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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
An in-depth introduction to the foundations of vibrations for students of mechanical engineering
For students pursuing their education in Mechanical Engineering, An Introduction to Mechanical Vibrations is a definitive resource. The text extensively covers foundational knowledge in the field and uses it to lead up to and include: finite elements, the inerter, Discrete Fourier Transforms, flow-induced vibrations, and self-excited oscillations in rail vehicles.
The text aims to accomplish two things in a single, introductory, semester-length, course in vibrations. The primary goal is to present the basics of vibrations in a manner that promotes understanding and interest while building a foundation of knowledge in the field. The secondary goal is to give students a good understanding of two topics that are ubiquitous in today's engineering workplace - finite element analysis (FEA) and Discrete Fourier Transforms (the DFT- most often seen in the form of the Fast Fourier Transform or FFT). FEA and FFT software tools are readily available to both students and practicing engineers and they need to be used with understanding and a degree of caution. While these two subjects fit nicely into vibrations, this book presents them in a way that emphasizes understanding of the underlying principles so that students are aware of both the power and the limitations of the methods.
In addition to covering all the topics that make up an introductory knowledge of vibrations, the book includes:
● End of chapter exercises to help students review key topics and definitions
● Access to sample data files, software, and animations via a dedicated website
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Veröffentlichungsjahr: 2020
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Table of Contents
Cover
Preface
About the Companion Website
1 The Transition from Dynamics to Vibrations
1.1 Bead on a Wire: The Nonlinear Equations of Motion
1.2 Equilibrium Solutions
1.3 Linearization
1.4 Summary
Exercises
Notes
2 Single Degree of Freedom Systems – Modeling
2.1 Modeling Single Degree of Freedom Systems
Exercises
Notes
3 Single Degree of Freedom Systems – Free Vibrations
3.1 Undamped Free Vibrations
3.2 Response to Initial Conditions
3.3 Damped Free Vibrations
3.4 Root Locus
Exercises
Notes
4 SDOF Systems – Forced Vibrations – Response to Initial Conditions
4.1 Time Response to a Harmonically Applied Force in Undamped Systems
Exercises
5 SDOF Systems – Steady State Forced Vibrations
5.1 Undamped Steady State Response to a Harmonically Applied Force
5.2 Damped Steady State Response to a Harmonically Applied Force
5.3 Response to Harmonic Base Motion
5.4 Response to a Rotating Unbalance
5.5 Accelerometers
Exercises
Notes
6 Damping
6.1 Linear Viscous Damping
6.2 Coulomb or Dry Friction Damping
6.3 Logarithmic Decrement
Exercises
Notes
7 Systems with More than One Degree of Freedom
7.1 2DOF Undamped Free Vibrations – Modeling
7.2 2DOF Undamped Free Vibrations – Natural Frequencies
7.3 2DOF Undamped Free Vibrations – Mode Shapes
7.4 Mode Shape Descriptions
7.5 Response to Initial Conditions
7.6 2DOF Undamped Forced Vibrations
7.7 Vibration Absorbers
7.8 The Method of Normal Modes
7.9 The Cart and Pendulum Example
7.10 Normal Modes Example
Exercises
Notes
8 Continuous Systems
8.1 The Equations of Motion for a Taut String
8.2 Natural Frequencies and Mode Shapes for a Taut String
8.3 Vibrations of Uniform Beams
Exercises
Notes
9 Finite Elements
9.1 Shape Functions
