A Consistent Approach to Problem Solving in Mechanical Vibrations

2018 ◽  
Author(s):  
Amir H. Danesh-Yazdi ◽  
Yi Wu ◽  
Oladipo Onipede
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Solution Method ◽  
Traditional Approach ◽  
Solution Technique ◽  
Mdof Systems ◽  
Solution Methods ◽  
Vibration Problems ◽  
The Mathematical Model ◽  
Consistent Approach

A consistent approach to solving problems in an undergraduate vibrations course in Mechanical Engineering is presented in this paper. The traditional approach of solving vibration problems involves several steps such as classifying the system according to degrees of freedom, free or forced vibrations and with or without damping. Based on the classification, an appropriate solution technique is applied and the results are obtained. Since the mathematical solution technique is strictly tied to the classification, students have to learn and apply a variety of solution methods based on the particular form of the mathematical model. The course was literally more like a math course rather than an engineering course. By introducing students to the state-space solution method early in the course and using it as the main/dominant solution method, students can focus more on learning both the physical modeling and mathematical modeling of the vibration systems as well as interpreting results in the engineering context. Since state-space computational solvers are readily available to students (MATLAB, Mathcad, etc.) and they can be applied to solve most (but not all) vibration problems including free or forced SDOF, 2DOF, MDOF systems with or without damping, it allows for consistency when teaching students how to solve vibration problems. State-space solvers can solve for either the time or the frequency response and provides a graphical solution. The students can go from modeling to visually exploring and interpreting results. The students’ response to this approach is also discussed.

Development of Mathematical Models, Simulating Vibration Control of Tracked Vehicle Weapon Dynamics

2017 ◽  
Vol 67 (4) ◽  
pp. 465 ◽  
Author(s):  
Ambarish Jakati ◽  
Saayan Banerjee ◽  
C. Jebaraj
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Control Strategies ◽  
Tracked Vehicle ◽  
Alternate Control ◽  
Space Matrix ◽  
Elevation Model ◽  
Dynamics Response ◽  
Vehicle Chassis ◽  
The Mathematical Model

Focuses upon development of the mathematical model, simulating the tracked vehicle weapon dynamics, integrated over a half car platform. Governing differential equations have been formulated for the weapon system using state space approach, simulating the elevation dynamics over a half vehicle chassis, and coded using Matlab. The elevation model of the weapon comprises 3 degrees of freedom, arising from the rotational dynamics of the drive, breech and muzzle, which has sequentially been coupled to the half car model. Thereafter, the backstepping, LQR and PID control techniques have been derived and incorporated into the state space matrix for the coupled dynamics model, in which the control parameters have been arrived at through various iterations. Comparative weapon dynamics response studies have been carried out between that obtained from the above control strategies and the passive model, over standard terrain conditions at specified speeds. The above study would form a very useful framework for implementation of alternate control strategies for weapon stabilisation in the full tracked vehicle.

scholarly journals A New Approach of Soft Joint Based on a Cable-Driven Parallel Mechanism for Robotic Applications

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1468
Author(s):  
Luis Nagua ◽  
Carlos Relaño ◽  
Concepción A. Monje ◽  
Carlos Balaguer
Keyword(s):  
Parallel Mechanism ◽  
Degrees Of Freedom ◽  
Kinematic Model ◽  
Experimental Tests ◽  
Humanoid Robots ◽  
Inverse Kinematic ◽  
Element Analysis ◽  
New Approach ◽  
Flexible Polymer ◽  
The Mathematical Model

A soft joint has been designed and modeled to perform as a robotic joint with 2 Degrees of Freedom (DOF) (inclination and orientation). The joint actuation is based on a Cable-Driven Parallel Mechanism (CDPM). To study its performance in more detail, a test platform has been developed using components that can be manufactured in a 3D printer using a flexible polymer. The mathematical model of the kinematics of the soft joint is developed, which includes a blocking mechanism and the morphology workspace. The model is validated using Finite Element Analysis (FEA) (CAD software). Experimental tests are performed to validate the inverse kinematic model and to show the potential use of the prototype in robotic platforms such as manipulators and humanoid robots.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1995 ◽  
Vol 117 (3) ◽  
pp. 582-588 ◽  
Author(s):  
L. N. Virgin ◽  
T. F. Walsh ◽  
J. D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations, which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions, and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors, which described the practical context of the work, free vibration, control aspects, and derivation of the mathematical model.

