Dynamic Equations of Motion and the Many Faces of DAlemberts Principle

Charles R. Thomas, Thomas E. Kirk, and R. Mark French Mechanical Engineering Technology, Purdue University

Knoy Hall of Technology, 401 N. Grant Street West Lafayette, IN 47907-2021

crthomas@tech.purdue.edu

ABSTRACT Plane motion kinetics frequently involves jargon terms such as equations of motion, dynamic equilibrium, free body diagrams, DAlemberts principle, kinetic diagram, mass-acceleration diagram (MAD), inertial response diagram (IRD), effective forces, reverse effective forces, inertia force, and inertia moment. Confusion is frequently due to the existence of three distinct solution methodologies in kinetics which may be applied at any one of four points. The equations of motion approach equates a diagram on the left of an equal sign set to a mathematical term on the right. The kinetic diagram method generates an equation by setting a diagram on the left equated to or equivalent to a diagram on the right. The dynamic equilibrium approach uses a diagram on the left set equal to zero on the right. The application of each method to specific problems yields the same results. A slender rod pinned at one end is used to illustrate each of the four moment summation points for each of the three methods. The two basic forms of DAlemberts principle are cited to add some depth and understanding to this fundamentally important principle. Some history of coverage and position shifts regarding DAlemberts principle is discussed. The seeming cycle from visual scalar kinetics to mathematical vector kinetics and back is explored. NOMENCLATURE a The vertical component of linear acceleration of the center of gravity

Aa The vertical or tangential component of linear acceleration of the arbitrary point A

na The normal component of acceleration of the center of gravity

ta The tangential component of linear acceleration of the center of gravity

xa The horizontal component of acceleration of the center of gravity

ya The vertical component of linear acceleration of the center of gravity A Arbitrary point b Rod length d The scalar portion (required perpendicular distance) of the position vector from point A to point G d The magnitude of the perpendicular distance between the new axis and the center of gravity axis

IF Inertia force

AIF Inertia force at arbitrary point A

OIF Inertia force at pin O g Gravity acceleration G The center of gravity

AI The mass moment of inertia for the arbitrary point A

GI The mass moment of inertia at the center of gravity

OI The mass moment of inertia for the fixed pivot, point O

m Mass IM Inertia moment

AIM Inertia moment at arbitrary point A

OIM Inertia moment at pin O O Fixed pivot point r Radius to the center of gravity Ar Radius to the arbitrary point A

XR The horizontal reaction at pin O

YR The vertical reaction at pin O W Rod weight x Horizontal axis y Vertical axis Angular acceleration Angular velocity

F Force summation x

F Horizontal force summation y F Vertical force summation

( )Y effF Effective vertical force summation AM Moment summation about an arbitrary point, point A GM Moment summation about the center of gravity, point G OM Moment summation about a fixed point, point O

( )A effM Effective moment summation about the arbitrary point, point A ( )G effM Effective moment summation about the center of gravity, G ( )O effM Effective moment summation about the fixed pivot, O INTRODUCTION A random look at plane motion kinetics in typical recent and classic first course engineering dynamics textbooks might reveal a diverse terminology such as equations of motion, dynamic equilibrium, DAlemberts principle, free-body-diagram equations, kinetic diagram, mass-acceleration diagram (MAD), inertial response diagram (IRD), effective forces, reverse effective forces, inertia force, and inertia moment. This rich collection of terminology spans a collection of three distinct, and sometimes confusing, solution methodologies in kinetics. Each solution methodology may be applied at any one of several point categories such as the center of gravity, the axis of rotation, or an arbitrary point with further strategy variations. Each methodology involves a unique mix of diagrams, procedures, and equations involved in the solution of dynamic motion problems. The three basic methods are visually summarized in Figure 1.

(a) Equations of Motion

(b) Free-Body-Diagram Equations

(c) Dynamic Equilibrium Figure 1 Schematic comparison of the three different motion equations approaches In Figure 1a, the equations of motion approach shows a diagram on the left set equal to mathematical terms on the right. The diagram on the left is a free body diagram (FBD) and the terms on the left sides of the equations are actually formulated from the external forces and external moments shown on the diagram. The mathematics on the right side of the equations must be available from previously derived results for specific points of moment summation based on points such as the center of gravity, a fixed pivot, an arbitrary point, and an arbitrary point with mass moment of inertia specifically evaluated at the arbitrary point. In Figure 1b, the free-body-diagram equations approach, with included kinetic diagram, shows a diagram on the left equated to or equivalent to a diagram on the right. The diagram on the left is a free body diagram (FBD) and the left sides of the resulting equations are again actually formulated from the external forces and external moments shown on the diagram. The diagram on the right is a kinetic diagram and the right sides of the equations are formulated from the kinetic terms shown on the diagram. All terms on both the left and right sides of the generated equations are drawn directly from the actual diagrams, there is no dependence on any previously derived mathematical terms in the development of the final underlying equations. In Figure 1c, the dynamic equilibrium approach shows a diagram on the left set equal to zero on the right. The terms on the left sides of the equations are actually formulated from the external forces, external moments, inertia forces, and inertia moments shown on the diagram. The right sides of the equations are set equal to zero. A single problem consisting of a slender rod pinned at one end is used to illustrate each of the four points, with appropriate additional variations, outlined above for each of the three methods outlined in Figure 1. Thus all cases discussed are comparable from the viewpoints of similarities, differences, and individual nuances. A variety of sources for statements of the two basic forms of DAlemberts principle are cited to add some depth and understanding to this fundamentally important principle. Appropriate terminology is introduced and defined. Some history of coverage and position shifts regarding DAlemberts principle will be traced through textbooks of great longevity. The driving factors in the shift from scalar dynamics to vector dynamics are discussed in terms of the Grinter report [1,2] and the advent of the 1957 Sputnik. The seeming cycle of visual scalar kinetics to mathematical vector kinetics with a current return to scalar visual kinetics is explored. TERMINOLOGY AND VARIABLES For the current focus, the comparison of methods at a variety of moment summation points, scalar equations for plane motion are exclusively considered to accentuate basic concepts and to promote visualization. All cases and strategies considered are demonstrated in terms of a slender rod pinned at one end. The slender rod is inherently simple but it still allows for a full discussion of all the differences, similarities, and nuances between and among the three approaches and the many variations of each approach. The rod is held horizontally as show in the Figure 2 below and is released from rest so that the initial angular velocity is zero, 0. =

