Multiparameter Identification in Linear Continuous Vibratory Systems

1971 ◽  
Vol 93 (1) ◽  
pp. 45-52 ◽  
Author(s):  
O. Bruce Dale ◽  
Raymond Cohen
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
System Model ◽  
State Variables ◽  
State Equations ◽  
Unknown Parameters ◽  
Response Data ◽  
State Variable ◽  
Finite Set ◽  
Arbitrary Location

A method is presented for identifying unknown parameters in linear continuous vibratory systems from frequency response data. The method is applicable to systems for which the steady-state equations of motion are reducible to a finite set of ordinary differential equations. After assuming a system model of partial differential equations and boundary conditions containing unknown parameters, initial value numerical integrations are performed to identify the unknown parameters. No analytic solution to the system model is required. For the identification of conservative or near conservative systems, only resonant frequency data are used and, for the identification of nonconservative systems, the data may consist of any state variable or combination of state variables at any arbitrary location within the system. The use of multiple response transducers is not required.

Multi-Parameter Optimization of Damped Linear Continuous Systems

1972 ◽  
Vol 94 (1) ◽  
pp. 1-7 ◽  
Author(s):  
O. B. Dale ◽  
R. Cohen
Keyword(s):  
Optimal Parameter ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
System Response ◽  
Continuous Systems ◽  
Frequency Interval ◽  
State Variables ◽  
Initial Value ◽  
Unknown Parameters ◽  
One Dimensional

A method is presented for obtaining and optimizing the frequency response of one-dimensional damped linear continuous systems. The systems considered are assumed to contain unknown constant parameters in the boundary conditions and equations of motion which the designer can vary to obtain a minimum resonant response in some selected frequency interval. The unknown parameters need not be strictly dissipative nor unconstrained. No analytic solutions, either exact or approximate, are required for the system response and only initial value numerical integrations of the state and adjoint differential equations are required to obtain the optimal parameter set. The combinations of state variables comprising the response and the response locations are arbitrary.

Computer Programming

SIMULATION ◽  
1965 ◽  
Vol 4 (5) ◽  
pp. 317-323 ◽  
Author(s):  
Joseph L. Hammond
Keyword(s):  
Differential Equations ◽  
Computer Program ◽  
Ordinary Differential Equations ◽  
Computer Programming ◽  
Characteristic Root ◽  
Analog Computer ◽  
State Variables ◽  
State Variable ◽  
Forcing Function ◽  
Derivatives Of

State variable techniques are reviewed and applied to analog computer programming. The concise rep resentation for ordinary differential equations made possible by this technique is used to formulate a gen eral program for all such equations. It is shown that an analog computer program based on state variables will not have redundant integrators. The fact that the use of state variables facilitates the choice of variables internal to an analog com puter program is illustrated by two techniques, namely, (1) a technique for avoiding derivatives of the forcing function in programming a large class of ordinary differential equations, and (2) a technique for simulating certain systems in such a way that the effect of each characteristic root is placed in evi dence.

An Ordinary Differential Equation Formulation for Multibody Dynamics: Nonholonomic Constraints

2016 ◽  
Vol 17 (1) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multibody Systems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Nonholonomic Constraints ◽  
State Variables ◽  
Well Conditioning ◽  
Fifth Order

A method is presented for formulating and numerically integrating ordinary differential equations of motion for nonholonomically constrained multibody systems. Tangent space coordinates are defined in configuration and velocity spaces as independent generalized coordinates that serve as state variables in the formulation, yielding ordinary differential equations of motion. Orthogonal-dependent coordinates and velocities are used to enforce constraints at position, velocity, and acceleration levels. Criteria that assure accuracy of constraint satisfaction and well conditioning of the reduced mass matrix in the equations of motion are used as the basis for updating local coordinates on configuration and velocity constraint manifolds, transparent to the user and at minimal computational cost. The formulation is developed for multibody systems with nonlinear holonomic constraints and nonholonomic constraints that are linear in velocity coordinates and nonlinear in configuration coordinates. A computational algorithm for implementing the approach is presented and used in the solution of three examples: one planar and two spatial. Numerical results using a fifth-order Runge–Kutta–Fehlberg explicit integrator verify that accurate results are obtained, satisfying all the three forms of kinematic constraint, to within error tolerances that are embedded in the formulation.

Sliding Mode Control of Vibration in Single-Degree-of-Freedom Fractional Oscillators

2017 ◽  
Vol 139 (11) ◽  
Author(s):  
Jian Yuan ◽  
Youan Zhang ◽  
Jingmao Liu ◽  
Bao Shi
Keyword(s):  
Differential Equations ◽  
Sliding Mode Control ◽  
Sliding Mode ◽  
Degree Of Freedom ◽  
State Variables ◽  
State Equations ◽  
Control Laws ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Mode Control

This paper proposes sliding mode control of vibration in three types of single-degree-of-freedom (SDOF) fractional oscillators: the Kelvin–Voigt type, the modified Kelvin–Voigt type, and the Duffing type. The dynamical behaviors are all described by second-order differential equations involving fractional derivatives. By introducing state variables of physical significance, the differential equations of motion are transformed into noncommensurate fractional-order state equations. Fractional sliding mode surfaces are constructed and the stability of the sliding mode dynamics is proved by means of the diffusive representation and Lyapunov stability theory. Then, sliding mode control laws are designed for fractional oscillators, respectively, in cases where the bound of the external exciting force is known or unknown. Furthermore, sliding mode control laws for nonzero initialization case are designed. Finally, numerical simulations are carried out to validate the above control designs.

