scholarly journals Impact of a Multiple Pendulum with a Non-Linear Contact Force

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1202
Author(s):  
Dan B. Marghitu ◽  
Jing Zhao
Keyword(s):  
Kinetic Energy ◽  
Contact Force ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Kinematic Chain ◽  
Double Pendulum ◽  
Lagrange Equations ◽  
Non Linear ◽  
Impact Kinetic Energy ◽  
The Impact

This article presents a method to solve the impact of a kinematic chain in terms of a non-linear contact force. The nonlinear contact force has different expressions for elastic compression, elasto-plastic compression, and elastic restitution. Lagrange equations of motion are used to obtain the non-linear equations of motion with friction for the collision period. The kinetic energy during the impact is compared with the pre-impact kinetic energy. During the impact of a double pendulum the kinetic energy of the non-impacting link is increasing and the total kinetic energy of the impacting link is decreasing.

Random Wave Response of Double Pendulum Articulated Off-Shore Tower

2003 ◽  
Author(s):  
Nazrul Islam ◽  
Suhail Ahmad
Keyword(s):  
Equations Of Motion ◽  
Linear Equations ◽  
Time Integration ◽  
Power Spectra ◽  
Integration Scheme ◽  
Random Wave ◽  
Double Pendulum ◽  
Non Linear ◽  
Time Histories ◽  
Time Integration Scheme

Present study investigates the non-linear dynamic behavior of Double Hinged Articulated Tower (DHAT) under long crested random Sea and directional random sea. The non-linearities due to time wise variation of submergence, buoyancy, added mass, instantaneous tower orientation and resulting hydrodynamic loading have been taken into account for modeling the forcing functions of equation of motion which is derived by Largrangian approach. A long crested random sea has been modeled by Monte-Carlo Simulation using P-M spectrum. The non-linear equations of motion are solved by an iterative time integration scheme using Newmark’s β integration scheme. Various important parameters such as heel angles, deck displacements, base share for double hinged articulated tower under long and short crested random sea are compared and presented in the form of time-histories and their respective PSDFs. Statistical studies of random time histories have been carried out and important characteristics like mean, maxima, minima, standard deviations etc. have been analyzed. The dynamic behaviors have been investigated in detail in terms of various parametric combinations. Effect of current, and significant wave height are also studied. Sub and super harmonic excitations are highlighted through power spectra. A multi-hinged articulated tower is found to be economical and suitable for various offshore activities in adverse environmental and deep sea conditions.

scholarly journals Numerical analysis of beam rail bridges - impact factors in the vertical deflection

2017 ◽  
Vol 2017 (9) ◽  
pp. 11-17 ◽  
Author(s):  
Monika Podwórna
Keyword(s):  
High Speed ◽  
Equations Of Motion ◽  
Impact Factors ◽  
Vertical Deflection ◽  
Lagrange Equations ◽  
High Speeds ◽  
Rail Vehicles ◽  
Non Linear ◽  
Random Track Irregularities ◽  
The Impact

The impact factors in the vertical deflection obtained in dynamic analysis of BTT systems - bridged / track structure / high speed train (BTT) - are discussed. The BTT system is one of 5 bridges spanning from 15 m to 27 m, modelled as simply supported beams loaded by ICE-3 trains traveling at high speeds. The two-dimensional, physically non-linear BTT model includes: viscoelastic suspension of rail vehicles on two independent axle bogies and non-linear one-sided wheel-rail contact springs according to Hertz theory, access zones for composite construction. The BTT system was divided into subsystems loaded with vertical interactions transmitted by elastic or viscoelastic and physically linear or nonlinear constraints. Using Lagrange equations and internal aggregation of subsystems, discretised according to the finite element method, matrix equations of motion of the subsystems were obtained, with explicit linear left sides and nonlinear implicit right sides, which were integrated numerically using the Newmark method with parameters βN=1/4, γN=1/2. The analysis focus on the effect of random track irregularities on the dynamic response of BTT systems.

