Showing papers in "Shock and Vibration in 1997"
TL;DR: In this paper, the generalized differential quadrature (GDQ) method has been applied in solutions to fluid dynamics and plate problems and has shown superb accuracy, efficiency, convenience, and great potential in solving differential equations.
Abstract: The analysis of cylindrical shells using an improved version of the differential quadrature method is presented. The generalized differential quadrature (GDQ) method has computa tional advantages over the existing differential quadrature method. The GDQ method has been applied in solutions to fluid dynamics and plate problems and has shown superb accuracy, efficiency, convenience, and great potential in solving differential equations. The present article attempts to apply the method to the solutions of cylindrical shell problems. To illustrate the implementation of the GDQ method, the frequencies and fundamental frequencies for simply supported-simply supported, clamped-clamped, and clamped-simply supported boundary conditions are determined. Results obtained are validated by comparing them with those in the literature. © 1997 John Wiley & Sons, Inc.
128 citations
TL;DR: In this paper, the application of coupled Lagrangian-Eulerian analysis to various types of underwater shock problems was investigated, with the verification and validation of this analysis approach in mind.
Abstract: The application of coupled Lagrangian—Eulerian analysis to various types of underwater shock problems was investigated, with the verification and validation of this analysis approach in mind. Analyses were conducted for a simple TNT detonation problem and for the classical problems of an infinite cylindrical shell and a spherical shell loaded by a plane acoustic step wave. The advantages, disadvantages, and limitations of this approach are identified and discussed.
52 citations
TL;DR: In this article, the pervasive shock response spectrum (SRS) and damage boundary methods for evaluating product fragility and designing external cushioning for shock protection are described in detail with references to the best available literature.
Abstract: The pervasive shock response spectrum (SRS) and damage boundary methods for evaluating product fragility and designing external cushioning for shock protection are described in detail with references to the best available literature. Underlying assumptions are carefully reviewed and the central message of the SRS is highlighted, particularly as it relates to standardized drop testing. Shortcomings of these methods are discussed, and the results are extended to apply to more general systems. Finally some general packaging and shock-mounting strategies are discussed in the context of protecting a fragile disk drive in a notebook computer, although the conclusions apply to other products as well. For example, exterior only cushioning (with low restitution to reduce subsequent impacts) will provide a slenderer form factor than the next best strategy: interior cushioning with a “dead” hard outer shell.
47 citations
TL;DR: Substructuring and component mode synthesis (CMS) is a very popular method of model reduction for large structural dynamics problems as mentioned in this paper. But it is not suitable for modeling large structural systems.
Abstract: Substructuring and component mode synthesis (CMS), is a very popular method of model reduction for large structural dynamics problems. Starting from the pioneering works on this technique in the early 1960s, many researchers have studied and used this technique in a variety of applications. Besides model reduction, CMS offers several other crucial advantages. The present work aims to provide a review of the available literature on this important technique.
43 citations
TL;DR: In this paper, it was shown that static deflection can be a more sensitive predictor of structural damage than frequency, and the reasons for this are illuminated within, and hinge on very fundamental issues about the very nature of structural response.
Abstract: Structural damage detection by nondestructive methods is highly desirable. Changes in modal parameters such as frequency, damping, and mode shape are particularly inviting. Evidence is presented here that reveals that static deflection can, in many cases, be a more sensitive predictor of structural damage than frequency. The reasons for this are illuminated within, and hinge on very fundamental issues about the very nature of structural response. Furthermore, static deflection measurements are often easier to make, with higher levels of accuracy than dynamic measurements. Comparisons are made between theoretical models and experimental results for simple structures, with extensions given to more complex structures.
38 citations
TL;DR: In this article, the authors used the instantaneous energy release model and the constant energy deposition rate (CER) model to measure the amount of data scatter for the peak overpressure.
Abstract: Gun blast data from a large variety of weapons are scaled and presented for both the instantaneous energy release and the constant energy deposition rate models. For both ideal explosion models, similar amounts of data scatter occur for the peak overpressure but the instantaneous energy release model correlated the impulse data significantly better, particularly for the region in front of the gun. Two parameters that characterize gun blast are used in conjunction with the ideal scaling models to improve the data correlation. The gun-emptying parameter works particularly well with the instantaneous energy release model to improve data correlation. In particular, the impulse, especially in the forward direction of the gun, is correlated significantly better using the instantaneous energy release model coupled with the use of the gun-emptying parameter. The use of the Mach disc location parameter improves the correlation only marginally. A predictive model is obtained from the modified instantaneous energy release correlation.
