The Practice of Engineering Dynamics
Description
The Practice of Engineering Dynamics includes end of chapter exercises and is accompanied by a website hosting a solutions manual.
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Person
DR. RONALD J. ANDERSON is a Professor in the Department of Mechanical and Materials Engineering, Queen's University at Kingston, Canada. He received his B.Sc.(Eng) from the University of Alberta in 1973, his M.Sc.(Eng) from Queen's University in 1974, and his Ph.D. from Queen's University in 1977. His doctoral research was in the field of road vehicle dynamics. From 1977 to 1979, he was a Defence Scientist with the Defence Research Establishment Atlantic where he was engaged in research on the dynamics of novel ships. From 1979 to 1981 he was Senior Dynamicist with the Urban Transportation Development Corporation where he worked on rail vehicle dynamics, particularly suspension design for steerable rail vehicles. He joined Queen's University in 1981 and, while conducting research into vehicle dynamics and multibody dynamics, has been teaching undergraduate courses on dynamics and vibrations and postgraduate courses on advanced dynamics and engineering analysis. Dr. Anderson has been the recipient of several departmental and faculty-wide teaching awards. He has also served the University in the academic administrative roles of Head of Department, Associate Dean (Research), and Dean of Graduate Studies.
Content
Preface xi
About the Companion Website xv
Part I Modeling: Deriving Equations of Motion 1
1 Kinematics 3
1.1 Derivatives of Vectors 3
1.2 Performing Kinematic Analysis 5
1.3 Two Dimensional Motion with Constant Length 6
1.4 Two Dimensional Motion with Variable Length 8
1.5 Three Dimensional Kinematics 10
1.6 Absolute Angular Velocity and Acceleration 13
1.7 The General Acceleration Expression 14
Exercises 16
2 Newton's Equations of Motion 19
2.1 The Study of Motion 19
2.2 Newton's Laws 19
2.3 Newton's Second Law for a Particle 20
2.4 Deriving Equations of Motion for Particles 21
2.5 Working with Rigid Bodies 25
2.6 Using F = ma in the Rigid Body Force Balance 26
2.7 Using F = dG/dt in the Rigid Body Force Balance 28
2.8 Moment Balance for a Rigid Body 30
2.9 The Angular Momentum Vector - HO 33
2.10 A Physical Interpretation of Moments and Products of Inertia 36
2.11 Euler's Moment Equations 40
2.12 Throwing a Spiral 41
2.13 A Two Body System 42
2.14 Gyroscopic Motion 48
Exercises 52
3 Lagrange's Equations of Motion 55
3.1 An Example to Start 55
3.2 Lagrange's Equation for a Single Particle 58
3.3 Generalized Forces 62
3.4 Generalized Forces as Derivatives of Potential Energy 64
3.5 Dampers - Rayleigh's Dissipation Function 65
3.6 Kinetic Energy of a Free Rigid Body 67
3.7 A Two Dimensional Example using Lagrange's Equation 70
3.7.1 The Kinetic Energy 70
3.7.2 The Potential Energy 71
3.7.3 The ¿¿¿¿ Equation 72
3.7.4 The ¿¿¿¿ Equation 73
3.8 Standard Form of the Equations of Motion 73
Exercises 74
Part II Simulation: Using the Equations of Motion 77
4 Equilibrium Solutions 79
4.1 The Simple Pendulum 79
4.2 Equilibrium with Two Degrees of Freedom 80
4.3 Equilibrium with Steady Motion 81
4.4 The General Equilibrium Solution 84
Exercises 85
5 Stability 87
5.1 Analytical Stability 87
5.2 Linearization of Functions 92
5.3 Example: A System with Two Degrees of Freedom 95
5.4 Routh Stability Criterion 99
5.5 Standard Procedure for Stability Analysis 103
Exercises 105
6 Mode Shapes 107
6.1 Eigenvectors 107
6.2 Comparing Translational and Rotational Degrees of Freedom 111
6.3 Nodal Points in Mode Shapes 115
6.4 Mode Shapes with Damping 116
6.5 Modal Damping 118
Exercises 122
7 Frequency Domain Analysis 125
7.1 Modeling Frequency Response 125
7.2 Seismic Disturbances 132
7.3 Power Spectral Density 133
7.3.1 Units of the PSD 138
7.3.2 Simulation using the PSD 139
Exercises 143
8 Time Domain Solutions 145
8.1 Getting the Equations of Motion Ready for Time Domain Simulation 146
8.2 A Time Domain Example 147
8.3 Numerical Schemes for Solving the Equations of Motion 149
8.4 Euler Integration 149
8.5 An Example Using the Euler Integrator 151
8.6 The Central Difference Method: An (h2) Method 153
8.7 Variable Time Step Methods 155
8.8 Methods with Higher Order Truncation Error 157
8.9 The Structure of a Simulation Program 159
