Analytical solution of the equation
=================================

When calculating the oscillator spectrum of an accelerogram [:ref:`R4.05.03 <R4.05.03>`], we have to solve the linear differential equation of the second order:


.. math::
 :label: eq-1

    \ ddot {q} +2\ mathrm {\ xi}\ mathrm {\ omega}\ dot {q} + {\ mathrm {\ omega}} ^ {2} q=\ mathrm {\ alpha}\ left (t\ right)


.. csv-table::

    "where", ":math:`q(t)` ", "is the relative displacement"
    "", ":math:`\alpha (t)` ", "is the acceleration of the movement imposed at the base"
    "", ":math:`\omega` ", "is the pulsation of the oscillator"
    "", ":math:`\xi` ", "is the reduced damping of the oscillator"


With initial conditions on :math:`q` and :math:`\dot{q}`. The solution to this equation is written in the form:


.. math::
 :label: eq-2

    q\ left (t\ right) = {\ int} _ {0} _ {0} ^ {t} h\ left (t-\ mathrm {\ tau}\ right)\ mathrm {.} \ mathrm {\ alpha}\ left (\ mathrm {\ tau}\ right) d\ mathrm {\ tau} +q\ left (0\ right) g\ left (t\ right) +\ dot {q}\ dot {q}\ left {q}\ left (q}\ left}\ left}\ left (t\ right)


where :math:`q(0)` and :math:`\dot{q}(0)` are the displacement and the speed at the initial instant.

We are going to give the expressions for :math:`h(t)` and :math:`g(t)` according to the value of the reduced depreciation :math:`\xi`.

If :math:`\xi <1` (subcritical damping), then:


.. math::
 :label: eq-3

    h\ left (t\ right) =\ frac {{e} ^ {} ^ {-\ mathrm {\ xi}\ mathrm {\ omega}}} {\ mathrm {\ omega}\ sqrt {1- {x}} ^ {x} ^ {2}} ^ {2}}}}\ text {2}}}}\ text {sin}}}}\ text {sin}\ left (\ mathrm {\ omega} t\ sqrt {1- {\ mathrm {\ x}}} ^ {2}}\ right)


And:


.. math::
: label: eq-4

    g\ left (t\ right) = {e} ^ {-\ mathrm {\ xi}\ mathrm {\ xi}\ mathrm {\ omega} t\ left (\ mathrm {\ omega} t\ sqrt {1- {\ mathrm {\ xi}}}\ mathrm {\ xi}}}\ right) +\ frac {\ mathrm {\ xi}} {\ sqrt {1- {\ mathrm {\ xi}}} ^ {2}}}}\ text {sin}}\ left (\ mathrm {\ omega} t\ sqrt {1- {\ mathrm {\ xi}}} ^ {2}}\ right)\ right]


If :math:`\xi =1` (critical damping), then:


.. math::
: label: eq-5

    h\ left (t\ right) = {\ text {te}}} ^ {-\ mathrm {\ omega} t}


And:


.. math::
 :label: eq-6

    g\ left (t\ right) =\ left (1-\ mathrm {\ omega}\ right) {e} ^ {\ mathrm {\ omega} t}


If :math:`\xi >1` (over-critical damping):


.. math::
 :label: eq-7

    h\ left (t\ right) =\ frac {{e} ^ {-\ mathrm {\ xi}\ mathrm {\ omega} t}} {\ mathrm {\ omega}\ sqrt {{\ mathrm {{e}} ^ {\ mathrm {\ xi}}} ^ {2} -1}}\ text {\ omega}}\ sqrt {{\ mathrm {\ omega}}\ sqrt {{\ mathrm {\ mathrm {\ xi}}} ^ {2} -1}}\ text {.} \ text {sh}\ left (\ mathrm {\ omega} t\ sqrt {{\ mathrm {\ xi}} ^ {2} -1} -1}\ right)


And:


.. math::
 :label: eq-8

    g\ left (t\ right) = {e} ^ {-\ mathrm {\ xi}\ mathrm {\ xi}\ mathrm {\ omega} t\ left (\ mathrm {\ omega} t\ sqrt {{\ mathrm {\ xi}}}} ^ {2} -1}\ right) +\ frac {\ mathrm {\ omega}}\ left (\ mathrm {\ omega}} t\ sqrt {\ omega}} t\ sqrt {\ mathrm {\ omega}} t\ sqrt {\ mathrm {\ xi}} {\ mathrm {\ xi}} ^ {2} -1}}}\ text {sh}\ left (\ mathrm {\ omega} t\ sqrt {{\ mathrm {\ xi}}} ^ {2} -1}}\ right)\ right]