Valéry Roy

A semi-analytic method for the computation of the effective properties of composites of two isotropic constituent materials

Tue, 15 Aug 2023 00:00:00 +0000

Proceedings of the Royal Society A

Published:16 August 2023

R. Valéry Roy

Abstract: We develop a computational framework for the semi-analytic representation of the effective transport properties of two-component composite materials, in the spirit of Bergman’s spectral representation. The components of the effective permittivity (or conductivity) tensor are expanded as power series in a contrast parameter between the two phases. The coefficients (the moments of a spectral measure) of these expansions are determined recursively through the numerical evaluations of integrals defined solely on the boundary between the constituent phases, and discretized by high-order (possibly exponentially converging) quadrature schemes. The high precision reached by these coefficients allows the series representation to be recast into Padé approximants (or continued fractions) to provide analytical expressions valid over large (possibly infinite) regions of the parameter complex plane. Examples, thus far limited to two-dimensional rhombic unit cells, demonstrate that the method is reliable for microgeometries and material parameters for which other methods are far less dependable. The analytical representations are particularly useful to study the optical or electrostatic resonant behaviour of some composite materials.

https://doi.org/10.1098/rspa.2023.0195

Numerical computation of the effective thermal conductivity of two-phase composite materials by digital image analysis

Thu, 01 Sep 2022 00:00:00 +0000

International Journal of Heat and Mass Transfer

Volume 197, 15 November 2022, 123377

Kelechi Ogbuanu and R. Valéry Roy

Abstract: We consider the numerical determination of the effective thermal conductivity of 2D, two-phase composite materials, whose microstructural geometries are described by digital images. More specifically, key microstructural data pertaining to the spatial distribution and volume fractions of the constituent phases in the material are extracted, along with the shapes of the interphase boundaries, using computer vision techniques. For inclusions of simple shapes (polygons, circles, ellipses), the extracted interphase boundary data is easily obtained and can be exploited immediately (e.g. polygonal vertices) or can be fitted with smooth curves, using least square error minimization techniques in order to reconstruct parametric representations in the case of circular or elliptical boundaries. For more intricate microstructure geometries characterized by clustered or agglomerated intersecting inclusions, piecewise cubic spline functions are adopted to obtain a least-square fitting of the interphase boundaries. The effective thermal conductivities are then calculated based on a boundary integral formulation defined over the reconstructed inclusion boundaries. They are expressed in terms of power series expansions of a conductivity contrast parameter of the constituent phases in the material. To accelerate the convergence of these expansions, nonlinear transformations (known as Padé Approximants) are used to yield rational approximations of the effective thermal conductivity tensor. Tests conducted on a variety of 2D, two-phase microstructures yielded accurate results over a wide range of microstructure image resolution, regardless of the contrast parameters of the constituent phases, thus, validating the efficacy and robustness of the proposed scheme.

https://doi.org/10.1016/j.ijheatmasstransfer.2022.123377

A three-dimensional numerical model for the motion of liquid drops by the particle finite element method

Thu, 05 May 2022 00:00:00 +0000

Elaf Mahrous, R. Valéry Roy, Alex Jarauta and Marc Secanell Physics of Fluids 34.5 (2022): 052120.

https://doi.org/10.1063/5.0091699

Abstract: Analysis of drop spreading and sliding on solid substrates is critical for many industrial applications, such as microfluidic devices, cooling towers, and fuel cells. A new three-dimensional model is proposed for droplet dynamics. Its numerical solution is obtained by the particle finite element method, based on an updated Lagrangian framework to accurately track the deformation of the droplet. The model hinges on boundary conditions at the solid–liquid interface to account for viscous dissipation and retention forces. These conditions are essential to obtain mesh-independent solutions and a realistic spatiotemporal evolution of the droplet deformation. Several numerical simulations are performed to assess the performance of the model for spreading and sliding drops, and results are compared to experimental data found in the literature. Good agreement is obtained with the available data. Simulations performed in two dimensions show striking discrepancies with the experimental data, thus demonstrating the need for three-dimensional simulations.

Analytical Representation and Efficient Computation of the Effective Conductivity of Two-Phase Composite Materials

Fri, 15 Oct 2021 00:00:00 +0000

International Journal for Numerical Methods in Engineering

Volume 123 (15), 15 August 2022, pp. 3567-3593

R. Valéry Roy

Abstract:

Many engineered materials display ordered or disordered microstructures. Such materials exhibit transport properties which are unmatched by their single-phase homogeneous counterparts. These properties are obtained by the mixture of two or more phases typically characterized by a large contrast in their properties. For the development of these materials, it is critical to develop a robust computational framework in order to provide a fundamental understanding of how microstructure affects performance. This hinges on predicting their macroscopic properties, given the constitutive laws and spatial distribution of their constituents. To this end, this work presents a computational framework based on formulating periodic conduction transport problems in terms of boundary integral equations whose kernel is expressed in terms of Weierstrass zeta-function. The components of the effective conductivity tensor are then sought in the form of power series expansions of a conductivity contrast parameter. To accelerate their convergence, these expansions are transformed into Padé approximants. Presently restricted to the case of two-dimensional, two-phase microstructures, this framework is shown to yield accurate results over the entire range of the contrast parameter. Representation of the kernel as a lattice sum allows the use the Fast Multipole Method, thereby making computations significantly more efficient.

https://doi.org/10.1002/nme.6980

A two-dimensional numerical model for the sliding motion of liquid drops by the particle finite element method

Thu, 30 Sep 2021 00:00:00 +0000

Elaf Mahrous, R. Valéry Roy, Alex Jarauta and Marc Secanell

Physics of Fluids 33, 032117 (2021); https://doi-org.udel.idm.oclc.org/10.1063/5.0039517

Liquid drops sliding on surfaces are ubiquitous both in the natural and industrial world. The prediction of such drop motions has far-reaching implications in many fields of application, including microfluidics, phase change heat transfer, or coating technology. We present a numerical model based on the particle finite element method for the prediction of the sliding motion of liquid drops. The model includes the effect of a retention force which acts in the vicinity of the drop’s contact line. This effect is found to be essential to obtain realistic spatiotemporal evolution of the drop. Thus far limited to two-dimensional simulations, the proposed model is validated by using experimental data found in the published literature, covering a wide range of drop size and physical properties. The numerical results are found to be mesh-independent and in good agreement with the experiments.

A particle finite element-based model for droplet spreading analysis

Mon, 30 Aug 2021 00:00:00 +0000

Elaf Mahrous, Alex Jarauta, Thomas Chan, Pavel Ryzhakov, Adam Z. Weber, R. Valéry Roy and Marc Secanell

Physics of Fluids 32, 042106 (2020); https://doi.org/10.1063/5.0006033

A particle finite element method-based model is proposed to analyze droplet dynamics problems, particularly droplet spreading on solid substrates (wetting). The model uses an updated Lagrangian framework to formulate the governing equations of the liquid. The curvature of the liquid surface is tracked accurately using a deforming boundary mesh. In order to predict the spreading rate of the droplet on the solid substrate and track the corresponding contact angle evolution, dissipative forces at the contact line are included in the formulation in addition to the Navier-slip boundary conditions at the solid–liquid interface. The inclusion of these boundary conditions makes it possible to account for the induced Young’s stress at the contact line and for the viscous dissipation along the solid–liquid interfacial region. These are found to be essential to obtain a mesh-independent physical solution. The temporal evolution of the contact angle and the contact line velocity of the proposed model are compared with spreading droplets and micro-sessile droplet injection experiments and are shown to be in good agreement.

A Novel Computational Framework For The Effective Transport Properties Of Heterogeneous Materials Reconstructed From Digital Images

Sun, 01 Aug 2021 00:00:00 +0000

ASME International Mechanical Engineering Congress and Exposition. Vol. 85680. American Society of Mechanical Engineers, 2021.

Kelechi O. Ogbuanu, R. Valéry Roy

In computational material science, Digital Image Processing and Big Data analysis play a crucial role in Microstructure Characterization and Reconstruction (MCR), especially in the estimation of structure-property relationships. In this work, we are interested in the calculation of the effective transport properties of composite materials from digital images. Because most MCR techniques are heavily statistical, they may suffer from significant microstructure information loss as they are incapable of reproducing exact images of microstructures. Here, we take advantage of pattern recognition algorithms to extract nearly exact morphological information pertaining to the interphase boundaries from digital microstructural images, thereby minimizing information loss. The data extracted then serves as the basis for our effective transport property module for calculating the effective properties of two-phase composite materials in a way that is automated, fast, stable, memory efficient, and accurate. Our current formulation is limited to circular or near-circular inclusions, with very large contrast properties. Preliminary numerical tests on four cases of 2D, two-phase microstructure images yielded relative errors ranging from 0.1% to 2.0%, for image pixel density around 1000 × 1000 pixels. These relative errors are perfectly acceptable without having to resort to unrealistic image sizes.

https://doi.org/10.1115/IMECE2021-70817

How to create a static website with Hugo Academic

Sat, 15 Feb 2020 00:00:00 +0000

In this post, I will show how you can build a website with Hugo using the academic theme. There are many useful guides out there, and you should first read Hugo website . Hugo claims to be “The world’s fastest framework for building websites”. However, it took me a while to make this work. This is the solution what worked for me on a Linux-based system. I keep modifying my website locally, and I can deploy the incremental changes within seconds on my web server.

