Acceleration Analysis

Acceleration Analysis#

This is the final and most complex step, as it requires the results from position and velocity analysis.

Relative Acceleration Method#

The fundamental equation is analogous to that of velocities, but with more components: \( \vec{a}_B = \vec{a}_A + \vec{a}_{B/A} \) As we saw in the introduction, each term of absolute acceleration (\(\vec{a}_A, \vec{a}_B\)) and relative acceleration (\(\vec{a}_{B/A}\)) is decomposed into a normal and a tangential component: \( (\vec{a}_B^n + \vec{a}_B^t) = (\vec{a}_A^n + \vec{a}_A^t) + (\vec{a}_{B/A}^n + \vec{a}_{B/A}^t) \) For each term:

  • The magnitude of the normal component depends on the angular velocity (calculated in the previous step): \(a^n = \omega^2 L\). Its direction always points towards the center of rotation.

  • The magnitude of the tangential component depends on the angular acceleration (which is usually the unknown): \(a^t = \alpha L\). Its direction is always perpendicular to the link.

The analysis (graphical or analytical) consists of setting up this complete vector equation and solving for the two scalar unknowns (usually, the angular accelerations \(\alpha_3\) and \(\alpha_4\)).

Coriolis Acceleration#

This acceleration component appears only under very specific conditions: when a point slides along a link that is, in turn, rotating. This is the case for mechanisms such as the slider-crank inversion (quick-return mechanism) seen in Example 3.

The relative acceleration equation between a point P on the slider and a coincident point Q on the rotating slotted link is: \( \vec{a}_P = \vec{a}_Q + (\vec{a}_{P/Q})_{rel} + \vec{a}_{Cor} \) Where \((\vec{a}_{P/Q})_{rel}\) is the sliding acceleration of point P along the slot. The Coriolis term is calculated as: \( \vec{a}_{Cor} = 2 \vec{\omega} \times \vec{v}_{rel} \) Where:

  • \(\vec{\omega}\) is the angular velocity of the link that contains the slot and rotates.

  • \(\vec{v}_{rel}\) is the sliding velocity of the slider along the slot.

The direction of the Coriolis vector is obtained by rotating the relative velocity vector \(\vec{v}_{rel}\) 90° in the same direction as \(\vec{\omega}\). The failure to include this term in mechanisms that require it is one of the most common errors in acceleration analysis.

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