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Constrained Multibody Dynamics With Python: From Symbolic Equation Generation to Publication
Abstract
Symbolic equations of motion (EOMs) for multibody systems are desirable for simulation, stability analyses, control system design, and parameter studies. Despite this, the majority of engineering software designed to analyze multibody systems are numeric in nature (or present a purely numeric user interface). To our knowledge, none of the existing software packages are 1) fully symbolic, 2) open source, and 3) implemented in a popular, general, purpose high level programming language. In response, we extended SymPy (an existing computer algebra system implemented in Python) with functionality for derivation of symbolic EOMs for constrained multibody systems with many degrees of freedom. We present the design and implementation of the software and cover the basic usage and workflow for solving and analyzing problems. The intended audience is the academic research community, graduate and advanced undergraduate students, and those in industry analyzing multibody systems. We demonstrate the software by deriving the EOMs of a N-link pendulum, show its capabilities for LATEX output, and how it integrates with other Python scientific libraries — allowing for numerical simulation, publication quality plotting, animation, and online notebooks designed for sharing results. This software fills a unique role in dynamics and is attractive to academics and industry because of its BSD open source license which permits open source or commercial use of the code.
... Exudyn models are created solely using the Python language. Python is also available in other multibody codes, such as ProjectChrono [18] or PyDy [8]. As a difference from the latter codes, Exudyn includes a large set of annotated examples. ...
... The expectation of the tests is to obtain a Python code from the LLM that can be directly processed in Python using the Exudyn package. 8 Tests are evaluated based on correct code syntax and on correct modeling of the dynamic system. In order to clearly distinguish between the two error types, syntax errors (counted by e syn ) are all errors that raise a Python error when the code is executed, except for Exudyn's solver failures due to modeling errors. ...
... However, looking at larger datasets available on HuggingFace [28], we find typical datasets 9 that are 7 https://platform.openai.com/tokenizer. 8 Due to the widely backwards compatibility of Exudyn with the version from September 2021, all tests are evaluated with Exudyn version 1.7.0. We did not observe syntax errors in the LLM outputs related to a change of version. ...
Computational models are conventionally created with input data, script files, programming interfaces, or graphical user interfaces. This paper explores the potential of expanding model generation, with a focus on multibody system dynamics. In particular, we investigate the ability of Large Language Model (LLM), to generate models from natural language. Our experimental findings indicate that LLM, some of them having been trained on our multibody code Exudyn, surpass the mere replication of existing code examples. The results demonstrate that LLM have a basic understanding of kinematics and dynamics, and that they can transfer this knowledge into a programming interface. Although our tests reveal that complex cases regularly result in programming or modeling errors, we found that LLM can successfully generate correct multibody simulation models from natural-language descriptions for simpler cases, often on the first attempt (zero-shot).
After a basic introduction into the functionality of LLM, our Python code, and the test setups, we provide a summarized evaluation for a series of examples with increasing complexity. We start with a single mass oscillator, both in SciPy as well as in Exudyn, and include varied inputs and statistical analysis to highlight the robustness of our approach. Thereafter, systems with mass points, constraints, and rigid bodies are evaluated. In particular, we show that in-context learning can levitate basic knowledge of a multibody code into a zero-shot correct output.
... We implemented an open-source version in Python using SymPy (SymPy, 2021), leveraging its mechanical toolbox. Alternative symbolic frameworks found in the literature are usually limited to rigid bodies (Verlinden et al., 2005;Kurz and Eberhard, 2009;Gede et al., 2013;Docquier et al., 2013) or are closed-source or using proprietary software (Reckdahl and Mitiguy, 1996;Kurtz et al., 2010;MotionGenesis, 2016;Lemmer, 2018). ...
... 2.3). We were also inspired by the package PyDy (Gede et al., 2013), which is a convenient tool to export the equations of motion to executable code and directly visualize the bodies in 3D. The core of our work consisted of implementing a class to define flexible bodies (FlexibleBody) and the corresponding Kane method for this class (Sect. ...
