## Mn Mm

Figure 3.4: An example of MDF system with two degrees of freedom at each mass

For a system of MDF system when subjected to ground acceleration in two directions and there are two degree of freedoms for each mass,as shown in figure 3.4,we can write the equation of motion based on the both degrees of freedom at each mass.For this system, we can write the displacement vector as:

Vi U2 V2

The same can be written for {U} and {U}. The equation of motion can be written as following:

In which {Ug} can be written as:

' 1 , |
0 | |

0 |
1 | |

1 |
0 | |

0 |
> + Ùgy< |
> |

1 |
0 | |

0 |
1 | |

1 |
0 | |

0 |
1 |

When Ugxand Ugy are the ground acceleration along the x and y direction, respectively. In the above equation of motion {f (t)} can be written as:

Which fix(t) and fiy (t) are the forces acting on the ith mass along x and y direction, respectively. In the case of concrete dams, {f (t)} can be seperated into hydrodynamic pressure {f} and resultant of all the other forces, {f1}, that act on the structure. Thus, the final form of the equation of the motion can be written as:

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