Rigid Body Research Articles

Internal dissipation in the Dzhanibekov effect

The Dzhanibekov effect is the phenomenon by which triaxial objects like a spinning wing bolt may continuously flip their rotational axis when initially spinning around the intermediate axis of inertia. This effect is closely related to the Tennis Racket theorem that establishes that the intermediate axis of inertia is unstable. Over time, however, dissipation ensures that a torque free spinning body will eventually rotate around its major axis, in a process called precession relaxation, which counteracts the Dzhanibekov effect. Euler’s equations for a rigid body effectively describe the Dzhanibekov effect, but cannot account for the precession relaxation effect. A dissipative generalization of Euler’s equations displays two dissipative mechanisms: orientational diffusion and viscoelasticity. Here we show through numerical simulations of the dissipative Euler’s equations that orientational diffusion, rather than viscoelasticity, primarily drives precession relaxation and effectively suppresses the Dzhanibekov effect.

Crystal Polymorph Search in the NPT Ensemble via a Deposition/Sublimation Alchemical Path.

The formulation of active pharmaceutical ingredients involves discovering stable crystal packing arrangements or polymorphs, each of which has distinct pharmaceutically relevant properties. Traditional experimental screening techniques utilizing various conditions are commonly supplemented with in silico crystal structure prediction (CSP) to inform the crystallization process and mitigate risk. Predictions are often based on advanced classical force fields or quantum mechanical calculations that model the crystal potential energy landscape but do not fully incorporate temperature, pressure, or solution conditions during the search procedure. This study proposes an innovative alchemical path that utilizes an advanced polarizable atomic multipole force field to predict crystal structures based on direct sampling of the NPT ensemble. The use of alchemical (i.e., nonphysical) intermediates, a novel Monte Carlo barostat, and an orthogonal space tempering bias combine to enhance the sampling efficiency of the deposition/sublimation phase transition. The proposed algorithm was applied to 2-((4-(2-(3,4-dichlorophenyl)ethyl)phenyl)amino)benzoic acid (Cambridge Crystallography Database Centre ID: XAFPAY) as a case study to showcase the algorithm. Each experimentally determined polymorph with one molecule in the asymmetric unit was successfully reproduced via approximately 1000 short 1 ns simulations per space group where each simulation was initiated from random rigid body coordinates and unit cell parameters. Utilizing two threads of a recent Intel CPU (a Xeon Gold 6330 CPU at 2.00 GHz), 1 ns of sampling using the polarizable AMOEBA force field can be acquired in 4 h (equating to more than 300 ns/day using all 112 threads/56 cores of a dual CPU node) within the Force Field X software (https://ffx.biochem.uiowa.edu). These results demonstrate a step forward in the rigorous use of the NPT ensemble during the CSP search process and open the door to future algorithms that incorporate solution conditions using continuum solvation methods.

Development of TPACK-Based Physics Magazine as Teaching Material for High Schools: A Study on Rotational Dynamics and Rigid Body Equilibrium

This study aims to develop a physics teaching magazine based on Technological Pedagogical and Content Knowledge (TPACK) for senior high school students, focusing on rotational dynamics and equilibrium of rigid bodies. Employing a Research and Development (R&D) approach with the 4-D model, the process encompassed definition, design, and development stages. The magazine's content and media aspects were validated by experts, achieving feasibility scores of 94% and 87.9%, respectively. The magazine was evaluated by students of SMAN 1 Lembah Seulawah Aceh Besar, with results indicating strong acceptance and effectiveness as a learning medium. While the study confirms the potential of TPACK-based interactive learning materials in science education, its limited scope and sample size suggest caution in generalizing the findings. Future research should expand the scope, involve larger samples, and integrate diverse technological tools. This research contributes to the discourse on digital competencies in science education, emphasizing the need for innovative materials and continuous professional development.

Numerical study to investigate the effect of sample size sensitivity on porous materials with respect to different geometrical parameters

Porous materials play a crucial role in various industrial applications, where precise determination of geometric parameters like dynamic tortuosity, as well as viscous and thermal characteristic lengths, is essential for optimizing acoustical performance. Up to this point, there has been no published size sensitivity test that recommends the ideal sample size for analyzing these parameters in an industrial environment. The present article analyzes different reconstructed porous material samples using conventional CFD and coupled FEA-CFD simulations and suggests optimal sample size and specific simulation setup optimized for industrial purposes, which grant high accuracy and reasonable computational cost. The present paper analyzed several samples, which were reconstructed by micro-CT and analyzed in Star-CCM+ simulation environment. It was found that the optimal sample size (1.5×1.5×1.5 mm or 2.5×2.5×2.5 mm) is different for specific average pore sizes. It was proven through various numerical simulations based on reconstructed porous material samples that running only one coupled CFD-FEM simulation is satisfactory to directly determine all investigated parameters. It was demonstrated that the computationally demanding numerical model could be simplified in specific cases using a rigid body and highlight the limitations.

