## Force Functions For A Centrobaric Body

Since for purposes of dealing with gravitational effects a centrobaric body (see Sec. 2.5) may be replaced with a particle situated at the mass center of the body, force functions suitable for dealing with the gravitational interaction of a centrobaric body and a particle, a centrobaric body and any body whatsoever, or a centrobaric body and a remote body can be obtained directly from Eqs. (2.10.2), (2.11.2), (2.11.5), (2.12.1), (2.13.12), or (2.13.14). Example The gravitational force F exerted...

## ML 2 2 0 y1 I i

U3(u2 + M3a,) 2 Ajs 0J - + 8 2 Ajs - 2 At, 6i + 2 (A,-,., - A,i)(0,_1 - 0,) where a y,, 5, are given by u2 + u3at - L 2 Aiku3+k + 2L 2 X A *M3+* 4.17 Figure P4.17 shows a planar double pendulum consisting of two identical, uniform, pin-connected rods, each of length L and mass m. Equal linear torsional springs of modulus a are attached as indicated and both springs are undeformed when the angles < , and q2 are equal to zero. Considering small motions during which the pendulum remains in a...

## Dynamical Equations For A Simple Gyrostat

J, G designates a simple gyrostat consisting of a rigid body A and an axisymmetric rotor B whose axis and mass center, B*, are fixed in A a,, a2, a3 form a dextral set of mutually perpendicular unit vectors fixed in A, each parallel to a centra principal axis of inertia of G. The axis of B is parallel to a unit vector fi, and G has central principal moments of inertia (, 2, j, while B has an axial moment of inertia J, After defining AwB, atit 3(, and M, as V Aft> (i (1) where...

## Info

1.7 A rigid body is subjected to successive rotations characterized by Rodrigues vectors pu p2, p3. Show that the Rodrigues vector p characterizing a single equivalent rotation can be expressed as P (Pi + P2 + Pi - Pi X Pi - P2 X Pi + Pi X Pi - Pi P3P1 + P3 P1P2 - Pi Pi Pi) (1 - Pi P2 - Pi Pi - Pi Pi + Pi * Pi X p3) 1.8 A rigid body B is brought into a desired orientation in a reference frame A by being subjected successively to an arrotation of amount 0U an a2-rotation of amount 02, and an...

## Angular Acceleration

The angular acceleration a of a rigid body B in a reference frame A is defined as the first time-derivative in A of the angular velocity < 0 of B in A (see Sec. 1.11) Frequently, it is convenient to resolve both < 0 and a into components parallel to unit vectors fixed in a reference frame C, that is, to express to and a as co ca> iC + Ca> 2c2 + cto3C3 (2) where c,, c2, c3 form a dextral set of orthogonal unit vectors. When this is done, ca, cw, + ft x w c, (i 1, 2, 3) (4) where ft is...

## O

One can express 01 (see Fig. 1.12.1) as + (a) o> 2 a2 ft)i)b3 (7) (1 g) Vijk (ocj ok - ak a> j) ( , 2, 3) (8) _ (q3 (O, - at ft)3)of3 ( 1 0)2 Q 2 < *> l)< *2 To perform the integration indicated in Eq. (1), introduce the angles 4> and p shown Fig. 1.12.2, noting that a then can be expressed as a cos < t> bi + sin < f> cos i > b2 + sin < j> sin 1p b3 (10) 1 cos (f> a2 sin < j> cos ip a3 sin < t> sin tp (11) da sin < p d< f> dtp (12) 4n6, (o,J j J sin...

## Reorientation Of A Torquefree Gyrostat Initially At Rest

Figure 3.9.1 represents a simple gyrostat G formed by a rigid body A that carries an axisymmetric rotor B whose axis and mass center B* are fixed in A. No forces act on A or B except those exerted by A on B and vice versa, and G has the following inertia properties A and B have masses mA and mB, respectively. A has a central inertia dyadic lA, and B has an axial moment of inertia J and a central transverse moment of inertia K. Subsequent to any instant at which A and B are at rest in a...

## Force Function Expression For The Moment Exerted On A Small Body By A Body

When the distance between the mass centers B* and B* of two bodies B and B exceeds the greatest distance from B* to any point of B, and a force function V(p) for the forces exerted by B on a particle of unit mass at a point P situated as shown in Fig. 2.16.1 is available, then the system of gravitational forces exerted by B on B produces a moment about B* that is given approximately by where I is the central inertia dyadic of B, R is the position vector of B* relative to B*, and V denotes...

## Angular Velocity And Rodrigues Parameters

If ai, a2, a3 and b,, b , b3 are two dextral sets of orthogonal unit vectors fixed respectively in reference frames or rigid bodies A and B which are moving relative to each other, one can use Eqs. (1.4.7) (1.4.9) to associate with each instant of time Rodrigues parameters p,, p2, and p3 and a Rodrigues vector p can then be formed by reference to Eq. (1.4.2). The angular velocity of B in A (see Sec. 1.11), expressed in terms of p, is given by Conversely, if to is known as a function of time,...