The Effects of Flare and Overhangs on the Motions of a Yacht in Head Seas

1993 ◽  
Author(s):  
John C. Kuhn ◽  
Eric C. Schlageter
Keyword(s):  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Order Effects ◽  
Third Order ◽  
Numerical Work ◽  
Strip Theory ◽  
Hull Forms ◽  
The Time Domain ◽  
The Impact

The coupled heave and pitch motions of hull forms with flare and overhangs are examined numerically. The presence of flare and overhangs is numerically modelled with nonlinear hydrostatic and Froude-Krylov forces based on integrals over the instantaneous wetted surface. Forces due to radiation and diffraction are computed with a linear strip-theory. These forces are combined in two coupled nonlinear differential equations of motion that are solved in the time domain with a fourth-order Runge-Kutta integration method. An assessment of the impact of flare and overhangs on motions is obtained by comparing these nonlinear solutions with solutions of the traditional linear equations of motion, which do not contain forces due to flare and overhangs. For an example based on an International America's Cup Class yacht design, it is found that the nonlinear heave and pitch motions are smaller than the linear motions. This is primarily due to reduced first-order response components, which are coupled with nonlinear response components. Comparisons of these results with towing tank data demonstrate that the nonlinear procedure improves prediction quality relative to linear results. In support of this numerical work, the hydrostatic and Froude­Krylov force integrals are expanded in Taylor series with respect to wave elevation. These results indicate how hydrostatic and Froude-Krylov forces change with changing flare and overhang angles, revealing that sectional slope has second and third-order effects on forces while sectional curvature and overhang angles produce third-order effects.

ANALYSIS OF ELASTIC IMPACT ON THIN SHELL STRUCTURES

2008 ◽  
Vol 22 (09n11) ◽  
pp. 1349-1354 ◽  
Author(s):  
SHIUH-CHUAN HER ◽  
CHING-CHUAN LIAO
Keyword(s):  
Differential Equation ◽  
Contact Force ◽  
Thin Shell ◽  
Time History ◽  
Dynamic Responses ◽  
Low Velocity Impact ◽  
Integro Differential Equation ◽  
Impact Process ◽  
Non Linear ◽  
The Impact

In this paper, a solution method for the response of a thin shell structure subjected to low velocity impact by a sphere is presented. The governing equation of the impact process is obtained by simultaneously solving the equations of motions for the sphere and shell. The derivation is based on the explicit expression of the displacement of the mid-surface of the shell under a single impulse load acting normal to apex of the shell. Incorporating the theory of convolution and Hertz contact law, a non-linear integro-differential equation in terms of the indentation of the contact, for the impact process is derived. The non-linear integro-differential equation is solved by the numerical scheme of Runge-Kutta method to obtain the time history of the contact force at the impact point of the shell. The contact force is then applied on the apex of the shell, the dynamic responses of the shell including the displacement and stress are obtained by the finite element method. The results are validated with the experimental test and numerical calculation published in the literatures. The effects of the radius and velocity of the impactor on the impact response is investigated through parametric study.

scholarly journals Modelling of Dynamics of a Wheeled Mobile Robot with Mecanum Wheels with the use of Lagrange Equations of the Second Kind

2017 ◽  
Vol 22 (1) ◽  
pp. 81-99 ◽  
Author(s):  
Z. Hendzel ◽  
Ł. Rykała
Keyword(s):  
Mathematical Model ◽  
Kinetic Energy ◽  
Mobile Robot ◽  
Mobile Robots ◽  
Equations Of Motion ◽  
Wheeled Mobile Robot ◽  
Dynamic Equations ◽  
Lagrange Equations ◽  
Wheeled Mobile Robots ◽  
New Type

Abstract The work presents the dynamic equations of motion of a wheeled mobile robot with mecanum wheels derived with the use of Lagrange equations of the second kind. Mecanum wheels are a new type of wheels used in wheeled mobile robots and they consist of freely rotating rollers attached to the circumference of the wheels. In order to derive dynamic equations of motion of a wheeled mobile robot, the kinetic energy of the system is determined, as well as the generalised forces affecting the system. The resulting mathematical model of a wheeled mobile robot was generated with the use of Maple V software. The results of a solution of inverse and forward problems of dynamics of the discussed object are also published.