Figure 2 The slender rod Points of interest on the rod include the center of gravity, point G , the fixed pivot point, point O , and the arbitrary point, point A . The rod has a length b 5 feet= and a weight W mg 40 pounds= = where m is the mass and 2g 32.2 ft / sec= . The angular acceleration is , the radius to the center of gravity is ( )r b / 2= , and the radius to the arbitrary point A is Ar b= . The horizontal or normal component acceleration of the center of gravity is ( )2 2x na a r b / 2 0 0= = = = , the vertical or tangential component of linear acceleration of the center of gravity is ( ) ( )y ta a a r b / 2 5 / 2= = = = = , and the vertical or tangential component of linear acceleration of the arbitrary point A is A Aa r b= = . Additional terminology includes the horizontal and vertical reactions XR and YR at pin O , inertia force IF , and inertia moment IM . An important distance, d , is the distance measured from the arbitrary point, point A , to the center of gravity, point G . This distance d may be positive or negative according to the coordinate axis or sign convention in use. Thus, the assumed sign convention must be used to determine if d is positive or negative. Distance d is actually the perpendicular moment arm measured from arbitrary point A to the vector ma which acts through the center of gravity. Distance d is the necessary scalar portion (required perpendicular distance) of the position vector from point A to point G . Several mass moments of inertia are frequently used. The mass moment of inertia at the center of gravity, point G , is

( ) 2 2 2G 1 40I 1/12 m b 5 2.588ft lb sec12 32.2 = = = . (1)

Using the parallel axis theorem, the mass moment of inertia for the fixed pivot, point O , is

( ) 22 2 2O G G 5 40I I d m I b / 2 m 2.588 10.35ft lb sec2 32.2 = + = + = + = , (2)

Where d is the magnitude of the perpendicular distance between the new axis and the center of gravity axis. In the current case, d is the magnitude of the distance between the center of gravity, point G , and the fixed axis, point O . Using the parallel axis theorem, the mass moment of inertia for the arbitrary point, point A , is

( ) 22 2 2A G G 5 40I I d m I b / 2 m 2.588 10.35ft lb sec2 32.2 = + = + = + = , (3)

where in this case, d is the magnitude of the distance between the center of gravity , point G , and the arbitrary point, point A . EQUATIONS OF MOTION APPROACH The equation of motion approach is equation driven; that is, the user must decide up front on the point at which the equation and moment summation are to be applied and then utilize a set of equations appropriate for that point. Thus a new and distinct equation is required for each point selected for moment summation. In 2005 Tongue and Sheppard [3] determined a set of four equations based on three points. They have detailed equations for moment summation about the center of gravity, about a fixed point, and about an arbitrary point, with two variations of this last case. In each of the four cases, the equation F = ma (4) is valid. The differences occur in the four derived , different moment equations. For moment summation about the center of gravity, point G , the moment equation is G GM = I (5) For moment summation about a fixed point, point O , the moment equation is O OM = I (6) For moment summation about an arbitrary point, point A , the moment equation is A GM I mad= + (7) For moment summation about an arbitrary point, point A , and in terms of AI , the moment equation is A AAM I ma d= + (8) The equations of motion approach is an equation based approach, the left side of these equations is developed using a FBD of the external applied forces and moments only. There is no diagram for the right side of the equations and there is no interpretation of the terms on the right side. The terms are not forces or moments. It is important to note that the linear and angular accelerations are generally directed in the positive direction according to the previously defined coordinate axis choice or the associated sign convention shown in Figure 2. These acceleration directions determine the directions of the right side equation terms. The equations of motion developed about the center of gravity, point G , may be expressed as F = ma (9) G GM = I (10) Figure 3 shows the appropriate FBD for the slender rod of Figure 2.

Figure 3 Equations of motion Free Body Diagram for a center of gravity ( G ) moment summation Performing force summation in component form, the horizontal force summation is xxF = ma = m(0) = 0 (11) This result will be true for all the future cases to be considered and hence will not be carried into those considerations. The vertical acceleration is

ba = r = 2

(12)

The vertical force summation y F = ma (13) results, after using the FBD to supply the left side of this equation, in

ybR - mg = m 2

(14)

The standard assumption for signs for the right hand side terms of the equations is that linear and angular accelerations and hence the terms based on these accelerations are assumed to be in the positive coordinate or sign convention directions with the right hand rule applying to the angular acceleration direction. This is, of course, consistent with the sign convention already chosen in Figure 3 for the left hand side of these equations. Plugging in the previously defined numerical values results in

y40 5R 40

32.2 2 = (15)

which is solved for yR as yR 40 3.106= + . (16) The moment equation for a summation about the center of gravity is G GM = I (17)

and using the FBD to supply the left side of this equation results in

y Gb- R = I 2

(18)

with an insertion of numerical values giving

y5- R = 2.5882

(19)

which with the force summation result yields two equations in two unknowns. A direct solution of these equations gives the numerical results 2 = - 9.66rad/sec and yR = 10 Lb. The equations of motion developed about the fixed pivot, point O , may be expressed as F = ma (20) O OM = I (21) Figure 4 shows the appropriate FBD for the slender rod of Figure 2.

