EXACT, SINGLE EQUATION, CLOSED-FORM SOLUTION OF VIBRATING SYSTEMS WITH PIECEWISE LINEAR SPRINGS

2001 ◽  
Vol 245 (2) ◽  
pp. 285-301 ◽  
Author(s):  
E. CHICUREL-UZIEL
Keyword(s):  
Closed Form ◽  
Piecewise Linear ◽  
Closed Form Solution ◽  
Single Equation ◽  
Form Solution ◽  
Vibrating Systems

Parametric Identification of a Vibratory System With a Clearance

1993 ◽  
Vol 115 (1) ◽  
pp. 25-32 ◽  
Author(s):  
R. M. Alexander ◽  
S. T. Noah ◽  
C. G. Franck
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Closed Form ◽  
Piecewise Linear ◽  
Closed Form Solution ◽  
Element Model ◽  
Form Solution ◽  
Degree Of Freedom ◽  
Vibratory System ◽  
Experimental Structure

An analytical and experimental investigation of a vibratory system with a clearance was conducted. A finite element model and an equivalent single-degree-of-freedom closed-form solution were used to determine the dynamic parameters and response of an experimental structure interacting with a gap. The closed-form solution is obtained by taking advantage of the piecewise linearity of the system. Results from these solution methods are in agreement with experimental data. The results also suggest that the closed-form solution approximates the response of the experimental structure with accuracy greater than that of the finite element model. The closed-form solution was also used to determine the gap size of the structure. The parameter identification procedure utilized in this study appears to be simple to use and can be readily extended to other types of piecewise-linear multi-degree-of-freedom systems.

Parametric Identification of a Vibratory System With a Clearance

1991 ◽  
Author(s):  
Richard M. Alexander ◽  
Sherif T. Noah ◽  
Charles G. Franck
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Closed Form ◽  
Piecewise Linear ◽  
Closed Form Solution ◽  
Parametric Identification ◽  
Element Model ◽  
Form Solution ◽  
Vibratory System ◽  
Experimental Structure

Abstract An analytical and experimental investigation of a vibratory system with a clearance was conducted. A finite element model and an equivalent single degree of freedom closed-form solution were used to determine the dynamic parameters and response of an experimental structure interacting with a gap. The closed-form solution is obtained by taking advantage of the piecewise linearity of the system. Results from these solution methods are in agreement with experimental data. The results also suggest that the closed-form solution approximates the response of the experimental structure with accuracy greater than that of the finite element model. The closed-form solution was also used to determine the gap size of the structure. The parameter identification procedure utilized in this study appears to be simple to use and can be readily extended to other types of piecewise-linear multidegree of freedom systems.

A Piecewise-Linear Approximation of the Canonical Spring-Loaded Inverted Pendulum Model of Legged Locomotion

2016 ◽  
Vol 11 (1) ◽  
Author(s):  
Zhuohua Shen ◽  
Justin Seipel
Keyword(s):  
Closed Form ◽  
Inverted Pendulum ◽  
Piecewise Linear ◽  
Closed Form Solution ◽  
Legged Locomotion ◽  
Form Solution ◽  
Piecewise Linear Approximation ◽  
Computational Time ◽  
Slip Model ◽  
Spring Loaded Inverted Pendulum

Here, we introduce and analyze a novel approximation of the well-established and widely used spring-loaded inverted pendulum (SLIP) model of legged locomotion, which has made several validated predictions of the center-of-mass (CoM) or point-mass motions of animal and robot running. Due to nonlinear stance equations in the existing SLIP model, many linear-based systems theories, analytical tools, and corresponding control strategies cannot be readily applied. In order to provide a significant simplification in the use and analysis of the SLIP model of locomotion, here we develop a novel piecewise-linear, time-invariant approximation. We show that a piecewise-linear system, with the only nonlinearity due to the switching event between stance and flight phases, can predict all the bifurcation features of the established nonlinear SLIP model over the entire three-dimensional model parameter space. Rather than precisely fitting only one particular solution, this approximation is made to quantitatively approximate the entire solution space of the SLIP model and capture all key aspects of solution bifurcation behavior and parametric sensitivity of the original SLIP model. Further, we provide an entirely closed-form solution for the stance trajectory as well as the system states at the end of stance, in terms of common functions that are easy to code and compute. Overall, the closed-form solution is found to be significantly faster than numerical integration when implemented using both matlab and c++. We also provide a closed-form analytical stride map, which is a Poincaré return section from touchdown (TD) to next TD event. This is the simplest closed-form approximate stride mapping yet developed for the SLIP model, enabling ease of analysis and numerical coding, and reducing computational time. The approximate piecewise-linear SLIP model presented here is a significant simplification over previous SLIP-based models and could enable more rapid development of legged locomotion theory, numerical simulations, and controllers.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

2021 ◽  
Vol 10 (7) ◽  
pp. 435
Author(s):  
Yongbo Wang ◽  
Nanshan Zheng ◽  
Zhengfu Bian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Simulated Data ◽  
Real Data ◽  
Point Clouds ◽  
Form Solution ◽  
Spatial Transformation ◽  
Dual Quaternions ◽  
Feature Based ◽  
Planar Feature

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

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