Updated Predictive Models for Permanent Seismic Displacement of Slopes for Greece and Their Effect on Probabilistic Landslide Hazard Assessment

Earthquake-triggered landslides have been widely recognized as a catastrophic hazard in mountainous regions. They may lead to direct consequences, such as property losses and casualties, as well as indirect consequences, such as disruption of the operation of lifeline infrastructures and delays in emergency response actions after earthquakes. Regional landslide hazard assessment is a useful tool to identify areas that are vulnerable to earthquake-induced slope instabilities and design prioritization schemes towards more detailed site-specific slope stability analyses. A widely used method to assess the seismic performance of slopes is by calculating the permanent downslope sliding displacement that is expected during ground shaking. Nathan M. Newmark was the first to propose a method to estimate the permanent displacement of a rigid body sliding on an inclined plane in 1965. The expected permanent displacement for a slope using the sliding block method is implemented by either selecting a suite of representative earthquake ground motions and computing the mean and standard deviation of the displacement or by using analytical equations that correlate the permanent displacement with ground motion intensity measures, the slope’s yield acceleration and seismological characteristics. Increased interest has been observed in the development of such empirical models using strong motion databases over the last decades. It has been almost a decade since the development of the latest empirical model for the prediction of permanent ground displacement for Greece. Since then, a significant amount of strong motion data have been collected. In the present study, several nonlinear regression-based empirical models are developed for the prediction of the permanent seismic displacements of slopes, including various ground motion intensity measures. Moreover, single-hidden layer Artificial Neural Network (ANN) models are developed to demonstrate their capability of simplifying the construction of empirical models. Finally, implementation of the produced modes based on Probabilistic Landslide Hazard Assessment is undertaken, and their effect on the resulting hazard curves is demonstrated and discussed.

Meso-scale simulation of moisture transport in concrete during wet-dry cycles using 3D RBSM conduit model with variable diffusion model

This paper introduces the 3D RBSM Conduit model, an improved simulation program based on the Rigid Body Spring Model (RBSM), designed to accurately depict the intricate phenomenon of moisture transfer within concrete. This is achieved by incorporating an expanded diffusion model into the simulation framework. The diffusion coefficient is dynamically adjusted at each location and at each time step to accurately represent the precise moisture transfer processes occurring in the concrete. This dynamic adjustment allows the program to efficiently simulate the rate of moisture transport in concrete based on the moisture degree and the dry or wet state of the material. By adapting the diffusion coefficient to the changing conditions in this way, the program achieves simulated moisture transfer rates that align closely with the real-world behavior of moisture in concrete. It offers a dynamic model of the changes in concrete transmission during the switch from drying and wetting conditions. Further, the program exhibits inherent advantages in addressing water transfer problems involving nonlinear boundary conditions and transfer equations. In this study, the program is used to simulate various processes including drying, wetting, local wetting, and combined wet-dry cycles. The results reveal the lag in wet-dry transitions between the interior and exposed surface during the wet-dry conversion process due to retained interior moisture. The simulation captures the moisture variation in the transitional zone.

Analyzing the spatial motion of a rigid body subjected to constant body-fixed torques and gyrostatic moment

This paper aims to explore the rotatory spatial motion of an asymmetric rigid body (RB) under constant body-fixed torques and a nonzero first component gyrostatic moment vector (GM). Euler's equations of motion are used to derive a set of dimensionless equations of motion, which are then proposed for the stability analysis of equilibrium points. Specifically, this study develops 3D phase space trajectories for three distinct scenarios; two of them are applied constant torques that are directed on the minor and major axes, while the third one is the action of applied constant torque on the body’s middle axis. Novel analytical and simulation results for both scenarios of constant torque applied along the minor and middle axes are provided in the context of separatrix surfaces, equilibrium manifolds, periodic or non-periodic solutions, and periodic solutions’ extreme. Concerning the scenario of a directed torque on the major axis, a numerical solution for the problem is presented in addition to a simulation of the graphed results for the angular velocities' trajectories in various regions. Moreover, the influence of GM is examined for each case and a full modeling for the body's stability has been present. The exceptional impact of these results is evident in the development and assessment of systems involving asymmetric RBs, such as satellites and spacecraft. It may serve as a motivating factor to explore different angles within the GM in similar cases, thereby influencing various industries, including engineering and astrophysics applications.

