16.14. Numerical Example: 2-DOF System

Mass matrix:

$[M] = \begin{bmatrix} m_1 & 0 \\ 0 & m_2 \end{bmatrix}$
Where $m_1$ and $m_2$ are the masses at the two degrees of freedom.

The mass matrix represents the masses associated with each degree of freedom in a two-degree-of-freedom (2-DOF) system. Each entry of the diagonal matrix corresponds to the mass at a specific point in the system. For instance, $m_1$ is the mass at the first DOF and $m_2$ is the mass at the second DOF, while off-diagonal entries are zero, indicating that there isn't mass coupling between the two DOFs.

Imagine you have two identical swinging pendulums attached at different heights; each pendulum's mass can be thought of as a point on a scale. The mass matrix tells you how heavy each pendulum is at its swinging point, irrespective of how they influence each other.

Stiffness matrix:

$[K] = \begin{bmatrix} k_1 + k_2 & -k_2 \\ -k_2 & k_2 \end{bmatrix}$
Where $k_1$ and $k_2$ are the stiffness constants of the system.

The stiffness matrix defines how both degrees of freedom resist deformation when forces are applied. The diagonal entries represent the total stiffness associated with each DOF, and the off-diagonal entries show how the stiffness of one DOF affects the other DOF. For example, $k_1$ and $k_2$ could represent stiffness from springs connected to each mass, where the negative off-diagonal terms indicate interactions between the two masses.

Think of two springs connected to each other. When one spring is compressed, it not only pushes back against its own deformation but also affects the other spring. The stiffness matrix is like a recipe that tells you how these springs will work together to resist forces.

Solving for:

• Natural frequencies
• Mode shapes
• Modal participation
• Response to ground motion (using modal superposition or response spectrum)

To analyze a 2-DOF system, engineers calculate the natural frequencies, which are the frequencies at which the system tends to oscillate when disturbed. The modal shapes describe how each mass moves during these oscillations. This analysis helps predict how the system will respond during events like earthquakes. Modal participation factors can be used to understand how effective each mode is in contributing to overall motion, and finally, the response to ground motions is calculated using modal superposition, which combines the effect of the oscillation modes to get a complete picture.

Imagine a swing (the natural frequency) at a playground where children play. If one child swings at their frequency, they create a different motion compared to when the swing is empty. Each child's unique way of swinging represents different modes of vibration. By analyzing how they swing together, you can foresee how the swings will behave when someone starts pushing them (like ground motion during an earthquake).

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