From beb037bb9514bf52b2209bc4c6c2d93609548643 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 3 Nov 2020 09:42:04 +0100 Subject: [PATCH] Update some typography --- index.html | Bin 80491 -> 73674 bytes index.org | 40 ++++++++++++++++++++++++++-------------- 2 files changed, 26 insertions(+), 14 deletions(-) diff --git a/index.html b/index.html index c7441110bdb02430321e9614989da1ea74c79e0d..f488f0341740f8cc997ed1da5a0b001253e4d7bb 100644 GIT binary patch delta 4481 zcmaJ_dvI3O5ucOCJ(4dWK}6okO~MxvLO$;M5o3@#$g>2Qjt=xQ;XZkp@CZR5egulu zPMtx*s$W$qQ2oQIsPvY}j3Q{K6$C}F8DFDVMzoIBIy`DyJ7jv!J@;M(JN+Y*J?HN3 z*|WQ6_qX4dmx_)bDcV`z-}Z+JQEU=qzSCNl;RUl<5uqB*g8=;k?yr=B-pWlVsH%M% zVbhgG&-US8lcr-^utnV!p*5{wQ+74o@?lfD5nH~kDXs_4q^DzDmla)5;Cy-k=SrgK zyD+nEewpXFsw#^$tzt&Yz#6L(4%Lkv?ihsGHrTERThevIRaG=R*m47kwHT&fthtefPDQbvV+LM;rRT#>S6LsdL-9XSKrPL72K zri{n3BWsT1!htCgCQ4mFv*3#q=m8A7l^|J3uX!nFo~k-c~bDGAS0|3FBzH|42n-8EE%4mi30S= zW3gvCw&ZBAM}8LT#Ceu4!y@GzC)=v-D$t_d!?9$BkKU_}$+LY`P^vADwL-5ZCTYHR zKcCS3+bdyVS20}HUc{y(o2q97hxOBlV~hNniHL_igBT8(+tKo1dNV3Yo(R#yvF8Hx zII5<z8}$tsI}tmV{q7#c~yM#uT|7Jz2PW_PDZGFh^ye zV%E4CKHI&bE0!+FdR=a>b?1s@T~l%;Eo)j!tS%RJ%)LG+pSh<1_RqSPct@UnZ=u5s zDt36H8qVG_ULZS>QMvDH>~GjLFI{o9UBStD51}h$8$;WBpl$y5 zWnt(19|}Xqk3TDq^T2Wh@<=84a#>|PNm;{G-CPOJcM=zcKFW3U63eJ|<}~t?soKUATfm_3$)3aqv?_&Uag6Izd{oXYZHi)9%yMJN|@K^aa76p zptJ4Mf~3yVhY{EO=wSp;EV@+0vEmPV_SY7(E)akH{PsK=?l`9HJG73sOtV^{ul;%) zZNcvLdjnyKU4U83@PHl>)3K6IvLy(6k}2D|0GICA7t_J+_TB;vI9f}Z_y3%%y_=OlkaYq!jDYy z8+&Bc6_Pz+0Fo^t8BB0a6(yM>%bMjIF3l8D_YBgo9H>}6{W_2)xC z+2|4La>SxA-prbH@X^baeGz-pcg(qVseu#@i6&jdd6gi$Leu}_APPF83o?7II)If6 zyAu{4o&l@dQZQ%5T9~@14oA(O28d%ch+S2?wiXv^83~$J&Os8~weoGO+YYJjCM;al zN>WH84$$GDRkiVce(gk5MXK>I`0J{JF$3M41V`^Yk~mSdNV}DM_<3g^$1nyQc(58r zaD!wDuw^@{BY14%y1O5tO_V|*mi!UjzdgMx*Fn0dV_34qcz>Du zJ2qrd(R?NxJh5d2fq@ZRqLE=Vv9RiQTMrc^SM_CC@?`~H`q@(nqM|P?LRUfm_PP{S z6-ic10}k&X_`x$ZM^)g=j{anS!viOh{nf&Z zJ(Ui|1Q`-C#?Cm_{{CYe0o1TU#xy;D_rZ?DUPmw#Pga9(4=znK1lGW|LL4DkAb#c; 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Let's express the kinetic energy $T$ and the potential energy $V$ of the mass $m$: @@ -111,7 +111,7 @@ F_{\text{ext}, x} &= F_u \cos{\theta} - F_v \sin{\theta}\\ F_{\text{ext}, y} &= F_u \sin{\theta} + F_v \cos{\theta} \end{align*} -By appling the Lagrangian equations, we obtain: +By applying the Lagrangian equations, we obtain: \begin{align} m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\ m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta} @@ -125,6 +125,11 @@ y & = d_u \sin{\theta} + d_v \cos{\theta} We obtain: \begin{align*} +\dot{x} & = \dot{d_u} \cos{\theta} - d_u\dot{\theta}\sin{\theta} - \dot{d_v}\sin{\theta} - d_v\dot{\theta} \cos{\theta} \\ +\dot{y} & = \dot{d_u} \sin{\theta} + d_u\dot{\theta}\cos{\theta} + \dot{d_v}\cos{\theta} - d_v\dot{\theta} \sin{\theta} +\end{align*} +and: +\begin{align*} \ddot{x} & = \ddot{d_u} \cos{\theta} - 2\dot{d_u}\dot{\theta}\sin{\theta} - d_u\ddot{\theta}\sin{\theta} - d_u\dot{\theta}^2 \cos{\theta} - \ddot{d_v} \sin{\theta} - 2\dot{d_v}\dot{\theta}\cos{\theta} - d_v\ddot{\theta}\cos{\theta} + d_v\dot{\theta}^2 \sin{\theta} \\ \ddot{y} & = \ddot{d_u} \sin{\theta} + 2\dot{d_u}\dot{\theta}\cos{\theta} + d_u\ddot{\theta}\cos{\theta} - d_u\dot{\theta}^2 \sin{\theta} @@ -133,21 +138,26 @@ We obtain: By injecting the previous result into the Lagrangian equation, we obtain: \begin{align*} - m \ddot{d_u} \cos{\theta} - 2m\dot{d_u}\dot{\theta}\sin{\theta} - m d_u\ddot{\theta}\sin{\theta} - m d_u\dot{\theta}^2 \cos{\theta} --m \ddot{d_v} \sin{\theta} - 2m\dot{d_v}\dot{\theta}\cos{\theta} - m d_v\ddot{\theta}\cos{\theta} + m d_v\dot{\theta}^2 \sin{\theta} + m \ddot{d_u} \cos{\theta} - 2m\dot{d_u}\dot{\theta}\sin{\theta} - m d_u\ddot{\theta}\sin{\theta} - m d_u\dot{\theta}^2 \cos{\theta} +- m \ddot{d_v} \sin{\theta} - 2m\dot{d_v}\dot{\theta}\cos{\theta} - m d_v\ddot{\theta}\cos{\theta} + m d_v\dot{\theta}^2 \sin{\theta} ++ c \dot{d_u} \cos{\theta} - c d_u\dot{\theta}\sin{\theta} - c \dot{d_v}\sin{\theta} - c d_v\dot{\theta} \cos{\theta} + k d_u \cos{\theta} - k d_v \sin{\theta} = F_u \cos{\theta} - F_v \sin{\theta} \\ m \ddot{d_u} \sin{\theta} + 2m\dot{d_u}\dot{\theta}\cos{\theta} + m d_u\ddot{\theta}\cos{\theta} - m d_u\dot{\theta}^2 \sin{\theta} + m \ddot{d_v} \cos{\theta} - 2m\dot{d_v}\dot{\theta}\sin{\theta} - m d_v\ddot{\theta}\sin{\theta} - m d_v\dot{\theta}^2 \cos{\theta} ++ c \dot{d_u} \sin{\theta} + c d_u\dot{\theta}\cos{\theta} + c \dot{d_v}\cos{\theta} - c d_v\dot{\theta} \sin{\theta} + k d_u \sin{\theta} + k d_v \cos{\theta} = F_u \sin{\theta} + F_v \cos{\theta} \end{align*} + Which is equivalent to: \begin{align*} - m \ddot{d_u} - 2m\dot{d_u}\dot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\ddot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\dot{\theta}^2 --m \ddot{d_v} \frac{\sin{\theta}}{\cos{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} + m d_v\dot{\theta}^2 \frac{\sin{\theta}}{\cos{\theta}} + m \ddot{d_u} - 2m\dot{d_u}\dot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\ddot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\dot{\theta}^2 +- m \ddot{d_v} \frac{\sin{\theta}}{\cos{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} + m d_v\dot{\theta}^2 \frac{\sin{\theta}}{\cos{\theta}} ++ c \dot{d_u} - c d_u\dot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - c \dot{d_v}\frac{\sin{\theta}}{\cos{\theta}} - c d_v\dot{\theta} + k d_u - k d_v \frac{\sin{\theta}}{\cos{\theta}} = F_u - F_v \frac{\sin{\theta}}{\cos{\theta}} \\ m \ddot{d_u} + 2m\dot{d_u}\dot{\theta}\frac{\cos{\theta}}{\sin{\theta}} + m d_u\ddot{\theta}\frac{\cos{\theta}}{\sin{\theta}} - m d_u\dot{\theta}^2 + m \ddot{d_v} \frac{\cos{\theta}}{\sin{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} - m d_v\dot{\theta}^2 \frac{\cos{\theta}}{\sin{\theta}} ++ c \dot{d_u} + c d_u\dot{\theta}\frac{\cos{\theta}}{\sin{\theta}} + c \dot{d_v}\frac{\cos{\theta}}{\sin{\theta}} - c d_v\dot{\theta} + k d_u + k d_v \frac{\cos{\theta}}{\sin{\theta}} = F_u + F_v \frac{\cos{\theta}}{\sin{\theta}} \end{align*} @@ -155,11 +165,11 @@ We can then subtract and add the previous equations to obtain the following equa #+begin_important #+NAME: eq:du_coupled \begin{equation} - m \ddot{d_u} + (k - m\dot{\theta}^2) d_u = F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} + m \ddot{d}_{u} + c \dot{d}_{u} + (k - m\dot{\theta}^2) d_u = F_u + 2 m\dot{d}_v\dot{\theta} + m d_v\ddot{\theta} + c d_{v} \dot{\theta} \end{equation} #+NAME: eq:dv_coupled \begin{equation} - m \ddot{d_v} + (k - m\dot{\theta}^2) d_v = F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta} + m \ddot{d}_{v} + c \dot{d}_{v} + (k - m\dot{\theta}^2) d_v = F_v - 2 m\dot{d}_u\dot{\theta} - m d_u\ddot{\theta} - c d_{u} \dot{\theta} \end{equation} #+end_important @@ -272,12 +282,13 @@ This is definitely negligible when using piezoelectric actuators. It may not be | Neg. Spring | 1381.7[N/m] | 0.9[N/m] | ** Limitations due to coupling +*** Equations To simplify, we consider a constant rotating speed $\dot{\theta} = \omega_0$ and thus $\ddot{\theta} = 0$. From equations [[eq:du_coupled]] and [[eq:dv_coupled]], we obtain: \begin{align*} - (m s^2 + (k - m{\omega_0}^2)) d_u &= F_u + 2 m {\omega_0} s d_v \\ - (m s^2 + (k - m{\omega_0}^2)) d_v &= F_v - 2 m {\omega_0} s d_u \\ + (m s^2 + c s + (k - m{\omega_0}^2)) d_u &= F_u + 2 m \omega_0 s d_v + c \omega_0 d_v \\ + (m s^2 + c s + (k - m{\omega_0}^2)) d_v &= F_v - 2 m \omega_0 s d_u - c \omega_0 d_v \end{align*} From second equation: @@ -303,10 +314,10 @@ The two previous equations can be written in a matrix form: #+NAME: eq:coupledplant \begin{equation} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = -\frac{1}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} +\frac{1}{(m s^2 + cs + (k - m{\omega_r}^2))^2 + (2 m {\omega_r} s + c \omega_r)^2} \begin{bmatrix} - ms^2 + (k-m{\omega_0}^2) & 2 m \omega_0 s \\ - -2 m \omega_0 s & ms^2 + (k-m{\omega_0}^2) \\ + ms^2 + cs + (k-m{\omega_r}^2) & 2 m \omega_r s + c \omega_r \\ + -2 m \omega_r s + c \omega_r & ms^2 + cs + (k-m{\omega_r}^2) \end{bmatrix} \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} @@ -411,6 +422,7 @@ From this analysis, we can determine the lowest practical stiffness that is poss | k min [N/m] | 2199 | 89 | ** Effect of rotation speed on the plant +*** Introduction :ignore: As shown in equation [[eq:coupledplant]], the plant changes with the rotation speed $\omega_0$. Then, we compute the bode plot of the direct term and coupling term for multiple rotating speed.