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Post by Br. Marius on Apr 10, 2015 17:48:14 GMT
For an articulated rotor in hovering flight, obtain the blade flapping equation under oscillatory pitch condition. For unsteady aerodynamic forces, use thin airfoil theory results in conjunction with a lift deficiency function of a typical section at 75% radius position. Assume a 4% hinge offset and the elastic axis located at 40% chord position.
Assume: \(\gamma=5\), \(\sigma=0.05\), \(R/c=20\), \(a=5.7\), \(\theta(\psi)=1^osin(4\psi)\)
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Post by Br. Marius on Apr 13, 2015 13:40:24 GMT
Going through the notes and trying to derive some of the expressions for myself, I don't see how you get from, say, the noncirculatory moment expression on pg 247 of the notes to the noncirculatory part of the moment expressed in example 2 on pg 249. I see how, in hover, you can express \(u_p\) at a given point (I think it is the aerodynamic center since that is what the aerodynamic moments are around) as a combination of heaving motion and pitching motion, but I can't see how to make the substitutions to get to the final expressions. Any thoughts?
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Post by matthorr on Apr 13, 2015 17:47:33 GMT
This is as far as I've got:
Given
\begin{align*}
\gamma &= 8 \\
\sigma &= 0.05 \\
\sfrac{R}{C} &= 20 \\
a &= 5.7 \\
\theta(\psi) &= 1\degree\sin4\psi \\
e &= 0.04R \\
x_{ea} &= 0.4c
\end{align*}
The reduced frequency from Theodorsen's Lift Deficiency Theorem is
\begin{equation}
k = \frac{\omega c}{2U}
\end{equation}
given \(\sfrac{R}{C} = 20\), \(U = 0.75\Omega R\) at 3/4 radius and \(\omega = p\Omega\) where \(p=4\) for a 4-per pitch input, \(k = \sfrac{2}{15}\), giving
\begin{equation}
|C(k)| = 0.85
\end{equation}
The flap equation with pitch coupling terms is
\begin{equation}
\SStar{\beta}+\nu_\beta^2\beta-\Star{I_x}\left(\SStar{\theta}+\theta\right) = \gamma \overline{M}_\beta
\end{equation}
For uniform blades
\begin{equation}
\Star{I_x} = \frac{3}{2}\frac{x_I}{R}
\end{equation}
where \(x_I\) is the distance of the blade cg axis behind the feather axis.
\begin{equation}
\overline{M}_\beta = \frac{1}{2}\int_e^1 x\frac{\delta F_z}{\left(\Omega R\right)^2} |C(k)|\dd x + \int_e^1 xL_{NC}\dd x
\end{equation}
\begin{equation}
\delta F_z = \frac{1}{2}\rho ca\left[\delta U_T\left(2U_T\theta-U_P\right)+\delta U_P\left(-U_T\right) + \delta\theta\left(U_T^2\right)\right]
\end{equation}
The velocity components in hover are given as
\begin{align}
\label{}
U_T &= \Omega r \\
U_P &= \lambda\Omega R + r\dot{\beta} - \left(\frac{c}{4}-a_h\frac{c}{2}\right)\dot{\theta} \nonumber \\
&= \lambda\Omega R + r\dot{\beta} - \left(\frac{c}{5}\right)\dot{\theta}
\end{align}
with \(a_h = 0.1\).
\begin{align}
\frac{U_T}{\Omega R} &= x \\
\frac{U_P}{\Omega R} &= \lambda \\
\frac{\delta U_T}{\Omega R} &= 0 \\
\frac{\delta U_P}{\Omega R} &= x\Star{\beta} - \frac{1}{5}\frac{c}{R}\Star{\theta} \\
\delta \theta &= 4\degree\Omega\cos4\psi
\end{align}
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Post by Br. Marius on Apr 13, 2015 18:18:55 GMT
I did the same for the perturbed aerodynamic moment and carried it through as follows. And no, I haven't figured out how to get a star instead of a dot (for shorthand derivatives)...I don't think that Mathjax (the site that is providing the LaTeX support for this forum) has the syntax for it. I've heard you can make LaTeX accept new commands, but I don't know how to do that. Do you? If so, send me the code. I have a bit of code that calls Mathjax (whenever the website sees the LaTeX delimiters) placed in the global header of the forum, so it would make sense that I would just have to put it in there or something.
Anyhow, that's about where I was too. I carried what you did a little further, nondimensionalizing the force and substituting in those nondimensionalized velocities.
