Lumped Models
-------------

The lumpedSystems module has only one system implemented, but this is an important one: 
The harmonic oscillator, the mother system of all vibro-acousticians! 

Not really useful in all technical application but helpful for understanding of damping and resonance effects.
A simple oscillator is created with::

    import pyva.systems.lumpedSystems as lSys

    # First example for oscillator
    mass   = 0.1   
    ks     = 98.696 # spring stiffness 

    # Undamped HO
    myHO = lSys.HarmonicOscillator(mass,ks)

With specific initial conditions the motion of the HarmonicOscillator can be calculated by the 
:meth:`~pyva.systems.lumpedSystems.HarmonicOscillator.displacement` method ::

    # evenly sampled time at 200ms intervals
    time = np.arange(0., 0.5, 0.001)

    # Initial conditions
    x0 = 0.1
    v0 = 1.4

    plt.plot(time, myHO.displacement(time,x0,v0))
    
Leading to the following graph

.. _fig-HO_oscillations:
    
.. figure:: ./images/HO_oscillations.*
   :align: center
   :width: 70%
   
   Oscillatory motion of undamped harmonic oscillator.
   
Interesting damped cases are created using the critical damping as reference constant ::

    # Derive all other constants from this
    c_vc   = myHO.critical_viscous_damping
    
    # Damped HOs
    cv1  = c_vc*3   # overdamped 
    cv2  = c_vc/10  # underdamped

    myHO_uD = lSys.HarmonicOscillator(mass,ks,cv2)
    myHO_oD = lSys.HarmonicOscillator(mass,ks,cv1)
    myHO_cD = lSys.HarmonicOscillator(mass,ks,c_vc)
    
Providing the following plot from ::

    plt.plot(time, myHO_uD.displacement(time,x0,v0),lw=2,label = 'underdamped')
    plt.plot(time, myHO_oD.displacement(time,x0,v0),lw=2,label = 'overdamped')
    plt.plot(time, myHO_cD.displacement(time,x0,v0),lw=2,label = 'critically damped')
   
.. _fig-HO_damped_oscillations:
    
.. figure:: ./images/HO_damped_oscillations.*
   :align: center
   :width: 70%
   
   Oscillatory motion of damped harmonic oscillators.
   
Forced harmonic motion can also be found with the :meth:`~pyva.systems.lumpedSystems.HarmonicOscillator.u_force` method ::

    force = 10.0
    omega = np.linspace(0,4*myHO.omega_mode,200)

    plt.plot(omega, np.abs(myHO_uD.u_force(omega,force)),lw=2,label = 'underdamped')
    plt.plot(omega, np.abs(myHO_oD.u_force(omega,force)),lw=2,label = 'overdamped')
    plt.plot(omega, np.abs(myHO_cD.u_force(omega,force)),lw=2,label = 'critically damped')
    
Providing the following amplitude slope over frequency

.. _fig-HO_forced_oscillations:
    
.. figure:: ./images/HO_forced_oscillations.*
   :align: center
   :width: 70%
   
   Amplitude over frequency of forced harmonic oscillators with damping.