... In this section, we present different wind energy applications of the symbolic framework. We focus on models with at least one flexible body because the rigid body formulation of SymPy has been well verified (Gede et al., 2013). For each example, the equations of motion are given and their results are compared with OpenFAST (Jonkman et al., 2021) simulations. ...
The article presents a symbolic framework (also called computer algebra program) that is used to obtain, in symbolic mathematical form, the linear and nonlinear equations of motion of a mid-fidelity multibody system including rigid and flexible bodies. Our approach is based on Kane's method and a nonlinear shape function representation for flexible bodies. The shape function approach does not represent the state of the art for flexible multibody dynamics but is an effective trade-off to obtain mid-fidelity models with few degrees of freedom, taking advantage of the separation of space and time. The method yields compact symbolic equations of motion with implicit account of the constraints. The general and automatic framework facilitates the creation and manipulation of models with various levels of complexity by adding or removing degrees of freedom. The symbolic treatment allows for analytical gradients and linearized equations of motion. The linear and nonlinear equations can be exported to Python code or dedicated software. There are multiple applications, such as time domain simulation, stability analyses, frequency domain analyses, advanced controller design, state observers, and digital twins. In this article, we describe the method we used to systematically generate the equations of motion of multibody systems and present the implementation of the framework using the Python package SymPy. We apply the framework to generate illustrative land-based and offshore wind turbine models. We compare our results with OpenFAST simulations and discuss the advantages and limitations of the method. The Python implementation is provided as an open-source project.
... Also, the use of generalized speeds to define the kinematics appears more natural for rotating systems rather than generalized coordinates. The method of generalized speeds is also suitable for implementation in a Computer Algebra System (CAS), assisting the bookkeeping of the equations, minimizing the risk for manual error and maintaining repeatability through coding which is highly desirable in an industrialized simulation process [14,15,16]. ...
... Even though Kane's method is suitable to implement in a Computer Algebra System (CAS), such as in Refs. [15,16], the equations of motion of the rotor pendulum system depicted in Figure 1 are manually derived to show the procedure of the method. ...
The multiple-order response of a rotor equipped with a centrifugal pendulum vibration absorber (CPVA) is investigated in this study. CPVAs have been used in the abatement of torsional vibrations since the 1930s and have been extensively researched but seldom with a focus on the full multiple-harmonic response. Although the firing order in reciprocating engines dominates in amplitude, the higher-order harmonics have impact on driveline-related noise issues. This investigation is conducted in order to understand if an higher-order amplification can stem from the design of the CPVA or if this phenomenon is due an interaction between other powertrain components. Therefore, an isolated rotor-CPVA system is studied. The investigated rotor is subjected to an oscillating torque consisting of single-and multiple-order harmonics. The equations of motion are derived by Kane's method and the multiple-harmonic response of the system is analyzed. The linear steady-state response is compared with numerical time integration of the equations of motion.
... the Euler-Lagrange equations of the three-link planar robot in Figure 1 can be written, with the aid of the symbolic multibody dynamics package PyDy [12], as M (θ, p) ∈ R 3×3 , M (θ, p) 0 is the symmetric mass matrix of the system with entries M 11 = I 1 + I 2 + I 3 + l 2 c1 m 1 + m 2 l 2 1 + 2l 1 l c2 c 2 + l 2 c2 + m 3 l 2 1 + 2l 1 l 2 c 2 + 2l 1 l c3 c 23 + l 2 2 + 2l 2 l c3 c 3 + l 2 c3 M 12 = I 2 + I 3 + l c2 m 2 (l 1 c 2 + l c2 ) + m 3 l 1 l 2 c 2 + l 1 l c3 c 23 + l 2 2 + 2l 2 l c3 c 3 + l 2 c3 M 13 = I 3 + l c3 m 3 (l 1 c 23 + l 2 c 3 + l c3 ) ...