Figure 4 Equations of motion Free Body Diagram for a fixed pivot ( O ) moment summation

The vertical force summation y F = ma (22) results, after using the FBD to supply the left side of this equation, in

ybR - mg = m 2

(23)

The moment equation for a summation about the pivot point O is O OM = I (24)

and using the FBD to supply the left side of this equation results in

Ob- mg I2

= (25) where it is already known that 2OI 10.35ft lb sec= which with the force and moment summation result yields two equations. A direct solution of these equation gives the numerical results 2 = - 9.66rad/sec and yR = 10 Lb. The equations of motion developed about an arbitrary point, point A , may be expressed as F = ma (26) A GM I mad= + (27) Figure 5 shows the appropriate FBD for the slender rod of Figure 2.

Figure 5 Equations of motion Free Body Diagram for an arbitrary point ( A ) moment summation using GI

The vertical force summation y F = ma (28) results, after using the FBD to supply the left side of this equation, in

ybR - mg = m 2

(29)

The moment equation for a summation about the arbitrary point, point A , is A GM I mad= + (30) and using the FBD to supply the left side of this equation results in

y Gb b bbR + mg I m2 2 2

= + (31) which with the force summation result yields two equations. A direct solution of these equations gives the numerical results 2 = - 9.66rad/sec and yR = 10 Lb. The equations of motion developed about an arbitrary point, point A , and involving the mass moment of inertia

AI may be expressed as F = ma (32) A AAM I ma d= + (33) Figure 6 shows the appropriate FBD for the slender rod of Figure 2.

Figure 6 Equations of motion Free Body Diagram for an arbitrary point ( A ) moment summation using AI

The vertical force summation y F = ma (34) results, after using the FBD to supply the left side of this equation, in

ybR - mg = m 2

(35)

The moment equation for a summation about the arbitrary point, point A , using AI is A AAM I ma d= + (36) and using the FBD to supply the left side of this equation results in

y Ab bbR + mg I m(b2 2

= + ) (37)

which with the force summation result yields two equations. A direct solution of these equations gives the numerical results 2 = - 9.66rad/sec and yR = 10 Lb. In addition to the sources discussed elsewhere, illustrations of applying the equations of motion are also discussed by Soutas-Little and Inman [4], Riley and Sturges [5], Sandor and Richter [6], Ginsberg and Genin [7], and McGill and King [8]. Some of the listed references do not include or cover all four of the moment summation points detailed by Tongue and Sheppard [3]. DALEMBERTS PRINCIPLE DAlemberts principle has been the subject of controversy for a very long time. Some detractors of the principle seem to react negatively because such action has been fashionable at the time. The applicability of the DAlembert principle to variational mechanics has led to a variety of statements by some strong champions and advocates of the principle. In 1970 Meirovitch [9]stated that The principle of virtual work is concerned with the static equilibrium of systems and by itself is not suitable for use in problems of dynamics. Its usefulness can be extended to dynamics, however, by means of a principle attributed to DAlembert. Although there is a difference of opinion about the form of the principle, there is general agreement about its far-reaching consequences. It turns out that interest lies primarily in these consequences. He further states that DAlemberts principle,

N

i i ii 1

(F p ) r 0=

= & ( i i ip m r= & ), (38) represents the most general formulation of the problems of dynamics, and all the various principles of mechanics, including Hamiltons principle, are derived from it. In 1977 Rosenberg [10]stated This book is aimed at a treatment of Lagrangean mechanics and that branch of mechanics could not exist without the central concept of the virtual displacement and the virtual velocity. Moreover, the transition from the Newtonian to the Lagrangean points of view could not be made without the much maligned and often misunderstood principle of dAlembert, nor without a careful and unambiguous classification of forces as internal or external, and as given or constrained forces. He further stated that While Lagrange was not the discoverer of the concept of virtual displacements, we owe to him the bold step of lifting it from the domain of statics and introducing it into dynamics by utilizing dAlemberts principle. Once this was accomplished, the general theory of constraints and that of generalized coordinates followed in a natural way. Rosenberg further stated that In the literature, the first form (of the fundamental equation) is often called Lagranges form of dAlemberts principle. However, regardless of the name attached to

N

s s s ss 1

(m u F ) u 0=

= && , (39) that equation has not only far-reaching consequences, but it may be regarded as occupying a central position in the theory of classical dynamics because all known principles of mechanics can be derived from it. In 1966 Lanczos [11] stated The force of inertia. With a stroke of genius the eminent French mathematician and philosopher dAlembert (1717-1783) succeeded in extending the applicability of the principle of virtual work from statics to dynamics. He further stated that DAlemberts principle introduces a new force, the force of inertia, defined as the negative of the product of mass times acceleration. If this force is added to the impressed forces we have equilibrium, which means that the principle of virtual work is satisfied. The principle of virtual work is thus extended from the realm of statics to the realm of dynamics. While there are two alternate forms of DAlemberts principle, the historically most widely know form is the second form or the dynamic equilibrium form. The dynamic equilibrium form is used in variational mechanics. It looks just