9.2 The Stiffness Matrix for an Elastic Rod
9.3 The Mass Matrix for an Elastic Rod
9.4 Using Multiple Elements
9.5 The Two‐noded Beam Element
9.6 Two‐noded Beam Element Vibrations Example
Exercises
Notes
10 The Inerter
10.1 Modeling the Inerter
10.2 The Inerter in the Equations of Motion
10.3 An Examination of the Effect of an Inerter on System Response
10.4 The Inerter as a Vibration Absorber
Exercises
Notes
11 Analysis of Experimental Data
11.1 Typical Test Data
11.2 Transforming to the Frequency Domain – The CFT
11.3 Transforming to the Frequency Domain – The DFT
11.4 Transforming to the Frequency Domain – A Faster DFT
11.5 Transforming to the Frequency Domain – The FFT
11.6 Transforming to the Frequency Domain – An Example
11.7 Sampling and Aliasing
11.8 Leakage and Windowing
11.9 Decimating Data
11.10 Averaging FFTs
Exercises
Notes
12 Topics in Vibrations
12.1 What About the Mass of the Spring?
12.2 Flow‐induced Vibrations
12.3 Self‐Excited Oscillations of Railway Wheelsets
12.4 What is a Rigid Body Mode?
12.5 Why Static Deflection is Very Useful
Exercises
Notes
Appendix A: Least Squares Curve Fitting
Appendix B: Moments of Inertia
B.1 Parallel Axis Theorem for Moments of Inertia
B.2 Moments of Inertia for Commonly Encountered Bodies
Notes
Index
End User License Agreement
List of Tables
Chapter 6
Table 6.1 Effective viscous damping coefficients.
Chapter 10
Table 10.1 Inertia coefficients.
Chapter 11
Table 11.1 Fourier Transforms Computational Effort.
Table 11.2 Sampled Data.
Table 11.3 DFT Coefficients.
Table 11.4 DFT Coefficients.
Chapter 12
Table 12.1 The Routh Table.
Table 12.2 The Routh Table for the wheelset.
Appendix A
Table A.1 Sample data points.
List of Illustrations
Chapter 1
Figure 1.1 A bead on a wire.
Figure 1.2 Free Body Diagram of a bead on a wire.
Figure 1.3 A 2D representation of the bead on a wire.
Figure 1.4 A linear viscous damper.
Figure 1.5 A simple pendulum.
Figure 1.6 Nonlinear structural element – Linearization and effective stiffn...
Figure E1.1
Figure E1.2
Figure E1.5
Figure E1.6
Chapter 2
Figure 2.1 A mass on a spring.
Figure 2.2 Independent front suspension.
Figure 2.3 Assembly of the mass/spring system.
Figure 2.4 FBD of the mass/spring system.
Figure 2.5 A spring element.
Figure 2.6 The linear spring constitutive relationship.
Figure 2.7 A mass on a spring.
Figure 2.8 A mass on a spring – FBD.
Figure 2.9 A simple pendulum.
Figure 2.10 Spring deflection due to a large angle of rotation.
Figure 2.11 A body with a rotational DOF: How to deal with a spring, a dampe...
Figure 2.12 Gravitational effects.
Figure E2.2
Figure E2.3
Figure E2.4
Figure E2.5
Figure E2.6
Chapter 3
Figure 3.1 Simple harmonic motion.
Figure 3.2 Simple harmonic motion with a phase shift.
Figure 3.3 Mass/spring/damper system.
Figure 3.4 Mass/spring/damper FBD.
Figure 3.5 The underdamped response.
Figure 3.6 The critically damped response.
Figure 3.7 The overdamped response.
Figure 3.8 The root locus.
Figure E3.1
Figure E3.2
Figure E3.3
Figure E3.4
Figure E3.5
Figure E3.6
Figure E3.7
Chapter 4
Figure 4.1 SDOF system with a harmonically applied force.
Figure 4.2 SDOF system with a harmonically applied force – FBD.
Figure 4.3 Widely separated frequencies.
Figure 4.4 The
Beating
phenomenon.
Figure 4.5 Resonance.
Figure E4.2
Chapter 5
Figure 5.1 SDOF system with a harmonically applied force.
Figure 5.2 Frequency response – SDOF undamped system.
Figure 5.3 Magnification Factor ‐ SDOF undamped system.
Figure 5.4 Damped SDOF system with a harmonically applied force.
Figure 5.5 Magnification Factor – SDOF damped system.
Figure 5.6 Damped SDOF system with harmonic base motion.