Exploring static responses, mode transitions, and feasible tunability of Kagome-based flexible mechanical metamaterials

We consider the static responses of the uniaxially compressed flexible mechanical metamaterials, which integrate soft hinges and rigid bodies, constructed from the Kagome lattice. First, we experimentally find that the static responses of the regular-Kagome-based structure significantly differ from those of the twisted-Kagome-based structure with a very small twisting angle. Following this, we establish a theoretical model, which is combined with the deflated continuation algorithm for bifurcation analysis, to delve into the static responses and potential bifurcation behavior of these structures. We then experimentally and numerically investigate the mode transitions between various stable modes, and systematically study the role of the twisting angle in the occurrence of bifurcations. Our findings indicate that mode transitions can be feasibly realized according to the calculated bifurcation diagrams. They also provide direct evidence of an interesting physical mechanism that the transition between different stable deformation states can occur through multiple pathways, possibly passing through unstable deformation states. Moreover, by introducing the twisting angle or stiff defects, the response of the structure can be modulated, thereby enhancing the programmability and tunability of Kagome-based flexible mechanical metamaterials. Our research also reveals that novel phenomena such as meta-beam buckling and multi-phase dominated deformations can be triggered within these flexible structures, which offers valuable insights for future metamaterial designs and applications.

Fluid–rigid body coupling simulations with the passively moving solid model based on a physically consistent particle method

Fluid–rigid body interaction is a significant topic in research on particle methods. This study developed a fluid–rigid body coupling method based on a physically consistent particle method, i.e., the moving particle hydrodynamics (MPH) method, incorporating the passively moving solid (PMS) model. When the discrete particle system satisfies the fundamental laws of physics, i.e., mass conservation, linear and angular momentum conservation, and the second law of thermodynamics, the method is asserted physically consistent, and this feature is important for robust dynamic calculations. The PMS model is a pioneering approach that is practical for particle methods in which fluid and rigid-body particles are initially calculated as a fluid. Then, only rigid-body particles are modified to restore the initial shape by applying rigid-body constraints. Thus, combining the MPH method and the PMS model realizes a fluid–rigid body coupling method that satisfies fundamental physical laws. The proposed method was first verified via the fundamental rigid body and fluid–rigid body coupling problems: the Dzhanibekov effect on a T-shaped rigid body, a floating rectangular solid, a floating cylinder, and water entry of a two-dimensional cylinder. Second, the proposed method was validated via calculating a cylinder rolling on a liquid film as a fluid–rigid body coupling problem with rotation. By using a potential-based surface tension model, the computed results showed reasonable agreement with the experimental data obtained in this study. Overall, it was confirmed that the proposed method is a promising fluid–rigid body coupling approach, in which the surface tension and wettability can be considered as well.

Enabling the transfer matrix method to model serial–parallel compliant mechanisms including curved flexure beams

AbstractCompliant mechanisms with curved flexure hinges/beams have potential advantages of small spaces, low stress levels, and flexible design parameters, which have attracted considerable attention in precision engineering, metamaterials, robotics, and so forth. However, serial–parallel configurations with curved flexure hinges/beams often lead to a complicated parametric design. Here, the transfer matrix method is enabled for analysis of both the kinetostatics and dynamics of general serial–parallel compliant mechanisms without deriving laborious formulas or combining other modeling methods. Consequently, serial–parallel compliant mechanisms with curved flexure hinges/beams can be modeled in a straightforward manner based on a single transfer matrix of Timoshenko straight beams using a step‐by‐step procedure. Theoretical and numerical validations on two customized XY nanopositioners comprised of straight and corrugated flexure units confirm the concise modeling process and high prediction accuracy of the presented approach. In conclusion, the present study provides an enhanced transfer matrix modeling approach to streamline the kinetostatic and dynamic analyses of general serial–parallel compliant mechanisms and beam structures, including curved flexure hinges and irregular‐shaped rigid bodies.

