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scaling.tex
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scaling.tex
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\documentclass{webquiz}
\usepackage[dvipdfmx]{graphicx}
\DeclareGraphicsExtensions{.png}
\title{Quiz 1: Estimation models - Scaling laws}
\UnitCode{MATH1001}
\UnitName{Differential Calculus}
\UnitURL{/u/UG/JM/MATH1001/}
\QuizzesURL{/u/UG/JM/MATH1001/Quizzes/}
\begin{document}
\begin{discussion}[Recall on scaling laws]\\
\newline
Assumptions :
\begin{minipage}[t]{.8\textwidth}
\begin{itemize}
\item Geometrical similarity
\item Material similarity
\item One dominant phenomena
\end{itemize}
\end{minipage}
Mathematical form: $y=kL^a $ \\
with k function of reference and a of physical effect
Notation: $y^*=L^*a$
Obtention ways:
\begin{minipage}[t]{.8\textwidth}
\begin{itemize}
\item direct manipulation of equations
\item dimensional analysis and Buckingham theorem
\item One dominant phenomena
\end{itemize}
\end{minipage}
Components: One main design driver express by a constant stress X*=1
\begin{center}
\includegraphics[height=70mm]{Picture1.png}
\end{center}
\end{discussion}
\begin{question}
\begin{center}
\large{Geometric similarity}
\end{center}
We assume to have similarity on all geometrical parameters : $r^* = d^* = …= l^*$ \\
\newline
Give evolutions of areas :
\begin{choice}
\incorrect $ l^*$
\incorrect $ l^{*^2}$
\incorrect $ l^{*^{-2}}$
\correct $ l^{*^{\frac{1}{2}}}$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Geometric similarity}
\end{center}
We assume to have similarity on all geometrical parameters : $r^* = d^* = …= l^*$ \\
\newline
Give evolutions of volumes :
\begin{choice}
\incorrect $ l^*$
\incorrect $ l^{*^2}$
\incorrect $ l^{*^3}$
\incorrect $ l^{*^{-3}}$
\correct $ l^{*^{\frac{1}{3}}}$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Geometric similarity}
\end{center}
We assume to have similarity on all geometrical parameters : $r^* = d^* = …= l^*$ \\
\newline
Give evolutions of masses :
\begin{choice}
\incorrect $ l^*$
\incorrect $ l^{*^2}$
\incorrect $ l^{*^3}$
\incorrect $ l^{*^{-3}}$
\correct $ l^{*^{\frac{1}{3}}}$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Geometric similarity}
\end{center}
We assume to have similarity on all geometrical parameters : $r^* = d^* = …= l^*$ \\
\newline
Give evolutions of intertias :
\begin{choice}
\incorrect $ l^{*^2}$
\incorrect $ l^{*^3}$
\incorrect $ l^{*^4}$
\incorrect $ l^{*^{5}}$
\correct $ l^{*^6}$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Main design drivers of components}
\end{center}
Mechanical stress $\sigma$ have a main influence on design of : \\
\begin{choice}
\incorrect Bearings
\incorrect Hydraulic jack
\correct Brushless motor
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Main design drivers of components}
\end{center}
Temperature $\theta$ and losses have a main influence on design of : \\
\begin{choice}
\incorrect Bearings
\correct Hydraulic jack
\incorrect Brushless motor
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Mechanics: stress}
\end{center}
\textbf{Recall on the Buckingham theorem} : the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n - k dimensionless parameters $\pi_1, \pi_2, ... , \pi_p$ constructed from the original variables where k is the number of independent physical dimensions involved \\
Thanks Buckingham theorem, find evolution of stress $\sigma$ with geometrical dimension for a rectangular sample under a load in a three-point bending setup :
\begin{minipage}[t]{.8\textwidth}
\begin{itemize}
\item F is the load (force) at the fracture point (N)
\item L is the length of the support span
\item b is width
\item d is thickness
\end{itemize}
\end{minipage}
\begin{center}
\includegraphics[height=50mm]{Picture2.png}
\end{center}
\begin{choice}
\incorrect $\sigma.L.F^{-1} = f(b.L^{-1},d.L^{-1})$
\correct $\sigma.L^2.F^{-1} = f(b.L^{-1},d.L^{-1})$
\incorrect $\sigma = f(F^{1}.L^{-2},b.L^{-1},d.L^{-1})$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Mechanics: stress}
\end{center}
Thanks strength theorem of material equation : $\sigma = \frac{3.F.L}{2.b.d^2}$ and geometrical similarity, find scaling laws of stress $\sigma^*$ with geometrical dimension $L^*$ :
\begin{choice}
\incorrect $\sigma^* = F^*.{L^*}^{-2}$
\correct $\sigma^* = F^*$
\incorrect $\sigma^* = F^*.{L^*}^2$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Mechanics: stress}
\end{center}
If $F^*=4$ (force x4) what should be the ratio $L^*$ :
\begin{choice}
\incorrect $L^* = 4$
\correct $L^* = 2$
\incorrect $L^* = 1$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Mechanics: stress $\&$ components example}
\end{center}
Estimate the diameter d of a rod-end with a static load $C_0 = 50 kN $ :
\begin{choice}
\incorrect $d = 5 mm$
\incorrect $d = 15 mm$
\correct $d = 20 mm$
\incorrect $d = 25 mm$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Mechanics: stress $\&$ components example}
\end{center}
Estimate the mass of a reducer of low speed axe torque 1150 N.m and reduction ratio 10 :
\begin{choice}
\incorrect $m = 5 kg$
\incorrect $m = 19,7 kg$
\correct $m = 34,5 kg$
\incorrect $m = 125,5 kg$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Mechanics: stress $\&$ components example}
\end{center}
Estimate the linear mass of the screw shaft characterized by a static load of 5,4 kN :
\begin{choice}
\incorrect $m = 0,12 kg$
\correct $m = 0,38 kg$
\incorrect $m = 0,64 kg$
\incorrect $m = 1,21 kg$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Mechanics: stiffness}
\end{center}
For components with a maximal constant stress, the stress-strain relationship gives : $\sigma = E.\epsilon \Rightarrow \frac{\Delta l^*}{l^*} = 1$
Estimate the torsional backlash of a reducer of low speed axe torque 1150 N.m and reduction ratio 10.
