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5.2 Dynamics of Rigid Body Motion

With the kinematics all sorted out, its time to put them to work. As in the case of particle motion, our goal will be to develop techniques that permit us, given details about the forces and moments acting on a rigid body, to predict its motion over a time interval.

5.2.1 Overview

Returning to the kinematics for a moment, what quantities do we need to determine from the force system to predict the motion of a rigid body? Recall that the translational velocity  and acceleration of reference B and the angular velocity and acceleration of the body determine the motion of an arbitrary point A of a rigid body:



Further, observe that velocities  and  can be determined by integrating accelerations  and  as,



Therefore, all we need to determine from the force system are the two accelerations: translational acceleration  and angular acceleration .

 
Figure 5.2.1: Determination of the motion of a rigid body from the active force system.

As we will discover shortly, owing to the decomposition of rigid motion into a translation and a rotation, this is a relatively simple task. With the center of mass C of the body taken as reference, the dynamics of a rigid body reduces to the independent analysis of the two components of the motion:

• Translational motion: Acceleration  from resultant force .
• Rotational motion: Angular acceleration  from resultant moment  about C.

You will be pleased to know that we have already encountered the tools needed to analyze the translational motion of the body. Recall from the consideration of systems of particles that the system responded to external forces as a single particle located at the center of mass of the system. As a rigid body is nothing but a system of particles constrained to remain equidistant, the acceleration  is related to the resultant external force applied to the body in a simple way:



where  is the linear momentum of the body.


Figure 5.2.2: Equivalence of the rigid body to a particle located at the center of mass.

A new physical principle, however, is needed for the analysis of the rotational motion. We have already had a glimpse of this when angular momentum principles for particles were considered. Recall that the moment of forces acting on a particle equaled the rate of change of its angular momentum. As will be seen, the rotational motion of a rigid body is determined by such an angular momentum principle:



where  is the moment of inertia of the rigid body about C and  is the angular momentum of the rigid body about C.

 
Figure 5.2.3: Determination of the rotational motion of the body by the resultant moment.

Observe that the two equations above are all that is needed to fix the three quantities describing motion: two components of the planar acceleration , and the scalar angular acceleration . Therefore, analyzing the dynamics of a rigid body is simply a matter of solving two force equations for the translational motion,



and a single moment equation for the rotational motion,



In the course of this discussion we have encountered analogies between translational and rotational quantities that will make the analyses to follow much easier: , , and  in rotational motion are akin to ,  and , respectively, in translational motion. This relationship between quantities related to the two components of rigid body motion will be seen to carry all the way through considerations of momentum and energy as well:

 

5.2.2 Preliminaries: Inertial Properties of Rigid Bodies

Before proceeding to the details of rigid body dynamics, we need to become familiar with techniques to determine certain inertial properties of a rigid body: the location of its center of mass  and its moment of inertia .

 
Figure 5.2.4: A planar rigid body and its idealization as a collection of particles.

To help motivate the integral expressions we will encounter, we will also provide analogous expressions for the body idealized as a collection of discrete particles. For instance, while the total mass of a collection of particles is computed as a discrete sum of individual particle masses ,



the total mass of a rigid body is the sum of elemental masses dm comprising the body, determined by an integral over the body:

 

5.2.2.1 Center of Mass

An expression for the center of mass of a rigid body is obtained by replacing the summation encountered previously for the center of mass of a collection of particles by an equivalent integral:

	Particles	Rigid bodies
Center of mass:

Alternately, in components :



Taking the rigid body to be a plate of thickness t with uniform density , substituting into the integral above,



where A () is the total cross-sectional area of the body in the X-Y plane. Therefore, the center of mass of a homogeneous planar rigid body is simply the centroid of its cross-sectional area,



 
Figure 5.2.5: Center of mass of a plate of thickness h.  

5.2.2.2 Moments of Inertia about an Axis

As we will see in the following, a rigid body will persist in a steady rotation about a fixed axis unless acted upon by an external moment. The moment of inertia of a body  about axis O is the measure of its resistance to angular acceleration about the axis due to a moment,



just as mass m is a measure of resistance to translational acceleration due to a force,




Figure 5.2.6: A plate rotating about an axis through O.

Expressions for the moment of inertia for a system of particles and a rigid body are simply:

	Particles	Rigid bodies
Moment of inertia:

where  and  are distances of particle  and elemental mass , respectively, from the axis of rotation O. Due to this dependence on distance from the axis of rotation, the moment of inertia of a body varies with the location and direction of the axis. For this reason, it is critical to define what axis of rotation the moment of inertia refers to. Using a subscript O to identify the rotation axis does this above. In the case of the planar motion considered here, as the axis must be normal to the plane of motion, location O of the axis defines it completely. If instead three-dimensional motion were considered, the direction of the axis would need to be prescribed as well.


Figure 5.2.7: The moment of inertia of a body varies with the rotation axis ().

While the term inertia in moment of inertia is derived from its representing a resistance to change, moment arises from the fact that in the definition of  the mass dm of each elemental particle is weighted by the square of its distance from the axis. This weighting makes rotational inertia depend not just on the mass of the body but on the distribution of the mass about the axis of rotation as well.

