Symposium on Mechanics of Slender Structures 2017

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Nonlinear Position and Vibration Control of a Rotating Smart Beam: Comparison of Three Different Techniques

André Fenili

Federal University of ABC, Aerospace Engineering / Mechanical Engineering,

Santo André, SP, Brazil

In this paper, position and vibration control of a smart rotating flexible beam-like structure is investigated. The nonlinear integro-partial differential governing equations of motion are derived via the Extended Hamilton’s Principle. The nonlinear control technique named SDRE is applied. Three different cases are considered for comparison: (1) control torque is applied at the rotating axis for angular position and vibration control (2) control torque is applied at the rotating axis for angular position control and a distributed moment is applied via piezoelectric actuator for vibration control and (3) control torque is applied at the rotating axis for angular position and vibration control and a distributed moment is applied via piezoelectric actuator for vibration control.


Atomic Force Microscopes study of hydrodynamics around micro- and nanofibers

A. Steinberger

University of Lyon

The description of the hydrodynamic interactions between a particle and the surrounding liquid, down to the nanometer scale, is of primary importance since confined liquids are ubiquitous in many natural and technological situations. Here, the hydrodynamics around micro- and nanocylinders is studied using three nonconventional Atomic Force Microscopes (AFM). These complementary methods allow the independent measurement of the friction and added mass terms over a large range of probe sizes, fluid viscosities, and solicitation conditions, corresponding to 7 orders of magnitude in Reynolds number. We show that the whole range of experimental data can be gathered in two master curves depending on a unique parameter, namely the ratio of the probe radius to the thickness of the viscous boundary layer. This behavior is well reproduced by an analytical hydrodynamic model. The model also provides a comprehensive description of the velocity field in the vicinity of the cylinder and yields the power laws associated with the asymptotic regime, which gives useful trends for the design of micro- and nanofluidic devices. In addition, our results validate the use of these AFM techniques for a quantitative study of hydrodynamics, and open the way to the investigation of other sources of dissipation in simple and complex fluids down to the submicron scale.



Energy efficient active vibration control strategies using electromagnetic linear actuators

Angel Torres-Perez

University of Northampton

Energy efficient current control methods in electromagnetic linear actuators are required to minimize the electrical power requirements imposed by active vibration control strategies. In this paper an efficient bidirectional buck-boost converter is discussed in two scenarios: an active vibration isolation system and an active dynamic vibration absorber ( ADVA ) using  a voice coil motor (VCM) actuator. An electrical analogous circuit of an experimental test platform is used as part of the simulation model. This test platform is based on a vibration shaker that provides the based excitation required for the single Degree of-Freedom (1DoF) vibration model under study.

The proposed bidirectional non-isolated buck-boost converter can recover the energy when the VCM acts as a generator and store it for future use. Simulation results prove that this type of topology is far more efficient than linear amplifiers typically used in active vibration control. Within the context of slender structures, this efficient current control method improves the viability of using active vibration control in flexible structures such as beams.



Elevator Rope Tension Analysis with Uneven Groove Wear of Sheave

Daisuke Nakazawa

Advanced Technology R&D Center, Mitsubishi Electric Corp

In the traction elevator system, the car is suspended by multiple ropes. Ideally, the tension of the ropes should be maintained as the balanced condition. However, the tension changes due to the elongation of the rope, the variation of rope stiffness or the uneven sheave groove wear. If the tension of the ropes become the unbalanced condition, the system design of the elevators deviates from the ideal condition. Therefore, an analytical method is required to evaluate the rope tension behavior.

 Since the uneven groove wear of the sheave causes a significant tension change in high-rise elevators even if the degree of the wear is small, it is more important to establish the tension analysis with the groove wear than other factors. In other words, it will be meaningful contribution to the system design to clarify the relationship between the groove wear, the tension and the travel.

 In this work, we describe the modeling and the mechanism of rope tension behavior with the uneven groove wear. An elevator vertical vibration model with multiple ropes is derived. The rope slip on the sheave is included, and the influence of the slip due to the unbalanced tension is evaluated.