1. First Steps in RStudio

I use an R package, called blogdown to create and update my website. This package takes advantage of R Markdown files and Hugo. I recommend you read Blogdown: Creating Websites with R Markdown by Yihui Xie and Amber Thomas. I am assuming that Rstudio is installed on your linux machine, and that you are familiar with this IDE. Markdown is a simple formatting syntax for authoring HTML, PDF, and other documents.

Open Rstudio from your Application menu (or launch it from a terminal). In the “File” menu, open “New Project…” and select the option “New Directory”, then in the “Project Type” window, select “Website using blogdown”. In the Dialog Window which immediately appears, fill in the name of the directory where you will build your website. Leave the “subdirectory” and “theme” blank. The final step is to click “Create Project”.

If all went well, you can now load the academic theme as follows:

blogdown::install_theme(theme = 'gcushen/hugo-academic',hostname="github.com",theme_example=T,update_config=T,update_hugo=T)

This should install the academic theme and update the config.toml file.

If the above steps failed, you can do this manually: launch rstudio in the empty directory where you first attempted to build your website, and install devtools and blogdown:

install.packages("devtools")
install.packages('blogdown')

then install Hugo

blogdown::install_hugo()

Finally install the academic theme:

blogdown::new_site(theme = 'gcushen/hugo-academic', theme_example = T)

The final step is to build the default website by typing:

blogdown:::serve_site()

or by selecting the option “Serve Site” in the “Addins” menu of RStudio. You will then see your website in the built-in Viewer of RStudio. It is better to view it in your browser at the URL:http://127.0.0.1:4321. Explore its features, and take a look at the structure of the home directory: you should see the file config.toml and five subdirectories: config, content, public, resources, static, themes. Most of the data resides in the content subdirectory.

2. Personalize your Website

There is a lot left to do to personalize your website.

The first step is to open the configuration file config.toml within Rstudio. It is located in the home directory of your website. Select this file to edit it in the file editor pane of RStudio (top left). This file allows you to control the configuration and appearance of your homepage.

You can set the “title” of your homepage, your “name”, your “role”, and your “organization”. Initially, the baseurl is set to “/”. You can change it after you deploy your website. Leave other variables unchanged (for now).

The second step is to select a picture of yourself and save it as content/authors/admin/avatar.jpg. Next, you want to edit the content of the file content/authors/admin/_index.md. The content of the header of this YAML file is self-explanatory. Edit the short biosketch which follows the header. The content of this file will start to considerably alter the look of your homepage. As you make these changes, reload the homepage in your browser.

You also want to edit the file config/_default/params.toml, mostly by updating your Contact details. Decide which personal information you want to reveal. Also, change the math = false to math = true.

The third step is to specify which of the widgets you wish to show in your homepage. These widgets are controlled by Markdown files in the content/home/ directory. For instance, you may want to turn off the content of the content/home/about.md file: just change the variable active = true to active = false in the header of this file.

If you turn off this widget, this will remove all your personal information section of your home page, including your picture and biosketch. Decide which of the widgets you want to keep in your homepage. You could start by deactivating all the widgets (except the About widget) and turning them on one at a time to decide which ones you want to keep. Edit all the selected markdown files (with the .md extension) located in the content/home/ directory.

In addition to the about widget, I only retained the post, people and contact widgets. You can order the appearance of each section by changing the variable weight of the widgets. At any time, you can re-activate new widgets.

The last step is to control the appearance of the menus at the top of your homepage. You would need to edit the file config/_default/menus.toml. Click on this file. The content is self-explanatory. You can remove or create menus. For instance, if you wish to create the Programs menu you would insert the following:

[[main]]
  name = "Programs"
  url = "programs/"
  weight = 10

You would then have a create a directory content/programs by selecting “New Folder” within the content directory in RStudio. The variable weight specifies the order of appearance of your menus. If you specify a menu such as

[[main]]
  name = "Posts"
  url = "#posts"
  weight = 20

clicking on the “Posts” menu will lead you to the “Recent Posts” section within your homepage (if you did not deactivate the widget). If you change it to

[[main]]
  name = "Posts"
  url = "/post"
  weight = 20

clicking on the “Posts” menu will lead you to the “Posts” webpage which would list all your posts.

3. Create Content

Suppose you want to create a new post. First, you should look at the content of the content/post/ directory. This gives you idea on how to create content. You will find a YAML file _index.md which you shoud not modify.

Use the “New Post” addin within Rstudio to create a new post or page, then edit the YAML metadata (the first lines between --- and ---), and finally start writing the content. The title of your new post will appear in the homepage, and you can click on the title to see the content. You can also click on the “post” menu to see a list of all your posts.

Markdown provides a simple formatting syntax for authoring HTML which is entirely made of punctuation characters. For more details on using R Markdown see http://rmarkdown.rstudio.com. You can also use HTML syntax if some markup is not covered by Markdown’s syntax. Here are the most basic syntax elements:

Headers: Place one or more hashtags at the start of a line to create a header or sub-header. For example, you can create a bold header with the syntax ### Section 1: Introduction. A single hashtag on a new line creates a header of the highest level.
Italicized and bold text: Surround italicized text with asterisks, *like this* to produce: like this. Surround bold text with two asterisks, **like this** to produce: like this.
Lists: Group lines into bullet points that begin with asterisks. Leave a blank line before the first bullet, such as this

* Item 1

* Item 2

* Item 3

Hyperlinks: Surround links with brackets, and then provide the link target in parentheses, such as this [Github](www.github.com).

Other content can be created in the content/ directory. For instance, create content in the content/project directory. This content must be accessible through the top menus of your website. There should already be a YAML file _index.md in this directory. Each of your projects can be created in separate markdown *md files in separate subdirectories.

Finally, you will find the static/ directory which stores static web files like pictures. If you store yhe image file static/python/pendulum.png, this picture can then be embedded in your post using the Markdown syntax ![This is a Pendulum](/python/pendulum.png). You can also use HTML code to do with more flexibility.

4. Technical Content

It is rather straightforward to include technical content in your webpage by using markdown files.

For instance, you can use special syntax to highlight code snippets. You can enable this feature by toggling the highlight option in your config/_default/params.toml file: set highlight = true. For instance, if you want to include python codes, you would write

```python
import matplotlib.pyplot as plt
import numpy as np
   
x = np.arange(0.0, 4.0*np.pi, 0.1) # array of x values from 0 to 4*pi with increment of 0.1
y = 1 + np.sin(x)  # array of y values
plt.plot(x, y) # plot y vs x with continuous line (default)
```

which would render as

import matplotlib.pyplot as plt
import numpy as np
 
x = np.arange(0.0, 4.0*np.pi, 0.1) # array of x values from 0 to 4*pi with increment of 0.1
y = 1 + np.sin(x)  # array of y values
plt.plot(x, y) # plot y vs x with continuous line (default)

It is also possible to render Latex math expressions in your webpage. You can enable this feature by toggling the math option in your config/_default/params.toml file. To render inline or block math, wrap your LaTeX math with $...$ or $$...$$, respectively. For instance, this inline math expression $J_n = \int_0^1 x(1-x)^n dx$ would render as $J_n = \int_0^1 x(1-x)^n dx$. This math block

$$
\newcommand{\bom}{\boldsymbol{\omega}}
\bom_{1/2}+ \bom_{2/3}+ \bom_{3/1} = {\bf 0}
$$

would be rendered successfully as \[ \newcommand{\bom}{\boldsymbol{\omega}} \bom_{1/2}+ \bom_{2/3}+ \bom_{3/1} = {\bf 0} \] I have had more problems rendering multiline equations such as this

$$
\newcommand{\bom}{\boldsymbol{\omega}} \newcommand{\bv}{\boldsymbol{v}}
\{ {\cal V}_{i/j} \} =
\begin{Bmatrix} \bom_{i/j} \\ \bv_{A\in i/j}
\end{Bmatrix}
$$

If you save your file with the extension .md, this will fail. However, if the file is saved with the extension .rmd, it renders successfully: \[ \newcommand{\bom}{\boldsymbol{\omega}} \newcommand{\bv}{\boldsymbol{v}} \{ {\cal V}_{i/j} \} = \begin{Bmatrix} \bom_{i/j} \\ \bv_{A\in i/j} \end{Bmatrix} \]

5. Deploy your Website

There are many ways to deploy (i.e. publish) your website. Most articles recommend deploying in Netlify through GitHub, and the process appears to be smooth.

In principle, all you need to do to deploy your site is to copy the public/ directory (by SFTP, WebDAV, Rsync, git push, …) to your web server. I have adopted the process described here. Since I have access to my department web host via SSH, the use of a simple rsync command makes the incremental deployment of my Hugo website a very simple matter. I highly recommend this method.

Note that if your previous generations are not removed by hugo. It is often a good thing to delete your previous public/ directory prior to running hugo. This will ensure that old deleted pages are no longer visible from your website.