... The parameter uncertainties lie within the interval [p, p] ⊆ R 12 in Table 1, which was calculated for a fluctuation of ±5% of the nominal weight of the user with anthropometric data from [21]. With a set P b ⊂ [p, p] of 500 parameters drawn from a Latin Hypercube, the first step in the analysis is to numerically solve the sensitivity equation (12) over the time horizon [0, 3.5] for all p ∈ P b . According to the sampling approach of the previous section, the sensitivity bounds S x , S x : [0, 3.5] → R 6×12 are then estimated by minimizing/maximizing the entries of the matrices (10) for each t ∈ T s as in (13). ...
A sensitivity-based approach for computing over-approximations of reachable sets, in the presence of constant parameter uncertainties and a single initial state, is used to analyze a three-link planar robot modeling a Powered Lower Limb Orthosis and its user. Given the nature of the mappings relating the state and parameters of the system with the inputs, and outputs describing the trajectories of its Center of Mass, reachable sets for their respective spaces can be obtained relying on the sensitivities of the nonlinear closed-loop dynamics in the state space. These over-approximations are used to evaluate the worst-case performances of a finite time horizon linear-quadratic regulator (LQR) for controlling the ascending phase of the Sit-To-Stand movement.
... 1 Due in part to this focus, it has become a popular language for scientific computing and data science, with a broad ecosystem of libraries (Oliphant, 2007). SymPy is itself used as a dependency by many libraries and tools to support research within a variety of domains, such as SageMath (The Sage Developers, 2016) (pure and applied mathematics), yt (Turk et al., 2011) (astronomy and astrophysics), PyDy (Gede et al., 2013) (multibody dynamics), and SfePy (Cimrman, 2014) (finite elements). ...
... Multibody Dynamics with Python (Gede et al., 2013). ...
SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
... This example is chosen from PyDy project [44]. A double pendulum is a pendulum with another pendulum attached to its end. ...
Cloud services have been widely employed in IT industry and scientific research. By using Cloud services users can move computing tasks and data away from local computers to remote datacenters. By accessing Internet-based services over lightweight and mobile devices, users deploy diversified Cloud applications on powerful machines. The key drivers towards this paradigm for the scientific computing field include the substantial computing capacity, on-demand provisioning and cross-platform interoperability. To fully harness the Cloud services for scientific computing, however, we need to design an application-specific platform to help the users efficiently migrate their applications. In this, we propose a Cloud service platform for symbolic-numeric computation - SNC. SNC allows the Cloud users to describe tasks as symbolic expressions through C/C++, Python, Java APIs and SNC script. Just-In-Time (JIT) compilation through using LLVM/JVM is used to compile the user code to the machine code. We implemented the SNC design and tested a wide range of symbolic-numeric computation applications (including nonlinear minimization, Monte Carlo integration, finite element assembly and multibody dynamics) on several popular cloud platforms (including the Google Compute Engine, Amazon EC2, Microsoft Azure, Rackspace, HP Helion and VMWare vCloud). These results demonstrate that our approach can work across multiple cloud platforms, support different languages and significantly improve the performance of symbolic-numeric computation using cloud platforms. This offered a way to stimulate the need for using the cloud computing for the symbolic-numeric computation in the field of scientific research.
... PyDy [30] was chosen as the framework for building the planar robot simulation as it uses a symbolic math backend that expresses equations of motion in a human-readable format. The kino-dynamic model of the planar robot was built using Kane's method [31]: a process for defining equations of motion with forces in different frames acting on rigid bodies with constraints. ...
