Double-beam Systems Research Articles

A State-Space Method for Vibration of Double-Beam Systems with Variable Cross Sections

A State-Space Method for Vibration of Double-Beam Systems with Variable Cross Sections

Analytical, semi-analytical and numerical solutions for global stability analysis in buildings: Double-Beam Systems Timoshenko

In this article, two analytical solutions, one semi-analytical solution, and one numerical solution are proposed to calculate the global critical buckling load of buildings employing a continuum model parallel coupling two Timoshenko beams. The parallel coupling of two Timoshenko beams represents the four types of building behavior: global bending, global shear, local bending, and local shear, the latter of which has been widely ignored in the literature. The model is associated with one translational kinematic field and two rotational fields. The equilibrium equations, constitutive laws, and boundary conditions are derived using a variational approach by applying Hamilton’s principle to the relevant Lagrangian function. Specifically, to solve the classical problem of buildings with uniform properties along their height, the first analytical solution proposes exact closed-form equations for a vertical load applied at the top and a closed-form analytical solution based on the decomposition of two subsystems for a polynomial vertical load uniformly applied along the height of the beam. Meanwhile, the semi-analytical solution exactly solves the differential equation associated with a polynomial vertical load profile and is based on coefficient iteration, converging with only two or three iterations. To address the challenge of buildings with variable properties along their height, the combined application of the continuous method and the transfer matrix method allows us to propose an efficient numerical method leading to a 6 × 6 transfer matrix, constant regardless of the number of discretizations and derived analytically, which results in a computational advantage by drastically reducing computational costs. The numerical results are encouraging and show good agreement with the results of the finite element method, suggesting that the proposed method can be used for engineering purposes in both the preliminary and final phases of the structural analysis of tall buildings.

Analytical and numerical solution of generalized static analysis of tall buildings: double-beam systems Timoshenko

Analytical and numerical solution of generalized static analysis of tall buildings: double-beam systems Timoshenko

Free Vibration and Buckling Analysis of Axially Loaded Double-Beam Systems with Generalized Boundary Conditions

Free Vibration and Buckling Analysis of Axially Loaded Double-Beam Systems with Generalized Boundary Conditions

Concurrent-Validity and Reliability of Photocells in Sport: A Systematic Review.

Specific physical qualities such as sprint running, change-of-direction or jump height are determinants of sports performance. Photocell systems are practical and easy to use systems to assess the time from point A to point B. In addition, these photoelectric systems are also used to obtain the time of vertically displaced movements. Knowing the accuracy and precision of photocell timing can be a determinant of ensuring a higher quality interpretation of results and of selecting the most appropriate devices for specific objectives. This systematic review aimed to identify and summarize studies that have examined the validity and reliability of photocells in sport sciences. A systematic review of PubMed, SPORTDiscus, and Web of Science databases was performed according to the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines. From the 164 studies initially identified, 16 were fully reviewed, and their outcome measures were extracted and analyzed. Photocells appear to have a strong agreement with force plates (gold standard), but are not interchangeable to measure the vertical jump. For monitoring horizontal displacement, double beam systems, compared to single beam systems, are more valid and reliable when it comes to avoiding false triggers caused by swinging arms or legs.

Numerical Analysis of Cracked Double-Beam Systems

Based on elasticity theory, this paper discusses the static analysis of a cracked double-beam system in the presence of a Winkler-type medium. It is further assumed that the double-beam system is constrained at both ends by elastically flexible springs with transverse and rotational stiffness. Using a variational formulation, the governing static equations are derived and solved using analytical and numerical approaches. In the first approach, closed-form solutions for the displacement functions are obtained based on the Euler–Bernoulli beam theory. In the second approach, the Cell Discretisation Method (CDM) is performed, whereby the two beams are reduced to a set of rigid bars connected by elastic constraints, in which the flexural stiffness of the bars is concentrated. The resulting stiffness matrix is easily deduced, and the governing equations of the static problem can be immediately solved. A comparative analysis is performed to verify the accuracy and validity of the proposed method. The study focuses on the effect of various parameters, including crack depth and position, boundary conditions, elastic medium and slenderness. The validity of the proposed analysis is confirmed by comparing the current results with those obtained from other approaches. In particular, the results obtained by closed-form solution and CDM are compared with the Finite Element Method (FEM). The accuracy of the results was assessed by making comparisons with results found in the literature and reported in the bibliography. It was shown that the proposed algorithm provides a simple and powerful tool for dealing with the static analysis of a double-beam system. Finally, some concluding remarks are made.

