Generalized Force Vector Research Articles

Simple formulations of imposing moments and evaluating joint reaction forces for rigid-flexible multibody systems

The Absolute-Coordinate-Based (ACB) method that combines the Natural Coordinate Formulation (NCF) describing rigid bodies and the Absolute Nodal Coordinate Formulation (ANCF) describing flexible bodies has been widely used to study the dynamics of rigid-flexible multibody system since it exhibits many good features, such as the constancy of the mass matrix of the derived dynamic equation, and the easy description and great simplification of the constraint conditions. In order to achieve these good features, both NCF and ANCF take the vectors, rather than rotational coordinates, to describe the rotation and deformation of the rigid-flexible bodies. In this study, the physical meaning of the components of the generalized force vector corresponding to the vector coordinates is revealed on the basis of both ANCF and NCF. Some new and simple formulations of imposing moments and evaluating joint reaction forces for rigid-flexible multibody system are proposed by making full use of the physical meaning of vector coordinates and Lagrange multipliers. All the proposed formulations are defined in the global frame so as to avoid the coordinate transformation. Hence, it can be directly applicable to various types of finite elements of ANCF, including the slope deficient elements. Finally, several typical and practical examples are used to verify the effectiveness of the proposed formulations.

Muscle discretization affects the loading transferred to bones in lower-limb musculoskeletal models

Modelling the mechanical effect of muscles is important in several research and clinical contexts. However, few studies have investigated the effect of different muscle discretizations from a mechanical standpoint. The present study evaluated the errors of a reduced discretization of the lower-limb muscles in reproducing the muscle loading transferred to bones. Skeletal geometries and a muscle data collection were derived from clinical images and dissection studies of two cadaver specimens. The guidelines of a general method previously proposed for a different anatomical district were followed. The data collection was used to calculate the mechanical effect of muscles, i.e. the generalized force vectors, and the errors between a large and a reduced discretization, in a reference skeletal pose and in the extreme poses of the range of motion of joints. The results showed that the errors committed using a reduced representation of muscles could be significant and higher than those reported for a different anatomical region. In particular, the calculated errors were found to be dependent on the individual anatomy and on the skeletal pose. Since different biomechanical applications may require different discretization levels, care is suggested in identifying the number of muscle lines of action to be used in musculoskeletal models.

Generalized force vectors for multi‐mode pushover analysis

AbstractA generalized pushover analysis (GPA) procedure is developed for estimating the inelastic seismic response of structures under earthquake ground excitations. The procedure comprises applying different generalized force vectors separately to the structure in an incremental form with increasing amplitude until a prescribed seismic demand is attained for each generalized force vector. A generalized force vector is expressed as a combination of modal forces, and simulates the instantaneous force distribution acting on the system when a given response parameter reaches its maximum value during dynamic response to a seismic excitation. While any response parameter can be selected arbitrarily, generalized force vectors in the presented study are derived for maximum interstory drift parameters. The maximum value of any other response parameter is then obtained from the envelope of GPAs results. Each nonlinear static analysis under a generalized force vector activates the entire multi‐degree of freedom effects simultaneously. Accordingly, inelastic actions develop in members with the contribution of all ‘instantaneous modes’ in the nonlinear response range. Target seismic demands for interstory drifts at the selected stories are calculated from the associated drift expressions. The implementation of the proposed GPA is simpler compared with nonlinear response history analysis, whereas it is less demanding in computational effort when compared with several multi‐mode adaptive nonlinear static procedures. Moreover, it does not suffer from the statistical combination of inelastic modal responses obtained separately. The results obtained from building frames have demonstrated that GPA is successful in estimating maximum member deformations and member forces with reference to the response history analysis. When the response is linear elastic, GPA and response spectrum analysis produce identical results. Copyright © 2010 John Wiley & Sons, Ltd.

An Analytical Complete Model of Tilting-Pad Journal Bearing Considering Pivot Stiffness and Damping

Abstract Considering the freedom of pad tilting and pad translation along preload orientation, an analytical complete model, as well as mathematical method, which contains 2n+2 degrees of freedom, is presented for calculating the dynamical characteristics of tilting-pad journal bearing. Based on the motion relationship of shaft and pad, the local coordinate system, the generalized displacement, and the generalized force vector are chosen. The concise transformation of generalized displacement, generalized force, and its Jacobian matrix between the local and global coordinate systems are built up in matrix form. A fast algorithm using the Newton–Raphson method for calculating the equilibrium position of journal and pads is proposed. The eight reduced stiffness and damping coefficients can be obtained assuming that the journal and all pads are subject to harmonic vibration. Numerical results show that the reduced damping coefficients and the threshold speed can be effectively enhanced by giving suitable pad pivot stiffness and damping simultaneously, and this analytical method can be applied to analyze dynamical behavior of the tilting-pad journal bearing rotor system.