On the Hinch–Kim dualism between singularity and Faxén operators in the hydromechanics of arbitrary bodies in Stokes flows

We generalize the multipole expansion and the structure of the Faxén operator in Stokes flows obtained for bodies with no-slip to generic boundary conditions, addressing the assumptions under which this generalization is conceivable. We show that a disturbance field generated by a body immersed in an ambient flow can be expressed, independently on the boundary conditions, as a multipole expansion, the coefficients of which are the moments of the volume forces. We find that the dualism between the operator giving the disturbance field of an nth order ambient flow and the nth order Faxén operator, referred to as the Hinch–Kim dualism, holds only if the boundary conditions satisfy a property that we call Boundary-Condition reciprocity (BC-reciprocity). If this property is fulfilled, the Faxén operators can be expressed in terms of the (m, n)th order geometrical moments of the volume forces (defined in the article). In addition, it is shown that in these cases, the hydromechanics of the fluid-body system is completely determined by the entire set of the Faxén operators. Finally, classical boundary conditions of hydrodynamic applications are investigated in light of this property: boundary conditions for rigid bodies, Newtonian drops at the mechanical equilibrium, porous bodies modeled by the Brinkman equations are BC-reciprocal, while deforming linear elastic bodies, deforming Newtonian drops, non-Newtonian drops, and porous bodies modeled by the Darcy equations do not have this property. For Navier-slip boundary conditions on a rigid body, we find the analytical expression for low order Faxén operators. By using these operators, the closed form expressions for the flow past a sphere with arbitrary slip length immersed in shear and quadratic flows are obtained.

The Robin boundary condition for modelling heat transfer

The heat exchange between a rigid body and a fluid is usually modelled by the Robin boundary condition saying that the heat flux through the interface is proportional to the difference between their temperatures. Such interface law describes only the unilateral heat exchange. The goal of this paper is to compare the Robin boundary condition starting with the transmission condition (the temperature and the flux continuity) using rigorous mathematical analysis. Our main results are the following. We first show that a generalized version of the Robin boundary condition can be justified. Second, we prove that replacing the generalized by the standard Robin condition can be justified for high convection velocity if the conductivity of the surrounding liquid is much lower than that of the body. On the other hand, if the fluid conducts much better than the body, then the effective boundary condition is shown not to be the Robin one, but it involves second-order derivatives. We strongly believe that those findings bring new insights to the physics of the heat exchange processes and, thus, could prove useful in engineering practice.

Berthing proximity maneuvers and trajectory planning of a space manipulator via Kane’s formulation

On-orbit servicing is one of the most challenging tasks of the recent space era. Space Manipulator Systems (SMS) can represent viable technological solutions for several space operations, such as in-situ refueling, repairing of operating spacecraft, or debris grappling. Regardless of the specific task, SMS must approach the target object and then establish a physical connection with it using a single or more robotic arms. This work considers a space vehicle equipped with a pair of two-link robotic arms, with the intent of efficiently designing and simulating the overall spacecraft dynamics until contact takes place. The links that compose each arm are modeled as rigid bodies, assuming relatively slow dynamics. Instead, flexibility is introduced in the model of the rotary joints. The overall dynamics is investigated using Kane's methodology, which is general, highly systematic, and joins the advantages of both the Newton/Euler and the Lagrange formulation. Both SMS and the target, assumed as noncooperative, are placed in low Earth orbit, in close proximity. Trajectory planning of the robotic arm is investigated, while considering the effect of the arm motion on the overall attitude dynamics. Precise attitude maneuvering is mandatory in order that grappling be successfully completed. To do this, a quaternion-based, globally stable attitude feedback law is designed, implemented, and tested. Actuation is demanded to an arrangement of control momentum gyroscopes. These devices are modeled with the inclusion of gyroscopic coupling effects, and are driven by means of suitable steering laws, for the purpose of pursuing the desired attitude control action. Accurate modeling of SMS, with also the inclusion of disturbances due to joint flexibility, in conjunction with the use of suitable feedback attitude control and steering laws for both the robotic arms and the actuation devices, allows obtaining the desired variables, i.e. (i) the motor torques (for steering both the joints and the attitude devices) and (ii) the constraint actions (forces and torques). A Monte Carlo analysis points out effectiveness of the control architecture proposed in this study in the challenging mission scenario of interest.

