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Overview
About the Author:
Bruce K. Donaldson, At the University of Maryland he became a professor of aerospace engineering and then a professor of civil engineering
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Meet the Author
Bruce K. Donaldson was first exposed to aircraft inertia loads when he was a carrierbased U.S. Navy antisubmarine pilot. He subsequently worked in the structural dynamics area at the Boeing Co. and at the Beech Aircraft Co. in Wichita, KS before returning to school and then embarking on an academic career in the area of structural analysis. He became a professor of Aerospace Engineering, and then a professor of Civil and Environmental Engineering at the University of Maryland. Professor Donaldson is the recipient of numerous teaching awards, and has maintained industrial contacts, working various summers at government agencies and for commercial enterprises, the last being Lockheed Martin at Fort Worth, Texas.
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Cambridge University Press
9780521865746  Introduction to structural dynamics  by Bruce K. Donaldson
Excerpt
1 The Lagrange Equations of Motion
1.1 Introduction
A knowledge of the rudiments of dynamics is essential to understanding structural dynamics. Thus this chapter reviews the basic theorems of dynamics without any consideration of structural behavior. This chapter is preliminary to the study of structural dynamics because these basic theorems cover the dynamics of both rigid bodies and deformable bodies. The scope of this chapter is quite limited in that it develops only those equations of dynamics, summarized in Section 1.10, that are needed in subsequent chapters for the study of the dynamic behavior of (mostly) elastic structures. Therefore it is suggested that this chapter need only be read, skimmed, or consulted as is necessary for the reader to learn, review, or check on (i) the fundamental equations of rigid/flexible body dynamics and, more importantly, (ii) to obtain a familiarity with the Lagrange equations of motion.
The first part of this chapter uses a vector approach to describe the motions of masses. The vector approach arises from the statement of Newton’s second and third laws of motion, which are the starting point for all the material in this textbook. These vector equations of motion are used only to prepare the way for the development of the scalar Lagrange equations of motion in the second part of thischapter. The Lagrange equations of motion are essentially a reformulation of Newton’s second law in terms of work and energy (stored work). As such, the Lagrange equations have the following three important advantages relative to the vector statement of Newton’s second law: (i) the Lagrange equations are written mostly in terms of point functions that sometimes allow significant simplification of the geometry of the system motion, (ii) the Lagrange equations do not normally involve either external or internal reaction forces and moments, and (iii) the Lagrange equations have the same mathematical form regardless of the choice of the coordinates used to describe the motion. These three advantages alone are sufficient reasons to use the Lagrange equations throughout the remaining chapters of this textbook.
1.2 Newton’s Laws of Motion
Newton’s three laws of motion can be paraphrased as (Ref. [1.1]):
These three laws are not the only possible logical starting point for the study of the dynamics of masses. However, (i) these three laws are at least as logically convenient as any other complete basis for the motion of masses, (ii) historically, they were the starting point for the development of the topic of the dynamics, and (iii) they are the one basis that almost all readers will have in common. Therefore they are the starting point for the study of dynamics in this textbook.
There are features of this statement of Newton’s laws that are not immediately evident. The first of these is that these laws of motion are stated for a single particle, which is a body of very, very small spatial dimensions, but with a fixed, finite mass. The mass of the j_{th} particle is symbolized as m_{j}. The second thing to note is that momentum, which means rectilinear momentum, is the product of the mass of the particle and its instantaneous velocity. Of course, mass is a scalar quantity, whereas velocity and force are vector quantities. Hence the second law is a vector equation. The third thing to note is that the second law, which includes the first law, is not true for all coordinate systems. The best that can be said is that there is a Cartesian coordinate system “in space” for which the second law is valid. Then it is easy to prove (see the first exercise) that the second law is also true for any other Cartesian coordinate system that translates at a constant velocity relative to the valid coordinate system. The second law is generally not true for a Cartesian coordinate system that rotates relative to the valid coordinate system. However, as a practical matter, it is satisfactory to use a Cartesian coordinate system fixed to the Earth’s surface if the duration of the motion being studied is only a matter of a few minutes. The explanation for this exception is that the rotation of the Cartesian coordinate system fixed at a point on the Earth’s surface at the constant rate of onequarter of a degree per minute, or 0.0007 rpm, mostly just translates that coordinate system at the earth’s surface in that short period of time. See Figure 1.1(a).
