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content/article/mcinroy02_model_desig_flexur_joint_stewar.md
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@ -115,7 +115,7 @@ where:
- \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass - \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
- \\(g\\) is the gravity vector - \\(g\\) is the gravity vector
By combining <eq:strut_dynamics_vec>, <eq:payload_dynamics> and <eq:generalized_force>, a single equation describing the dynamics of a flexure jointed hexapod can be found: By combining \eqref{eq:strut\_dynamics\_vec}, \eqref{eq:payload\_dynamics} and \eqref{eq:generalized\_force}, a single equation describing the dynamics of a flexure jointed hexapod can be found:
\begin{equation} \begin{equation}
{}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega) \label{eq:eom\_fjh} {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega) \label{eq:eom\_fjh}
@ -200,7 +200,7 @@ In order to get dominance at low frequencies, the hexapod must be designed so th
This puts a limit on the rotational stiffness of the flexure joint and shows that as the strut is made softer (by decreasing \\(k\\)), the spherical flexure joint must be made proportionately softer. This puts a limit on the rotational stiffness of the flexure joint and shows that as the strut is made softer (by decreasing \\(k\\)), the spherical flexure joint must be made proportionately softer.
By satisfying <eq:cond_stiff>, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode: By satisfying \eqref{eq:cond\_stiff}, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
\\[ \omega \ll \sqrt{\frac{k\_r}{m\_s l^2}} \rightarrow x\_{\text{gain}\_\omega} \approx \frac{k}{k\_r/l^2} \gg 1 \\] \\[ \omega \ll \sqrt{\frac{k\_r}{m\_s l^2}} \rightarrow x\_{\text{gain}\_\omega} \approx \frac{k}{k\_r/l^2} \gg 1 \\]
At frequencies much above the strut's resonance mode, \\(f\_p\\) is not dominated by its \\(x\\) component: At frequencies much above the strut's resonance mode, \\(f\_p\\) is not dominated by its \\(x\\) component:
\\[ \omega \gg \sqrt{\frac{k}{m\_s}} \rightarrow x\_{\text{gain}\_\omega} \approx 1 \\] \\[ \omega \gg \sqrt{\frac{k}{m\_s}} \rightarrow x\_{\text{gain}\_\omega} \approx 1 \\]
@ -225,14 +225,14 @@ In this case, it is reasonable to use:
<div class="important"> <div class="important">
By designing the flexure jointed hexapod and its controller so that both <eq:cond_stiff> and <eq:cond_bandwidth> are met, the dynamics of the hexapod can be greatly reduced in complexity. By designing the flexure jointed hexapod and its controller so that both \eqref{eq:cond\_stiff} and \eqref{eq:cond\_bandwidth} are met, the dynamics of the hexapod can be greatly reduced in complexity.
</div> </div>
## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space} ## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
Equation <eq:eom_fjh> is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\). Equation \eqref{eq:eom\_fjh} is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
This section will derive this implicit relationship. This section will derive this implicit relationship.
Let denote: Let denote:

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content/article/mcinroy99_dynam.md
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@ -53,7 +53,7 @@ In order to provide low frequency passive vibration isolation, the hard actuator
<a id="table--tab:mcinroy99-strut-model"></a> <a id="table--tab:mcinroy99-strut-model"></a>
<div class="table-caption"> <div class="table-caption">
<span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a>:</span> <span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a>:</span>
Definition of quantities on <a href="#orgffe7e8f">2</a> Definition of quantities on <a href="#org1f8da5d">2</a>
</div> </div>
| **Symbol** | **Meaning** | | **Symbol** | **Meaning** |
@ -142,7 +142,7 @@ where:
- \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass - \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
- \\(g\\) is the gravity vector - \\(g\\) is the gravity vector
By combining <eq:strut_dynamics_vec>, <eq:payload_dynamics> and <eq:generalized_force>, a single equation describing the dynamics of a flexure jointed hexapod can be found: By combining \eqref{eq:strut\_dynamics\_vec}, \eqref{eq:payload\_dynamics} and \eqref{eq:generalized\_force}, a single equation describing the dynamics of a flexure jointed hexapod can be found:
\begin{aligned} \begin{aligned}
& {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u\\\\ & {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u\\\\

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content/book/du19_multi_actuat_system_contr.md
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@ -206,7 +206,7 @@ is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="<span class=\"figure-number\">Figure 6: </span>Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}} {{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="<span class=\"figure-number\">Figure 6: </span>Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
Equation [ 1](#orgc94b8e5) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\). Equation [ 1](#orgcf76ccd) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
One form of \\(W(s)\\) is taken as One form of \\(W(s)\\) is taken as
\begin{equation} \begin{equation}
@ -339,7 +339,7 @@ A decoupled control structure can be used for the three-stage actuation system (
The overall sensitivity function is The overall sensitivity function is
\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\] \\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [ 1](#org5626095) and with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [ 1](#org40d0f02) and
\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\] \\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
Denote the dual-stage open-loop transfer function as \\(G\_d\\) Denote the dual-stage open-loop transfer function as \\(G\_d\\)

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content/book/ewins00_modal.md
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@ -221,7 +221,7 @@ This process itself falls into two stages:
Most of the effort goes into this second stage, which is widely referred to as "modal parameter extraction", or simply as "modal analysis". Most of the effort goes into this second stage, which is widely referred to as "modal parameter extraction", or simply as "modal analysis".
We have seen that we can predict the form of the FRF plots for a multi degree-of-freedom system, and that these are directly related to the modal properties of that system. We have seen that we can predict the form of the FRF plots for a multi degree-of-freedom system, and that these are directly related to the modal properties of that system.
The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation <eq:frf_modal> to the measured FRF and thereby finding the appropriate modal parameters. The great majority of the modal analysis effort involves **curve-fitting** an expression such as equation \eqref{eq:frf\_modal} to the measured FRF and thereby finding the appropriate modal parameters.
A completely general curve-fitting approach is possible but generally inefficient. A completely general curve-fitting approach is possible but generally inefficient.
Mathematically, we can take an equation of the form Mathematically, we can take an equation of the form
@ -477,11 +477,11 @@ where \\(\eta\\) is the **structural damping loss factor** and replaces the crit
#### Alternative Forms of FRF {#alternative-forms-of-frf} #### Alternative Forms of FRF {#alternative-forms-of-frf}
So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force <eq:receptance>. So far we have defined our receptance frequency response function \\(\alpha(\omega)\\) as the ratio between a harmonic displacement response and the harmonic force \eqref{eq:receptance}.
This ratio is complex: we can look at its **amplitude** ratio \\(|\alpha(\omega)|\\) and its **phase** angle \\(\theta\_\alpha(\omega)\\). This ratio is complex: we can look at its **amplitude** ratio \\(|\alpha(\omega)|\\) and its **phase** angle \\(\theta\_\alpha(\omega)\\).
We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function <eq:mobility>. We could have selected the response velocity \\(v(t)\\) as the output quantity and defined an alternative frequency response function \eqref{eq:mobility}.
Similarly we could use the acceleration parameter so we could define a third FRF parameter <eq:inertance>. Similarly we could use the acceleration parameter so we could define a third FRF parameter \eqref{eq:inertance}.
<div class="definition"> <div class="definition">
@ -588,7 +588,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
##### Real part and Imaginary part of reciprocal FRF {#real-part-and-imaginary-part-of-reciprocal-frf} ##### Real part and Imaginary part of reciprocal FRF {#real-part-and-imaginary-part-of-reciprocal-frf}
It can be seen from the expression of the inverse receptance <eq:dynamic_stiffness> that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping. It can be seen from the expression of the inverse receptance \eqref{eq:dynamic\_stiffness} that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
[ 5](#org-target--fig-inverse-frf-mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\). [ 5](#org-target--fig-inverse-frf-mixed) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
@ -629,7 +629,7 @@ This makes the Nyquist plot very effective for modal testing applications.
#### Free Vibration Solution - The modal Properties {#free-vibration-solution-the-modal-properties} #### Free Vibration Solution - The modal Properties {#free-vibration-solution-the-modal-properties}
For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form <eq:undamped_mdof>. For an undamped MDOF system, with \\(N\\) degrees of freedom, the governing equations of motion can be written in matrix form \eqref{eq:undamped\_mdof}.
<div class="important"> <div class="important">
@ -645,7 +645,7 @@ where \\([M]\\) and \\([K]\\) are \\(N\times N\\) mass and stiffness matrices, a
We shall consider first the free vibration solution by taking \\(f(t) = 0\\). We shall consider first the free vibration solution by taking \\(f(t) = 0\\).