Again, specializing for a homogeneous rigid body of density  and thickness t we have,



Observe that the quantity,



is the area moment of inertia of the body. Therefore, the moment of inertia of a homogeneous body follows simply from its area moment of inertia.
 

5.2.2.3 Radius of Gyration


Figure 5.2.8: The radius of gyration of a body about an axis.

The radius of gyration of the body about axis O is defined as the quantity:

   

or 



Observe that the second expression suggests that the moment of inertia of a body about an axis equals that of a particle of mass m located at distance  from the axis. In this way,  is a measure of the distribution of mass of the body about the axis. For the same mass, bodies with a greater radius of gyration have mass located further away from the axis.
 

5.2.2.4 Parallel-Axis Theorem


Figure 5.2.9: Schematic demonstrating the parallel-axis theorem.

We have seen that the moment of inertia of a body varies with its axis of rotation. Using the parallel-axis theorem, the moment of inertia of a body about any axis can be determined in a simple way from that associated with a parallel axis passing through the center of mass C. In particular, the moment of inertia  about an axis through an arbitrary point O is related to the moment of inertia  about a parallel axis through the center of mass as,



where d is the distance between the two axes measured in the plane of rotation. Alternately, introducing expressions for the radius of gyration,



Observe from the expressions that the moment of inertia is always smaller for an axis through the center of mass than about any other parallel axis.

 

5.2.3 Equations of Motion for Systems of Particles

Recall that a rigid body is simply a collection of particles constrained to maintain a constant separation during motions of the body. In what follows, we will exploit this property to derive expressions describing the dynamics of rigid bodies by first examining the dynamics of a collection of particles.


Figure 5.2.10: A system of two particles  and .

To keep the algebra simple, we consider a system of two particles  and  with masses  and , respectively, as shown above. Let  be the resultant of all external forces applied to particle  and  be the internal force exerted by particle  on particle . Similarly, let  be the resultant external force applied to  and  be the internal force exerted by  on .

Then, applying Newton's law to each of the particles yields relations for the particle accelerations:



We proceed to derive an expression for the balance of linear momentum of the system by summing the two relations:



Similarly, to derive an expression for the balance of angular momentum of the system, we take cross products of the two equations with  and , respectively, and sum,



To simplify the terms on the left-hand side, let  be the resultant external force applied on the system, and  be the resultant external moment about acting on the system. Further, by Newton's third law, internal forces  and  must be equal and opposite and have the same line of action,



Applying this to the left-hand side of the second equation,



as the vector  is collinear with force .

We turn next to relate the expressions on the right hand side to the linear momentum  of the system,



and the angular momentum  of the system about ,



Accordingly, differentiate to obtain,



and,



Substituting, we obtain simplified expression for the balance of linear and angular momentum of the system:





or, simply that the rate of change of linear momentum of the system equals the resultant external force applied to the system, and the rate of change of angular momentum about O of the system equals the resultant moment about O applied to the system.

The structure of the two balance relations is simplified considerably when they are expressed relative to the center of mass C of the system. Recall that the position  of the center of mass is the weighted sum of the particle masses,



We differentiate this relationship to obtain a simple expression for the linear momentum of the system,



where  is the velocity of the center of mass. Repeating the differentiation once more and substituting the result into the balance of linear momentum of the system yields a relationship we have encountered previously,



where  is the acceleration of the center of mass of the system. Thus, the balance of linear momentum of a system of particles has a form identical to Newton's law except mass m is the total mass of the system of particles,  is the resultant of all external forces applied to the system, and  is the acceleration of the center of mass of the system. In effect, the center of mass of the system moves under the external forces as if all mass was concentrated at C and all forces were applied at C.

An analogous expression for the balance of angular momentum relative to the center of mass follows by observing that the angular momentum of the system about O can be expressed in terms of the angular momentum relative to C as,


Figure 5.2.11: Schematic demonstrating the relationship between angular momenta about O and C.



with,



where  is the position of particle 1 relative to C, and  is the velocity of particle 1 relative to C.

 
Figure 5.2.12: Describing particle positions relative to the center of mass.

Differentiating, and substituting into the expression for the balance of angular momentum derived previously, we obtain,



Taking the reference to coincide with the center of mass ( or ), the first term on the right hand side drops out yielding an alternate expression for the balance of angular momentum,



where  is the resultant external moment about C and  is the angular momentum of the system relative to C.


Figure 5.2.13: Dynamics of a system of particles reduced to linear momentum balance at C, and angular momentum balance about fixed point O or C.

 

5.2.4 Equations of Motion for Rigid Bodies

Returning now to our main focus, we proceed to mold the balance relations for systems of particles into equations of motion for rigid bodies. Recall that rigid bodies are systems of particles with particles restricted to move in a way that ensures inter-particle distances remain unchanged. Therefore, all we need to do in what follows is to specialize the balance relations for systems of particles by introducing rigid body kinematics to restrict the motion of individual particles.