Figure 1.1. (a) Valid and invalid coordinate systems for Newton’s second law, both moving at constant speed. (b) Illustration of the righthand rule for r x F = M = r F sin αn.
As is derived below, when Newton’s second law is extended to a mass m of finite spatial dimensions, which is subjected to a net external force of magnitude^{1} F, then Newton’s second law can be written in vector form as follows:
where P = mv is the momentum vector, v is the velocity vector of the total mass m relative to the valid coordinate origin, t is time, and a is the acceleration vector, which of course is the time derivative of the velocity vector. The velocity vector is not the velocity of all points within the mass m relative to the valid coordinate system. Rather, it is the velocity of the one point called the center of mass, which is defined below. Further, note that the mass of the system of particles whose motion is described by this equation is the mass of a fixed collection of specific mass particles. Hence, even though the boundary surface that encloses these specified mass particles may change considerably over time, the mathematical magnitude of the mass term is a constant. Those mass particles that are included within the mass, or alternately, enclosed by the boundary surface of the mass system, are defined by the analyst as the “mass system under study.”
The above basic result, Eq. (1.1), can be derived as follows. Consider a collection of, that is, a specific grouping of, N particles of total mass m = Σm_{j}, where all such sums run from j = 1 to j = N, where N can be a very large number. Again, it is not essential that there be any particular geometric relationship between the N particles. Newton’s second law applies to each of these N particles. To write Newton’s second law in a useful way, let each of these N particles be located by means of its own position vector r_{j}(t) originating at the origin of a valid coordinate system. Note that if the timevarying spatial position of the ith particle in terms of the valid Cartesian coordinates is [x_{i}(t), y_{i}(t), z_{i}(t)], then the position vector can be written as r_{i}(t) = x_{i}(t)i + y_{i}(t)j + z_{i}(t)k. Since the differential quantity dr_{i} is tangent to the path of the ith particle, the velocity vector is always tangent to the particle path. However, because the forces applied to the particle are not necessarily tangent to the particle path, neither is the acceleration vector, d^{2}r/dt^{2}. Thus the path of the particle need not be straight.
The statement of the second law for the individual ith particle now can be written as
where F_{i}^{ex} is the vector sum of all the forces acting on the ith particle that originate from sources outside of this collection of N particles (to be called the net external force acting on the ith particle), and F_{i}^{in} is the vector sum of all the forces acting on ith particle that originate from interactions with the other N1 particles (i.e., the net internal force acting on the ith particle). From Newton’s third law, each of the N1 components of the net internal force acting on the ith particle can be associated with an equal and opposite force acting on one of the other particles in the collection of N particles. Hence, summing all such Eqs. (1.2) for the N particles leads to the cancellation of all the internal forces between the N particles, with the result
Again, the total mass m is defined simply as the scalar sum of all the m_{i}. That is m = Σm_{j}. The location of the center of mass of the total mass m is identified by introducing the center of mass position vector, r(t) (without a subscript). Since this position vector goes from the coordinate origin to the center of mass, this vector alone fully describes the path traveled by the center of mass as a function of time. The center of mass position vector r at any time t is defined so that
This definition means that the center of mass position vector is a massweighted average of all the mass particle position vectors. This definition can also be viewed as an application of the mean value theorem. Differentiating both sides of the definition of the center of mass position vector with respect to time twice and then substituting into the previous equation immediately yields Eq. (1.1): F = mr ≡ ma. Again, the force vector F, without superscripts and subscripts, is the sum of all the external forces. Note that external forces can arise from only one of two sources: (i) the direct contact of the boundary surface of the N particles under study with the boundary of other masses or (ii) the distant action of other masses, in which case they are called field forces. Gravitational forces are an example of the latter type of action.
