ME 232 Classroom Problems

Three Dimensional Dynamics of Rigid Bodies

Governing Equations: S F = dG/dt S M = dH/dt

In an earlier lesson we commented on Newton's knowledge of sports and his observation that if linear momentum was to be changed, then a net external force must be applied. This was stated as the following rule for any mechanical system:

S F = dG/dt|_I
I = Fixed or inertial coordinate system
G = Linear momentum of the system relative to the fixed coordinate system I.
dG/dt|_I = Time derivative with respect to the fixed coordinate system I of the linear momentum G of the system.
S F = Net sum of all external forces acting on the mechanical system

The above rule has been stated for mechanical systems which are closed systems that do not exchange mass with their surroundings. It is important to note that a similar rule can be stated for open systems that do exchange mass with their surroundings (fluids, thermodynamics). The utility of this rule in the analysis of the motion of mechanical systems cannot be overstated. Virtually everything we understand about how and why things move begins with the above rule! This rule is given in our textbook as Equation 4/6 on page 275 and is discussed in some detail. Don't ever forget this rule!

However we have learned that there is more to life than linear momentum. There is spin, rotation, and angular momentum. Does there exist an analogous formula for rotational motion? Fortunately yes! Moments are to angular momentum as forces are to linear momentum. The rule in its angular form can be stated in two ways. The time derivative with respect to an inertial coordinate system of the angular momentum about a fixed point relative to the inertial coordinate system of a mechanical system is equal to the net sum of the external moments about that fixed point (another mouthful). Alternatively the rule can be stated using the system mass center. The time derivative with respect to an inertial coordinate system of the angular momentum about the mass center relative to the inertia coordinate system of a mechanical system is equal to the net sum of the external moments about the system mass center. Note that this second application of the rule may well involve a moving reference point, the system mass center. Thus for any mechanical system (closed, no mass exchange with its surroundings) these verbose rules may be stated more compactly mathematically as the following equations:

S M_O = dH_O/dt|_I
I = Fixed or inertial coordinate system.
O = Point fixed in the fixed coordinate system I.
H_O= Angular momentum of the mechanical system about the fixed point O relative to the fixed coordinate system I.
dH_O/dt|_I = Time derivative with respect to the fixed coordinate system I of the aforementioned angular momentum vector.
S M_O = Net sum of external moments about the fixed point O applied to the mechanical system.

S M_G = dH_G/dt|_I
I = Fixed or inertial coordinate system.
G = Mass center of the mechanical system.
H_G= Angular momentum of the mechanical system about its mass center G relative to the fixed coordinate system I.
dH_G/dt|_I = Time derivative with respect to the fixed coordinate system I of the aforementioned angular momentum vector.
S M_G = Net sum of external moments about the mass center G applied to the mechanical system.

These rules are exceptionally useful in the analysis of systems exhibiting rotational motion. They fit perfectly with our beloved four step procedure. At step 3a, we need to evaluate the sum of the external forces acting on the system and the sum of the external moments acting on the system about either the system mass center or about a fixed point. At step 3b, we need to evaluate the time derivative of the linear momentum of the system and the time derivative of the angular momentum of the system (about either the system mass center or the fixed point as appropriate). At step 4, we equate the results of step 3a to those of step 3b and try to solve. This may be no small task as it involves solving equations that include derivatives of vectors. However we are ready for the challenge. To the battle we take the four step procedure and our knowledge of forces and moments. Victory shall be ours! Extensions exist for open system exchanging mass with their surroundings. These rules are given as Equations 4/7 and 4/9 on pages 276 and 277 of our text and are discussed in some detail. They are also merit commitment to your long term memory. Never forget what it takes to stop spinning!

A subtle issue regarding coordinate systems arises in the use of the equations stated above. These equations involve derivatives of vectors. It is mandated that these derivatives be those derivatives that would be obtained by an observer in a fixed or inertial coordinate system. At first this seems to be a stringent restriction. However our second fundamental formula provides a powerful tool for relating the derivatives of different observers. Thus if we calculate the derivatives in a rotating system but add the appropriate compensating terms, we can still obtain the required derivatives. Thus although the derivatives must be those obtained relative to an inertial coordinate system, the unit vectors used in expressing the various vectors need not be fixed in an inertial system. It is perfectly acceptable to express the forces, moments, linear momentum, or angular momentum relative to a rotating coordinate system. As long as one appropriately evaluates the momentum derivatives, unit vectors that are themselves rotating are perfectly acceptable. It is frequently the case that a solution to a problem can be practically obtained only by using rotating coordinate systems.

Recalling that we have developed "simple" expressions for the angular momentum of rigid bodies, the above governing equations provide us with powerful tools for analyzing the rotations of rigid bodies. From frisbees to universal joints, let them spin. We're ready!

The equations are beautiful in their simplicity. But engineers need do more than appreciate beauty. We need to be able to understand how things work. To investigate an application of these formulas to understanding how things work, click here!