Exercises 163
Part III Working with Experimental Data 165
9 Experimental Data - Frequency Domain Analysis 167
9.1 Typical Test Data 167
9.2 Transforming to the Frequency Domain - The CFT 169
9.3 Transforming to the Frequency Domain - The DFT 172
9.4 Transforming to the Frequency Domain - A Faster DFT 174
9.5 Transforming to the Frequency Domain - The FFT 175
9.6 Transforming to the Frequency Domain - An Example 176
9.7 Sampling and Aliasing 179
9.8 Leakage and Windowing 184
9.9 Decimating Data 187
9.10 Averaging DFTs 189
Exercises 189
A Representative Dynamic Systems 193
A.1 System 1 193
A.2 System 2 193
A.3 System 3 194
A.4 System 4 194
A.5 System 5 195
A.6 System 6 195
A.7 System 7 196
A.8 System 8 197
A.9 System 9 197
A.10 System 10 198
A.11 System 11 198
A.12 System 12 199
A.13 System 13 200
A.14 System 14 200
A.15 System 15 201
A.16 System 16 201
A.17 System 17 202
A.18 System 18 202
A.19 System 19 203
A.20 System 20 203
A.21 System 21 204
A.22 System 22 204
A.23 System 23 205
B Moments and Products of Inertia 207
B.1 Moments of Inertia 207
B.2 Parallel Axis Theorem for Moments of Inertia 208
B.3 Parallel Axis Theorem for Products of Inertia 210
B.4 Moments of Inertia for Commonly Encountered Bodies 210
C Dimensions and Units 213
D Least Squares Curve Fitting 215
Index 219
1
Kinematics
Kinematics is defined as the study of motion without reference to the forces that cause the motion. A proper kinematic analysis is an essential first step in any dynamics problem. This is where the analyst defines the degrees of freedom and develops expressions for the absolute velocities and accelerations of the bodies in the system that satisfy all of the physical constraints. The ability to differentiate vectors with respect to time is a critical skill in kinematic analysis.
1.1 Derivatives of Vectors
Vectors have two distinct properties - magnitude and direction. Either or both of these properties may change with time and the time derivative of a vector must account for both.
The rate of change of a vector with respect to time is therefore formed from,
- The rate of change of magnitude .
- The rate of change of direction .
Figure 1.1 A vector changing with time.
Figure 1.1 shows the vector that changes after a time increment, , to .
The difference between and can be defined as the vector shown in Figure 1.1 and, by the rules of vector addition,
(1.1)or,
(1.2)Then, using the definition of the time derivative,
(1.3)Imagine now that Figure 1.1 is compressed to show only an infinitesimally small time interval, . The components of for the interval are shown in Figure 1.1. They are,
- A component aligned with the vector . This is a component that is strictly due to the rate of change of magnitude of . The magnitude of is where is the rate of change of length (or magnitude) of the vector . The direction of is the same as the direction of . Let be designated1 as .
- A component that is perpendicular to the vector . That is, a component due to the rate of change of direction of the vector. Terms of this type arise only when there is an angular velocity. The rate of change of direction term arises from the time rate of change of the angle in Figure 1.1 and is the magnitude of the angular velocity of the vector. The rate of change of direction therefore arises from the angular velocity of the vector. The magnitude of is where is the length of . By definition the rate of change of the angle (i.e. ) has the same positive sense as the angle itself. It is clear that is the "tip speed" one would expect from an object of length rotating with angular speed .
The angular velocity is itself a vector quantity since it must specify both the angular speed (i.e. magnitude) and the axis of rotation (i.e. direction). In Figure 1.1, the speed of rotation is and the axis of rotation is perpendicular to the page. This results in an angular velocity vector,
(1.4)where the right handed set of unit vectors, , is defined in Figure 1.2. Note that it is essential that right handed coordinate systems be used for dynamic analysis because of the extensive use of the cross product and the directions of vectors arising from it. If there is a right handed coordinate system , with respective unit vectors , then the cross products are such that,
Figure 1.2 Even 2D problems are 3D.