Robotic manipulators are critical in many applications but are known to degrade over time. This degradation is influenced by the nature of the tasks performed by the robot. Tasks with higher severity, such as handling heavy payloads, can accelerate the degradation process. One way this degradation is reflected is in the position accuracy of the robot's end-effector. In this paper, we present a prognostic modeling framework that predicts a robotic manipulator's Remaining Useful Life (RUL) while accounting for the effects of task severity. Our framework represents the robot's position accuracy as a Brownian motion process with a random drift parameter that is influenced by task severity. The dynamic nature of task severity is modeled using a continuous-time Markov chain (CTMC). To evaluate RUL, we discuss two approaches -- (1) a novel closed-form expression for Remaining Lifetime Distribution (RLD), and (2) Monte Carlo simulations, commonly used in prognostics literature. Theoretical results establish the equivalence between these RUL computation approaches. We validate our framework through experiments using two distinct physics-based simulators for planar and spatial robot fleets. Our findings show that robots in both fleets experience shorter RUL when handling a higher proportion of high-severity tasks.
... Exudyn models are created solely using the Python language. Python is also available in other multibody codes, such as ProjectChrono [18] or PyDy [8]. As a difference to the latter codes, Exudyn includes a large set of annotated examples. ...
Computational models are conventionally created with input data, script files, programming interfaces, or graphical user interfaces. This paper explores the potential of expanding model generation, with a focus on multibody system dynamics. In particular, we investigate the ability of Large Language Models (LLMs), to generate models from natural language. Our experimental findings indicate that LLMs, some of them having been trained on our multibody code Exudyn, surpass the mere replication of existing code examples. The results demonstrate that LLMs have a basic understanding of kinematics and dynamics, and that they can transfer this knowledge into a programming interface. Although our tests reveal that complex cases regularly result in programming or modeling errors, we found that LLMs can successfully generate correct multibody simulation models from natural language descriptions for simpler cases, often on the first attempt (zero-shot). Next to a basic introduction into the functionality of LLMs, our Python code, and the test setups, we provide a summarized evaluation for a series of examples with increasing complexity. We start with a single mass oscillator, both in SciPy as well as in Exudyn, and include varied inputs and statistical analysis to highlight the robustness of our approach. Hereafter, systems with mass points, constraints, and rigid bodies are evaluated. In particular, we show that in-context learning can levitate basic knowledge of a multibody code into a zero-shot correct output.
... A different selection of system states would result in distinct expressions for joint moments and would also lead to Equations of Motion (EoMs) of different shape. Symbolic EoMs were obtained using Kane's Method (Kane and Levinson [5]) implementation of SymPy (Meurer et al. [6]) and evaluated numerically with the use of PyDy (Gede et al. [7]). ...
This paper presents a modeling approach using lumped element multibody systems to describe aeroe-lastic dynamics of aircraft. The model is shown to describe geometric nonlinearities arising from large structural displacements and oscillations such as that of a very flexible wing bending under aerodynamic loads. The presented approach is applied to a Goland's wing to determine the flutter speed resulting from coupling the multibody structure to a quasi-steady aerodynamic model. Flutter velocities are comparable to those obtained using modal representations for structural dynamics. Finally, the modeling appoach is shown to be scalable, parametrized and modular, all of which could make it valuable in iterative environments such as during conceptual design phases of very flexible aircraft.
... The manual derivation of the equations of motion are given here. However, the process and bookkeeping may be facilitated by means of CAS such as SymPy [31] . ...
A model of the normal-force dependant friction loss between the rotor and pendula is developed for bifilar centrifugal pendulum vibration absorbers (CPVAs). The normal-force is dominantly dependant on the rotor rotational velocity but also dependant on the pendulum path. Simulation results of the pendulum with the proposed normal-force friction model agree well with novel experimental results. Order-response simulations of the rotor-CPVA model with the proposed friction model reveal accuracy improvements on the prediction of the rotor oscillation amplitude at different rotational velocities in comparison to existing friction formulations. Derivation of the equations of motion of the rotor-CPVA is performed by Kane’s method and solved numerically. The developed experimental setup is used to validate the model parameters such as friction coefficients, path parameters and relative pendulum rotation without necessitating a special test-apparatus other than standard vibration measurement equipment. CPVAs are commonly used to reduce torsional vibration created by reciprocating engines. To reduce emissions, heavy-duty vehicles manufactures are downsizing and downspeeding the combustion engine while maintaining the power output. Unfortunately, this gives rise to increased torsional vibration in the powertrain. The CPVA is a device that can reduce the torsional vibration and thus help fulfill environmental goals.