1.3 Newton’s Equations for Rotations
A knowledge of the motion of the center of mass can tell the analyst a lot about the overall motion of the mass system under study. However, that information is incomplete because it tells the analyst nothing at all about the rotations of the mass particles about the center of mass. Since rotational motions can be quite important, this aspect of the overall motion needs investigation.
Just as the translational motion of the center of mass can be viewed as determined by forces, rotational motions are determined by moments of forces. Recall that the mathematical definition of a moment about a point, when the moment center is the origin of the valid coordinate system, is
Recall that reversing the order of a vector cross product requires a change in sign to maintain an equality. Further note that it is immaterial where this position vector intercepts the line of action of the above force vector because the product of the magnitude of the r vector and the sine of the angle between the r and F vectors is always equal to the perpendicular distance between (i) the line of action of the force and (ii) the moment center.
Structural engineers are more familiar with moments about Cartesian coordinate axes than moments about points. The relation between a moment about a point and a moment about such an axis can be understood by reference to Figure 1.1. (b). This figure illustrates that the moment resulting from the cross product of the r vector and the F vector, by the rules of vector algebra, is in the direction of the unit vector n, which is perpendicular to the plane formed by the r and F vectors. The positive direction of n is determined by the thumb of the right hand after sweeping the other four fingers of the right hand from the direction of r, the first vector of the cross product, through to the direction of F. In terms of α, the angle between these two vectors in the plane formed by the two vectors
Like any other vector, the vector M has components along the Cartesian coordinate axes. In terms of the components of the force F and the position vector r, the moment about a point can be written, using vector algebra, as follows:
Considering the last equation, it is clear that moments about axes are simply components of moments about points.
When describing the rotation of the mass m, it is often convenient to consider a reference point P that is other than the valid coordinate origin, which is here called the point O. See Figure 1.2. Let the this new reference point P move in an arbitrary fashion relative to the coordinate origin, point O, in a fashion defined by the position vector r_{P}(t). Introduce the vector quantity L_{Pj}(t) which is to be called the angular momentum about point P, or, more descriptively, the moment of momentum of the mass particle m_{j} about the arbitrary point P. That is, the angular momentum about point P of the jth mass particle is defined as the vector cross product of (i) the position vector from point P to the particle m_{j} and (ii) the momentum vector of m_{j} where the associated velocity vector is that relative to point P rather than the origin of the coordinate system, point O. Thus, in mathematical symbols, relative to point P, the angular momentum of the jth particle, and the angular momentum of the total mass m are, respectively,
Figure 1.2. Vectors relevant to the rotational motion of a mass. Point P has an arbitrary motion relative to point O.
Differentiating both sides of the total angular momentum with respect to time, and noting that the cross product of the relative velocity vector (r_{j}  r_{P}) with itself is zero, yields the following result:
From the original statement of Newton’s second law, it is possible to substitute in the above equation the net external and internal forces on the jth particle for m_{j}(d^{2}/dt^{2})r_{j}. The result is
The term involving the net internal forces sums to zero because all the component internal forces are not only equal and oppositely directed, but, by the strong form of Newton’s third law, they are also collinear. See Exercise . The remaining portion of the first term, that involving the net external forces on the N particles, sums to M_{P}, called the moment about point P of all the external forces acting on the mass m. The last term in the above sum can be simplified by noting that
Thus the final result for the time derivative of the angular momentum of the mass m is
In other words, with reference to Figure 1.2,
Clearly, if point P is coincident with the center of mass (called the center of mass or CG case, where r_{P} = r), or if the relative position vector r_{P}r and the acceleration vector (d^{2}/dt2)r_{P} are collinear (unimportant because it is unusual), or if point P is moving at a constant or zero velocity with respect to point O (called, for simplicity, the fixed point or FP case), then the rotation equation reduces to simply
Note that the above vector equation is the origin of the static equilibrium equations, which state that “the sum of the moments about any axis is zero.” That is, when the angular momentum relative to the selected point P is zero or a constant, then the three orthogonal components of the total moment vector of the external forces acting on the system about point P are zero. These three orthogonal components are the moments about any three orthogonal axes.