Screws (Part 7): An Energy Theorem

Sun, 19 Jan 2020 00:00:00 +0000

1. Introduction

Recall that the Principle of Virtual Power states that the sum of the virtual power of body and contact forces acting on a material system $\Sigma$ and of the virtual power of inertial forces relative to a Newtonian referential ${\cal E}$ adds to $0$: \[ \int_\Sigma{\bf f}^g _{\bar{\Sigma}\to \Sigma} (P) \cdot \, {\bf v}_P ^* \, dV(P) + \int_{\partial\Sigma}{\bf f}^c _{\bar{\Sigma}\to \Sigma} (Q) \cdot \, {\bf v}_Q ^* \, dA(Q) = \int_\Sigma{\bf a}_{P/{\cal E}} \, \cdot \, {\bf v}_P^* \, dm(P) \qquad\qquad (1) \] for all rigidifying virtual velocity fields $P\mapsto {\bf v}_P^*$, that is, vector fields satisfying ${\bf v}_Q^*={\bf v}_P^*+{\bf V}\times{\bf r}_{PQ}$ (screws).

We are now assuming that system $\Sigma$ is a rigid body ${\cal B}$ of mass $m$, mass center $G$ and inertia operator ${\cal I}_B$ (about a particular point $B$ of the body).

We will choose a particular virtual field as the actual velocity field $P\in{\cal B}\mapsto {\bf v}_{P/{\cal E}}$. This is possible since this field defines the kinematic screw $\{{\cal V}_{{\cal B}/{\cal B}} \}$ of the rigid body.

2. Power Generated by Body/Contact Forces

Recall that the power generated by a force ${\bf F}$ acting on a particle $P$ of velocity ${\bf v}_{P/{\cal E}}$ is the instantaneous scalar quantity defined as ${\bf v}_{P/{\cal E}} \cdot {\bf F}$.

The total power generated by the external (gravitational) body forces acting on body ${\cal B}$ is then naturally defined as the quantity \[ \mathbb{P}^g_{\bar{{\cal B}}\to {\cal B}/{\cal E}} = \int_{{\cal B}} {\bf f}^g _{\bar{{\cal B}}\to {\cal B}} (P) \cdot \, {\bf v}_{P/{\cal E}} \, dV(P) \] Similarly, the total power generated by the contact forces acting on body ${\cal B}$ is defined as the quantity \[ \mathbb{P}^c_{\bar{{\cal B}}\to {\cal B}/{\cal E}} = \int_{\partial{\cal B}} {\bf f}^c _{\bar{{\cal B}}\to {\cal B}} (Q) \cdot \, {\bf v}_{Q/{\cal E}} \, dA(Q) \] Recall that the body and contact forces define the action screws $\{{\cal A}^g_{\bar{{\cal B}}\to {\cal B}}\}$ and $\{{\cal A}^c_{\bar{{\cal B}}\to {\cal B}}\}$, of resultant ${\bf F}^g_{\bar{{\cal B}}\to{\cal B}}$ and ${\bf F}^c_{\bar{{\cal B}}\to {\cal B}}$, respectively. Hence, in practice, the integrals which define the powers generated by the body and contact forces can be expressed in the form of the dot product of the kinematic screw with the corresponding action screw: \[ \mathbb{P}^g_{\bar{{\cal B}}\to {\cal B}/{\cal E}} = \{ {\cal V}_{\bar{{\cal B}}/{\cal E}} \} \cdot \{{\cal A}^g_{\bar{{\cal B}}\to {\cal B}}\} = {\bf v}_{A \in {\cal B}/{\cal E}}\cdot {\bf F}^g_{\bar{{\cal B}}\to {\cal B}} + \boldsymbol{\omega}_{{\cal B}/{\cal E}}\cdot {\bf M}^g_{A,\bar{{\cal B}}\to{\cal B}} \] \[ \mathbb{P}^c_{\bar{{\cal B}}\to {\cal B}/{\cal E}} = \{ {\cal V}_{\bar{{\cal B}}/{\cal E}} \} \cdot \{{\cal A}^c_{\bar{{\cal B}}\to {\cal B}}\} = {\bf v}_{A\in {\cal B}/{\cal E}} \cdot {\bf F}^c_{\bar{{\cal B}}\to {\cal B}} + \boldsymbol{\omega}_{{\cal B}/{\cal E}}\cdot {\bf M}^c_{A,\bar{{\cal B}}\to{\cal B}} \]

3. The Kinetic Energy Theorem

Let us now apply the Principle of Virtual Power (1) by using ${\bf v}^*_p = {\bf v}_{P/{\cal E}}$, that is, by substituting the true velocity field in the integrals of equation (1) in place of the virtual velocity field. Then, we recognize the integrals of the left-hand-side of equation (1) (virtual powers) as the powers $\mathbb{P}^g_{\bar{{\cal B}}\to {\cal B}/{\cal E}}$ and $\mathbb{P}^c_{\bar{{\cal B}}\to {\cal B}/{\cal E}}$ generated by the body and contact forces.

The integral of the right-hand-side of (1) can be expressed in terms of the kinetic energy of the body since ${\bf a}_{P/{\cal E}}\cdot {\bf v}_{P/{\cal E}}= \frac{d}{dt}(\frac{1}{2} {\bf v}^2 _{P/{\cal E}})$: \[ \mathbb{K}_{{\cal B}/{\cal E}} = \int_{\cal B}\frac{1}{2} {\bf v}_{P/{\cal E}}^2 \, dm(P) \]

This yields the expression of the Kinetic Energy Theorem: \[ \frac{d}{dt}\mathbb{K}_{{\cal B}/{\cal E}} \, = \, \mathbb{P}^g_{\bar{{\cal B}}\to {\cal B}/{\cal E}} \, + \, \mathbb{P}^c_{\bar{{\cal B}}\to {\cal B}/{\cal E}} \qquad\qquad (2) \] which shows that the time rate of change of the kinetic energy of a rigid body is equal to the total power of the external body and contact forces acting on the body (relative to a Newtonian referential).

In practice, the kinetic energy of the body is found from the knowledge of $\{{\cal V}_{{\cal B}/{\cal E}}\}$ and the inertia operator ${\cal I}_B$: \[ \mathbb{K}_{{\cal B}/{\cal E}} = \frac{1}{2} m {\bf v}^{2}_{B\in{\cal B}/{\cal E}} + m {\bf v}_{B\in{\cal B}/{\cal E}} \cdot (\boldsymbol{\omega}_{{\cal B}/{\cal E}} \times {\bf r}_{BG})+ \frac{1}{2} \boldsymbol{\omega}_{{\cal B}/{\cal E}} \cdot {\cal I}_B (\boldsymbol{\omega}_{{\cal B}/{\cal E}}) \]
If we choose the mass center $G$ for point $B$, we find the expression \[ \mathbb{K}_{{\cal B}/{\cal E}} = \frac{1}{2} m {\bf v}^{2}_{G/{\cal E}} + \frac{1}{2} \boldsymbol{\omega}_{{\cal B}/{\cal E}} \cdot {\cal I}_G (\boldsymbol{\omega}_{{\cal B}/{\cal E}}) \]

4. Example

Consider the disk $1$ of center $G$, radius $R$ and mass $m$ first treated in Part 4. It rolls on a horizontal plane $(O,\boldsymbol{\hat{\imath}},\boldsymbol{\hat{\jmath}})$ of a referential $0(O,\boldsymbol{\hat{\imath}},\boldsymbol{\hat{\jmath}},\boldsymbol{\hat{k}})$. Assume that the contact at $I$ is without slip, so that ${\bf v}_{I\in 1/0}= {\bf 0}$. The position of $1$ relative to $0$ is defined by the Cartesian coordinates $(x,y)$ of $I$ and the (Euler) angles $(\psi,\theta,\phi)$ as shown below.

We want to apply the Kinetic Energy Theorem to body $1$ (assuming referential $0$ Newtonian).

