An Accurate Spatial Discretization and Substructure Method With Application to Moving Elevator Cable-Car Systems—Part I: Methodology

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
W. D. Zhu ◽  
H. Ren
Keyword(s):  
Differential Algebraic Equations ◽  
Dynamic Responses ◽  
Distributed Parameter ◽  
Spatial Discretization ◽  
Matching Conditions ◽  
Substructure Method ◽  
Axial Forces ◽  
Geometric Matching ◽  
Dependent Variables ◽  
Elevator Cable

A spatial discretization and substructure method is developed to accurately calculate dynamic responses of one-dimensional structural systems, which consist of length-variant distributed-parameter components, such as strings, rods, and beams, and lumped-parameter components, such as point masses and rigid bodies. The dependent variable of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge if the dependent variables are smooth enough. Spatial derivatives of the dependent variables, which are related to internal forces/moments of the distributed-parameter components, such as axial forces, bending moments, and shear forces, can be accurately calculated. Combining component equations that are derived from Lagrange's equations and geometric matching conditions that arise from continuity relations leads to a system of differential algebraic equations (DAEs). When the geometric matching conditions are linear, the DAEs can be transformed to a system of ordinary differential equations (ODEs), which can be solved by an ODE solver. The methodology is applied to several moving elevator cable-car systems in Part II of this work.

An Accurate Spatial Discretization and Substructural Method With Application to Moving Elevator Cable-Car Systems: Part I—Methodology

2011 ◽  
Author(s):  
H. Ren ◽  
W. D. Zhu
Keyword(s):  
Differential Algebraic Equations ◽  
Distributed Parameter ◽  
Algebraic Equations ◽  
Spatial Discretization ◽  
Matching Conditions ◽  
Generalized Coordinates ◽  
Axial Forces ◽  
Linear Algebraic Equations ◽  
Dependent Variables ◽  
Elevator Cable

A spatial discretization and substructure method is developed to calculate the dynamic responses of one-dimensional systems, which consist of length-variant distributed-parameter components such as strings, rods, and beams, and lumped-parameter components such as point masses and rigid bodies. The dependent variable, such as the displacement, of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from the boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge. The spatial derivatives of the dependent variables, which are related to the internal forces/moments, such as the axial forces, bending moments, and shear forces, can be accurately calculated. Assembling the component equations and the geometric matching conditions that arise from the continuity relations leads to a system of differential algebraic equations (DAEs). When some matching conditions are linear algebraic equations, some generalized coordinates can be represented by others so that the number of the generalized coordinates can be reduced. The methodology is applied to moving elevator cable-car systems in Part II of this work.

Dynamic Analysis of a Rotating Planar Timoshenko Beam Using an Accurate Global Spatial Discretization Method

2019 ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu ◽  
H. Zhu
Keyword(s):  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Timoshenko Beam ◽  
Slope Angle ◽  
Dynamic Responses ◽  
Discretization Method ◽  
Spatial Discretization ◽  
Dependent Variables ◽  
Spatial Derivatives ◽  
Shear Strains

Abstract A new formulation is developed for dynamic analysis of a rotating planar Timoshenko beam. The configuration of Timoshenko beam is described using its slope angle and axial and shear strains; hence, the shear locking problem can be naturally avoided. While six boundary conditions are needed for choices of trial functions of three dependent variables, there are only four boundary conditions that can be determined and two boundary conditions are undetermined. An accurate global spatial discretization method is used, where dependent variables are divided into internal and boundary-induced terms. Internal terms only need to satisfy homogeneous boundary conditions, which can be easily chosen as trigonometric functions. Boundary-induced terms are interpolated using dependent variables at boundaries that are taken as generalized coordinates. When the hub rotates at a constant angular velocity, nonlinear governing equations can be linearized for vibration analysis. Frequency veering and mode shift phenomena occur. Nonlinear dynamic responses of the system are then calculated and compared with those from the commercial software ADAMS, and they are in good agreement. Axial and shear strains of the beam and their spatial derivatives are also calculated. Since trial functions in the assumed modes method cannot satisfy undetermined boundary conditions, inaccurate results of strains and their spatial derivatives are obtained using the assumed modes method. Hence, use of the accurate global spatial discretization method in the current formulation is essential.