A homotopy analysis method for forced transverse vibrations of simply supported double-beam systems with a nonlinear inner layer

The present study introduces a novel algorithm based on the homotopy analysis method (HAM) to efficiently solve the equation of motion of simply supported transversely and axially loaded double-beam systems. The original HAM was developed for single partial differential equations (PDEs); the current formulation applies to systems of PDEs. The system of PDEs is derived by modeling two prismatic beams interconnected by a nonlinear inner layer as Euler–Bernoulli beams. We employ the Bubnov–Galerkin technique to turn the PDEs’ system into a system of ordinary differential equations that is further solved with the HAM. The flexibility and straightforwardness of the HAM in computing time-dependent components of the system’s transverse deflection and natural frequencies, in conjunction with the observed fast convergence, offer a robust semi-analytical method for analyzing such systems. Finally, the transverse deflection is built through the modal superposition principle. Thanks to a judicious and high-flexibility selection of initial guesses and convergence control parameters, numerical examples confirm that at most six iterations are needed to achieve convergence, and the results are consistent with the selected benchmark cases.

A closed-form solution of forced vibration of a double-curved-beam system by means of the Green's function method

Double-curved-beam systems (DCBSs) are usually seen in many engineering fields. Compared to straight double-beam systems, DCBSs are more efficient in noise and vibration control. This paper aims to obtain a closed-form solution of steady-state forced vibration of a DCBS. The classical Euler-Bernoulli beam model is employed in this work to model vibration equations of the DCBS. Green's function and Laplace transform methods are successively used to obtain the fundamental solution of vibration equations of the DCBS. The fundamental solution is the general solution and can be used for any boundary conditions. In the numerical result section, the present solution is verified by comparing its results with some results in references. Effects of some important geometric and physical parameters on vibration responses and interaction between the stiffness of the elastic layer and the DCBS are discussed. Results show that the DCBS can be degenerated to a straight double-beam system by setting two radii of curvature of the beams to infinity, and the DCBS can also be reduced to a double-beam system whose one upper or lower beam is a straight beam and the other is a curved beam.

Forced Vibration Analysis of Euler-Bernoulli Double-Beam Systems by Means of Green’s Functions

Double-curved-beam (DCB) systems are usually seen in many engineering fields. Compared to straight double-beam systems, DCB systems are more efficiency in noise and vibration control problems. This work aims to obtain closed-form solutions of steady-state forced vibrations of DCBs. The classical Euler-Bernoulli curved beam (ECB) model is employed in this work to modelling vibration equations of the DCB systems. The Green’s function and Laplace transform methods are successively used to obtain fundamental solutions for the vibration equations of the DCB systems. In this work, the fundamental solutions are general solutions and can be used for any boundary conditions. In numerical section, the present solutions are verified by comparing to some results in references. Effects of some important geometric and physical parameters on vibration responses and the interaction between the stiffness of the elastic layer and the DCB system are discussed. The results show that the DCB system can be degenerated to the straight double-beam system by setting two radiuses to infinity, and the DCB system also can be reduced to a double-beam system whose one upper or lower beam is a straight beam and the other is a curved beam.

Vibrations of Timoshenko Double-Beam Systems with Arbitrary Intermediate Supports and Axial Loads

Vibrations of Timoshenko Double-Beam Systems with Arbitrary Intermediate Supports and Axial Loads

Buckling and postbuckling behaviors of symmetric/asymmetric double-beam systems

Double-beam systems composed of two interconnected parallel beams have become basic mechanical components of modern engineering structures. Focusing on the asymmetric features in boundary conditions and other structural parameters, this paper studies the buckling and postbuckling behaviors of a double-beam system supported on an elastic foundation. The nonlinear governing differential equations of the system are derived by using the Hamilton's principle, and discretized via the differential quadrature method. Then the pseudo-arc length algorithm is adopted to overcome difficulties of numerical continuation around bifurcation points. Compared with single beam systems, double-beam systems exhibit some peculiarities like multivalued critical buckling loads, in-phase and out-of-phase postbuckling paths. Based on the proposed procedures, these buckling and postbuckling behaviors are investigated in detail. For symmetric boundary cases of the two sub-beams, the multivalued critical buckling loads of the system compose a cluster of critical curves that intersect at a point determined only by the critical loads of single beams. Hence a simple criterion based on single beam critical loads is proposed to quickly estimate whether or not the axial loads would trigger the buckling of the system. However for asymmetric boundary cases, the cluster of critical curves locate in a region that depends on the intrinsic double-beam characteristics, and the criterion is revised accordingly. Also for asymmetric systems, some unique features of the out-of-phase postbuckling paths are detected. It is found that their stabilities are quite different from those of symmetric systems, and that strengthening the system stiffness may even trigger the out-of-phase buckling more easily.