Recall that the expression of the kinematic screw of the body is given by \[ \{ {\cal V}_{1/0} \} = \begin{Bmatrix} \boldsymbol{\omega}_{1/0} \\ {\bf v}_{I\in 1/0} \end{Bmatrix} = \begin{Bmatrix} \dot{\psi}\boldsymbol{\hat{k}}+ \dot{\theta}\boldsymbol{\hat{u}}+ \dot{\phi}\boldsymbol{\hat{w}} \\ {\bf 0} \end{Bmatrix}_I \] leading to ${\bf v}_{G/0}= \boldsymbol{\omega}_{1/0}\times {\bf r}_{IG}= (\dot{\psi}\boldsymbol{\hat{k}}+ \dot{\theta}\boldsymbol{\hat{u}}+ \dot{\phi}\boldsymbol{\hat{w}}) \times R \boldsymbol{\hat{v}}= -R(\dot{\psi}\cos\theta +\dot{\phi}) \boldsymbol{\hat{u}}+ R\dot{\theta}\boldsymbol{\hat{w}}$. Since ${\bf v}_{G/0}$ is also given by $\frac{d}{dt}(x\boldsymbol{\hat{\imath}}+y\boldsymbol{\hat{\jmath}}+ R\boldsymbol{\hat{v}}) = \dot{x}\boldsymbol{\hat{\imath}}+\dot{y}\boldsymbol{\hat{\jmath}}+R (\dot{\theta}\boldsymbol{\hat{w}}-\dot{\psi}\cos\theta\boldsymbol{\hat{u}})$, we find the non-holonomic equations \[ \dot{x}= -R \dot{\phi}\cos\psi, \qquad \dot{y}= - R\dot{\phi}\sin\psi \] The kinetic energy of the body is obtained by using inertia operator ${\cal I}_G$ on basis $(\boldsymbol{\hat{u}},\boldsymbol{\hat{v}},\boldsymbol{\hat{w}})$: \[ \mathbb{K}_{0/1} = \frac{1}{2} m {\bf v}^{2}_{G/0} + \frac{1}{2} \boldsymbol{\omega}_{1/0} \cdot {\bf H}_G = \frac{1}{2} m R^2 \big( (\dot{\phi}+\dot{\psi}\cos\theta)^2 + \dot{\theta}^2 \big) +\frac{1}{4}mR^2 \big( \dot{\theta}^2 + \dot{\psi}^2 \sin^2\theta +2 (\dot{\phi}+\dot{\psi}\cos\theta)^2 \big) \] which gives \[ \mathbb{K}_{0/1} = \frac{1}{4} m R^2 \big( 3\dot{\theta}^2 + \dot{\psi}^2 \sin^2\theta +4 (\dot{\phi}+\dot{\psi}\cos\theta)^2 \big) \] The total action screw on the body can be expressed as the sum of two contributions \[ \{ {\cal A}_{\bar{1}\to 1} \} = \begin{Bmatrix} -mg \boldsymbol{\hat{k}} \\ {\bf 0} \end{Bmatrix}_G + \begin{Bmatrix} N\boldsymbol{\hat{k}}+ {\bf F} \\ {\bf 0} \end{Bmatrix}_I \] with ${\bf F}\cdot \boldsymbol{\hat{k}}=0$. We can easily find the power generated by the action screw: \[ \mathbb{P}_{\bar{1}\to 1/0} = \begin{Bmatrix} -mg \boldsymbol{\hat{k}} \\ {\bf 0} \end{Bmatrix}_G \cdot \begin{Bmatrix} \dot{\psi}\boldsymbol{\hat{k}}+ \dot{\theta}\boldsymbol{\hat{u}}+ \dot{\phi}\boldsymbol{\hat{w}} \\ {\bf v}_{G/0} \end{Bmatrix}_G + \begin{Bmatrix} N\boldsymbol{\hat{k}}+ {\bf F} \\ {\bf 0} \end{Bmatrix}_I \cdot \begin{Bmatrix} \dot{\psi}\boldsymbol{\hat{k}}+ \dot{\theta}\boldsymbol{\hat{u}}+ \dot{\phi}\boldsymbol{\hat{w}} \\ {\bf 0} \end{Bmatrix}_I = -mg \boldsymbol{\hat{k}}\cdot {\bf v}_{G/0} = -mg R \dot{\theta}\cos\theta \] Finally, application of the Kinetic Energy Theorem gives \[ \frac{d}{dt} \mathbb{K}_{1/0} = -mgR \dot{\theta}\cos\theta = -\frac{d}{dt} (mg R \sin\theta) \] which gives the first integral \[ \frac{1}{4} m R^2 \big( 3\dot{\theta}^2 + \dot{\psi}^2 \sin^2\theta +4 (\dot{\phi}+\dot{\psi}\cos\theta)^2 \big) + mgR \sin\theta = \text{Constant} \]

5. Relationship between the Kinetic Energy Theorem and the Fundamental Theorem of Dynamics

Let us write the Fundamental Theorem of Dynamics for rigid body ${\cal B}$: \[ \{{\cal D}_{{\cal B}/{\cal E}} \} = \{ {\cal A}_{\bar{{\cal B}}\to{\cal B}} \} \qquad\qquad (3) \] We can take the dot product of (3) with the kinematic screw of body ${\cal B}$ to obtain \[ \{{\cal D}_{{\cal B}/{\cal E}} \} \cdot \{{\cal V}_{{\cal B}/{\cal E}} \} = \{ {\cal A}_{\bar{{\cal B}}\to{\cal B}} \} \cdot \{{\cal V}_{{\cal B}/{\cal E}} \} \] The right-hand side is nothing but the power $\mathbb{P}_{\bar{{\cal B}}\to {\cal B}/{\cal E}}$.

The left-hand side is given by $\int_{\cal B}{\bf v}_{P/{\cal E}} \cdot {\bf a}_{P/{\cal E}} dm$: it is recognized as the time rate of change of the kinetic energy of the body.

We have recovered the Kinetic Energy Theorem.

6. Power of Interaction

The Kinetic Energy Theorem can also be derived for a system $\Sigma$ of $N$ rigid bodies. Since the bodies are interconnected, there are inevitably internal actions between the bodies. The question is then whether the contributions of these internal actions cancel each other out in the context of power. To answer this question, consider two bodies $i$ and $j$ of $\Sigma$ and the corresponding powers $\mathbb{P}_{i\to j/0}$ and $\mathbb{P}_{j\to i/0}$ generated by the action $\{{\cal A}_{i\to j}\}$ and the reaction $\{{\cal A}_{j\to i}\}$ relative to referential $0$. These powers are given by \[ \mathbb{P}_{i\to j/0}= \{{\cal V}_{j/0} \} \cdot \{{\cal A}_{i\to j}\} , \qquad \mathbb{P}_{j\to i/0}= \{{\cal V}_{i/0} \} \cdot \{{\cal A}_{j\to i}\} \] Addition of these two terms gives, upon using $\{{\cal A}_{j\to i}\}= -\{{\cal A}_{i\to j}\}$: \[ \mathbb{P}_{i\to j/0} \, + \, \mathbb{P}_{j\to i/0}= ( \{{\cal V}_{j/0} \} - \{{\cal V}_{i/0} \}) \cdot \{{\cal A}_{i\to j}\} = \{{\cal V}_{j/i} \} \cdot \{{\cal A}_{i\to j}\} \qquad\qquad (4) \] This result shows that the sum of the powers generated by the action and the reaction, which we call power of interaction between the two bodies and denote $\mathbb{P}_{i\leftrightarrow j}$, does not vanish. In fact, it is independent of Newtonian referential $0$, and is only a function of the relative motion between the two bodies. In practice, it is found by using (4): \[ \mathbb{P}_{i\leftrightarrow j} \, = \, {\bf v}_{A\in j/i}\cdot {\bf F}_{i\to j} \, + \, \boldsymbol{\omega}_{j/i}\cdot {\bf M}_{A, i\to j} \]

7. The Kinetic Energy Theorem for a System of Rigid Bodies

To derive the Kinetic Energy Theorem for a system $\Sigma$ of $N$ rigid bodies in motion relative to a Newtonian referential $0$, we apply (1) for bodies $1$, $2$, $\ldots$, $N$ and sum these $N$ equations to obtain: \[ \sum_{i=1}^N \frac{d}{dt}\mathbb{K}_{i/0} = \sum_{i=1}^N \mathbb{P}_{\bar{i} \to i /0} = \underbrace{\sum_{i=1}^N \mathbb{P}_{\bar{\Sigma}\to i/0}}_{\text{external}} + \underbrace{\sum_{i,j=1}^N \mathbb{P}_{i \to j/0}}_{\text{internal}} \] To obtain the last terms of this expression, we have split the actions on body $i$ into contributions external to the system and contributions internal to the system.

The sum of the left-hand side is the time rate of change kinetic energy of the system $\mathbb{K}_{\Sigma/0}$.

The first sum of the left hand side is the contribution to the power of all actions external to the system $\mathbb{P}_{\bar{\Sigma}\to \Sigma/0}$.

The second sum of the left hand side is the contribution to the power of all actions internal to the system: we have learned in the previous section that this power term does not vanish: it amounts to all pairwise powers of interaction $\mathbb{P}_{i \leftrightarrow j}$.

In conclusion, the Kinetic Energy Theorem applied to the system takes the following form \[ \frac{d}{dt}\mathbb{K}_{\Sigma/0} = \mathbb{P}_{\bar{\Sigma}\to \Sigma/0}+ \sum_{1\leq i<j \leq N} \mathbb{P}_{i \leftrightarrow j} \]

Reference: Advanced Engineering Dynamics, R. Valéry Roy, Hyperbolic Press (2015).

Screws (Part 6): Newton-Euler Formalism

Sat, 18 Jan 2020 00:00:00 +0000

In construction,….

1. Introduction

2.

3.

4.

Reference: Advanced Engineering Dynamics, R. Valéry Roy, Hyperbolic Press (2015).

