Fatigue data obtained under biaxial loading conditions for adhesively bonded joints are used to plot S-N type diagrams to assess the effects of biaxiality in loading. Independently Loaded Mixed-Mode Specimens (ILM MS) are used for data collection purposes. These specimens are basically two (steel) beams bonded to be fatigue loaded under cantilever (opening) mode while a simultaneous but physically separate in-plane (static) shear load is also induced with the aid of a small hydraulic piston embedded in the specimen. Application of such static shear loads results in different S-N behavior for the bonded joint. The model adhesives used are Metlbond 1113-2 and Metlbond 1113 solid film thermosetting adhesives similar to those commonly used in aircraft and aerospace industries. The former is an elastomer-modified epoxy adhesive and the latter is identical except that it containes a synthetic earner cloth. Thus, the effects of carrier cloth in adhesive’s S-N behavior is also assessed. Analytically, the classical linear log-log representation of the adhesive S-N data is explored and modifications necessary to reflect the effects of biaxiality in loading and also the presence of a carrier cloth are assessed. The fatigue failure results are also compared with results obtained under monotonic biaxial loading conditions.

The partially covered, sandwich-type cantilever with concentrated mass at the free end is studied. The equations of motion for the system modeled via Euler beam theory are derived and the resonant frequency and loss factor of the system are analyzed. The variations of resonance frequency and system loss factor for different geometrical and physical parameters are also discussed. Variation of these two parameters are found to strongly depend on the geometrical and physical properties of the constraining layers and the mass ratio.

Structural identification and vibration control of flexible systems have drawn much attention in recent years. This article presents an analytical study on a distributed piezoelectric sensor and a distributed actuator coupled with a flexible plate. The integrated piezoelectric sensor/actuator can monitor the oscillation as well as actively control the structural vibration by the direct/converse piezoelectric effects, respectively. Based on Love assumptions, theories on distributed sensing and active vibration control of a plate using piezoelectric materials are derived. By employing the finite element technique, the integrated structure is further discretized. Applications to the dynamic measurement is demonstrated and the dynamic performance of a cantilever plate is also evaluated.

A new linearized two degree of freedom model of an industrial press feed mechanism, containing an RSSR linkage, a bent coupler, an overrunning sprag clutch, a feed strip and a brake, is presented. By introducing a double cantilever model of the coupler with an assumed quarter sine shape function, simplifying certain terms of secondary importance and replacing the non-linear clutch spring by a linear torsional spring with a deflection dependent stiffness, it was possible to develop a set of two linear differential equations for the all important feed stroke, which could be fully solved in an analytic manner for the dynamic responses of the coupler strain and the clutch windup angle.

In this paper, structural topology optimization is addressed through Genetic Algorithms. A set of designs is evolved following the Darwinian survival-of-fittest principle. The standard crossover and mutation operators are tailored for the needs of 2D topology optimization. The genetic algorithm based on these operators is experimented on plane stress problems of cantilever plates: the goal is to optimize the weight of the structure under displacement constraints. The main advantage of this approach is that it can both find out alternative optimal solutions, as experimentally demonstrated on a problem with multiple solutions, and handle different kinds of mechanical model: some results in elasticity with large displacements are presented. In that case, the nonlinear geometrical effects of the model lead to non viable solutions, unless some constraints are imposed on the stress field.

The dynamic stability of a non-uniform rectangular cantilever plate on a Pasternak foundation under the action of a pulsating inplane load is reported in this paper. The small deflection theory of the thin plate is used Hamilton’s principle is used to derive the time variables while the dynamic stability is solved by the harmonic balance method. Natural frequencies and buckling properties are presented at first. Then, regions of instability which contain simple parametric resonances and combination resonances are discussed for various parameters of a Pasternak foundation, non-uniform cross section and thermal gradient.

This paper presents 4-, 5-, 6- and 7-node isoparametric Timoshenko beam elements for modeling constant and varying cross-section thick and thin beams with various boundary conditions. Numerical integration is employed to determine the mass and stiffness matrices to facilitate modeling of varying cross-section beams. The accuracy of each proposed element is illustrated by determining the natural frequencies of: Thick and thin beams with constant and varying cross-section; linearly tapered cantilever circular tube with small wall thickness; chimney structure; and of a four-bar mechanism. In all examples, the results obtained using the proposed elements are compared, whenever available, with exact solution and with solutions determined when other elements available in the literature are used. The proposed elements did not cause shear locking when tested on thin beams of aspect ratio 500.

In the paper the natural frequencies and modal loss factors of a three-layered rectangular cantilever plate with high-modulus composite face layers and a viscoelastic mid-layer are discussed. Two analysis approaches are considered, the Ritz method and the finite element method. The Ritz analysis is based on an appropriate 2-D plate model. Algebraic polynomials are used which satisfy the necessary geometric boundary conditions. The discretized eigenvalue problem is then solved for complex eigenvalues, from which damped frequencies and modal loss factors are extracted. Both frequencies and loss factors are shown to converge with an increasing number of the polynomials used. Apart from the Ritz method, frequencies and modal loss factors are calculated by the finite element method based on the so-called strain energy method. Comparison between the two approaches is then given.

