@@ -4,7 +4,7 @@ I would not be where I am today without countless hours of wrong turns and blind charges. -I have some people to thank for their criticisms at one time or another. +I have some people to thank for their criticisms at one time or another. Richard Kelso (2004): "[This] leaves me wondering whether you are just juggling magnets." (Turns out that, yes, I was.) Paraphrasing Nobua Tanaka (2005): "A spring made with magnets is still a spring!" Will Robertson (2007): "What have I been doing for the last four years?" acknowledgements.tex @@ -1,8 +1,8 @@ %!TEX root = thesis.tex \addtocontents{toc}{% - \protect \ifshorttoc - \protect \null + \protect \ifshorttoc + \protect \null \protect \vspace{3ex}% \protect \fi } conclusion.tex @@ -1,13 +1,13 @@ %!TEX root = thesis.tex -\setupshorttoc +\setupshorttoc -\tableofcontents -\clearpage +\tableofcontents +\clearpage -\setupparasubsecs -\setupmaintoc -\tableofcontents +\setupparasubsecs +\setupmaintoc +\tableofcontents \clearpage \listoftables contents.tex @@ -48,14 +48,14 @@ with displacement $x_1$, velocity $x_2$, stiffness $\stiffnessLinear$, damping $ and mass $\mass$. This system can be shown to be stable by defining the Lyapunov function \begin{dmath} - \lyapunov = \half \lyaGainA x_1^2 + \half \lyaGainB x_2^2 + \lyapunov = \half \lyaGainA x_1^2 + \half \lyaGainB x_2^2 \end{dmath}, and showing that its time derivative is negative semi-definite: \begin{dmath} \dot\lyapunov - = \lyaGainA x_1 \dot x_1 + \lyaGainB x_2 \dot x_2 - = -\lyaGainB(\damping/\mass) x_2^2 + - (\lyaGainA-\lyaGainB\stiffnessLinear/\mass) x_1 x_2 + = \lyaGainA x_1 \dot x_1 + \lyaGainB x_2 \dot x_2 + = -\lyaGainB(\damping/\mass) x_2^2 + + (\lyaGainA-\lyaGainB\stiffnessLinear/\mass) x_1 x_2 \end{dmath}. The first term here is always negative semi-definite, but the second term is a little more ambiguous. One of the arts in Lyapunov functions is to recognise @@ -70,21 +70,21 @@ A linear spring has equivalent state equation $\dot x = -\stiffnessLinear x$. To show stability, the Lyapunov function $L=\half x^2$ with derivative $\dot L = x \dot x$ is used here and below: \begin{dmath} - \dot L = x \dot x = -\stiffnessLinear x^2 \le 0 \forall x . + \dot L = x \dot x = -\stiffnessLinear x^2 \le 0 \forall x . \end{dmath} A quadratic spring (for example, as created in a zero stiffness magnetic system) is only stable for $x<0$: \begin{dmath} \dot x = \stiffnessQuad x^2 , \\ - \dot L = x \dot x = \stiffnessQuad x^3 \le 0 \text{~for~} x\le 0 . + \dot L = x \dot x = \stiffnessQuad x^3 \le 0 \text{~for~} x\le 0 . \end{dmath} On the other hand, a cubic spring (a zero stiffness mechanical spring) is stable: \begin{dmath} \dot x = -\stiffnessDuffing x^3 , \\ - \dot L = x \dot x = -\stiffnessDuffing x^4 \le 0 \forall x . + \dot L = x \dot x = -\stiffnessDuffing x^4 \le 0 \forall x . \end{dmath} \fixme{draw accompanying graphs} @@ -96,15 +96,15 @@ quasi-zero stiffness in all degrees of freedom. Could a virtual cubic stiffness be applied with a controller to stabilise this system while ensuring quasi-zero stiffness? The state equation is \begin{dmath} - \dot x = \stiffnessQuad x^2 -\stiffnessDuffing x^3 + \dot x = \stiffnessQuad x^2 -\stiffnessDuffing x^3 \end{dmath} with $\stiffnessQuad > 0$ and $\stiffnessDuffing$ freely able to be chosen in the control law. The derivative of the Lyapunov function is \begin{dmath} - \dot L = \stiffnessQuad x^3 -\stiffnessDuffing x^4 + \dot L = \stiffnessQuad x^3 -\stiffnessDuffing x^4 = x^2 \gp{\stiffnessQuad x -\stiffnessDuffing x^2} . \end{dmath}. -Since $x^2\ge0$, for stability +Since $x^2\ge0$, for stability \begin{dmath}[compact] \stiffnessQuad x -\stiffnessDuffing x^2 \le 0 \implies x\gp{\gamma-x} \le 0 @@ -117,7 +117,7 @@ the controller can have the effect of moving this global maxima below zero. The state equation now contains the augmenting linear term $\stiffnessLinear$ that can also be freely chosen in the control law: \begin{dmath} - \dot x = -\stiffnessLinear x + \stiffnessQuad x^2 - \stiffnessDuffing x^3. + \dot x = -\stiffnessLinear x + \stiffnessQuad x^2 - \stiffnessDuffing x^3. \end{dmath} Similarly to before (\fixme{crossref??}), for stability \begin{dmath} @@ -304,7 +304,7 @@ load-bearing, direction is stable. \begin{figure}[p] \begin{subfigure} \grf{Active2006/work/eps/active-vrepl-kz} - \caption{Vertical stiffness (stable).\figlabel{K-V-repl-Z}} + \caption{Vertical stiffness (stable).\figlabel{K-V-repl-Z}} \end{subfigure} \hfill \begin{subfigure} @@ -333,13 +333,13 @@ large design space is possible with appropriately chosen parameters. \begin{figure}[p] \begin{subfigure} \grf{Active2006/work/eps/active-vzerok-kz} - \caption{Stiffness in the vertical direction + \caption{Stiffness in the vertical direction (stable for negative displacements).\figlabel{K-V-zerok-Z}} \end{subfigure} \hfill \begin{subfigure} \grf{Active2006/work/eps/active-vzerok-ky} - \caption{Stiffnesses in the horizontal directions + \caption{Stiffnesses in the horizontal directions (stable for positive displacements).\figlabel{K-V-zerok-Y}} \end{subfigure} \caption{Stiffnesses of the zero stiffness spring (\figref{spring-vcomb}).} @@ -352,7 +352,7 @@ large design space is possible with appropriately chosen parameters. \caption{Force/displacement curves for the magnetic arrangement shown in \figref{spring-vcomb}. 20\,mm cube magnets are used with various gaps. - \figlabel{active-zerok-fxrange}} + \figlabel{active-zerok-fxrange}} \end{subfigure} \par \begin{subfigure} @@ -360,7 +360,7 @@ large design space is possible with appropriately chosen parameters. \caption{Supporting force/gap curves for various magnet sizes at zero displacement. In each case, the gap range is 1--2 magnet dimensions. Increasing force results from increasing magnet size. - \figlabel{active-zerok-fgap}} + \figlabel{active-zerok-fgap}} \end{subfigure} \caption{Dependence of the supporting force on geometry of the zero stiffness spring.} @@ -422,13 +422,13 @@ design for a linear controller might proceed from here by (Jacobian) linearisation around the operating point; linear stiffness $k = \partial F/\partial x |_{x=0}$. This linearised system in state space form is \begin{dmath}[label=linearised-dynamics] -\inlinematr{ \ddot x ; - \dot x } = \inlinematr{ 0 , 0 ; - 1 , 0 } - \inlinematr{ \dot x ; +\inlinematr{ \ddot x ; + \dot x } = \inlinematr{ 0 , 0 ; + 1 , 0 } + \inlinematr{ \dot x ; x } - + \inlinematr{ 1/m ; - 0 } u + + \inlinematr{ 1/m ; + 0 } u \end{dmath}. A standard linear controller designed around this model fails to stabilise the actual system robustly, due to the large gains that @@ -494,7 +494,7 @@ The approximate nonlinear system dynamics for the zero stiffness spring may be written in the following form: \begin{dgroup} \begin{dmath}[label=xdot] -\dot \x = \y +\dot \x = \y \end{dmath}, \begin{dmath} \dot \y = k\xv^2 + u @@ -531,22 +531,22 @@ is negative definite: The virtual state error term, $\z=\y-\yy$, is now \begin{dgroup} \begin{dmath*} - \z = \y+\cc\x + \z = \y+\cc\x \end{dmath*} , \begin{dmath*} - \dot\z = k\xv^2+u+\cc\y + \dot\z = k\xv^2+u+\cc\y \end{dmath*} . \end{dgroup} Backstepping one integrator to incorporate $\dot\z$, and hence the input $u$, a second control Lyapunov function is defined: \begin{dgroup} \begin{dmath} - V_2 \eqdef V_1 + \half\z^2 + V_2 \eqdef V_1 + \half\z^2 \end{dmath}, \begin{dmath} - \dot V_2 = \dot V_1 + \z\dot\z - = \group{\x\z + \x\yy} + \z\group{ u+k\xv^2+\cc\y } - = -\cc\x^2 + \z\group{ u+\x+k\xv^2+\cc\y } + \dot V_2 = \dot V_1 + \z\dot\z + = \group{\x\z + \x\yy} + \z\group{ u+k\xv^2+\cc\y } + = -\cc\x^2 + \z\group{ u+\x+k\xv^2+\cc\y } \end{dmath}. \end{dgroup} The simplest route to stability is taken when the nonlinearities @@ -593,13 +593,13 @@ identical closed loop dynamics (\eqref{cl-dynamics}). Referring to U_2 = -mx\group{\cc\d+1} - m\dot x\group{\cc+\d}-\Q\xv^2 \end{dmath}, \begin{dmath}[label=controller2] - U_1 = \underbrace{ + U_1 = \underbrace{ -mx\group{\cc\d+1} - m\dot x\group{\cc+\d} }_{\text{Controller dynamics}} - - + - \underbrace{ \q3\xv^3-\q2\xv^2 - \q1\xv - }_{\text{Cancelation of open loop dynamics}} + }_{\text{Cancelation of open loop dynamics}} \end{dmath}. \end{dgroup} @@ -610,11 +610,11 @@ Recall, \begin{dmath} m\ddot x = K(x-y)^2 + u \end{dmath}, -\begin{dmath} -u = \{K(x-y)^2\}_{\text{est}} + k_c x + c_c \dot x +\begin{dmath} +u = \{K(x-y)^2\}_{\text{est}} + k_c x + c_c \dot x \end{dmath}, \begin{dmath} -m\ddot x = e[x,y] + k_c x + c_c \dot x +m\ddot x = e[x,y] + k_c x + c_c \dot x \end{dmath}. \end{dgroup} So the final vibration of the support, $x$, is independent of $y$ @@ -701,13 +701,13 @@ response. \begin{figure} \begin{wide} \begin{subfigure} - \grf{Active2006/work/eps/active-sim5-disp} + \grf{Active2006/work/eps/active-sim5-disp} \caption{Vertically repelling spring.\figlabel{active-sim5-disp}} \end{subfigure} \hfill \begin{subfigure} \grf{Active2006/work/eps/active-sim2-disp} - \caption{Zero stiffness spring.\figlabel{active-sim2-disp}} + \caption{Zero stiffness spring.\figlabel{active-sim2-disp}} \end{subfigure} \end{wide} \caption{Displacement traces of the floating magnet and the base @@ -721,14 +721,14 @@ response. \begin{wide} \begin{subfigure} \grf{Active2006/work/eps/active-sim5-force} - \caption{Vertically repelling spring (control - force $U_1$ in \eqref{controller2}).\figlabel{active-sim5-force}} + \caption{Vertically repelling spring (control + force $U_1$ in \eqref{controller2}).\figlabel{active-sim5-force}} \end{subfigure} \hfill \begin{subfigure} \grf{Active2006/work/eps/active-sim2-force} - \caption{Zero stiffness spring (control force - $U_2$ in \eqref{controller1}).\figlabel{active-sim2-force}} + \caption{Zero stiffness spring (control force + $U_2$ in \eqref{controller1}).\figlabel{active-sim2-force}} \end{subfigure} \end{wide} \caption{Force traces for the two active springs with the lowest @@ -1032,9 +1032,9 @@ with stabilising functions\footnote{The notation \end{multline} \vspace{-2\baselineskip} \begin{multline} -\alpha _2=\varrho _2 \left(-\vartheta _3 x_1^2-\vartheta _2 +\alpha _2=\varrho _2 \left(-\vartheta _3 x_1^2-\vartheta _2 x_1-x_1+y-\vartheta _1-x_2 \vartheta _4-c_2z_2\nl --\left(x_1^4+x_1^2+x_2^2+1\right) \kappa _2 z_2+x_2 \delta +-\left(x_1^4+x_1^2+x_2^2+1\right) \kappa _2 z_2+x_2 \delta \left(\alpha _1,x_1\right)\right), \end{multline} \vspace{-2\baselineskip} @@ -1090,7 +1090,7 @@ function of the gaps between the fixed and floating magnets. However, this calibration cannot be performed in open loop, because the desired location of the spring is in a position of marginal stability; therefore, the controller must be active while the fixed -magnet positions are adjusted. +magnet positions are adjusted. The first simulation results demonstrate the stability and convergence properties of the controller. The system parameters used in this @@ -1170,8 +1170,8 @@ $\Gamma = \operatorname{diag}(1,0.001,0.0001,1,0.01,0.0001)$ \hfill \begin{subfigure} \grf{ICSV14/work/eps/icsv-ideal-params} - \caption{Parameter estimates, normalised with their exact values - (that is, $\vartheta_i/\theta_i$, $\beta_2/b_2$, $\rho_i/p_i$ are plotted).\figlabel{converge-params}} + \caption{Parameter estimates, normalised with their exact values + (that is, $\vartheta_i/\theta_i$, $\beta_2/b_2$, $\rho_i/p_i$ are plotted).