相対運動の解析 - 加速度

A slider-crank mechanism converts rotational motion from the crank into linear motion of the slider, or vice versa. It consists of three main parts: the crank, the connecting rod, and the slider. The fluctuating angle between the crank and connecting rod creates general plane motion, making the slider's motion non-uniform and dependent on the crank's rotational position.

The slider-crank motion is non-uniform because the angle between the crank and connecting rod varies continuously as the mechanism operates. This changing geometry means the slider's velocity and acceleration are not constant. The motion results from a complex interplay of the crank's rotation, the connecting rod's orientation, and the slider's linear constraints.

Two reference frames are used: a fixed reference system at point O and a translating frame of reference at point A, located at the slider. The fixed frame provides absolute motion measurements, while the translating frame captures relative motion between the slider and the crank. This dual-frame approach simplifies the analysis of complex motion in the mechanism.

The absolute acceleration of point B is expressed as the vector sum of the absolute acceleration of point A and the relative acceleration of point B with respect to point A. Since point B moves in a circular path relative to point A, the relative acceleration has normal and tangential components. This decomposition allows engineers to analyze each motion component separately using relative motion analysis using rotating axes acceleration.

The relative acceleration of point B consists of normal and tangential components because point B moves in a circular path relative to point A. The normal component is directed toward the center of the circular path and depends on angular velocity. The tangential component is perpendicular to the radius and depends on angular acceleration of point B with respect to point A.

The motion of point B results from three factors: the linear acceleration of point A, the angular acceleration of point B with respect to point A, and the angular velocity of point B with respect to point A. These three components combine to produce the total motion observed at point B. Understanding each factor separately enables accurate prediction of the mechanism's behavior.

Taking time derivatives of velocity equations yields acceleration equations. The absolute velocity of point B equals the vector sum of absolute velocity at point A and relative velocity of point B with respect to point A. When time derivatives are applied, this relationship becomes the absolute acceleration equation, allowing engineers to determine how quickly velocities change throughout the mechanism's motion cycle.