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Studying mechanical system motion
A mechanical system initially at rest comes into operation by gravity; the initial position of the system is represented at the Fig.1-4. Take into consideration dry friction acting on body A (variants 5,7, 15-17, 24) and resistance to rolling of the body B (variants 1-16, 18-19, 21, 23, 24, 29). Neglect other resistance forces. The body rolls without slipping, cords are ideally flexible inextensible weightless.
Force of elasticity of linear spring obeys Hook’s law , where is stiffness of the spring, is absolute deformation of the spring.
Moment of elasticity of spiral spring obeys Hook’s law , where is stiffness of the spiral spring, is angle of the spring winding.
In the initial position of the system springs are undeformed.
Couple moments shown in variants 2, 4, 8, 12, 18, 20, 22, 26, 28, 30 are constant values.
Determine: the body A velocity when the distance traveled is .
deferential equation of the system motion.
are masses of the bodies correspondingly;
are radii of big and small circles;
is the body C radius of gyration about fixed axis passing through the centers of gravity;
is the body B radius of gyration about moving axis passing through the centers of gravity;
and are angles of inclines;
is a coefficient of dry kinetic friction;
is a coefficient of rolling resistance.
The given for each variant data are denoted by plus in the table 1.
Pulleys and rollers, which radiuses of gyration aren’t given at the table 1 can be considered as uniform cylinders.
Inclined parts of cords are parallel to corresponding inclines. Neglect sag of the cords.