like static equilibrium with just a few new, extra terms. It historically allowed for the application of existing static graphical techniques. While there has been some historical usage of the first form; a recently introduced usage methodology has made it exceedingly popular, although some of the modern users do not realize that they are applying and working with DAlemberts principle. In 1914, Maurer [12] gave a solid but brief overview of D Alemberts Principle as follows: D Alemberts Principle, not heretofore discussed. The resultant of all the forces acting on any particle of a body is called the effective force for that particle. Its magnitude equals the product of the mass and acceleration of the particle; its direction is the same as that of the acceleration. The group of effective forces for all the particles of a body is called the effective system (of forces) for the body. It should be noted that these forces are fictitious or imaginary, equivalent respectively to the actual forces acting upon the particles. The principle may be stated in two forms: - (a) The external system of the forces and the effective system are equivalent, and (b) the external system and the reverse effective system jointly balance, or are in equilibrium. Pletta [13] in 1951 stated that The relationship between the resultant of the external forces F and the resultant of the effective forces ma leads to DAlemberts principle and may be stated in either of two following ways. I. The resultant (F ) of the external forces on a body is identical with the resultant of the effective forces ( ma ) on all particles. II. If the resultant of the effective forces ( ma ) is reversed, it will balance the external forces and hold the body in static equilibrium. The second form enables designers to consider kinetics problems as equivalent statics problems. Many of the classic sources, as above, make detailed statements both forms of DAlemberts principle for forces and then apply the concept without fanfare or careful statement to moments. Girvin [14]stated in 1938 that There are two general methods of procedure for solving problems involving translation of a rigid body (a) The resultant effective force method (b) The reverse resultant effective force method. He further states for moments that According to the DAlemberts Principle, the resultant of the effective forces for all particles of a given body is identical with the resultant of the external forces acting on the body. Therefore by the principle of moments, the algebraic sum of the moments of all the effective forces for all the particles must be equal to the algebraic sum of the moments of the external forces. An anecdote of interest for dynamics instructors from the preface in Girvin states For many years Applied Mechanics has generally been considered a subject which should be taught during the junior year of the various engineering courses after the student had completed the required work in Physics and Calculus. During the last four or five years many colleges have rearranged their engineering curricula. At most institutions this has resulted in Applied Mechanics becoming more or less of a sophomore subject. Shigley [15] in 1961 summarized the inertia or dynamic equilibrium form of DAlemberts principle very succinctly as The vector sum of all the external force and inertia forces acting upon a rigid body is zero. The vector sum of all the external moments and the inertia torques acting upon a rigid body is also separately zero. Two dynamics textbooks of great longevity chronicle the shift in emphasis in the form of DAlemberts principle and perhaps more importantly the resilience of the DAlembert principle. Historically the second form, the reverse effective force form or dynamic equilibrium form was popular and in common use. In 1952, Meriam [16] documents the second form of DAlemberts Principle with the statements Equation ( F ma = ) for the motion of a particle may be written as F ma 0 = which has the same form as the force equilibrium equation, where the sum of a number of terms equals zero. Thus, if a fictitious force equal to ma is applied to the particle in the direction opposite to the acceleration, the particle may be considered to be in equilibrium, and the equation F 0 = may be used where one of the terms is the fictitious ma force. This hypothetical force which, if applied, would provide equilibrium with the actual forces is known as the inertia force, since it is proportional to the mass or inertia of the particle. This artificial state of equilibrium is known as dynamic equilibrium. The resultant of the actual forces is equal to ma and is known as the effective force for the particle. Hence the equal and opposite inertia force is often called the reversed effective force. This viewpoint is due to DAlembert, and the transformation of a problem in dynamics to an equivalent problem in statics is known as DAlemberts principle. He also states that The essential advantage of the dynamic equilibrium method is that a zero moment sum may be taken about any point. Further, in 1952, Meriam illustrates two solution methods in several of the included sample problems. The first method, frequently labeled as Solution I, used the traditional

equations of motion approach with a left side free body diagram contribution to the equations and a mathematics right side to the equations. The second method, frequently labeled as Solution II, used the traditional dynamic equilibrium approach with a single left side free body diagram including inertia effects set equal to zero. But in 2002, Miriam and Kraige [17]discuss the kinetic diagram concept and in a Helpful Hints comment note that the left-hand side of the equation is evaluated from the free-body diagram, and the right-hand side from the kinetic diagram. While some included solved problems may be direct applications of the equations of motion, other solved problems clearly involve effective forces drawn from the included kinetic diagrams. Perhaps a key to the transition and the modern incorporation of the DAlembert principle is the careful treatment of effective and reverse effective forces by Beer and Johnston. In 1957, Beer and Johnston [18] state DAlemberts principle as the external forces acting on the system of particles are equivalent to the effective forces of the various particles of the system and they further state that The resultant obtained may be presented in an alternate form if we consider the inertia vectors ma of sense opposite to the acceleration of the particles. Adding these vectors, also called reverse effective forces, to the external forces, we obtain a system equivalent to zero. Similar results are stated for moments. In 1957, Beer and Johnston solved most sample problems using the revered effective force or the dynamic equilibrium approach. In 2004, Beer, Johnston, Clausen, and Staab [19] state that By placing the emphasis on free-body-diagram equations rather than on the standard algebraic equations of motion, a more intuitive and more complete understanding of the fundamental principles of dynamics can be achieved. This approach, which was first introduced in 1962 in the first edition of Vector Mechanics for Engineers [20], has now gained wide acceptance among mechanics teachers in this country. It is, therefore, used in preference to the method of dynamic equilibrium and to the equations of motion in the solution of all sample problems in this book. This statement seems to mark the origins of the current usage of the first form of DAlemberts principle with the equating of a free body diagram on the left and a kinetic diagram on the right to generate the corresponding mathematical equations. FREE-BODY-DIAGRAM EQUATIONS The free-body-diagram equations emphasis of Beer, Johnston, Clausen, and Staab [19] is an application of DAlemberts principle first form utilizing effective forces. In this effective force approach, a FBD of external forces and moments is set on the left side of an equal sign and is equated to a right side kinetic diagram consisting of effective forces and moments. The equations to be analyzed are generated thru a careful translation of the two diagrams into corresponding equations. For the case of moment summation about the center of gravity, point G , Figure 7, a FDB is shown on the left of the equal sign and a kinetic diagram is shown on the right side of the equal sign.