 By using the testbed with the two ropes, the derived model is validated by comparison with the experimental data. Furthermore, influence of the car position and the degree of wear on tension are investigated by the model. We clarify the mechanism of the tension hysteresis about the car position by the rope slip. Moreover, it is shown that the tension variation can be formulated as a function of the initial tension and the car position. As we can clarify the mechanism of the tension variation by the uneven groove wear, we can put the analytical method to practical elevator system design.

Experimental analysis of primary stability of prosthetic elements implanted in femur bone in traumatology surgery

Evelyn García Nieto

Universidad Politécnica de Madrid

 The human femur bone is a notorious case of a slender biological structure. The strong flexocompression stresses to be withstood, are absorbed with a combination of geometries and distributions of heterogeneous and anisotropic tissue, as well as with an amazing capacity of in vivo remodelling against different patterns of loads.

Traumatological surgery restore the lost or limited functionality due to pathologies or fractures. Its success in the medium and long term depends to a large extent on the primary stability, being an important line of investigation the development of experimental methods that allow the evaluation of stability. This work presents an in vitro dynamic test procedure that, through a stress of free oscillations and instrumentation with resistive strain gauge sensors, allows the evaluation of the perturbation of the natural state of stress caused by the implantation of a prosthesis, as well as the detection and identification of relative micromovements that may be responsible for the failure of the osseointegration processes, and the reduction of the durability of the implant.

The test meets the conditions of simplicity, ease of assembly and repeatability, is non-destructive, allows a rigorous control of parameters and provides direct information of the state of deformations at any point and direction. The results recording and processing system gives a direct and debugged visualization of the response signals of the sensors in the time and frequency domains. The test is suitable for the detection and characterization of clearances and micromovements between the femur and the implanted prosthesis.

The method has been successfully applied to two types of uncemented intramedullary nails: femoral stems from total hip arthroplasty and gamma nails. The methodology can be generalized as a procedure to evaluate the quality of the design of commercial femoral implants and their clinic implantation techniques.


The Influence of geometric parameters on the Vibratory Response of Variable Beams

D. Rodriguez

University of Extremadura

Pure analytical solution for complex variable beams subjected to bending concerning both its natural frequencies and mode shapes is presented in this paper. Following the Euler-Bernouilli scheme for slender light cross section variation beams and Henri Bouasses’s analytical procedure the in-plane linear motion differential equation is deduced. Taking into account the bending plane and the variable beam’s geometry a generalized differential equation is derived and characterized by only four parameters: the cross area parameter and the moment of inertia parameter and two exponents depending on the variation polynomial law for the sections with respect to the longitudinal distance to the extreme section of the bar. 

The differential equation is analytically solved using the symbolic Wolfram Mathematica module and then applied to study how those parameters influences the natural frequencies and mode shapes of the beam. Results within the more typical range of the parameters are plotted and compared to those for uniform cross section beams, resulting in a sensitive analysis on how the beam geometry changes its solid’s vibratory behavior.

Results will also be compared - for the case of linear variation - to those given by computer-based numerical analysis and an experiment conducted in the laboratory.



On stationary motion of drive belt as nonlinear elastic rod

A.K. Belyaev

Russian Academy of Sciences

We consider the stationary motion of a drive belt on two pulleys. The belt initial configuration is a circle. Before fitting the belt on the pulleys, they are just tangent to the belt. Then the distance between the pulleys increases, the belt deforms creating the contact zones whose boundaries are unknown a priori and solution dependent.

The belt rod model is more difficult than the most used string model. However this model is necessary for describing the elastic microslip – the most important effect in belt drives which is caused by the friction. Because deformations are not small, the model is geometrically nonlinear. In the elastic microslip modelling we need to account for the transverse shear, and that requires the consideration of the extension in the nonlinear model. The shear is important because of the distributed moments of the friction forces acting on the belt unilaterally. Besides, in the frictionless static contact the contact reactions are significantly different for the cases with shear and without it: in the unshearable model we have the lumped contact forces.