Coupled vibrations of rope-guided hoisting system with tension difference between two guiding ropes

2016 ◽  
Vol 232 (2) ◽  
pp. 231-244 ◽  
Author(s):  
Guohua Cao ◽  
Jinjie Wang ◽  
Zhencai Zhu
Keyword(s):  
Lateral Displacement ◽  
Current Method ◽  
Differential Algebraic Equations ◽  
Lagrangian Multiplier ◽  
Torsional Vibrations ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Matching Conditions ◽  
Geometric Matching ◽  
Hoisting System

The flexibility of the guiding rope and the tension difference between two guiding ropes cause the lateral and torsional vibrations of the hoisting conveyance in the rope-guided hoisting system, respectively, which are theoretically investigated with two different cases in this paper. The assumed modes method is used to discretize the hoisting rope and two guiding ropes, and Lagrange equations of the first kind is adopted to derive the equations of motion, while the geometric matching conditions at the interfaces of the ropes are accounted for by the Lagrangian multiplier. Considering all the geometric matching conditions are approximately linear, the differential algebraic equations are transformed to a system of ordinary differential equations. The current method can obtain not only the accurate lateral displacements of two guiding ropes, but also the constraint forces between the hoisting conveyance and two guiding ropes. Further, the current method is verified by the ADAMS simulation. Finally, the effects of various parameters on the lateral and torsional vibrations of the hoisting conveyance are analyzed and results indicate that the appropriate tension difference and distance difference could decrease the maximum lateral displacement, which is useful to design super deep rope-guided hoisting system for the decrease of the vibration.

Robustness of Natural Controls of Distributed-Parameter Systems

1996 ◽  
Vol 118 (1) ◽  
pp. 56-63 ◽  
Author(s):  
Jai Hyuk Hwang ◽  
Doo Man Kim ◽  
Kyoung Ho Lim
Keyword(s):  
Closed Loop ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Control Force ◽  
Spatial Discretization ◽  
Actual System ◽  
Discretization Errors

In this paper, the effect of parameter and spatial discretization errors on the closed-loop behavior of distributed-parameter systems is analyzed for natural controls. If the control force designed on the basis of the postulated system with the parameter and discretization errors is applied to control the actual system, the closed-loop performance of the actual system will be degraded depending on the degree of the errors. The extent of deviation of the closed-loop performance from the expected one is derived and evaluated using operator techniques. It has been found that the extent of the deviation is proportional to the magnitude of the parameter and discretization errors, and that the proportional coeffecient depends on the structures of the natural controls.

A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Continuous System ◽  
Dynamic Responses ◽  
Kirchhoff Plate ◽  
Continuous Systems ◽  
Discretization Method ◽  
Two Dimensional ◽  
Spatial Discretization ◽  
Arbitrary Boundary ◽  
Discretization Methods

A new global spatial discretization method (NGSDM) is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional (2D) continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a 2D continuous system is separated into a 2D internal term and a 2D boundary-induced term; the latter is interpolated from one-dimensional (1D) boundary functions that are further divided into 1D internal terms and 1D boundary-induced terms. The 2D and 1D internal terms are chosen to satisfy prescribed boundary conditions, and the 2D and 1D boundary-induced terms use additional degrees-of-freedom (DOFs) at boundaries to ensure satisfaction of all the boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a 2D continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply supported boundaries and one free boundary with an attached Euler–Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Advantages of the new method over local spatial discretization methods are much fewer DOFs and much less computational effort, and those over the assumed modes method (AMM) are better numerical property, a faster calculation speed, and much higher accuracy in calculation of bending moments and transverse shearing forces that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

Modeling of a Finger-Follower Cam System With Verification in Contact Forces

1993 ◽  
Author(s):  
Wensyang Hsu ◽  
Albert P. Pisano
Keyword(s):  
Constraint Equation ◽  
Contact Forces ◽  
Valve Train ◽  
Dynamic Responses ◽  
Valve Lift ◽  
Distributed Parameter ◽  
Contact Points ◽  
Frictional Forces ◽  
Contact Position ◽  
Force Data

Abstract A lumped/distributed-parameter, dynamic model is developed to investigate the dynamic responses of a finger-follower valve train with the effects of an oscillating pivot, frictional forces between sliding surfaces, and a hydraulic lash adjuster. Based on the measured force data at low speed, an algorithm is derived to determine the dynamic Coulomb friction coefficients around maximum valve lift simultaneously at three contact points. A constraint equation is formulated to find the contact position between the cam and the follower kinematically. This makes it possible for the model to simulate the dynamic response of the cam system when the pivot is moving. A hydraulic lash adjuster acting as the pivot of the follower is also modeled with the effects of oil compressibility and oil refill mechanism. The model is numerically integrated and shown to have good agreement between simulation results and experimental data of contact forces at three different speeds. The maximum operating speed is limited by valve toss, loss contact between components. The model predicts toss between the hydraulic lash adjuster and the follower at 2535 rpm, and experiment indicates toss starting at 2520 rpm of camshaft speed.

