Who is Samuel J Ling William Moebs?# Samuel J Ling William Moebs Popular Books

Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called nutation. In physics, there are two types of precession: torque-free and torque-induced.
In astronomy, precession refers to any of several slow changes in an astronomical body's rotational or orbital parameters. An important example is the steady change in the orientation of the axis of rotation of the Earth, known as the precession of the equinoxes.
Torque-free
Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the angular momentum is a constant, but the angular velocity vector changes orientation with time. What makes this possible is a time-varying moment of inertia, or more precisely, a time-varying inertia matrix. The inertia matrix is composed of the moments of inertia of a body calculated with respect to separate coordinate axes (e.g. x, y, z). If an object is asymmetric about its principal axis of rotation, the moment of inertia with respect to each coordinate direction will change with time, while preserving angular momentum. The result is that the component of the angular velocities of the body about each axis will vary inversely with each axis' moment of inertia.
The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows:
where ωp is the precession rate, ωs is the spin rate about the axis of symmetry, Is is the moment of inertia about the axis of symmetry, Ip is moment of inertia about either of the other two equal perpendicular principal axes, and α is the angle between the moment of inertia direction and the symmetry axis.When an object is not perfectly solid, internal vortices will tend to damp torque-free precession, and the rotation axis will align itself with one of the inertia axes of the body.
For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix R that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal moment of inertia tensor I0 and fixed external angular momentum L, the instantaneous angular velocity is
Precession occurs by repeatedly recalculating ω and applying a small rotation vector ω dt for the short time dt; e.g.:
for the skew-symmetric matrix [ω]×. The errors induced by finite time steps tend to increase the rotational kinetic energy:
this unphysical tendency can be counteracted by repeatedly applying a small rotation vector v perpendicular to both ω and L, noting that
Torque-induced
Torque-induced precession (gyroscopic precession) is the phenomenon in which the axis of a spinning object (e.g., a gyroscope) describes a cone in space when an external torque is applied to it. The phenomenon is commonly seen in a spinning toy top, but all rotating objects can undergo precession. If the speed of the rotation and the magnitude of the external torque are constant, the spin axis will move at right angles to the direction that would intuitively result from the external torque. In the case of a toy top, its weight is acting downwards from its center of mass and the normal force (reaction) of the ground is pushing up on it at the point of contact with the support. These two opposite forces produce a torque which causes the top to precess.
The device depicted on the right (or above on mobile devices) is gimbal mounted. From inside to outside there are three axes of rotation: the hub of the wheel, the gimbal axis, and the vertical pivot.
To distinguish between the two horizontal axes, rotation around the wheel hub will be called spinning, and rotation around the gimbal axis will be called pitching. Rotation around the vertical pivot axis is called rotation.
First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (around the wheelhub) is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a torque around the gimbal axis.
In the picture, a section of the wheel has been named dm1. At the depicted moment in time, section dm1 is at the perimeter of the rotating motion around the (vertical) pivot axis. Section dm1, therefore, has a lot of angular rotating velocity with respect to the rotation around the pivot axis, and as dm1 is forced closer to the pivot axis of the rotation (by the wheel spinning further), because of the Coriolis effect, with respect to the vertical pivot axis, dm1 tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis. Section dm2 of the wheel is moving away from the pivot axis, and so a force (again, a Coriolis force) acts in the same direction as in the case of dm1. Note that both arrows point in the same direction.
The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some spinning is added to rotation around a vertical axis.
It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous.
In the discussion above, the setup was kept unchanging by preventing pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over, the spinning top just pitches a little. This pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity – via the pitching motion – elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) rather than causing the spinning top to fall to its side.
Precession or gyroscopic considerations have an effect on bicycle performance at high speed. Precession is also the mechanism behind gyrocompasses.
Classical (Newtonian)
Precession is the change of angular velocity and angular momentum produced by a torque. The general equation that relates the torque to the rate of change of angular momentum is:
where τ {\displaystyle {\boldsymbol {\tau }}} and L {\displaystyle \mathbf {L} } are the torque and angular momentum vectors respectively.
Due to the way the torque vecto.... Discover the Samuel J Ling William Moebs popular books. Find the top 100 most popular **Samuel J Ling William Moebs books**.