Exact Closed-Form Solutions for Free Vibration of Double-Beam Systems Interconnected by Elastic Supports Under Axial Forces

This paper aims to present the exact closed-form solutions for the free vibration of double-beam systems composed of two parallel beams connected by an arbitrary number of discrete elastic supports. The general solutions of the mode shapes of the double-beam system are derived employing the Laplace transform method from a perspective of the entire domain of beams without enforcement of any segmentation. A unified strategy applied to various boundary conditions is proposed to determine the independent constants involved in the general solutions, as well as the frequency equation. Numerical calculations are performed to verify the present solutions by comparing the results from the previous literature and finite element simulation, and to discuss the effects of support parameters (stiffness, location, and number) on the modal characteristics of the double-beam system in detail. Outcomes show that the support location plays a pivotal role in regulating the modal characteristics of the double-beam system; for each-order mode, there are one or more potential optimal positions to maximize the effect of the elastic support. The mode veering phenomenon is detected as the support parameters change. It is highlighted that, by introducing an amplitude similarity index, the proximity degree for the mode shapes of the two beams influenced by the support parameters can be evaluated quantitatively. The present analysis is greatly helpful to the optimal design, health monitoring, and vibration control of the double-beam system.

Coupled dynamics of double beams reinforced with bidirectional functionally graded carbon nanotubes

The coupled dynamics of a double beam system connected via an elastic layer is investigated. The double beam is reinforced with bidirectional functionally graded carbon nanotubes. Independent transverse and axial motions for each beam, as well as their couplings, are considered in the model. The two interconnected beams are functionally graded with patterns of UD, FGV, FGO, and FGX in the thickness direction, and a power-law function in the longitudinal direction, and a power-law function in both the thickness and longitudinal directions. The results suggested that increasing the carbon nanotube volume fraction in the thickness or length or both the directions of the two beams leads to an increase in both series of the axial and transverse natural frequencies. The study also identifies that when the double beams are reinforced with UD, FGO, or FGX patterns, with the same volume fraction, which are symmetrical about the transverse axis, they possess the same axial natural frequencies. It is also revealed that increasing the stiffness of the elastic layer place in-between the two beams, causes an increase in the second series of the transverse natural frequency, for all cases, whether it is reinforced by carbon nanotube in the thickness or in the length or in both thickness and length directions of the beams. These results will form crucial considerations when designing double beam systems to improve reliability and safety.

A unified method for in-plane vibration analysis of double-beam systems with translational springs

Double-beam structures with many translational springs are a class of significant engineering structures, which have been widely used in civil, aerospace, and mechanical engineering. The dynamic characteristics of such structures is significant to their structural design, vibration abatement and condition assessment. Since the motion differential equation of a double-beam system is a system of complex fourth-order variable coefficient partial differential equations. The existing analytical methods either need to simplify the structural mechanics model or make some prior assumptions about the form of the solution of the equation, which is convenient for the simplification and solution of the equation, and finally reduces the universality of the whole analysis process, and has many limitations in the practical application. An alternative solution is the finite element method, but it can only give an approximate solution to distributed parameter systems. In view of this, this article developed a unified approach for the free vibration of the double-beam structures by employing the dynamic stiffness method (DSM), which has no restrictions on the geometric characteristics and boundary conditions of the structure, and can obtain the exact dynamic characteristics of the system without any approximation. The correctness of the proposed method is validated by comparison with literatures and the finite element solutions. Finally, a comprehensive parametric study is performed to investigate the influence of key parameters on modal frequencies and mode shapes.

Free and Forced Vibrations of an Undamped Double-Beam System Carrying a Tip Mass with Rotary Inertia

AbstractMany civil and mechanical engineering structures can be simplified as double-beam systems, i.e., a primary beam and a secondary beam connected to the primary beam. Many studies have investi...