We derive the ODE for the rod moving along the constant trajectory (contour motion). Here we use both the Lagrangian (with material coordinate) and the Eulerian (with spatial arc coordinate) descriptions of motion. We have the full contact instead of the point-wise contact between the belt and pulleys and use Coulomb’s law.

We use Wolfram Mathematica to solve the nonlinear BVP and present the results for several special cases.



Effects of muscle length and physiological cross sectional area on muscle force production: a comparative study

F. Romero

University of Extremadura

The estimation of muscular forces is useful in several areas such as biomedical or rehabilitation engineering. The analysis of the forces that produce a given movement (inverse dynamics, ID) or the movement induced by a set of muscle forces or activations (forward dynamics, FD) are typical problems that need the description of muscle mechanical properties. Moreover, to solve the indeterminacy problem in the ID analysis, optimization schemes are required. Several optimization methods (static optimization, dynamic optimization, augmented static optimization) and optimization criteria (minimum metabolical cost of transport, minimum sum of muscle stresses, time-integral cost of activations, torque-tracking) are available in the literature to that end. In all the aforementioned problems muscle properties, such as muscle length or physiological cross sectional area (PCSA), play a key role in the development of consistent models to perform specific tests. In this work, a comparative study of the effects on muscle force production in muscles with different ratios between length and PSA is presented. Its relevance in the traditional problems faced in biomechanics is also studied.



Design of slender and lightweight rehabilitation orthotics based on Lines of Non Extension

J. Barrios-Muriel

University of Extremadura

The development of dynamic orthoses is a fast-growing field of research and has resulted in many different devices, where all of them try to solve or correct some motion pathology. As these devices are held on the human skin, is behaviour must be a source of inspiration for the safety design of wearable devices. Furthermore, to design a comfortable orthosis the joint motion must be considered. Based on this, the approach of this work for the design of rehabilitation orthotics is to analyse the skin strain field during a prescribed rehabilitation motion. Then, based on anatomical lines with minimum deformation, also called, Lines of non-extension (LoNES) it is possible to design the structural parts of the orthosis. This LoNEs can be obtained from the recorded motion by Digital Image Correlation and different optimization criteria. The structural parts of the orthotic can be designed over these lines, and consequently these parts must be considered as slender structures. In this work the complete process to obtain a functional orthotic device is presented. The design of the slender contours of the orthotic and the optimization (material selection, thicknes, etc.) based on finite element analysis is also described. Different prototypes are analysed to determine the areas with major strain. The resulting prototype is optimize the design based on this information.



Free vibration of sailplane wings: a parametric investigation using bending and torsional rigidities

Huijuan Su

University of Northampton

The research focuses on a detailed parametric investigation for free vibration behaviour of two different types of sailplane wings by altering their bending and torsional rigidities. The formulation used is based on the dynamic stiffness method with particular reference to the application of the Wittrick-Williams algorithm as solution technique. Each of the wings is idealised as an assembly of a number of bending-torsional coupled beams for which the frequency dependent dynamic stiffness matrix is well established. A computer program is developed to obtain the natural frequencies and mode shapes of the baseline sailplane wings with cantilever boundary conditions in the first instance. In the next stage of the investigation, the bending and torsional rigidities of each type of the wings are varied in steps of ±5% between +25% and -25% and their subsequent effects on the natural frequencies and mode shapes are examined. Such a detailed parametric study based on the variations of bending and torsional rigidities shows some interesting results which can be of practical help in the design of sailplane wings. In particular, the investigation reveals that significant modal interchanges between bending and torsional deformation (and vice-versa) can occur due to the changes in the bending and torsional stiffnesses. From an aeroelastic point of view this may have profound consequences. The illustrative examples that are chosen for the baseline wings are representative of existing sailplanes. The results are discussed and the paper concludes with some remarks. 