Figure 7 Free body diagram equated to kinetic diagram for center of gravity ( G ) moment summation The Kinetic effects, ma and GI , on the right side kinetic diagram are assumed positive by the sign convention to remain consistent with the equations of motion approach; an equally valid assumption would be for one or both kinetic effects to be assumed negative. For vertical force summation, the basic principle is

( )Y Y effF F= (40) and using appropriate terms from Figure 7, the free body diagram on the left yields the terms YR mg and the kinetic diagram on the right yields the terms ma m(b / 2)= and thus the force summation equation is

YbR mg ma m2

= = . (41) For moment summation about the center of gravity, G , the basic principle is ( )G G effM M= (42)

and using appropriate terms from Figure 7, the free body diagram on the left yields the term Yb R2

and the kinetic diagram on the right yields the term GI and thus the moment summation equation is

Y Gb R I2

= , (43) solving the force and moment equations, 29.66 rad / sec = and YR 10 Lb= . For the case of moment summation about the fixed pivot, point O , Figure 8, a FDB is shown on the left of the equal sign and a kinetic diagram is shown on the right side of the equal sign.

Figure 8 Free body diagram equated to kinetic diagram for fixed pivot ( O ) moment summation For vertical force summation, the basic principle is ( )Y Y effF F= (44) and using appropriate left side free body diagram and right side kinetic diagram terms from Figure 8, the force summation equation is

YbR mg ma m2

= = . (45) For moment summation about the fixed pivot, O , the basic principle is

( )O O effM M= (46) and using appropriate left side free body diagram and right side kinetic diagram terms from Figure 8, the moment summation equation is

G Gb b b bmg I ma I m2 2 2 2

= + = + , (47) solving the resulting force and moment equations, 29.66 rad / sec = and YR 10 Lb= . The effective force and moment system terms in the previous example were evaluated about the center of gravity, point G . Since the equations developed were based on appropriate diagrams, a moment summation about the fixed pivot, point O , was a valid approach. If an analysis which mirrors the equations of motion approach, which placed effective force and moment system terms at the pivot point O , is desired, an equivalent effective force-moment system at point O is possible.

Figure 9 Equivalent effective force-moment system at the pivot point O . Adding and subtracting ma vectors at point O and isolating ma opposing vectors as shown in Figure 9 results in a ma vector at pivot O and a moment OM defined as

2 2

O G G Gb b bM I ma I m I m2 2 2

= + = + = + (48)

letting b / 2 d= results in

( )( )2O GM I d m= + . (49) Applying the parallel axis theorem,

( )2O GI I d m= + (50) and thus O OM I= . (51)

Figure 10 Free body diagram equated to a kinetic diagram for fixed point ( O ) moment summation using mass moment of inertia OI For vertical force summation, the basic principle is ( )Y Y effF F= (52) and using appropriate left side free body diagram and right side kinetic diagram terms from Figure 10, the force summation equation is

YbR mg ma m2

= = . (53) For moment summation about the fixed axis, O , the basic principle is ( )O O effM M= (54) and using appropriate left side free body diagram and right side kinetic diagram terms from Figure 10, the moment summation equation is

Ob mg I2

= , (55) solving the resulting force and moment equations, 29.66 rad / sec = and YR 10 Lb= .

Figure 11 Free body diagram equated to a kinetic diagram for arbitrary point ( A ) moment summation For vertical force summation, the basic principle is

( )Y Y effF F= (56) and using appropriate left side free body diagram and right side kinetic diagram terms from Figure 11, the force summation equation is

YbR mg ma m2

= = . (57) For moment summation about the arbitrary point, A , the basic principle is ( )A A effM M= (58) and using appropriate left side free body diagram and right side kinetic diagram terms from Figure 11, the moment summation equation is

Y Gb b bbR mg I m2 2 2

+ = , (59) solving the resulting force and moment equations, 29.66 rad / sec = and YR 10 Lb= . For arbitrary point ( A ) moment summation using the mass moment of inertia AI , a determination of an equivalent moment as illustrated in Figure 12 is required.

Figure 12 Equivalent force-moment system a the arbitrary point ( A ) Consider point A in the original diagram where ma is applied at the center of gravity G and the moment GI acts about point G . Vectors ma are added and subtracted at point A as shown in the second diagram. The third diagram shows an equivalent system comprised of vector ma acting at point A and moment ( )b / 2 ma acting about point A . This equivalent force moment system may be represented as

2

A G G Gb b b bM I ma I m m2 2 2 2

= = = (60)

If d b / 2= is used in the parallel axis theorem, then

2

2

A G GbI I d m I m2

= + = + (61) Solving this equation for GI results in

2

G AbI I m2

= (62) This equation for GI is substituted in the bracket term in the above AM equation which results in

2 2 2

2G A A

b b b 2I m I m m I b m2 2 2 4

= = (63)

some additional manipulation of this expression along with setting d b / 2= results in

2

G A Ab bI m I bm I bmd2 2

= = + (64)

Substituting this value into the previous AM equation and realizing that Aa b= yields ( )A A AM I bmd mdb ma d = + = + = + (65) or thus in summary, A AM ma d= + (66) with this equivalent moment show in the last diagram in the Figure above.