\figlabel{converge-params}} \end{subfigure} \caption{Example control with system parameters as shown in \tabref{params}.} control.tex @@ -5,7 +5,7 @@ \label{titlepage} \bookmark[dest=titlepage]{Title page} \calccentering{\unitlength} -\begin{adjustwidth*}{\unitlength}{-\unitlength} +\begin{adjustwidth*}{\unitlength}{-\unitlength} \setlength{\parindent}{0pt} \begin{flushright} @@ -33,7 +33,7 @@ Ph.D.\ Thesis \vfill \vfill - + \begin{tabular}{@{}ll} Supervisors: & Assoc.\,Prof.\ Ben Cazzolato \\ & Assoc.\,Prof.\ Anthony Zander @@ -44,14 +44,14 @@ Supervisors: & Assoc.\,Prof.\ Ben Cazzolato \\ \thispagestyle{empty} \null \vfill -\begin{quote} +\begin{quote} \LARGE \makebox[0pt][r]{`}\textit{Table that floats on magnets}' \vfill \normalsize \raggedright - Copyright \textcopyright\ 2003--2009 Will Robertson + Copyright \textcopyright\ 2003--2009 Will Robertson and The University of Adelaide \bigskip @@ -69,12 +69,12 @@ Supervisors: & Assoc.\,Prof.\ Ben Cazzolato \\ preparation system. Page margins have been chosen as a trade-off between achieving the - optimal number of characters per line for ease of reading, and trying - to fit the typeblock onto the poorly-sized (for books), although + optimal number of characters per line for ease of reading, and trying + to fit the typeblock onto the poorly-sized (for books), although convenient, \acro{A4} stock. \note{For reference, see \textit{The Elements of Typographic - Style} by Robert Bringhurst.} - pdf\/\TeX's margin kerning is used to ensure optical straightness of these + Style} by Robert Bringhurst.} + pdf\/\TeX's margin kerning is used to ensure optical straightness of these margins. If you are reading this electronically, \PDF\ hyperlinks have front.tex @@ -180,7 +180,7 @@ The dynamic response of the isolated mass is \end{dmath}. First assume that there is no input force; taking the Laplace tranform and rearranging produced the transmissibility $\transmissibility$ of the system in the frequency domain: \begin{dmath}[compact,label=simple-isolation-freq] - \transmissibility = \frac{\laplaceMass}{\laplaceBase} = + \transmissibility = \frac{\laplaceMass}{\laplaceBase} = \frac{\ii\freq\dampingRel + \stiffnessRel} {-\massMass\freq^2 + \ii\freq\dampingRel + \stiffnessRel}. \end{dmath} @@ -190,7 +190,7 @@ A force sensor can also be used for feedback purposes, but this yields results f Absolute velocity $\velMass$ and absolute displacement $\dispMass$ of the mass can be estimated by integrating the acceleration, and for completeness we also consider the relative acceleration between the mass and the base ($\accMass-\accBase$) as a possible signal for feedback control. The generalised feedback force can then be represented by \begin{dmath} - \forceIn = + \forceIn = \gainDisp\gp{\dispMass-\dispBase} + \gainVel \gp{\velMass -\velBase} + \gainAcc \gp{\accMass -\accBase} + @@ -212,7 +212,7 @@ Substituting this force equation into \eqref{simple-isolation} yields \end{dmath}. The transfer function between base and mass displacement for this generalised feedback case is \begin{dmath}[label=tf-genfeedback] - \frac\laplaceMass\laplaceBase = + \frac\laplaceMass\laplaceBase = \frac{\gainDisp+\stiffness+s \gp{\damping+\gainVel+\gainAcc s}}{\gp{\gainAcc+\gainSkymass+\mass} s^2+\gp{\damping+\gainSkyhook+\gainVel} s+\gainDisp+\gainSkyspring+\stiffness} \end{dmath}. From \eqref{isolation-feedback} it can be seen that there is an exact equivalence between the feedback gains of three of the different signals and a corresponding physical parameter of the system. @@ -284,29 +284,29 @@ In both cases the resonance peak is lowered; the relative velocity feedback corr \psfragfig{\phdpath Simulations/Springs/fig/sdof-vel}\hfil \psfragfig{\phdpath Simulations/Springs/fig/sdof-sky} \end{wide} - \caption{Relative and absolute velocity feedback control on the + \caption{Relative and absolute velocity feedback control on the system shown in \figref{simple-isolation}.} \figlabel{vel-vs-sky} \end{figure} Velocity feedback is stable for \begin{dmath} - \damping+\gainSkyhook+\gainVel \pm + \damping+\gainSkyhook+\gainVel \pm \Real{\sqrt{\gp{\damping+\gainSkyhook+\gainVel}^2-4 \stiffness \mass}}>0 \end{dmath}, which is true for $\gainSkyhook+\gainVel>-\damping$. While increased damping is usually desired, it is possible to reduce the effective damping in the system with negative velocity feedback gain, with the effect of increasing the amplitude of the resonance peak. Active damping reduction has been performed to aid the efficiency of vibration neutralisers \cite{kidner1998}. -The absolute and relative velocity feedback results may be compared by calculating the \RMS\ transmissibilities over a frequency range of interest ($\sqrt{\Int{\transmissibility}{\freq,\freq_1,\freq_2}}$) as a function of increasing feedback gain for the two cases. +The absolute and relative velocity feedback results may be compared by calculating the \RMS\ transmissibilities over a frequency range of interest ($\sqrt{\Int{\transmissibility}{\freq,\freq_1,\freq_2}}$) as a function of increasing feedback gain for the two cases. This is shown in \figref{rms-transmissibility}, where the relative feedback \RMS\ transmissiblity has a local minimum whereas the absolute feedback case continuously decreases. It is clear in the ideal case that skyhook damping is more efficacious at reducing the total vibration of a system. (The maximum frequency in this case was chosen to be much greater than the resonance frequency; $[\freq_1,\freq_2]=[0,\SI{1000}{rad/s}]$.) \begin{figure} \psfragfig{\phdpath Simulations/Springs/fig/rms-transmissibility} - \caption{\RMS\ transmissibility versus feedback gain of relative and - absolute velocity feedback control on the system shown in + \caption{\RMS\ transmissibility versus feedback gain of relative and + absolute velocity feedback control on the system shown in \figref{simple-isolation}.} \figlabel{rms-transmissibility} \end{figure} @@ -408,7 +408,7 @@ Adaptive vibration control in six degrees of freedom for broadband noise has als \paragraph{Optimal control} \textcite{balandin1998} review the field of optimal control as applied to shock and vibration isolation problems. -\note{Their comment that the \enquote{number of papers is so great that there is little incentive to discuss them here} does not bode well for any attempts by me to even summarise their review.} +\note{Their comment that the \enquote{number of papers is so great that there is little incentive to discuss them here} does not bode well for any attempts by me to even summarise their review.} They differentiate shock and vibration isolation succintly: \begin{quote} The operating quality of shock isolators is usually described in terms of certain characteristics of the transient motion of the body being isolated, whereas the quality of vibration isolators is determined by the characteristics of steady-state forced oscillations. @@ -435,13 +435,13 @@ Leave the heuristic approaches to systems that cannot be modelled (or even simul While skyhook damping has been introduced above in \secref{skyhook-intro} as force feedback proportional to the absolute velocity of the base, others have considered the case where the influence of the ground velocity is actively reduced. In both cases, the ratio of the influence of mass velocity to base velocity is being increased. In the ideal sense, the dynamic equation of motion \begin{dmath} - \massMass\accMass + \dampingRel\gp{\velMass-\velBase} + + \massMass\accMass + \dampingRel\gp{\velMass-\velBase} + \stiffness\gp{\dispMass-\dispBase} = \forceIn \end{dmath} can be transformed into \begin{dmath} - \massMass\accMass + \dampingRel\velMass + - \stiffness(\dispMass-\dispBase) = 0 + \massMass\accMass + \dampingRel\velMass + + \stiffness(\dispMass-\dispBase) = 0 \end{dmath}. The adaptive sliding mode approach used by \textcite{zuo2004} was targetted towards vehicle suspension, in which ground vibration cannot be measured. Sliding mode techniques has been recently shown to be more effective than classical techniques for an example of vehicle suspension control \cite{dong2009}. @@ -477,11 +477,11 @@ These tend to be inertial electromagnetic actuators, also known as `proof-mass' This is shown schematically in \figref{vibration-absorb}. \begin{figure} - \asyfig{Systems/vibration-inertial} - \caption{An inertial force $\forceAbsorb$ designed to reduce the vibration + \asyfig{Systems/vibration-inertial} + \caption{An inertial force $\forceAbsorb$ designed to reduce the vibration response $\dispMass$ due to disturbance $\dispBase$. - The inertial actuator - has dynamics of its own ($\stiffnessAbsorb$, $\dampingAbsorb$) that + The inertial actuator + has dynamics of its own ($\stiffnessAbsorb$, $\dampingAbsorb$) that influence the overall vibration of the structure.} \figlabel{vibration-absorb} \end{figure} @@ -506,7 +506,7 @@ These self-sensing devices (perhaps obviously, since the \backemf/ is such a noi One method of reducing vibration on a supported mass is to attach a supplementary mass that resonanates in concert with the disturbance; this has the effect of adding an anti-resonance to the original system at the frequency of interest. These are known under various names as `tuned mass dampers', `vibration neutralisers', `dynamic vibration absorbers', and so on. - \note{No effort has been made to compile an exhaustive list.} + \note{No effort has been made to compile an exhaustive list.} The descriptions involving such terms as `damper' and `absorber' are not strictly accurate on the grounds that these devices do not act as energy dissipators; rather, they direct energy into a subsystem for which continuous disturbance is not undesirable. In this thesis, the term `\vibneut/' is used, following \textcite{kidner1998} and others. @@ -612,7 +612,7 @@ The cross-section of introductory concepts and literature shown in this section %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -This section composes a general overview of the applications of magnetic fields: +This section composes a general overview of the applications of magnetic fields: \begin{description} \item[\Secref{magnetic-apps}] Introducing the underlying mechanisms and showing (non-exhaustive) examples in the literature of interesting or novel uses of magnets and magnetic fields. @@ -645,8 +645,8 @@ Uniform & Zeeman splitting & Magnetic resonance imaging \\ & Hall effect, magnetoresistance & Sensors, read-heads \\ & Force on conductor & Dynamic Motors, actuators, loudspeakers \\ & Induced emf & Generators, microphones \\ -Nonuniform & Force on charged particles & Beam control, -radiation sources %(microwave, ultra-violet; X-ray) +Nonuniform & Force on charged particles & Beam control, +radiation sources %(microwave, ultra-violet; X-ray) \\ & Force on magnet & Bearings, couplings, Maglev \\ & Force on paramagnet & Mineral separation \\ @@ -656,7 +656,7 @@ Time varying & Varying field & Dynamic Magnetometers \\ \bottomrule \end{tabular} \end{wide} -\caption{Applications of permanent magnet materials, +\caption{Applications of permanent magnet materials, adapted from \textcite{coey2002}.