... Kane's dynamical equation is of the formF r +F * r = 0, whereF r andF * r represents generalized active forces and generalized inertia forces, respectively, as described in Kane and Levinson (1985) (chapter 6, page 159). The equations in the form necessary for simulation is obtained using python libraries, the details of which are given in Gede et al. (2013). Symbols m 1 , m 2 represent the mass of the body and swing leg, respectively, as shown in Figure 2. The length of both the legs is represented by l. ...
Freezing is an involuntary stopping of gait observed in late-stage Parkinson's disease (PD) patients. This is a highly debilitating symptom lacking a clear understanding of its causes. Walking in these patients is also associated with high variability, making both prediction of freezing and its understanding difficult. A neuromechanical model describes the motion of the mechanical (motor) aspects of the body under the action of neuromuscular forcing. In this work, a simplified neuromechanical model of gait is used to infer the causes for both the observed variability and freezing in PD. The mathematical model consists of the stance leg (during walking) modeled as a simple inverted pendulum acted upon by the ankle-push off forces from the trailing leg and pathological forces by the plantar-flexors of the stance leg. We model the effect on walking of the swing leg in the biped model and provide a rationale for using an inverted pendulum model. Freezing and irregular walking is demonstrated in the biped model as well as the inverted pendulum model. The inverted pendulum model is further studied semi-analytically to show the presence of horseshoe and chaos. While the plantar flexors of the swing leg push the center of mass (CoM) forward, the plantar flexors of the stance leg generate an opposing torque. Our study reveals that these opposing forces generated by the plantar flexors can induce freezing. Other gait abnormalities nearer to freezing such as a reduction in step length, and irregular walking patterns can also be explained by the model.
... The following Kane's method integrator for equations of motion is due to VanderPlas [5] and Gede et al. [6]. Thus, we implement a Simulation() method which will apply the aforementioned procedure to determine the average oscillation period for each pendulum bob and then take an average over all masses. ...
We present the Euler--Langrage equations for a many-body system of coupled planar pendulums. Hence, imposing initial condition data, the equations of motion are linearized and later developed in an idealized model for the pseudo-periodicity of the system as a function of the number of pendulums N. The result is empirically corroborated by comparing the model with data obtained via a numerical simulation, and by employing Kane's Method integrator in Python.
... Dallali [445] studied compliant humanoid robot dynamics by a symbolic modeling method. Gede [446] extended the SymPy computer algebra system to deduce symbolic equations of motion for constrained MSs with many DOFs. Hall [447] proposed a graph-theoretic symbolic multibody modeling environment of MapleSim. ...
Flexible multibody system dynamics (MSD) is one of the hot spots and difficulties in modern mechanics. It provides a powerful theoretical tool and technical support for dynamic performance evaluation and optimization design of a large number of complex systems in many engineering fields, such as machinery, aviation, aerospace, weapon, robot and biological engineering. How to find an efficient accurate dynamics modeling method and its stable reliable numerical solving algorithm are the two core problems of flexible MSD. In this paper, the research status of modeling methods of flexible MSD in recent years is summarized first, including the selection of reference frames, the flexible body’s kinematics descriptions, the deductions of dynamics equation, the model reduction techniques and the modeling methods of the contact/collision, uncertainty and multi-field coupling problems. Then, numerical solution technologies and their latest developments of flexible MSD are discussed in detail. Finally, the future research directions of modeling and numerical computation of flexible MSD are briefly prospected.