\begin{choice}
\correct $\Delta \theta = 3 arcmin$
\feedback Car $\theta = \frac{\Delta L}{L} = 1$
\incorrect $\Delta \theta = 7,6 arcmin$
\incorrect $\Delta \theta = 1,2 arcmin$
\incorrect $\Delta \theta = 0,5 arcmin$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Mechanics: stiffness}
\end{center}
For components with a maximal constant stress, the stress-strain relationship gives : $\sigma = E.\epsilon \Rightarrow \frac{\Delta l^*}{l^*} = 1$
Estimate the torsional stiffness of a reducer of low speed axe torque 1150 N.m and reduction ratio 10.
\begin{choice}
\incorrect $K_t = 530 000 Nm/rad$
\correct $K_t = 1 325 000 Nm/rad$
\feedback Car $K = \frac{T}{\Delta \theta} = T$
\incorrect $K_t = 212 000 Nm/rad$
\incorrect $K_t = 84 000 Nm/rad$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Mechanics: resonance modes }
\end{center}
Flexural resonance frequency of plane is given by : $f_r = \alpha_r.\frac{e}{L^2}.\sqrt{\frac{E}{\rho}} $
\begin{center}
\includegraphics[height=50mm]{Picture3.png}
\end{center}
In case of geometrical similarity, resonance frequency evolution is given by :
\begin{choice}
\incorrect $f_r^* = L^*$
\incorrect $f_r^* = {L^*}^{-2}$
\incorrect $f_r^* = {L^*}^{\frac{-1}{2}}$
\correct $f_r^* = {L^*}^{-1}$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Electrotechnics: induction and heat transfer }
\end{center}
For a magnetic circuit including permanent magnets, such as those encountered in brushless motors, the magnetic field can be assumed constant : $B = B_r.\frac{1}{1+\frac{e}{L_m}} \Rightarrow B^* = 1$
\begin{center}
\includegraphics[height=50mm]{Picture4.png}
\end{center}
In a brushless motor, give the scaling law which links torque $T^*$ to current density $J^*$ and dimensions $L^*$ (geometric similarty assumption) :
\begin{choice}
\incorrect $T^* = J^*.L^*$
\incorrect $T^* = {J^*}^{2}.{L^*}^2$
\incorrect $T^* = {J^*}.{L^*}^3$
\correct $T^* = {J^*}.{L^*}^4$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Electrotechnics: induction and heat transfer }
\end{center}
If convective is the main issue for heat transfer of brushless motors, link winding temperature $\theta^*$ to current density $J^*$ and dimensions $L^*$ :
\begin{choice}
\incorrect $\theta^* = J^*.L^*$
\incorrect $\theta^* = {J^*}^{2}.{L^*}^2$
\incorrect $\theta^* = {J^*}^2.{L^*}$
\correct $\theta^* = {J^*}.{L^*}^2$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Electrotechnics: brushless motors}
\end{center}
The maximum temperature $\theta$ should be constant throughout a range of products, giving : $\theta^* = 1$
Estimate the mass of a brushless motor of 8 N.m :
\begin{choice}
\incorrect $M = 2,1 kg$
\correct $M = 6,8 kg$
\incorrect $M = 4,2 kg$
\incorrect $M = 18 kg$
\end{choice}
\end{question}
\begin{question}
\begin{center}
\large{Electrotechnics: brushless motors}
\end{center}
Estimate the inertia of a brushless motor of 8 N.m :
\begin{choice}
\incorrect $J = 8.10^{-5} kg.m^2$
\incorrect $J = 22.10^{-5} kg.m^2$
\correct $J = 57.10^{-5} kg.m^2$
\incorrect $J =79.10^{-5} kg.m^2$
\end{choice}
\end{question}
\end{document}