22 citations
TL;DR: In this paper, the authors present several methods for the generation of stationary realizations of sampled time histories with non-Gaussian distributions and introduce a new method which can be used to control the cross-spectral density matrix and the probability density functions (pdfs) of the multiple input problem.
Abstract: The paper reviews several methods for the generation of stationary realizations of sampled time histories with non-Gaussian distributions and introduces a new method which can be used to control the cross-spectral density matrix and the probability density functions (pdfs) of the multiple input problem. Discussed first are two methods for the specialized case of matching the auto (power) spectrum, the skewness, and kurtosis using generalized shot noise and using polynomial functions. It is then shown that the skewness and kurtosis can also be controlled by the phase of a complex frequency domain description of the random process. The general case of matching a target probability density function using a zero memory nonlinear (ZMNL) function is then covered. Next methods for generating vectors of random variables with a specified covariance matrix for a class of spherically invariant random vectors (SIRV) are discussed. Finally the general case of matching the cross-spectral density matrix of a vector of inputs with non-Gaussian marginal distributions is presented.
21 citations
TL;DR: In this paper, the free vibrations of the bottom plate of an otherwise rigid circular cylindrical tank filled with liquid are studied, considering only the bulging modes (when the amplitude of the plate displacement is predominant with respect to that of the free surface).
Abstract: In this article the free vibrations of the bottom plate of an otherwise rigid circular cylindrical tank filled with liquid are studied, considering only the bulging modes (when the amplitude of the plate displacement is predominant with respect to that of the free surface). The tank axis is vertical, thus the free liquid surface is orthogonal to the tank axis. The liquid is assumed to be inviscid, and the contribution of the free surface waves to the dynamic pressure on the free liquid surface is neglected. Wet and dry mode shapes of the plate are assumed to be the same, so that the natural frequencies are obtained by using the nondimensionalized added virtual mass incremental (NA VMI) factors and the modal properties of dry plates. This simplifies computations compared to other existing theoretical approaches. NAVMI factors express the nondimensionalized ratio between the reference kinetic energy of the liquid and that of the plate and have the advantage that, due to their nondimensional form, they can be computed once and for all. Numerical results for simply supported and clamped bottom plates, as well as for supported plates with an elastic moment edge constraint are given. For more accurate results, and to exceed the limits of the assumed modes approach, the Rayleigh-Ritz method is applied and results are compared to those obtained by using the NAVMI factors and other existing methods in the literature.
20 citations
TL;DR: In this article, fine element analyses of explosion gas bubbles were conducted at various stand-off distances from simple rigid and constant pressure surface boundaries, and results from these analyses serve to characterize the behavior of the bubbles for the charge type, charge weight, and hydrostatic pressure used in the analyses.
Abstract: Finite element analyses of explosion gas bubbles show that including the compressibility of the surrounding media leads to appreciable differences in key areas of the bubble’s behavior. In order to more fully understand the behavior of bubbles created by detonations near simple boundaries, analyses incorporating fluid compressibility were conducted at various stand-off distances from simple rigid and constant pressure surface boundaries. The results from these analyses serve to characterize the behavior of the bubbles for the charge type, charge weight, and hydrostatic pressure used in the analyses.
16 citations
TL;DR: In this paper, a homogeneous, isotropic, elastic bar of cylindrical shape with uniformly distributed spherical cavities under alternating uniform axial stress was considered and the analytical calculation of the dynamic characteristics of the porous structure yielded the damping factor of the bar and the material damping factors.
Abstract: When a material is subjected to an alternating stress field, there are temperature fluctuations throughout its volume due to the thermoelastic effect. The resulting irreversible heat conduction leads to entropy production that in turn is the cause of thermoelastic damping. An analytical investigation of the entropy produced during a vibration cycle due to the reciprocity of temperature rise and strain yielded the change of the material damping factor as a function of the porosity of the material. A homogeneous, isotropic, elastic bar of cylindrical shape is considered with uniformly distributed spherical cavities under alternating uniform axial stress. The analytical calculation of the dynamic characteristics of the porous structure yielded the damping factor of the bar and the material damping factor. Exsperimental results on porous metals are in good correlation with an analysis.