As outlined previously, our goal is to determine the linear acceleration of the center of mass  and the angular acceleration  of the body from the external force system. Fortunately, half of this task has already been accomplished as the balance of linear momentum for systems of particles applies unchanged to rigid bodies:



On the other hand, molding the expressions for the balance of angular momentum for systems of particles into a useful form needs additional work. As this will lead us into somewhat unfamiliar territory, we will first examine the simpler situation of fixed axis rotation. This will amount to considering the balance of angular momentum with the axis of rotation taken as reference O:



Following this we will proceed to the increased clutter of general plane motion. There the balance of angular momentum about the center of mass C for the angular component of the motion,



will supplement the balance of linear momentum for the translational component of the motion.
 

5.2.4.1 Fixed Axis Rotation

Proceeding then, recall that as the kinematics of fixed axis rotation are described by angular quantities  and  alone,



all we need to describe the dynamics of a rotating rigid body is a relationship between the external force system and angular acceleration .


Figure 5.2.14: A rigid body rotating about a fixed axis at O.

We do this by considering a rigid body rotating about a fixed axis through O with angular velocity  and angular acceleration . The body is assumed to be comprised of a collection of particles  at distances  away from the axis. By the balance of angular momentum about axis O,



where  is the external moment about O applied to the body and  is the angular momentum of the body about O. Introducing the kinematics of fixed axis rotation into the expression for angular momentum,



Observe that the summation  is simply the moment of inertia  of the body about O,



Consequently, we obtain an expression for the angular momentum of the body,



or the angular momentum of a rigid body is simply the product of the moment of inertia with the angular velocity. Another analogy between translational and rotational quantities becomes apparent when this is compared with the expression for linear momentum:

Both linear and angular momentum are simply products of the appropriate inertia with a velocity.

Differentiating, and substituting into the balance of angular momentum yields the desired relationship between the external force system and the angular acceleration of the body,



or the external moment about O equals the product of the moment of inertia with the angular acceleration. Again, by recalling the balance of linear momentum, we observe that the external forces (or moments) required to drive either a translational or rotational motion are simply the product of the appropriate inertia with the acceleration:

Figure 5.2.15: Reduction of the dynamics of fixed axis rotation to a relationship linking the resultant external moment about the axis to the angular acceleration of the body.  

With that, we have all the tools necessary to analyze the dynamics of rigid bodies rotating about a fixed axis. As problems can sometimes be a little tricky, the following sequence of steps is recommended:

• Kinematics: Identify the axis of rotation. Select a fixed reference frame with origin located at the axis.
• Free Body Diagram: Next, draw a free body diagram of the body and label all known and unknown quantities.
• Equations of Motion: Finally, apply the equations of motion, taking care to be consistent about the convention adopted for rotations.

 

For the rotation of a rigid body about axis O:  where is the external moment about O,  is the moment of inertia about O and  is the angular acceleration.

 

5.2.4.2 General Plane Motion

Recall that unlike fixed axis rotation, a description of general plane motion requires both, the translational motion of a reference, and the rotational motion of the body:



Accordingly, in the following, we set out to derive expressions relating the linear acceleration  and angular acceleration  to the force system acting on the body.


Figure 5.2.16: A rigid body with center of mass C undergoing general plane motion.

Consider a rigid body moving in the plane with angular velocity  and angular acceleration . The body is assumed to be composed of a collection of particles  located at positions  relative to the fixed reference at O. In the following, we will observe the motion of the particles relative to a translating reference attached to the center of mass C. Accordingly, for particle  we have,



where  is the position of C relative to the fixed frame at O, and  is the position of the particle relative to C.

The balance of linear momentum for a system of particles provides the required relationship linking the force system to the acceleration  of the center if mass,



where  is the resultant external force exerted on the body. To derive an expression for the angular acceleration of the body, we turn to the balance of angular momentum about C,



where  is the external moment about C applied to the body and  is the angular momentum of the body relative to C. Introducing the kinematics of general plane motion into the expression for angular momentum,



Recall that the summation  is the moment of inertia of the body about C,



Substituting, yields a simple expression for the angular momentum of the body about C,



The required relationship between the force system and the angular acceleration of the body follows by introducing this expression for angular momentum into the balance relation above,



or the external moment about C equals the product of the moment of inertia with the angular acceleration.

In this way, we have the expressions necessary to analyze the dynamics of general plane motion: linear momentum balance for the acceleration of the center of mass;



and angular momentum balance for the angular acceleration of the body,





Figure 5.2.17: Reduction of the dynamics of general plane motion to relationships linking external forces to the acceleration of the center of mass, and external moments about the center of mass to the angular acceleration of the body.

As the analysis of general plane motion is often complex, it is sensible to approach problems systematically. The following sequence of steps is recommended:

• Kinematics: First, identify the class of motion and solve for any linear and angular accelerations that can be determined solely from kinematical information provided. Determine if the motion is constrained or unconstrained.
• Free Body Diagram: Next, draw a free body diagram of the body to be analyzed. Assign a convenient fixed reference frame and label all known and unknown quantities.
• Equations of Motion: Finally, apply the equations of motion, taking care to be consistent about the convention adopted for rotations. If the motion is constrained, the equations should be supplemented by the kinematical relationships determined previously. Count the unknowns remaining to ensure you have enough equations.