Screws (Part 5): The Action Screw

Tue, 14 Jan 2020 00:00:00 +0000

1. Introduction

The notion of screws is most suitable to model mechanical actions. It is a well-known principle of statics that equilibrium of a rigid body is achieved by guaranteeing that the sums of both forces and moments applied to the body vanish. The action of a force ${\bf F}_A =F_A \boldsymbol{\hat{u}}$ whose line of action passes through a point $A$ is adequately modeled as a slider of axis $(A,\boldsymbol{\hat{u}})$: this defines a screw, called action screw, which takes the form \[ \begin{Bmatrix} {\bf F}_A \\ {\bf 0} \end{Bmatrix}_A \] It is also possible model a mechanical action on $\Sigma$ as a system of forces whose resultant is zero, yet gives rise to a non-zero moment $\boldsymbol{C}$ (a torque): this action is modeled as a couple \[ \begin{Bmatrix} {\bf 0} \\ \boldsymbol{C} \end{Bmatrix} \] More generally, a mechanical action caused by a material system $\Sigma_1$ on system $\Sigma$ can always be modeled as an action screw, characterized by a resultant force and a resultant moment about a particular point. It will be denoted as \[ \{{\cal A}_{\Sigma_1 \to \Sigma}\} = \begin{Bmatrix} {\bf F}_{\Sigma_1 \to \Sigma} \\\\ {\bf M}_{A, \Sigma_1 \to \Sigma} \end{Bmatrix} \]

2. Actions At-a-Distance

Action at-a-distance do not require contact between two material systems. A typical example is given by the gravitational action caused by a celestial body. Such actions are modeled by a distribution of forces within the whole interior of a material system: they are defined in terms of a vector field $P\in \Sigma\mapsto {\bf f}_{\Sigma_1\to \Sigma} (P)$ of volumetric forces. The global action of material system $\Sigma_1$ on $\Sigma$ is then modeled by the action screw \[ \{{\cal A}_{\Sigma_1 \to \Sigma}\} = \begin{Bmatrix} \int_\Sigma{\bf f}_{\Sigma_1 \to \Sigma} (P) dV \\\\ \int_\Sigma{\bf r}_{AP}\times {\bf f}_{\Sigma_1 \to \Sigma} (P) dV \end{Bmatrix} \] For instance, the gravitational action screw $\{{\cal A}^g_{\Sigma_1 \to \Sigma}\}$ due to $\Sigma_1$ on $\Sigma$ is obtained with \[ {\bf f}_{\Sigma_1 \to \Sigma} (P) = - \int_{\Sigma_1} G \frac {\rho(P)\rho(P_1)}{|{\bf r}_{P_1 P}|^3} {\bf r}_{P_1 P} \, dV(P_1) \] where $G$ is the universal gravitational constant, $\rho (P)$ is the volumetric mass density at $P\in\Sigma$, and $\rho(P_1)$ is the volumetric mass density at $P_1\in\Sigma_1$. A variety of (approximate) expressions can be derived for the action screw under particular assumptions. If body $\Sigma_1$ is assumed of spherical shape, and with a spherical mass distribution, it is well-known that ${\bf f}_{\Sigma_1\to \Sigma}$ is given by \[ {\bf f}_{\Sigma_1\to \Sigma} (P) = -M_1 G \rho(P) \frac {{\bf r}_{O_1 P}}{|{\bf r}_{O_1 P}|^3} \] where $O_1$ is the center of $\Sigma_1$.

3. Contact Actions

If material system $\Sigma_1$ is in direct physical contact with $\Sigma$, then at all point $Q$ of the boundary $\partial\Sigma$ of $\Sigma$, a force per unit area due to $\Sigma_1$ can be written as a sum of normal and tangential component: \[ {\bf f}^c _{\Sigma_1 \to \Sigma} (Q) = N (Q) \boldsymbol{\hat{n}} (Q) + \boldsymbol{T} (Q) \] where $\boldsymbol{\hat{n}} (Q)$ is the unit outward normal vector to $\partial\Sigma$ at $Q$. The contact action screw due to $\Sigma_1$ on $\Sigma$ is represented by \[ \{{\cal A}^c_{\Sigma_1 \to \Sigma}\} = \begin{Bmatrix} \int_{\partial\Sigma} {\bf f}^c_{\Sigma_1 \to \Sigma} (Q) dA \\ \\ \int_{\partial\Sigma} {\bf r}_{AQ}\times {\bf f}^c_{\Sigma_1 \to \Sigma} (Q) dA \end{Bmatrix} \] Contact action screws pause significant problems for the solution of rigid body dynamics problems, since they are generally unknown, unless empirical constitutive laws are used. With modeling simplifications and/or empirical models, rigid body dynamics problems would remains indeterminate, that is, characterized by more unknowns than available equations. For instance, in the case of point-contact between two rigid bodies $1$ and $2$, we usually model the corresponding action as follows (denoting as $I$ the contact point) \[ \{{\cal A}^c_{1 \to 2}\} = \begin{Bmatrix} N_{1\to 2}\boldsymbol{\hat{n}}_{12} + \boldsymbol{T}_{1\to 2} \\ \\ {\bf 0} \end{Bmatrix}_I \] where $N_{1\to 2}\boldsymbol{\hat{n}}_{12}$ is the normal reaction force and $\boldsymbol{T}_{1\to 2}$ is the friction force. If the contact is dry, the two surfaces are characterized by a static friction coefficient $\mu_s$ and a kinetic friction coefficient $\mu_k$. Coulomb law of (dry) friction give empirical relationships between the normal and tangential component of the reaction force. Two cases must be considered:

Case 1: Slip at $I$. If ${\bf v}_{I\in 2/1} \neq {\bf 0}$, then the friction force is opposed to the slip velocity and its magnitude is proportional to the normal reaction: \[ \boldsymbol{T}_{1\to 2}=- \mu_k |N_{1\to 2}| \frac{{\bf v}_{I\in 2/1}}{|{\bf v}_{I\in 2/1}|} \]
Case 2: No slip at $I$ (${\bf v}_{I\in 2/1} ={\bf 0}$). Then the friction force satisfies the following bound: \[ |\boldsymbol{T}_{1\to 2}| \leq \mu_s |N_{1\to 2}| \]

4. Frictionless Joint Between Two Rigid Bodies

In most mechanical systems, rigid bodies are connected by joints. Suppose that bodies $i$ and $j$ of a system $\Sigma$ are interconnected and that the kinematics of body $j$ relative to body $i$ is characterized by the following kinematic screw \[ \{ {\cal V}_{j/i} \} = \begin{Bmatrix} \boldsymbol{\omega}_{j/i} \\ {\bf v}_{A\in j/i} \end{Bmatrix} \] Assume the action of body $i$ on body $j$ gives rise to the following contact action screw \[ \{{\cal A}^c_{i \to j}\} = \begin{Bmatrix} {\bf F}^c_{i\to j} \\ \\ {\bf M}^c_{A, i\to j} \end{Bmatrix}_I \] Then the joint between body $i$ and $j$ is said to be frictionless if and only if the following condition is satisfied: \[ {\bf v}_{A\in j/i}\cdot {\bf F}^c_{i\to j} + \boldsymbol{\omega}_{j/i}\cdot {\bf M}^c_{A, i\to j} =0 \qquad\qquad (1) \] for all arbitrary motions allowed by the joints.

In equation (1), point $A$ is arbitrary: indeed it is straightforward to show that the scalar ${\bf v}_{A\in j/i}\cdot {\bf F}^c_{i\to j}+ \boldsymbol{\omega}_{j/i}\cdot {\bf M}^c_{A, i\to j}$ is an invariant, that is, is independent of the choice of $A$.
Equation (1) expresses that the power generated by the contact action for all possible motions of body $j$ relative to body $i$ is zero at all time.
If the joint $j/i$ has $m$ degrees of freedom, equation (1) will yield $m$ scalar relationships.
Frictionless joints represent an idealized concept. However, this concept is valuable to resolve indeterminations which are inevitable in rigid body dynamics. This is achieved by addition of equations stemming from (1) to the mathematical model of the system.
Example 1: Frictionless Pivot. If the joint between body $1$ and body $2$ is a pivot of axis $(O_2,\boldsymbol{\hat{z}}_1)$, then the corresponding kinematic screw is given by \[ \{ {\cal V}_{2/1} \} = \begin{Bmatrix} \omega_{2/1}\boldsymbol{\hat{z}}_1 \\ {\bf 0} \end{Bmatrix}_{O_2} \] The pivot is frictionless if $\omega_{2/1}\boldsymbol{\hat{z}}_1 \cdot {\bf M}^c_{O_2, 1\to 2} = 0$ according to (1). Since this must be true for all values of $\omega_{2/1}$, this imposes the condition $\boldsymbol{\hat{z}}_1 \cdot {\bf M}^c_{O_2, 1\to 2} = 0$. A frictionless pivot is thus characterized by 5 components (3 components for the resultant force, and 2 components for the resultant moment about $O_2$). Note that point $O_2$ can be replaced by any point on the axis of the pivot.

Example 2: Frictionless Helical Joint. If the joint between body $1$ and body $2$ is helical of axis $(O_2,\boldsymbol{\hat{z}}_1)$ and pitch $p$, then the corresponding kinematic screw is given by \[ \{ {\cal V}_{2/1} \} = \begin{Bmatrix} \omega_{2/1}\boldsymbol{\hat{z}}_1 \\ \frac{p}{2\pi} \omega_{2/1}\boldsymbol{\hat{z}}_1 \end{Bmatrix}_{O_2} \] The helical joint is frictionless if $\frac{p}{2\pi}\omega_{2/1}\boldsymbol{\hat{z}}_1 \cdot {\bf F}_{1\to 2}^c + \omega_{2/1}\boldsymbol{\hat{z}}_1 \cdot {\bf M}^c_{O_2, 1\to 2} = 0$ according to (1). Since this must be true for all values of $\omega_{2/1}$, this imposes the condition \[ \frac{p}{2\pi} \boldsymbol{\hat{z}}_1 \cdot {\bf F}_{1\to 2}^c + \boldsymbol{\hat{z}}_1 \cdot {\bf M}^c_{O_2, 1\to 2} = 0 \]

Many other cases of joints can be treated in a similar manner.