11 citations
TL;DR: In this paper, an analysis of the dynamic behaviour of an interferometric measurement system based on the counting technique with regard to the measurement errors due to deterministic and stochastic disturbing quantities is presented.
Abstract: A uniform description is given of a method of measurement using a Michelson interferometer for measuring the linear motion quantities acceleration, velocity and displacement, and a diffraction grating interferometer for measuring the circular motion quantities angular acceleration, angular velocity and rotation angle. The paper focusses on an analysis of the dynamic behaviour of an interferometric measurement system based on the counting technique with regard to the measurement errors due to deterministic and stochastic disturbing quantities. The error analysis and description presented are aimed at giving some rules, mathematical expressions and graphical presentations that have proved to be helpful in recognizing the errors in interferometric measurements of motion quantities, optimizing the measurement conditions (e.g., filter settings), obtaining corrections and estimating the uncertainty of measurement.
TL;DR: In this article, a fluid surface damping (FSD) technique for vibration suppression of beam-like structures is proposed, where two viscoelastic surface layers containing fluid-filled cavities are attached symmetrically to the opposite surfaces of the beam.
Abstract: A fluid surface damping (FSD) technique for vibration suppression of beamlikestructures is proposed. The technique is a modification of the surface layer damping method. Two viscoelastic surface layers containing fluid-filled cavities are attached symmetrically to the opposite surfaces of the beam. The cavities on one side are attached to the corresponding cavities on the other side via connection passages. As the beam vibrates, the fluid is pumped back and forth through the connecting passages. Therefore, in addition to the viscoelastic damping provided by the surface layers, the technique offers viscous damping due to the fluid flow through the passage. A mathematical model for the proposed technique is developed, normalized, and solved in the frequency domain to investigate the effect of various parameters on the vibration suppression of a cantilever beam. The steady-state frequency response for a base white-noise excitation is calculated at the beam's free tip and over a frequency range containing the first five resonant frequencies. The parameters investigated are the flow-through passage viscous resistance, the length and location of the layers, the hydraulic capacitance of the fluid-filled cavities, and inertia of the moving fluid (hydraulic inertance). Results indicate that the proposed technique has promising potential in the field of vibration suppression of beamlike structures. With two FSD elements, all peak vibration amplitudes can be well suppressed over the entire frequency spectrum studied.
TL;DR: In this article, the authors focus on the problem of guide-vane vibrations of reversible pump-turbines, especially in the pump mode, and present mathematical models for theoretical torsional vibration prediction formulated using linear and nonlinear differential equations.
Abstract: This article focuses on the problem of guide-vane vibrations of reversible pump–turbines, especially, in the pump mode. These vibrations are transmitted to the guide-vane shaft torque. The guide-vane vibrations are caused by the impeller exit flow, which has a turbulent and partly nondeterministic property. Experimentally determined flow velocities at the impeller exit are given. The mathematical models for theoretical torsional vibration prediction formulated using linear and nonlinear differential equations are presented. The results of theoretical calculations are compared with measurement results. The possibility of transferring the parameters from the model to the prototype is discussed.
TL;DR: In this article, two extensions of the force reconstruction method, the sum of weighted accelerations tech-nique (SWAT), are presented, which may be applied to structures whose mode shapes are not available.
Abstract: Two extensions of the force reconstruction method, the sum of weighted accelerations tech nique (SWAT), are presented in this article. SWAT requires the use of the structure's elastic mode shapes for reconstruction of the applied force. Although based on the same the ory, the two new techniques do not rely on mode shapes to reconstruct the applied force and may be applied to structures whose mode shapes are not available. One technique uses the measured force and acceleration responses with the rigid body mode shapes to calculate the scalar weighting vector, so the technique is called SWAT-CAL (SWAT using a calibrated force input). The second technique uses the free-decay time response of the structure with the rigid body mode shapes to calculate the scalar weighting vector and is called SWAT-TEEM (SWAT using time eliminated elastic modes). All three methods are used to reconstruct forces for a simple structure.
TL;DR: In this paper, a first-principles-based model is presented for calculating the hole diameter resulting from the normal hypervelocity impact of a spherical aluminum projectile on a thin aluminum plate.