Analysis of the Non-Linear Dynamic Responses of an Elastic Rotor With a Slant Crack

2012 ◽  
Author(s):  
Xiangmin Zhang ◽  
Changping Chen ◽  
Liming Dai
Keyword(s):  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Rotor System ◽  
Dynamic Responses ◽  
Cracked Rotor ◽  
Lagrange Equations ◽  
Linear Dynamic ◽  
Slant Crack ◽  
Non Linear ◽  
Line Spring Model

Considering a rotor system with a slant crack, and using an equivalent line-spring model to simulate the slant crack of the rotor, the flexibility model of the slant-cracked rotor is derived. Then considered the geometric non-linearity and based on the Lagrange equations, the non-linear dimensionless differential equations of motion for the slant-cracked rotor are obtained. Further the non-linear dynamic responses of the single rotor system with a slant crack are discussed by the Galerkin method and the harmonic balance method. It’s detailedly studied that the angle, the depth and the position of the slant crack on the rotor all affect on the non-linear dynamic responses of the rotor system, and the conclusion is very significant to the design of the rotor system in the practical reference aspect.

scholarly journals Dynamics and rigidity of a manipulator with three DOFs

2018 ◽  
Vol 226 ◽  
pp. 01020 ◽  
Author(s):  
Serikbay Kosbolov ◽  
Yerlan Yeleukulov ◽  
Alfiya Atalykova ◽  
Algazy Zhauyt ◽  
Gulsara Yestemessova ◽  
...  
Keyword(s):  
Potential Energy ◽  
Cross Section ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Kinematic Chain ◽  
System Of Linear Equations ◽  
Anthropometric Data ◽  
Elastic Potential Energy ◽  
System Equations ◽  
Linear Equations Of Motion

The problem of dynamic elastic four-link initial kinematic chain (IKC) of the load-bearing manipulator, which is the basis for various modifications are considered. Using the Lagrange operator for this system, equations of motion in matrix form are obtained. To determine the potential energy of an elastic four-link IKC manipulator, we use the formula for the elastic potential energy of a rectilinear homogeneous rod of length l. The cross-section of the rod is considered annular or circular. Solving the system of linear equations of motion on a computer using the ADAMS program, the results of the movement of links and cargo were obtained. Kinematics and dynamics are presented for a generic 3 DOFs Initial Kinematic Chain; with anthropometric data and the dynamics equations, simulations were performed to understand its behavior.

Meteoroid trajectories from BRAMS data

2021 ◽  
Author(s):  
Hervé Lamy
Keyword(s):  
Time Delays ◽  
Specular Reflection ◽  
Linear Equations ◽  
Radio Interferometer ◽  
Additional Information ◽  
Sensitivity Pattern ◽  
Non Linear ◽  
Using Data ◽  
The Impact ◽  
Non Linear Equations

<p>BRAMS (Belgian RAdio Meteor Stations) is a Belgian radio network using forward scatter observations to detect and characterize meteoroids. A dedicated transmitter located in south of Belgium emits a CW signal with no modulation at a frequency of 49.97 MHz and with a power of 130 W. The network comprises currently 35 similar receiving stations located in Belgium and neighboring countries. They use Yagi antennas with a wide sensitivity pattern which therefore provide no information about the directivity of the meteor echoes. One of these stations is however a radio interferometer using the classical Jones configuration and is able to retrieve the direction of the meteor echoes.</p><p>We discuss here a general method to retrieve meteoroid trajectories based solely on time delays measured between meteor echoes recorded at multiple receiving stations. It is based on solving at least 6 non-linear equations to solve for the position of one specular reflection point (3 unknowns) and the 3 components of the speed. This method has also been described recently in Mazur et al (2020) and applied to CMOR data. However, specificities of the CMOR configuration has allowed simplifications that cannot be made with the BRAMS network. In order to maximize the number of meteoroid trajectories with at least 6 stations detecting meteor echoes, a number of additional stations geographically close to each other have been installed in the Limburg province in 2020. Another method to retrieve meteoroid trajectories using data from the radio interferometer and from 3 other stations is also presented.</p><p>We show preliminary results from both methods using also complementary data from the optical CAMS Benelux network.  The CAMS trajectories are used to select specific meteor echoes in the BRAMS data. The time delays between them are computed and used to solve the set of non-linear equations to retrieve the meteoroid trajectory and speed, which are then compared to the CAMS values. This allows us to assess the accuracy of both methods.</p><p>Finally we simulate the impact of using additional information, not currently available but that might become in a near future. This includes data from a monostatic system (a radar nearby our BRAMS transmitter is currently built), from a second radio interferometer (to be located in Limburg and/or near the transmitter), or the total range traveled by the radio wave if a coded CW transmitter such as in Vierinen et al (2016) is used.</p>

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