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Thomas Dehaeze
2024-12-17 15:37:17 +01:00
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content/zettels/mass_spring_damper_systems.md
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@@ -13,7 +13,7 @@ Tags
### Model and equation of motion {#model-and-equation-of-motion}
Let's consider Figure [1](#figure--fig:mass-spring-damper-system) where:
Let's consider [Figure 1](#figure--fig:mass-spring-damper-system) where:
- \\(m\\) is the mass in [kg]
- \\(k\\) is the spring stiffness in [N/m]
@@ -86,7 +86,7 @@ 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 [4](#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>
@@ -154,7 +154,7 @@ 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 [5](#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
@@ -163,7 +163,7 @@ From Figure [5](#figure--fig:mass-spring-damper-2dof-x0-bode-plots), we can see
{{< 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 [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)).
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\\).
@@ -171,7 +171,7 @@ However, the low frequency gain is now equal to \\(1/k\_1\\).
{{< 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 [7](#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\\)