Inertial body frame

The assumption that the only allowable relative motion between the two bodies is a rotation about the bond connecting them implies a relationship between the angular velocity Cj of their respective centers of mass measured in an inertial ( lab ) frame ... 

As a prelude to the formulation of the rotations of asymmetric molecules, we review the description of the Euler angles involved in going from the laboratory or inertial frame XYZ to the molecular or body frame xyz. 

Here, we call the attention of the reader that our Eq. (24) in the previous interlude correspond to Eqs. (32) in Ref. [3] for the cartesian components of the angular momentum in the body frame and the inertial frame, respectively, in terms of the Euler angles. Notice that the angles if and

commutation rules from Eq. (22) in Ref. [3], for the analysis of the rotations of asymmetric molecules are as follows ... 

In the equation of motion for the endolymph fluid, the force Tf dA acts on the fluid at the fluid-otoconial layer interface (Figure 64.2). This shear stress tf is responsible for driving the fluid flow. The linear Navier-Stokes equation for an incompressible fluid are used to describe this endolymph flow. Expressions for the pressure gradient, the flow velocity of the fluid measured with respect to an inertial reference frame, and the force due to gravity (body force) are substituted into the Navier-Stokes equation for flow in the x-direction yielding... 

Second law Given O is a fixed point on the inertial reference frame, the rate of change of the angular momentum of the body about O is equal to net moment of forces acting on the body about O. 

FIGURE 6.14 Two bodies, A and B, shown articulating at a joint. Body A is fixed in an inertial reference frame, and body B moves relative to it. The path of a generic muscle is represented by the origin S on B, the insertion N on A, and three intermediate via points P, Q, and R.QandR are via points arising from contact of the muscle path with body B. Via point P arises from contact of die muscle path with body A. The ISA of B relative to A is defined by the angular velocity vector of B in A [Modified from Pandy (1999). ... 

The e, 62, 63 coordinate system is defined to generalize the discussion of the angular velocity derivations and represents the inertial frame of reference. The 3-1-2 transformation follows an initial rotation about the third axis, Cj, by an angle of to yield the e[, 2, ej coordinate system. Then a second rotation is performed about the e, axis by an angle of yielding the e", e, e, system. Finally, a third rotation is performed about the axis by to yield the final e", e" body frame of reference. This defines the transformation from the ej, C2,63 system to the ef, e, e" system. To supplement the kinematics tables, an expression for the angular velocity vector is defined from this transformation as... 

This section will develop the bond graph of a mechanical elements system. This system represents the free movement of a rigid body, with a local reference system designated as 1, which has its center of mass at point G. A generic point P will also be defined to serve as an example of the junction point with another rigid body. Finally, an inertial reference frame centered at point 0 will also be defined (Fig. 9.7). The kinematic relations for the velocity of a generic point P of a body are... 

In this section we present efficient recursive solutions for the position, velocity and acceleration problems of multibody systems composed of an arbitrary number of joints and bodies. In addition to a global (or inertial) reference frame, consider a moving reference frame rigidly attached to each body. TTie position of the multibody system is characterized by a vector x, composed of... 

Figure 1 shows the undeformed and deformed (shadowed) states of a flexible body k. r is the position vector of an arbitary point M, while a is the position vector of the mass center. Both r and a are defined on the deformable body. X is the vector from the mass center (Gd) to an arbitrary point M on the deformed body and Y is the vector from the mass center (Go) to an arbitrary point Mq on the undeformed body. The following formulations use an inertial reference frame (Ro) and a body fixed frame (R[Pg.63]

Once the identification number k for each body in the system has been properly assigned, the topology of the system is uniquely defined by identification numbers of the proximal bodies. The notation Pr[k] refers to the set containing only the proximal body of the B. Pr is called the proximal body array and the convention that Pr[k] = 0 is used if is a base body where bodies 0 and N are synonymous labels for the inertial reference frame. It is also useful to introduce an additional set Dist[k] to characterize the system s topology. The notation Dist[k] refers to a set of body numbers defined as... 

The system consists of a chain of rigid bodies with each end of the chain connected to a point fixed in an inertial reference frame. The y bodies of the chain axe connected to each other and to points fixed in the inertial frame by two degree of freedom Hooke s joints. 

Here b means the IMU body frame e denotes the ECEF frame i indicates the inertial frame Cl is the Direction Cosine Matrix (DCM) from body frame to ECEF frame, ft is the skew-symmetric matrix for angular rate measurements is the vector of acceleration measurements from the accelerometers. F is the system matrix applied in the ECEF frame is the distance from the earth geometric center to the earth surface g is the local gravity is the position of the IMU in ECEF. The noise vector w contains, in the indicated order, gyroscope bias, acceleration bias, acceleration noise, angular rate noise, receiver clock error and receiver clock rate noise. These noise terms are described by the error covariance matrix Q in the Kalman filter routine ... 

Coriolis Acceleration The Coriohs acceleration arises in a rotating frame, which has no parallel in an inertial frame. When a body moves at a linear velocity u in a. rotating frame with angular speed H, it experiences a Coriolis acceleration with magnitude ... 

In order to outline the main features of measuring the gravitational field with the help of ballistic gravimeter imagine that a small body falls inside a vacuum cylinder under the action of the gravitational field only. In accordance with Newton s second law in the inertial frame we have... 

Inertial frame. A non-accelerating coordinate system. One in which F = ma holds, where Fis the sum of all real forces acting on a body of mass m whose acceleration is a. In classical mechanics,... 

Consider two bodies, i and j, connected by a bond of fixed length hij. We make the assumption that the only allowable relative motion between the two bodies is a rotation about hij. Let f) and rj locate (with respect to an inertial frame) the center of mass of body i and j respectively, and let Sy and Sji define the points of attachment for each body with respect to its center of mass. The position of the center of mass of body j with respect to that of body i is simply = fj — f). Finally, the scalar qij measures the relative angle of rotation about the bond hij. 

A comparison of equations (7.103) and (7.104) shows that the Newton s second law of motion in the inertial frame O is identical in form to that in O except that the latter formulation contains several additional fictitious body forces. The term —2mil x v is the Coriolis force, and —mQ x (f2 x r) designates the centrifugal force. No name is in general use for the term — x r. The acceleration —ao compensates for the translational acceleration of the frame. 

We consider first a body of negligible mass moving around a body of finite mass m in an elliptic orbit. It can be proved (Murray and Dermott, 1999) that the action-angle variables of the two-body problem in the inertial frame, in the plane, are the Delaunay variables defined by... 

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