} \tablabel{magnet-applications} \end{table} @@ -669,7 +669,7 @@ For example, noncontact sensing of material properties that involve variable con Medical uses: \begin{itemize} -\item +\item Brain imaging \cite{sekino2005,gjini2005,lu2008-ietm,demachi2008} \item Measurements of the health of the heart \cite{lim2009-ietm}. @@ -683,7 +683,7 @@ Six \dof/ remote localisation within the human body \cite{yang2009-ietm}. Robotics: \begin{itemize} -\item +\item I don't know if I care, but \textcite{vanwest2007} created a haptic interface for manuipulating small objects with magnetic levitation. \item Wireless motion capture device \cite{hashi2005}. @@ -908,7 +908,7 @@ Both \textcite{nijsse2001} and \textcite{mizuno2003a} (with related publications) talk about using a combination of a positive and negative stiffness springs to achieve affects like \begin{dmath*}[compact] - k_T = \frac{ k_1 k_2 }{ k_1 + k_2 } = \infty + k_T = \frac{ k_1 k_2 }{ k_1 + k_2 } = \infty \condition{when $ k_2 = -k_1 $.} \end{dmath*} @@ -932,7 +932,7 @@ Interesting results have been shown using nonlinear springs to attach the vibrat \fixme{Why would you do this?!} \textcite{mann2008} performed a preliminary investigation into the use of a nonlinear vibration mount for energy harvesting, using a magnetic suspension of repulsive magnets to create a Duffing-like oscillator. -Large damping ensured that the nonlinear regimes were only realised at large excitation amplitudes, but the idea is that highly nonlinear resonances have a much broader resonance peak through the higher branch around the jump phenomenon. +Large damping ensured that the nonlinear regimes were only realised at large excitation amplitudes, but the idea is that highly nonlinear resonances have a much broader resonance peak through the higher branch around the jump phenomenon. \fixme{reference this elsewhere} A similar idea is explored by \textcite{shahruz2008} for an energy scavenging cantilever beam that uses an arrangement of attracting magnets to shape the force characteristic of the response. @@ -1028,12 +1028,12 @@ Most obviously, the shape of the frequency response is not independent of the am Consider the stable single \dof/ system \begin{dmath} -m \ddot x + b \dot x + \stiffnessDuffing (x+s)^3 = 0, +m \ddot x + b \dot x + \stiffnessDuffing (x+s)^3 = 0, \end{dmath} where $s$ is an induced displacement disturbance. At the operating position $x=0$, the nonlinear spring stiffness is $3kx^2|_{x=0}=0$. For a disturbance $s$, the spring is perturbed and generates a reaction force of $ks^3$ on the mass. -The stiffness here is $3ks^2$; \ie, dependent on the amplitude of disturbance. +The stiffness here is $3ks^2$; \ie, dependent on the amplitude of disturbance. The ramifications of this nonlinear force on the vibratory response of the system are not exactly straightforward. \textcite{tentor2001} analysed a spring generated by repulsion magnets which behaved as a Duffing oscillator for large amplitude vibrations. @@ -1051,7 +1051,7 @@ Rather than perform a nonlinear analysis on the system above (which is known in (The transfer function is not examined because it removes nonlinear components of the original signals.) The results are compared with the system \begin{dmath} -m \ddot x + b \dot x + k_{\text{lin}}(x+s) = 0, +m \ddot x + b \dot x + k_{\text{lin}}(x+s) = 0, \end{dmath} which has a linear stiffness $k_{\text{lin}}=3kS^2$ equivalent to the stiffness of the nonlinear spring at the variance displacement. intro.tex @@ -5,7 +5,7 @@ This chapter uses the theory presented earlier to discuss simple permanent magnetic systems and how they can be used in spring-like fashions. First cover various arrangements that produce forces in one direction or - another, keeping track of where the stabilities and instabilities are in the + another, keeping track of where the stabilities and instabilities are in the translational degrees of freedom. This is followed by a discussion on rotational stability; this is more of a design problem as geometric placement of the magnets has more influence. @@ -114,9 +114,9 @@ includes the force curve of the vertical spring. The graph was produced from a finite element analysis performed by \ANSYS/, using half-inch neodymium rare-earth cube magnets. \note{Remanence $B_r=\SI{1.2}{T}$ and coercivity - $H_c\approx\SI{900}{kA\cdot m^{-1}}$} -The effect of varying the gap between the fixed and floating magnets for the -horizontal spring is also demonstrated; the further away the fixed magnets + $H_c\approx\SI{900}{kA\cdot m^{-1}}$} +The effect of varying the gap between the fixed and floating magnets for the +horizontal spring is also demonstrated; the further away the fixed magnets are, the weaker the forces are. \begin{figure} @@ -325,8 +325,8 @@ direction. \begin{figure} \centering \grf{Simulations/Single_magnets/5_magnet_spring/eps/zhvspring-move-z} - \caption{Individual and combined vertical forces on the ZHV spring for - displacement in the \z\ direction. For this case of $\x=\y=0$, all + \caption{Individual and combined vertical forces on the ZHV spring for + displacement in the \z\ direction. For this case of $\x=\y=0$, all horizontal forces are zero.} \figlabel{zhvspring-move-z} \end{figure} @@ -530,7 +530,7 @@ The augmented horizontal spring is used as a basis, as the least unstable spring presented thus far with a positive vertical stiffness. For point load support, this spring is unstable both in the $y$-direction and around the $x$-direction. For planar load support, -with a coupled set of these springs, the instability is now in $y$-direction +with a coupled set of these springs, the instability is now in $y$-direction (as before) and around the $z$-direction (in contrast). The method of eliminating one of these rotational instabilities involves adding supplementary weak magnets that apply small translational @@ -556,7 +556,7 @@ geometry; as shown below, it is possible to exploit geometry to stabilise a magnetic system in all rotational degrees of freedom. The main idea is that a magnetic system can be -stabilised in all but one translational direction, and +stabilised in all but one translational direction, and supplementary magnets can be added to the design to stabilise the rotation directions. @@ -608,7 +608,7 @@ And for the moments: M_{\text{total}} = - r F_{\text{inner}} + R F_{\text{outer}} \end{dmath}, \begin{dmath} - R\cdot F_{\text{outer}} > r\cdot F_{\text{inner}} + R\cdot F_{\text{outer}} > r\cdot F_{\text{inner}} \condition{for $M_{\text{total}} > 0$} \end{dmath}, \intertext{therefore,} @@ -625,7 +625,7 @@ that between $d$~and $D$. \begin{subfigure} \grf{Figures/Bearings/delamare-forces} \caption{ - The added magnets are spaced farther away, + The added magnets are spaced farther away, so they do not affect the translational stability. \figlabel{delamare-forces}} @@ -730,7 +730,7 @@ appropriate equations.) simple-stable-rotation-verify-moments} \caption{Moments around the \x-axis produced by rotation only.} \figlabel{simple-stable-rotation-verify-moments} - \end{subfigure} + \end{subfigure} \caption{Forces and moments \fixme{what?}} \end{figure} @@ -842,8 +842,8 @@ small. \begin{figure} \psfragfig{\phdpath Simulations/Coupling/fig/mag-coupling-equil} - \caption{Equilibrium position of a suspended magnet bearing a range of - masses varying with horizontal displacement. + \caption{Equilibrium position of a suspended magnet bearing a range of + masses varying with horizontal displacement. The displacement is one quarter of the magnet width.} \figlabel{mag-coupling-equil} \end{figure} @@ -855,8 +855,8 @@ shown in \figref{mag-couplingratio-equil}. \begin{figure} \psfragfig{\phdpath Simulations/Coupling/fig/mag-couplingratio-equil} - \caption{Ratio of vertical to horizontal displacement (`coupling ratio') of - a suspended magnet bearing a range of masses varying with horizontal + \caption{Ratio of vertical to horizontal displacement (`coupling ratio') of + a suspended magnet bearing a range of masses varying with horizontal displacement. The displacement is one quarter of the magnet width.} \figlabel{mag-couplingratio-equil} \end{figure} @@ -877,7 +877,7 @@ vibrating platform; a non-contact coil, rather than a classical shaker, say, was chosen for its zero stiffness properties; without driving current, the coil does not add stiffness to the structure (although it does add damping via velocity-induced eddy currents). Additionally, the actuator does not add mass -to the vibratory system. \fixme{discuss `zero impendance' instead of +to the vibratory system. \fixme{discuss `zero impendance' instead of zero stiffness; added `mass' via induced eddy currents, etc.} The study also investigates the possibility of building large, low-cost @@ -904,7 +904,7 @@ The study is split into three parts, as described above: \subsection{Derivation of the force equations} -In this section, the equations of the force between a single cylindrical coil and a +In this section, the equations of the force between a single cylindrical coil and a cylindrical magnet are established; this forms the mathematical basis for the three parts of the study. The theory in this section is based on the theory given by @@ -932,7 +932,7 @@ the `out-of-page' direction. \small \asyfig{Coil/integral} \caption{ - Cross section schematic of the system with coil cross section + Cross section schematic of the system with coil cross section $\surfCoil$ and magnet cross section $\surfMag$. } \figlabel{int-geometry} @@ -943,11 +943,11 @@ in cylindrical \compound{co}{ordinates} $\distra=\distra{1,2,3}$ as the displace vector for the point in space at which the magnetic field is being calculated (inside the volume of the magnet), and $\distrb=\distrb{1,2,3}$ as the displacement vector to a differential volume of the coil. The magnetic -field $\magB$ from a current source $\magJ$ is given by: +field $\magB$ from a current source $\magJ$ is given by: \cite[][\S3.3]{furlani2001} \begin{dmath} \magB\fn{\distra} = \magconst\Int{\frac{\magJ\fn{\distrb}\cross\gp{\distra - -\distrb}}{\Abs{\distra-\distrb}^3}}{\diffvolCoil,\volCoil} +\distrb}}{\Abs{\distra-\distrb}^3}}{\diffvolCoil,\volCoil} \end{dmath}, where $\permVac$ is the permeability of free space and $\cross$ is the vector cross product. @@ -983,7 +983,7 @@ number of turns of wire, $I$ is the current flowing in the coil, and $A=\lengthCoil\gp{\iradiusCoil-\oradiusCoil}$ is the cross sectional area of the electromagnet. Therefore, \begin{dmath} -\magJ\fn{\distrb{1},\distrb{2},\distrb{3}}\cross\gp{\distra-\distrb} = +\magJ\fn{\distrb{1},\distrb{2},\distrb{3}}\cross\gp{\distra-\distrb} = NI/A\cdot\inlinevect{\distra{3}-\distrb{3},0,\distra{1}-\distrb{1}} \end{dmath}, resulting in the final magnetic field equation @@ -1001,7 +1001,7 @@ resulting in the final magnetic field equation {\distrb{3},\iradiusCoil,\oradiusCoil} \end{dmath} or more applicably here (since it is the axial force that is of interest), -\begin{dmath} +\begin{dmath} \magB{z}(\distra{1},\distra{2},\distra{3}) = \magconst\cdot\frac{NI}{A} \Int{ @@ -1037,10 +1037,10 @@ cylindrical magnet, \begin{dmath} \normn = \left\{ \begin{array}{@{}c@{\quad}l} - \pm\hat{\vect{z}} & + \pm\hat{\vect{z}} & \distra{3} = \pm \half\lengthMag, - \quad \distra{1}\leq \oradiusMag \\ - \hat{\vect{r}} & + \quad \distra{1}\leq \oradiusMag \\ + \hat{\vect{r}} & -\half\lengthMag < \distra{3} < \half\lengthMag, \quad \distra{1}=\oradiusMag \end{array}\right. @@ -1050,7 +1050,7 @@ $\hat{\vect{r}}$ their dot product is zero, and \begin{dseries} \begin{math} \magM\dotprod\normn = \pm M -\end{math}, +\end{math}, \begin{math} \distra{3}=\pm \half\lengthMag \end{math} @@ -1060,18 +1060,18 @@ which leads to the final expression \eqlabel{Fds} \force = \left.