Design and Experimental Validation of a LQG Control System for the Stabilization of a 2DOF Commercial Gimbal Using Physical Modelling Techniques

2020 ◽  
Author(s):  
Julián Andres Gómez Gómez ◽  
Camilo E. Moncada Guayazán ◽  
Sebastián Roa Prada ◽  
Hernando Gonzalez Acevedo
Keyword(s):  
Mathematical Model ◽  
Control System ◽  
State Space ◽  
Transient Response ◽  
Space Form ◽  
Angular Position ◽  
Physical Modelling ◽  
Motor Shaft ◽  
Brushless Dc ◽  
The Mathematical Model

Abstract Gimbals are mechatronic systems well known for their use in the stabilization of cameras which are under the effect of sudden movements. Gimbals help keeping cameras at previously defined fixed orientations, so that the captured images have the highest quality. This paper focuses on the design of a Linear Quadratic Gaussian, LQG, controller, based on the physical modeling of a commercial Gimbal with two degrees of freedom (2DOF), which is used for first-person applications in unmanned aerial vehicle (UAV). This approach is proposed to make a more realistic representation of the system under study, since it guarantees high accuracy in the simulation of the dynamic response, as compared to the prediction of the mathematical model of the same system. The development of the model starts by sectioning the Gimbal into a series of interconnected links. Subsequently, a fixed reference system is assigned to each link body and the corresponding homogeneous transformation matrices are established, which will allow the calculation of the orientation of each link and the displacement of their centers of mass. Once the total kinetic and potential energy of the mechanical components are obtained, Lagrange’s method is utilized to establish the mathematical model of the mechanical structure of the Gimbal. The equations of motion of the system are then expressed in state space form, with two inputs, two outputs and four states, where the inputs are the torques produced by each one of the motors, the outputs are the orientation of the first two links, and the states are the aforementioned orientations along with their time derivatives. The state space model was implemented in MATLAB’s Simulink environment to compare its prediction of the transient response with the prediction obtained with the representation of the same system using MATLAB’s SimMechanics physical modelling interface. The mathematical model of each one of the three-phase Brushless DC motors is also expressed in state space form, where the three inputs of each motor model are the voltages of the corresponding motor phases, its two outputs are the angular position and angular velocity, and its four states are the currents in two of the phases, the orientation of the motor shaft and its rate of change. This model is experimentally validated by performing a switching sequence in both the simulation model and the physical system and observing that the transient response of the angular position of the motor shaft is in accordance with the theoretical model. The control system design process starts with the interconnection of the models of the mechanical components and the models of the Brushless DC Motor, using their corresponding state space representations. The resulting model features six inputs, two outputs and eight states. The inputs are the voltages in each phase of the two motors in the Gimbal, the outputs are the angular positions of the first two links, and the states are the currents in two of the phases for each motor and the orientations of the first two links, along with their corresponding time derivatives. An optimal LQG control system is designed using MATLAB’s dlqr and Kalman functions, which calculate the gains for the control system and the gains for the states estimated by the observer. The external excitation in each of the phases is carried out by pulse width modulation. Finally, the transient response of the overall system is evaluated for different reference points. The simulation results show very good agreement with the experimental measurements.

scholarly journals Robust and Optimal Output-Feedback Control for Interval State-Space Model: Application to a Two-Degrees-of-Freedom Piezoelectric Tube Actuator

2018 ◽  
Vol 141 (2) ◽  
Author(s):  
Mounir Hammouche ◽  
Philippe Lutz ◽  
Micky Rakotondrabe
Keyword(s):  
State Space ◽  
Output Feedback ◽  
Degrees Of Freedom ◽  
Closed Loop ◽  
State Space Model ◽  
Output Feedback Control ◽  
Optimal Solution ◽  
Loop System ◽  
Closed Loop System ◽  
Optimal Output

The problem of robust and optimal output feedback design for interval state-space systems is addressed in this paper. Indeed, an algorithm based on set inversion via interval analysis (SIVIA) combined with interval eigenvalues computation and eigenvalues clustering techniques is proposed to seek for a set of robust gains. This recursive SIVIA-based algorithm allows to approximate with subpaving the set solutions [K] that satisfy the inclusion of the eigenvalues of the closed-loop system in a desired region in the complex plane. Moreover, the LQ tracker design is employed to find from the set solutions [K] the optimal solution that minimizes the inputs/outputs energy and ensures the best behaviors of the closed-loop system. Finally, the effectiveness of the algorithm is illustrated by a real experimentation on a piezoelectric tube actuator.

scholarly journals Limiting Vibroisolation Control of an Oscillating String on a Moving Base

1995 ◽  
Vol 2 (2) ◽  
pp. 163-171 ◽  
Author(s):  
Dmitry V. Balandin
Keyword(s):  
Solution Method ◽  
Solution Technique ◽  
Quadratic Functional ◽  
Numerical Example ◽  
Guaranteed Quality ◽  
Moving Base ◽  
Elastic Object

The vibroisolating capability of an elastic object that is assumed to be protected against a class of excitations is studied. It is proposed that this capability be estimated by a quadratic functional. The solution method gives an estimate of the optimal isolation with a criterion of minimum guaranteed quality. A numerical example of the solution technique is presented.

