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Palmgren-Miner linear damage model
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The function `palmgren_miner_linear_damage` uses the Palmgren-Miner linear damage hypothesis to find the outputs listed below.

.. admonition:: API Reference

   For inputs and outputs see the `API reference <https://reliability.readthedocs.io/en/latest/API/PoF/palmgren_miner_linear_damage.html>`_.

In the following example, we consider a scenario in which ball bearings fail after 50000 hrs, 6500 hrs, and 1000 hrs, after being subjected to a stress of 1kN, 2kN, and 4kN respectively. If each load cycle involves 40 mins at 1kN, 15 mins at 2kN, and 5 mins at 4kN, how long will the ball bearings last?

.. code:: python
    
    from reliability.PoF import palmgren_miner_linear_damage
    palmgren_miner_linear_damage(rated_life=[50000,6500,1000], time_at_stress=[40/60, 15/60, 5/60], stress=[1, 2, 4])
    
    '''
    Palmgren-Miner Linear Damage Model results:
    Each load cycle uses 0.01351 % of the components life.
    The service life of the component is 7400.37951 load cycles.
    The amount of damage caused at each stress level is:
    Stress =  1 , Damage fraction = 9.86717 %.
    Stress =  2 , Damage fraction = 28.463 %.
    Stress =  4 , Damage fraction = 61.66983 %.
    '''

**References:**

- Probabilistic Physics of Failure Approach to Reliability (2017), by M. Modarres, M. Amiri, and C. Jackson. pp. 33-37