EBOOK - Dynamics of Cyclic Machines (Foundations of Engineering Mechanics) - Losif Vulfson | Cộng đồng Kỹ thuật cơ điện Việt Nam EBOOKBKMT

The book focuses on the methods of dynamic analysis and synthesis of machines, comprising cyclic action mechanisms, such as linkages, cams, steppers, etc. This book presents the modern methods of oscillation analysis in machines, including cyclic action mechanisms (linkage, cam, stepper, etc.). Basically, the intention is to build up a bridge between the classic theory of oscillations and its practical application in dynamic problems for cyclic machines.
Intensification of production processes always requires the growth of operating speeds, which in turn dictates the need for more in-depth and comprehensive accounting of the dynamic factors.
Obviously, problems of machine dynamics are discussed in a large number of textbooks and monographs, since this section of engineering science concerns both the wide variety of tasks and the various levels of coverage of each problem. The latter is associated both with the variety of interests pursued by the solution of a concrete engineering problem and with a large number of conditions and factors that determine the final outcome. Therefore, ready-made recipes are not very suitable for the formation of approaches to solve problems of this class.

Experience shows that the solution of scientific and engineering problems of machine dynamics depends on overcoming certain illusions. One of them is related to an assumption regarding the classical theory of mechanisms and machines, which presumes the absolute rigidity of links. Meanwhile, practical experience of machine operations shows that under modern operating speeds this assumption is acceptable only as afirst approximation, but in some cases even leads to incorrect
orientation in the analysis of complex dynamic processes and the selection of areas of further machine improvement. For instance, the indisputable influence of the geometric characteristics of cyclic mechanisms (position function, transfer functions, angles of pressure, etc.) on dynamic processes is sometimes wrongly perceived as the opportunity to solve a dynamic problem by purely geometric means.
In this respect, the wide range of modifications of the so-called optimal laws of motion, which are credited with the capacity to eliminate the oscillations of the output links, irrespective of a system’s frequency characteristics, is highly indicative. Thus, it is impossible to design modern machines without due regard to the oscillatory processes that in many ways define the productivity, quality of
production, durability, and reliability of the equipment, as well as the working conditions of a human operator.

1 Cyclic Mechanisms................................... 1
1.1 General Information About Cyclic Mechanisms . . . . . . . . . . . 1
1.1.1 Functional Features of Cyclic Mechanisms. . . . . . . . . 1
1.1.2 Position Function and Geometric Transfer
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Simplest Criteria for Dynamic Synthesis . . . . . . . . . . 5
1.2 Program Motion of the Links of Cyclic Mechanisms . . . . . . . . 6
1.2.1 Methods for Obtaining Program Motion . . . . . . . . . . 6
1.2.2 Structure of Law of Motion. Dimensionless
Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Dimensionless Constants of Laws of Motion . . . . . . . 10
1.2.4 Typical Problems of Synthesis of Motion Law . . . . . . 10
2 Dynamic Models of Cyclic Mechanical Systems.............. 17
2.1 Main Objectives of Machine Vibrations Analysis . . . . . . . . . . 17
2.2 Main Stages of Dynamic Analysis . . . . . . . . . . . . . . . . . . . . 18
2.3 Classification of Mechanical Vibrations . . . . . . . . . . . . . . . . . 19
2.4 Initial Data and the Principles of Dynamic Model Creation . . . 21
2.5 Typical Dynamic Models of Cyclic Systems and Their
Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Elements of the Dynamic Model and Their Reduction . . . . . . . 30
2.6.1 Inertial Characteristics . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.2 Characteristics of Elastic Elements . . . . . . . . . . . . . . 32
2.6.3 Parameters of Dissipation. . . . . . . . . . . . . . . . . . . . . 35
3 Mathematical Model.................................. 41
3.1 Some Information About Analytical Mechanics,
Applicable to the Analysis of Vibrations
in Cyclic Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Constraints, Implemented in Mechanisms. . . . . . . . . . 41
3.1.2 Presentation of Kinetic and Potential Energy
in the Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . 43
3.1.3 Lagrange Equations of the Second Kind . . . . . . . . . . 44
3.1.4 Special Form of the Second-Kind Lagrange
Equations with Redundant Coordinates . . . . . . . . . . . 45
3.1.5 Generation of the Appell’s Differential
Equations for Holonomic Systems. . . . . . . . . . . . . . . 46
3.1.6 Generation of Differential Equations by Using
