Abstract
A common automatic seatbelt inertial sensor design, comprised of a constrained spherical pendulum, is modeled to study its motions and possible unintentional release during vehicle emergency maneuvers. The kinematics are derived for the system with the most general inputs: arbitrary pivot motions. The influence of forces due to gravity and constraint torque functions is developed. The equations of motion are then derived using Kane's method. The equations of motion are used in a numerical simulation with both actual and hypothetical automobile crash data.
Highlights
This article addresses the motions of a spherical pendulum in a gravitational field in response to arbitrary motions of its pivot point
The complete equations of motion of a spherical pendulum have been derived for the conditions of a movable pivot point and restraining forces at that point
The model and methods presented are general and can be used for further investigations
Summary
This article addresses the motions of a spherical pendulum in a gravitational field in response to arbitrary motions of its pivot point. The belt locking mechanism is activated by the pendulum's polar angle relative to the vehicle (q2) to prevent the unreeling of the seatbelt during rapid deceleration or overturning. The equations of motion are formed using Kane's equations, a form of Newton's equations, in which nonworking forces of constraint are "automatically" eliminated; the analysis considers only the actual degrees of freedom Numerical simulation of these equations for actual and hypothetical inputs is used to predict conditions under which the retractor mechanism might fail to lock or be unlocked, for example due to rebound, oscillations of the pendulum, or motions of the vehicle
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