Null Space Method of Differential Equation Type for Motion Analysis of Multibody Systems

2017 ◽  
Author(s):  
Keisuke Kamiya ◽  
Yusaku Yamashita
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Null Space ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Accurate Analysis ◽  
Dependent Variables ◽  
Space Matrix ◽  
Space Method

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the constraint Jacobian. In previous reports, one of the authors presented methods which use the null space matrix. In the procedure to obtain the null space matrix, the inverse of a matrix whose regularity may not be always guaranteed. In this report, a new method is proposed in which the null space matrix is obtained by solving differential equations that can be always defined by using the QR decomposition, even if the constraints are redundant. Examples of numerical analysis are shown to validate the proposed method.

Modeling of a Finger-Follower Cam System With Verification in Contact Forces

1996 ◽  
Vol 118 (1) ◽  
pp. 132-137 ◽  
Author(s):  
Wensyang Hsu ◽  
A. P. Pisano
Keyword(s):  
Coulomb Friction ◽  
Constraint Equation ◽  
Contact Forces ◽  
Dynamic Responses ◽  
Distributed Parameter ◽  
Contact Points ◽  
Frictional Forces ◽  
Contact Position ◽  
Force Data ◽  
Good Agreement

A lumped/distributed-parameter dynamic model is developed to investigate the dynamic responses of a finger-follower cam system by considering a hydraulic lash adjuster with an oscillating pivot, and frictional forces between sliding surfaces. The measured force data at low speed are employed to derive an algorithm to determine the dynamic Coulomb friction coefficients at contact points. The contact position between the cam and the follower with moving pivot is determined by a constraint equation method. A hydraulic lash adjuster acting as the pivot of the follower is also modeled with the effects of oil compressibility and oil refill mechanism. Simulated contact forces at three different speeds are shown to have good agreement with experimental data. The separation between the hydraulic lash adjuster and the follower is predicted at a camshaft speed of 2535 rpm, and experiment indicates at 2520 rpm.

Dynamic Analysis of an Integrated Train–Bridge–Foundation–Soil System by the Substructure Method

2018 ◽  
Vol 18 (05) ◽  
pp. 1850069 ◽  
Author(s):  
Hong Qiao ◽  
He Xia ◽  
Xianting Du
Keyword(s):  
Dynamic Analysis ◽  
Shear Velocity ◽  
Rigid Body Dynamics ◽  
Dynamic Responses ◽  
Superposition Method ◽  
Foundation Soil ◽  
Soil System ◽  
Substructure Method ◽  
Soil Foundation ◽  
Bridge Foundation

The substructure method is applied to the dynamic analysis of a train–bridge system considering the soil–structure interaction. With this method, the integrated train–bridge–foundation–soil system is divided into the train–bridge subsystem and the soil–foundation subsystem. Further, the train–bridge subsystem is divided into the train and bridge components. The frequency-dependent impedance function of the soil–foundation subsystem is transformed into time domain by rational approximation and simulated by a high-order lumped-parameter model with masses. The equations of motion of the train and bridge components are established by the rigid-body dynamics method and the modal superposition method, respectively. Finally, the dynamic responses of the two subsystems are obtained by iterative procedures, with the influence of the soil shear velocity studied. The case study reveals that it is important to consider the effect of soil–foundation interaction in the dynamic analysis of train–bridge systems, but with the increase of the shear velocity of the soil, such influence becomes weaker.

Continuous Null Space Method for Constrained Mechanical Systems

2011 ◽  
Author(s):  
Keisuke Kamiya ◽  
Makoto Sawada ◽  
Yuji Furusawa
Keyword(s):  
Lagrange Multipliers ◽  
Null Space ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Accurate Analysis ◽  
Constrained Mechanical Systems ◽  
Dependent Variables ◽  
Space Matrix ◽  
Space Method

The governing equations for multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. It is desirable for efficient and accurate analysis to eliminate the Lagrange multipliers and dependent variables. As a method to solve the DAEs by eliminating the Lagrange multipliers, there is a method called the null space method. In this report, first, it is shown that using the null space matrix one can eliminate the Lagrange multipliers and reduce the number of velocities to that of the independent ones. Then, a new method to obtain the continuous null space matrix is presented. Finally, the presented method is applied to four-bar linkages.

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