On establishing equations of motion of mechanical vibration systems placed on moving bases

2017 ◽  
Vol 45 (3) ◽  
pp. 209-227
Author(s):  
M Gürgöze ◽  
F Terzioğlu
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Main Idea ◽  
Vertical Axis ◽  
Constant Angular Velocity ◽  
Vibration System ◽  
Reaction Forces ◽  
The Subject ◽  
Vibrating Systems

The first author has been teaching the postgraduate course, “The Dynamics of Mechanical Systems” in The ITU Faculty of Mechanical Engineering for nearly 20 years. He has observed that students frequently have problems in obtaining the equations of motion of the vibrating systems which were placed on moving bases. Starting from this observation, he has found that the homework stated below, which was given to the students occasionally, was very helpful in learning the subject. The main idea of the homework is the derivation of the equations of motion, with the help of formulating the Lagrange’s equations with respect to a moving set of axis for a vibration system with two degrees of freedom which consists of a horizontal table rotating with a constant angular velocity around a vertical axis. The students were also asked to solve the same problem with a different method of their choice and to determine the reaction forces as well. We want to share this problem with the reader, which we have assessed as very instructive and appropriate from the viewpoint of applicability of different methods.

Influence of Bistable Plunge Stiffness On Nonlinear Airfoil Flutter

2021 ◽  
Author(s):  
Renan F. Corrêa ◽  
Flávio D. Marques
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Nonlinear Behavior ◽  
Limit Cycle Oscillation ◽  
Restoring Force ◽  
Aeroelastic System ◽  
Technical Literature ◽  
Practical Applications

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry’s influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell, and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation amplitudes and also delays the occurrence of the snap-through.

Computational Analysis of Multibody Dynamic Systems Using Nonlinear Recursive Formulation

2009 ◽  
Vol 419-420 ◽  
pp. 289-292
Author(s):  
Yunn Lin Hwang ◽  
Shen Jenn Hwang ◽  
Zi Gui Huang ◽  
Ming Tzong Lin ◽  
Yen Chien Mao ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Recursive Formulation ◽  
Loosely Coupled ◽  
Multibody Dynamic ◽  
Joint Coordinates ◽  
Joint Reaction Forces ◽  
Reaction Forces ◽  
Strong Dynamic ◽  
Joint Reaction

. In this paper the computer implementation of the nonlinear recursive formulation in multibody dynamics systems is described. The organization of the computer algorithm which is used to automatically construct and numerically solve the system of loosely coupled dynamic equations expressed in terms of the absolute and joint coordinates is discussed. The inertia projection schemes used in most existing recursive formulations for the dynamic analysis of deformable mechanisms lead to dense coefficient matrices in the equations of motion. Consequently, there are strong dynamic couplings between the joint and elastic coordinates. By using the inertia matrix structure of deformable mechanical systems and the fact that the joint reaction forces associated with the elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent of the number of elastic degrees of freedom is obtained. This system can be solved for the joint accelerations as well as the joint reaction forces. The multibody flexible four-bar system is used as an example to demonstrate the use of the procedure discussed in this paper.

scholarly journals Minimization of dynamic joint reaction forces of the 2-DOF serial manipulators based on interpolating polynomials and counterweights

2015 ◽  
Vol 42 (4) ◽  
pp. 249-260 ◽  
Author(s):  
Slavisa Salinic ◽  
Marina Boskovic ◽  
Radovan Bulatovic
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Lagrange Equations ◽  
Symbolic Form ◽  
Serial Manipulators ◽  
Joint Reaction Forces ◽  
Reaction Forces ◽  
Inertia Forces ◽  
Joint Reaction ◽  
Selection Of

This paper presents two ways for the minimization of joint reaction forces due to inertia forces (dynamic joint reaction forces) in a two degrees of freedom (2-DOF) planar serial manipulator. The first way is based on the optimal selection of the angular rotations laws of the manipulator links and the second one is by attaching counterweights to the manipulator links. The influence of the payload carrying by the manipulator on the dynamic joint reaction forces is also considered. The expressions for the joint reaction forces are obtained in a symbolic form by means of the Lagrange equations of motion. The inertial properties of the manipulator links are represented by dynamical equivalent systems of two point masses. The weighted sum of the root mean squares of the magnitudes of the dynamic joint reactions is used as an objective function. The effectiveness of the two ways mentioned is discussed.

scholarly journals ELECTROMAGNETIC DEFLECTION OF SPINNING PARTICLES

1994 ◽  
Vol 09 (03) ◽  
pp. 461-473 ◽  
Author(s):  
JOHN P. COSTELLA ◽  
BRUCE H.J. MCKELLAR
Keyword(s):  
Magnetic Dipole ◽  
Magnetic Dipole Moment ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Radiation Reaction ◽  
Exact Form ◽  
Dipole Force ◽  
Self Consistent ◽  
Recent Debate ◽  
The Subject