... the Euler-Lagrange equations of the three-link planar robot in Fig. 1 can be written, with the aid of the symbolic multibody dynamics package PyDy [5], as Fig. 1: Three-link planar robot for modeling a powered lower limb orthosis and the interaction with its user during a Sit-to-Stand (STS) movement. ...
This study presents a technique to safely control the Sit-to-Stand movement of powered lower limb orthoses in the presence of parameter uncertainty. The weight matrices used to calculate the finite time horizon linear-quadratic regulator (LQR) gain in the feedback loop are chosen from a pool of candidates as to minimize a robust performance metric involving induced gains that measure the deviation of variables of interest in a linear time-varying (LTV) system, at specific times within a finite horizon, caused by a perturbation signal modeling the variation of the parameters. Two relevant Sit-to-Stand movements are simulated for drawing comparisons with the results documented in a previous work.
... We have a closed form expression for the position as a function of the curvatures q and of the curvilinear coordinates s = (s 1 , s 2 ): r = r(q, s). Since this expression is relatively complex we, at first, used Sympy and PyDy [120] to generate symbolic expressions for M (q), ∂M (q) ∂q i and F g (q). Then from the symbolic expressions C code is generated using the codegen module of the PyDy package (around 8000 lines of C code are generated). ...
This dissertation focuses on the numerical modelling of thin elastic structures in contact. Many objects around us, either natural or man-made, are slender deformable objects. Curve-like objects such as industrial cables, helicopter blades, plant stems and hair can be modelled as thin elastic rods. While surface-like objects such as paper, boat sails, leaves and clothes can be modelled as thin elastic shells. The numerical study of the mechanical response of such structures is important in many applications of engineering, bio-mechanics, computer graphics and other fields. In this dissertation we treat rods and shells as finite dimensional multibody systems.When a multibody system is subject to frictional contact constraints, a problem often arises. In some configurations there may exist no contact force which can prevent the system from violating its contact constraints. This is known as the Painlev'e paradox. In the first part of this manuscript we analyze the contact problem (whose unknowns are the accelerations and the contact forces) and we derive computable upper bounds on the friction coefficients at each contact, such that if verified, the contact problem is well-posed and Painlev'e paradoxes are avoided.Some rod-like structures may easily bend and twist but hardly stretch and shear, such structures can be modelled as Kirchhoff rods. In the second part of this manuscript we consider the problem of computing the stable static equilibria of Kirchhoff rods subject to different boundary conditions and frictionless contact constraints. We formulate the problem as an Optimal Control Problem, where the strains of the rod are interpreted as control variables and the position and orientation of the rod are interpreted as state variables. Employing direct methods of numerical Optimal Control then leads us to the proposal of new spatial discretization schemes for Kirchhoff rods. The proposed schemes are either of the strain-based type, where the main degrees of freedom are the strains of the rod, or of the mixed type, where the main degrees of freedom are both the strains and the generalized displacements.Very much like for Kirchhoff rods, thin surface-like structures such as paper can hardly stretch or shear at all. One of the advantages of the strain based approach is that the no extension and no shear constraints of the Kirchhoff rod are handled intrinsically, without the need of stiff repulsion forces, or of further algebraic constraints on the degrees of freedom. In the third part of this dissertation we propose an extension of this approach to model the dynamics of inextensible and unshearable shells. We restrict our study to the case of a shell patch with a developable mid-surface. We use as primary degrees of freedom the components of the second fundamental form of the shell's mid-surface. This also leads to an intrinsic handling of the no shear and no extension constraints of the shell.
... This example is chosen from PyDy project [44]. A double pendulum is a pendulum with another pendulum attached to its end. ...
Cloud services have been widely employed in IT industry and scientific research. By using Cloud services users can move computing tasks and data away from local computers to remote datacenters. By accessing Internet-based services over lightweight and mobile devices, users deploy diversified Cloud applications on powerful machines. The key drivers towards this paradigm for the scientific computing field include the substantial computing capacity, on-demand provisioning and cross-platform interoperability. To fully harness the Cloud services for scientific computing, however, we need to design an application-specific platform to help the users efficiently migrate their applications. In this, we propose a Cloud service platform for symbolic-numeric computation– SNC. SNC allows the Cloud users to describe tasks as symbolic expressions through C/C++, Python, Java APIs and SNC script. Just-In-Time (JIT) compilation through using LLVM/JVM is used to compile the user code to the machine code.