Abstract: A first-principles-based model is presented for calculating the hole diameter resulting from the normal hypervelocity impact of a spherical aluminum projectile on a thin aluminum plate. One-dimensional shock theory is used to predict the creation and attenuation of Hugoniot pressures along the plate surface. Pressures are translated into the plate thickness by calculating intersecting positions of advancing shock fronts and centered-fan rarefaction waves. The radial position at which the shock pressure equals a predetermined value is defined to be the hole diameter. The model was calibrated by determining this critical value for aluminum-an-aluminum impacts using several hundred data points. A residuals analysis indicated some inherent problems with the model. Two empirical factors were added to account for thin plate and two-dimensional shock dissipation effects. The predictions of the adjusted model are shown to compare well with predictions of several empirical hole diameter models.
TL;DR: In this paper, a diagnostic method using the two-sided directional power spectra of complex-valued engine vibration signals is presented and tested with four-cylinder compression and spark ignition engines for the diagnosis of cylinder power faults.
Abstract: A diagnostic method, which uses the two-sided directional power spectra of complex-valued engine vibration signals, is presented and tested with four-cylinder compression and spark ignition engines for the diagnosis of cylinder power faults. As spectral estimators, the maximum likelihood and FFT methods are compared, and the multi-layer neural network is employed for pattern recognition. Experimental results show that the success rate for identifying the misfired cylinder is much higher with the use of two-sided directional power spectra than conventional one-sided power spectra.
TL;DR: In this paper, the synchronization of two eccentric rotors driven by hydraulic motors (TERDHM) in one vibrating system is studied, and the synchronous and stable rotating conditions of TERDHM are established.
Abstract: In this article the synchronization of two eccentric rotors driven by hydraulic motors (TERDHM) in one vibrating system is studied The differential equations of TERDHM motion are derived, and the synchronous and stable rotating conditions of TERDHM are established It is found from the analysis and simulation that the angular velocities of TERDHM act on each other and both vary in a small range approaching their mean value The synchronous rotation speed is equal to the average angular velocity of TERDHM, and the phase difference between them depends on the difference of the flow rates of the two hydraulic motors
TL;DR: In this paper, a single actuator placed at the beam center was found to be sufficient to suppress excessive vibration of a beam traversed by the moving load with a reasonable amount of control input.
Abstract: Optimal independent modal space control for vibration suppression of beam structures traversed by a moving concentrated force was examined. Two control methodologies, optimal linear quadratic tracking and an optimal linear quadratic regulator, were utilized, with the former approach taking into account the disturbance due to the moving load and the latter one simply ignoring that disturbance. One single actuator placed at the beam center was found to be sufficient to suppress excessive vibration of a beam traversed by the moving load with a reasonable amount of control input. This study shows that excessive vibration of the beam structure induced by the moving load can be more effectively suppressed using the tracking control approach than using the regulator design, even with ±50% variation of the moving load magnitude or speed from that designed for the tracking control system.
TL;DR: In this paper, a nonlinear three dimensional (3D) single rack model and a non-linear 3D whole pool multi-rack model are developed for the spent fuel storage racks of a nuclear power plant (NPP) to determine impacts and frictional motion responses when subjected to 3D exci- tations from the supporting building floor.
Abstract: A nonlinear three dimensional (3D) single rack model and a nonlinear 3D whole pool multi-rack model are developed for the spent fuel storage racks of a nuclear power plant (NPP) to determine impacts and frictional motion responses when subjected to 3D exci tations from the supporting building floor. The submerged free standing rack system and surrounding water are coupled due to hydrodynamic fluid-structure interaction (FSI) using potential theory. The models developed have features that allow consideration of geomet ric and material nonlinearities including (I) the impacts of fuel assemblies to rack cells, a rack to adjacent racks or pool walls, and rack support legs to the poolfloor; (2) the hydro dynamic coupling offuel assemblies with their storing racks, and of a rack with adjacent racks, pool walls, and the poolfloor; and (3) the dynamic motion behavior of rocking, twist ing, and frictional sliding of rack modules. Using these models 3D nonlinear time history dynamic analyses are performed per the U.S. Nuclear Regulatory Commission (USNRC) criteria. Since few such modeling, analyses, and results using both the 3D single and whole pool multiple rack models are available in the literature, this paper emphasizes descrip tion of modeling and analysis techniques using the SOLVIA general purpose nonlinear finite element code. Typical response results with different Coulomb friction coefficients are presented and discussed.