The above rotational motion equation, Eq. (1.3b) is not as useful as Eq. (1.1), the corresponding translational motion equation. In Eq. (1.1), the three quantities force, mass, and acceleration are individually quantifiable. In Eq. (1.3b), while the moment term is easily understood, the time rate of change of the angular momentum needs further refinement so that perhaps it too can be written as some sort of fixed mass type of quantity multiplied by some sort of acceleration. Recall that for the mass system m, the total angular momentum relative to point P, is defined as the sum of the moments of the momentum of all the particles that comprise the mass m. That is, again
From the previous development, that is, Eqs. (a,b), there are two simplifying choices for the reference point P: the FP (socalled fixed point) case and the CG (center of mass) case, where the time derivative of the angular momentum is equal to just the moment about point P of all the external forces. First consider the FP case, where point P has only a constant velocity relative to the coordinate origin, point O. Then, from Exercise , either point P or point O is the origin of a valid Cartesian coordinate system. Since these two points are alike, for the sake of simplicity, let the reference point P coincide with the origin of the coordinate system, point O. Again, this placement of point P at point O does not compromise generality within the FP case because when point P is only moving at a constant velocity relative to point O, point P can also be an origin for a valid coordinate system. Then with r_{P} = 0, and because the e_{i} vectors of Figure 1.2. originate at the center of mass, the total angular momentum becomes
To explain why the second and third terms of the above second line are zero, recall the definition of the center of mass position vector, r. That mean value definition is mr ≡ Σm_{i} r_{i}. Since r_{i} = r+e_{i}, mr ≡ Σ m_{i}r +Σ m_{i}e_{i}. Since r is not affected by the summation over the N particles, it can be factored out of the first sum on the above righthand side. The result is mr ≡ mr + Σmiei or 0 = Σ m_{i}e_{i}. Furthermore, because the mass value of each particle is a constant, the time derivative of this last equation shows that 0 = Σ m_{i}e_{i}. This is just an illustration of the general fact that first moments, that is, multiplications by distances raised to the first power, of mass or area, or whatever, about the respective mean point are always zero. Multiplications of mass by distances with exponents other than one lead to terms which are generally not zero.
In the above FP equation, Eq. (1.4a), for the angular momentum, the first term depends only on the motion of the center of mass relative to the Cartesian coordinate origin. Even if the mass is not rotating relative to the Cartesian coordinate origin, this term is generally not zero. The second part of the angular momentum exists even if the center of mass is not moving. This second part accounts for the spin of the mass about its own center of mass.
The CG case is where the reference point P is located at the center of mass, point C, rather than at the coordinate origin, point O, as in the FP case. In this CG case, r=r_{P} and r_{i}r_{P}=e_{i}. Substituting these vector relationships into the expression for L_{P} immediately leads to the same result for the angular momentum, as was obtained for the FP case, except that the first of those two terms is absent. Hence the mathematics of the CG case are included within that of the FP case, and therefore the CG case does not need a parallel development.