the Inverse Method . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Specifics of Composition of Differential Equations
for Drives with Cyclic Mechanisms. . . . . . . . . . . . . . . . . . . . 48
4 Dynamic Models with Constant Parameters................. 63
4.1 Models with One Degree of Freedom . . . . . . . . . . . . . . . . . . 63
4.1.1 General Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.2 Solving for Steady Regimes . . . . . . . . . . . . . . . . . . . 65
4.1.3 Vibration Activity and Dynamic Errors . . . . . . . . . . . 74
4.2 Forced Vibrations of the Systems, with Finite Number
of Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.1 Harmonic Excitation . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.2 Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Dynamic Unloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4 Synthesis of the Cyclic Oscillatory Systems
with Quasi-Constant Amplitude-Frequency Characteristic. . . . . 91
4.4.1 Dynamic Model and Its Modifications . . . . . . . . . . . . 91
4.4.2 Systems with One Degree of Freedom. . . . . . . . . . . . 92
4.4.3 Systems with Two Degrees of Freedom. . . . . . . . . . . 98
5 Dynamic Models with Variable Parameters................. 103
5.1 Linearization of the Geometric Characteristics of the Cyclic
Mechanism in the Vicinity of the Program Motion . . . . . . . . . 103
5.2 Method of the Conditional Oscillator. . . . . . . . . . . . . . . . . . . 107
5.2.1 General Information About the Method
of the Conditional Oscillator . . . . . . . . . . . . . . . . . . 107
5.2.2 Analytical Method of Solving for Steady-State
Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.3 Numerical-Analytical Method of Solving
for Steady-State Regimes . . . . . . . . . . . . . . . . . . . . . 117
5.2.4 Systems with Many Degrees of Freedom . . . . . . . . . . 118
5.3 Dynamic Synthesis of Cyclic Mechanisms with Slow
Changing Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3.1 Stability Within an Arbitrary Time Interval . . . . . . . . 120
5.3.2 Ways to Vibration Activity Reduce and Some
Dynamic Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4 Conditions of Dynamic Stability in the Areas of Parametric
Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.1 General Information About the Parametric
Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.2 Methods of Elimination of the Parametric
Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4.3 Estimation of Resonant Amplitudes Under
the Joint Action of the Disturbing Force
and Parametric Excitation . . . . . . . . . . . . . . . . . . . . 128
5.5 Parametric Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.5.1 Single Parametric Impulse . . . . . . . . . . . . . . . . . . . . 129
5.5.2 Dynamic Effect Due to Action of Periodic
Parametric Impulses . . . . . . . . . . . . . . . . . . . . . . . . 133
5.6 Cyclic Mechanisms with Force Closure . . . . . . . . . . . . . . . . . 133
5.6.1 The Influence of the Drive’s Vibrations
on Conditions of Force Closure . . . . . . . . . . . . . . . . 133
5.6.2 Longitudinal Oscillations of the Closing Springs. . . . . 136
5.6.3 Transverse Vibrations of Closing Springs. . . . . . . . . . 141
5.7 Some Problems of Dynamics of Cyclic Mechanisms,
Schematized as Chain Systems with Variable Parameters . . . . . 145
5.7.1 Dynamic Errors of the Operating Members
with Increased Dimensions. . . . . . . . . . . . . . . . . . . . 145
5.7.2 Vibrations in the Differential Mechanism
with Built-In Cyclic Mechanism . . . . . . . . . . . . . . . . 150
5.7.3 Bending Vibrations of the Actuator,
Schematized as a Cantilever Beam
with Variable Length. . . . . . . . . . . . . . . . . . . . . . . . 158
5.7.4 Vibrations of the Drives of Cyclic Machine,
Taking into Account the Dynamic Characteristics
of the Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6 Nonlinear Dissipative Forces............................ 169
6.1 Accounting of Nonlinear Dissipative Forces in Case
of Mono-harmonic Oscillations . . . . . . . . . . . . . . . . . . . . . . . 169
6.1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1.2 Equivalent Linearization of Dissipative
Forces in the Oscillatory System with One
Degree of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.1.3 Equivalent Linearization of Dissipative Forces
in Oscillatory Systems with Many Degrees
of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2 Reduced Characteristics of Elasto-Dissipative Elements
of Machine Drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Contents xiii
6.3 Accounting of Nonlinear Dissipative Forces in Case
of Multi-frequency Oscillations. . . . . . . . . . . . . . . . . . . . . . . 180
6.3.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . 180
6.3.2 Free Vibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.3.3 Analytical Description of Coefficients of Dissipation
in Case of Multi-frequency Regimes . . . . . . . . . . . . . 183
6.3.4 Resonant Oscillations . . . . . . . . . . . . . . . . . . . . . . . 187