We show that it is possible to obtain self-consistent and physically acceptable relativistic classical equations of motion for a pointlike spin-half particle possessing an electric charge and a magnetic dipole moment, directly from a manifestly covariant Lagrangian, if the classical degrees of freedom are appropriately chosen. It is shown that the equations obtained encompass the well-tested Lorentz force and Thomas-Bargmann-Michel-Telegdi spin equations, as well as providing a definite specification of the classical magnetic dipole force, whose exact form has been the subject of recent debate. Radiation reaction — the force and torque on an accelerated particle due to its self-interaction — is neglected at this stage.

scholarly journals Revolving fluid in the atmosphere

1917 ◽  
Vol 94 (656) ◽  
pp. 34-52 ◽  
Keyword(s):  
Equations Of Motion ◽  
Vertical Axis ◽  
Inviscid Fluid ◽  
Present Purpose ◽  
Dynamical Aspect ◽  
Starting Point ◽  
The Subject ◽  
Definition Of

In a paper read on December 8, 1916, Lord Rayleigh makes an important contribution to the dynamics of revolving fluids, taking as “the starting point of part of his investigation,” the paper by Dr. Aitken on “The Dynamics of Cyclones and Anticyclones.” After setting out the general equations of motion of an inviscid fluid Lord Rayleigh says “for the present purpose we assume symmetry with respect to the axis of z so that u, v, w and P (assumed to be single valued) are independent of θ .” I take that to be the definition of a revolving fluid for the purpose of the subject under consideration, and it is that form of motion, that is to say, motion which is symmetrical with regard to a vertical axis, that Lord Rayleigh had in mind when he wrote the opening sentence of the paper: “So much of meteorology depends ultimately upon the dynamics of revolving fluid that it is desirable to formulate as clearly as possible such simple conclusions as are within our reach.” For most ordinary readers, meteorology in its dynamical aspect is quite rightly regarded merely as a useful synonym for “Cyclones and Anticyclones.” Dr. Aitken illustrates his view's of the nature of cyclones and anticyclones by many interesting experiments in the dynamics of revolving masses, and when Lord Rayleigh is moved by Dr. Aitken’s paper to set down clearly the conclusions that can be drawn from the theory of revolving fluids it is apparently with the hope that those conclusions may find their application in the phenomena exhibited by cyclones and anticyclones.

Comportement dynamique d'une éolienne à axe vertical (développement théorique)

1988 ◽  
Vol 15 (3) ◽  
pp. 355-368
Author(s):  
Mario Veilleux ◽  
René Tinawi
Keyword(s):  
Degrees Of Freedom ◽  
Dynamic Instability ◽  
Equations Of Motion ◽  
Vertical Component ◽  
Vertical Axis ◽  
Nonlinear Static Analysis ◽  
Mode Shapes ◽  
Vertical Axis Wind Turbine ◽  
Qr Algorithm ◽  
Gyroscopic Effects

Complex frequencies and mode shapes are evaluated and presented for a guyed vertical axis wind turbine to detect any dynamic instability for a given speed of rotation. The equations of motion are developed in the rotating system of axes of the rotor to eliminate the time dependent terms. These equations take into account gyroscopic effects by evaluating the Coriolis and Circulatory (softening) matrices. The guys are replaced by nonlinear springs. The geometric stiffness matrix is also considered by performing a nonlinear static analysis that includes centrifugal effects and gravity loads, as well as compression from the vertical component of the guys. A reduction of the dynamic degrees of freedom is performed using the Rayleigh–Ritz technique. The complex frequencies and mode shapes are obtained using the QR algorithm. A program developed on a microcomputer was used to evaluate the dynamic instabilities of the ÉOLE Project. These results are described in the following paper. Key words: Vertical axis wind turbines, guys, complex frequencies, complex mode shapes, centrifugal forces, Coriolis forces, numerical software.

Vibration of a Spring-Supported Body

1965 ◽  
Vol 7 (2) ◽  
pp. 185-192 ◽  
Author(s):  
P. Grootenhuis ◽  
D. J. Ewins
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Vertical Axis ◽  
The Body ◽  
Stiffness Ratio ◽  
Six Degrees Of Freedom ◽  
Centre Of Gravity ◽  
Longitudinal Stiffness ◽  
The Matrix