\(\frac{\delta M_{aero}}{R^2(\Omega R)^2}=\LARGE\int_e^1\normalsize\frac{1}{2}\rho ca[-r^2\Star\beta-\frac{rc}{4R}\Star\theta+r^2\delta\theta][C(k)](r-e)dr\)
\(\frac{\delta M_{aero}}{R^2(\Omega R)^2}=\frac{\rho ac[C(k)](e-1)^2}{48R}[2c\Star\theta-6\theta R+6R\Star\beta+ce\Star\theta-4Re\theta+4Re\Star\beta-2Re^2\theta+2Re^2\Star\beta]\)
This simplifies down to:
\(\bar{\delta M_{aero}}=0.016[0.007cos(4\psi)-0.108sin(4\psi)+6.163\Star\beta\)
In case it is as not noticeable as I think, there is a bar over the \(\delta M_{aero}\) to show that it's nondimensional.
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Post by Br. Marius on Apr 13, 2015 18:35:49 GMT
This attachment may be easier and more clear than LaTex right now. It contains all my work for problem 4, including the noncirculatory moment solution. I'm not happy with that solution it runs into the same issue I brought up in problem 3 (units don't work)...but it's what I got right now. Attachments:HW8P4.pdf (312.22 KB)
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Post by matthorr on Apr 14, 2015 1:33:06 GMT
Here are a few things I put in the header file to my LaTeX files:
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{xfrac}
\usepackage{xcolor}
\usepackage{gensymb}
\usepackage{float}
\usepackage{hhline}
\usepackage{dcolumn}
\usepackage{bm}
\newcommand{\dd}{\; \mathrm{d}}
\newcommand{\sstar}{\;\star\star}
\newcommand{\Star}[1]{\stackrel{\star}{#1}}
\newcommand{\SStar}[1]{\stackrel{\sstar}{#1}}
\newcommand{\ihat}{\boldsymbol{\;\hat{\imath}}}
\newcommand{\jhat}{\boldsymbol{\;\hat{\jmath}}}
\newcommand{\khat}{\boldsymbol{\;\hat{k}}}
You would need to have these in the header file for the website for them to work here. I forget now what packages make what command work.
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Post by Br. Marius on Apr 14, 2015 13:13:16 GMT
Got it--thanks! I'm looking into it...Mathjax supports the AMSmath and AMSsymbols extensions by default, but it is proving difficult to find a list of javascript files that can enable other extensions. I put the other packages into the header, but I don't seem to be calling them correctly as there are still errors on this page. On the other hand, I'm not sure how to insert the \newcommand declarations into the HTML header...so that may be the issue too. You can declare it locally and it'll work...but that's all I got right now. Below, I call the command before and after I declare it in 3 separate LaTeX math areas as a test to see how Mathjax processes.
\(\Star\beta\)
\(\newcommand{\SStar}[1]{\stackrel{\star\star}{#1}} \newcommand{\Star}[1]{\stackrel{\star}{#1}} \Star\beta+\SStar\beta\)
\(\newcommand{\SStar}[1]{\stackrel{\star\star}{#1}} \newcommand{\Star}[1]{\stackrel{\star}{#1}} \Star\beta+\SStar\beta\)
\(\Star\beta\)
It would appear that once you declare the new command once it should carry through the rest of your post.
Just saw this now: are you trying to show a double-dot derivative with \dd? If so, use, \ddot instead. Ex: \(\ddot\beta+\dddot a\)
Here is an implementation of the rest of the declarations you wrote--do they work as intended?
\(\newcommand{\dd}{\; \mathrm{d}}
\newcommand{\sstar}{\;\star\star}
\newcommand{\ihat}{\boldsymbol{\;\hat{\imath}}}
\newcommand{\jhat}{\boldsymbol{\;\hat{\jmath}}}
\newcommand{\khat}{\boldsymbol{\;\hat{k}}}\)
\(\ihat\beta,\jhat\beta,\khat\beta,\dd\beta,\sstar\beta\)
\(\ihat\beta,\jhat\beta,\khat\beta,\dd\beta,\sstar\beta\)
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Post by Br. Marius on Apr 14, 2015 13:54:06 GMT
\(\Star\beta\) -- and it looks like once one person declares the command, everyone else who posts after them is good to go as well. I'll just try to declare the commands in the initial posting for the time being.
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Post by matthorr on Apr 14, 2015 22:49:02 GMT
I use \dd for the derivative at the end of an integral - it uses the Roman typeface so the d doesn't look like a variable. I used the \sstar with \stackrel before I figured out how to pass variables to \newcommand and replaced it with \SStar.
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