In this case, we assume that a solution exists of the form \\(\\{x(t)\\} = \\{X\\} e^{i \omega t}\\) where \\(\\{X\\}\\) is an \\(N \times 1\\) vector of time-independent amplitudes. In this case, we assume that a solution exists of the form \\(\\{x(t)\\} = \\{X\\} e^{i \omega t}\\) where \\(\\{X\\}\\) is an \\(N \times 1\\) vector of time-independent amplitudes.
Substitution of this condition into <eq:undamped_mdof> leads to Substitution of this condition into \eqref{eq:undamped\_mdof} leads to
\begin{equation} \label{eq:free\_eom\_mdof} \begin{equation} \label{eq:free\_eom\_mdof}
\left( [K] - \omega^2 [M] \right) \\{X\\} e^{i\omega t} = \\{0\\} \left( [K] - \omega^2 [M] \right) \\{X\\} e^{i\omega t} = \\{0\\}
@ -655,7 +655,7 @@ for which the non trivial solutions are those which satisfy
\\[ \det \left| [K] - \omega^2 [M] \right| = 0 \\] \\[ \det \left| [K] - \omega^2 [M] \right| = 0 \\]
from which we can find \\(N\\) values of \\(\omega^2\\) corresponding to the undamped system's **natural frequencies**. from which we can find \\(N\\) values of \\(\omega^2\\) corresponding to the undamped system's **natural frequencies**.
Substituting any of these back into <eq:free_eom_mdof> yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency. Substituting any of these back into \eqref{eq:free\_eom\_mdof} yields a corresponding set of relative values for \\(\\{X\\}\\): \\(\\{\psi\\}\_r\\) the so-called **mode shape** corresponding to that natural frequency.
<div class="important"> <div class="important">
@ -792,7 +792,7 @@ An alternative means of deriving the FRF parameters is used which makes use of t
\\[ [K] - \omega^2 [M] = [\alpha(\omega)]^{-1} \\] \\[ [K] - \omega^2 [M] = [\alpha(\omega)]^{-1} \\]
Pre-multiply both sides by \\([\Phi]^T\\) and post-multiply both sides by \\([\Phi]\\) to obtain Pre-multiply both sides by \\([\Phi]^T\\) and post-multiply both sides by \\([\Phi]\\) to obtain
\\[ [\Phi]^T ([K] - \omega^2 [M]) [\Phi] = [\Phi]^T [\alpha(\omega)]^{-1} [\Phi] \\] \\[ [\Phi]^T ([K] - \omega^2 [M]) [\Phi] = [\Phi]^T [\alpha(\omega)]^{-1} [\Phi] \\]
which leads to <eq:receptance_modal>. which leads to \eqref{eq:receptance\_modal}.
<div class="important"> <div class="important">
@ -802,7 +802,7 @@ Receptance FRF matrix - Modal Properties;
[\alpha(\omega)] = [\Phi] \left[ \bar{\omega}\_r^2 - \omega^2 \right]^{-1} [\Phi]^T \label{eq:receptance\_modal} [\alpha(\omega)] = [\Phi] \left[ \bar{\omega}\_r^2 - \omega^2 \right]^{-1} [\Phi]^T \label{eq:receptance\_modal}
\end{equation} \end{equation}
Equation <eq:receptance_modal> permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula Equation \eqref{eq:receptance\_modal} permits us to compute any individual FRF parameters \\(\alpha\_{jk}(\omega)\\) using the following formula
\begin{align} \begin{align}
\alpha\_{jk}(\omega) &= \sum\_{r=1}^N \frac{\phi\_{jr} \phi\_{kr}}{\bar{\omega}\_r^2 - \omega^2}\\\\ \alpha\_{jk}(\omega) &= \sum\_{r=1}^N \frac{\phi\_{jr} \phi\_{kr}}{\bar{\omega}\_r^2 - \omega^2}\\\\
@ -816,7 +816,7 @@ where \\({}\_rA\_{jk}\\) is called the **modal constant**.
<div class="important"> <div class="important">
It is clear from equation <eq:receptance_modal> that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**. It is clear from equation \eqref{eq:receptance\_modal} that the receptance matrix \\([\alpha(\omega)]\\) is **symmetric** and this will be recognized as the **principle of reciprocity**.
This principle of reciprocity applies to many structural characteristics. This principle of reciprocity applies to many structural characteristics.
@ -954,7 +954,7 @@ From this full matrix equation, we have:
Having derived an expression for the general term in the frequency response function matrix \\(\alpha\_{jk}(\omega)\\), it is appropriate to consider next the analysis of a situation where the system is **excited simultaneously at several points**. Having derived an expression for the general term in the frequency response function matrix \\(\alpha\_{jk}(\omega)\\), it is appropriate to consider next the analysis of a situation where the system is **excited simultaneously at several points**.
The general behavior for this case is governed by equation <eq:force_response_eom> with solution <eq:force_response_eom_solution>. The general behavior for this case is governed by equation \eqref{eq:force\_response\_eom} with solution \eqref{eq:force\_response\_eom\_solution}.
However, a more explicit form of the solution is However, a more explicit form of the solution is
\begin{equation} \label{eq:ods} \begin{equation} \label{eq:ods}
@ -977,7 +977,7 @@ The properties of the normal modes of the undamped system are of interest becaus
</div> </div>
We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in <eq:ods> but one is zero. We are seeking an excitation vector \\(\\{F\\}\\) such that the **response** \\(\\{X\\}\\) **consists of a single modal component** so that all terms in \eqref{eq:ods} but one is zero.
This can be attained if \\(\\{F\\}\\) is chosen such that This can be attained if \\(\\{F\\}\\) is chosen such that
\\[ \\{\phi\_r\\}^T \\{F\\}\_s = 0, \ r \neq s \\] \\[ \\{\phi\_r\\}^T \\{F\\}\_s = 0, \ r \neq s \\]
@ -1059,7 +1059,7 @@ where \\(\omega\_r\\) is the **natural frequency** and \\(\xi\_r\\) is the **cri
When the modes \\(r\\) and \\(q\\) are a complex conjugate pair: When the modes \\(r\\) and \\(q\\) are a complex conjugate pair:
\\[ s\_r = \omega\_r \left( -\xi\_r - i\sqrt{1 - \xi\_r^2} \right); \quad \\{\psi\\}\_q = \\{\psi\\}\_r^\* \\] \\[ s\_r = \omega\_r \left( -\xi\_r - i\sqrt{1 - \xi\_r^2} \right); \quad \\{\psi\\}\_q = \\{\psi\\}\_r^\* \\]
From equations <eq:viscous_damping_orthogonality>, we can obtain From equations \eqref{eq:viscous\_damping\_orthogonality}, we can obtain
\begin{align} \begin{align}
2 \omega\_r \xi\_r &= \frac{\\{\psi\\}\_r^H [C] \\{\psi\\}\_r}{\\{\psi\\}\_r^H [M] \\{\psi\\}\_r} = \frac{c\_r}{m\_r} \\\\ 2 \omega\_r \xi\_r &= \frac{\\{\psi\\}\_r^H [C] \\{\psi\\}\_r}{\\{\psi\\}\_r^H [M] \\{\psi\\}\_r} = \frac{c\_r}{m\_r} \\\\
@ -1352,7 +1352,7 @@ One these two series are available, the FRF can be defined at the same set of fr
##### Analysis via Fourier transform {#analysis-via-fourier-transform} ##### Analysis via Fourier transform {#analysis-via-fourier-transform}
For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from <eq:fourier_transform>. For most transient cases, the input function \\(f(t)\\) will satisfy the **Dirichlet condition** and so its Fourier Transform \\(F(\omega)\\) can be computed from \eqref{eq:fourier\_transform}.
\begin{equation} \label{eq:fourier\_transform} \begin{equation} \label{eq:fourier\_transform}
F(\omega) = \frac{1}{2 \pi} \int\_{-\infty}^\infty f(t) e^{i\omega t} dt F(\omega) = \frac{1}{2 \pi} \int\_{-\infty}^\infty f(t) e^{i\omega t} dt
@ -1500,10 +1500,10 @@ However, the same equation can be transform to the frequency domain
\tcmbox{ S\_{xx}(\omega) = \left| H(\omega) \right|^2 S\_{ff}(\omega) } \tcmbox{ S\_{xx}(\omega) = \left| H(\omega) \right|^2 S\_{ff}(\omega) }
\end{equation} \end{equation}
Although very convenient, equation <eq:psd_input_output> does not provide a complete description of the random vibration conditions. Although very convenient, equation \eqref{eq:psd\_input\_output} does not provide a complete description of the random vibration conditions.