\Int {M\magB(\distra{1},\distra{2},\distra{3})}{\diffsurfMag,\surfMag}\right|_{\distra{3}=\lengthMag/2} - +\left.\Int{-M\magB(\distra{1},\distra{2},\distra{3})}{\diffsurfMag,\surfMag}\right|_{\distra{3}=-\lengthMag/2} -= + +\left.\Int{-M\magB(\distra{1},\distra{2},\distra{3})}{\diffsurfMag,\surfMag}\right|_{\distra{3}=-\lengthMag/2} += \Int{M\magB(\distra{1},\distra{2},\lengthMag/2) \distra{1}} {\distra{1},\iradiusMag,\oradiusMag} {\distra{2},0,2\pi} -\Int{M\magB(\distra{1},\distra{2},-\lengthMag/2) \distra{1}} {\distra{1},\iradiusMag,\oradiusMag} - {\distra{2},0,2\pi} + {\distra{2},0,2\pi} \end{dmath}, and more applicably, \begin{dmath}[label=coilFz] -F_z = +F_z = \Int{M\magB{z}(\distra{1},\distra{2},\lengthMag/2) \distra{1}} {\distra{1},\iradiusMag,\oradiusMag} {\distra{2},0,2\pi} @@ -1100,7 +1100,7 @@ made between differently sized magnets and coils. The precision of the numerical integration needs to be selected for an appropriate compromise between accuracy and computation time. Varying Mathematica's \verb|PrecisionGoal| from 1--4 (which is equivalent to the -number of `correct' significant figures), a typical force curve \vs\ +number of `correct' significant figures), a typical force curve \vs\ displacement calculation ranges in processing time as shown in \tabref{numerical-integration-times}. Eleven integrations were performed over a range of displacements from zero to \SI{20}{mm} in steps of \SI{2}{mm}. The @@ -1129,7 +1129,7 @@ Time (s) & \num{0.25} & \num{1.6} & \num{18} & \num{187} \\ Magnet & Remanence & $\remanence $ & \SI{1.2}{T} \\ & Radius & $\oradiusMag $ & \SI{6.35}{mm} \\ & Length & $\lengthMag $ & \SI{6.35}{mm} \\ -\midrule +\midrule Coil & turns & $\turnsWire $ & \num{400} \\ & Current & $\current $ & \SI{1}{A} \\ & Inner radius & $\iradiusCoil$ & \SI{7}{mm} \\ @@ -1144,10 +1144,10 @@ figures are shown in \figref{voice-coil-integration-precision} accompanied by the error of the curves with respect to the values calculated with a precision of four significant figures. The -error was calculated with -\begin{dmath*} - 100 \times \Abs{\frac{F_4 - F_i}{F_4}} -\end{dmath*} +error was calculated with +\begin{dmath*} + 100 \times \Abs{\frac{F_4 - F_i}{F_4}} +\end{dmath*} where $F_i$ is the force calculated with a precision of $i$. Reasonable results are obtained with a precision of two significant figures (error less than 5\%), and very accurate results (less than 0.1\% error) are obtained with a precision of three. @@ -1174,7 +1174,7 @@ It is also desirable to calculate forces for asymmetric magnet positions, for reasons of either evaluating the effects of unbalanced load, or for building more complex geometries. It is the latter application that will be investigated here, with an analysis of the forces produced with a ring of a -number of smaller magnets compared to that of using a solid ring magnet. +number of smaller magnets compared to that of using a solid ring magnet. The integrals used in \eqref{Fds} to calculate the forces have variables of integration $\dee\diffsurfMag=\distra{1}\dee{\distra{1}}\dee{\distra{2}}$ and @@ -1318,7 +1318,7 @@ magnet is examined. \fixme{Tabulate coil simulation parameters/} -\paragraph{On comparisons} +\paragraph{On comparisons} The various graphs presented later are generated with different geometries of the coil(s) and magnet(s). This makes comparisons between separate results @@ -1356,8 +1356,8 @@ decreases with increasing magnet lengths. \psfragfig{\coilpath voicecoil3-vary-length-2d} \end{subfigure} \lofcaption{Forces between various coil and magnet pairs of equal but - varying length.}{ The magnet radius is $\oradiusMag=\SI{6.4}{mm}$ and the - inner coil radius is $\iradiusCoil=\SI{7}{mm}$. Note the change in + varying length.}{ The magnet radius is $\oradiusMag=\SI{6.4}{mm}$ and the + inner coil radius is $\iradiusCoil=\SI{7}{mm}$. Note the change in outer coil radius as the coil length varies in order to retain a constant resistance.} \figlabel{voicecoil3-vary-length-2d} @@ -1380,8 +1380,8 @@ increases the force due to the coil, this is only true in the second case. \psfragfig{\coilpath voicecoil3-vary-radii-2d} \end{subfigure} \lofcaption{Forces between various coil and magnet pairs of fixed length - but varying diameters.}{ The coil and magnet lengths are - $\lengthCoil=\SI{7}{mm}$ and $\lengthMag=\SI{12.7}{mm}$, + but varying diameters.}{ The coil and magnet lengths are + $\lengthCoil=\SI{7}{mm}$ and $\lengthMag=\SI{12.7}{mm}$, and the inner radius of the coil is $\iradiusCoil=\oradiusMag+\SI{1}{mm}$. The `glitch' at \SI{0.25}{mm} for the $\oradiusMag=\SI{20}{mm}$ case is due to numerical imprecision in the computation of the integral.} @@ -1405,7 +1405,7 @@ length of the magnet increases. \end{subfigure} \lofcaption{ Forces between varius coil and magnet pairs with fixed coil length.}{ -The coil length is $\lengthCoil=\SI{20}{mm}$, +The coil length is $\lengthCoil=\SI{20}{mm}$, the magnet radius is $\oradiusMag=\SI{6.4}{mm}$, and the coil inner radius is $\iradiusCoil=\SI{7}{mm}$.} \figlabel{vary-mag-length} @@ -1414,11 +1414,11 @@ the coil inner radius is $\iradiusCoil=\SI{7}{mm}$.} Conversely, \figref{vary-coil-length} shows that \emph{decreasing} the length of the coil increases the force created on a fixed-size magnet. Recall that as the coil geometry is chosen to ensure a constant -resistance and hence power consumption. This is shown in the schematics of +resistance and hence power consumption. This is shown in the schematics of \figref{vary-coil-length}: as the coil length increases, its height decreases. It is interesting to compare the positions of maximum force -between the cases shown in \figref{vary-mag-length,vary-coil-length}; +between the cases shown in \figref{vary-mag-length,vary-coil-length}; which is depicted in the schematics above each graph. It can be seen that the maximum force is obtained, approximately, when the shorter of the magnet or coil is axially centred at an edge of the longer. @@ -1432,7 +1432,7 @@ the shorter of the magnet or coil is axially centred at an edge of the longer. \end{subfigure} \lofcaption{ Forces between varius coil and magnet pairs with fixed magnet length.}{ -The magnet width is $\lengthMag=\SI{20}{mm}$, +The magnet width is $\lengthMag=\SI{20}{mm}$, the magnet radius is $\oradiusMag=\SI{6.4}{mm}$, and the coil inner radius is $\iradiusCoil=\SI{7}{mm}$.} \figlabel{vary-coil-length} @@ -1443,8 +1443,8 @@ force/displacement curves for these various systems. The shorter coil gives greater forces, but at the expense of a smaller `width' of force close to the maximum. When used as shakers, these coils are assumed to have a proportional relationship between current and force, which is only valid -around the region of maximum force for a certain displacement range. -Practically, this limits the lower frequency (which has larger displacements) +around the region of maximum force for a certain displacement range. +Practically, this limits the lower frequency (which has larger displacements) at which such a device could impart vibrations into a structure. \paragraph{Air gap} @@ -1489,7 +1489,7 @@ occur as the magnet draws near, which follows from results seen previously in \end{subfigure} \lofcaption{ Forces between a coil and a magnet with varying radial eccentricity, - $\offsetCoilMag$\periodifnocomma}{, shown in \SI{}{mm}. + $\offsetCoilMag$\periodifnocomma}{, shown in \SI{}{mm}. The inner radius of the coil is $\iradiusCoil=\SI{21}{mm}$, and the radius of the magnet is $\oradiusMag=\SI{10}{mm}$ Note that while the offset @@ -1528,7 +1528,7 @@ r = \frac{R\Sin{\pi/\COILnmag}-d/2}{\Sin{\pi/\COILnmag}+1}. This expression can be used to calculate the radius of \COILnmag\ smaller magnets that can be used to fill up the volume of a larger one. In the first case, the larger magnet is approximated with a concentric -arrangement of disc magnets, approximating ring magnetisation. +arrangement of disc magnets, approximating ring magnetisation. Such a configuration can be modelled with the same expressions derived for the simple cases in the previous section. Superposition can be used to extrapolate the force from a single small eccentric magnet to @@ -1619,7 +1619,7 @@ for the nineteen magnet case. \figlabel{coils-gaps} \end{figure} -\paragraph{Cost considerations} +\paragraph{Cost considerations} While greater forces can be obtained from using a single large magnet, the costs of magnetic material is not proportional to volume. As @@ -1640,7 +1640,7 @@ construction complexity. \begin{table} \caption[Approximate magnet prices.] -{Approximate prices for magnets of thickness \SI{6.4}{mm} that could be used +{Approximate prices for magnets of thickness \SI{6.4}{mm} that could be used to construct magnet systems as shown in \figref{coils-gaps}. Data obtained from \url{http://www.kjmagnetics.com}, 2007/\textsc{may}/08.} @@ -1677,7 +1677,7 @@ that does not have the side-effects of effectively reducing the force that is being generated. By wiring two coils in series and in opposite winding directions, with the permanent magnet in the gap between them, greater forces can be achieved than with a single coil of the same -resistance. This design is also better at dissipating heat, as the effective +resistance. This design is also better at dissipating heat, as the effective surface area is greater (or, conversely, it is harder to cool thicker coils). \begin{figure} @@ -1703,7 +1703,7 @@ respects. For a fixed dual-coil geometry, this equation allows an investigation of the force/\-displacement curves while varying the distance between the coils and the position of the permanent magnet. -Less comprehensive parameter variations are performed to analyse the dual coil +Less comprehensive parameter variations are performed to analyse the dual coil arrangement, as many of the results carry over from the single coil cases. \Figref{dual-coil} shows a dual-coil design with increasing coil gaps. The results are evident: if the coils are too far apart, they magnet-design.tex @@ -81,9 +81,9 @@ aligned microscopic orbiting electrons is given by: \begin{dmath} \vect{J}_m = \curl\magM. \end{dmath} -This is a good beginning for describing the effects of an \emph{external} -current density ($\vect{J}$) acting on the magnet. To separate the effects of -induced magnetisation and that caused spontaneously by magnetic material, a +This is a good beginning for describing the effects of an \emph{external} +current density ($\vect{J}$) acting on the magnet. To separate the effects of +induced magnetisation and that caused spontaneously by magnetic material, a new term is created: the magnetic field strength, $H$: \begin{dmath} \vect{J} = \curl\magH. @@ -112,7 +112,7 @@ fields. The \define{relative permeability} $\permrel$ is the ratio of the permeability to $\permVac$. \begin{dgroup} \begin{dmath} - \perm = \frac{B}{H} + \perm = \frac{B}{H} \end{dmath}, \begin{dmath} \permrel = \frac{\perm}{\permVac} @@ -132,7 +132,7 @@ being discontinuous. \caption{The magnetic field, $\magB$, both inside and outside a magnet.