraft attitude control is discussed and results for a particular system are presented. T is a well-known fact1 that for a free, triaxial, rigid body, the condition of uniform rotation about its axis of in- termediate, principal moment of inertia is unstable. This in- stability manifests itself when small disturbances in the angular velocity components about the other principal axes occur; for then, the once-stationary axis of intermediate moment of inertia performs large-angle motions with respect to a nonrotating reference frame. Recently, Beachley and Dicker2 and Beachley,3 have suggested that this characteristic of rigid body rotational motion might be used to advantage to "control" a spinning spacecraft in the sense that predictable large-angle motions of an axis fixed in the spacecraft might be induced by changing the mass distribution of the spacecraft.% A control system based on this idea might be more economical than a mass expulsion system which would rotate the angular momentum vector of the spacecraft. Changes in the mass distribution of the spacecraft may be accomplished by internal motion of control masses. The ef- fects of such internal mass motion on the rotational motion of a free system of bodies have been studied by several authors and a review of some of the work in this area is given by Lorell and Lange,5 who propose an automatic mass-trim system for spinning spacecraft which utilizes movable control masses. The primary purpose of this paper is not to study the effects of internal mass motion on the rotational motion of a free system of bodies, although these effects cannot be ignored en- tirely and will be considered. Our main goal is to obtain an analytical description of the spacecraft's rotational motion af- ter a redistribution of mass has occurred. Such a description has previously been obtained only in a limited way by numerical integration.3 We shall also provide geometrical in- terpretation of the analytical results and consider the use of simple mass redistribution as a means for spacecraft attitude control. In this paper we shall consider a system similar to those discussed in Ref. 3 and employ some results from the classical theory for the rotational motion of a free, triaxial, rigid body.

Free vibration analysis of the neutrally buoyant inflated cantilevers, made of plastic sandwiched films, is presented, accounting for the added inertia and nonlinear hydrodynamic drag. The significant feature of the analysis is the reduction of the shell equations (the membrane, Fliigge's, and Herrmann-Armenakas') into a single equation which is similar in form to that for a vibrating beam with rotary inertia effects. The natural frequencies obtained are compared with the experimental results and those predicted by the Rayleigh-Ritz method in conjunction with the Washizu and membrane shell theories. The analyses show, and the experimental program confirms, that Fltigge's shell equation in reduced form is capable of predicting free vibration behavior quite accurately. However, the reduction technique should be applied with care, since in several cases it leads to misleading results, e.g., in the case of the Herrmann-Armenakas theory, generally accepted to be one of the most elaborate.

The influence of different parameters on the impact behavior of laminated composite plates is considered analytically. A Rayleigh-Ritz energy method was used to spatially discretize the time-varying boundary value problem and a set of coupled, ordinary differential equations in time were obtained based on the discretized system Lagrangian. The effects of shearing deformation, bending-twisting coupling, and nonlinear contact behavior were included in the model. The resulting equations were integrated using the implicit Newmark beta method without the effects of rotary inertia. The results indicate that the effective mass of the plate is often an important effect in the response to impact events. In general, the influence of the constitutive behavior dominates for very low velocity impact, whereas the target mass properties become more important as the impactor velocity increases. This importance of mass clearly shows that impactor kinetic energy is not sufficient to characterize the impactor as the impactor mass is shown to have a large influence on the resulting dynamic behavior. In addition to these parameters, the effects of preload and material properties are considered and discussed.

A theoretical analysis is presented for the supersonic flutter of a simply-supported isotropic sandwich panel. The aerodynamic loading is described by a static approximation valid for a wide range of Mach number. Small deflection theory for sandwich plates which considers transverse shear deformations and rotary inertia in addition to bending is used. The purpose of this study is to demonstrate the effect of the rotary inertia parameter on the critical flutter speed of simply-suppor ted panels. Previous investigators have neglected this effect and results obtained are of an anomalous nature for certain plate parameters. The effect of such plate parameters as rotary inertia, transverse shear, and midplane stress resultants on the critical flutter speed are presented in graphical form.

HE classical method of separation of variables combined with the Mindlin-Goodman procedure1 is employed to analyze the dynamic response of an infinitely long simplysupported composite plate under a uniform dynamic pressure, P0, on the upper surface of the plate. Two cases of dynamic pressures are considered; 1) the magnitude of the uniform pressure remains unchanged, and 2) the magnitude of the uniform pressure increases linearly as a function of time and then remains constant. Numerical results for eight layer granite/epoxy and glass/expoxy composites with (0/0/0/ — 0) s and (#/ — 0/0/0)5 stacking sequences are evaluated. Dynamic response characterized in terms of normal displacement, bending stress, interlaminar shear stress, and interlaminar normal stress are evaluated and compared with the corresponding quantities obtained statically (i.e., a dynamic load factor is established). It is observed that the dynamic values are consistently twice the corresponding static values for Case 1, and varies from one to two times the corresponding static values for Case 2. Contents The equations of motion, including transverse shear deformation and rotary inertia, for an infinitely long laminated composite plate under cylindrical bending are given in Ref. 2. The solution to the governing equations in terms of separation of variables in conjunction with the Mindlin-Goodman procedure] is also presented in Ref. 2. In the present paper the interlaminar shear stress TXZ is determined by integrating the equation of motion (1) with respect to z using the condition Txz(±h/2,t) = 0 and the condition of continuity at the interfaces. Once rxz(k) is determined the interlaminar normal stress ozz (k) can also be determined by integrating the equations of motion

only transverse shear and in the analysis the flexural rigidities of the face sheets about their own midplanes have been included. Results for co/o>0 were again computed by omitting the effect of the rotary inertia from the governing differential equations. These results can be generated by redefining the four coefficients A5, A6, A8, and A9 in Eqs. (7) and then computing the roots of Eq. (14) again. For lower frequencies, the values obtained for o)/co0 by dropping the rotary inertia terms and the corresponding values shown in Fig. 2 are essentially the same. This conclusion is in agreement with the one for rectangular sandwich plates and for homogeneous circular plates. Rotary inertia terms would, however, play significant role for higher modes. Figure 3 shows modal shapes for first three modes for which the frequency distribution is given in Fig. 2.

[ Y] = column matrix of deflections [EI] = diagonal matrix of flexural rigidities of the beam at n — 1 equally spaced points [M] = diagonal matrix of mass per unit length at n — 1 equally spaced points h = L/n, the distance between two adjacent points co = circular frequency [B] = finite-difference operator for d/dx In the case of simply supported beam, y0 = vn = 0; and [B] is an (w — 1) X (n — 1) square matrix as follows