Update Content - 2023-04-15

content/zettels/mass_spring_damper_systems.md

@@ -1,5 +1,6 @@
 +++
 title = "Mass Spring Damper Systems"
+author = ["Dehaeze Thomas"]
 draft = false
 +++
 
@@ -43,6 +44,14 @@ with:
 -   \\(\omega\_0 = \sqrt{k/m}\\) the natural frequency in [rad/s]
 -   \\(\xi = \frac{1}{2} \frac{c}{\sqrt{km}}\\) the damping ratio [unit-less]
 
+A quality factor \\(Q\\) can also be defined:
+
+\begin{equation}
+Q = \frac{1}{2\xi}
+\end{equation}
+
+This corresponds to the amplification at the natural frequency \\(\omega\_0\\).
+
 
 ### Matlab model {#matlab-model}
 
@@ -69,7 +78,7 @@ Gw = (c*s + k)/(m*s^2 + c*s + k);
 
 <a id="figure--fig:mass-spring-damper-1dof-transmissibility"></a>
 
-{{< figure src="/ox-hugo/mass_spring_damper_1dof_transmissibility.png" caption="<span class=\"figure-number\">Figure 1: </span>1dof Mass spring damper system - Transmissibility" >}}
+{{< figure src="/ox-hugo/mass_spring_damper_1dof_transmissibility.png" caption="<span class=\"figure-number\">Figure 3: </span>1dof Mass spring damper system - Transmissibility" >}}
 
 
 ## Two Degrees of Freedom {#two-degrees-of-freedom}
@@ -77,11 +86,11 @@ Gw = (c*s + k)/(m*s^2 + c*s + k);
 
 ### Model and equation of motion {#model-and-equation-of-motion}
 
-Consider the two degrees of freedom mass spring damper system of Figure [1](#figure--fig:mass-spring-damper-2dof).
+Consider the two degrees of freedom mass spring damper system of Figure [4](#figure--fig:mass-spring-damper-2dof).
 
 <a id="figure--fig:mass-spring-damper-2dof"></a>
 
-{{< figure src="/ox-hugo/mass_spring_damper_2dof.png" caption="<span class=\"figure-number\">Figure 1: </span>2 DoF Mass Spring Damper system" >}}
+{{< figure src="/ox-hugo/mass_spring_damper_2dof.png" caption="<span class=\"figure-number\">Figure 4: </span>2 DoF Mass Spring Damper system" >}}
 
 We can write the Newton's second law of motion to the two masses:
 
@@ -145,30 +154,30 @@ G_F1_to_d2 = -m2*s^2/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s +
 G_F2_to_d2 = (m1*s^2 + c1*s + k1)/((m1*s^2 + c1*s + k1)*(m2*s^2 + c2*s + k2) + m2*s^2*(c2*s + k2));
 ```
 
-From Figure [1](#figure--fig:mass-spring-damper-2dof-x0-bode-plots), we can see that:
+From Figure [5](#figure--fig:mass-spring-damper-2dof-x0-bode-plots), we can see that:
 
 -   the low frequency transmissibility is equal to one
 -   the high frequency transmissibility to the second mass is smaller than to the first mass
 
 <a id="figure--fig:mass-spring-damper-2dof-x0-bode-plots"></a>
 
-{{< figure src="/ox-hugo/mass_spring_damper_2dof_x0_bode_plots.png" caption="<span class=\"figure-number\">Figure 1: </span>Transfer functions from x0 to x1 and x2 (Transmissibility)" >}}
+{{< figure src="/ox-hugo/mass_spring_damper_2dof_x0_bode_plots.png" caption="<span class=\"figure-number\">Figure 5: </span>Transfer functions from x0 to x1 and x2 (Transmissibility)" >}}
 
-The transfer function from \\(F\_1\\) to the mass displacements (Figure [1](#figure--fig:mass-spring-damper-2dof-F1-bode-plots)) has the same shape than the transmissibility (Figure [1](#figure--fig:mass-spring-damper-2dof-x0-bode-plots)).
+The transfer function from \\(F\_1\\) to the mass displacements (Figure [6](#figure--fig:mass-spring-damper-2dof-F1-bode-plots)) has the same shape than the transmissibility (Figure [5](#figure--fig:mass-spring-damper-2dof-x0-bode-plots)).
 
 However, the low frequency gain is now equal to \\(1/k\_1\\).
 
 <a id="figure--fig:mass-spring-damper-2dof-F1-bode-plots"></a>
 
-{{< figure src="/ox-hugo/mass_spring_damper_2dof_F1_bode_plots.png" caption="<span class=\"figure-number\">Figure 1: </span>Transfer functions from F1 to x1 and x2" >}}
+{{< figure src="/ox-hugo/mass_spring_damper_2dof_F1_bode_plots.png" caption="<span class=\"figure-number\">Figure 6: </span>Transfer functions from F1 to x1 and x2" >}}
 
-The transfer functions from \\(F\_2\\) to the mass displacements are shown in Figure [1](#figure--fig:mass-spring-damper-2dof-F2-bode-plots):
+The transfer functions from \\(F\_2\\) to the mass displacements are shown in Figure [7](#figure--fig:mass-spring-damper-2dof-F2-bode-plots):
 
 -   the motion \\(x\_1\\) is smaller than \\(x\_2\\)
 
 <a id="figure--fig:mass-spring-damper-2dof-F2-bode-plots"></a>
 
-{{< figure src="/ox-hugo/mass_spring_damper_2dof_F2_bode_plots.png" caption="<span class=\"figure-number\">Figure 1: </span>Transfer functions from F2 to x1 and x2" >}}
+{{< figure src="/ox-hugo/mass_spring_damper_2dof_F2_bode_plots.png" caption="<span class=\"figure-number\">Figure 7: </span>Transfer functions from F2 to x1 and x2" >}}
 
 
 ## Bibliography {#bibliography}