Further, it is clear that **is could not be used to determine the FRF** from measurement of excitation and response because it **contains only the modulus** of \\(H(\omega)\\), the phase information begin omitted from this formula. Further, it is clear that **is could not be used to determine the FRF** from measurement of excitation and response because it **contains only the modulus** of \\(H(\omega)\\), the phase information begin omitted from this formula.
A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained <eq:cross_relation_alternatives>. A second equation is required and this may be obtain by a similar analysis, two alternative formulas can be obtained \eqref{eq:cross\_relation\_alternatives}.
<div class="important"> <div class="important">
@ -1517,8 +1517,8 @@ A second equation is required and this may be obtain by a similar analysis, two
##### To derive FRF from random vibration signals {#to-derive-frf-from-random-vibration-signals} ##### To derive FRF from random vibration signals {#to-derive-frf-from-random-vibration-signals}
The pair of equations <eq:cross_relation_alternatives> provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test. The pair of equations \eqref{eq:cross\_relation\_alternatives} provides the basic of determining a system's FRF properties from the measurements and analysis of a random vibration test.
Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities <eq:H1> <eq:H2>. Using either of them, we have a simple formula for determining the FRF from estimates of the relevant spectral densities \eqref{eq:H1} \eqref{eq:H2}.
<div class="important"> <div class="important">
@ -1555,7 +1555,7 @@ Then in [ 13](#org-target--fig-frf-feedback-model) is given a more detailed and
| width=\linewidth | width=\linewidth | | width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply. In this configuration, it can be seen that there are two feedback mechanisms which apply.
We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities <eq:H3>. We then introduce an alternative formula which is available for the determination of the system FRF from measurements of the input and output quantities \eqref{eq:H3}.
<div class="important"> <div class="important">
@ -1694,7 +1694,7 @@ First, if we have a **modal incompleteness** (\\(m<N\\) modes included), then we
However, if we have **spatial incompleteness** (only \\(n<N\\) DOFs included), then we cannot express any orthogonality properties at all because the eigenvector matrix is not commutable with the system mass and stiffness matrices. However, if we have **spatial incompleteness** (only \\(n<N\\) DOFs included), then we cannot express any orthogonality properties at all because the eigenvector matrix is not commutable with the system mass and stiffness matrices.
In both reduced-model cases, it is not possible to use equation <eq:spatial_model_from_modal> to re-construct the system mass and stiffness matrices. In both reduced-model cases, it is not possible to use equation \eqref{eq:spatial\_model\_from\_modal} to re-construct the system mass and stiffness matrices.
First of all because the eigen matrices are generally singular and even if it is not, the obtained mass and stiffness matrices produced have no physical significance and should not be used. First of all because the eigen matrices are generally singular and even if it is not, the obtained mass and stiffness matrices produced have no physical significance and should not be used.
@ -2699,7 +2699,7 @@ Then, the full \\(6 \times 6\\) mobility matrix can be measured, however this pr
Other methods for measuring rotational effects include specially developed rotational accelerometers and shakers. Other methods for measuring rotational effects include specially developed rotational accelerometers and shakers.
However, there is a major problem that is encountered when measuring rotational FRF: the translational components of the structure's movement tends to overshadow those due to the rotational motions. However, there is a major problem that is encountered when measuring rotational FRF: the translational components of the structure's movement tends to overshadow those due to the rotational motions.
For example, the magnitude of the difference in equation <eq:rotational_diff> is often of the order of \\(\SI{1}{\\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous. For example, the magnitude of the difference in equation \eqref{eq:rotational\_diff} is often of the order of \\(\SI{1}{\\%}\\) of the two individual values which is similar to the transverse sensitivity of the accelerometers: potential errors in rotations are thus enormous.
### Multi-point excitation methods {#multi-point-excitation-methods} ### Multi-point excitation methods {#multi-point-excitation-methods}
@ -2738,7 +2738,7 @@ The two vectors are related by the system's FRF properties as:
\\{X\\}\_{n\times 1} = [H(\omega)]\_{n\times p} \\{F\\}\_{p\times 1} \\{X\\}\_{n\times 1} = [H(\omega)]\_{n\times p} \\{F\\}\_{p\times 1}
\end{equation} \end{equation}
However, it is not possible to derive the FRF matrix from the single equation <eq:mpss_equation>, because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix. However, it is not possible to derive the FRF matrix from the single equation \eqref{eq:mpss\_equation}, because there will be insufficient data in the two vectors (one of length \\(p\\), the other of length \\(n\\)) to define completely the \\(n\times p\\) FRF matrix.
What is required is to make a series of \\(p^\prime\\) measurements of the same basic type using different excitation vectors \\(\\{F\\}\_i\\) that should be chosen such that the forcing matrix \\([F]\_{p\times p^\prime} = [\\{F\\}\_1, \dots, \\{F\\}\_p]\\) is non-singular. What is required is to make a series of \\(p^\prime\\) measurements of the same basic type using different excitation vectors \\(\\{F\\}\_i\\) that should be chosen such that the forcing matrix \\([F]\_{p\times p^\prime} = [\\{F\\}\_1, \dots, \\{F\\}\_p]\\) is non-singular.
This can be assured if: This can be assured if:
@ -3226,13 +3226,13 @@ For any frequency \\(\omega\\), we have the following relationship:
\end{aligned} \end{aligned}
\end{equation} \end{equation}
From <eq:modal_circle_tan>, we obtain: From \eqref{eq:modal\_circle\_tan}, we obtain:
\begin{equation} \label{eq:modal\_circle\_omega} \begin{equation} \label{eq:modal\_circle\_omega}
\omega^2 = \omega\_r^2 \left(1 - \eta\_r \tan\left(\frac{\theta}{2}\right) \right) \omega^2 = \omega\_r^2 \left(1 - \eta\_r \tan\left(\frac{\theta}{2}\right) \right)
\end{equation} \end{equation}
If we differentiate <eq:modal_circle_omega> with respect to \\(\theta\\), we obtain: If we differentiate \eqref{eq:modal\_circle\_omega} with respect to \\(\theta\\), we obtain:
\begin{equation} \begin{equation}
\frac{d\omega^2}{d\theta} = \frac{-\omega\_r^2 \eta\_r}{2} \frac{\left(1 - (\omega/\omega\_r)^2\right)^2}{\eta\_r^2} \frac{d\omega^2}{d\theta} = \frac{-\omega\_r^2 \eta\_r}{2} \frac{\left(1 - (\omega/\omega\_r)^2\right)^2}{\eta\_r^2}
@ -3317,10 +3317,10 @@ The sequence is:
Then we obtain the **center** and **radius** of the circle and the **quality factor** is the mean square deviation of the chosen points from the circle. Then we obtain the **center** and **radius** of the circle and the **quality factor** is the mean square deviation of the chosen points from the circle.
3. **Locate natural frequency, obtain damping estimate**. 3. **Locate natural frequency, obtain damping estimate**.
The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced. The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced.
At the same time, an estimate of the damping is derived using <eq:estimate_damping_sweep_rate>. At the same time, an estimate of the damping is derived using \eqref{eq:estimate\_damping\_sweep\_rate}.
A typical example is shown on [Figure 29](#figure--fig:circle-fit-natural-frequency). A typical example is shown on [Figure 29](#figure--fig:circle-fit-natural-frequency).
4. **Calculate multiple damping estimates, and scatter**. 4. **Calculate multiple damping estimates, and scatter**.
A set of damping estimates using all possible combination of the selected data points are computed using <eq:estimate_damping>. A set of damping estimates using all possible combination of the selected data points are computed using \eqref{eq:estimate\_damping}.
Then, we can choose the damping estimate to be the mean value. Then, we can choose the damping estimate to be the mean value.
We also look at the distribution of the obtained damping estimates as is permits a useful diagnostic of the quality of the entire analysis: We also look at the distribution of the obtained damping estimates as is permits a useful diagnostic of the quality of the entire analysis:
- Good measured data should lead to a smooth plot of these damping estimates, any roughness of the surface can be explained in terms of noise in the original data. - Good measured data should lead to a smooth plot of these damping estimates, any roughness of the surface can be explained in terms of noise in the original data.