Using this definition of the angular velocity, the motion of the tip of vector , resulting from the angular change in time , can be determined from the cross product
which, by the rules of the vector cross product, has magnitude,
and a direction that, according to the right hand rule2 used for cross products, is perpendicular to both and and, in fact, lies in the direction of .
Combining these two terms to get and substituting into Equation 1.3 results in,
(1.5)The time derivative of any vector, , can therefore be written as,
(1.6)It is important to understand that the angular velocity vector, , is the angular velocity of the coordinate system in which the vector, , is expressed. There is a danger that the rate of change of direction terms will be included twice if the angular velocity of the vector with respect to the coordinate system in which it measured is used instead. The example presented in Section 1.3 shows a number of different ways to arrive at the derivative of a vector which rotates in a plane.
1.2 Performing Kinematic Analysis
Before proceeding with examples of kinematic analyses we state here the steps that are necessary in achieving a successful result. This first step in any dynamic analysis is vitally important. The goal is to derive expressions for the absolute velocities and accelerations of the centers of mass of the bodies making up the system being analyzed. In addition, expressions for the absolute angular velocities and angular accelerations of the bodies will be required. It is at this first step of the analysis that degrees of freedom are defined and constraints on relative motion between bodies are satisfied.
For this general description of kinematic analysis, we assume that we are analyzing a system that has multiple bodies connected to each other by joints and that we are attempting to derive an expression for the acceleration of the center of mass of a body that is not the first in the assembly.
The procedure is as follows.
- Find a fixed point (i.e. one having no velocity or acceleration) in the system from which you can begin to write relative position vectors that will lead to the centers of mass of bodies in the system.
- Define a position vector that goes from the fixed point, through the first body, to the next joint in the system. This is the position of the joint relative to the fixed point.
- Determine how many degrees of freedom, both translational and rotational, are required to define the motion of the relative position vector just defined. The degrees of freedom must be chosen to satisfy the constraints imposed by the joint that connects this body to ground.
- Define a coordinate system in which the relative position vector will be written and determine the angular velocity of the coordinate system.
- Repeat the previous three steps as you go from joint to joint in the system, always being careful to satisfy the joint constraints by defining appropriate degrees of freedom.
- When the desired body is reached, define a final relative position vector from the joint to the center of mass.
- The sum of all the relative position vectors will be the absolute position of the center of mass and the derivatives of the sum of vectors will yield the absolute velocity and acceleration of the center of mass.
1.3 Two Dimensional Motion with Constant Length
Figure 1.3 shows a rigid rod of length, , rotating about a fixed point, , in a plane. An expression for the velocity of the free end of the rod, , relative to point is desired.
Figure 1.3 A rigid rod rotating about a fixed point.
By definition, the velocity of relative to is the time derivative of the position of relative to . This position vector is designated and is shown in the figure.
In order to differentiate the position vector, we must have an expression for it and this means we must first choose a coordinate system3 in which to work. For a start, we can choose a right handed coordinate system fixed in the ground. The set of unit vectors is such a system. The angular velocity of this coordinate system is zero (i.e. ) since it is fixed in the ground.
An expression for the position of relative to in this system is,
(1.7)We apply Equation 1.6 to to get,
(1.8)In this coordinate system, it is clear that there is a rate of change of magnitude of the vector only and the velocity of point relative to after performing the simple differentiation is,
(1.9)Another derivation of the velocity of relative to might use the system of unit vectors that are fixed in the rod. The advantage of using this system is that the position vector is easily expressed as,
(1.10)Note that the length of this vector is a constant so that the total derivative must come from its rate of change of direction. The angular velocity of the coordinate system is equal to the angular velocity of the rod since the coordinate system is fixed in the rod. That is,
(1.11)and the velocity of relative to is therefore4,
(1.12)Since is constant, , and the final result is,
(1.13)We now have two expressions for (Equations 1.9 and 1.13). Since there can only be one value of this relative velocity, the two expressions must be equal to each other. However, the use of two different coordinate systems makes them look different. In order to compare them, we must be able to transform results from one coordinate system to the other.
Keep in mind that sequential sets of unit vectors are related to each other by simple plane rotations. Also note that the unit vectors are not related to any point in the system - they simply express directions. Given these two facts, we can relate the two sets of unit vectors we have been using by noting that...