Beams formed by long fiber composite materials have certain internal damping torque. A mathematical model for the displacement of this type of beams in cantilever configuration is the following initial-boundary value problem of an integro-differential equation [1, 14]: (1) ρ ( x ) w t t ( x , t ) − 2 ( ∫ 0 L h ( x , y ) [ w t x ( x , t ) − w t x ( y , t ) ] d y ) x + ( E I w x x ( x , t ) ) x x = f ( x , t ) , (2) w ( 0 , t ) = 0 , w x ( 0 , t ) = 0 , (3) w x x ( L , t ) = b l 1 ( t ) , (4) − ( E I w x x ( x , t ) ) x | x = L + 2 ∫ 0 L h ( L , y ) [ w t x ( L , t ) − w t x ( y , t ) ] d y = b l 2 ( t ) , (5) w ( x , 0 ) = w 0 ( x ) , w t ( x , 0 ) = w 1 ( x ) , where L is length of the beam, w ( x , t ) is the transverse displacement of the beam at time t and position x , ρ( x ) is the mass density, EI is the stiffness parameter. The interaction integral kernel h ( x , ξ) is introduced in this model by considering a restoring torque which comes from spatially variable bending of the beam. This kernel h ( x , ξ) depends on the material properties of the beam. Choosing a different material (different h ( x , ξ)) can realize a different damping effect for the beam.

The sensitivity method is applied to update finite element models of welded and bolted joints, and the boundary condition of a cantilever plate. Careful parameterisation is found to be critical in updating the joints and boundary conditions. Updating geometric parameters has considerable potential in updating. The use of nodal offset dimensions results in an updated model of the welded joint with physical interpretation. Similarly the ‘rigid’ boundary in a cantilever plate is successfully updated using the effective length of the elements closest to the joint. In all cases an improvement on the analytical natural frequencies is demonstrated.

We experimentally investigated nonlinear combination resonances in a graphite-epoxy cantilever plate having the configuration (–75/75/75/ – 75/75/ – 75) s . As a first step, we compared the natural frequencies and mode shapes obtained from the finite-element and experimental modal analyses. The largest difference in the obtained frequencies was 2.6%. Then, we transversely excited the plate and obtained force-response and frequency-response curves, which were used to characterize the plate dynamics. We acquired time-domain data for specific input conditions using an A/D card and used them to generate time traces, power spectra, pseudo-state portraits, and Poincaré maps. The data were obtained with an accelerometer monitoring the excitation and a laser vibrometer monitoring the plate response. We observed the external combination resonance Ω ≈ 1 2 ( ω 2 + ω 5 ) and the internal combination resonance Ω ≈ ω 8 ≈ 1 2 ( ω 2 + ω 13 ) , where the ω i are the natural frequencies of the plate and Ω is the excitation frequency. The results show that a low-amplitude high-frequency excitation can produce a high-amplitude low-frequency motion.

Cantilever hook features are commonly molded into plastic parts as a inexpensive method of assembling several parts without separate fasteners. Insertion force, insertion dynamic strain, and retention force represent the critical performance data needed to design a cantilever hook for given loading conditions. This paper explores the performance of this feature under both insertion and retention using numerical and experimental methods. A finite element model using contact and friction surface elements was used to simulate the actual insertion and retention processes of hooks. The design space for a hook was explored by using a design of experiments approach. Sensitivity information was obtained by tabulating main effects and interactions. For the goal of minimizing insertion force and maximizing retention strength, a balanced design was found which was sensitive to differing factors. Based on this data, generalized design rules for designing hooks were formulated.

Traditional snap-fit design methods focus exclusively on design of an individual locking feature such as a cantilever hook or an annular lock. Location and orientation of other significant features on the two joined parts are not considered. This paper presents: (1) a comprehensive design methodology for nesting plastic parts via snap-fits and, (2) the concept of a part nesting table to encourage good design practices. This nesting approach is based on relatively new methodologies and guidelines for arranging features on a plastic part. The authors advocate a nesting which is a statically determinate assembly and minimizes the use of snap-fit features. The advantages of such an assembly include robustness with respect to tolerance and warpage concerns, maximum utilization of existing “natural” part features, and a reduction in the number of locking features needed. The entire process is presented as an improved paradigm for attachment design and assembly. It presents the concept of a part nesting table to help designers produce nested plastic assemblies. In this approach, degrees-of-motion to be removed are represented as columns in a table, and individual features are entered as rows. Advantages of this structured approach include recognition of under constrained and weakly constrained assemblies, identification of over constrained assemblies, and first-level optimization of lock location and assembly direction. Work is continuing on computer implementation of this approach so that force and moment values can be deduced at the feature level.