Reference: Advanced Engineering Dynamics, R. Valéry Roy, Hyperbolic Press (2015).

Screws (Part 4): The Kinetic & Dynamic Screws of a Material System

Sun, 12 Jan 2020 00:00:00 +0000

1. The Kinetic Screw

Consider a material system $\Sigma$ (not necessarily a rigid body) of mass $m$ and mass center $G$ in motion relative to a referential ${\cal E}$. Each particle $P$ of mass $dm$ of $\Sigma$ has a linear momentum $dm{\bf v}_{P/{\cal E}}$. The angular momentum about a (arbitrary) point $A$ of particle $P$ is ${\bf r}_{AP}\times dm {\bf v}_{P/{\cal E}}$. It is straightforward to show that the vector field $A\mapsto {\bf H}_{A, \Sigma/{\cal E}} = \int_\Sigma{\bf r}_{AP}\times {\bf v}_{P/{\cal E}} \, dm$ defines a screw. This quantity, which we can simply denote as ${\bf H}_A$, is the angular momentum of system $\Sigma$ about A. It satisfies the property: ${\bf H}_B = {\bf H}_A + {\bf r}_{BA}\times m{\bf v}_{G/{\cal E}}$. The corresponding screw, called kinetic screw of system $\Sigma$ relative to referential ${\cal E}$ is denoted as follows \[ \{ {\cal H}_{\Sigma/{\cal E}} \} = \begin{Bmatrix} m {\bf v}_{G/{\cal E}} \\ {\bf H}_{A, \Sigma/{\cal E}} \end{Bmatrix} \]

In practice, the angular momentum of a rigid body ${\cal B}$ is found by determining its inertia operator about a point $B$ defined as the mapping ${\bf u}\mapsto {\cal I}_B ({\bf u}) = \int_{\cal B}{\bf r}_{BP}\times ({\bf u}\times {\bf r}_{BP}) dm$. After choosing a basis $(\boldsymbol{\hat{b}}_1,\boldsymbol{\hat{b}}_2,\boldsymbol{\hat{b}}_3)$ attached to ${\cal B}$, the inertia operator is characterized by the body’s moments $(I_{Bxx}, I_{Byy}, I_{Bzz} )$ and products of inertia $(I_{Bxy}, I_{Byz}, I_{Bxz})$ about the axes $(B,\boldsymbol{\hat{b}}_1,\boldsymbol{\hat{b}}_2,\boldsymbol{\hat{b}}_3)$. For instance, the angular momentum about mass center $G$ can be found as \[ {\bf H}_G = {\cal I}_G (\boldsymbol{\omega}) = \begin{pmatrix} I_{Bxx} & I_{Bxy} & I_{Bxz}\\ I_{Bxy} & I_{Byy} & I_{Byz}\\ I_{Bxz} & I_{Byz} & I_{Bzz} \end{pmatrix} \begin{pmatrix} \omega_1\\ \omega_2\\ \omega_3 \end{pmatrix} \] where $\boldsymbol{\omega}= \omega_1 \boldsymbol{\hat{b}}_1 +\omega_2 \boldsymbol{\hat{b}}_2 +\omega_3 \boldsymbol{\hat{b}}_3$ is the angular velocity of body ${\cal B}$ relative to ${\cal E}$. It is also possible to derive a more general expression of angular momentum ${\bf H}_A$ about an arbitrary point $A$ in terms of ${\cal I}_B$.
The kinetic screw of a system $\Sigma$ of $N$ rigid bodies ${\cal B}_1$, ${\cal B}_2$, $\ldots$, ${\cal B}_N$ is found to be given by \[ \{ {\cal H}_{\Sigma/{\cal E}} \} =\{ {\cal H}_{{\cal B}_1 /{\cal E}} \} + \{ {\cal H}_{{\cal B}_2 /{\cal E}} \} + \cdots + \{ {\cal H}_{{\cal B}_N /{\cal E}} \} \]

2. The Dynamic Screw

The dynamic moment about $A$ of system $\Sigma$ is the vector ${\bf D}_{A, \Sigma/{\cal E}} = \int_\Sigma{\bf r}_{AP}\times {\bf a}_{P/{\cal E}} \, dm$, where ${\bf a}_{P/{\cal E}}$ is the acceleration vector of point $P$. It is straightforward to show that the vector field $A\mapsto {\bf D}_{A, \Sigma/{\cal E}} = \int_\Sigma{\bf r}_{AP}\times {\bf v}_{P/{\cal E}} \, dm$ defines a screw: it satisfies the property ${\bf D}_B = {\bf D}_A + {\bf r}_{BA}\times m{\bf a}_{G/{\cal E}}$. This screw, called dynamic screw of system $\Sigma$ relative to referential ${\cal E}$ is formally the time-derivative of the kinetic screw. It is denoted as follows \[ \{ {\cal D}_{\Sigma/{\cal E}} \} = \frac{d}{dt} \{ {\cal H}_{\Sigma/{\cal E}} \} = \begin{Bmatrix} m {\bf a}_{G/{\cal E}} \\ {\bf D}_G = \frac{d}{dt} {\bf H}_G \end{Bmatrix} = \begin{Bmatrix} m {\bf a}_{G/{\cal E}} \\ {\bf D}_A = \frac{d}{dt} {\bf H}_A + {\bf v}_A \times m {\bf v}_G \end{Bmatrix} \] The dynamic screw of a system $\Sigma$ of $N$ rigid bodies ${\cal B}_1$, ${\cal B}_2$, $\ldots$, ${\cal B}_N$ is found to be given by \[ \{ {\cal D}_{\Sigma/{\cal E}} \} =\{ {\cal D}_{{\cal B}_1 /{\cal E}} \} + \{ {\cal D}_{{\cal B}_2 /{\cal E}} \} + \cdots + \{ {\cal D}_{{\cal B}_N /{\cal E}} \} \]

3. Example

Consider the disk $1$ of mass center $G$, radius $R$, uniform mass $m$, in contact with plane $(O,\boldsymbol{\hat{\imath}},\boldsymbol{\hat{\jmath}})$ of referential $0(O,\boldsymbol{\hat{\imath}},\boldsymbol{\hat{\jmath}},\boldsymbol{\hat{k}})$. Assume that the contact at $I$ is without slip, so that ${\bf v}_{I\in 1/0}= {\bf 0}$. The position of $1$ relative to $0$ is defined by the Cartesian coordinates $(x,y)$ of $I$ and the (Euler) angles $(\psi,\theta,\phi)$ as shown below.

We want to find the expression of the kinetic screw of body $1$.

We first derive the velocity of center $G$ by determining the expression of the kinematic screw of the body: \[ \{ {\cal V}_{1/0} \} = \begin{Bmatrix} \boldsymbol{\omega}_{1/0} \\ {\bf v}_{I\in 1/0} \end{Bmatrix} = \begin{Bmatrix} \dot{\psi}\boldsymbol{\hat{k}}+ \dot{\theta}\boldsymbol{\hat{u}}+ \dot{\phi}\boldsymbol{\hat{w}} \\ {\bf 0} \end{Bmatrix}_I \] which gives ${\bf v}_{G/0}= \boldsymbol{\omega}_{1/0}\times {\bf r}_{IG}= (\dot{\psi}\boldsymbol{\hat{k}}+ \dot{\theta}\boldsymbol{\hat{u}}+ \dot{\phi}\boldsymbol{\hat{w}}) \times R \boldsymbol{\hat{v}}= -R(\dot{\psi}\cos\theta +\dot{\phi}) \boldsymbol{\hat{u}}+ R\dot{\theta}\boldsymbol{\hat{w}}$. Given the expression of inertia operator ${\cal I}_G$ on basis $(\boldsymbol{\hat{u}},\boldsymbol{\hat{v}},\boldsymbol{\hat{w}})$ we obtain the angular momentum of the body about $G$: \[ {\bf H}_G = {\cal I}_G (\boldsymbol{\omega}) = mR^2 \begin{pmatrix} \frac{1}{4} & 0 & 0\\ 0 & \frac{1}{4} & 0\\ 0 & 0 & \frac{1}{2} \end{pmatrix} \begin{pmatrix} \dot{\theta}\\ \dot{\psi}\sin\theta\\ \dot{\phi}+\dot{\psi}\cos\theta \end{pmatrix} =\frac{1}{4}mR^2 \Big( \dot{\theta}\boldsymbol{\hat{u}}+ \dot{\psi}\sin\theta \boldsymbol{\hat{v}}+2 (\dot{\phi}+\dot{\psi}\cos\theta)\boldsymbol{\hat{w}}\Big) \]

We can now give the expression of the kinetic screw: \[ \{ {\cal H}_{1/0} \} = \begin{Bmatrix} -mR(\dot{\psi}\cos\theta +\dot{\phi}) \boldsymbol{\hat{u}}+ mR\dot{\theta}\boldsymbol{\hat{w}} \\ \frac{1}{4}mR^2 \Big( \dot{\theta}\boldsymbol{\hat{u}}+ \dot{\psi}\sin\theta \boldsymbol{\hat{v}}+(\dot{\phi}+\dot{\psi}\cos\theta)\boldsymbol{\hat{w}}\Big) \end{Bmatrix}_G \]

Reference: Advanced Engineering Dynamics, R. Valéry Roy, Hyperbolic Press (2015).

