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Thomas Dehaeze
2024-12-17 16:38:55 +01:00
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content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
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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:
\\[ \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">
@@ -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.
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.
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.
#### 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 \\]
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}\\))
- \\(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.
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}
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{bmatrix}D\\\S\end{bmatrix}
@@ -748,7 +748,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</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{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" >}}
Equation <eq:piezo_stack_eq> can be inverted to obtain
Equation \eqref{eq:piezo\_stack\_eq} can be inverted to obtain
\begin{equation}
\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" >}}
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}
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
\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>