} \figlabel{BHM} \end{figure} - + This equivalence in air is essentially the reason that there is often confusion between $\magB$ and $\magH$. It can be seen that within a magnet, however, their relationship is more complex and important. The @@ -215,10 +215,10 @@ covers permanent magnet design for a wide range of uses. \begin{enumerate} \item Flux lines follow the path of least resistance. This means that they will -travel through the shortest path possible, +travel through the shortest path possible, through the material with the \emph{greatest} permeability---so they will travel more readily through -magnetic or ferrous material than air, and more readily through air +magnetic or ferrous material than air, and more readily through air (although only slightly) than diamagnetic material. \item Flux lines travelling in the same direction repel each other. This means @@ -269,19 +269,19 @@ magnetic field. \toprule & \multicolumn{6}{c}{Magnet type}\\ \cmidrule{2-6} - Property & \multicolumn{2}{c}{Ferrite} - & \multicolumn{2}{c}{Alnico} + Property & \multicolumn{2}{c}{Ferrite} + & \multicolumn{2}{c}{Alnico} & \multicolumn{2}{c}{Neodymium} \\ \midrule Max.\ temperature (°C) & \num{400} & \num{500} & \num{800} & \num{900} & \num{ 80} & \num{200} \\ Remanence (T) & \num{0.2} & \num{0.4} & \num{0.5} & \num{1.3} & \num{ 1} & \num{1.3} \\ Coercivity (\si{kA/m}) & \num{100} & \num{200} & \num{50 } & \num{160} & ~~~\num{800} & \num{900} \\ - Max.\ energy product + Max.\ energy product (\si{kJ/m^3}) & \num{6} & \num{33} & \num{10} & \num{80} & \num{200} & \num{300} \\ \bottomrule \end{tabular} \caption[Typical values for various permanent magnets.] - {Typical values for various permanent magnets. + {Typical values for various permanent magnets. Adapted from information from \url{http://www.magtech.com.hk/}.} \tablabel{magnets} \end{table} @@ -321,7 +321,7 @@ be precise, the forces exerted by two magnets upon each other remain equal so long as the sum of their angles of magnetisation remain constant.} So, for example, the two bearings shown in \figref{equalbearings}, despite having different magnetisation directions, have the same stiffness (due to the forces -being equal). +being equal). \begin{figure}[hbtp] \grf[scale = 0.7]{Figures/Bearings/equalbearings} @@ -378,8 +378,8 @@ of a fixed and volume-constant magnetisation hold very well. In the first step, the integration takes place over the region of the first magnet: \begin{dmath} -\magB_1\fn{\pos_2} = - \magconst\Int{ +\magB_1\fn{\pos_2} = + \magconst\Int{ \gp{-\Div\magM_1} \frac{\pos_2-\pos_1}{\Abs{\pos_2-\pos_1}^3}}{v_1,V_1} + \magconst\oint\limits_{S_1} @@ -392,9 +392,9 @@ In the second step, the integration of the function of the magnetic field of the first magnet takes place over the region of the second magnet: \begin{dmath} -\force = - \int\limits_{V_2} - \gp{-\Div\magM_2} +\force = + \int\limits_{V_2} + \gp{-\Div\magM_2} \magB_1\fn{\pos_2} \dee v_2 + \oint\limits_{S_2} \gp{\magM_2\dotprod\normn} @@ -404,8 +404,8 @@ magnet: Torque is similar. \begin{dmath} \torque = - \int\limits_{V_2} - \gp{-\Div\magM_2} + \int\limits_{V_2} + \gp{-\Div\magM_2} \gp{ \lever\crossprod\magB_1\fn{\pos_2} } \dee v_2 + \oint\limits_{S_2} \gp{ \magM_2\dotprod\normn } @@ -419,14 +419,14 @@ earth magnetic material. The equations above can therefore be simplified. \begin{dmath} -\magB_1\fn{\pos_2} = +\magB_1\fn{\pos_2} = \magconst\oint\limits_{S_1} \gp{ \magM_1\dotprod\normn } \frac{\pos_2-\pos_1}{\Abs{\pos_2-\pos_1}^3} \dee s_1 \end{dmath}, \begin{dmath} -\force = +\force = \oint\limits_{S_2} \gp{ \magM_2\dotprod\normn } \magB_1\fn{\pos_2} \dee s_2 @@ -443,13 +443,13 @@ simplified. \torque = \oint\limits_{S_2} \gp{ \magM_2\dotprod\normn } - \gp{ \lever\crossprod\magB_1\fn{\pos_2} } - \dee s_2 + \gp{ \lever\crossprod\magB_1\fn{\pos_2} } + \dee s_2 = \magconst \oint\limits_{S_2} \gp{ \magM_2 \dotprod \normn_{s_2} } - \gp{ + \gp{ \lever\crossprod \gp{ \oint\limits_{S_1} @@ -457,8 +457,8 @@ simplified. \frac{\pos_2-\pos_1}{\Abs{\pos_2-\pos_1}^3} \dee s_1 } - } - \dee s_2 + } + \dee s_2 \end{dmath}. \subsection{Analytical force expressions} @@ -479,16 +479,16 @@ magnetisations \cite{akoun1984,nagaraj1988,bonisoli2006}. More complex geometries can be realised through superposition of the solutions \cite{bancel1999}. \citeauthor{akoun1984}'s expression is \begin{dmath}[label=akoun] -\vect F = \frac{JJ'}{4\pi\permVac} - \sum_{(i,j,k,l,p,q)\in\{0,1\}^6} +\vect F = \frac{JJ'}{4\pi\permVac} + \sum_{(i,j,k,l,p,q)\in\{0,1\}^6} \vect f\fn{u_{ij},v_{kl},w_{pq},r} \end{dmath}, \begin{dgroup} \begin{dmath} -f_x = \half\gp{v^2-w^2}\Log{r-u}+uv\Log{r-v}+vw\ArcTan{\frac{uv}{rw}}+\half ru +f_x = \half\gp{v^2-w^2}\Log{r-u}+uv\Log{r-v}+vw\ArcTan{\frac{uv}{rw}}+\half ru \end{dmath}, \begin{dmath} -f_y = \half\gp{u^2-w^2}\Log{r-v}+uv\Log{r-u}+uw\ArcTan{\frac{uv}{rw}}+\half rv +f_y = \half\gp{u^2-w^2}\Log{r-v}+uv\Log{r-u}+uw\ArcTan{\frac{uv}{rw}}+\half rv \end{dmath}, \begin{dmath} f_z = -uw\Log{r-u}-vw\Log{r-v}+uv\ArcTan{\tfrac{uv}{rw}}-rw @@ -719,18 +719,18 @@ So, what is this equation that is so useful for me? Given two magnets located in \threeD/ space, of sizes $\inlinevect{A,B,C}$ and $\inlinevect{a,b,c}$, with the plane of the second rotated by $\theta$ around the \x-axis and their origins offset by $\inlinevect{x_0,y_0,z_0}$, the forces in the \y- and -\z-directions ($F_y$ and $F_z$) between the two magnets can be calculated. +\z-directions ($F_y$ and $F_z$) between the two magnets can be calculated. Forces in both the $y$ and $z$ direction use the $f_3$ function in their calculations. \begin{dgroup*} \begin{dmath} F_y = \frac{J^2}{4\pi\permVac} - \sum_{i_{a,b,c,A,B,C}\in\{0,1\}} + \sum_{i_{a,b,c,A,B,C}\in\{0,1\}} f_{y_2}\cdot\gp{-1}^{i_a+i_b+i_c+i_A+i_B+i_C} \end{dmath}, \begin{dmath} -F_z\bigg|_{\theta\neq k\pi} = +F_z\bigg|_{\theta\neq k\pi} = \frac{-J^2}{4\pi\permVac}\cdot\smash{\sum_{i_{a,b,c,A,B,C}\in\{0,1\}}} f_{z_2}\cdot\gp{-1}^{i_a+i_b+i_c+i_A+i_B+i_C} \end{dmath}, @@ -754,21 +754,21 @@ v_1 = -v_2\cos\theta - w_2\sin\theta w_1 = v_2\sin\theta - w_2\cos\theta \end{dmath}, \begin{dmath} -v_2 = y_0-Ci_C\sin\theta +v_2 = y_0-Ci_C\sin\theta \end{dmath}, \begin{dmath} -w_2 = z_0-ci_c+Ci_C\cos\theta +w_2 = z_0-ci_c+Ci_C\cos\theta \end{dmath}. \end{dgroup*} This is the auxiliary function used in the above. All dashed variables are local to this function. \begin{dgroup*} \begin{dmath} -f_3\fn{u',v',w',\theta',c',C'} = +f_3\fn{u',v',w',\theta',c',C'} = u' f_5\gp{\Log{f_4-u'}-1} +\half\gp{f_6^2-{u'}^2}\Log{f_4+f_5} +\half u' \pi \Sign{f_5}\Abs{f_6} - +u' f_6 \ArcTan{\frac{u' f_4 - u^2 -f_6^2}{f_5 f_6}} + +u' f_6 \ArcTan{\frac{u' f_4 - u^2 -f_6^2}{f_5 f_6}} +\half f_4 f_5 \end{dmath}, \begin{dmath} @@ -778,7 +778,7 @@ f_4 = \sqrt{{u'}^2+f_5^2+f_6^2} f_5 = \gp{v'-b i_b}\cos\theta'+\gp{w'-c'}\sin\theta'+B i_B \end{dmath}, \begin{dmath} -f_6 = -\gp{v'-b i_b}\sin\theta'+\gp{w'-c'}\cos\theta'+C' +f_6 = -\gp{v'-b i_b}\sin\theta'+\gp{w'-c'}\cos\theta'+C' \end{dmath}. \end{dgroup*} @@ -787,13 +787,13 @@ magnet is not rotated around its axis ($f_{z_2}$ has a singularity at $\theta=k\pi$). \begin{dgroup} \begin{dmath} -F_z\bigg|_{\theta=k\pi} = +F_z\bigg|_{\theta=k\pi} = \cos\theta\cdot\frac{-J^2}{4\pi\permVac} \sum_{i_{a,b,c,A,B,C}\in\{0,1\}}f_{z_1} \cdot\gp{-1}^{i_a+i_b+i_c+i_A+i_B+i_C} \end{dmath}, -\begin{dmath} -f_{z_1} = +\begin{dmath} +f_{z_1} = \half uw\Log{\frac{r+u}{r-u}}+\half vw\Log{\frac{r+v}{r-v}}+ uv\ArcTan{\frac{uv}{wr}-wr} \end{dmath}, @@ -831,7 +831,7 @@ are consistent and give reasonable results. \Figref{charp-rotate} shows the forces produced between two \SI{10}{mm} cube magnets as a function of rotation angle $\theta$ of the second magnet, with a \SI{20}{mm} offset between their centres. Two cases are shown: in the first , the magnets are displaced -vertically; in the second, the magnets are displaced horizontally. +vertically; in the second, the magnets are displaced horizontally. \begin{figure} \begin{wide} @@ -847,7 +847,7 @@ vertically; in the second, the magnets are displaced horizontally. \hfil \null \end{wide} - \caption{Vertical and horizontal forces on a rotating magnet + \caption{Vertical and horizontal forces on a rotating magnet as a function of rotation angle for fixed horizontal and vertical displacements.} \figlabel{charp-rotate} \end{figure} @@ -892,7 +892,7 @@ multipole arrays with arbitrary numbers of magnet per wavelength. \section{Multipole} -Applications: +Applications: artificial hearts \parencite{finocchiaro2008,samiappan2008} wave power generation \parencite{kimoulakis2008} wind power generation \parencite{liu2008-ietm} @@ -972,8 +972,8 @@ passengers. \begin{figure} \grf{Figures/Multipole/halbach} \caption[Magnetic flux lines of a Halbach array.]{% - The magnetic flux lines of a Halbach array made from magnets with \ang{45} - magnetisation increments (indicated by the arrowheads). The magnetic field + The magnetic flux lines of a Halbach array made from magnets with \ang{45} + magnetisation increments (indicated by the arrowheads). The magnetic field can be seen to be much stronger above the array than below it.} \figlabel{halbach} \end{figure} @@ -1086,8 +1086,8 @@ superposition. \fixme{crossref} \begin{figure} \grf{Figures/Multipole/pa-cho} \caption[Novel planar array shown in the literature.]{% - The planar array by \textcite{cho2001}. Flux travels out of the page - from the north- to south-faced magnets, and back through the \emph{array} + The planar array by \textcite{cho2001}. Flux travels out of the page + from the north- to south-faced magnets, and back through the \emph{array} in the triangular magnets.} \figlabel{pa-cho} \end{figure} @@ -1163,8 +1163,8 @@ $\forceEddy$ due to these eddy currents is \end{dmath} Now, \begin{dmath} - \magJeddy \cross \magflux - \propto + \magJeddy \cross \magflux + \propto \gp{\velocity \cross \magflux}\cross \magflux , \end{dmath} which is anti-parallel to $\velocity$ \fixme{small figure}. Due to the @@ -1172,7 +1172,7 @@ cross terms, the maximum force is obtained for magnetic fields perpendicular to the motion of the conductor. Associatively, it is only the component of magnetic field in the perpendicular direction that influences the eddy force. This has implications on the -arrangement of eddy current dampers for vibrating structures. +arrangement of eddy current dampers for vibrating structures. Two configurations of eddy current dampers are shown in \figref{eddy}. The two systems behave somewhat differently. A magnet-theory.tex @@ -10,7 +10,7 @@ a little funny, and inconclusive. I use the theory introduced in \chapref{magnet-theory} to investigate the forces between multiple arrays which are composed of magnets with rotating polarisations; this approximates a magnet with sinusoidal - polarisation. The amount of discretisation and the spatial wavelength of + polarisation. The amount of discretisation and the spatial wavelength of magnetisation of the arrays both influence the resulting forces. } @@ -86,7 +86,7 @@ the magnetisation in the facing direction: ${\hat M}_{z_2} = -\hat M_z$. \subsection{Simple arrays} To begin, the simple arrays are examined and the vertical forces between two -facing arrays solved statically for a range of displacements. +facing arrays solved statically for a range of displacements. For opposing homogeneous arrays, equivalent to a single magnet, the magnetisations do not vary: @@ -234,7 +234,7 @@ over a smaller range. \centering \grf{Simulations/Magnet_arrays/Halbach_pitch/eps/vary-magnets-3} \grf{Simulations/Magnet_arrays/Halbach_pitch/eps/vary-wavelength-3} - \caption{Old ANSYS plots. First has wavelength as twice the height. + \caption{Old ANSYS plots. First has wavelength as twice the height. Second has four magnets per wavelength.