Figure 5.7 Free body diagram – damped SDOF system with harmonic base motion.
Figure 5.8 Magnification Factor – Harmonic base motion.
Figure 5.9 Force transmissibility – Harmonic base motion.
Figure 5.10 Damped SDOF system with a rotating unbalance.
Figure 5.11 Free body diagram – Damped SDOF system with a rotating unbalance...
Figure 5.12 Amplitude Ratio – SDOF rotating unbalance.
Figure 5.13 Schematic layout of a typical piezoelectric accelerometer.
Figure 5.14 Model of a typical piezoelectric accelerometer.
Figure 5.15 Free body diagram of the accelerometer model.
Figure 5.16 Piezoelectric accelerometer – % error.
Figure E5.1
Figure E5.3
Figure E5.8
Figure E5.9
Chapter 6
Figure 6.1 A linear viscous damper.
Figure 6.2 Test setup for energy removed by a linear viscous damper.
Figure 6.3 Hysteresis loop for a linear viscous damper.
Figure 6.4 Experimental hysteresis loops for two shock absorbers.
Figure 6.5 Friction force magnitude and direction.
Figure 6.6 A system with Coulomb friction.
Figure 6.7 Response with Coulomb friction.
Figure 6.8 The underdamped response.
Figure E6.1
Figure E6.3
Chapter 7
Figure 7.1 A 2DOF undamped system.
Figure 7.2 Free body diagram for the 2DOF undamped system.
Figure 7.3 First mode shape for the 2DOF undamped system.
Figure 7.4 Second mode shape for the 2DOF undamped system.
Figure 7.5 Another sample system.
Figure 7.6 A 2DOF undamped, forced, system.
Figure 7.7 A model of a machine experiencing large amplitudes.
Figure 7.8 The machine with the vibration absorber mounted.
Figure 7.9 The cart and pendulum system.
Figure 7.10 The cart and pendulum system – FBD.
Figure E7.1
Figure E7.2
Figure E7.3
Figure E7.5
Figure E7.6
Figure E7.7
Chapter 8
Figure 8.1 A taut string.
Figure 8.2 The first five string modes.
Figure 8.3 Deflections of a uniform beam.
Figure 8.4 A beam element.
Figure 8.5 A cantilever beam.
Figure 8.6 The first three cantilever beam modes.
Chapter 9
Figure 9.1 A beam element.
Figure 9.2 A non‐uniform beam.
Figure 9.3 An elastic rod.
Figure 9.4 An elastic rod element.
Figure 9.5 An element of the element.
Figure 9.6
versus
for the element.
Figure 9.7 The rod element with nodal forces applied.
Figure 9.8 A statically loaded rod modeled with one element.
Figure 9.9 The velocity of the element of the element.
Figure 9.10 Two rod elements in an assembly.
Figure 9.11 Free body diagrams of the two rod elements.
Figure 9.12 The global mass matrix.
Figure 9.13 The global stiffness matrix.
Figure 9.14 The two‐noded beam element.
Figure 9.15
versus
for the element.
Figure 9.16 Node and element numbering.
Figure 9.17 Node and element numbering.
Figure 9.18 Externally applied forces and moments.
Figure 9.19 The harmonically varying applied load.
Figure E9.3
Figure E9.4
Chapter 10
Figure 10.1 An inerter implementation.
Figure 10.2 A screw.
Figure 10.3 The inerter symbol.
Figure 10.4 Single degree of freedom system with an inerter.
Figure 10.5 Free Body Diagram for the single degree of freedom system with a...
Figure 10.6 Two degree of freedom system with inerters.
Figure 10.7 Simplified two degree of freedom system with an inerter.
Figure 10.8 A single degree of freedom system with harmonic ground motion.
Figure 10.9 Free body diagram for the single degree of freedom system with h...
Figure 10.10 A single degree of freedom system with an inerter and harmonic ...
Figure 10.11 Free body diagram for the single degree of freedom system with ...