Mechanical response of a human femoral diaphyseal stabilized fracture using implant plate

Ignacio Herrera

University of Extremadura

The stabilization of a human femoral diaphyseal comminuted fracture using a fixation plate and screw system is performed.  A diaphyseal comminuted fracture represents a disruption in the middle part of the femur with more than two bony fragments and bone loss. As a result, the osteosynthesis implant is subjected to a very high stress. The fracture pattern of an in-vivo implant plate is analysed. An Artificial Vision Programme was developed to build a 3D finite element model by scanning real human femur tomography images and it was then applied to calculate the mechanical response of this lower limb. A special interest represents the mechanical static stresses and displacements that occurred in the screws, plate and fractured bone using the finite element method produced by the forces applied to the femoral head by the hip and to the diaphyseal cortex by tendons. Different cortical screws were studied, varying the diameter and screw pitch. A number of different fixation plate configurations and external bone locations were analysed as well as different mechanical bone properties taking into account heterogeneity and anisotropy of cortical and medullary bone.  Recommendations about optimal plate material strength and plate location for the patient standing up stability and load bearing are concluded.



The design of a fretting wear test machine for thin steel wires

Iñigo Llavori

Mondragon University

Steel wire ropes experience fretting wear damage when the rope runs over a sheave promoting an oscillatory motion between the wires. Consequently, wear scars appear between the contacting wires leading to an increase of the stress field and the following rupture of the wires. That is why the understanding and prediction of the fretting wear phenomena of thin wires is fundamental in order to improve the performance of steel wire ropes.

The present research deals with the design of a self-made fretting wear teste machine for thin steel wires. The test apparatus is designed for testing thin steel wires with a maximum diameter of 0.45 mm, at slip amplitudes ranging from 5 to 300 μm, crossing angle between 0-90º, and contacting force ranging from 0,5 to 5 N. The working principle of displacement amplitude and contacting force as well as the crossing angle between the wires are described. Preliminary studies for understanding the fretting wear characteristics were conducted on 0.45 mm diameter cold-drawn eutectoid carbon steel (0.8% C) with a tensile strength higher than 3000 MPa.



Tests of instability in small structures for laboratory practices of Civil Engineering students

J. Torrecilla

University of Extremadura

The effects of instability are a topic within the structural mechanics that is not always easily assimilated by the students of this discipline. Moving away from the linear behavior of the structures and entering into load regimes in which the effects are conditioned by geometrical aspects is somewhat complicated for those who approach for the first time to the study of such structures.

In the laboratory of structures from the Polytechnic School of Cáceres, we developed a multipurpose device in which students are encouraged to carry out different tests of simple structures, so that they can check how the structures have a predictable behavior through the equations of traditional mechanics. We implemented a trial system of low-cost, fully monitored, for the verification of non-linear effects in slender simple structures, such as beams and small trusses. This work describes test protocol, monitoring system, the practices that can be done by using our device and the results obtained.



The local frame formulation of geometrically nonlinear slender structures

Olivier Brüls

University of Liège

In geometrically exact beam formulations, a local frame is generally defined at every point of the centerline to represent the motion of the cross-section. Similarly, in shell formulations, the local frame represents the director at any point of the reference surface. In this work, we show that the equations of motion of nonlinear beams and shells can be conveniently expressed in this local frame. This approach departs from usual approaches in computational mechanics which are based on the representation of the equilibrium in the inertial frame. It is also very different from corotational formulations or floating fame of reference formulations as it does not involve the definition of an intermediate frame that represent the mean motion of a element or of a flexible body.

A consistent finite element discretization is then proposed using interpolation schemes on the special Euclidean group. Translations and rotations variables are thus treated in a coupled manner and helicoidal shape functions are obtained. Similarly, the time discretization is performed using the Lie group generalized-alpha method. Resorting to differential geometry concepts, the formulation is completely independent of the selected representation of the translation and rotation field.

The proposed local frame formulation exhibits several favorable properties such as the absence of shear locking (without the need to resort to a reduced integration procedure) and the strict independence of the internal elastic forces, tangent stiffness matrix and mass matrix with respect to an arbitrary rigid body motion. In the presentation, these properties will be examplified for several numerical tests involving finite motions of beams and shells.