6.3.5 Refined Conditions of Dynamic Stability in Case
of the Main Parametric Resonance . . . . . . . . . . . . . . 189
6.3.6 The Influence of the High-Frequency Impacts
on the Resonant Vibrations, in Case of Joint
Action of Force and Parametric Excitations . . . . . . . . 190
6.3.7 Refined Conditions for the Emergence
of the Sub-Harmonic Resonances . . . . . . . . . . . . . . . 191
6.3.8 The Influence of High-Frequency Disturbances
on the Emergence of Stick-Slip Frictional
Self-excited Oscillations. . . . . . . . . . . . . . . . . . . . . . 196
6.4 Nonlinear Resonance Oscillations on the Frequency
of the Amplitude Modulation Caused by the High-Frequency
Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.5 Vibrations in the Systems with Intermediate Friction
Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.5.1 Dynamic Model with Finite Number
of Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . 205
6.5.2 Study of the Forced Vibrations of the Drive
Using the Model with Distributed Elastic-Friction
Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7 Clearances......................................... 219
7.1 Dynamic Effects and Mathematical Description . . . . . . . . . . . 219
7.2 Excitation of Vibrations Due to Impacts in the Kinematic
Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.3 Excitation of Vibrations During Shockless Reversals
in Clearance—Joint. Pseudo-Impact . . . . . . . . . . . . . . . . . . . 226
7.4 Mathematical Models for the Study of Vibrations,
Excited by Pseudo-Impacts in the Clearances . . . . . . . . . . . . . 232
7.4.1 Crank-and-Rocker Mechanism . . . . . . . . . . . . . . . . . 232
7.4.2 Slider-Crank Mechanism . . . . . . . . . . . . . . . . . . . . . 238
7.4.3 Spatial Crank-and-Rocker Mechanism . . . . . . . . . . . . 240
7.5 Some Criteria of Efficiency of Dynamic Unloading,
Taking into Account the Clearances . . . . . . . . . . . . . . . . . . . 241
8 Vibration Analysis of Cyclic Machines Using Modified
Transition Matrices.................................. 247
8.1 Modified Transition Matrices . . . . . . . . . . . . . . . . . . . . . . . . 247
8.2 Determination of“Natural”Frequencies and Non-stationary
Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.3 Forced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.4 Frequency and Modal Analysis of Systems
with Complex Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
8.5 Joint Accounting of Dynamic Characteristics
of the Motor and the Machine Drive . . . . . . . . . . . . . . . . . . . 263
9 Regular Torsional Cyclic Systems with Branched Structure..... 267
9.1 Overview of Regular Systems. . . . . . . . . . . . . . . . . . . . . . . . 267
9.2 Model with Lumped Parameters . . . . . . . . . . . . . . . . . . . . . . 270
9.3 Model with Distributed Parameters . . . . . . . . . . . . . . . . . . . . 276
10 Regular Cyclic Systems with Ring and Branched-Ring
Structure.......................................... 281
10.1 Model of Ring Structure, with Lumped Parameters . . . . . . . . . 281
10.2 Model of a Ring Structure, with Absolutely Rigid
Main Shaft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10.3 Model of Branched-Ring Structure, with Lumped
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
10.4 Model of Multisection Drive with Branched-Ring
Structure and Distributed Parameters . . . . . . . . . . . . . . . . . . . 304
11 Regular Cyclic Systems with Translational Motion
of the Actuator...................................... 315
11.1 Dynamic Model of the General Form . . . . . . . . . . . . . . . . . . 315
11.1.1 Frequency and Modal Analysis. . . . . . . . . . . . . . . . . 315
11.1.2 Forced Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 323
11.2 Bending Vibrations of the Actuator, Mounted
on the Output Links of Identical Cyclic Mechanisms. . . . . . . . 328
11.3 Vibrations of Multisection Drives for Moving
the Massive Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
11.4 Torsion-Bending Vibrations of Branched-Ring
Structured Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
12 Energy Exchange in the Regular Cyclic Oscillatory
Systems. Spatial Localization of Vibrations................. 349
12.1 Brief Information About the Energy Transfer
in Oscillatory Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
12.2 Computer Simulation of Vibrations in Regular
Cyclic Systems, Taking into Account the Clearances
and Dissipative Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
12.2.1 Torsion System of Ring Structure . . . . . . . . . . . . . . . 356
12.2.2 Torsion-Bending System of Branched-Ring
Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
12.3 Spatial Localization of Vibrations . . . . . . . . . . . . . . . . . . . . . 362
12.3.1 Parametric Analysis of the Results of the Computer
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
12.3.2 Analysis of the Factors Influencing Spatial
Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

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