Chapter 11
Figure 11.1 A measured variable
plotted versus time.
Figure 11.2 The square of
plotted versus time.
Figure 11.3 The example function,
, plotted versus time.
Figure 11.4 The DFT amplitudes of the example function,
, plotted versus fr...
Figure 11.5 Aliasing.
Figure 11.6 The folding frequency.
Figure 11.7 Aliased DFT results.
Figure 11.8 The DFT for the first example.
Figure 11.9 The DFT for the second example.
Figure 11.10 CFT approximation to the square wave.
Figure 11.11 The Hanning window.
Figure 11.12 The data from Equation 11.83.
Figure 11.13 The windowed data.
Figure 11.14 The DFT for the second example with windowing.
Figure 11.15 The DFT before downsampling.
Figure 11.16 The DFT after downsampling.
Figure 11.17 The time series before and after digital filtering.
Figure 11.18 The DFT after decimating.
Figure 11.19 A low‐pass filter circuit.
Figure 11.20 Low‐pass filter frequency response in dB on a logarithmic scale...
Figure 11.21 Low‐pass filter frequency response on a linear scale.
Figure 11.22 Exponential moving average smoothed data.
Figure 11.23 Exponential moving average DFT.
Figure 11.24 Low‐pass digital filter frequency response.
Figure 11.25 Low‐pass filter frequency response near the cut‐off frequency.
Figure 11.26 Noisy time signal.
Figure 11.27 Noisy time signal zoomed.
Figure 11.28 DFT – 1 average.
Figure 11.29 DFT – 10 averages.
Figure 11.30 DFT – 20 averages.
Figure E11.5
Chapter 12
Figure 12.1 A mass on a spring.
Figure 12.2 A spring connecting two masses.
Figure 12.3 Lift and drag in a wind tunnel.
Figure 12.4 Typical lift and drag forces versus angle of attack.
Figure 12.5 A suspended airfoil in steady flow.
Figure 12.6 Free body diagram of the airfoil.
Figure 12.7 Representation of a von Karman vortex street in a wake.
Figure 12.8 Helical strakes on tall chimneys.
Figure 12.9 A railway truck supported by its two wheelsets.
Figure 12.10 Parameters for a railway wheelset.
Figure 12.11 Back‐to‐back cones forming a railway wheelset.
Figure 12.12 Wheelset degrees of freedom.
Figure 12.13 Creep forces.
Figure 12.14 Wheelset free body diagram.
Figure 12.15 Wheelset instability.
Figure 12.16 System with a rigid body mode.
Figure 12.17 The three modes.
Figure 12.18 A spring/mass system.
Figure 12.19 The model of a double wishbone suspension.
Figure E12.1
Figure E12.3
Appendix A
Figure A.1 Three data points and two Least Squares curve fits.
Appendix B
Figure B.1 Parallel Axis Theorem.
Guide
Cover
Table of Contents
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Introduction to Mechanical Vibrations
Ronald J. Anderson
Queen’s UniversityKingstonCanada
This edition first published 2020© 2020 John Wiley & Sons Ltd
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The right of Ronald J. Anderson to be identified as the author of this work has been asserted in accordance with law.
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To June
Preface
When I first studied vibrations, as an undergraduate student, its importance was clear to our class because it was a required course for mechanical engineers. A few years later, when I started teaching vibrations and new topics were entering the field of mechanical engineering, a course on vibrations was no longer seen as being important enough to be a required so it became an elective. Now, although “mechanical engineering” is still used as an umbrella term, the students who graduate are mechanical engineers with a specialization. Students in the specialized streams do not have time to cover all of the topics that used to be expected of mechanical engineers so some graduate without thermodynamics, others without vibrations, and so on. Specialization like this is inevitable given the expanding scope of knowledge in engineering and the limited time available to undergraduate students but it means even fewer students are learning about vibrations and other important topics. While preparing this introduction to vibrations, I kept in mind the need for undergraduate students to have a better understanding of two topics that are ubiquitous in today's engineering workplace – finite element analysis (FEA) and fast Fourier transforms (FFT). FEA and FFT software tools are readily available to both students and practicing engineers and they need to be used with understanding and a degree of caution.