Cables with transported discrete mass-points

Robert Winkler

Leopold Franzens Universität Innsbruck

The accurate modeling and simulation of ropes and cables includes geometrically nonlinear effects as well as coupled axial, bending and torsional deformation. For the simulation of moving cables, in which the overall axial velocity is almost constant, such as in belt drives, cranes, or ropeway systems, different models exist. Simple models considering discrete mass-spring-damper systems are highly efficient and can run in real-time. They are, however, not applicable for rope-sheave contact modeling. Alternatively, a transient simulation using thin beam elements is possible at high computational costs. Thus, the so-called Arbitrary Lagrange-Eulerian (ALE) formulation can be applied, such that the cable finite elements, e.g. in a belt drive, do not move at the overall speed of the belt drive. Using adaptive meshes, this standard ALE approach can make the simulation highly efficient. However, if discrete masses need to be transported along the rope, such as in ropeway systems or in cranes, the standard ALE formulation cannot be applied.

The present paper attempts to extend existing ALE formulations for moving continuous cables with respect to discrete moving masses. Therefore, the cables are modeled in an ALE manner, while a Lagrangian approach is applied to model the discrete masses. The cables and the discrete masses are coupled by means of constraint conditions and an adaptive length of the cable elements. In such a way, the mesh size can be finer e.g. at the rope-sheave contact and coarser at regions which have less influence on the overall behavior. The paper includes a derivation of the formulation based on the absolute nodal coordinate formulation and numerical results for two and three dimensional benchmark examples in order to show the accuracy and performance of the method.



On vibration models of axially moving slender continua with time-varying length

S. Kaczmarczyk

University of Northampton

Elastic axially moving slender continua such as ropes and belts are used in many engineering systems, as suspension/ power transmission members and mass compensation means in lifting installations, for example. Their response can be predicted by modelling their dynamic behaviour and interactions by representing them as moving strings or beams. The equations of motion are then expressed as linear or nonlinear partial differential equations (PDE) defined in a time-dependent spatial domain spanning the length of the system. The dynamic characteristics change with the length variations rendering the system nonstationary. The response of the system can be analysed in a moving frame with the displacements expressed in terms of a Lagrangian (material) spatial coordinate, or in the classical non-moving (inertial) frame with an Eulerian coordinate applied. Only in simple cases an approximate analytical solution of the PDE model can be obtained, typically by perturbation methods. In more involved problems, with the boundary conditions reflecting practical applications, an approximate solution to the problem can be sought by the Galerkin discretization method so that the problem is expressed by a set of ordinary differential equations (ODE). In engineering applications the most effective and convenient approach is using numerical techniques to solve the resulting problems. In this paper modelling vibrations of an axially moving slender continuum of varying length with concentrated inertia elements are considered. The length of the continua very with time and the boundaries are often unknown. The models are formulated in the moving frame and non-moving of reference, using the Eulerian and Lagrangian formulation, respectively. The correspondence between the two formulations is discussed. It is shown that in many practical applications the length can be considerd as slowly varying meaning that the change over a period corresponding to the instantaneous fundamental frequency of the system is small. The time derivative of length is then proportional to a small parameter which simplifies the model and the solution strategy.



Wind-Induced Vibrations of Lift Ropes in Open-Structure Shafts

S. Kaczmarczyk

University of Northampton

In many installations lifts operate in open structure hoistways (shafts). In such arrangements suspension ropes and compensating cables my suffer from wind-induced vibrations. In this paper wake-induced flutter of a bundle of suspension/ compensating ropes is discussed. The mechanism of wake-induced flutter of a bundle of lift ropes is fundamentally similar to wake-induced instabilities of a small group of cylinders in cross-flow. This type of instability occurs in fundamental mode and may lead to large lateral motions of the ropes causing structural damage to either the ropes themselves or to the hoistway equipment. It can also occur for long enough periods to cause fatigue problems in the ropes. In order to predict the behaviour of the ropes subject to cross-flow excitation a simplified model of a small group of ropes subject to wake-induced flutter is developed. A twin-rope bundle is first considered which comprises a windward rope and leeward rope. In this arrangement the windward rope is assumed to be aerodynamically isolated from the leeward rope. The model can be extended to accommodate a more complex configuration, with comprehensive treatment of the aerodynamic forces acting upon the ropes as well as with their structural dynamics accounted for. Numerical tests are then conducted to predict and to analyse the behaviour of the system. This facilitates the development of relevant control strategies to mitigate the effects of the wake-induced instabilities taking place in open-shaft systems.