scholarly journals Distribution-Based Identification of Yield Coefficients in a Baker’s Yeast Bioprocess

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Dorin Sendrescu
Keyword(s):  
Algebraic System ◽  
Linear Equations ◽  
Distribution Theory ◽  
Baker’S Yeast ◽  
Baker's Yeast ◽  
State Variables ◽  
Algebraic Equations ◽  
Test Functions ◽  
State Equations ◽  
Unknown Parameters

A distribution-based identification procedure for estimation of yield coefficients in a baker’s yeast bioprocess is proposed. This procedure transforms a system of differential equations to a system of algebraic equations with respect to unknown parameters. The relation between the state variables is represented by functionals using techniques from distribution theory. A hierarchical structure of identification is used, which allows obtaining a linear algebraic system of equations in the unknown parameters. The coefficients of this algebraic system are functionals depending on the input and state variables evaluated through some test functions from distribution theory. First, only some state equations are evaluated throughout test functions to obtain a set of linear equations in parameters. The results of this first stage of identification are used to express other parameters by linear equations. The process is repeated until all parameters are identified. The performances of the method are analyzed by numerical simulations.

Lyapunov Functions and Sliding Mode Control for Two Degrees-of-Freedom and Multidegrees-of-Freedom Fractional Oscillators

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
Youan Zhang ◽  
Jian Yuan ◽  
Jingmao Liu ◽  
Bao Shi
Keyword(s):  
Differential Equations ◽  
Sliding Mode Control ◽  
Lyapunov Functions ◽  
Degrees Of Freedom ◽  
Sliding Mode ◽  
Equations Of Motion ◽  
Control Design ◽  
State Equations ◽  
Two Degrees Of Freedom ◽  
Mode Control

This paper addresses the Lyapunov functions and sliding mode control design for two degrees-of-freedom (2DOF) and multidegrees-of-freedom (MDOF) fractional oscillators. First, differential equations of motion for 2DOF fractional oscillators are established by adopting the fractional Kelvin–Voigt constitute relation for viscoelastic materials. Second, a Lyapunov function candidate for 2DOF fractional oscillators is suggested, which includes the potential energy stored in fractional derivatives. Third, the differential equations of motion for 2DOF fractional oscillators are transformed into noncommensurate fractional state equations with six dimensions by introducing state variables with physical significance. Sliding mode control design and adaptive sliding mode control design are proposed based on the noncommensurate fractional state equations. Furthermore, the above results are generalized to MDOF fractional oscillators. Finally, numerical simulations are carried out to validate the above control designs.

scholarly journals Dual Faceted Linearization of Nonlinear Dynamical Systems Based on Physical Modeling Theory

2018 ◽  
Vol 141 (2) ◽  
Author(s):  
H. Harry Asada ◽  
Filippos E. Sotiropoulos
Keyword(s):  
Dynamical System ◽  
Nonlinear System ◽  
Differential Equations ◽  
Physical Modeling ◽  
Nonlinear Dynamical System ◽  
State Variables ◽  
Auxiliary Variables ◽  
State Equations ◽  
Nonlinear Dynamical ◽  
Modeling Theory

A new approach to modeling and linearization of nonlinear lumped-parameter systems based on physical modeling theory and a data-driven statistical method is presented. A nonlinear dynamical system is represented with two sets of differential equations in an augmented space consisting of independent state variables and auxiliary variables that are nonlinearly related to the state variables. It is shown that the state equation of a nonlinear dynamical system having a bond graph model of integral causality is linear, if the space is augmented by using the output variables of all the nonlinear elements as auxiliary variables. The dynamic transition of the auxiliary variables is investigated as the second set of differential equations, which is linearized by using statistical linearization. It is shown that the linear differential equations of the auxiliary variables inform behaviors of the original nonlinear system that the first set of state equations alone cannot represent. The linearization based on the two sets of linear state equations, termed dual faceted linearization (DFL), can capture diverse facets of the nonlinear dynamics and, thereby, provide a richer representation of the nonlinear system. The two state equations are also integrated into a single latent model consisting of all significant modes with no collinearity. Finally, numerical examples verify and demonstrate the effectiveness of the new methodology.

On solvability of initial boundary-value problems of micropolar elastic shells with rigid inclusions

2022 ◽  
pp. 108128652110731
Author(s):  
Victor A Eremeyev ◽  
Leonid P Lebedev ◽  
Violetta Konopińska-Zmysłowska
Keyword(s):  
Weak Solution ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Equations Of Motion ◽  
Elastic Shell ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Finite Set ◽  
Initial Boundary Value ◽  
Small Deformations

The problem of dynamics of a linear micropolar shell with a finite set of rigid inclusions is considered. The equations of motion consist of the system of partial differential equations (PDEs) describing small deformations of an elastic shell and ordinary differential equations (ODEs) describing the motions of inclusions. Few types of the contact of the shell with inclusions are considered. The weak setup of the problem is formulated and studied. It is proved a theorem of existence and uniqueness of a weak solution for the problem under consideration.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

Sign in / Sign up

Export Citation Format

Share Document