Figure 13 Free body diagram equated to a kinetic diagram for arbitrary point ( A ) moment summation using AI For vertical force summation, the basic principle is

( )Y Y effF F= (67) and using appropriate left side free body diagram and right side kinetic diagram terms from Figure 13, the force summation equation is

YbR mg ma m2

= = . (68) For moment summation about the arbitrary point, A , the basic principle is ( )A A effM M= (69) and using appropriate left side free body diagram and right side kinetic diagram terms from Figure 13, the moment summation equation is

Y A AbbR mg I ma d2

+ = + , (70) solving the resulting force and moment equations, 29.66 rad / sec = and YR 10 Lb= . The free-body-diagram equations approach terminology, with or without the inclusion of the kinetic diagram terminology, is far from universal. Of course, the free-body-diagram equations terminology comes directly from Beer, Johnston, Clausen, and Staab [19]. In 1999, Pytel and Kiusalaas [21] termed the method the force-mass-acceleration (FMA) method and equate a free-body diagram (FBD) on the left to a mass-acceleration diagram (MAD) on the right. It is not clear if the (MAD) term was inspired by the (mad) term that was previously used in the equation of motion approaches. In 2005, Tongue and Sheppard [3] use the concept of a free body diagram (FBD) on the left set equal to an Inertial Response Diagram (IRD) on the right. The diagrams seem to compliment an approach that is clearly based on the equations of motion. It is striking how the words Inertial Response Diagram elicit a near echo of the dynamic equilibrium terms inertia force and inertia moment. In 2004, Hibbeler [22] includes some solutions which incorporate kinetic diagrams, but the approach most frequently used is the equations of motion approach. The actual use of the kinetic diagram to generate the right side of the equations seems occasional. As previously discussed, in 2002, Meriam and Kraige [17] include the concept of the kinetic diagram in some but not all solved problems; some solved problems clearly involve effective forces drawn from the included kinetic diagrams. DYNAMIC EQUILIBRIUM APPROACH In the dynamic equilibrium approach, a free body diagram (FBD) on the left is set equal to zero on the right. The left sides of the equations are actually formulated from the external forces, external moments, inertia forces, and inertia moments shown on the diagram. The right sides of the equations are set equal to zero. It is a fairly common practice to place the directions of the linear acceleration, a , and the angular acceleration, , onto the Free Body Diagram to aid in establishing the sense and directions of the inertia effects. The linear acceleration and the angular acceleration are both assumed positive by the sign convention to remain consistent with the equations of motion approach and the free-body-diagram equations approach; an equally valid assumption would be for one or both accelerations to be assumed negative. Inertia effects are shown in the

directions or senses opposite to the accelerations. As a result, the inertia force is directed in a sense opposite to the linear acceleration, a , and the inertia moment is directed in a sense opposite to the angular acceleration, . While shown on the diagram, the linear acceleration, a , and the angular acceleration, , must not be included in the force and moment summations. For the case of moment summation about the center of gravity, point G , Figure 14, a FDB is shown for the diagram on the left of the equal sign and it includes both an inertia force and an inertia moment evaluated about the center of gravity point G .

Figure 14 Dynamic equilibrium free body diagram for inertia effects for point ( G ) moment summation Inertia force IF is found as

IW W b 5F ma a 3.106g g 2 2

40 = = = = = 32.2 (71) and inertia moment IM is found to be I GM I 2.588= = (72) For vertical force summation, the basic principle is YF 0= (73) and using appropriate left side free body diagram terms from Figure 14, the force summation equation is Y IR mg F 0 = . (74) For moment summation about the center of gravity, G , the basic principle is GM 0= (75) and using appropriate left side free body diagram terms from Figure 14, the moment summation equation is

Y Ib R M 02

= , (76) solving the force and moment equations, 29.66 rad / sec = and YR 10 Lb= .

For the case of moment summation about the fixed pivot, point O , Figure 15, a FDB is shown for the diagram on the left of the equal sign and it includes both an inertia force and an inertia moment evaluated about the center of gravity point G .

Figure 15 Dynamic equilibrium free body diagram for inertia effects for point ( G ) with moment summation about point ( O ) Inertia force IF is found as

IW W b 5F ma a 3.106g g 2 2

40 = = = = = 32.2 (77) and inertia moment IM is found to be I GM I 2.588= = (78) For vertical force summation, the basic principle is YF 0= (79) and using appropriate left side free body diagram terms from Figure 15, the force summation equation is Y IR mg F 0 = . (80) For moment summation about the fixed point, O , the basic principle is OM 0= (81) and using appropriate left side free body diagram terms from Figure 15, the moment summation equation is

I Ib bmg F M 02 2

= , (82) solving the resulting force and moment equations, 29.66 rad / sec = and YR 10 Lb= . The inertia terms in the previous example were evaluated about the center of gravity, point G . Since the equations just develop were based on appropriate diagrams, a moment summation about the fixed pivot, point O , was a valid approach. If an analysis which mirrors the equations of motion approach, which placed effective force and moment system terms at the pivot point O , is desired, an equivalent set of inertia terms evaluated at point O

is possible. The Figure 9 development of an equivalent effective force-moment system at the pivot point O can be directly applied to the determination of the required inertia force and the inertia moment.

Figure 16 Dynamic equilibrium free body diagram for inertia effects for point ( O ) with moment summation about point ( O ) Inertia force

OIF is found as

OI

W W b 5F ma a 3.106g g 2 2

40 = = = = = 32.2 (83) and inertia moment

OIM is found to be

OI OM I 10.35= = (84)

For vertical force summation, the basic principle is YF 0= (85) and using appropriate left side free body diagram terms from Figure 16, the force summation equation is

OY IR mg F 0 = . (86)

For moment summation about the fixed point, O , the basic principle is OM 0= (87) and using appropriate left side free body diagram terms from Figure 16, the moment summation equation is

OI

b mg M 02

= , (88) solving the resulting force and moment equations, 29.66 rad / sec = and YR 10 Lb= . For the case of moment summation about the arbitrary point ( A ) with inertia effects evaluated at point ( G ), Figure 17, a FDB is shown for the diagram on the left of the equal sign and it includes both an inertia force and an inertia moment evaluated about the center of gravity point G .