@ -3426,7 +3426,7 @@ we now have sufficient information to extract estimates for the four parameters
3. Plot graphs of \\(m\_R(\Omega)\\) vs \\(\Omega^2\\) and of \\(m\_I(\Omega)\\) vs \\(\Omega^2\\) using the results from step 1., each time using a different measurement points as the fixing frequency \\(\Omega\_j\\) 3. Plot graphs of \\(m\_R(\Omega)\\) vs \\(\Omega^2\\) and of \\(m\_I(\Omega)\\) vs \\(\Omega^2\\) using the results from step 1., each time using a different measurement points as the fixing frequency \\(\Omega\_j\\)
4. Determine the slopes of the best fit straight lines through these two plots, \\(n\_R\\) and \\(n\_I\\), and their intercepts with the vertical axis \\(d\_R\\) and \\(d\_I\\) 4. Determine the slopes of the best fit straight lines through these two plots, \\(n\_R\\) and \\(n\_I\\), and their intercepts with the vertical axis \\(d\_R\\) and \\(d\_I\\)
5. Use these four quantities, and equation <eq:modal_parameters_formula>, to determine the **four modal parameters** required for that mode 5. Use these four quantities, and equation \eqref{eq:modal\_parameters\_formula}, to determine the **four modal parameters** required for that mode
This procedure which places more weight to points slightly away from the resonance region is likely to be less sensitive to measurement difficulties of measuring the resonance region. This procedure which places more weight to points slightly away from the resonance region is likely to be less sensitive to measurement difficulties of measuring the resonance region.
@ -3491,7 +3491,7 @@ From the sketch, it may be seen that within the frequency range of interest:
- the first term tends to approximate to a **mass-like behavior** - the first term tends to approximate to a **mass-like behavior**
- the third term approximates to a **stiffness effect** - the third term approximates to a **stiffness effect**
Thus, we have a basis for the residual terms and shall rewrite equation <eq:sum_modes>: Thus, we have a basis for the residual terms and shall rewrite equation \eqref{eq:sum\_modes:}
\begin{equation} \begin{equation}
H\_{jk}(\omega) \simeq -\frac{1}{\omega^2 M\_{jk}^R} + \sum\_{r=m\_1}^{m\_2} \left( \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} \right) + \frac{1}{K\_{jk}^R} H\_{jk}(\omega) \simeq -\frac{1}{\omega^2 M\_{jk}^R} + \sum\_{r=m\_1}^{m\_2} \left( \frac{{}\_rA\_{jk}}{\omega\_r^2 - \omega^2 + i \eta\_r \omega\_r^2} \right) + \frac{1}{K\_{jk}^R}
@ -3551,7 +3551,7 @@ We can write the receptance in the frequency range of interest as:
In the previous methods, the second term was assumed to be a constant in the curve-fit procedure for mode \\(r\\). In the previous methods, the second term was assumed to be a constant in the curve-fit procedure for mode \\(r\\).
However, if we have good **estimates** for the coefficients which constitutes the second term, for example by having already completed an SDOF analysis, we may remove the restriction on the analysis. However, if we have good **estimates** for the coefficients which constitutes the second term, for example by having already completed an SDOF analysis, we may remove the restriction on the analysis.
Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in <eq:second_term_refinement>, we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\). Indeed, suppose we take a set of measured data points around the resonance at \\(\omega\_r\\), and that we can compute the magnitude of the second term in \eqref{eq:second\_term\_refinement}, we then subtract this from the measurement and we obtain adjusted data points that are conform to a true SDOF behavior and we can use the same technique as before to obtain **improved estimated** to the modal parameters of more \\(r\\).
This procedure can be repeated iteratively for all the modes in the range of interest and it can significantly enhance the quality of found modal parameters for system with **strong coupling**. This procedure can be repeated iteratively for all the modes in the range of interest and it can significantly enhance the quality of found modal parameters for system with **strong coupling**.
@ -3610,7 +3610,7 @@ If we further increase the generality by attaching a **weighting factor** \\(w\_
is minimized. is minimized.
This is achieved by differentiating <eq:error_weighted> with respect to each unknown in turn, thus generating a set of as many equations as there are unknown: This is achieved by differentiating \eqref{eq:error\_weighted} with respect to each unknown in turn, thus generating a set of as many equations as there are unknown:
\begin{equation} \begin{equation}
\frac{d E}{d q} = 0; \quad q = {}\_1A\_{jk}, {}\_2A\_{jk}, \dots \frac{d E}{d q} = 0; \quad q = {}\_1A\_{jk}, {}\_2A\_{jk}, \dots
@ -3661,7 +3661,7 @@ leading to the modified, but more convenient version actually used in the analys
\end{equation} \end{equation}
In these expressions, only \\(m\\) modes are included in the theoretical FRF formula: the true number of modes, \\(N\\), is actually one of the **unknowns** to be determined during the analysis. In these expressions, only \\(m\\) modes are included in the theoretical FRF formula: the true number of modes, \\(N\\), is actually one of the **unknowns** to be determined during the analysis.
Equation <eq:rpf_error> can be rewritten as follows: Equation \eqref{eq:rpf\_error} can be rewritten as follows:
\begin{equation} \begin{equation}
\begin{aligned} \begin{aligned}
@ -3715,7 +3715,7 @@ where \\([X], [Y], [Z], \\{G\\}\\) and \\(\\{F\\}\\) are known measured quantiti
\end{equation} \end{equation}
Once the solution has been obtained for the coefficients \\(a\_k, \dots , b\_k, \dots\\) then the second stage of the modal analysis can be performed in which the required **modal parameters are derived**. Once the solution has been obtained for the coefficients \\(a\_k, \dots , b\_k, \dots\\) then the second stage of the modal analysis can be performed in which the required **modal parameters are derived**.
This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations <eq:frf_clasic> and <eq:frf_rational>: This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations \eqref{eq:frf\_clasic} and \eqref{eq:frf\_rational:}
- the denominator is used to obtain the natural frequencies \\(\omega\_r\\) and damping factors \\(\xi\_r\\) - the denominator is used to obtain the natural frequencies \\(\omega\_r\\) and damping factors \\(\xi\_r\\)
- the numerator is used to determine the complex modal constants \\(A\_r\\) - the numerator is used to determine the complex modal constants \\(A\_r\\)
@ -3943,7 +3943,7 @@ First, we note that from a single FRF curve, \\(H\_{jk}(\omega)\\), it is possib
Now, although this gives us the natural frequency and damping properties directly, it does not explicitly yield the mode shape: only a modal constant \\({}\_rA\_{jk}\\) which is formed from the mode shape data. Now, although this gives us the natural frequency and damping properties directly, it does not explicitly yield the mode shape: only a modal constant \\({}\_rA\_{jk}\\) which is formed from the mode shape data.
In order to extract the individual elements \\(\phi\_{jr}\\) of the mode shape matrix \\([\Phi]\\), it is necessary to make a series of measurements of specific FRFs including, especially, the point FRF at the excitation position. In order to extract the individual elements \\(\phi\_{jr}\\) of the mode shape matrix \\([\Phi]\\), it is necessary to make a series of measurements of specific FRFs including, especially, the point FRF at the excitation position.
If we measure \\(H\_{kk}\\), then by using <eq:modal_model_from_frf>, we also obtain the specific elements in the mode shape matrix corresponding to the excitation point: If we measure \\(H\_{kk}\\), then by using \eqref{eq:modal\_model\_from\_frf}, we also obtain the specific elements in the mode shape matrix corresponding to the excitation point:
\begin{equation} \begin{equation}
H\_{kk}(\omega) \longrightarrow \omega\_r, \eta\_r, {}\_rA\_{jk} \longrightarrow \phi\_{kr}; \quad r=1, m H\_{kk}(\omega) \longrightarrow \omega\_r, \eta\_r, {}\_rA\_{jk} \longrightarrow \phi\_{kr}; \quad r=1, m
@ -4417,7 +4417,7 @@ In this respect, the demands of the response model are more stringent that those
#### Synthesis of FRF curves {#synthesis-of-frf-curves} #### Synthesis of FRF curves {#synthesis-of-frf-curves}
One of the implications of equation <eq:regenerate_full_frf_matrix> is that **it is possible to synthesize the FRF curves which were not measured**. One of the implications of equation \eqref{eq:regenerate\_full\_frf\_matrix} is that **it is possible to synthesize the FRF curves which were not measured**.