1.4 Simplifications for Rotations
Since Newton’s second law is a vector equation, it has been convenient to derive its rotational corollaries by use of vector algebra in threedimensional space. However, it is no longer convenient to pursue the subject of rotations using threespace vector forms because, in general, the rotations themselves about axes in three dimensions (as opposed to moments about axes in three dimensions) are not vector quantities. For a quantity be classified as a vector, the order of an addition has to be immaterial; that is, it is necessary that A + B = B + A, which is called the commutative law for vector addition. In contrast, as Figure 1.3. illustrates, the order of addition of rotations in threespace can greatly change the final orientation of the mass whenever the rotational angles involved are large, like the 90° angles selected for Figure 1.3. There are two simple ways of circumventing this difficulty. The first simplifying approach is to restrict the rotational motion equations to a single plane. In a single plane, all rotations simply add or subtract as scalar quantities. This is a wholly satisfactory approach for most of the illustrative pendulum problems considered in the next chapter. The second option for simplification is to retain rotations about more than one orthogonal axis but limit all those rotations to being small. Here “small” means that the tangent of the angle is closely approximated by the angle itself.^{2} As is explained in Ref. [1.2], p. 271, in contrast to larger angles, angles about orthogonal axes of these small magnitudes can be added to each other as vector quantities. This approach of restricting the rotations to either being small or lying in a single plane would not be adequate for formulating a general analysis of the motion of bodies of finite size, which is not a present concern. However, this is a satisfactory approach for almost all structural dynamics problems because structural rotations due solely to the vibrations of a flexible structure are almost always less than 10° or 12°. Therefore, to repeat and thus underline this important point, for the present purposes of structural dynamics, it is often satisfactory only to look at rotations in a single plane or restrict the analysis to small rotations, which can be added vectorially.
Figure 1.3. Proof that, generally, rotations are not vectors because the order of the rotations is not irrelevant.
To further the discussion, consider all rotations confined to a single plane that, for the sake of explicitness, is identified as the z plane. To reflect the change from three to two dimensions, the notation FP for a fixed point in threedimensional space, transitions to FA for a fixed axis perpendicular to the z plane. This simplification from a general state of rotations to those only about an axis paralleling the z axis allows the introduction of a pair of convenient unit vectors in the z plane called p_{1} and q_{1} such that p_{1} is directed from the origin toward the center of mass and q_{1} is rotated 90° counterclockwise from p_{1}. These two unit vectors rotate in the z plane as the center of mass moves in that plane. In terms of the fixedinspace Cartesian coordinate unit vectors, i, j, as shown in Figure 1.4. (a),
Figure 1.4. (a) The relationship between the rotating unit vectors and the fixed unit vectors, i and j. (b) Use of unit vectors to locate the ith mass particle.
Again, even though p_{1} and q_{1} have a fixed unit length, they have time derivatives because their orientation in the z plane varies with time as the angle ϕ changes with time. The above equations show that the time derivatives of these rotating unit vectors are
This unit vector pair p,q can be used with both the position vector for the center of mass and the vector from the center of mass to the ith mass particle. That is, as illustrated in Figure 1.4. (b),
As the final limitation on the dynamics equations to be developed, let the geometry of the total mass be restricted to small changes in overall shape so that the rotation angle for the jth mass about the center of mass differs so little from that average rotation that the average rotation ϕ_{2} can be used as the rotation angle about the center of mass for all the mass particles that are included within the boundary of the total mass. This is a rather minor limitation, if any at all, for almost all structures.
