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Calculations/rendered/Spring-Dimensioning.md

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+```python
+import math
+import numpy as np
+from pint import UnitRegistry
+unit = UnitRegistry()
+unit.formatter.default_format = "~"
+
+
+# Parameters
+spring_constant = 1.1 * 4 * unit.N / unit.mm
+spring_length_resting = 112 * unit.mm
+
+weight_total = 14 * unit.kg
+
+dampening = 1 * unit.N / (unit.m / unit.s)
+
+def spring_length_at(weight):
+    return (weight * unit.standard_gravity / spring_constant + spring_length_resting).to(unit.mm)
+
+def resonant_freq_at(weight):
+    return (1 / (2 * math.pi) * np.sqrt(spring_constant / weight)).to(unit.Hz)
+
+spring_length = spring_length_at(weight_total)
+f0 = resonant_freq_at(weight_total)
+
+print(f"Length: {spring_length:~.1f}")
+print(f"Freq:   {f0:~.3f}")
+```
+
+    Length: 143.2 mm
+    Freq:   2.822 Hz
+
+
+
+```python
+def lehr_dampening_factor(d, k, m):
+    return d / (2 * np.sqrt(m * k))
+
+lehr_dampening = lehr_dampening_factor(dampening, spring_constant, weight_total)
+lehr_dampening.ito_reduced_units()
+lehr_dampening
+```
+
+
+
+
+0.00201455741006345
+
+
+
+
+```python
+def amplitude_ratio(lehr, f0, f):
+    eta = f / f0
+    return 1 / np.sqrt((1 - eta**2)**2 + (2 * eta * lehr)**2)
+
+f_in = np.geomspace(0.5, 90, 100) * unit.Hz
+ratio = amplitude_ratio(lehr_dampening, f0, f_in)
+
+import matplotlib.pyplot as plt
+plt.plot(f_in, ratio)
+
+def transmissibility_plot_setup(plt):
+    plt.xscale('log')
+    xticks = [1, 3, 10, 50]
+    plt.xticks(xticks, [f"{t} Hz" for t in xticks], minor=False)
+    plt.xlabel("Excitation Frequency")
+    
+    plt.ylim(0.001, 10)
+    plt.yscale('log')
+    plt.ylabel("Transmissibility Ratio")
+    
+    plt.grid(True, "both")
+
+transmissibility_plot_setup(plt)
+```
+
+    /usr/lib/python3.13/site-packages/matplotlib/cbook.py:1355: UnitStrippedWarning: The unit of the quantity is stripped when downcasting to ndarray.
+      return np.asarray(x, float)
+
+
+
+    
+![png](Spring-Dimensioning_files/Spring-Dimensioning_2_1.png)
+    
+
+
+
+```python
+def ratio_for_dkm(d, k, m, f_in):
+    f0 = (1 / (2 * math.pi) * np.sqrt(k / m)).to(unit.Hz)
+    lehr = lehr_dampening_factor(d, k, m)
+    return amplitude_ratio(lehr, f0, f_in)
+
+d = dampening
+k = [
+    spring_constant,
+    spring_constant / 4,
+]
+m = [
+    weight_total,
+    24 * unit.kg,
+]
+
+import matplotlib.pyplot as plt
+for k_ in k:
+    m_ = m[0]
+    plt.plot(f_in, ratio_for_dkm(d, k_, m_, f_in), label=f"{k_:~.1f}, {m_:~.1f}")
+for m_ in m[1:]:
+    k_ = k[0]
+    plt.plot(f_in, ratio_for_dkm(d, k_, m_, f_in), label=f"{k_:~.1f}, {m_:~.1f}")
+
+ref_thorlabs_f = [3, 4, 4.42, 5, 6,   7,   8,   9,    10,   40]
+ref_thorlabs_r = [2, 5, 15.0, 4, 1.5, 0.7, 0.5, 0.35, 0.28, 0.018]
+plt.plot(ref_thorlabs_f, ref_thorlabs_r, label="Thorlabs PTP702 (Passive)")
+
+ref_thorlabs_f = [1, 1.35, 2,   3,   5,   9,    20,   27,     30]
+ref_thorlabs_r = [2, 3,    0.9, 0.3, 0.1, 0.02, 0.009, 0.007, 0.0023]
+plt.plot(ref_thorlabs_f, ref_thorlabs_r, label="Thorlabs PTS601 (Active)")
+    
+plt.legend(loc="upper right")
+transmissibility_plot_setup(plt)
+```
+
+    /usr/lib/python3.13/site-packages/matplotlib/cbook.py:1355: UnitStrippedWarning: The unit of the quantity is stripped when downcasting to ndarray.
+      return np.asarray(x, float)
+
+
+
+    
+![png](Spring-Dimensioning_files/Spring-Dimensioning_3_1.png)
+    
+
+
+
+```python
+
+```