Screws (Part 3): Application to Mechanisms

Fri, 10 Jan 2020 00:00:00 +0000

1. Kinematic loop formula

Consider 3 rigid bodies which may or may not be physically interconnected. To simplify notations, we denote them by the numerals 1, 2, and 3. Hence, we denote $\{{\cal V}_{i/j}\}$ the kinematic screw of body $i$ relative to body $j$. Recall that it defines the vector field $P\in i\mapsto {\bf v}_{P\in i/j}$. If $\boldsymbol{\omega}_{i/j}$ is the angular velocity of body $j$ relative to body $i$, then we can write \[ \{ {\cal V}_{i/j} \} = \begin{Bmatrix} \boldsymbol{\omega}_{i/j} \\ {\bf v}_{A\in i/j} \end{Bmatrix} \] The notation ${\bf v}_{A\in i/j}$ emphasizes that point $A$ is attached to body $i$, even it is so only instantaneously.

We can show that the pairwise relationships formed between bodies 1, 2, and 3 are not independent from the following identities: \[ \boldsymbol{\omega}_{1/2}+ \boldsymbol{\omega}_{2/3}+ \boldsymbol{\omega}_{3/1} = {\bf 0}, \qquad {\bf v}_{P\in 1/2}+ {\bf v}_{P\in 2/3}+ {\bf v}_{P\in 3/1} = {\bf 0} \] These two equations can be assembled into a single formula, known as the kinematic loop formula \[ \boxed{ \{ {\cal V}_{1/2} \}+\{ {\cal V}_{2/3} \}+\{ {\cal V}_{3/1} \} = \{0\} } \]

As a special case, we obtain $\{ {\cal V}_{i/j} \} = - \{ {\cal V}_{j/i} \}$.
This formula can be generalized to $n$ bodies.
The velocity loop formula can be shown by using the change of referential formula for a particle $P$ in motion relative to 2 referentials ${\cal E}$ and ${\cal F}$: ${\bf v}_{P/{\cal E}}= {\bf v}_{P/{\cal F}} + {\bf v}_{P\in{\cal F}/{\cal E}}$.
This formula is very useful for the kinematic analysis of closed-loop mechanisms, that is, mechanims constituted of rigid bodies such that each body is connected with 2 or more bodies.

2. Example of a closed loop mechanism

In the planar mechanism shown below, 4 rigid bodies form a single closed loop:

The joint 1/0 is slider along axis $(C,\boldsymbol{\hat{\jmath}})$: $\{ {\cal V}_{1/0} \} = \begin{Bmatrix} {\bf 0}\\ \dot{y}_B \boldsymbol{\hat{\jmath}}\end{Bmatrix}$.
The joint 2/1 is a no-slip point-contact at $I$: $\{ {\cal V}_{2/1} \} = \begin{Bmatrix} \omega_{2/1}\boldsymbol{\hat{k}}\\ {\bf 0}\end{Bmatrix}_I$.
The joint 3/2 is a pivot of center $A$: $\{ {\cal V}_{3/2} \} = \begin{Bmatrix} \omega_{3/2}\boldsymbol{\hat{k}}\\ {\bf 0}\end{Bmatrix}_A$.
The joint 3/0 is a pivot of center $O$: $\{ {\cal V}_{3/0} \} = \begin{Bmatrix} \dot{\theta}_3 \boldsymbol{\hat{k}}\\ {\bf 0}\end{Bmatrix}_O$.

The loop equation $\{ {\cal V}_{1/0} \}+\{ {\cal V}_{2/1} \}+\{ {\cal V}_{3/2} \} +\{ {\cal V}_{0/3} \} = \{0\}$ gives \[ \begin{Bmatrix} {\bf 0}\\ \dot{y}_B \boldsymbol{\hat{\jmath}}\end{Bmatrix} +\begin{Bmatrix} \omega_{2/1}\boldsymbol{\hat{k}}\\ {\bf 0}\end{Bmatrix}_I +\begin{Bmatrix} \omega_{3/2}\boldsymbol{\hat{k}}\\ {\bf 0}\end{Bmatrix}_A -\begin{Bmatrix} \dot{\theta}_3 \boldsymbol{\hat{k}}\\ {\bf 0}\end{Bmatrix}_O =\begin{Bmatrix} {\bf 0}\\ {\bf 0}\end{Bmatrix} \] To properly guarantee this equality, we resolve all screws about the same point (we choose $A$): \[ \begin{Bmatrix} {\bf 0}\\ \dot{y}_B \boldsymbol{\hat{\jmath}}\end{Bmatrix}_A +\begin{Bmatrix} \omega_{2/1}\boldsymbol{\hat{k}}\\ \omega_{2/1}\boldsymbol{\hat{k}}\times {\bf r}_{IA} \end{Bmatrix}_A +\begin{Bmatrix} \omega_{3/2}\boldsymbol{\hat{k}}\\ {\bf 0}\end{Bmatrix}_A -\begin{Bmatrix} \dot{\theta}_3 \boldsymbol{\hat{k}}\\ \dot{\theta}_3\boldsymbol{\hat{k}}\times {\bf r}_{OA} \end{Bmatrix}_A =\begin{Bmatrix} {\bf 0}\\ {\bf 0}\end{Bmatrix} \] This gives the following equations: \[ \omega_{2/1}+\omega_{3/2}=\dot{\theta}_3, \qquad \dot{y}_B\boldsymbol{\hat{\jmath}}+ \omega_{2/1}\boldsymbol{\hat{k}}\times r\boldsymbol{\hat{\jmath}}=\dot{\theta}_3 \boldsymbol{\hat{k}}\times l(\cos\theta_3 \boldsymbol{\hat{\imath}}+\sin\theta_3 \boldsymbol{\hat{\jmath}}) \] These equations allow for the determination of the kinematic outputs $(\dot{\theta}_3,\omega_{2/1},\omega_{3/2})$ in terms of input $\dot{y}_B$: \[ \dot{\theta}_3 =\frac{\dot{y}_B}{l\cos\theta_3} , \qquad \omega_{2/1}= \frac{\dot{y}_B}{r}\tan\theta_3, \qquad \omega_{3/2} = \frac{\dot{y}_B}{l\cos\theta_3} (1- \frac{l}{r} \sin\theta_3) \]

Reference: Advanced Engineering Dynamics, R. Valéry Roy, Hyperbolic Press, (2015).

Screws (Part 2): Application to Kinematics

Wed, 08 Jan 2020 00:00:00 +0000

1. Definition: Kinematic Screw

Since the velocity field $P\in {\cal B} \mapsto {\bf v}_{P/{\cal E}}$ of a rigid body ${\cal B}$ satisfied ${\bf v}_{Q/{\cal E}} = {\bf v}_{P/{\cal E}} + \boldsymbol{\omega}_{{\cal B}/{\cal E}}\times {\bf r}_{PQ}$, it defines a screw, called kinematic screw of body ${\cal B}$ relative to referential ${{\cal E}}$. Recall that $\boldsymbol{\omega}_{{\cal B}/{\cal E}}$ is the angular velocity of ${\cal B}$ relative to ${{\cal E}}$, defined as the unique vector satisfying \[ \Big(\frac{d{\bf U}}{dt}\Big)_{\cal E}= \boldsymbol{\omega}_{{\cal B}/{\cal E}}\times {\bf U} \] for all vectors $\bf U$ fixed relative to ${\cal B}$.

Notation: \[ \{ {\cal V}_{{\cal B}/{\cal E}} \} = \begin{Bmatrix} \boldsymbol{\omega}_{{\cal B}/{\cal E}} \\ {\bf v}_{A/{\cal E}} \end{Bmatrix} \] Recall that:

the scalar $\boldsymbol{\omega}_{{\cal B}/{\cal E}}\cdot{\bf v}_{A/{\cal E}}$ is an invariant, that is, it is independent of the choice of point $A\in{\cal B}$.
the velocity field is equiprojective: ${\bf v}_{A/{\cal E}}\cdot{\bf r}_{AB} = {\bf v}_{B/{\cal E}}\cdot{\bf r}_{AB}$ for any two points $A$ and $B$ of ${\cal B}$.

2. Examples

A body ${\cal B}$ in translational motion relative to ${\cal E}$ is characterized by the following kinematic screw (a couple in language of screws) \[ \{ {\cal V}_{{\cal B}/{\cal E}} \} = \begin{Bmatrix} {\bf 0} \\ {\bf v}_A \end{Bmatrix} \] A body ${\cal B}$ in rotational motion relative ${\cal E}$ is characterized by the following kinematic screw (a slider in language of screws) \[ \{ {\cal V}_{{\cal B}/{\cal E}} \} = \begin{Bmatrix} \boldsymbol{\omega} \\ {\bf 0} \end{Bmatrix}_A \] The axis $\Delta (A, \boldsymbol{\omega})$ represents the axis of rotation of the body. If this representation is only valid at a specific time, the body will be said to be in instantaneous rotation.