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its “exural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to the two integro-partial-differential equations of Crespo da Silva and Glynn. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. The modulation equations exhibit the symmetry property found by Feng and Leal by analytically manipulating the interaction coefficients in the modulation equations obtained by Nayfeh and Pai by applying the method of multiple scales to the governing integro-partial-differential equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.

In this paper, a general method of regular perturbation for linear eigenvalue problems is presented, in which the orders of perturbation terms are extended to infinity. The method of regular perturbation is applied to study vibration mode localization in randomly disordered weakly coupled two-dimensional cantilever-spring arrays. Localization factors, which characterize the average exponential rates of decay or growth of the amplitudes of vibration, are defined in terms of the angles of orientation. First-order approximate results of the localization factors are obtained using a combined analytical-numerical approach. For the systems under consideration, the direction in which vibration is originated corresponds to the smallest localization factor; whereas the “diagonal” directions correspond to the largest rate of decay or growth of the amplitudes of vibration. When plotted in the logarithmic scale, the vibration modes are of a hill shape with the amplitudes of vibration decaying linearly away from the cantilever at which vibration is originated.

A method is proposed for probabilistically model updating an initial deterministic finite element model using measured statistical changes in natural frequencies and mode shapes (i.e., modal parameters). The approach accounts for variations in the modal properties of a structure (due to experimental errors in the test procedure). A perturbation of the eigenvalue problem is performed to yield the relationship between the changes in eigenvalues and in the global stiffness matrix. This stiffness change is represented as a sum over every structural member by a product of a stiffness reduction factor and a stiffness submatrix. Monte Carlo simulations, in conjunction with the variations of the structural modal parameters, are used to determine the variations of the stiffness reduction factors. These values will subsequently be used to estimate statistics for the corrected stiffness parameters. The effectiveness of the proposed technique is illustrated using simulated data on an aluminum cantilever Euler-Bernoulli beam.

In this paper, a technique is presented for improving the efficiency of the Craig-Bampton method of Component Mode Synthesis (CMS). An eigenanalysis is performed on the partitions of the CMS mass and stiffness matrices that correspond to the so-called constraint modes. The resultant eigenvectors are referred to as “characteristic constraint modes,” since they represent the characteristic motion of the interface between the component structures. By truncating the characteristic constraint modes, a CMS model with a highly-reduced number of degrees of freedom may be obtained. An example of a cantilever plate is considered. It is shown that relatively few characteristic constraint modes are needed to yield accurate approximations of the lower natural frequencies. This method also provides physical insight into the mechanisms of vibration transmission in complex structures.

A Dynamic Finite Element (DFE) for vibrational analysis of rotating assemblages composed of beams is presented in which the complexity of the acceleration, due to the presence of gyroscopic, or Coriolis forces, is taken into consideration. The dynamic trigonometric shape functions of uncoupled bending and axial vibrations of an axially loaded uniform beam element are derived in an exact sense. Then, exploiting the Principle of Virtual Work together with the nodal approximations of variables, based on these dynamic shape functions, leads to a single frequency dependent stiffness matrix which is Hermitian and represents both mass and stiffness properties. A Wittrick-Williams algorithm, based on a Sturm sequence root counting technique, is then used as the solution method. The application of the theory is demonstrated by two illustrative examples of vertical and radial beams where the influence of Coriolis forces on natural frequencies of the clamped-free rotating beams is demonstrated by numerical results.

The 3-d dynamics of a cantilever beam undergoing large displacements under a sinusoidally varying, concentrated, vertical force at its free end is analyzed. The pde’s of the motion are obtained by using the Hamilton principle and then a reduced 3 degree-of-freedom model is obtained using in a Galerkin discretization, three eigenfunctions of the linearized model. A path following procedure is used to describe the global dynamic behavior in the excitation control parameter plane. The results obtained using the simple 3 d.o.f. analytical model are then compared with those of an experimental steel model of the cantilever. The regions of instability of the unimodal planar solution in which the nonlinear modal coupling excites the torsional component are studied; the planar motion usually looses stability via Hopf bifurcations after which quasi-periodic and/or chaotic motions are found. Inside the regions in which the system shows a complex time-evolution the complexity-level of the response is analyzed using, for the experimental model, a time series analysis tool.

This article introduces a new design for a cantilever type snap-fit fastener. This new design is configured to provide a positive interference between the mating parts even in the assembled position. The legs are therefore not in a relaxed position at the end of assembly. This results in higher levels of alignment quality and part fit in a completed assembly. The positive interference is achieved by the use of angular legs as opposed to orthogonal legs in a traditional snap-fit. Issues of material strength, fastener geometry, part stiffness, attachment strategy and manufacturability are addressed in the paper.