Time Domain Hydroelastic Analysis of a Flexible Marine Structure Using State-Space Models

2007 ◽  
Author(s):  
Reza Taghipour ◽  
Tristan Perez ◽  
Torgeir Moan
Keyword(s):  
Mathematical Model ◽  
State Space ◽  
Time Domain ◽  
State Space Model ◽  
Realization Theory ◽  
Regular Waves ◽  
Alternative State ◽  
Marine Structure ◽  
Hydroelastic Analysis ◽  
The Mathematical Model

This article deals with time-domain hydroelastic analysis of a marine structure. The convolution terms in the mathematical model are replaced by their alternative state-space representations whose parameters are obtained by using the realization theory. The mathematical model is validated by comparison to experimental results of a very flexible barge. Two types of time-domain simulations are performed: dynamic response of the initially inert structure to incident regular waves and transient response of the structure after it is released from a displaced condition in still water. The accuracy and the efficiency of the simulations based on the state-space model representations are compared to those that integrate the convolutions.

scholarly journals IASI-METOP and MIPAS-ENVISAT data fusion

2010 ◽  
Vol 10 (1) ◽  
pp. 183-211 ◽  
Author(s):  
S. Ceccherini ◽  
U. Cortesi ◽  
S. Del Bianco ◽  
P. Raspollini ◽  
B. Carli
Keyword(s):  
Data Fusion ◽  
Degrees Of Freedom ◽  
Solution Method ◽  
Information Gain ◽  
A Priori ◽  
A Priori Information ◽  
Priori Information ◽  
Single Instrument ◽  
Fusion Products ◽  
Temporal Coverage

Abstract. The combination of data obtained with different sensors (data fusion) is a powerful technique that can provide target products of the best quality in terms of precision and accuracy, as well as spatial and temporal coverage and resolution. In this paper the results are presented of the data fusion of measurements of ozone vertical profile performed by two space-borne interferometers (IASI on METOP and MIPAS on ENVISAT) using the new measurement-space-solution method. With this method both the loss of information due to interpolation and the propagation of possible biases (caused by a priori information) are avoided. The data fusion products are characterized by means of retrieval errors, information gain, averaging kernels and number of degrees of freedom. The analysis is performed both on simulated and real measurements and the results demonstrate and quantify the improvement of data fusion products with respect to measurements of a single instrument.

Dynamical Model Updating Based on Gradient Regularization Method

2013 ◽  
Vol 351-352 ◽  
pp. 118-121
Author(s):  
He Long Xu ◽  
Jun Xiao ◽  
Yu Xin Zhang
Keyword(s):  
Regularization Method ◽  
Solution Method ◽  
Model Updating ◽  
Input Parameter ◽  
Mode Shapes ◽  
Problem Solution ◽  
Finite Element Program ◽  
Element Program ◽  
Important Input ◽  
The Mathematical Model

Modulus of elasticity is an important input parameter in all kinds of structural analyses. The mathematical model used to identify the structural elastic modulus with measured Frequencies and mode shapes at several points is thusly built up in this paper, and then Gradient-Regularization method, an inverse problem solution method, is employed to solve the problem. General finite element program is compiled, and numerical examples have proved that the method of this thesis is efficient. The issues such as the choice of model error and the choice of measuring points are discussed as well.

Transient Response of MDOF Systems With Inerters to Nonstationary Stochastic Excitation

2017 ◽  
Vol 84 (10) ◽  
Author(s):  
Sami F. Masri ◽  
John P. Caffrey ◽  
Hui Li
Keyword(s):  
Degrees Of Freedom ◽  
Broad Band ◽  
Primary System ◽  
Mean Square ◽  
Stochastic Excitation ◽  
Mass Ratio ◽  
Tuning Parameters ◽  
Mdof Systems ◽  
Mean Square Response ◽  
Arbitrary Location

Explicit, closed-form, exact analytical expressions are derived for the covariance kernels of a multi degrees-of-freedom (MDOF) system with arbitrary amounts of viscous damping (not necessarily proportional-type), that is equipped with one or more auxiliary mass damper-inerters placed at arbitrary location(s) within the system. The “inerter” is a device that imparts additional inertia to the vibration damper, hence magnifying its effectiveness without a significant damper mass addition. The MDOF system is subjected to nonstationary stochastic excitation consisting of modulated white noise. Results of the analysis are used to determine the dependence of the time-varying mean-square response of the primary MDOF system on the key system parameters such as primary system damping, auxiliary damper mass ratio, location of the damper-inerter, inerter mass ratio, inerter node choices, tuning of the coupling between the damper-inerter and the primary system, and the excitation envelope function. Results of the analysis are used to determine the dependence of the peak transient mean-square response of the system on the damper/inerter tuning parameters, and the shape of the deterministic intensity function. It is shown that, under favorable dynamic environments, a properly designed auxiliary damper, encompassing an inerter with a sizable mass ratio, can significantly attenuate the response of the primary system to broad band excitations; however, the dimensionless “rise-time” of the nonstationary excitation substantially reduces the effectiveness of such a class of devices (even when optimally tuned) in attenuating the peak dynamic response of the primary system.

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