3. Instantaneous Screw Axis:

If angular velocity $\boldsymbol{\omega}_{{\cal B}/{\cal E}}\neq {\bf 0}$ at a given instant, then the kinematic screw can be characterized by the set $\Delta$ of points of ${\cal B}$ which satisfy ${\bf v}_{Q/{\cal E}} = p \boldsymbol{\omega}_{{\cal B}/{\cal E}}$: we call this set the instantaneous screw axis of the body. The pitch $p$ is given by $p = \boldsymbol{\omega}_{{\cal B}/{\cal E}}\cdot{\bf v}_{A/{\cal E}}/\boldsymbol{\omega}_{{\cal B}/{\cal E}}^2$.

Then kinematic screw $\{ {\cal V}_{{\cal B}/{\cal E}} \}$ can be uniquely represented as the sum \[ \{ {\cal V}_{{\cal B}/{\cal E}} \} = \underbrace { \begin{Bmatrix} {\bf 0} \\ p \, \boldsymbol{\omega}_{{\cal B}/{\cal E}} \end{Bmatrix} }_{\text{translation along }\Delta} +\, \underbrace { \begin{Bmatrix} \boldsymbol{\omega}_{{\cal B}/{\cal E}} \\ {\bf 0} \end{Bmatrix}_{H\in\Delta} }_{\text{rotation about }\Delta} \qquad\qquad (1) \] This sum shows that body ${\cal B}$ is in instantaneous helical motion about $\Delta$. Axis $\Delta$ is neither fixed in ${\cal B}$ nor in ${\cal E}$. Hence, this characterization of the motion of ${\cal B}$ is only true for the velocity field (rather that the acceleration field). At any given time, the distribution of velocities of points located in a plane $\Pi$ perpendicular to $\Delta$ can be sketched in accordance to (1):

This distribution is invariant under a translation along $\Delta$ or under a rotation about $\Delta$.

Reference: Advanced Engineering Dynamics, R. Valéry Roy, Hyperbolic Press, (2015).

Screws (Part 1): A Powerful Tool for Rigid Body Mechanics

Tue, 07 Jan 2020 00:00:00 +0000

Screws form a special class of vector fields (known as fields of moments). The fields of velocity, angular momentum, dynamic moment, and moments of mechanical actions all define screws. Screws provide a simple formalism which unifies all aspects of rigid body mechanics.

1. Definition: What is a screw?

A screw is a mathematical representation of a vector field $P \in {\cal E} \to \boldsymbol{v}_P$ which satisfies the relationship

\[ \boldsymbol{v}_Q = \boldsymbol{v}_P + \boldsymbol{V} \times {\bf r}_{PQ}\]

where vector $\boldsymbol{V}$ is independent of position. Such fields play an important role in rigid body mechanics. We can take advantage of the fact that they are entirely defined by the knowledge of $\boldsymbol{V}$ and $\boldsymbol{v}_A$ (where $A$ is a particular point) to use the following notation: \[ \{ {\cal V} \} = \begin{Bmatrix} \boldsymbol{V} \\ \boldsymbol{v}_A \end{Bmatrix} \qquad\qquad\quad (1) \] Vector $\boldsymbol{V}$ is referred to as the resultant of screw $\{ {\cal V} \}$. Vector $\boldsymbol{v}_A$ is referred to as the moment of screw $\{ {\cal V} \}$ about point $A$. We can then define operations on the sets of screws (which is a vector space of dimension 6). Note that all vector fields satisfying $\boldsymbol{v}_A\cdot{\bf r}_{AB}=\boldsymbol{v}_B\cdot{\bf r}_{AB}$ (equiprojectivity) necessarily define screws (the converse is of course true).

It is possible to define operations on the set of screws (addition and multiplication by a scalar) which can be shown to be a vector space (of dimension 6).

Note that the scalar quantity $\boldsymbol{V}\cdot \boldsymbol{v}_A$ is invariant, that is, it is independent of the choice of point $A$.

2. Example:

Given a right-handed basis $(\boldsymbol{\hat{\imath}},\boldsymbol{\hat{\jmath}},\boldsymbol{\hat{k}})$ of unit vectors of $\cal E$, the screw $\{ {\cal V} \}$ defined by \[ \{ {\cal V} \} = \begin{Bmatrix} \boldsymbol{\hat{\imath}}-\boldsymbol{\hat{k}} \\ 2\boldsymbol{\hat{\jmath}} \end{Bmatrix}_O \] is the vector field $P \in {\cal E} \to \boldsymbol{v}_P = 2\boldsymbol{\hat{\jmath}}+ (\boldsymbol{\hat{\imath}}-\boldsymbol{\hat{k}})\times {\bf r}_{OP}$. Note that we must indicate the point (here point $O$) about which the screw has been resolved.

3. Two Special Classes of Screws:

Two types of screws (1) play a special role in mechanics:

Couples: A couple $\{ {\cal C} \}$ is a screw with vanishing resultant: \[ \{ {\cal C} \} = \begin{Bmatrix} \boldsymbol{0} \\ \boldsymbol{v}_O \end{Bmatrix} \] This represents the uniform field: $\boldsymbol{v}_P= \boldsymbol{v}_O$ for all points.
Sliders: A slider $\{ {\cal S} \}$ is a screw of resultant $\boldsymbol{S}\neq 0$ and with a vanishing moment about a particular point $A$: \[ \{ {\cal S} \} = \begin{Bmatrix} \boldsymbol{S} \\ \boldsymbol{0} \end{Bmatrix}_A \] All points of line $(A,\boldsymbol{S})$ have vanishing moment. This line is referred to as the axis of slider $\{ {\cal S} \}$.

4. Sum of Sliders:

It is possible to show that the sum of two sliders is a slider if and only if their axes intersect or are parallel.

5. General Representation of Screws

It can be shown that any screw $\{ {\cal V} \}$ of non-zero resultant can be decomposed in a unique way as the sum of a slider $\{ {\cal S} \}$ and a couple $\{ {\cal C} \}$. The geometric object which defines $\{ {\cal S} \}$ and $\{ {\cal C} \}$ is the set of points $Q$ whose moments are colinear to the resultant $\boldsymbol{V}$ of screw $\{ {\cal V} \}$: this set is in fact a straight line $\Delta$ called screw axis whose direction is defined by vector $\boldsymbol{V}$. All points $Q$ of $\Delta$ satisfy $\boldsymbol{v}_Q = p \, \boldsymbol{V}$ where $p$ (called pitch of the screw) is the scalar given by \[ p = \frac{\boldsymbol{V}\cdot \boldsymbol{v}_O}{\boldsymbol{V}\cdot\boldsymbol{V}} \] where point $O$ is arbitrary (the scalar $\boldsymbol{V}\cdot \boldsymbol{v}_O$ is an invariant). Then screw $\{ {\cal V} \}$ can uniquely expressed as the sum \[ \{ {\cal V} \} = \underbrace{ \begin{Bmatrix} \boldsymbol{0} \\ p \, \boldsymbol{V} \end{Bmatrix}} _{\{ {\cal C} \}} \,+\, \underbrace{ \begin{Bmatrix} \boldsymbol{V} \\ \boldsymbol{0} \end{Bmatrix}_{H\in\Delta} }_{\{ {\cal S} \}} \qquad\qquad(2) \]

Note that, when $p=0$, the screw is necessarily a slider and the screw axis is then the set of points $Q$ satisfying $\boldsymbol{v}_Q ={\bf 0}$.

Equation (2) yields a simple geometric representation of the values of the field $P \in {\cal E} \to \boldsymbol{v}_P$ in relation to its screw axis $\Delta$ as shown in this sketch:

There are many other mathematical properties satisfied by screws. In the next post, we will see the role played by screws in kinematics.

Reference: Advanced Engineering Dynamics, R. Valéry Roy, Hyperbolic Press, (2015).

Freefall Without Secured Rope

Sat, 04 Jan 2020 00:00:00 +0000

A spectacular video entitled ``Freefall Without Secured Rope'' (Fritt fall uten sikret tau) is well worth watching:

In a carefully planned stunt to test the laws of physics, Norwegian physicist and television host Andreas Wahl is seen hanging 15 meters above the ground, attached by his harness to a rope which is allowed to slide over a horizontal, cantilevered pole. The other strand of the rope is attached horizontally to a secured hook, while supporting a counterweight of only a few kilograms. Upon release of the rope from its attachment point, Wahl is seen to drop while the counterweight winds around the horizontal pole. After a few turns, the rope quickly tightens around the pole, decelerating the drop and leading Wahl to a complete stop, safely above the ground. Watch the video to get a sense of the experiment and to appreciate the risks involved (no safety net can be seen).

We can identify at least three principles of mechanics at play in this “experiment”:

conservation of angular momentum: as the rope winds around the horizontal mast, the small counterweight is seen to accelerate rotationally. The effect of the centrifugal inertial force becomes increasingly dominant over that of gravity.
the friction force exerted on the rope around the support: the rope quickly stops slipping after a few turns thanks to exponentially increasing friction (Euler-Eytelwein equation).
the elasticity of the rope: the stuntman is seen bouncing at the end of the drop, which is a good thing. Without an elastic rope, he would probably not survive the sudden stop.

This would be an excellent problem to model mathematically. As with most problems in mechanics, a solution will require a numerical model.

An example preprint / working paper

Sun, 07 Apr 2019 00:00:00 +0000

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Tue, 01 Sep 2015 00:00:00 +0000

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An example conference paper

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Lecture 10

Mon, 01 Jan 0001 00:00:00 +0000

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