} \figlabel{ansys} \end{figure} @@ -251,11 +251,11 @@ Taking another look at Paden's paper, I realised I had the Fourier expansion of the magnetisation all wrong in my MATLAB simulations for such square wave magnetisation in multipole arrays (the results of which were funny and which hadn't been used for anything). Fixing this up (see page~112 of my workbook), -the graphs now match up with FEA pretty well. +the graphs now match up with FEA pretty well. \begin{dmath*} P_y = \frac{4B_r}{\pi^2\permVac}\sum^\infty_{n=1,3,5,\dots} \frac{1}{n^2}\exp(-k_n -g)\gp{1-\exp(-k_n d)}^2 -\end{dmath*} +g)\gp{1-\exp(-k_n d)}^2 +\end{dmath*} The results are shown in \figref{square-magnetisation-comparison}, but are not compared to a magnetic nodes solution because at this stage I'm happy enough with this graph. @@ -263,8 +263,8 @@ nodes solution because at this stage I'm happy enough with this graph. \begin{figure} \centering \grf{Simulations/Magnet_arrays/Halbach_pitch/eps/square-magnetisation-comparison} - \caption{Comparison between analytical and FEA solutions to two repulsive - eight magnet multipole arrays with half-inch cube magnets alternating + \caption{Comparison between analytical and FEA solutions to two repulsive + eight magnet multipole arrays with half-inch cube magnets alternating in magnetisation.} \figlabel{square-magnetisation-comparison} \end{figure} @@ -301,10 +301,10 @@ sinusoidally magnetised plates. Main features of the work: that is, varying sinusoidally with \z, magnetised in the same direction as the gap; \item magnetisation of the second plate is phase shifted, possibly, - with an offset: - \begin{dmath} - \magM_2 = (0, 0, \overline M\Cos{k(y-y_0)}). - \end{dmath} + with an offset: + \begin{dmath} + \magM_2 = (0, 0, \overline M\Cos{k(y-y_0)}). + \end{dmath} \end{enumerate} To get the forces between them, we start off by looking at the @@ -450,7 +450,7 @@ thick, that is two magnet widths wider than the array itself. Nominal energy product & \SI{382}{kJ/m^3} \\ Intrinsic coercivity & \SI{875}{kA/m} \\ Array height & \SI{12.7}{mm} \\ - Array width & \SI{114.3}{mm} + Array width & \SI{114.3}{mm} ($9\times\SI{12.7}{mm}$) \\ Array depth & \SI{21.3}{mm} \\ Former depth & \SI{1.6}{mm} \\ multipole.tex @@ -42,7 +42,7 @@ resonance at \SI{5}{Hz} requires a static displacement of approximately relationship that can exhibit a low \emph{dynamic} stiffness without the need for an associated large static deflection. -An example of a system with such behaviour is that of a cubic force \vs\ +An example of a system with such behaviour is that of a cubic force \vs\ displacement characteristic; for a dynamic force characteristic of, say, $\func{f}{x}=f_0+x^3$, localised zero stiffness occurs at zero deflection (\ie, $\func{f'}{0}=0$), which is termed `\qzs/'. The `quasi' qualifier is @@ -95,12 +95,12 @@ position. \begin{wide} \begin{subfigure} \asyfig{ZKS/schematic} - \caption{Inclined springs in their unloaded, uncompressed state, corresponding + \caption{Inclined springs in their unloaded, uncompressed state, corresponding to a vertical displacement of $\ZKSdisp=\ZKSheight$.\figlabel{zks-dia}} \end{subfigure} \begin{subfigure} \asyfig{ZKS/schematic-qzs} - \caption{Inclined springs at a position of maximum negative stiffness, + \caption{Inclined springs at a position of maximum negative stiffness, corresponding to a vertical displacement of $\ZKSdisp=0$.\figlabel{zks-qzs}} \end{subfigure} \end{wide} @@ -116,7 +116,7 @@ $\ZKSstiffvert=\ZKSkratio\ZKSstiffincl$. The stiffness and deflection properties of the springs are summarised in \tabref{spring-properties}. \begin{table} - \caption{Properties of the springs in the \qzs/ inclined spring system + \caption{Properties of the springs in the \qzs/ inclined spring system defining stiffness ratio $\ZKSkratio$ and length ratio $\ZKSlengthratio$.} \tablabel{spring-properties} \begin{tabular}{@{}lcc@{}} @@ -124,7 +124,7 @@ defining stiffness ratio $\ZKSkratio$ and length ratio $\ZKSlengthratio$.} Spring & Stiffness & Undeflected length \\ \cmidrule(r){1-1}\cmidrule(l){2-3} Inclined & $\ZKSstiffincl$ & $\ZKSlength = \sqrt{\ZKSheight^2+\ZKSwidth^2}$ \\ - Vertical & $\ZKSstiffvert = \ZKSkratio\ZKSstiffincl$ + Vertical & $\ZKSstiffvert = \ZKSkratio\ZKSstiffincl$ & $\ZKSvlength = \ZKSlengthratio\ZKSlength$ \\ \bottomrule \end{tabular} @@ -170,15 +170,15 @@ forces due to each spring individually. The force due to the inclined spring \end{dmath}. Assuming only vertical displacement ($\ZKSshift=0$), -the vertical component of this inclined spring force is +the vertical component of this inclined spring force is \begin{dmath}[label=ZKSforceinclV,compact] \ZKSforceinclV = \ZKSforceincl{\ZKSdisp,0} \frac{x}{\ZKSlengthcomp{1,0}} % from LaTeXPrint[Fv, x, 0]; bring stiffincl to the front: -= \ZKSdisp \ZKSstiffincl += \ZKSdisp \ZKSstiffincl \gp{ \frac {\sqrt{\ZKSheight^2+\ZKSwidth^2}} {\sqrt{\ZKSwidth^2+\ZKSdisp^2} } -1 - } + } \end{dmath}. It is convenient to normalise this result by representing the lengths and @@ -188,7 +188,7 @@ $\ZKSratio=\ZKSwidth/\ZKSheight$, the inclined spring force in the vertical direction can be written in non-dimensional form as \begin{dmath}[label=ZKSforceinclVnorm] - \frac{\ZKSforceinclV{\ZKSnorm}}{\ZKSheight \ZKSstiffincl} = + \frac{\ZKSforceinclV{\ZKSnorm}}{\ZKSheight \ZKSstiffincl} = \ZKSnorm \gp{\sqrt{\frac{\ZKSratio^2+1}{\ZKSratio^2+\ZKSnorm^2}}-1} \end{dmath}. @@ -222,8 +222,8 @@ with \eqref{ZKSforcetotalVnorm}. Minimum displacements are calculated from spring being compressed to zero length.\figlabel{ZKSforcetotalVnorm}} \end{subfigure} \end{wide} -\caption{Vertical forces due to the inclined springs. $\ZKSratioQZS$ is the -value of $\ZKSratio$ for which \qzs/ is achieved at $\ZKSnorm$, calculated +\caption{Vertical forces due to the inclined springs. $\ZKSratioQZS$ is the +value of $\ZKSratio$ for which \qzs/ is achieved at $\ZKSnorm$, calculated from \eqref{ZKSratioQZS}.} \figlabel{ZKSforce-both} \end{figure} @@ -233,14 +233,14 @@ calculated from combining \eqref{ZKSforceinclV} for each inclined spring with the force due to the vertical spring: \begin{dmath}[label=ZKSforcetotalV] -\ZKSforcetotalV +\ZKSforcetotalV = 2\ZKSforceinclV + \ZKSforcevertV \end{dmath}. For vertical displacements, the force due to the vertical spring is given by \begin{dmath}[label=ZKSforcevertV] -\ZKSforcevertV +\ZKSforcevertV = \gp{\ZKSheight-x} \ZKSstiffvert \end{dmath}, @@ -248,7 +248,7 @@ and the total force in the vertical direction can be nondimensionally represented by \begin{dmath}[label=ZKSforcetotalVnorm] -\frac{\ZKSforcetotalV}{\ZKSheight \ZKSstiffincl} = -\ZKSnorm \ZKSkratio+\ZKSkratio+2 +\frac{\ZKSforcetotalV}{\ZKSheight \ZKSstiffincl} = -\ZKSnorm \ZKSkratio+\ZKSkratio+2 \ZKSnorm \gp{\sqrt{\frac{\ZKSratio^2+1}{\ZKSratio^2+\ZKSnorm^2}}-1} \end{dmath}, @@ -326,7 +326,7 @@ the relation \begin{dmath}[label=ZKSratioQZS] \ZKSratioQZS = - \frac{2}{\sqrt{\ZKSkratio^2+4 \ZKSkratio}} + \frac{2}{\sqrt{\ZKSkratio^2+4 \ZKSkratio}} \end{dmath} which is used as the reference value of $\ZKSratio$ in @@ -363,17 +363,17 @@ order to retain stability. \begin{subfigure} \psfragfig{\phdpath ZKS/fig/ZKSoffqzs} \caption{\figlabel{ZKSoff} -The stable and unstable equilibrium points for $\ZKSoff\in\{-0.1, 0, 0.1\}$. +The stable and unstable equilibrium points for $\ZKSoff\in\{-0.1, 0, 0.1\}$. The rest position will move from the unstable point to the stable point of equilibrium.} \end{subfigure} \begin{subfigure} \psfragfig{\phdpath ZKS/fig/ZKSratioErr} \caption{\figlabel{ZKS-ratioErr} -The stiffness at equilibrium as $\ZKSoff$ varies; as the stiffness becomes +The stiffness at equilibrium as $\ZKSoff$ varies; as the stiffness becomes negative, the stiffness shown corresponds to the stable point of equilibrium shown in the figure adjacent.} \end{subfigure} \end{wide} -\caption{Force and stiffness of the inclined spring system near \qzs/, showing +\caption{Force and stiffness of the inclined spring system near \qzs/, showing the effect of unstable equilibrium.} \end{figure} @@ -482,7 +482,7 @@ stiffness in order to remain at \qzs/. \begin{figure} \psfragfig{\phdpath ZKS/fig/ZKS-hk-ratio} - \caption{Relationship between $\ZKSkratio$ and $\ZKSlengthratio$ at \qzs/ + \caption{Relationship between $\ZKSkratio$ and $\ZKSlengthratio$ at \qzs/ both horizontally and vertically.} \figlabel{ZKS-hk-ratio} \end{figure} @@ -520,12 +520,12 @@ decreases with $\ZKSlengthratio$). \begin{wide} \begin{subfigure} \psfragfig{\phdpath ZKS/fig/ZKSstiffnessV-QZS} - \caption{\figlabel{ZKSstiffnessV-QZS} + \caption{\figlabel{ZKSstiffnessV-QZS} Normalised vertical stiffness of the system.} \end{subfigure} \begin{subfigure} \psfragfig{\phdpath ZKS/fig/ZKSstiffnessH-QZS} - \caption{\figlabel{ZKSstiffnessH-QZS} + \caption{\figlabel{ZKSstiffnessH-QZS} Normalised horizontal stiffness of the system.} \end{subfigure} \end{wide} @@ -712,10 +712,10 @@ length: \begin{dmath} \ndisp = \gamma/\mdim . \end{dmath} -After some manipulation of the original equation given these simplifying +After some manipulation of the original equation given these simplifying assumptions, the force $\force$ on the second magnet in attraction (\ie, for magnets with polarisation in the same direction) -can be shown to be directly proportional to the facing area of the magnets, +can be shown to be directly proportional to the facing area of the magnets, $\mdim^2$, for a fixed normalised displacement, $\ndisp$, between the magnets: \begin{dmath}[label=force] \magforce = \mdim^2 \nforce . @@ -725,7 +725,7 @@ expression for the normalised force $\nforce$ is given in \eqref{nforce} in the appendix. The $\mdim^2$ relationship shown in \eqref{force} is interesting because it is not evident from Akoun~and Yonnet's original equations that such a simplification (for various subsets of magnet geometries -such as the one considered here) is possible. +such as the one considered here) is possible. \fixme(That stuff is not in an appendix in the thesis? Crossref to MTHEORY.) @@ -751,7 +751,7 @@ as normalised magnet gap and normalised magnet displacement, respectively. The force due to the lower magnet in repulsion is \begin{align} \frepl &= -\magforce{\mdim,\mdim\ngap+\mdim\ndispZ} \\ - &= -\mdim^2\nforce{\ngap+\ndispZ}, + &= -\mdim^2\nforce{\ngap+\ndispZ}, \end{align} and the force due to the upper magnet in attraction is \begin{align} @@ -762,7 +762,7 @@ The total force on the floating magnet, $\magforceZ$, is a superposition of $\frepl$ and $\fattr$, yielding \begin{align} \magforceZ &= \frepl+\fattr \\ - &= \mdim^2\gp{-\nforce{ \ngap+\ndispZ} + &= \mdim^2\gp{-\nforce{ \ngap+\ndispZ} +\nforce{-\ngap+\ndispZ}} \\ &\eqdef \mdim^2 \nforceZ . \eqlabel{nforceZ} \end{align} @@ -770,7 +770,7 @@ The stiffness of the system can be similarly expressed as \begin{align} \magstiffnessZ &= \mdim\nstiffnessZ , \\ \intertext{where} - \nstiffnessZ &= -\nstiffness{ \ngap+\ndispZ} + \nstiffnessZ &= -\nstiffness{ \ngap+\ndispZ} +\nstiffness{-\ngap+\ndispZ} . \eqlabel{nstiffnessZ} \end{align} The force $\nforceZ$ and stiffness $\nstiffnessZ$ of the magnetic spring are @@ -805,13 +805,13 @@ the normalised force $\nforce$ can be found by numerically fitting the constant coefficients $\ccA$, $\ccB$, and possibly $\nn$ in the empirical approximation for the forces between two magnets \begin{dmath}[label=nnfit] - \nforce \approx \frac{\ccA}{\gp{\ccB+\ndisp}^\nn}. + \nforce \approx \frac{\ccA}{\gp{\ccB+\ndisp}^\nn}. \end{dmath} \textcite{xu1993} used the more complicated approximation \begin{dmath} \nforce \approx \ccD\gp{\frac{\ccA}{\ccB+\ndisp}}^\nn+\ccC \end{dmath} -although in this case the additional complexity does not justify +although in this case the additional complexity does not justify the slight increase in accuracy this expression may offer. \textcite{bonisoli2007-mssp,bonisoli2007-mrc} used $\nn=3$ in their work, and @@ -858,11 +858,11 @@ those shown in \figref{nnfit} and have been omitted for clarity. \bottomrule \end{tabular} \caption{ - Best fit parameters for \eqref{nnfit}. - Fixed integer values of $\nn$ were chosen - for the first two cases, and the latter - value best fits the model by varying all - three parameters. Note that these are + Best fit parameters for \eqref{nnfit}. + Fixed integer values of $\nn$ were chosen + for the first two cases, and the latter + value best fits the model by varying all + three parameters. Note that these are are unitless parameters.} \tablabel{nnfit} \end{table} @@ -880,7 +880,7 @@ positive for magnets in repulsion and negative for magnets in attraction. the magnetic system using \eqref{nnfit} to calculate the force due to the repelling and attracting magnets separately: \begin{dmath}[label=nnfitz] - \nforceZ \approx \ccA\gp{\ccB+\ngap+\ndispZ}^{-\nn} + \nforceZ \approx \ccA\gp{\ccB+\ngap+\ndispZ}^{-\nn} +\ccA\gp{\ccB+\ngap-\ndispZ}^{-\nn} , \end{dmath} where $\ccA$, $\ccB$, and $\nn$ are the best-fit parameters previously @@ -892,11 +892,11 @@ be discussed in the following sections. The normalised stiffness can be approximated by differentiating \eqref{nnfitz} with respect to $\ndispZ$: (as shown previously in \eqref{stiffness}) \begin{dmath}[label=nnfitzk] - \nstiffnessZ \approx {n\ccA}{\gp{\ccB+\ngap+\ndispZ}^{-\nn-1}} + \nstiffnessZ \approx {n\ccA}{\gp{\ccB+\ngap+\ndispZ}^{-\nn-1}} + {n\ccA}{\gp{\ccB+\ngap-\ndispZ}^{-\nn-1}}. \end{dmath} -\begin{figure} +\begin{figure} {% \let\labelsize\footnotesize \def\LBL#1{\colorbox{white}{#1}}% @@ -991,7 +991,7 @@ required to support larger loads. \caption{ Regions of $\mdim$ and $\ngap$ satisfying the static deflection criterion - of \eqref{strong} for a range of masses. + of \eqref{strong} for a range of masses. Darker sections denote overlap of the regions in the overlay plot.} \figlabel{cons-strong} @@ -1052,7 +1052,7 @@ $\dmax$, limits the lower size of the magnet. {\psfragfig{\phdpath QZS/fig/cons-saturateN2}} {\psfragfig{\phdpath QZS/fig/cons-saturateN3}} {\psfragfig{\phdpath QZS/fig/cons-saturateNall}} - \caption{Regions of $\mdim$ and $\ndispZ$ satisfying the + \caption{Regions of $\mdim$ and $\ndispZ$ satisfying the maximum displacement criterion of \eqref{saturate} for a range of disturbance displacements $\dmax$ and a mass $\mass=\SI{0.5}{kg}$. Darker @@ -1324,12 +1324,12 @@ their nonlinear `frequency response', for which they calculate a {\color{gray}\tfrac{1}{T}}\Integrate{\ddot z_i(t)^2}{t,0,T} \middle/ {\color{gray}\tfrac{1}{T}}\Integrate{z_{r_i}(t)^2}{t,0,T} - \right.} + \right.} \condition*{i=1,2,\dots,N} \end{dmath}, for input signals \begin{dmath} -z_{r_i}(t) = A \Sin{\omega_i t} +z_{r_i}(t) = A \Sin{\omega_i t} \condition*{t\in[0,T]} \end{dmath}, and measured output signals $\ddot z_i(t)$. Note that the $1/T$ terms (above, qzs.tex @@ -4,11 +4,11 @@ To add coherency to the bibliography and to aid browsing, references are ordered alphabetically by first author and grouped by major topic: -\begin{description} - \item[-- Vibrations] on page \secpageref{refvib}, - \item[-- Levitation] on page \secpageref{reflev}, - \item[-- Forces] on page \secpageref{refforce}, and - \item[-- Other] (none of the above) on page \secpageref{refother}. +\begin{description} + \item[-- Vibrations] on page \secpageref{refvib}, + \item[-- Levitation] on page \secpageref{reflev}, + \item[-- Forces] on page \secpageref{refforce}, and + \item[-- Other] (none of the above) on page \secpageref{refother}. \end{description} Some citations fall into more than one category (this very thesis is an @@ -35,17 +35,17 @@ citation numbers will be out-of-order. maxnames=9,minnames=9, heading=vibrations, keyword=Vibrations] - + \printbibliography[ maxnames=9,minnames=9, heading=levitation, keyword=Levitation] - + \printbibliography[ maxnames=9,minnames=9, heading=forces, keyword=Forces] - + \printbibliography[ maxnames=9,minnames=9, heading=other, references.tex @@ -16,7 +16,7 @@ {\orig@Integrate{#1}{#2}}} \let\Int\Integrate %\renewcommand\D[3][]{% -% \text{\textsc{d}}_{#3^{#1}}\,#2} +% \text{\textsc{d}}_{#3^{#1}}\,#2} %%%%%%%%%%%%%%%%% \Style{LogParen=p,ArcTrig=arc,ImagParen=p,RealParen=p} %%%%%%%%%%%%%%%%% @@ -108,7 +108,7 @@ \newvariable\freq{\omega} \newvariable\massBase{M} -\newvariable\massMass{m} +\newvariable\massMass{m} \newsignal\dispBase\laplaceBase{x_1} \newsignal\dispMass\laplaceMass{x_2} \newvariable\velMass{\expandafter\dot\dispMass} @@ -313,15 +313,15 @@ nom=Compression of the vertical spring at \qzs/]{\ZKScompress_\ZKSvert} \newvariable\ZKSlengthratio[ - nom=Ratio between inclined and vertical + nom=Ratio between inclined and vertical spring lengths, $\ZKSlength$ and $\ZKSlength$] {\eta} \newvariable\ZKSratio[ - nom=Ratio between the inclined spring + nom=Ratio between the inclined spring width and height] {\gamma} \newvariable\ZKSkratio[ - nom=Ratio between the inclined and + nom=Ratio between the inclined and vertical spring stiffnesses] {\alpha} @@ -332,15 +332,15 @@ \newvariable\ZKSoff[nom=Percentage difference between $\ZKSratio$ and $\ZKSratioQZS$]{\epsilon} \newvariable\ZKSlengthratioQZS[ - nom=Value of $\ZKSlengthratio$ corresponding + nom=Value of $\ZKSlengthratio$ corresponding to \qzs/ in both directions] {\ZKSQZS{\eta}} \newvariable\ZKSkratioQZS[ - nom=Value of $\ZKSkratio$ corresponding to + nom=Value of $\ZKSkratio$ corresponding to \qzs/ in both directions] {\ZKSQZS{\alpha}} \newvariable\ZKSratioQZS[ - nom=Value of $\ZKSratio$ for which \qzs/ in + nom=Value of $\ZKSratio$ for which \qzs/ in the vertical direction is achieved] {\ZKSQZS{\ZKSratio}} @@ -401,7 +401,7 @@ \newvariable\LengthBeam{\VariableBeam L} \newvariable\HeightBeam{\VariableBeam h} \newvariable\WidthBeam{\VariableBeam w} -\newvariable\ThicknessBeam{\VariableBeam t} +\newvariable\ThicknessBeam{\VariableBeam t} \newvariable\DensityAluminium{\rho_{\text{a}}} thesis-maths.sty @@ -181,85 +181,85 @@ %%%%%%%%%%%%%%%%%%%%%%%% % See <http://groups.google.com/group/comp.text.tex/msg/4111d98fc7627748?dmode=source> %%%%%%%%%%%%%%%%%%%%%%%% -%%% page check for even/odd -% Stolen from komascript. -\def\mh@newevenodd@label@nn#1#2{% - \expandafter\gdef\csname mh@evenodd@#1\endcsname{#2}% -} - -\def\mh@getevenodd@n #1{% - \@ifundefined{mh@evenodd@#1}{\value{page}}% - {\@nameuse{mh@evenodd@#1}}} -\def\mh@evenodd{0} -\DeclareRobustCommand\IfOddPageTF{% - \begingroup - \@tempcnta=\mh@evenodd\relax - \advance\@tempcnta by\@ne - \xdef\mh@evenodd{\the\@tempcnta}% - \@bsphack - \protected@write\@auxout{}{% - \string\mh@newevenodd@label@nn{\mh@evenodd}{\noexpand\the\value{page}}}% - \@esphack - \count@\mh@getevenodd@n{\mh@evenodd}\relax - \expandafter\endgroup - \ifodd\count@ - \expandafter\@firstoftwo - \else - \expandafter\@secondoftwo - \fi -} - -%%% -% recall the boxes can't be 0pt, otherwise TeX might put it on a -% separate line so we make them almost 0pt. -\def\mh@print@in@outer@margin#1{% - \if@twoside - \IfOddPageTF +%%% page check for even/odd +% Stolen from komascript. +\def\mh@newevenodd@label@nn#1#2{% + \expandafter\gdef\csname mh@evenodd@#1\endcsname{#2}% +} + +\def\mh@getevenodd@n #1{% + \@ifundefined{mh@evenodd@#1}{\value{page}}% + {\@nameuse{mh@evenodd@#1}}} +\def\mh@evenodd{0} +\DeclareRobustCommand\IfOddPageTF{% + \begingroup + \@tempcnta=\mh@evenodd\relax + \advance\@tempcnta by\@ne + \xdef\mh@evenodd{\the\@tempcnta}% + \@bsphack + \protected@write\@auxout{}{% + \string\mh@newevenodd@label@nn{\mh@evenodd}{\noexpand\the\value{page}}}% + \@esphack + \count@\mh@getevenodd@n{\mh@evenodd}\relax + \expandafter\endgroup + \ifodd\count@ + \expandafter\@firstoftwo + \else + \expandafter\@secondoftwo + \fi +} + +%%% +% recall the boxes can't be 0pt, otherwise TeX might put it on a +% separate line so we make them almost 0pt. +\def\mh@print@in@outer@margin#1{% + \if@twoside + \IfOddPageTF {\makebox[1sp][l]{% \hspace{\tagmarginsep}% - \makebox[\marginparwidth][c]{#1}}}% + \makebox[\marginparwidth][c]{#1}}}% {\makebox[1sp][r]{% \makebox[\marginparwidth][c]{#1}% \hspace{\textwidth}% - \hspace{\tagmarginsep}}}% - \else + \hspace{\tagmarginsep}}}% + \else {\makebox[1sp][l]{% \hspace{\tagmarginsep}% - \makebox[\marginparwidth][c]{#1}}}% - \fi -} + \makebox[\marginparwidth][c]{#1}}}% + \fi +} % 1sp = 5.363 nanometres, so I think we'll be okay -\newlength\tagmarginsep -\setlength\tagmarginsep{\marginparsep} -\def\print@eqnum{\mh@print@in@outer@margin{\tagform@\theequation}} -\def\make@df@tag@@#1{% - \gdef\df@tag{\mh@print@in@outer@margin - {\maketag@@@{#1}\def\@currentlabel{#1}}}} - -\def\make@df@tag@@@#1{% - \gdef\df@tag{% - \mh@print@in@outer@margin{\tagform@{#1}}% - \toks@\@xp{\p@equation{#1}}% - \edef\@currentlabel{\the\toks@}% - }% -} - -\def\endmathdisplay@a{% - \if@eqnsw - \gdef\df@tag{% - \mh@print@in@outer@margin{\tagform@\theequation}% - }% - \fi - \if@fleqn \@xp\endmathdisplay@fleqn - \else \ifx\df@tag\@empty \else \veqno \alt@tag \df@tag \fi - \ifx\df@label\@empty \else \@xp\ltx@label\@xp{\df@label}\fi - \fi - \ifnum\dspbrk@lvl>\m@ne - \postdisplaypenalty -\@getpen\dspbrk@lvl - \global\dspbrk@lvl\m@ne - \fi -} +\newlength\tagmarginsep +\setlength\tagmarginsep{\marginparsep} +\def\print@eqnum{\mh@print@in@outer@margin{\tagform@\theequation}} +\def\make@df@tag@@#1{% + \gdef\df@tag{\mh@print@in@outer@margin + {\maketag@@@{#1}\def\@currentlabel{#1}}}} + +\def\make@df@tag@@@#1{% + \gdef\df@tag{% + \mh@print@in@outer@margin{\tagform@{#1}}% + \toks@\@xp{\p@equation{#1}}% + \edef\@currentlabel{\the\toks@}% + }% +} + +\def\endmathdisplay@a{% + \if@eqnsw + \gdef\df@tag{% + \mh@print@in@outer@margin{\tagform@\theequation}% + }% + \fi + \if@fleqn \@xp\endmathdisplay@fleqn + \else \ifx\df@tag\@empty \else \veqno \alt@tag \df@tag \fi + \ifx\df@label\@empty \else \@xp\ltx@label\@xp{\df@label}\fi + \fi + \ifnum\dspbrk@lvl>\m@ne + \postdisplaypenalty -\@getpen\dspbrk@lvl + \global\dspbrk@lvl\m@ne + \fi +} %%%%%%%%%%%%%%%%%%%%%%%%% @@ -278,7 +278,7 @@ \DeclareMathSymbol{9}{0}{letters}{'071} %%%%%%%%%%%%% -%! FLOATS +%! FLOATS \renewcommand{\topfraction}{.8} \renewcommand{\bottomfraction}{.3} \renewcommand{\textfraction}{.15} @@ -302,7 +302,7 @@ %% Floats & captions \captionnamefont{\scshape} -%\setlength\captionwidth{0.9\linewidth} +%\setlength\captionwidth{0.9\linewidth} %\changecaptionwidth \newcommand\lofcaption[2]{\caption[#1]{#1#2}} @@ -385,24 +385,24 @@ \setlength{\headheight}{\baselineskip} \setlength{\headwidth}{\textwidth+\marginparsep+0.5\marginparwidth} \makepagestyle{wspr} -\makepsmarks{wspr}{% +\makepsmarks{wspr}{% \let\@mkboth\markboth% \def\chaptermark##1{% \markboth{% - \if@mainmatter + \if@mainmatter \textsc{Chapter \thechapter:} % \fi - ##1}{##1}}% + ##1}{##1}}% \def\sectionmark##1{% \markright{% - \if@mainmatter - \textsc{\S\thesection:} % + \if@mainmatter + \textsc{\S\thesection:} % \fi ##1}}} \makerunningwidth{wspr}{\headwidth} -\makeheadposition{wspr}{flushright}{flushleft}{}{} -\makeevenhead{wspr}{\makebox[0pt][c]{\thesispage}} {} {\small\leftmark} -\makeoddhead{wspr} {\small\rightmark} {} {\thesispage} +\makeheadposition{wspr}{flushright}{flushleft}{}{} +\makeevenhead{wspr}{\makebox[0pt][c]{\thesispage}} {} {\small\leftmark} +\makeoddhead{wspr} {\small\rightmark} {} {\thesispage} \pagestyle{wspr} @@ -424,9 +424,9 @@ \renewcommand{\precistoctext}[1]{% \nobreak \begin{quote} - \leftskip = 0cm plus 0.5fil - \rightskip = 0cm plus -0.5fil - \parfillskip = 0cm plus 1fil + \leftskip = 0cm plus 0.5fil + \rightskip = 0cm plus -0.5fil + \parfillskip = 0cm plus 1fil \footnotesize ##1 \end{quote} } @@ -460,7 +460,7 @@ \else \noindent \fi - \ignorespaces + \ignorespaces {\cftsubsectionfont ##1}~{\cftsubsectionpagefont##2}% \let\numberline\oldnumberline\expandafter\ignorespaces \fi thesis-preamble.sty @@ -62,7 +62,7 @@ the ideas behind the \qzs/ arrangement analysed in detail in \secref{qzs}. Beam width & $\WidthBeam$ & \SI{ 40}{mm} \\ Beam thickness & $\ThicknessBeam$ & \SI{ 2}{mm} \\ Beam vertical offset & $\OriginBeam$ & \SI{ 82}{mm} \\ -\midrule +\midrule Magnet support height & $\HeightRigMagnets$ & \SI{105 }{mm} \\ Magnet support lever arm & $\LengthRigMagnets$ & \SI{300 }{mm} \\ Magnet support mass & $\MassRigMagnets$ & \SI{ 87 }{g} \\ @@ -234,7 +234,7 @@ certain wire diameter and resistance. \caption{Normalised force \vs\ displacement curve for the dual coil electromagnet. Zero displacement is as shown in \figref{rig-dual-coil}, and negative displacements show symmetric behaviour to that shown here. \figlabel{2coils-normdesign}} -\end{sidefigure} +\end{sidefigure} \end{wide} \end{figure} @@ -288,7 +288,7 @@ enumerated below. See \textcite{boehm1993} for a more detailed overview. An inductive, or Hall effect, sensor works by exciting a coil with a high frequency sinusoidal current which induces eddy currents in the target. - \note{An example of a commercial inductive sensor is + \note{An example of a commercial inductive sensor is \url{http://www.microstrain.com/ncdvrt.aspx}} These eddy currents may be measured very accurately, but the whole effect is very dependent on a lack of magnetic noise. This makes these type of @@ -307,7 +307,7 @@ enumerated below. See \textcite{boehm1993} for a more detailed overview. `\acro{lvdt}' sensors use a moving probe attached to an electrical circuit that exhibits a measureable effect which varies with displacement - of the probe. + of the probe. \note{An example of a commercial \acro{lvdt} sensor is \url{http://www.microstrain.com/dvrt.aspx}.} Differential Variable Reluctance Transducers @@ -353,7 +353,7 @@ Relevant operating properties are listed in \tabref{wenglor}. The measured output of the sensor was used infer a magnet displacement, \DispMag, (referenced from the \qzs/ position) with the following transformation: \begin{dmath} - \DispMag\fn{\DisplSensor,\OriginMagHigh} = + \DispMag\fn{\DisplSensor,\OriginMagHigh} = \OriginMagDisp\fn{\DisplSensor} - \OriginQZS\fn{\OriginMagHigh} \end{dmath}, where $\OriginQZS$ is the height of the \qzs/ location, given by @@ -403,7 +403,7 @@ For the beam, a \BnK/ 4367 accelerometer was used to measure the `output' signal \end{table} For open loop measurements, these sensors were used to measure acceleration directly; for closed loop control, the charge amplifier was used to integrate the measured signals and estimate the velocities. -Standard methods for interpreting the accelerometer involve high-pass filtering the measured signal to avoid drift as integration errors accumulate. +Standard methods for interpreting the accelerometer involve high-pass filtering the measured signal to avoid drift as integration errors accumulate. The corner frequency for the integration must be reasonably high, as it is not possible to distinguish between drift and low frequency components in the signal. An alternative is the use of `drift-free' integrators \cite{gavin1998}, in which the drift is compensated for using alternative signal processing methods, but this approach is only advantageous for measuring low-frequency periodic signals; for transient or wide-band signals, the typical approach is better suited. @@ -433,7 +433,7 @@ radius of the coil, the greater the forces imparted by the coil on the magnet A simplified geometry of the moving magnet arrangement is used to calculate the minimum tolerance required to avoid contact with the coil as a function -of the vertical motion of the magnets. This geometry is shown in +of the vertical motion of the magnets. This geometry is shown in \figref{horiz-tolerance}, where the L-shaped magnet support is shown in the `rest' position $\overline{OAB}$ and in a rotated position $\overline{OA'B'}$. @@ -474,7 +474,7 @@ avoid contact. \section{Experimental results} A number of measurements were performed using the experimental apparatus; -in the sections following, data recorded is presented for +in the sections following, data recorded is presented for \begin{enumerate} \item Magnet gap versus beam displacement; \item Open loop frequency responses for a range of magnet gaps; and, @@ -501,10 +501,10 @@ gap between the magnets at \qzs/: \half\gp{\HeightRig-\OriginMagLow-\OriginMagHigh-\HeightRigMagnets} +\HeightMag+\HeightBuffer \end{dmath}, -where geometrical properties are described in \tabref{rigprop} and +where geometrical properties are described in \tabref{rigprop} and $\HeightBuffer=\SI{2.5}{mm}$ is a little extra height to account for extra space taken up by the beam shell thickness and the shell of the magnet mount. -The normalised magnet gap is $\ngap=\OriginMagGap/\HeightMag$, where +The normalised magnet gap is $\ngap=\OriginMagGap/\HeightMag$, where $\HeightMag$ is the height of the magnets. \begin{figure} @@ -537,7 +537,7 @@ control, as discussed in \secref{vibes-feedback}. \FFT\ points & $2^{16}$ \\ Sample time & $\approx$\SI{17.5}{min} \\ Average overlap & $0.75$ \\ - Number of non-overlapping averages & $16$ \\ + Number of non-overlapping averages & $16$ \\ \bottomrule \end{tabular} \caption{Parameters used in the signal and spectrum analysis for the @@ -564,7 +564,7 @@ between the accelerometer measurements of the base and magnet-supported beam: \begin{figure}[p] \psfragfig{\phdpath XPMT/latex/ol-undamped-frflin} \caption{Open loop measurements with the electromagnetic coil connected in - an open circuit as a function of normalised gap~$\ngap$; + an open circuit as a function of normalised gap~$\ngap$; no additional damping is added to the system.} \figlabel{ol-undamped-frflin} \end{figure} @@ -572,9 +572,9 @@ between the accelerometer measurements of the base and magnet-supported beam: \begin{figure}[p] \psfragfig{\phdpath XPMT/latex/ol-damped-frflin} \caption{Open loop measurements with the electromagnetic coil connected - in a closed circuit as a function of normalised gap~$\ngap$. + in a closed circuit as a function of normalised gap~$\ngap$. The coil adds damping to the system, which can - be seen by the reduction in height of the resonant peaks in + be seen by the reduction in height of the resonant peaks in comparison to \figref{ol-undamped-frflin}.} \figlabel{ol-damped-frflin} \end{figure} @@ -585,7 +585,7 @@ From the measurements shown in the previous section, data fitting of the frequen While more sophisticated techniques are possible \cite{chen2009}, fitting the data to a known exact frequency response function yielded acceptable results in this case since the peaks are quite simple. % doublecheck how I want to say this -The model used to fit the data was a simple vibration isolation system +The model used to fit the data was a simple vibration isolation system given by \eqref{simple-isolation-freq} in terms of the natural frequency, $\natfreq = \sqrt{\stiffness/\mass}$, and damping ratio $\dampingratio = \damping / (2\sqrt{\stiffness\mass})$. The resulting expression is @@ -595,7 +595,7 @@ given by \eqref{simple-isolation-freq} in terms of the natural frequency, $\natf {-\freq^2 + 2\ii\dampingratio\freq\natfreq + \natfreq^2}, \end{dmath} where $\freq$ is the frequency at which to calculate the transmissibility -$\transmissibility$. The data was fit +$\transmissibility$. The data was fit \note{Using \Matlab/'s \code{fminsearch}.}% check to \eqref{fit-model} between $\half\natfreq \le \freq \le 2\natfreq$, returning good estimates of the natural frequency and @@ -702,7 +702,7 @@ feeding back the velocity signal for the controller. The overall improvement to the vibration isolation can be shown by calculating the root-mean-square of the transmissibility over a certain frequency range. \begin{dmath} - \RMSof\transmissibility = + \RMSof\transmissibility = \sqrt{\Sum{\transmissibility^2}{\freq,\freq_1,\freq_2}} \end{dmath} The lower frequency limit is defined as $\freq_1=\SI{1}{Hz}$, as the results @@ -748,12 +748,12 @@ accelerometer measurements used for velocity feedback control. In the time domain, the response of this linear system is given by \begin{dmath} -\massMass \accMass = +\massMass \accMass = \forceIn - \damping\gp{\velMass-\velBase} - \stiffness\gp{\dispMass-\dispBase} \end{dmath}, which can be re-written in the Laplace domain as \begin{dmath}[label=simple-isolator-laplace] -s^2 \laplaceMass \underbrace{\gp{\massMass+\damping/s+\stiffness/s^2}}_{\gp{\Block[2]}^{-1}} = +s^2 \laplaceMass \underbrace{\gp{\massMass+\damping/s+\stiffness/s^2}}_{\gp{\Block[2]}^{-1}} = \laplaceForce + s^2 \laplaceBase \underbrace{\gp{\damping/s+\stiffness/s^2}}_{\Block[1]} \end{dmath}. This is shown as a block diagram in \figref{simple-isolator-block}. If the @@ -794,9 +794,9 @@ of the high pass filter in the charge amplifier. Using \eqref{cl-filter-controller} in \eqref{cl-generic} gives the final transfer function between the mass and base states, \begin{dmath}[label=cl-filter] - \frac\laplaceMass\laplaceBase = + \frac\laplaceMass\laplaceBase = \frac{ \gp{\damping s+\stiffness} \gp{s+\freqHPfilter}^2 } - { + { \gain s^3 + \gp{ \massMass s^2 + \damping s + \stiffness } \gp{ s+\freqHPfilter }^2 xpmt.tex 004c952169212213dac6df24cad79e3340b1a61a Will Robertson wspr81@gmail.com http://github.com/wspr/thesis/commit/1451438b5ad06dab62523351aa5a86f8e2349902 1451438b5ad06dab62523351aa5a86f8e2349902 2009-09-30T00:38:01-07:00 2009-09-30T00:38:01-07:00 removing trailing whitespace git will be happy with me hope i didn't break anything (how could i?) 42bfaf952d14674796f0aee337968037768852c7 Will Robertson wspr81@gmail.com