TL;DR: In this paper, a procedure for the analytical synthesis of optimal vibration isolation for a hand-arm system subjected to stochastic excitation is presented and some numerical examples illustrating the procedure have also been included.
Abstract: In this article a procedure is presented for the analytical synthesis of optimal vibration isolation for a hand-arm system subjected to stochastic excitation. A general approach is discussed for a selected vibration isolation criterion. The general procedure is illustrated by analytical examples for different hand-arm systems described by their driving-point impedances. The influence of particular forms of excitation and the structure of the vibroisolated hand-arm systems on the resultant vibration isolation is then discussed. Some numerical examples illustrating the procedure have also been included.
TL;DR: In this article, a numerical method is presented to investigate the dynamic response of uniform orthotropic beams subjected to an impact of a mass, and the effects of boundary conditions on the contact force, contact duration, stress distributions, and beam deflection are discussed.
Abstract: A numerical method is presented to investigate the dynamic response of uniform orthotropic beams subjected to an impact of a mass. Higher order shear deformation and rotary inertia are included in the analysis of the beams. The impactor and laminated composite beam are treated as a system. The nonlinear differential governing equations of motion are then derived based on the Lagrange principle and modified nonlinear contact law, and solved numerically. The solution procedure is applicable to arbitrary boundary conditions. Numerical results are compared with those available in the literature to demonstrate the validity of the method, and very good agreement is achieved. The effects of boundary conditions on the contact force, contact duration, stress distributions, and beam deflection are discussed.
TL;DR: In this paper, simple models representing a shaker and a test object are used to illustrate changes in test object response due to shaker dynamics and differences between the test and service environment.
Abstract: Simple models representing a shaker and a test object are used to illustrate changes in test object response due to shaker dynamics and differences between the test and service environment. The degree of coupling is quantified in terms of ratios of the natural frequencies and the masses. Regions of overstress can depend on reproducing absolute rather than relative motion in a test. Shaker tests reprodusing output spectra observed in service, when shakerlji.xture impedance is higher than the impedance in service, is shown to cause overtest at frequencies below natural frequencies of the service environment.
TL;DR: In this paper, an application of the dynamic condensation in modal testing is presented, which requires measurements of natural frequencies and a limited number of components of modal shapes only.
Abstract: This article addresses an application of the dynamic condensation in modal testing, which requires measurements of natural frequencies and a limited number of components of modal shapes only. The measured frequencies can be employed as the initial estimates in the iterated dynamic condensation to effectively find an exact solution for the original eigenvalue problem and determine the transformation matrix between the testing and nontesting coordinates. Complete modal shapes including the nontesting coordinates can then be estimated based on the limited measurements. Numerical results are presented to illustrate the feasibility and efficiency of the iterated dynamic condensation. A comparison with the experimental results of a multiple disk system is also provided. © 1997 John Wiley & Sons, Inc.
TL;DR: In this article, a method for thoroughly examining the dynamic characteristics of a tube under the influence of either the internal flow or the external cross flow is presented, where the tube is modeled as a thin cylindrical shell whose governing equations are derived from an energy method.
Abstract: This article presents a method for thoroughly examining the dynamic characteristics of a tube under the influence of either the internal flow or the external cross flow. The tube is modeled as a thin cylindrical shell whose governing equations are derived from an energy method. The effects due to internal flow are introduced into the system through initial stress. Galerkin’s method in conjunction with the method of multiple scales is employed for obtaining the stability of the tube vibration. According to the results, instability can occur under certain conditions of resonance. Regarding the effects of the external cross flow, a numerical approach is initially employed to interpolate the experimental data of the pressure distributions due to the flow. The dynamic characteristics of the tube under steady flows and flows with small time variation are then investigated. Stability of the solution is also discussed.
TL;DR: In this paper, a single strut with a single clevis-tang pinned joint was characterized using a force-state mapping technique, and the strut was subjected to axial dynamic loads.
Abstract: As part of a research effort to study the microgravity dynamics of a truss with pinned joints, a single strut with a single clevis-tang pinned joint was characterized. Experimental data was collected using a force-state mapping technique. The strut was subjected to axial dynamic loads and the response of the strut was measured. The force-state map aids visualization of the strut dynamics. Finite element modeling of the response was explored. An example is presented that uses a method of manual determination of the finite element model parameters. The finite element model results correspond well with the measured strut response. © 1997 John Wiley & Sons, 1nc.