Figure 17 Dynamic equilibrium free body diagram for inertia effects for point ( G ) with moment summation about arbitrary point (A) Inertia force IF is found as

IW W b 5F ma a 3.106g g 2 2

40 = = = = = 32.2 (89) and inertia moment IM is found to be I GM I 2.588= = (90) For vertical force summation, the basic principle is YF 0= (91) and using appropriate left side free body diagram terms from Figure 17, the force summation equation is Y IR mg F 0 = . (92) For moment summation about the arbitrary point, A , the basic principle is AM 0= (93) and using appropriate left side free body diagram terms from Figure 17, the moment summation equation is

Y I Ib bbR mg F M 02 2

+ + = , (94) solving the resulting force and moment equations, 29.66 rad / sec = and YR 10 Lb= . For arbitrary point ( A ) moment summation using the mass moment of inertia AI , the determination of an equivalent moment as illustrated in Figure 12 resulted in A AM ma d= + (95)

Figure 18 Dynamic equilibrium free body diagram for arbitrary point ( A ) moment summation using AI Inertia force

AIF is found as

AI

W W b 5F ma a 3.106g g 2 2

40 = = = = = 32.2 (96) and inertia moment

AIM is found to be

AI A AM M ma d= = + (97)

Using Aa b= and d b / 2= along with the other known values results in

AIM 5.18= . (98)

For vertical force summation, the basic principle is YF 0= (99) and using appropriate left side free body diagram terms from Figure 18, the force summation equation is

AY IR mg F 0 = . (100)

For moment summation about the arbitrary point, A , the basic principle is AM 0= (101) and using appropriate left side free body diagram terms from Figure18, the moment summation equation is

AY I

bbR mg M 02

+ = , (102) solving the resulting force and moment equations, 29.66 rad / sec = and YR 10 Lb= . Common variations of the dynamic equilibrium approach use the terms ma , I , mad , etc. directly instead of the inertia terms IF and IM . Some individuals used broken rather than solid lines to delineate the various inertia effects.

In addition to the sources previously discussed, illustrations of applying dynamic equilibrium or the inertia approach are discussed by Shelly [23], Jensen and Chenoweth [24], Boresi and Schmidt [25], Bedford and Fowler [26], Shigley [15], Goodman and Warner [27], and Shames [28]. DISCUSSION While many students at the time probably considered the 1957 Sputnik launching and the cold war as the main reasons for the major engineering curriculum and program changes of the late 1950s and early 1960s, engineering faculty of that time probably understood the changes to be the follow up of the 1955 Grinter report (Grinter [1] and Harris [2]). Dynamics, under the heading of Mechanics of solids, was specifically included under The Engineering Sciences heading in the Grinter report. This move to an engineering science thrust for dynamics may well have the impetus for the move from scalar dynamics textbooks of the time to vector editions in the 1960s. At this juncture, some textbook authors may have made the final efforts to distance themselves from the pictorial and graphical dynamic equilibrium approached often favored in scalar textbooks to the mathematical approach of the equations of motion. The dynamic equilibrium approach and Dalemberts principle had long been the object of controversy and had thus faded into disuse, except perhaps by those faculty who saw the utility of the approach as a teaching tool. But the equations of motion approach favored by the theoreticians, was an often dry, vague, and difficult to understand or teach to first time dynamics students. In some sense, the luxury of having two forms of DAlemberts principle was the saving grace, the free-body-diagram equations form, with a free body diagram set equal to a kinetic diagram and utilizing effective forces and effective moments has evolved into a common and a popular teaching tool. Perhaps a future revelation will be that a single diagram with a few reversed terms is also an outstanding teaching tool. Such a single diagram looks just like statics and all the rules learned in statics still apply. Perhaps either form of DAlemberts principle should hold equal respect in the eyes of the dynamics teaching community. Maybe the current trends to eradicate the concept of the equations of dynamic equilibrium in teaching dynamics should be softened. The concept of a best way to teach the principles of kinetics is probably just a viewpoint of the individual dynamics instructor. The three basic methods; the equations of motion, free-body-diagram equations, and dynamic equilibrium have been shown above to be equally applicable and versatile in the solution of kinetics problems. Which choice to follow is really just a decision of personal preference and, in reality, the existence of an appropriate textbook. Should modern textbooks go in the direction followed by some classical textbooks; that is, to show several problems worked by alternate methods and thus letting the individual instructors decide on their preference of method? The recent trend of making available separate computer software supplements for MATLAB, Mathcad, Maple, and Mathematica to be used with dynamics textbooks might apply to kinetic methods. Perhaps separate sample problem textbook supplements for each of the three kinetic approaches along with a synopsis of each specific method would be appropriate. Regardless of which of the three solution strategies is followed, a clear understanding the different approaches, within the chosen strategy, hinges on examples of non-zero moment summations. Surprisingly, many textbooks have only one or two worked examples of non-zero moment summations. It would seem highly desirable that an emphasis on a clear solution strategy would hinge on consistent and frequent application of non-zero moment summations in worked textbook examples. CONCLUSIONS In the equations of motion approach, a Free Body Diagram (FBD), involving external forces and moments, is appropriate for the left side of the chosen governing equation. The left side of the equation may be developed with the aid of the FBD. However, a diagram for the right side of the chosen governing equation is not appropriate. The terms on the right side of the equations of motion are just that, they are mathematical terms. In the equations of motion approach, the user needs to be discerning in the original choice equations for the case desired, a mismatch of the point for moment summation to analytic equation naturally leads to an error. It is not possible to peek at the right side of a diagram to modify equations; a new set of equations is needed for the chosen point for the evaluation of moment.