The modelling and prediction of dynamic responses of long slender continua deployed in tall structures under long period seismic excitations

S. Kaczmarczyk

University of Northampton

Tall buildings are susceptible to large sway motions when subjected to earthquake excitations. They are particularly affected by long period earthquake ground motions. These low frequency seismic waves resonate with the fundamental mode of the building structure which in turn causes resonance interactions with long slender continua such as lift suspension ropes, compensating ropes and overspeed governor ropes deployed in modular lift installations that provide vertical transportation service in the buildings. Damage due to large resonance motions of suspension/ compensating ropes and cables during earthquake are one of the most common modes of failure in high-rise lift installations. In this paper an analytical model to predict the dynamic responses of suspension/ compensating/ governor rope system installed in tall buildings under seismic conditions is presented. The model is then used to predict the dynamic performance of the system under long period earthquake excitations. The predictions can then be used to develop suitable mitigating strategies and protective measures to minimize the earthquake damage.



Vertical Vibration of Elevator Compensating Sheave due to Brake Activation of Traction Machine

Seiji Watanabe

Advanced Technology R&D Center, Mitsubishi Electric Corp.

Most elevators applied to tall buildings include compensating ropes to satisfy the balanced rope tension between the car and the counter weight. The compensating ropes receive tension by the compensating sheave, which is installed at the bottom space of the elevator shaft. The compensating sheave is only suspended by the compensating ropes, therefore, the sheave can move vertically during the car traveling.

It is important to evaluate the vertical displacement of the compensating sheave, because the displacement is one of the key factor to determine the pit depth of the elevator shaft.

This paper shows the static displacement and the vertical vibration of the compensating sheave. Firstly, an elevator system model is proposed to evaluate the vertical motion of each component. The derived simulation model indicates that the static displacement depends on the car position and the car loading condition. Based on the simulation results, we can produce a simplified mathematical formulation to evaluate the static displacement.

Secondary, the vertical vibration induced by the brake of the traction machine is evaluated numerically. As the simulation results correspond with the experimental ones, we can investigate the worst condition of the vibration by changing the elevator's system parameters. It is concluded that the maximum vibration occurs when the vertical vibration induced by the car stopping synchronizes the phase of the vibration induced by the brake activation. By the result, we can also introduce a mathematical formulation to evaluate the maximum vibration.

Finally, we evaluate the relation between the vertical vibration and the building height. By the derived equation, we can conclude that the amplitude of vertical vibration induced by the brake is linear to the building height.

In the end, the derived simulation model and mathematical formulations contribute to the elevator's optimal design, especially for the pit depth evaluation.


Non-smooth dynamics of a multi-cable driven parallel suspension platform with both bilateral and unilateral constraints

Yandong Wang

China University of Mining and Technology

In this paper, an accurate numerical procedure is proceeded to simulate the non-smooth dynamical responses in a multi-cable driven parallel suspension platform system. For such systems, the cables might become slack due to external excitations and due to the fact that cables can only bear unilateral loads in longitudinal direction. While in lateral and torsional directions, the constraints between the cables and platform are bilateral. This paper will deal with the non-smooth cable vibrations by taking both the unilateral and bilateral properties of the cables into consideration. Firstly, the Lagrange equation with constraints is employed to derive the equations of motion of the multi-cable suspension platform considering both the unilateral and bilateral constraints. Then, by modifying the complementary condition by means of a non-smooth generalized-α scheme, the dynamic equations can be solved numerically. Finally, the numerical results are compared with an ADAMS simulation, and the two results agree well with each other. Moreover, the results in this paper significantly improve the numerical results used in dynamics for multi-cable systems which usually neglect the lateral properties of the cables.



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