Modelling of damage accumulation and failure of structural members subjected to strong seismic actions

Following the total Lagrangian approach, an incremental formulation for three-dimensional Timoshenko beam element taking into account large displacements and rotations is developed. For the failure analysis of reinforced concrete structural members, subjected to extreme loads, a new elastoplastic damage constitutive model is proposed on the level of cross-sectional variables. The model is based on the concept of the yield surface and associated flow rule. The effects of softening and strength deterioration are accounted for by the introduction of damage variables. To assure the objectivity of the numerical simulation a non-local treatment of damage variables is implemented. Comparison to different experimental results on biaxial cyclic tests is performed. Numerical results demonstrate that the proposed model effectively reproduces softening, strength deterioration, coupling between different components of the generalized force vector and other nonlinear effects accompanying the inelastic structural response under three-dimensional seismic loading.

Three-port impedance model of a piezoelectric bar element: Application to generation and damping of extensional waves

A straight bar element containing piezoelectric members is viewed as a linear system with one electrical and two mechanical ports where it can interact with external electrical and mechanical devices through voltage, current, forces and velocities. A generalized force vector, with one voltage and two forces as elements, is expressed as the product of an impedance matrix and a generalized velocity vector, with one current and two velocities, as elements. Due to symmetry and reciprocity, this matrix is defined by four of its nine elements. Two applications are considered for a piezoelectric bar element (PBE) that constitutes a part of a long elastic or viscoelastic bar, viz. generation and damping of extensional waves in the bar. In the first, the PBE is driven by a given input voltage or by the output voltage from a linear power amplifier. In the second, the PBE supplies an output voltage to an external load. In numerical simulations carried out for a specific laminated PBE, an elastic bar, a serial RL load and a bell-shaped incident wave, the highest fraction of wave energy dissipated was 8.1%. This is much less than the 50% achievable for a harmonic wave under condition of electrical impedance matching.

Separated-Form Equations of Motion of Controlled Flexible Multibody Systems

This paper introduces a method leading to separated-form formulation of dynamic equations of multibody systems subject to control. The algorithm is derived from the virtual work principle and includes the moving base effects. The elastic members are treated as Euler-Bernoulli beams. Equations of motion are expanded using generalized coordinate partitioning and a Jacobian matrix expansion. The formulation of each physical term is separated, i.e., the inertia matrix, nonlinear coupling vector, generalized force vector and base motion-induced terms are established individually. The formulation is implemented on a workstation using MAPLE. Nonlinear and linearized equations with control are generated in FORTRAN format. The control design adopts second-order models directly. Several examples including a spin-up cantilever beam, an elastic vehicle with active suspensions and an elastic slider-crank mechanism are given. Numerical results for nonlinear and linear spin-up beam models are provided. Simulation for the active vehicle model using second-order control theory is presented.

Vibration of Rotationally Periodic Structures

The purpose of this paper is to determine eigensolutions of a rotationally periodic structure P of period N in terms of those of the i th substructure S (i) of P. Let y (i) be the generalized displacement vector and f (i) be the generalized force vector at interface I (i) , where substructures S (i) and S (i - 1) are connected. Transfer function formulation of S (i) implies that there exists a linear operator G mapping y (t) and f (i) at interface I (i) to y (i + 1) and f (i + 1) at interface I (i + 1). In addition, the operator G depends on transfer functions (and therefore eigensolutions) of S (i) . The periodicity of P then requires that G N be an identity map. This results in N self-adjoint Fredholm integral equations, the non-trivial solutions of which predict eigensolutions of P. As a consequence, a periodic structure P with period N will have exactly N eigenvalues lying between two consecutive eigenvalues of the substructure S (i), if the interface I (i) contains only one degree of freedom. Finally, three examples are illustrated. The first example considers anti-plane strain vibration of a linear elastic solid containing four slots of infinitesimal width. The second example derives exact eigensolutions of a string-mass periodic structure. The third example illustrates how eigensolutions of an axisymmetric structure can be recovered under the formulation of rotationally periodic structures.

Parallel Processing for Real-Time Dynamic System Simulation

A parallel processing algorithm based on the recursive dynamics formulation presented in a companion paper [1] is developed for multiprocessor implementation. Lagrange multipliers associated with cut-joint constraints for closed loop systems are eliminated, resulting in a minimal set of equations of motion. Concurrent generation of the system inertia matrix and the generalized force vector using the algorithm of Ref. 1 is shown to yield finer grain parallelism than earlier recursive algorithms. A new computational structure for dynamic analysis is proposed for high speed parallel processing. Real-time simulation of a vehicle is demonstrated on an eight processor parallel computer to illustrate efficiency and effectiveness of the algorithm, even for interactive operator-in-the-loop simulation.

Sound radiation from building elements

The directivity patterns of typical building elements, a large concrete panel and a glazed window, are calculated. The two elements are treated as homogeneous plates. The Galerkin method is used to calculate the forced vibration patterns of these plates due to exposure to a uniform sound field. Rayleigh's far field approximation is then used to calculate the sound level due to the plate vibrations at distant points and a directivity pattern is thus built up. The effect on the radiation patterns of varying the loss factor of internal damping, the frequency of excitation and the boundary conditions is investigated. Whilst the effect of varying any of the parameters is to alter the detailed radiation pattern, since sinusoidal excitation was assumed in the calculation, the overall shapes of the directivity patterns remain constant. In order to investigate the effect of a reverberant sound field a trial function is introduced to modify the generalized force vector. Predicted directivity effects are then compared with measurements made on a 1:20 scale model.