TL;DR: In this paper, the angular velocity vector of each sensor was transformed to local link axes and integrated to obtain the incremental change in angular position for each time step using the angular position time histories, a transformation matrix between the reference frame of each link was calculated using an axis system defined for the joint.
Abstract: High speed rotary motion of complex joints were quantified with triaxial angular rate sensors. Angular rate sensors were mounted to rigid links on either side of a joint to measure angular velocities about three orthogonal sensor axes. After collecting the data, the angular velocity vector of each sensor was transformed to local link axes and integrated to obtain the incremental change in angular position for each time step. Using the angular position time histories, a transformation matrix between the reference frame of each link was calculated. Incremental Eulerian rotations from the transformation matrix were calculated using an axis system defined for the joint. Summation of the incremental Eulerian rotations produced the angular position of the joint in terms of the standard axes. This procedure is illustrated by applying it to joint motion of the ankle, the spine, and the neck of crash dummies during impact tests. The methodology exhibited an accuracy of less than 5% error, improved flexibility over photographic techniques, and the ability to examine 3-dimensional motion.
TL;DR: In this article, the existence of the nonlinear term is determined by means of the time varying characteristics of the estimated linear parameters using the linear model and the data from a time invariant nonlinear system.
Abstract: An on-line least squares algorithm has previously been successfully applied to linear vibration systems in order to identify time varying parameters. In this article the limitations of the approach and the factors affecting the identification are further examined. The existence of the nonlinear term is determined by means of the time varying characteristics of the estimated linear parameters using the linear model and the data from a time invariant nonlinear system. The identification of the time varying linear parameters is also examined in accordance with the linear model by using the data with nonlinear elements.
TL;DR: In this paper, statistical methodologies were employed for measuring and analyzing the explosively induced transient responses of a flat steel plate excited with shock, and the results indicated that these techniques aid in understanding the pyroshock phenomenon with respect to the effects and interrelationships of explosive-charge weight and location on the laser Doppler and contract accelerometer recording systems.
Abstract: Statistical methodologies were employed for measuring and analyzing the explosively induced transient responses of a flat steel plate excited with shock. The application of design of experiment methodology was made to structure and test a Taguchi L9(32) full factorial experimental matrix (which uses nine tests to study two factors, with each factor examined at three levels) in which a helium-neon laser Doppler vibrometer and two piezocrystal accelerometers were used to monitor explosively induced vibrations ranging from 10 to 105 Hz on a 96 × 48 × 0.25 in. flat steel plate. Resulting conclusions were drawn indicating how these techniques aid in understanding the pyroshock phenomenon with respect to the effects and interrelationships of explosive-charge weight and location on the laser Doppler and contract accelerometer recording systems.
TL;DR: In this paper, an in situ full-scale test is conducted to measure the dynamic response of a long cantilever wall that retains backfill soil, and the recorded modal parameters of this retaining wall exhibited significant similarity to those of a clamped cantilevers plate (rather than those of the cantileve beam or plane-strain analysis).
Abstract: An in situ full-scale test is conducted to measure the dynamic response of a long cantilever wall that retains backfill soil. The recorded modal parameters of this retaining wall exhibited significant similarity to those of a clamped cantilever plate (rather than those of a cantilever beam or plane-strain analysis). Such a three-dimensional (3-D) response pattern is not accounted for by current analysis procedures. A simple 3-D finite element model is employed to further analyze the observed resonant configurations. The results indicate that such configurations play an important role in the seismic response of wall backfill soil systems of variable height, such as wing walls supporting highway approach ramps.
TL;DR: In this article, an eigenvalue sensitivity formula for a combined beam structure due to the variation of its shape or joint coordinates is proposed based on the Lagrange multiplier technique for a free vibration equation of the structure.
Abstract: An eigenvalue sensitivity formula for a combined beam structure due to the variation of its shape or joint coordinates is proposed based on the Lagrange multiplier technique for a free vibration equation of the structure. The shape variation is decomposed into the length and the orientation variation of each beam to obtain individual effects on the total sensitiv ity. The sensitivity equations due to the length and the orientation variations are expressed as an energy density form and the cross product of joint forces and displacements respec tively. Several numerical examples are also presented to validate the proposedformulation and to show how to implement the idea.