Formulation of the equations of motion: Single-degree-of-freedom systems

2012 ◽  
pp. 47-96
Keyword(s):  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree

A Comparison of the Effects of Nonlinear Damping on the Free Vibration of a Single-Degree-of-Freedom System

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Bin Tang ◽  
M. J. Brennan
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Nonlinear Damping ◽  
Degree Of Freedom ◽  
Damping Force ◽  
Single Degree Of Freedom ◽  
Sdof System ◽  
Single Degree ◽  
Dependent Term ◽  
Quadratic Damping

This article concerns the free vibration of a single-degree-of-freedom (SDOF) system with three types of nonlinear damping. One system considered is where the spring and the damper are connected to the mass so that they are orthogonal, and the vibration is in the direction of the spring. It is shown that, provided the displacement is small, this system behaves in a similar way to the conventional SDOF system with cubic damping, in which the spring and the damper are connected so they act in the same direction. For completeness, these systems are compared with a conventional SDOF system with quadratic damping. By transforming all the equations of motion of the systems so that the damping force is proportional to the product of a displacement dependent term and velocity, then all the systems can be directly compared. It is seen that the system with cubic damping is worse than that with quadratic damping for the attenuation of free vibration.

An Approximate Method for the Dynamic Analysis of Elastic Linkages

1977 ◽  
Vol 99 (2) ◽  
pp. 449-455 ◽  
Author(s):  
A. Midha ◽  
A. G. Erdman ◽  
D. A. Frohrib
Keyword(s):  
Exact Analytical Solution ◽  
Equations Of Motion ◽  
Numerical Procedure ◽  
Degree Of Freedom ◽  
Suggested Approach ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
System A ◽  
Suggested Technique ◽  
Particular Solution

A new numerical procedure based on an iterative technique is progressively developed in this paper for obtaining an approximate particular solution from the equations of motion of an elastic linkage with small damping and at subresonant speeds. The method is introduced by employing a simple vibrating system, a single degree-of-freedom mass-dashpot-spring model under both harmonic forcing and periodic forcing. A harmonically excited two degree-of-freedom model is also solved by the suggested approach. Error functions are developed for each case to give an estimation of the order of error between the exact analytical solution and the approximate technique. The suggested technique is then extended to solve an elastic linkage problem where the uncoupled equations of motion are treated as a series of single degree-of-freedom problems and solved. These are retransformed into the physical coordinate system to obtain the particular solution. The first and second derivatives of the forcing functions (involving rigid-body inertia) are approximated utilizing the finite difference method.

Tow-Cable Snap Loads

1965 ◽  
Vol 2 (01) ◽  
pp. 42-49
Author(s):  
Lewis Schneider ◽  
L. Grady Burton ◽  
Thomas Mahan
Keyword(s):  
Large Amplitude ◽  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Ship Motions ◽  
Body Density ◽  
Dynamical Equations ◽  
Dynamic Stresses ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Tension Recovery

When towing submerged bodies such as sonar vehicles or oceanographic instrument packages, the tow cable may be subjected to severe loadings because of large-amplitude ship motions. These motions first cause the cable to become slack and subsequently subject it to impact stresses at the instant of tension recovery. A typical body, cable, and oscillating tow-point assemblage is idealized as a single-degree-of-freedom system and the dynamical equations of motion are solved over a broad range of forcing amplitudes and periods and system compliance. The effects of such parameters as body density and cable compliance in attenuating the dynamic stresses are discussed.

Fast Simulation of a Single Degree-of-Freedom System Consisting of An Iwan Element Using the Method of Averaging

2020 ◽  
Vol 142 (5) ◽  
Author(s):  
Drithi Shetty ◽  
Matthew Allen
Keyword(s):  
Equations Of Motion ◽  
Element Model ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Integration Schemes ◽  
Current Implementation ◽  
Modal Model ◽  
Stiffness And Damping

Abstract While Iwan elements have been used to effectively model the stiffness and energy dissipation in bolted joints, integrating the equations of motion of these elements is fairly expensive since implicit schemes, such as Newmark’s methods, need to be used. This paper presents a method of simulating dynamic systems containing nonlinear Iwan elements that significantly reduce the computation cost by using closed-form expressions for stiffness and damping in the microslip regime and an averaging method for regions of time in which no external force is applied. The proposed algorithm is demonstrated on a single degree-of-freedom (SDOF) system to evaluate the range over which it retains accuracy and the improvement in performance it offers. Although the current implementation is limited to SDOF systems, it can be used to simulate the response of each mode in structures exhibiting weak nonlinearity that can be modeled using the modal Iwan approach. To verify this, the dynamic response of a finite element model of a beam assembly, integrated using the Newmark-β method, has been compared with its equivalent modal model integrated using the proposed algorithm. The results show that the algorithm accurately predicts the response in a fraction of the time taken by implicit integration schemes, so long as the modes remain uncoupled and weakly nonlinear.

The Study of a Single Degree of Freedom System with Time Dependent Parameters

1967 ◽  
Vol 71 (678) ◽  
pp. 439-440 ◽  
Author(s):  
B. V. Dasarathy ◽  
P. Srinivasan
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Time Dependent ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stochastic Nature ◽  
Restoring Forces

The equations of motion of many mechanical systems reduce to ordinary differential equations with time dependent parameters. A single degree of freedom system, with the damping and restoring forces varying with time according to the law … m (t+k)nhas been studied using the WKBJ approximation. The general solution of the differential equation can be used to arrive at the response of such time dependent systems, subjected to excitations which are specified to be of either deterministic or stochastic nature.

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