This arises because if we measured three individual FRF such as \\(H\_{ik}(\omega)\\), \\(H\_{jk}(\omega)\\) and \\(K\_{kk}(\omega)\\), then modal analysis of these yields the modal parameters from which it is possible to generate the FRF \\(H\_{ij}(\omega)\\), \\(H\_{jj}(\omega)\\), etc. This arises because if we measured three individual FRF such as \\(H\_{ik}(\omega)\\), \\(H\_{jk}(\omega)\\) and \\(K\_{kk}(\omega)\\), then modal analysis of these yields the modal parameters from which it is possible to generate the FRF \\(H\_{ij}(\omega)\\), \\(H\_{jj}(\omega)\\), etc.
However, it must be noted that there is an important **limitation to this procedure** which is highlighted in the example below. However, it must be noted that there is an important **limitation to this procedure** which is highlighted in the example below.
@ -4541,7 +4541,7 @@ from which is would appear that we can write
\end{aligned} \end{aligned}
\end{equation} \end{equation}
However, equation <eq:m_k_from_modes> is **only applicable when we have available the complete \\(N \times N\\) modal model**. However, equation \eqref{eq:m\_k\_from\_modes} is **only applicable when we have available the complete \\(N \times N\\) modal model**.
It is much more usual to have an incomplete model in which the eigenvector matrix is rectangle and, as such, is non-invertible. It is much more usual to have an incomplete model in which the eigenvector matrix is rectangle and, as such, is non-invertible.
One step which can be made using the incomplete data is the construction of "pseudo" flexibility and inverse-mass matrices. One step which can be made using the incomplete data is the construction of "pseudo" flexibility and inverse-mass matrices.

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@ -252,7 +252,7 @@ where:
#### Eigenvalues / Characteristic Equation {#eigenvalues-characteristic-equation} #### Eigenvalues / Characteristic Equation {#eigenvalues-characteristic-equation}
Re-injecting normal modes <eq:principal_mode> into the equation of motion <eq:tdof_eom> gives the eigenvalue problem: Re-injecting normal modes \eqref{eq:principal\_mode} into the equation of motion \eqref{eq:tdof\_eom} gives the eigenvalue problem:
\begin{equation} \begin{equation}
(\bm{k} - \omega\_i^2 \bm{m}) \bm{z}\_{mi} = 0 (\bm{k} - \omega\_i^2 \bm{m}) \bm{z}\_{mi} = 0
@ -310,7 +310,7 @@ Virtual interpretation of the eigenvectors are shown in [Figure 5](#figure--fig:
#### Modal Matrix {#modal-matrix} #### Modal Matrix {#modal-matrix}
The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. <eq:modal_matrix> The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. \eqref{eq:modal\_matrix}
\begin{equation} \begin{equation}
\bm{z}\_m = \begin{bmatrix} \bm{z}\_m = \begin{bmatrix}
@ -873,7 +873,7 @@ If modes have some damping:
\frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \label{eq:general\_add\_tf\_damp} \frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \label{eq:general\_add\_tf\_damp}
\end{equation} \end{equation}
Equations <eq:general_add_tf> and <eq:general_add_tf_damp> shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\). Equations \eqref{eq:general\_add\_tf} and \eqref{eq:general\_add\_tf\_damp} shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
</div> </div>

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@ -182,7 +182,7 @@ And we see that \\(H\\) has units of amperes per meter.
The magnetic field of a solenoid is shown in [Figure 7](#figure--fig:morrison16-solenoid). The magnetic field of a solenoid is shown in [Figure 7](#figure--fig:morrison16-solenoid).
The field intensity inside the solenoid is nearly constant, while outside its intensity falls of rapidly. The field intensity inside the solenoid is nearly constant, while outside its intensity falls of rapidly.
Using Ampere's law <eq:ampere_law>: Using Ampere's law \eqref{eq:ampere\_law:}
\begin{equation} \begin{equation}
\oint H dl \approx n I l \oint H dl \approx n I l
@ -244,12 +244,12 @@ V = n^2 A k \mu\_0 \frac{dI}{dt} = L \frac{dI}{dt}
where \\(k\\) relates to the geometry of the coil. where \\(k\\) relates to the geometry of the coil.
Equation <eq:inductance_coil> states that if \\(V\\) is one volt, then for an inductance of one henry, the current will rise at the rate of one ampere per second. Equation \eqref{eq:inductance\_coil} states that if \\(V\\) is one volt, then for an inductance of one henry, the current will rise at the rate of one ampere per second.
### The energy stored in an inductance {#the-energy-stored-in-an-inductance} ### The energy stored in an inductance {#the-energy-stored-in-an-inductance}
One way to calculate the work stored in a magnetic field is to use Eq. <eq:inductance_coil>. One way to calculate the work stored in a magnetic field is to use Eq. \eqref{eq:inductance\_coil}.
The voltage \\(V\\) applied to a coil results in a linearly increasing current. The voltage \\(V\\) applied to a coil results in a linearly increasing current.
At any time \\(t\\), the power \\(P\\) supplied is equal to \\(VI\\). At any time \\(t\\), the power \\(P\\) supplied is equal to \\(VI\\).
Power is the rate of change of energy or \\(P = d\bm{E}/dt\\) where \\(\bm{E}\\) is the stored energy in the inductance. Power is the rate of change of energy or \\(P = d\bm{E}/dt\\) where \\(\bm{E}\\) is the stored energy in the inductance.

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@ -331,9 +331,9 @@ If we left multiply the equation by \\(\Phi^T\\) and we use the orthogonalily re
If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case: If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case:
\\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\] \\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\]
One can verify that the Rayleigh damping <eq:rayleigh_damping> complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\). One can verify that the Rayleigh damping \eqref{eq:rayleigh\_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
And we obtain decoupled modal equations <eq:modal_eom>. And we obtain decoupled modal equations \eqref{eq:modal\_eom}.
<div class="cbox"> <div class="cbox">
@ -366,15 +366,15 @@ Typical values of the modal damping ratio are summarized on table <tab:damping_r
The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large. The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
If one accepts the assumption of classical damping, the only difference between equation <eq:general_eom> and <eq:modal_eom> lies in the change of coordinates. If one accepts the assumption of classical damping, the only difference between equation \eqref{eq:general\_eom} and \eqref{eq:modal\_eom} lies in the change of coordinates.
However, in physical coordinates, the number of degrees of freedom is usually very large. However, in physical coordinates, the number of degrees of freedom is usually very large.
If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation <eq:modal_eom> can often be restricted to theses modes. If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation \eqref{eq:modal\_eom} can often be restricted to theses modes.
Therefore, the number of degrees of freedom contribution effectively to the response is **reduced drastically** in modal coordinates. Therefore, the number of degrees of freedom contribution effectively to the response is **reduced drastically** in modal coordinates.
#### Dynamic Flexibility Matrix {#dynamic-flexibility-matrix} #### Dynamic Flexibility Matrix {#dynamic-flexibility-matrix}
If we consider the steady-state response of equation <eq:general_eom> to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by: If we consider the steady-state response of equation \eqref{eq:general\_eom} to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
\\[ X = G(\omega) F \\] \\[ X = G(\omega) F \\]
Where \\(G(\omega)\\) is called the **Dynamic flexibility Matrix**: Where \\(G(\omega)\\) is called the **Dynamic flexibility Matrix**:
@ -634,7 +634,7 @@ With:
- \\(T\_{me}\\) is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input (in \\(\si{\newton\per\ampere}\\)) - \\(T\_{me}\\) is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input (in \\(\si{\newton\per\ampere}\\))
- \\(Z\_m\\) is the mechanical impedance measured when \\(i=0\\) - \\(Z\_m\\) is the mechanical impedance measured when \\(i=0\\)
Equation <eq:gen_trans_e> shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals. Equation \eqref{eq:gen\_trans\_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained. Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
To do so, the bridge circuit as shown on [Figure 15](#figure--fig:bridge-circuit) can be used. To do so, the bridge circuit as shown on [Figure 15](#figure--fig:bridge-circuit) can be used.