In the free-body-diagram equations approach, a diagram on the left is equated to or equivalent to a diagram on the right. The diagram on the left is a free body diagram (FBD) and the diagram on the right is a kinetic diagram. All terms on both the left and right sides of the generated equations are drawn directly from the actual diagrams, there is no dependence on any previously derived mathematical terms in the development of the final underlying equations. In the free-body-diagram equations approach, the user has an open choice of solution method that is independent of underlying equations. However, the user must decide on a specific solution method prior to generating and drawing both the free body diagram and the kinetic diagram. Missing, duplicate, or extra terms on either of the diagrams must be avoided at all costs. In the dynamic equilibrium approach, a diagram on the left are set equal to zero on the right. The diagram on the left is a free body diagram (FBD) which now additionally includes inertia forces and inertia moments. . All terms on the left side of the generated equations are drawn directly from the actual diagram, there is no dependence on any previously derived mathematical terms in the development of the final underlying equations. The right sides of the equations are set equal to zero. In the dynamic equilibrium approach, the user has an open choice of solution method that is independent of underlying equations. However, the user must decide on a specific solution method prior to generating and drawing the free body diagram. Missing or extra terms on the diagram must be avoided at all costs. Many authors currently reject, trivialize, or distance themselves from the second form dynamic equilibrium or inertia form of DAlemberts Principle but then completely embrace the first form free-body-diagram equations or effective force/moment form of DAlemberts Principle. Perhaps some personal reflection is appropriate in these cases. If the primary goal in studying kinetics is mathematical formulations, the equation of motion approach is the most obvious path. If visualization and an emphasis on concepts and/or a teaching approach to kinetics is the primary goal, then one of the two forms based on DAlemberts Principle may be a more fruitful path. If there is an application of DAlemberts Principle, whether or not the action is declared, the solution process still works because DAlemberts Principle makes the action correct! REFERENCES [1] Grinter, L. E. (1955). Report of the Committee on Evaluation of Engineering Education. Journal of Engineering Education, 46(1), 25-60. [2] Harris, J. G. (1994). Journal of Engineering Education Round Table: Reflections on the Grinter Report. Journal of Engineering Education, 83(1), 69-94. [3] Tongue, B. H., & Sheppard, S. D. (2005). Dynamics Analysis and Design of Systems in Motion. New York: John Wiley & Sons, Inc. [4] Soutas-Little, R. W., & Inman, D. J. (1999). Engineering Mechanics Dynamics. New Jersey: Prentice- Hall, Inc. [5] Riley, W. F., & Sturges, L. D. (1996). Engineering Mechanics Dynamics (2nd ed.). New York: John Wiley & Sons, Inc. [6] Sandor, B. I., & Richter, K. J. (1987). Engineering Mechanics Statics and Dynamics (2nd ed.). New Jersey: Prentice-Hall, Inc. [7] Ginsberg, J. H., & Genin, J. (1977). Statics and Dynamics Combined Version. New York: John Wiley & Sons, Inc.

[8] McGill, D. J., & King, W. W. (1989). Engineering Mechanics Statics and an Introduction to Dynamics (2nd ed.). Boston: PWS-KENT Publishing Company. [9] Meirovitch, L. (1970). Methods of Analytical Dynamics. New York: McGraw-Hill Book Company. [10] Rosenberg, R. M. (1977). Analytical Dynamics of Discrete Systems. New York: Plenum Press. [11] Lanczos, C. (1966). The Variational Principles of Mechanics (3rd ed.). Toronto: University of Toronto Press. [12] Maurer, E. R. (1914). Technical Mechanics Statics and Dynamics (3rd ed.). New York: John Wiley & Sons, Inc. [13] Pletta, D. H. (1951). Engineering Statics and Dynamics. New York: The Ronald Press Company. [14] Girvin, H. F., (1938). Applied Mechanics. Scranton: International Textbook Company. [15] Shigley, J. E. (1961). Dynamic Analysis of Machines. New York: McGraw-Hill Book Company [16] Meriam, J. L. (1952). Mechanics Part II Dynamics. New York: John Wiley & Sons, Inc. [17] Meriam, J. L., & Kraige, L. G. (2002). Engineering Mechanics - Dynamics (5th ed.). New York: John Wiley & Sons, Inc. [18] Beer, F. P., & Johnston, E. R., Jr. (1957). Mechanics for Engineers Dynamics. New York: McGraw-Hill Book Company, Inc. [19] Beer, F. P., & Johnston, E. R., Jr., Clausen, W. E., & Staab, G. H. (2004). Vector Mechanics for Engineers Dynamics (7th ed.). New York: McGraw-Hill Book Company, Inc. [20] Beer, F. P., & Johnston, E. R., Jr. (1962). Vector Mechanics for Engineers Dynamics. New York: McGraw-Hill Book Company, Inc. [21] Pytel, A., & Kiusalaas, J. (1999). Engineering Mechanics Dynamics (2nd ed.). New York: Brooks/Cole Publishing Company. [22] Hibbeler, R. C. (2004). Engineering Mechanics Dynamics (10th ed.). New Jersey: Pearson Prentice Hall. [23] Shelly, J. F. (1980). Engineering Mechanics Dynamics. New York: McGraw-Hill Book Company. [24] Jensen, A., & Chenoweth, H. H. (1983). Applied Engineering Mechanics (4th ed.). New York: McGraw-Hill Book Company. [25] Boresi, A. P., & Schmidt, R. J. (2001). Engineering Mechanics: Dynamics. New York: Brooks/Cole Publishing Company. [26] Bedford, A., & Fowler, W. (1999). Engineering Mechanics Dynamics (2nd ed.). California: Addison Wesley Longman, Inc. [27] Goodman, L. E., & Warner, W. H. (1961). Dynamics. California: Wadsworth Publishing Company, Inc. [28] Shames, I. H. (1960). Engineering Mechanics Statics and Dynamics. New Jersey: Prentice-Hall, Inc.

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