We implemented the SNC design and tested a wide range of symbolic-numeric computation applications (including nonlinear minimization, Monte Carlo integration, finite element assembly and multibody dynamics) on several popular cloud platforms (including the Google Compute Engine, Amazon EC2, Microsoft Azure, Rackspace, HP Helion and VMWare vCloud). These results demonstrate that our approach can work across multiple cloud platforms, support different languages and significantly improve the performance of symbolic-numeric computation using cloud platforms. This offered a way to stimulate the need for using the cloud computing for the symbolic-numeric computation in the field of scientific research.
A nonlinear centrifugal pendulum vibration absorber (CPVA) with normal-force dependant friction loss is investigated in a torsional model of a downspeeded powertrain of a heavy-truck. The engine model includes gas-pressure excitation and the existing pendulum model is extended to include a continuous formulation of end-stops at the end of the pendulum-path. Furthermore, the friction loss of the pendulum is experimentally determined. A pendulum-path parameter-study in the complete powertrain model is conducted to consider the effects of the system dynamics on the CPVA. It is shown that the performance of the CPVA is affected by the powertrain system-dynamics and thus important to consider in the design of the CPVA. Downspeeding of the engine by appropriate gearing of the driveline is a measure to decrease the CO2 emissions. However, downspeeding increases the torsional vibration and noise of the powertrain with conventional torsional vibration reduction methods. The CPVA can be used to reduce the torsional vibration and thus facilitate to reach environmental goals.
Dynamics is a core course in all engineering disciplines. Problem‐solving using computers may be done using programs specific for a problem or class of problems, or by using versatile and professional multibody dynamics packages such as ADAMS. Although such packages are good motivators, they can hardly enhance the student's knowledge of Dynamics. They are also uneconomical in terms of investment and computer infrastructure, unless used for more advanced purposes. So, the objective of the present paper is to propose an alternative by using an open‐source, lower level, multibody dynamics package like PyDy. This is not only economical but also demands knowledge of dynamics and an introductory knowledge of multibody dynamics. Multibody Dynamics being programmable makes the teaching and learning of dynamics more systematic. The approach followed in the present paper is to introduce some basic principles of multibody dynamics to an undergraduate student of conventional dynamics with the objective of exhibiting its systematic approach. With elementary knowledge of multibody dynamics, the students can write small programs for problem‐solving, with say Python, in their early years of engineering education, even before using such packages. Furthermore, this freeware PyDy allows generation of symbolic equations, which further benefits the students. Subsequently, numerical solution along with animation also fulfils the purpose of motivating students. The response of students toward queries connected with willingness to use the package and their perspective about its effectiveness as a supportive tool in learning dynamics has been found to be quite encouraging.
Recent advances in laminate manufacturing techniques have driven the development of new classes of millimeterscale sensorized medical devices, robots capable of terrestrial locomotion and sustained fight, and new techniques for sensing and actuation. Recently, the analysis of laminate micro-devices has focused more manufacturability concerns and not on mechanics. Considering the nature of such devices, we draw from existing research in composites, origami kinematics, and finite element methods in order to identify issues related to sequential assembly and self-folding prior to fabrication as well as the stiffness of composite folded systems during operation. These techniques can be useful for understanding how such devices will bend and ex under normal operating conditions, and when added to new design tools like popupCAD, will give designers another means to develop better devices throughout the design process.
We present an open source software implementation of a popular mathematical method developed by M.R. Yeadon for calculating the body and segment inertia parameters of a human body. The software is written in a high level open source language and provides three interfaces for manipulating the data and the model: a Python API, a command-line user interface, and a graphical user interface. Thus the software can fit into various data processing pipelines and requires only simple geometrical measures as input.
