IMPROVED ANALYSIS OF A PROPPED CANTILEVER UNDER LATERAL VIBRATION

Continuous systems are sometimes analysed as lumped masses connected by massless elements. This reduces the structure’s degree of freedom and therefore simplifies the analysis. However this over simplification introduces an error in the analysis and the results are therefore approximate. In this work sections of the vibrating beam were isolated and the equations of the forces causing vibration obtained using the Hamilton’s principle. These forces were applied to the nodes of an equivalent lumped mass beam and the stiffness modification needed for it to behave as a continuous beam obtained. The beam’s stiffness was modified using a set of stiffness modification factors to . It was observed that by applying these factors in the dynamic analysis of the beam using the Lagrange’s equation, we obtain the exact values of the fundamental frequency irrespective of the way the mass of the beam was lumped. From this work we observed that in order to obtain an accurate dynamic response from a lumped mass beam there is need to modify the stiffness composition of the system and no linear modification of the stiffness distribution of lumped mass beams can cause them to be dynamically equivalent to the continuous beams. This is so because the values of the modification factors obtained for each beam segment were not equal. The stiffness modification factors were obtained for elements at different sections of the beam

Mathematical Modelling of Tank Wagon Vibrations Considering Partially Filling of the Tank with Liquid Cargo

Mathematical modelling of processes of motion makes it possible to assess the dynamic characteristics of a wagon at the stage of its design. However, it is necessary to consider the type of cargo transported, the movement of which affects the values of these features.The paper considers a mathematical model of an eight-axle railway tank wagon developed using the Lagrange’s equation of the second kind. The considered mathematical model suggests an approach based on the consideration of the influence of the energy of a liquid cargo in a steady state of motion. This influence was considered by evaluating the kinetic and potential energies of vibrations of the transported liquid cargo.Differential equations of vibration compiled for the model under consideration represent the liquid cargo as a solid. The approach for considering the effect of liquid cargo during vibrations of a tank wagon assumes that the total volume of the displaced liquid approximately corresponds to the volume of the layer of the fluid determined by displacement of bouncing, or in the case of galloping, with an angular displacement of one end section of the tank wagon, the second section rises by the same value, in other words, we observe the system of communicating vessels. Based on these assumptions, energy additions are obtained that consider movement of a liquid cargo under steady-state modes of motion.According to the proposed approach, preliminary calculations were performed, and the results obtained were assessed. The results obtained showed satisfactory convergence with the calculations carried out using other approaches to modelling of the processes of movement of railway tank wagons.

Time-Delayed Acceleration Feedback Control of a Single-Link Flexible Manipulator Using Kalman Filter

The design problem of a discrete controller with time delay and acceleration feedback for a single-link flexible manipulator system is addressed in this paper. The dynamical model of a single-link flexible manipulator system is presented by the adoption of the finite element method and Lagrange’s equation. Based on the random-walk process and the discrete reduction method, an augmented discretized delay-free state derivate space equation containing the random noise is established. An acceleration-based Kalman filtering method is developed in order to estimate the system state and external excitation necessary for the controller design. In light of the estimated augmented states, a hybrid controller that combines a feedback control algorithm and a feedforward control algorithm is designed according to optimal control theory and Moore–Penrose theory. Numerical simulation results show that the proposed controller can damp out the vibration response of the flexible manipulator system effectively upon external excitations. Moreover, it is further revealed that the control performance of the presented method can be improved by adding the time delay appropriately.

A Virtual Prototyping Model on Dynamic Behaviors of Prefabricated Bridges

With loads on prefabricated bridges becoming more and more heavier, their dynamic behaviors should be paid more attention to. A virtual prototyping model with the object-oriented technology and the mode synthesis method is presented in MATLAB to analyze dynamic behaviors of prefabricated bridges. Using structural characteristics of the bridges properly, substructures can be looked on as objects which can be encapsulated with the mode synthesis method. The compatibility conditions on interfaces, the Lagrange’s equation and Ritz’s variational principle play the role of messages transmitting among objects. Simulation results indicate that the present model can easily study dynamic behaviors of bridges in the same model.

Fault-tolerant control of flexible satellite with magnetic actuation and reaction wheel

The attitude fault-tolerant control of a flexible satellite actuated by reaction wheels and magnetic torquer bars is investigated in this article. A low earth orbit is considered for moment perturbations such as drag and gravity gradient. Furthermore, the flexible panels attached to a rigid central body are modeled through the assumed mode approach by a finite set of bending modal motion. The ordinary differential equations of their generalized coordinates are found using Lagrange’s equation, and the resulting dynamical model is validated by comparing its simulation results to the NX Siemens software results. Finally, a fault-tolerant controller based on sliding mode control is suggested and tested in different scenarios. It is showed that the proposed control method tolerates the actuators’ faults and controls the satellite’s attitude while desaturating the reaction wheels.