Influence of Exhaust Aftertreatment System on Powertrain Vibration Behavior

<div>NVH refinement of commercial vehicles is the key attribute for customer acceptance. Engine and road irregularities are the two major factors responsible for the same. During powertrain isolators’ design alone, the mass and inertia of the powertrain are usually considered, but in practical scenarios, a directly coupled subsystem also disturbs the boundary conditions for design. Due to the upgradation in emission norms, the exhaust aftertreatment system of modern automotive vehicles becomes heavier and more complex. This system is further coupled to the powertrain through a flexible joint or fixed joint, which results in the disturbance of the performance of the isolators.</div> <div>Therefore, to address this, the isolators design study is done by considering a multi-body dynamics model of vehicles with 16 DOF and 22 DOF problems, which is capable to simulate static and dynamic real-life events of vehicles. Design indicators are thoroughly analyzed and validated through the rigid body modes and real field events of the vehicle. As most of the research is done for four-point mounting powertrain systems by considering 6 DOF or 12 DOF in commercial vehicles but a novel approach with a 22 DOF model is proposed in this study to predict the impact of the inclusion of the exhaust aftertreatment system on torque roll axis and rigid body modes decoupling.</div> <div>The results of the proposed system show that the rigid body mode decoupling of the powertrain system improved and consequently the overall NVH performance of the vehicle in real life is further improved. Therefore, it is suggested from the study that ignoring the inclusion of the exhaust aftertreatment system in the powertrain mounting system design reduces the NVH performance of the vehicle, hence it is recommended to include it in the early phase of design.</div>

Resonant response of a flexible semi-submersible floating structure: experimental analysis and second-order modelling

The dynamics and nonlinear wave forcing of a flexible floating structure are investigated experimentally and numerically. The floater was designed to match sub-harmonic rigid-body natural frequencies of typical floating wind turbine substructures, with the addition of a flexible bending mode. Experiments were carried out for three sea states with phase-shifted input signals to allow harmonic separation of the measured response. We find for the weakest sea states that sub-harmonic rigid-body motion is driven by even-harmonic difference frequency forcing, and by linear forcing for the strongest sea state. The flexible mode was tested in a soft, linearly forced layout, and a stiff layout, forced by second-, third- and fourth-harmonic frequency content, for increasing severity of the sea state. Further insight is gained by analysis of the amplitude scaling of the resonant response. A new simplified approach is proposed and compared with the recent method of Orszaghova et al. (J. Fluid Mech., vol. 929, 2021, A32). We find that resonant surge and pitch motions are dominated by even-harmonic potential-flow forcing and that odd-harmonic response is mainly potential-flow driven in surge and mainly drag driven in pitch. The measured responses are reproduced numerically with second-order forcing and quadratic drag loads, using a recent and computationally efficient calculation method, extended here for the heave, pitch and flexible motions. We are able to reproduce the response statistics and power spectra for the measurements, including the subharmonic pitch and heave modes and the flexible mode. Deeper analysis reveals that inaccuracies in the even-harmonic forcing content can be compensated by the odd-harmonic loads.

Partial stabilization of an orbiting satellite model with a flexible attachment

We consider a mathematical model of an orbiting satellite comprising a perfectly rigid carrier body and a flexible boom operating under the influence of the orbital moment of the gravity gradient. This model is represented by a nonlinear control system which includes ordinary differential equations governing the carrier body’s angular velocity and attitude quaternion coupled with the Euler – Bernoulli equations that describe the vibration of the flexible component. We propose an explicit feedback design aimed at guaranteeing the partial stability of the closed-loop system in an appropriate Hilbert space.

Maximizing ollie height by optimizing control strategy and skateboard geometry using direct collocation

The ollie is the base aerial human–board maneuver, foundational to most modern skateboarding tricks. We formulate and solve an optimal control problem of a two-dimensional simplified human model and a rigid body skateboard with the objective of maximizing the height of the ollie. Our solution simultaneously discovers realistic human-applied force trajectories and optimal board geometry. We accomplish this with a direct collocation formulation using a null seed initial guess by carefully modeling the discontinuous aspects of board–ground impact and foot–board friction. This leads to efficient and robust solutions that are 10 times more computationally efficient than prior work on similar problems. The solutions show that ollie height can increase 3% by decreasing the wheelbase and that a smaller board with a back-foot-dominated force strategy can give 12% higher ollies. Our model can be used to inform jump strategy and the effects of changes to the essential board geometry.

