Equations of Motion of Dynamic System

Author(s):  
Ehsan Sarshari ◽  
Nastaran Vasegh ◽  
Mehran Khaghani ◽  
Saeid Dousti

Ziegler’s pendulum is an appropriate model of a non-conservative dynamic system. By considering gravity effect, new equations of motion are extracted from Newton’s motion laws. The instability of equilibriums is determined by linear stability analysis. Chaotic behavior of the model is shown by numerical simulations. Sliding mode controller is used for eliminating chaos and for stabilizing the equilibriums.


1981 ◽  
Vol 103 (4) ◽  
pp. 805-810
Author(s):  
H. M. Chen ◽  
S. B. Malanoski

This paper presents a simplified analysis procedure to provide initial assessment and guidance on fan rotor dynamics including the foundation interaction. For purposes of early-design decision-making (or trouble-shooting), the interaction of a rotor-bearing dynamic system and a foundation-soil/piling dynamic system is viewed approximately for the vertical, horizontal, and rocking modes of vibration. The equations of motion are written in matrix form and include the pertinent parameters. A numerical example is presented to guide in the interpretation of the analysis; this example considers the unbalance response of the entire system as measured at the bearings.


Author(s):  
Ismail Akharas ◽  
Michael P. Hennessey ◽  
Eric J. Tornoe

Abstract This paper introduces a novel method for playing dynamic animations of rigid body assemblies with internal motions in virtual reality (VR). Through previous research over a decade ago, an inexpensive, relatively straight-forward process has been developed that entailed using SolidWorks, MATLAB/Simulink, and movie player software to permit one to view 2D MP4 files, such as on a laptop, smart phone, etc. Inspired by the usefulness of these previous results, the approach presented here targets a VR environment, clearly representing a technological leap over viewing 2D MP4 files. It’s made possible by recent advances in VR & gaming software (e.g. Unity) along with some unique software interfacing, including use of CADLink, to permit importation of CAD files, such as from SolidWorks, into Unity. Those interested in VR visualization of their dynamic system can use the step-by-step process presented as a manual to guide them through the hardware and software setup and ultimately learn how to use SolidWorks, MATLAB/Simulink, and Unity interactively to visualize their simulations in VR. Another key point is that the analyst has considerable control and access over each step in the process, including the dynamic modeling, unlike that commonly found in large, structured dynamic simulation software packages. As an example to illustrate the process, a dynamic simulation of a classic pendulum/slider system was created using MATLAB/Simulink, which in effect numerically solves the ordinary differential equations of motion. The time-dependent displacement data for both the slider’s lateral movement and the pendulum’s angle, along with a time vector in incremental difference form, was saved as an Excel file. In turn, it was read by a C# script residing within Unity to permit an animation playback scenario of the SolidWorks CAD model of the entire pendulum/slider system (previously brought into Unity via CADLink with some reassembly), viewed more generally as an assembly with internal motions. Unity, a popular open-source piece of VR game development software used to produce both 2D and 3D video games and simulations, then serves as a platform to access the virtual world with the aid of an Oculus Rift (or Quest) VR headset and two hand controllers. In the end, the VR viewer can physically move around in the VR environment while at the same time view the playback motion of the pendulum/slider system from varying vantage points, just as one would expect in the real world. This work significantly advances the typical visualization experience with respect to dynamic system simulation & animation in addition to being widely applicable to generic mechanical assemblies.


1996 ◽  
Vol 118 (1) ◽  
pp. 107-114
Author(s):  
R. Ganesan

Parametric steady-state vibrations of an asymmetric rotor while passing through primary resonance and the associated stability behavior are analyzed. The undamped case is considered and the equations of motion are rewritten in a from suitable for applying the method of multiple scales. Sensitivity to the bearing as well as shaft asymmetries of the oscillations due to unbalance excitation is evaluated. Expressions for amplitude and frequency modulation functions are obtained and are specialized to yield the steady-state solutions near primary resonance. Frequency-amplitude relationships that result from combined parametric and mass unbalance excitations are derived. Stability regions in the parameter space are obtained based on the time evolution of the amplitude and phase of the steady-state motions. The effects of bearing asymmetry on the amplitude and phase of the resonant oscillations are brought out. The sensitivity of vibrational and stability characteristics to various rotor-dynamic system parameters is illustrated through a numerical investigation.