I was never able to find a textbook that covered just enough, and the right, material for a semester length introductory course in vibrations. I used many textbooks over the years but there was never a fit with what I thought should be in an introduction to vibrations. I was looking for something student‐friendly in that it should be readable, almost conversational, but still be mathematically rigorous. What I found on the market were mainly “reference books” as opposed to “teaching books”. Many of the textbooks I tried are very good at covering, in depth, a broad range of topics in vibrations, but students have difficulty using them as a first text in the subject, mainly because of the overwhelming amount of material presented.
This book grew from my attempt to accomplish two things in a single course in vibrations. The primary goal is, of course, to present the basics of vibrations in a manner that promotes understanding and interest while building a foundation of knowledge in the field. To do this, I have had to give only brief coverage of many important topics with the hope that some students will go on to expand their knowledge in these areas if their interest is piqued. As mentioned earlier, a secondary goal is to give students a good understanding of finite element analysis and Fourier transforms. While these two subjects fit nicely into vibrations, this book presents them in a way that emphasizes understanding of the underlying principles so that students are aware of both the power and the limitations of the methods.
Chapter 1 addresses the way in which a student has to think about previous undergraduate dynamics knowledge in order to make the transition to analysis of vibrating systems. It introduces the idea of small motions about a stable equilibrium state and addresses the details of linearization. Lagrange's Equations are introduced here and students take to them very quickly as an alternative to Newton's Laws.
Chapter 2 considers the details of analyzing single degree of freedom systems. While much of this material is obvious to those skilled in vibrations, it is vital material for developing the students' abilities. It covers topics such as preloads in springs and why gravitational forces don't need to be included because they are canceled out by the constant preloads. It looks at the constitutive relationship for a spring and shows how to draw free body diagrams consistently and accurately.
Chapter 3 is about free vibrations of single degree of freedom systems. It covers systems with and without damping and tries to make sense of what it means to solve a second‐order, linear differential equation without being too prescriptive about it.
Chapter 4 looks at time response when applying a harmonic forcing function to an undamped single degree of freedom system, thereby introducing the phenomena of beating and resonance. This is a short chapter although the subject of time response, if presented in detail, could make for a very long one. Time response is an area that I see as being of secondary importance in an introduction to vibrations.
Chapter 5 considers steady state forced vibrations, covering harmonic forcing functions, harmonic base motion, systems with a rotating unbalance, and accelerometer design. This is really the essence of vibrational analysis and is covered in detail.
Chapter 6 is devoted to the very important subject of damping. Linear viscous damping is discussed and the concept of modeling other energy removal devices as “equivalent linear viscous dampers” is introduced. Coulomb damping is covered. The concept of logarithmic decrement is introduced.
Chapter 7 recognizes that systems often have more than one degree of freedom. Deriving the equations of motion for systems with many degrees of freedom is discussed. The concept of multiple natural frequencies, each associated with a different mode shape, is covered in detail. Description of mode shapes is given a lot of time because of its importance in the field. Forced vibrations, vibration absorbers, and the method of normal modes are covered.
Chapter 8 moves on into the study of continuous systems and uses vibrations of a taut string and a cantilever beam as examples of two continuous systems where solutions can be found. The concept of infinitely many degrees of freedom is introduced.
Chapter 9 recognizes that solutions cannot always be found for continuous systems so the finite element method is introduced as an alternate way to get solutions. Shape functions, element mass and stiffness matrices, and assembly of global mass and stiffness matrices are covered, as well as application of boundary conditions and applied forces. Derivations here are handled using Lagrange's Equation because the students are familiar with that approach by the time we get to finite elements. This is certainly not the approach taken by experts in finite elements but it is a useful and appropriate way to get the students to understand the assumptions made in using FEA.