Influence of centroid acceleration acquisition and filtering class on head injury criterion evaluation

BackgroundAlthough the Head Injury Criteria (HIC) has been widely applied to assess head impact injuries, it faces two outstanding problems: 1) HIC is affected strongly by the cut-off frequency when processing acceleration signals. And these cut-off frequencies are experiential and lack unified guidelines; 2) If the head was impacted on a different part, should the corresponding HIC threshold be the same? If these problems are not resolved, it could potentially lead to a critical misinterpretation of the safety assessment. MethodsFinite element method was used to reconstruct head impacts. The head model includes tissues like skull, brainstem, cerebrospinal fluid, etc. The head model was impacted in the frontal, occipital, parietal or lateral direction with different impact velocities. Acceleration signals of the head model were extracted directly from the skull and the head centroid node. To obtain a robust HIC, the filtering class of acceleration signals were analyzed carefully. Then, the relation between rigid body HIC and the centroid node HIC were studied systematically. ResultsWhen the filtering class of rigid body acceleration and centroid node acceleration reached the cut-off frequency, the corresponding derivative of HIC tended to change smoothly. Using these cut-off frequencies, robust HICs were obtained. The rigid body HIC far exceeded that of centroid node HIC, such as 8, 9, 14 and 31 times exceeded in the frontal, occipital, parietal and lateral impact conditions, respectively. Moreover, approximate linear relations were found between the rigid body HIC and the centroid node HIC in different impact directions, respectively. From these relations, the injury thresholds of rigid body HIC of various directions were given quantitatively. ConclusionsThe rational filtering class like CFC 800 and CFC 700 were given for rigid body HIC and centroid node HIC, respectively. The rigid body HIC had a significant discrepancy from the centroid node HIC. Linear relations between the rigid body HIC and centroid node HIC were found, and their slopes changed with impact directions. From these relations, we can adjust the injury thresholds reasonably if the head receives different impacts. These findings can effectively enhance the applicability of HIC.

A FE2 shell model with periodic boundary conditions for thin and thick shells

AbstractIn this article a FE2 shell model for thin and thick shells within a first order homogenization scheme is presented. A variational formulation for the two‐scale boundary value problem and the associated finite element formulation is developed. Constraints with 5 or 9 Lagrange parameters are derived which eliminate both rigid body movements and dependencies of the shear stiffness on the size of the representative volume elements (RVEs). At the bottom and top surface of the RVEs which extend through the total thickness of the shell stress boundary conditions are present. The periodic boundary conditions at the lateral surfaces are applied in such a way that particular membrane, bending and shear modes are not restrained. This is shown by means of a homogeneous RVE. The first of all linear formulation is extended to finite strain problems introducing transformation relations for the stress resultants and the material matrix. The transformations are performed at the Gauss points on macro level. Several boundary value problems including large deformations, stability and inelasticity are computed and compared with 3D reference solutions.

BROM: An extension of beam theory through model order reduction

Beam analysis played a crucial role in the design of structures throughout modern history. Classical beam theories rely on analytical derivations under certain a priori kinematic assumptions (and are accurate with low computational cost), but they are not amenable to, for example, orthotropic materials and arbitrary cross sections. On the other hand, full 3D linear elasticity theory solved via the Finite Element (FE) method provides solutions for a wide range of engineering problems at the expense of a high computational cost. Reduced-order models try to provide a theory as general as 3D linear elasticity but as efficient as classical beam theories by assuming the solution to be represented in a low-dimensional basis. Among the several options available in the literature, the domain decomposition and hyperreduction via Empirical Cubature Method (ECM), which we name ddROM [1], exhibits this compelling feature. However, ddROM applied to beam structures, may provide more Degrees of Freedom (DoF) per node than most standard commercial FE codes which only consider rigid body motion of the cross section. This paper presents a methodology, called beam Reduced Order Model (bROM), to integrate ddROM in standard FE codes by means of a condensation process followed by a regression procedure. Condensation gives the stiffness matrix, required by FE commercial codes, of a super-element of a given length. This is achieved by assembling several elements and finding the relationship between applied displacements and reactions (Lagrange multipliers). The condensation process is then used to create a database of stiffness matrices as a function of the beam length over which a regression is performed. Some numerical examples are first presented to validate the proposed methodology against the classical beam theory for the case of isotropic beams, and finally, two orthotropic beam showing the utility of the method in comparison with FE.