Solution to Vibrations of Double-Beam Systems under General Boundary Conditions

The problem of vibrations of double-beam systems is of technological interest and has been extensively studied in the literature. However, a general solution is yet to be developed for the general boundary conditions. In this paper, a general solution of the free and forced vibrations of double-beam systems under classical, nonclassical, and mixed boundary conditions is derived. The obtained mode-shape solutions are exact and explicit. The characteristic frequency equation of a double-beam system is of order-4 and order-8 for the classical boundary conditions and the nonclassical boundary conditions, respectively. The solution of forced vibrations is developed with the classical modal expansion technique. In addition to the theory, applications of the proposed method are illustrated by numerical examples. The obtained frequencies for special problems are compared with the literature, and the agreement between the proposed method and the literature is remarkably good. This shows that the proposed shape function method may be applied reliably to determine the transverse vibration mode shapes and natural frequencies of the double-beam system under the general boundary conditions. The solutions of free and forced vibrations of the double-beam systems presented in the paper could easily be programmed into a computer for vibration analyses.

Free and forced vibration analysis of double-beam systems with concentrated masses

Free and forced vibration analysis of double-beam systems with concentrated masses

State-space method for dynamic responses of double beams with general viscoelastic interlayer

Due to their effective engineering applications, double-beam systems have intensively been studied in recent decades. The interlayer between beams in the system plays a key role in the structural vibration and energy dissipation. However, existing research efforts only observe the pure elastic interlayer or simplest viscoelastic interlayer such as Kelvin-model interlayer. In this paper, a general viscoelastic interlayer that can consider more complex stiffness and damping effects is investigated. A novel state-space approach with the proposed mode-shape constant is presented to determine the transverse vibration of the double-beams. The given integrals of mode shapes can effectively decouple the governing equations, making it possible to analyze the complex viscoelastic interlayer. Numerical examples demonstrate that the proposed method is accurate and reliable. The viscoelastic interlayer affects the natural frequencies of asynchronous vibration mode more apparently than those of synchronous vibration mode. The damping coefficients of the viscoelastic interlayer have a significant effect on the damping characteristic of the double-beam systems while reducing the dynamic responses of the beams in the forced vibration. The proposed method can be useful to simulate the dynamic responses of double-beam systems with various viscoelastic interlayers in engineering practices and applications.

Development of Optimal Design Method for Steel Double-Beam Floor System Considering Rotational Constraints

Under high gravity loads, steel double-beam floor systems need to be reinforced by beam-end concrete panels to reduce the material quantity since rotational constraints from the concrete panel can decrease the moment demand by inducing a negative moment at the ends of the beams. However, the optimal design process for the material quantity of steel beams requires a time-consuming iterative analysis for the entire floor system while especially keeping in consideration the rotational constraints in composite connections between the concrete panel and steel beams. This study aimed to develop an optimal design method with the LM (Length-Moment) index for the steel double-beam floor system to minimize material quantity without the iterative design process. The LM index is an indicator that can select a minimum cross-section of the steel beams in consideration of the flexural strength by lateral-torsional buckling. To verify the proposed design method, the material quantities between the proposed and code-based design methods were compared at various gravity loads. The proposed design method successfully optimized the material quantity of the steel double-beam floor systems without the iterative analysis by simply choosing the LM index of the steel beams that can minimize objective function while satisfying the safety-related constraint conditions. In particular, under the high gravity loads, the proposed design method was superb at providing a quantity-optimized design option. Thus, the proposed optimal design method can be an alternative for designing the steel double-beam floor system.

Closed-form solutions for forced vibrations of a cracked double-beam system interconnected by a viscoelastic layer resting on Winkler–Pasternak​ elastic foundation

A double-beam system, which consists of two parallel beams connected by a viscoelastic layer, has found broad application in practical engineering such as continuous dynamic absorbers and floating-slab tracks. During the structural service period, the crack is one of the most common defects, which poses a great threat to its normal operation and safety. As a first endeavor, this paper strives to obtain the closed-form solutions for steady-state forced vibrations of a cracked double-beam system resting on the Winkler–Pasternak elastic foundation subjected to harmonic loads. The mechanical properties of cracked cross-sections are characterized by the local stiffness model. Due to the existence of cracks, the cracked double-beam system is artificially divided into several intact segments, where fundamental dynamic responses of each segment are achieved by the Green’s functions method. Subsequently, the transfer matrix method is employed to obtain the steady-state responses of the whole cracked system via the compatibility conditions of cracked cross-sections and boundary conditions. Numerical calculations are performed to check the validity of the present solutions and to discuss the influences of some important parameters, such as crack geometries and connecting layer stiffness, on dynamic behaviors of cracked double-beam systems. Significant effects of the crack depth and location are revealed on the natural frequency and dynamic responses of the system. It is highlighted that based on the bending moment diagram of the mode shape of the intact double-beam system, the effect of the crack location on dynamic behaviors of its cracked system can be effectively predicted.