Chapter 10 is devoted to a relatively new device called the “inerter”. This device provides a force that is proportional to the relative acceleration across it. A concept for how to construct such a device and analyses showing its effects on vibrations are presented. This is presented to give the students a look at developing technology with which they can have some fun while realizing that pat answers coming from what they have learned about vibrations to this point may not apply to this device.
Chapter 11 presents a detailed description of how to analyze experimental data in studying vibrations. Topics covered include: Discrete Fourier transforms; sampling and aliasing; leakage and windowing. The approach I have taken here is very old‐fashioned in that it treats the Discrete Fourier Transform as something that is derived from a least squares curve fit. This harks back to methods used more than fifty years ago but it enhances students' understanding of what lies behind the transformation to the frequency domain. This is a long chapter that presents methods that the students can program themselves so they don't have to be tied down to packaged FFT programs. Once they have their software working, they can experiment with data sets that clearly demonstrate aliasing, leakage, and so on.
Chapter 12 discusses a variety of topics in vibrations such as how to handle the mass of a spring, flow‐induced vibrations, self‐excited vibrations in rail vehicles, rigid body modes, and things you can determine from the static deflection of a system. I find that I am usually able to get somewhere into this chapter before the semester is over. The material in this chapter is interesting but certainly doesn't need to be covered to have a complete introduction to vibrations.
It is my hope that this book strikes the right balance for professors teaching introductory vibrations and for their students. I wish them all well.
Ronald J. Anderson
September, 2019
Kingston, Canada
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1The Transition from Dynamics to Vibrations
Introductory undergraduate courses on dynamics typically consider large scale motions of systems of particles and/or rigid bodies and instantaneous solutions to their nonlinear, governing equations. You may recall working on dynamics problems where a system of bodies starts from rest at a prescribed position and your task was to determine, for example, the angular acceleration of a body or the forces acting on some part of the system. Solutions like this, while having some utility, provide only part of the understanding of the system that is required for a successful design. In most cases, the derived governing equations are complete enough but the “snapshot” solutions don't help much with the design process.
There are, in fact, many things that can be done with the equations governing the dynamic motion of the system. Briefly, they can be used to
Find where the bodies in the system would be if the system were at rest. These are the
Equilibrium States
.
Determine whether the equilibrium states are stable or unstable.
Determine how the system behaves for small motions away from a stable equilibrium state.
Determine the response of the system in the time domain through the use of numerical simulations. This is the most complex type of analysis and, perhaps surprisingly, gives the least information to the designer until the design has reached the fine tuning phase. The simulations are the analog of “cut and try” experiments where an unsuccessful result gives little information on what to change in order to improve the design.
While going through the material presented in this book, you will be concentrating on very small motions of systems about stable equilibrium states. In doing so, you will see connections to topics you may have covered in courses on statics, on dynamics, and on control systems. You will become very familiar with the linearized, differential, equations of motion for dynamic systems moving around stable equilibrium states and methods for deriving and solving them. This is the essence of Vibrations.
To get started and as a review of sorts we begin with the dynamic analysis to a relatively simple system – a bead sliding on a rotating semicircular wire.
1.1 Bead on a Wire: The Nonlinear Equations of Motion
First courses on the subject of Dynamics, whether for particles or rigid bodies, are primarily concerned with teaching the basics of kinematics, free body diagrams, and applications of Newton's Laws of Motion. Applying these three concepts sequentially will lead to a set of simultaneous force and moment balance equations that take account of kinematic constraints.
There are different ways of approaching these problems. One can use a formal vector‐based approach and we will start with that here because it gives a complete set of governing equations including solutions for all constraint forces that are required to enforce kinematic constraints on the motion. A shorthand version of this approach which may be called an “informal vector approach” is often used in practice and that will be the second method addressed here. It typically works with two‐dimensional views and leads to the governing equations of motion without necessarily solving for all constraint forces. The third approach will see the equations of motion derived using Lagrange's Equations. This is a work/energy approach that leads to the nonlinear differential equation of motion with minimal effort on the part of the analyst. The kinematic constraint forces are automatically eliminated as the governing equations are derived, leaving a designer with no information about forces acting on elements of the system unless extra work is done to find them. Lagrange's Equations are not typically introduced to undergraduate engineers as often as Newton's Laws are, so extra effort is made in this chapter to introduce the procedures for applying Lagrange's Equations to mechanical systems.