@ -706,7 +706,7 @@ With:
#### Constitutive Relations of a Discrete Transducer {#constitutive-relations-of-a-discrete-transducer} #### Constitutive Relations of a Discrete Transducer {#constitutive-relations-of-a-discrete-transducer}
The set of equations <eq:piezo_eq> can be written in a matrix form: The set of equations \eqref{eq:piezo\_eq} can be written in a matrix form:
\begin{equation} \begin{equation}
\begin{bmatrix}D\\\S\end{bmatrix} \begin{bmatrix}D\\\S\end{bmatrix}
@ -748,7 +748,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</div> </div>
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see [Figure 17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating <eq:piezo_eq_matrix_bis> over the volume of the transducer: If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see [Figure 17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo\_eq\_matrix\_bis} over the volume of the transducer:
\begin{equation} \begin{equation}
\begin{bmatrix}Q\\\\Delta\end{bmatrix} \begin{bmatrix}Q\\\\Delta\end{bmatrix}
@ -773,7 +773,7 @@ where
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="<span class=\"figure-number\">Figure 17: </span>Piezoelectric linear transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="<span class=\"figure-number\">Figure 17: </span>Piezoelectric linear transducer" >}}
Equation <eq:piezo_stack_eq> can be inverted to obtain Equation \eqref{eq:piezo\_stack\_eq} can be inverted to obtain
\begin{equation} \begin{equation}
\begin{bmatrix}V\\\f\end{bmatrix} \begin{bmatrix}V\\\f\end{bmatrix}
@ -801,7 +801,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="<span class=\"figure-number\">Figure 18: </span>Discrete Piezoelectric Transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="<span class=\"figure-number\">Figure 18: </span>Discrete Piezoelectric Transducer" >}}
By integrating equation <eq:piezo_work> and using the constitutive equations <eq:piezo_stack_eq_inv>, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer: By integrating equation \eqref{eq:piezo\_work} and using the constitutive equations \eqref{eq:piezo\_stack\_eq\_inv}, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
\begin{equation} \begin{equation}
W\_e(\Delta, Q) = \frac{Q^2}{2 C (1 - k^2)} - \frac{n d\_{33} K\_a}{C(1-k^2)} Q\Delta + \frac{K\_a}{1-k^2}\frac{\Delta^2}{2} W\_e(\Delta, Q) = \frac{Q^2}{2 C (1 - k^2)} - \frac{n d\_{33} K\_a}{C(1-k^2)} Q\Delta + \frac{K\_a}{1-k^2}\frac{\Delta^2}{2}
@ -853,7 +853,7 @@ And one can see that
\frac{z^2 - p^2}{z^2} = k^2 \frac{z^2 - p^2}{z^2} = k^2
\end{equation} \end{equation}
Equation <eq:distance_p_z> constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement ([Figure 20](#figure--fig:piezo-admittance-curve)). Equation \eqref{eq:distance\_p\_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement ([Figure 20](#figure--fig:piezo-admittance-curve)).
<a id="figure--fig:piezo-admittance-curve"></a> <a id="figure--fig:piezo-admittance-curve"></a>

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@ -4447,7 +4447,7 @@ The peak value is close to 6, meaning that even with 6 times less uncertainty, t
We here consider the relationship between \\(\mu\\) for RP and the condition number of the plant or of the controller. We here consider the relationship between \\(\mu\\) for RP and the condition number of the plant or of the controller.
We consider unstructured multiplicative uncertainty (i.e. \\(\Delta\_I\\) is a full matrix) and performance is measured in terms of the weighted sensitivity. We consider unstructured multiplicative uncertainty (i.e. \\(\Delta\_I\\) is a full matrix) and performance is measured in terms of the weighted sensitivity.
With \\(N\\) given by <eq:n_delta_structure_clasic>, we have: With \\(N\\) given by \eqref{eq:n\_delta\_structure\_clasic}, we have:
\begin{equation\*} \begin{equation\*}
\overbrace{\mu\_{\tilde{\Delta}}(N)}^{\text{RP}} \le [ \overbrace{\overline{\sigma}(w\_I T\_I)}^{\text{RS}} + \overbrace{\overline{\sigma}(w\_P S)}^{\text{NP}} ] (1 + \sqrt{k}) \overbrace{\mu\_{\tilde{\Delta}}(N)}^{\text{RP}} \le [ \overbrace{\overline{\sigma}(w\_I T\_I)}^{\text{RS}} + \overbrace{\overline{\sigma}(w\_P S)}^{\text{NP}} ] (1 + \sqrt{k})
@ -5439,8 +5439,8 @@ for a specified \\(\gamma > \gamma\_\text{min}\\), is given by
L &= (1-\gamma^2) I + XZ L &= (1-\gamma^2) I + XZ
\end{align} \end{align}
The Matlab function `coprimeunc` can be used to generate the controller in <eq:control_coprime_factor>. The Matlab function `coprimeunc` can be used to generate the controller in \eqref{eq:control\_coprime\_factor}.
It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from <eq:gamma_min_coprime> we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem. It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from \eqref{eq:gamma\_min\_coprime} we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
#### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic-hinf-loop-shaping-design-procedure} #### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic-hinf-loop-shaping-design-procedure}
@ -6273,7 +6273,7 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy
Consider the conventional cascade control system in [ 7](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\). Consider the conventional cascade control system in [ 7](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\). The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\).
From <eq:partial_control>, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\). From \eqref{eq:partial\_control}, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
These arguments particularly apply at high frequencies. These arguments particularly apply at high frequencies.
More precisely, we want the input-output controllability of \\([P\_u\ P\_r]\\) with disturbance model \\(P\_d\\) to be better that of the plant \\([G\_{11}\ G\_{12}]\\) with disturbance model \\(G\_{d1}\\). More precisely, we want the input-output controllability of \\([P\_u\ P\_r]\\) with disturbance model \\(P\_d\\) to be better that of the plant \\([G\_{11}\ G\_{12}]\\) with disturbance model \\(G\_{d1}\\).
@ -6289,7 +6289,7 @@ A set of outputs \\(y\_1\\) may be left uncontrolled only if the effects of all
</div> </div>
To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in <eq:partial_control_partitioning> and <eq:partial_control>, and compute \\(P\_d\\) using <eq:tight_control_y2>. To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in \eqref{eq:partial\_control\_partitioning} and \eqref{eq:partial\_control}, and compute \\(P\_d\\) using \eqref{eq:tight\_control\_y2}.
#### Measurement Selection for Indirect Control {#measurement-selection-for-indirect-control} #### Measurement Selection for Indirect Control {#measurement-selection-for-indirect-control}
@ -6459,7 +6459,7 @@ We then derive **necessary conditions for stability** which may be used to elimi
For decentralized diagonal control, it is desirable that the system can be tuned and operated one loop at a time. For decentralized diagonal control, it is desirable that the system can be tuned and operated one loop at a time.
Assume therefore that \\(G\\) is stable and each individual loop is stable by itself (\\(\tilde{S}\\) and \\(\tilde{T}\\) are stable). Assume therefore that \\(G\\) is stable and each individual loop is stable by itself (\\(\tilde{S}\\) and \\(\tilde{T}\\) are stable).
Using the **spectral radius condition** on the factorized \\(S\\) in <eq:S_factorization>, we have that the overall system is stable (\\(S\\) is stable) if Using the **spectral radius condition** on the factorized \\(S\\) in \eqref{eq:S\_factorization}, we have that the overall system is stable (\\(S\\) is stable) if
\begin{equation} \begin{equation}
\rho(E\tilde{T}(j\omega)) < 1, \forall\omega \rho(E\tilde{T}(j\omega)) < 1, \forall\omega
@ -6684,7 +6684,7 @@ When a reasonable choice of pairings have been made, one should rearrange \\(G\\
|1 + L\_i| > \max\_{k,j}\\{|\tilde{g}\_{dik}|, |\gamma\_{ij}|\\} |1 + L\_i| > \max\_{k,j}\\{|\tilde{g}\_{dik}|, |\gamma\_{ij}|\\}
\end{equation} \end{equation}
To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by <eq:decent_contr_one_loop> is achievable. To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by \eqref{eq:decent\_contr\_one\_loop} is achievable.
Note that RHP-zeros in the diagonal elements may limit achievable decentralized control, whereas they may not pose any problems for a multivariable controller. Note that RHP-zeros in the diagonal elements may limit achievable decentralized control, whereas they may not pose any problems for a multivariable controller.