The equations of motion for a rigid body supported on four springs are derived for the general case of the centre-of-gravity being anywhere within the body and allowing for the sideways as well as the longitudinal stiffnesses of the springs. This constitutes a six-degrees-of-freedom case with three degrees of asymmetry. Coupling between motions in all directions occurs even when the centre-of-gravity is at the geometric centre with the exception then of vertical oscillations and rotation about the vertical axis. Any number of additional springs can be allowed for by adding terms to the expression for the potential energy stored in the springs. Allowance is made in the expression for kinetic energy for the products of inertia which arise with an offset centre-of-gravity. The real case is simulated for purposes of analysis by replacing the rigid body by a rectangular box with a light framework and all the mass concentrated at the eight corners. The matrix solution is changed into dimensionless parameters and the effect of an offset centre-of-gravity upon the eigenvalues and eigenvectors studied. Only the proportions of the box and the stiffness ratio between sideways to longitudinal stiffness of the springs remain as factors. The numerical example given is for proportions of height to width to length of 3/4/5 and for a stiffness ratio of 5. Small amounts of offset of the centre-of-gravity from the geometric centre do not alter the dynamic behaviour of the system much but displacing the total mass towards either a lower or an upper corner has marked effects. Some of the natural frequencies associated with motion in rotation when the system is symmetric become less than the frequencies connected with motion in translation for the centre-of-gravity being close to a corner connected to a spring. A large region free from any natural frequency arises when the centre-of-gravity is moved towards a corner furthest removed from the plane containing the springs. The asymptotic conditions for the position of the centre-of-gravity are also considered.

scholarly journals Diversity and Regularity of Periodic Impact Motions of a Mechanical Vibration System with Multiple Rigid Stops

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Fengwei Yin ◽  
Guanwei Luo ◽  
Xueming Wang
Keyword(s):  
Dynamic Characteristics ◽  
Simulation Analysis ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Low Frequency ◽  
Model Parameters ◽  
Vibration System ◽  
Parametric Plane ◽  
Parameter Matching ◽  
Dynamical Parameters

The mechanical model of a two-degree-of-freedom vibration system with multiple rigid stops was established, and the effects of the multiple rigid stops to dynamic characteristics of two mass blocks of the system were studied. The judgment conditions and differential equations of motion of the system masses impacting rigid stops were analyzed. Based on the multiparameter and multiobjective collaborative simulation analysis, the correlation between the dynamic characteristics of the vibration system and the model parameters is studied. The basic periodic and subharmonic impact motions are analyzed with emphasis on the influences of dynamical parameters on the mode diversity and the distribution characteristics, and the law of emergence and competition of various periodic impact motions on the parametric plane is revealed. The singular points, the hysteresis transition domains, and the accompanying codimension-two bifurcations, caused by the irreversibility of the transition between adjacent basic periodic impact motions in the low-frequency domain, are analyzed. The reasonable parameter matching range, associated with dynamic characteristic optimization of the system, is determined.

scholarly journals Improving Inverse Dynamics Accuracy in a Planar Walking Model Based on Stable Reference Point

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Alaa Abdulrahman ◽  
Kamran Iqbal ◽  
Gannon White
Keyword(s):  
Human Body ◽  
Reference Point ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
The Body ◽  
Sensor Position ◽  
Reaction Forces ◽  
The Cost

Physiologically and biomechanically, the human body represents a complicated system with an abundance of degrees of freedom (DOF). When developing mathematical representations of the body, a researcher has to decide on how many of those DOF to include in the model. Though accuracy can be enhanced at the cost of complexity by including more DOF, their necessity must be rigorously examined. In this study a planar seven-segment human body walking model with single DOF joints was developed. A reference point was added to the model to track the body’s global position while moving. Due to the kinematic instability of the pelvis, the top of the head was selected as the reference point, which also assimilates the vestibular sensor position. Inverse dynamics methods were used to formulate and solve the equations of motion based on Newton-Euler formulae. The torques and ground reaction forces generated by the planar model during a regular gait cycle were compared with similar results from a more complex three-dimensional OpenSim model with muscles, which resulted in correlation errors in the range of 0.9–0.98. The close comparison between the two torque outputs supports the use of planar models in gait studies.

scholarly journals Development of Multi-DOF Model of Automotive LED Headlamp Assembly for Force Transmission Prediction Using MATLAB GUI

2020 ◽  
Vol 10 (17) ◽  
pp. 5906
Author(s):  
Muhammad Moghees Ud Din ◽  
Byeongil Kim
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
System Of Equations ◽  
Vibration System ◽  
Force Transmission ◽  
Light Emitting ◽  
The Past ◽  
Impact Forces ◽  
Led Headlamp ◽  
Angular Displacements

The use of Light-emitting diodes (LEDs) in automobile headlamps began two decades ago. Since then, several design and efficiency improvements have been made. However, the reliability and durability of these LED systems remain uncertain. There are several approaches for reliability analysis, e.g., thermal, electrical, optical, or structural. The first three issues have been studied in the past, but there has been minimal focus on structural and dynamic durability. Uneven road conditions and impact forces acting on the headlamp module can damage components and misalign the aiming mechanism. Moreover, the functionality could be disturbed, thereby decreasing the efficiency of the system. To determine the forces acting inside the module on each component, this study proposes a simulation technique for predicting the magnitude of forces transmitted from the automobile chassis to the headlamp module under even and uneven road conditions. A vibration system with 23 degrees-of-freedom is developed and equations of motion are derived using Newton’s second law of motion. Solving this system of equations with Simulink and MATLAB provided the linear and angular displacements of each element, which were then utilized to calculate the forces transmitted through these elements. Two forcing conditions were compared and the locations with maximum forces are highlighted.

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