As an example, consider Figure 1.1. The figure shows a small bead with mass, , sliding on a frictionless semicircular wire that rotates about a vertical axis with a constant angular velocity, . The wire has radius . Gravity acts to pull the mass to the bottom of the semicircle while centripetal effects try to move it to the top. The single angular degree of freedom, , is sufficient to describe the motion of the bead on the wire.
Figure 1.1 A bead on a wire.
1.1.1 Formal Vector Approach using Newton's Laws
Using the formal vector approach, the first step in the kinematic analysis is to choose a coordinate system (i.e. a set of unit vectors) that is convenient for expressing the vectors that will be used. The coordinate system may be fixed or rotating with some known angular velocity. In this case, we will use the (, , ) system shown in Figure 1.1. This is a rotating system fixed in the wire so that and stay in the plane of the wire and is perpendicular to the plane. Furthermore, and remain horizontal and is always vertical. The angular velocity of the coordinate system is .
We use the general approach to differentiating vectors, as follows, where can be a position vector, a velocity vector, an angular momentum vector, or any other vector.
It is important to understand that the angular velocity vector, , is the absolute 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 relative to the coordinate system in which it is measured is used instead.
We start the kinematic analysis by locating a fixed point, in this case point , and writing an expression for the position vector that locates with respect to .
The absolute velocity of is
Then, using Equation 1.1 and recognizing that since is a fixed point and that since the radius of a semicircle is constant,
which can be simplified to
The absolute acceleration of is then
which simplifies to
Once an expression for the absolute acceleration has been found, the kinematic analysis is complete and we move on to drawing a Free Body Diagram (FBD). For this example, the FBD is shown in Figure 1.2.
Figure 1.2 Free Body Diagram of a bead on a wire.
Constraints are taken into account when showing the forces acting on the bead. The forces shown and the rationale behind them are:
= the weight of the body acting vertically downward. This is the effect of gravity.
= one component of the normal force that the wire transmits to the mass. Since
is perpendicular to the plane of the wire, there can be a normal force in that direction.
= the other component of the normal force. We let it have an unknown magnitude
and align it with the radial direction since that direction is normal to the wire.
Note that there is no friction force because the system is frictionless. If there were, we would need to show a friction force acting in the direction that is tangential to the wire.
Once the FBD is complete, we can proceed to write Newton's Equations of Motion by simply summing forces in the positive coordinate directions and letting them equal the mass multiplied by the absolute acceleration in that direction. The result is three scalar equations as follows
At this point in the majority of undergraduate Dynamics courses we would count the number of unknowns that we have in the three equations to see if there is sufficient information to solve the problem. We would find five unknowns
and say that we are unable to solve this without further information since we have only three equations. A typical textbook problem would say, for example, that the mass is released from rest (i.e. ) at a specified angle, , thereby removing two of the unknowns and letting you solve for , and .
This solution gives an instantaneous look at the system that really doesn't point out the value of the equations derived. Equations do not have five unknowns. They have two unknown constraint forces, and , and a group of variables (, , ) that are related by differentiation. Rather than counting five unknowns as we did earlier, we should say that there are three unknowns
and three equations.
We can combine the three equations to eliminate and and we will be left with a single differential equation containing , , and . This nonlinear, ordinary differential equation is the equation of motion for the system. Given initial conditions for and , we can solve the equation of motion as a function of time and predict the angle, its derivatives, and the two normal forces at any time. The solution of nonlinear differential equations is not a trivial exercise but can be handled fairly easily using numerical techniques.
The equation of motion for this system can be found by multiplying Equation 1.8
