By Dean Karnopp
This reference deals a scientific method of the dynamics and balance of cars resembling vehicles, bicycles, trailers, bikes, airplanes, and trains—showing how mathematical types of various levels of complexity can be utilized to indicate layout guidance for coverage of auto balance and together with a bankruptcy on lively balance enhancement.
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11), and the angular momentum law, Eq. 12), the basic dynamic equations are uncoupled. However, the use of a rotating coordinate frame requires that the components of the angular velocity vector must be used to compute the changes in the linear and angular momenta as shown in Eqs. 40). The gyrator rings in both parts of Fig. 3 are modulated by the angular velocity components so they are indeed coupled. The gyrator rings shown in Fig. 3 are sometimes called Euler junction structures because they reﬂect the manipulations necessary to ﬁnd absolute accelerations or changes in momenta when using a rotating coordinate system.
YÁ À b/ À U/ ¼ 0 ð1:17Þ (It is true but perhaps not completely obvious at ﬁrst that if the two rear wheels cannot move sideways but can only roll in the direction that they are pointed instantaneously, then any point on the rear axle also cannot have any sideways velocity. This means that the kinematic constraint of Eq. ) FIG. 3 Velocity components at the center of the rear axle. Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved. Now combining Eqs. 14), one can eliminate F yielding a single dynamic equation.
1, it is clear that the marble will tend to roll back toward the bottom point if it is displaced away from the bottom point. This is then called a statically stable equilibrium point. Note that the idea of static stability only has to do with the tendency of the marble to move after it has been displaced from equilibrium, not with the question of whether or not the marble would settle down back to the equilibrium point after some time. In terms of a simple diﬀerential equation such as that for a mass– spring system, as shown in Fig.