Since with decentralized control, we usually want to use simple controllers, the achievable bandwidth in each loop will be limited by the frequency where \\(\angle g\_{ii}\\) is \\(\SI{-180}{\degree}\\) Since with decentralized control, we usually want to use simple controllers, the achievable bandwidth in each loop will be limited by the frequency where \\(\angle g\_{ii}\\) is \\(\SI{-180}{\degree}\\)
6. Check for constraints by considering the elements of \\(G^{-1} G\_d\\) and make sure that they do not exceed one in magnitude within the frequency range where control is needed. 6. Check for constraints by considering the elements of \\(G^{-1} G\_d\\) and make sure that they do not exceed one in magnitude within the frequency range where control is needed.
@ -6700,7 +6700,7 @@ When a reasonable choice of pairings have been made, one should rearrange \\(G\\
If the plant is not controllable, then one may consider another choice of pairing and go back to Step 4. If the plant is not controllable, then one may consider another choice of pairing and go back to Step 4.
If one still cannot find any pairing which are controllable, then one should consider multivariable control. If one still cannot find any pairing which are controllable, then one should consider multivariable control.
7. If the chosen pairing is controllable, then <eq:decent_contr_one_loop> tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be. 7. If the chosen pairing is controllable, then \eqref{eq:decent\_contr\_one\_loop} tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
This can be used as a basis for designing the controller \\(k\_i(s)\\) for loop \\(i\\) This can be used as a basis for designing the controller \\(k\_i(s)\\) for loop \\(i\\)
@ -6720,7 +6720,7 @@ Thus sequential design may involve many iterations.
#### Conclusion on Decentralized Control {#conclusion-on-decentralized-control} #### Conclusion on Decentralized Control {#conclusion-on-decentralized-control}
A number of **conditions for the stability**, e.g. <eq:decent_contr_cond_stability> and <eq:decent_contr_necessary_cond_stability>, and **performance**, e.g. <eq:decent_contr_cond_perf_dist> and <eq:decent_contr_cond_perf_ref>, of decentralized control systems have been derived. A number of **conditions for the stability**, e.g. \eqref{eq:decent\_contr\_cond\_stability} and \eqref{eq:decent\_contr\_necessary\_cond\_stability}, and **performance**, e.g. \eqref{eq:decent\_contr\_cond\_perf\_dist} and \eqref{eq:decent\_contr\_cond\_perf\_ref}, of decentralized control systems have been derived.
The conditions may be useful in **determining appropriate pairings of inputs and outputs** and the **sequence in which the decentralized controllers should be designed**. The conditions may be useful in **determining appropriate pairings of inputs and outputs** and the **sequence in which the decentralized controllers should be designed**.

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@ -131,7 +131,7 @@ Define a new input and a new output:
u\_1 = J^T f\_m, \quad y = J^{-1} (l - l\_r) u\_1 = J^T f\_m, \quad y = J^{-1} (l - l\_r)
\end{equation} \end{equation}
Equation <eq:hexapod_eq_motion> can be rewritten as: Equation \eqref{eq:hexapod\_eq\_motion} can be rewritten as:
\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_1} \begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_1}
\begin{split} \begin{split}

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@ -335,17 +335,17 @@ These eigenvectors have the following orthogonality properties, or can always be
\phi\_i^T M \phi\_j = 0 \quad (i \neq j) \phi\_i^T M \phi\_j = 0 \quad (i \neq j)
\end{equation} \end{equation}
For \\(i=j\\) the result of the multiplication according to equation <eq:eigenvector_orthogonality_mass> yields a non-zero result, which is normally indicated as **modal mass** \\(\mathit{m}\_i\\): For \\(i=j\\) the result of the multiplication according to equation \eqref{eq:eigenvector\_orthogonality\_mass} yields a non-zero result, which is normally indicated as **modal mass** \\(\mathit{m}\_i\\):
\begin{equation} \label{eq:modal\_mass} \begin{equation} \label{eq:modal\_mass}
\phi\_i^T M \phi\_i = \mathit{m}\_i \phi\_i^T M \phi\_i = \mathit{m}\_i
\end{equation} \end{equation}
Because only the direction but not the length of an eigenvector is defined, several scaling methods are used, all based on equation <eq:modal_mass>: Because only the direction but not the length of an eigenvector is defined, several scaling methods are used, all based on equation \eqref{eq:modal\_mass:}
- \\(|\phi\_i| = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its length is equal to \\(1\\). The modal mass are then calculated from equation <eq:modal_mass>. - \\(|\phi\_i| = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its length is equal to \\(1\\). The modal mass are then calculated from equation \eqref{eq:modal\_mass}.
- \\(\max(\phi\_i) = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its largest element is equation to \\(1\\). The modal mass is then calculated from equation <eq:modal_mass>. - \\(\max(\phi\_i) = 1\\). Each eigenvector \\(\phi\_i\\) is scaled such that its largest element is equation to \\(1\\). The modal mass is then calculated from equation \eqref{eq:modal\_mass}.
- \\(m\_i = 1\\). The modal mass \\(\mathit{m}\_i\\) is set to \\(1\\). The scaling of the mode vector \\(\phi\_i\\) follows from equation <eq:modal_mass>. - \\(m\_i = 1\\). The modal mass \\(\mathit{m}\_i\\) is set to \\(1\\). The scaling of the mode vector \\(\phi\_i\\) follows from equation \eqref{eq:modal\_mass}.
The orthogonality properties also apply to the stiffness matrix \\(K\\): The orthogonality properties also apply to the stiffness matrix \\(K\\):
@ -450,7 +450,7 @@ In such an analysis one is typically interested in the transfer function between
Applying the principle of modal decomposition, any transfer function can be derived by first calculating the behavior of the individual modes, and then **summing all modal contributions**. Applying the principle of modal decomposition, any transfer function can be derived by first calculating the behavior of the individual modes, and then **summing all modal contributions**.
The contribution of one single mode \\(i\\) to the transfer function \\(x\_l/f\_k\\) can be derived by first considering the response of the modal DoF \\(q\_i\\) to a force vector \\(f\\) with only one non-zero component \\(f\_k\\). The contribution of one single mode \\(i\\) to the transfer function \\(x\_l/f\_k\\) can be derived by first considering the response of the modal DoF \\(q\_i\\) to a force vector \\(f\\) with only one non-zero component \\(f\_k\\).
In that case, equation <eq:eoq_modal_i> is reduced to: In that case, equation \eqref{eq:eoq\_modal\_i} is reduced to:
\begin{equation} \begin{equation}
m\_i \ddot{q}\_i(t) + k\_i q\_i(t) = \phi\_{ik} f\_k(t) m\_i \ddot{q}\_i(t) + k\_i q\_i(t) = \phi\_{ik} f\_k(t)
@ -481,7 +481,7 @@ The overall transfer function can be found by summation of the individual modal
### Graphical Representation {#graphical-representation} ### Graphical Representation {#graphical-representation}
Due to the equivalence with the differential equations of a single mass spring system, equation <eq:eoq_modal_i> is often represented by a single mass spring system on which a force \\(f^\prime = \phi\_i^T f\\) acts. Due to the equivalence with the differential equations of a single mass spring system, equation \eqref{eq:eoq\_modal\_i} is often represented by a single mass spring system on which a force \\(f^\prime = \phi\_i^T f\\) acts.
However, this representation implies an important loss of information because it neglects all information about the mode-shape vector. However, this representation implies an important loss of information because it neglects all information about the mode-shape vector.
Consider the system in [Figure 8](#figure--fig:rankers98-mode-trad-representation) for which the three mode shapes are depicted in the traditional graphical representation. Consider the system in [Figure 8](#figure--fig:rankers98-mode-trad-representation) for which the three mode shapes are depicted in the traditional graphical representation.
@ -516,7 +516,7 @@ The resulting moment of inertia \\(J\_i\\) of the i-th modal lever then is:
J\_i = \sum\_{k=1}^n m\_k \phi\_{ik}^2 J\_i = \sum\_{k=1}^n m\_k \phi\_{ik}^2
\end{equation} \end{equation}
This result is identical to the modal mass \\(m\_i\\) found with Equation <eq:modal_mass>, because the mass matrix \\(M\\) is a diagonal matrix of physical masses \\(m\_k\\), and consequently the expression for the modal mass \\(m\_i\\) yields: This result is identical to the modal mass \\(m\_i\\) found with Equation \eqref{eq:modal\_mass}, because the mass matrix \\(M\\) is a diagonal matrix of physical masses \\(m\_k\\), and consequently the expression for the modal mass \\(m\_i\\) yields:
\begin{equation} \begin{equation}
m\_i = \phi\_j^T M \phi\_j = \sum\_{k=1}^n m\_k \phi\_{ik}^2 m\_i = \phi\_j^T M \phi\_j = \sum\_{k=1}^n m\_k \phi\_{ik}^2
@ -637,7 +637,7 @@ whereas the modal stiffnesses follow from \\(k\_i = \omega\_i^2 m\_i\\).