Dynamic Simulation of Reeving Systems With the Extension of the Modal Approach in the Axial Direction

In this work, the simulation of reeving systems has been studied by including axial modes using the Arbitrary Lagrangian-Eulerian (ALE) description. The reeving system is considered as a deformable multibody system in which the rigid bodies are connected by the elastic wire ropes through sheaves and reels. A set of absolute nodal coordinates and modal coordinates is employed to describe the motion and deformation in the axial direction. This new method allows the analysis of elements with non-constant axial strain along its length. In addition, modal coordinates are employed to describe the dynamic motion in the transverse direction. The non-constant axial displacement within the wire rope is computed in terms of the absolute position coordinates, longitudinal material coordinates, and modal deformation coordinates. To derive the governing equations of motion, Lagrange’s equation is employed. The formulation is validated for a simple pendulumlike motion actuated by an initial velocity. The simulation results are provided to trace the movements of the payload. It can be seen that by adding modal coordinates, the axial force within the element changes. Moreover, the effects of modal coordinates in the axial direction are presented for a different number of nodes, and the resulting axial forces are compared with reference solution.

Potential Energy as Design Criterion in Planar Multistable Mechanisms

Toggle mechanisms are used throughout engineering to accomplish various tasks, for example residential electrical switching. The design of toggle mechanisms can be broken into three categories: determination of a topology, geometric parameterization, and optimization. While topological determination and optimization have well established processes for use in design, geometric parameterization which includes defining link lengths and spring stiffness has largely been left to engineering judgement. This paper presents a design methodology using potential energy graphs which informs the engineering decisions made in choosing mechanism parameters, giving designers higher confidence in the design. A kinematic analysis coupled with Lagrange’s equation determines the relationship between the mechanism parameters and the potential energy curve. Plotting the potential energy with respect to the generalized coordinate yields a graph with a slope that is the generalized force or moment. The relationships between parameters and their effects on the mechanism are difficult to observe in the equations of motion, but potential energy plots readily provide information pertinent to the design of toggle mechanisms and decouple their effects. The plots also allow design by position rather than time which makes the design process faster. The design process is applied to three examples: a simple toggle mechanism, a compliant mechanism, and a reconfigurable mechanism to show the nuances of the approach.

A new pendulum motion with a suspended point near infinity

In this paper, a pendulum model is represented by a mechanical system that consists of a simple pendulum suspended on a spring, which is permitted oscillations in a plane. The point of suspension moves in a circular path of the radius (a) which is sufficiently large. There are two degrees of freedom for describing the motion named; the angular displacement of the pendulum and the extension of the spring. The equations of motion in terms of the generalized coordinates $$\varphi$$ φ and $$\xi$$ ξ are obtained using Lagrange’s equation. The approximated solutions of these equations are achieved up to the third order of approximation in terms of a large parameter $$\varepsilon$$ ε will be defined instead of a small one in previous studies. The influences of parameters of the system on the motion are obtained using a computerized program. The computerized studies obtained show the accuracy of the used methods through graphical representations.

Study on Dynamic Response of Straddle-Type Monorail Vehicle with Single-Axle Bogie Under Curve Condition

This paper is concerned with the dynamic response of straddle-monorail with single-axle bogie under curve condition. A 15 degrees-of-freedom(DOF) dynamic model is established for straddle-type monorail vehicle with single-axle bogie, which consists driving wheels, steering wheels and stabilizing wheels. The motion equations of the straddle-type monorail vehicle are derived using the Lagrange's equation, and the wheel-rail contact model and the curving track beam model are created. Compared with the test results, the accuracy of the method is verified. Finally, the influence of curve radius, curve superelevation rate, number of passengers and stiffness of driving wheels on dynamic response is discussed.

Characteristic Analysis and Optimization of a Four-bar Compliant Mechanism with Single Flexible Member

The characteristics and optimization analysis of a four-bar compliant mechanism with one flexible member (or flexible joint) are carried out. Firstly, based on the pseudo-rigid body model theory of the compliant mechanism, the kinematics relationship, system kinetic energy and potential energy of the general four-bar compliant mechanism are analyzed, and the dynamic model of the four-bar compliant mechanism is established by using Lagrange's equation. Secondly, through the creation of the energy equation of the four-bar compliant mechanism and the analysis of the first and second derivatives of the input variables, the parametric conditions for the existence of the bistable characteristics of the four-bar compliant mechanism with one flexible member are proposed. Then, taking the compliant bistable switch as an example, the internal relations between the driving characteristics of the four-bar compliant mechanism, the initial motion position of the mechanism and the parameters of the flexible member are explored. Finally, based on the improvement of the performance of the compliant bistable switch, the optimization analysis of the maximization of the motion range of the driving link and the maximization of the deformation energy of the flexible member are carried out. The research provides a theoretical basis for the product development and control of a bistable four-bar compliant mechanism.