## Spacecraft With Continuous Elastic Components

When a spacecraft consists in part of continuous elastic components supported by rigid bodies, small motions of the components relative to the rigid bodies generally are governed by partial differential equations that cannot be solved by the method of separation of variables. But it can occur that the partial differential equations governing certain motions of the components can be solved by this method, as is the case, for example, when a component is a uniform cantilever beam and one...

## Lumped Mass Models Of Spacecraft

Certain spacecraft can be modeled as sets of elastically connected particles. For example, when a spacecraft is an unrestrained truss formed by a relatively large number of prismatic members, its motions may be analyzed by considering a set of particles placed at the joints of the structure, each particle having a mass equal to one-half the sum of the masses of all truss members meeting at the joint and the particles being connected to each other with massless springs whose stiffnesses reflect...

## P

Finally, if the successive rotations are a bi rotation of amount 0i, a b2 rotation of amount 02, and again a bi rotation, but this time of amount 03, then SiS2 -S1C2S3 + C3C1 S1C2C3 - S3C1 C1S2 CiC2S3 + C3S1 CiC2C3 S3S1 and, if Cn 1, 0,, 02, and 03 may be found by taking 02 cos1 C 0 < 02 < 7T a sin1 C32 - < a < (38) 01 I 77 - a if C22 < 0 (39) The matrices in Eqs. (1) and (11) are intimately related to each other either one may be obtained from the other by replacing 0, with -0, ( 1,...

## Auxiliary Reference Frames

The angular velocity of a rigid body B in a reference frame A (see Sec. 1.11) can be expressed in the following form involving n auxiliary reference frames A1, , An A< aB AtoA + Ala> A2 + + A-'(aA -I- A < aB (1) This relationship, the addition theorem for angular velocities, is particularly useful when each term in the right-hand member represents the angular velocity of a body performing a motion of simple rotation (see Sec. 1.1) and can, therefore, be expressed as in Eq. (1.11.5)....

## To The Reader

Each of the four chapters of this book is divided into sections. A section is identified by two numbers separated by a decimal point, the first number referring to the chapter in which the section appears, and the second identifying the section within the chapter. Thus, the identifier 2.13 refers to the thirteenth section of the second chapter. A section identifier appears at the top of each page. Equations are numbered serially within sections. For example, the equations in Sees. 2.12 and 2.13...

## Dynamics

Kane, Stanford University Peter W. Likins, Lehigh University David A. Levinson, Lockheed Palo Alto Research Laboratory Previously published by McGraw-Hill Book Company, 1993 Now published by The Internet-First University Press This manuscript is among the initial offerings being published as part of a new approach to scholarly publishing. The manuscript is freely available from the Internet-First University Press repository within DSpace at Cornell University at The online version of...

## N

Where M is the mass of B, a* is the acceleration of the mass center B* of B in A, m, is the mass of a generic particle P of B, N is the number of particles comprising B, r, is the position vector from B* to P, and a* is the acceleration of P in A. T* is called the inertia torque for B in A, and (Fr*)B can be written (Fr*)B tor T* + vr R* (r 1, . . . , n) (5) where vr is the rth partial velocity of B* in A, and < or is the rth partial angular velocity of B in A (see Sec. 1.21). The utility of...

## Influence Of Orbit Eccentricity On The Rotational Motion Of An Axisymmetric Rigid Body

Under the circumstances described at the beginning of Sec. 3.2, B* can move also in an elliptic orbit that is fixed in N, has an eccentricity e and major semidiameter a, and is traversed in a time T related to x and a by n 2ttT'1 ( uj3)1 2 (1) The polar coordinates, R and 6, of B* (see Fig. 3.3.1) satisfy the equation R a(l - e2)(l + e cos 0)1 (2) and 8 is governed by the differential equation while the rotational motion of B in N proceeds in such a way that 2ei 2(0*3 s + ft) e3o> 2 + e4o)...

## Centrobaric Bodies

As noted_in Sec. 2.2, the center of gravity B' of a body B for an attracting particle P does not in general coincide with the mass center B of B. However, there exist bodies for which the center of gravity and center of mass necessarily coincide. Such bodies are called centrobaric.'f Thus, a body B of mass m is centrobaric if the force F exerted on B by every particle P of mass m is given by where G is the universal gravitational constant, R is the distance between P and B , and at is a unit...

## Dynamical Equations

Given a system S possessing n degrees of freedom in a Newtonian reference frame N, let , . . . , un be generalized speeds for S n N see Sec. 1.21 , and form Ft, . . . , F , the associated generalized active forces for 5 in N see Sec. 4.1 , and Fi , . . . , F , the associated generalized inertia forces for 5 in N see Sec. 4.3 . Then all motions of 5 are governed by the equations Fr F 0 r 1, . . . , ri 1 These equations are called Kane's dynamical equations. Derivation Regard 5 as composed of v...