We present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally symmetri ideally hinged parts: two wheels, a frame and a front assembly. The wheels are also axisymmetric and make ideal knife-edg rolling point contact with the ground level. The mass distribution and geometry are otherwise arbitrary. This conservativ non-holonomic system has a seven-dimensional accessible configuration space and three velocity degrees of freedom parametrize by rates of frame lean, steer angle and rear wheel rotation. We construct the terms in the governing equations methodicall for easy implementation. The equations are suitable for e.g. the study of bicycle self-stability. We derived these equation by hand in two ways and also checked them against two nonlinear dynamics simulations. In the century-old literature, severa sets of equations fully agree with those here and several do not. Two benchmarks provide test cases for checking alternativ formulations of the equations of motion or alternative numerical solutions. Further, the results here can also serve as check for general purpose dynamic programs. For the benchmark bicycles, we accurately calculate the eigenvalues (the root of the characteristic equation) and the speeds at which bicycle lean and steer are self-stable, confirming the century-ol result that this conservative system can have asymptotic stability.
This book is ideal for teaching students in engineering or physics the skills necessary to analyze motions of complex mechanical systems such as spacecraft, robotic manipulators, and articulated scientific instruments. Kane's method, which emerged recently, reduces the labor needed to derive equations of motion and leads to equations that are simpler and more readily solved by computer, in comparison to earlier, classical approaches. Moreover, the method is highly systematic and thus easy to teach. This book is a revision of Dynamics: Theory and Applications by T. R. Kane and D. A. Levinson and presents the method for forming equations of motion by constructing generalized active forces and generalized inertia forces. Important additional topics include approaches for dealing with finite rotation, an updated treatment of constraint forces and constraint torques, an extension of Kane's method to deal with a broader class of nonholonomic constraint equations, and other recent advances.
I anatomize a successful open-source project, fetchmail, that was run as a deliberate test of some surprising theories about software engineering suggested by the history of Linux. I discuss these theories in terms of two fundamentally different development styles, the "cathedral" model of most of the commercial world versus the "bazaar" model of the Linux world. I show that these models derive from opposing assumptions about the nature of the software-debugging task. I then make a sustained argument from the Linux experience for the proposition that "Given enough eyeballs, all bugs are shallow", suggest productive analogies with other self-correcting systems of selfish agents, and conclude with some exploration of the implications of this insight for the future of software.
This book takes a traditional approach to the development of the methods of analytical dynamics. After a review of Newtonian dynamics, the basic concepts of analytical dynamics - classification of constraints, classification of forces, virtual displacements, virtual work and variational principles - are introduced and developed. Next, Langrange's equations are derived and their integration is discussed. The Hamiltonian portion of the book covers Hamilton's canonical equations, contact transformations, and Hamilton-Jacobi theory. Also included are chapters on stability of motion, impulsive forces, and the Gibbs-Appell equation. Two types of examples are used throughout the book. The first type is intended to illustrate key results of the theoretical development, and these are deliberately kept as simple as possible. The other type is included to show the application of the theoretical results to complex, real-life problems. These examples are often quite lengthy, comprising an entire chapter in some cases. © 2005 Kluwer Academic / Plenum Publishers. All rights reserved.
I anatomize a successful open-source project, fetchmail, that was run as a deliberate test of some theories about software engineering suggested by the history of Linux. I discuss these theories in terms of two fundamentally different development styles, the "cathedral" model, representing most of the commercial world, versus the "bazaar" model of the Linux world. I show that these models derive from opposing assumptions about the nature of the software-debugging task. I then make a sustained argument from the Linux experience for the proposition that "Given enough eyeballs, all bugs are shallow," suggest productive analogies with other self-correcting systems of selfish agents, and conclude with some exploration of the implications of this insight for the future of software.
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