1978 ◽  
Vol 1 (2) ◽  
pp. 245-253
Author(s):  
Munawar Hussain

In this paper the classical theorem “a conservative holonomic dynamic system is invariantly connected with a certain differential form” is generalized to group variables. This general theorem is then used to reduce the order of a Hamiltonian system by the use of the integral of energy. Equations of motion of the reduced system so obtained are derived which are the so-called generalized Whittaker's equations. Finally an example is given as an application of the theory.


2012 ◽  
Vol 91 (105) ◽  
pp. 19-24
Author(s):  
Aleksandar Baksa

This article deals with a dynamic system whose motion is constrained by nonholonomic, reonomic, affine constraints. The article analyses the geometrical properties of the ?reactions" of nonholonomic constraints in Voronets?s equations of motion. The analysis shows their link with the torsion of the Ehresmann connection, which is defined by the nonholonomic constraints.


2020 ◽  
Vol 18 (2) ◽  
pp. 81
Author(s):  
Angga Septiyana

This paper discusses the stability, control and observation of the dynamic system of the Lapan Surveillance UAV 05-NG (LSU 05-NG) aircraft equation. This analysis is important to determine the performance of aircraft when carrying out missions such as photography, surveillance, observation and as a scientific platform to test communication based on satellite. Before analyzing the dynamic system, first arranged equations of motion of the plane which includes the force equation, moment equation and kinematics equation. The equation of motion of the aircraft obtained by the equation of motion of the longitudinal and lateral directional dimensions. Each of these equations of motion will be linearized to obtain state space conditions. In this state space, A, B and C is linear matrices will be obtained in the time domain. The results of the analysis of matrices A, B and C show that the dynamic system in the LSU 05-NG motion equation is a stable system on the longitudinal dimension but on the lateral dimension directional on the unstable spiral mode. As for the analysis of the control of both the longitudinal and lateral directional dimensions, the results show that the system is controlled.


2016 ◽  
Vol 2016 (5) ◽  
pp. 71-83 ◽  
Author(s):  
Danuta Bryja ◽  
Dawid Prokopowicz

The paper presents the computational model of the pantograph – overhead contact line (OCL), which uses the theory of cable vibrations and Lagrange – Ritz approximation method to derive equations of motion of the overhead contact line subjected to moving pantographs. The pantograph is modelled as a dynamic system of two degrees of freedom describing the motion of two masses replacing a collector head and an articulating frame. The overhead contact line is defined as a catenary system with continuously distributed mass. It consists of a multi-span cable characterized by a curvilinear route (catenary wire) and a straight cable (contact wire) connected with a catenary wire by elastic droppers. The main objective of the paper is to present principal ideas of the computational model, with a particular emphasis on formulating the equation of motion of a pre-tensioned multi-span cable with non-negligible static sag. Much attention is paid to the description of dynamic interaction between the pantograph and overhead contact line. The model allows computer simulation of catenary vibrations induced by two pantographs passing over the contact line, as well as a simulation of dynamic increments of the contact force.


2009 ◽  
Vol 09 (04) ◽  
pp. 649-660 ◽  
Author(s):  
Y. WEN ◽  
Q. Y. ZENG

A high-order finite element model is presented to perform the vibration analysis of beams. The equations of motion are formulated by applying the principle of total potential energy in elastic dynamic system and the "set-in-right-position" rule for the construction of system matrices first proposed by the author. The primary advantage of the principle and rule lies in its simplicity and efficiency in solving the modeling problem of complex dynamic system. The requirement of strain continuity has certainly not being met at element interfaces with the use of conventional cubic Hermitian formulation. Hence, it is difficult to predict the dynamic responses of beams accurately. In order to overcome this problem, a beam element with simple higher-order interpolation function is chosen as the analysis model. Although the number of nodal degrees of freedom is increased herein, usually a coarse mesh will suffice. The present formulation is able to provide results of high accuracy with low computational effort. For the purpose of illustration, the dynamic characteristics analysis and dynamic response analysis are carried out on beam models. The solutions obtained for all the examples are in good agreement with the exact solutions found by fundamental theory of vibration.


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