© Cambridge University Press
Table of Contents
Preface for the Student xi
Preface for the Instructor xv
Acknowledgments xvii
List of Symbols xix
The Lagrange Equations of Motion 1
Introduction 1
Newton's Laws of Motion 2
Newton's Equations for Rotations 5
Simplifications for Rotations 8
Conservation Laws 12
Generalized Coordinates 12
Virtual Quantities and the Variational Operator 15
The Lagrange Equations 19
Kinetic Energy 25
Summary 29
Exercises 33
Further Explanation of the Variational Operator 37
Kinetic Energy and Energy Dissipation 41
A Rigid Body Dynamics Example Problem 42
Mechanical Vibrations: Practice Using the Lagrange Equations 46
Introduction 46
Techniques of Analysis for Pendulum Systems 47
Example Problems 53
Interpreting Solutions to Pendulum Equations 66
Linearizing Differential Equations for Small Deflections 71
Summary 72
**Conservation of Energy versus the Lagrange Equations** 73
**Nasty Equations of Motion** 80
**Stability of Vibratory Systems** 82
Exercises 85
The LargeDeflection, Simple Pendulum Solution 93
Divergence and Flutter in Multidegree of Freedom, Force Free Systems 94
Review of the Basics of the Finite Element Method for Simple Elements 99
Introduction 99
Generalized Coordinates for Deformable Bodies 100
Element and Global Stiffness Matrices 103
More Beam Element Stiffness Matrices 112
Summary 123
Exercises 133
A Simple TwoDimensional Finite Element 138
The Curved Beam Finite Element 146
FEM Equations of Motion for Elastic Systems 157
Introduction 157
Structural Dynamic Modeling 158
Isolating Dynamic from Static Loads 163
Finite Element Equations of Motion for Structures 165
Finite Element Example Problems 172
Summary 186
**Offset Elastic Elements** 193
Exercises 195
Mass Refinement Natural Frequency Results 205
The Rayleigh Quotient 206
The Matrix Form of the Lagrange Equations 210
The Consistent Mass Matrix 210
A Beam Cross Section with Equal Bending and Twisting Stiffness Coefficients 211
Damped Structural Systems 213
Introduction 213
Descriptions of Damping Forces 213
The Response of a Viscously Damped Oscillator to a Harmonic Loading 230
Equivalent Viscous Damping 239
Measuring Damping 242
Example Problems 243
Harmonic Excitation of Multidegree of Freedom Systems 247
Summary 248
Exercises 253
A Real Function Solution to a Harmonic Input 260
Natural Frequencies and Mode Shapes 263
Introduction 263
Natural Frequencies by the Determinant Method 265
Mode Shapes by Use of the Determinant Method 273
**Repeated Natural Frequencies** 279
Orthogonality and the Expansion Theorem 289
The Matrix Iteration Method 293
**Higher Modes by Matrix Iteration** 300
Other Eigenvalue Problem Procedures 307
Summary 311
**Modal Tuning** 315
Exercises 320
Linearly Independent Quantities 323
The Cholesky Decomposition 324
Constant Momentum Transformations 326
Illustration of Jacobi's Method 329
The GramSchmidt Process for Creating Orthogonal Vectors 332
The Modal Transformation 334
Introduction 334
Initial Conditions 334
The Modal Transformation 337
Harmonic Loading Revisited 340
Impulsive and Sudden Loadings 342
The Modal Solution for a General Type of Loading 351
Example Problems 353
Random Vibration Analyses 363
Selecting Mode Shapes and Solution Convergence 366
Summary 371
**Aeroelasticity** 373
**Response Spectrums** 388
Exercises 391
Verification of the Duhamel Integral Solution 396
A Rayleigh Analysis Example 398
An Example of the Accuracy of Basic Strip Theory 399
Nonlinear Vibrations 400
Continuous Dynamic Models 402
Introduction 402
Derivation of the Beam Bending Equation 402
Modal Frequencies and Mode Shapes for Continuous Models 406
Conclusion 431
Exercises 438
The Long Beam and Thin Plate Differential Equations 439
Derivation of the Beam Equation of Motion Using Hamilton's Principle 442
SturmLiouvilie Problems 445
The Bessel Equation and Its Solutions 445
Nonhomogeneous Boundary Conditions 449
Numerical Integration of the Equations of Motion 451
Introduction 451
The Finite Difference Method 452
Assumed Acceleration Techniques 460
PredictorCorrector Methods 463
The RungeKutta Method 468
Summary 474
**Matrix Function Solutions** 475
Exercises 480
Answers to Exercises 483
Solutions 483
Solutions 486
Solutions 494
Solutions 498
Solutions 509
Solutions 516
Solutions 519
Solutions 525
Solutions 529
Fourier Transform Pairs 531
Introduction to Fourier Transforms 531
Index 537