add structure beam class

src/fvr/structure.py

@@ -0,0 +1,107 @@
+import math
+
+class beam:
+  """Euler-Bernoulli beam.
+  """
+  def __init__(self, E, I, A, L, rho):
+    """
+    Args:
+      E: Elastic modulus / Young's modulus
+      I: Second moment of area of the beam's cross-section
+      A: Cross-section
+      L: Length
+      rho: Density
+    """
+    self.E = E
+    self.I = I
+    self.A = A
+    self.L = L
+    self.rho = rho
+
+  @property
+  def V():
+    return self.A * self.L
+
+  @property
+  def mu():
+    """Mass per unit length (or the product of density and cross-section)"""
+    return self.rho * self.A
+
+  @property
+  def m():
+    return self.mu * self.L
+    # return self.rho * self.V
+
+  def eigenfrequency(self, n:int, support : str = 'fixed-free') -> float:
+    r"""Natural frequencies of the beam.
+
+    Args:
+      n: Mode number (1 for first mode, ...)
+      support: only 'fixed-free' atm.
+
+    Returns:
+      n-th natural frequencies of vibration
+      !!! Currently only the first four frequencies can be calculated.
+
+    Dynamic beam equation, the Euler-Lagrange equation, of an Euler-Bernoulli
+    beam. The governing differential equation of motion, with :math:`\mu` the
+    mass per unit length and :math:`\delta` the viscous damping per unit length.
+
+    .. math::
+
+      \frac{\partial^2}{\partial x^2} \left( EI(x)
+        \frac{\partial^2 y}{\partial x^2} \right) +
+        \mu(x)\frac{\partial^2 y}{\partial t^2} +
+        \delta (x)\frac{\partial y}{\partial t} = f(x, t)
+
+    Consider the undamped mode in bending vibration of the beam with uniform
+    sectional property. The natural frequencies and mode shapes are obtained
+    considering the homogeneous solution of the beam vibration equation. The
+    Ansatz is
+
+    .. math::
+
+      y(x, t) = \phi(x)\sin(\omega t)
+
+    with :math:`\phi` being mode shape functions and :math:`\omega` the angular
+    natural frequency. Substituting gives
+
+    .. math::
+
+      \frac{\mathrm{d}^4\phi}{\mathrm{d}x^4} - a^4 \phi(x) = 0
+
+    with :math:`a^4:=\mu\omega^2/(EI)` or
+
+    .. math::
+
+      a_n := \left(\frac{\mu\,\omega_n^2}{E\,I}\right)^{1/4}
+
+    The general solution is
+
+    .. math::
+
+      \phi(x) = A\sin{ax} + B\cos{ax} + C\sinh{ax} + D\cosh{ax}
+
+    with :math:`A,B,C,D` being integration constants and defined by the coundary
+    conditions.
+
+    ``fixed-free`` support (free vibration of a cantilever beam). Characteristic
+    equations:
+
+    .. math::
+
+      \cos{aL}\,\cosh{aL}+1=0
+
+    :math:`a_n` is a numerically solved value: :math:`a_1 L = 0.596864...\pi`,
+    :math:`a_2 L = 1.49418...\pi`, :math:`a_3 L = 2.50025...\pi`, :math:`a_4 L =
+    3.49999...\pi`, ...
+
+    .. math::
+
+      \omega_n = a_n^2\sqrt{\frac{E\,I}{\mu}} \quad,\quad
+      f_n = \frac{a_n^2}{2\pi}\sqrt{\frac{E\,I}{\mu}}
+
+    """
+    a_nLopi = [0.596864, 1.49418, 2.50025, 3.49999]
+    a_n = a_nLopi[n-1]*math.pi/self.L if n < len(a_nLopi) else 0
+    return a_n**2*math.sqrt(self.E*self.I/self.mu)/(2*math.pi)