Newtonian Dynamics

A clear and full understanding of Newtonian dynamics is essential to understanding of advanced techniques such as Lagrangian analysis. Furthermore, the Newtonian approach is attractive in many situations because it automatically yields important information not readily available from other approaches, such as internal constraint forces and moments. In this section we present the essence of the Newtonian approach and illustrate it using the SPENDULAP.

Description of Newtonian formulation. The first step in obtaining the equations of motion for a multi-body mechanical system is to draw a free body diagram for each body, including external (applied) and internal (constraint) forces and torques as shown in Figure 4.


Figure 4. Free body diagram of a rigid body.

Using Figure 4, three-dimensional force equations are given by F = m aC, where F = Fxi + Fyj + Fzk is a vector of external and internal forces applied to the body, m is the mass of the body, and aC = aCx i + aCy j + aCz k is the acceleration of the center of mass obtained from a kinematic analysis. Note that the vector F is expressed in terms of the unit vectors of the moving reference frame xyz, and the acceleration is an absolute quantity measured relative to the inertial reference frame XYZ but expressed in terms of the unit vectors of the moving reference frame xyz.

Similarly, using Figure 4, three-dimensional moment equations are given by , where is the vector of external and internal moments acting on the body, is the angular velocity of the body, and is the angular momentum vector, the components of which are given by

where

is the inertia tensor. Also, , where

In the particular case where the axes of the moving reference frame xyz coincide with the principal axes of the rigid body, the inertia tensor elements are simplified as follows: Ixx = Ix, Iyy = Iy, Izz = Iz, and Ixy = Iyz = Ixz = 0.

In summary, and using the previous derivations, the translational and rotational equations of motion can be obtained for each rigid body as follows:

(9a)

(9b)

(9c)

(10a)

(10b)

(10c)

Newtonian SPENDULAP Formulation. In the case of the SPENDULAP, a free body diagram of the pendulum is shown in Figure 5. For this case (kinematics case ii), frame xyz constitutes principal axes of the pendulum and,

(11)

and

(12)

From (1), (2), and (10a)

(13)


Figure 5. Free body diagram of SPENDULAP pendulum.

The moment Mx is due to the pendulum rod and bob weight, therefore

(14)

where mr is the rod mass and mb is the pendulum bob mass. Combining (13) and (14), and noting (11), the pendulum angular acceleraction can be written

(15)

Next, from (1), (2), and (10b),

(16)

and from (10c), (11), and (12)

(17)

From (11), (16), and (17), the moment about the frame rotation axis is

so

(18)

Also,

(19)

where IFZ is the rotating frame moment of inertia about the rotation axis. Using similar reasoning,

Equating (18) and (19), the second equation of motion is

(20)

The equations of motion [(15) and (20)] can be rewritten as

(21)

An important advantage of using the Newtonian approach is that it yields the constraint forces as a by-product of the equations of motion. For the spherical pendulum system, we can easily compute the constraint forces using (3), (9a), (9b), and (9c). From (9a), (9b), and (9c), the components of the forces acting on the pendulum rod at O in the xyz frame are


In the XYZ frame,
 FX = Fx,
and
We can also compute the reactions at the ends of the swivel pin (A and B). The reactions at A and B in the X direction depend on the construction of the bearings and the frame deflection. We assume FAX = FBX. To compute the reactions in the Y direction, we solve the linear system

(22)

Similarly, we compute the reactions in the Z direction by solving the linear system

(23)

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