{{< figure src="/ox-hugo/rankers98_example_2dof_modal.png" caption="<span class=\"figure-number\">Figure 15: </span>Graphical representation of modes and modal parameters of the two mass spring system" >}} {{< figure src="/ox-hugo/rankers98_example_2dof_modal.png" caption="<span class=\"figure-number\">Figure 15: </span>Graphical representation of modes and modal parameters of the two mass spring system" >}}
From these results, the effective modal parameters for each mode, and for each individual DoF can be defined using equations <eq:m_modal_eff> and <eq:k_modal_eff>. From these results, the effective modal parameters for each mode, and for each individual DoF can be defined using equations \eqref{eq:m\_modal\_eff} and \eqref{eq:k\_modal\_eff}.
The results are summarized in [Table 2](#table--tab:2dof-example-modal-params-eff). The results are summarized in [Table 2](#table--tab:2dof-example-modal-params-eff).
<a id="table--tab:2dof-example-modal-params-eff"></a> <a id="table--tab:2dof-example-modal-params-eff"></a>
@ -716,7 +716,7 @@ f\_{\text{new},i}(\Delta k) &= \frac{1}{2\pi}\sqrt{\frac{k\_{\text{eff},i} + \De
Let's use the two mass spring system in [Figure 14](#figure--fig:rankers98-example-2dof) as an example. Let's use the two mass spring system in [Figure 14](#figure--fig:rankers98-example-2dof) as an example.
In order to analyze the effect of an extra mass at \\(x\_2\\), the effective modal mass at that DoF needs to be known for both modes (see [Table 2](#table--tab:2dof-example-modal-params-eff)). In order to analyze the effect of an extra mass at \\(x\_2\\), the effective modal mass at that DoF needs to be known for both modes (see [Table 2](#table--tab:2dof-example-modal-params-eff)).
Then using equation <eq:sensitivity_add_m>, one can estimate the effect of an extra mass \\(\Delta m = 1 kg\\) added to \\(m\_2\\). Then using equation \eqref{eq:sensitivity\_add\_m}, one can estimate the effect of an extra mass \\(\Delta m = 1 kg\\) added to \\(m\_2\\).
To estimate the influence of extra stiffness between the two DoF, one needs to calculate the effective modal stiffness that corresponds to the relative motion between \\(x\_2\\) and \\(x\_1\\). To estimate the influence of extra stiffness between the two DoF, one needs to calculate the effective modal stiffness that corresponds to the relative motion between \\(x\_2\\) and \\(x\_1\\).
This can be graphically done as shown in [Figure 19](#figure--fig:rankers98-example-sensitivity-2dof): This can be graphically done as shown in [Figure 19](#figure--fig:rankers98-example-sensitivity-2dof):
@ -726,7 +726,7 @@ k\_{\text{eff},1,(2-1)} &= 0.46 \cdot 10^7 / 0.07^2 = 93.9 \cdot 10^7 N/m \\\\
k\_{\text{eff},2,(2-1)} &= 1.23 \cdot 10^7 / 1.1^2 = 1.0 \cdot 10^7 N/m k\_{\text{eff},2,(2-1)} &= 1.23 \cdot 10^7 / 1.1^2 = 1.0 \cdot 10^7 N/m
\end{align} \end{align}
And using equation <eq:sensitivity_add_m>, the effect of additional stiffness on the frequency of the two modes can be computed. And using equation \eqref{eq:sensitivity\_add\_m}, the effect of additional stiffness on the frequency of the two modes can be computed.
The results are summarized in [Table 3](#table--tab:example-sensitivity-2dof-results). The results are summarized in [Table 3](#table--tab:example-sensitivity-2dof-results).
@ -860,13 +860,13 @@ Let's introduce a variable \\(\alpha\\), which **relates the high-frequency cont
\alpha = \frac{\frac{\phi\_{i,\text{servo}} \phi\_{i,\text{force}}}{m\_i}}{\frac{1}{m}} \alpha = \frac{\frac{\phi\_{i,\text{servo}} \phi\_{i,\text{force}}}{m\_i}}{\frac{1}{m}}
\end{equation} \end{equation}
which simplifies equation <eq:effect_one_mode> to: which simplifies equation \eqref{eq:effect\_one\_mode} to:
\begin{equation} \label{eq:effect\_one\_mode\_simplified} \begin{equation} \label{eq:effect\_one\_mode\_simplified}
\boxed{\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{\alpha}{m s^2 + m \omega\_i^2}} \boxed{\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{\alpha}{m s^2 + m \omega\_i^2}}
\end{equation} \end{equation}
Equation <eq:effect_one_mode_simplified> will be the basis for the discussion of the various patterns that can be observe in the frequency response functions and the effect of resonances on servo stability. Equation \eqref{eq:effect\_one\_mode\_simplified} will be the basis for the discussion of the various patterns that can be observe in the frequency response functions and the effect of resonances on servo stability.
Three different types of intersection pattern can be found in the amplitude plot as shown in [Figure 25](#figure--fig:rankers98-frf-effect-alpha). Three different types of intersection pattern can be found in the amplitude plot as shown in [Figure 25](#figure--fig:rankers98-frf-effect-alpha).
Depending on the absolute value of \\(\alpha\\) one can observe: Depending on the absolute value of \\(\alpha\\) one can observe:

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@ -87,13 +87,13 @@ Here is some inline mathematics: \\(z = 2\\).
Unumbered equation: Unumbered equation:
\\[ F(x) = \int\_0^x f(t) dt \\] \\[ F(x) = \int\_0^x f(t) dt \\]
Using the `equation` environment in Eq. <eq:numbered>. Using the `equation` environment in Eq. \eqref{eq:numbered}.
\begin{equation} \label{eq:numbered} \begin{equation} \label{eq:numbered}
F(s) = \int\_0^\infty f(t) e^{-st} dt F(s) = \int\_0^\infty f(t) e^{-st} dt
\end{equation} \end{equation}
Using the `align` environment Equations <eq:align_1> and <eq:align_2>. Using the `align` environment Equations \eqref{eq:align\_1} and \eqref{eq:align\_2}.
\begin{align} \begin{align}
\mathcal{F}(a) &= \frac{1}{2\pi i}\oint\_\gamma \frac{f(z)}{z - a}\\,dz \label{eq:align\_1} \\\\ \mathcal{F}(a) &= \frac{1}{2\pi i}\oint\_\gamma \frac{f(z)}{z - a}\\,dz \label{eq:align\_1} \\\\

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@ -2,7 +2,7 @@
title = "Second Blog Post" title = "Second Blog Post"
author = ["Dehaeze Thomas"] author = ["Dehaeze Thomas"]
date = 2021-05-01T00:00:00+02:00 date = 2021-05-01T00:00:00+02:00
lastmod = 2024-12-17T15:36:59+01:00 lastmod = 2024-12-17T16:38:28+01:00
tags = ["hugo", "org"] tags = ["hugo", "org"]
categories = ["emacs"] categories = ["emacs"]
draft = false draft = false

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@ -465,12 +465,12 @@ Numbering can be continued by using `+n` option as shown below.
``` ```
<div class="src-block-caption"> <div class="src-block-caption">
<span class="src-block-number"><a href="#code-snippet--lst:tikz-test-general-control-names">Code Snippet 4</a>:</span> <span class="src-block-number"><a href="#code-snippet--lst:tikz-test-general-control-names">Code Snippet 4</a>:</span>
Tikz code that is used to generate <a href="#org3d7f4fc">4</a> Tikz code that is used to generate <a href="#orga43b285">4</a>
</div> </div>
<a id="figure--fig:test-general-control-names"></a> <a id="figure--fig:test-general-control-names"></a>
{{< figure src="/ox-hugo/test_general_control_names.png" caption="<span class=\"figure-number\">Figure 3: </span>General Control Configuration. With some mathematics in the caption: \\(z = 2\\)" >}} {{< figure src="/ox-hugo/test_general_control_names.png" caption="<span class=\"figure-number\">Figure 3: </span>General Control Configuration. With some mathematics in the caption: \\(z^2, \sqrt{y}\\)" >}}
```md ```md
#+name: fig:general_control_names #+name: fig:general_control_names