In this post, we will see the book Elements Of The Applied Theory Of Elastic Vibration by Ya. Panovko.

About the book

We may distinguish at least the following live sufficiently independent categories of vibratory processes differing in their nature:

free vibrations, i.e., vibrations which are performed by a mechanical system having no energy supply from outside if the system is disturbed from its position of equilibrium and then released;

critical states of rotating shafts and rotors which consist in a sudden increase in the deflections of their axes at definite speeds of rotation (or in definite ranges of speeds);

forced vibrations which result when the mechanical system is acted on by fluctuating external forces (driving forces);

parametric vibrations caused by periodic variations of the para meters of a system (for example, its stiffness);

self-excited vibrations, i.e., vibratory processes which are maintained by constant sources of energy of a non-vibratory nature.

Each of these categories of vibratory processes is discussed in the appropriate chapter.

The book was translated from the Russian by M. Konyaeva and was published by Mir in 1971.

Many thanks to *Akbar Azimi* for providing the raw scan 2-in-1 page scans. We cleaned the book with OCR.

**Contents**

Notation. 7

Introduction. 9

**CHAPTER I. FUNDAMENTALS. 11 **

1. Number of Degrees of Freedom of an Elastic System. 11

2. Classification of Forces. 15

3. Methods for Setting Up Equations of Motion in the General Case. 22

**CHAPTER II. Free vibrations . 24 **

4. Linear Systems of One Degree of Freedom Without Inelastic Resistances. 24

5. Effect of Inelastic Resisting Forces on Free Vibrations

of Linear Systems of One Degree of Freedom. 56

6. Undamped Systems of One Degree of Freedom with Non-Linear Restoring Forces. 68

7. Linear Systems of Several Degrees of Freedom. 87

8. Vibrations of Bars of Uniform Section (Exact Solution) 118

9. Vibrations of Ban of Variable Section. 138

10. Two-Dimensional Vibrations of Disks. 147

11. Flexural Vibrations of Disks. 153

12. Flexural Vibrations of Rectangular Plates. 157

**CHAPTER III. CRITICAL STATES OF ROTATING SHAFTS AND ROTORS 161 **

13. Shaft with One Disk. 161

14. Effect of Friction. 174

15. Automatic Balancing of Rotating Shafts. 185

16. Critical States of Helicopter Rotors. 187

17. Shalt with Several Disks. Rigid Rotor on Elastic Supports 190

**CHAPTER IV. FORCED VIBRATIONS. 193 **

18. Linear Systems of One Degree of Freedom without Inelastic Resistances. 193

19. Linear Systems of One Degree of Freedom with Inelastic Resisting Forces. 226

20. Systems with Non-Linear Restoring Forces (Single Degree of Freedom). 241

21. Linear Systems of Several Degrees of Freedom. 249

22. Linear Systems with Distributed Parameters. 269

**CHAPTER V. PARAMETRIC VIBRATIONS. 279 **

23. Basic Equation. 279

24. Cases of Periodic Variation of Stiffness. 283

25. Cases of Periodic Variation of Parametric Loads. 285

26. Pendulum with a Vibrating Point of Suspension. 288

27. Cases of Periodic Variation of the Inertia of a System. 289

**CHAPTER VI. SELF-EXCITED FRICTIONAL VIBRATIONS. 293**

28. Nature of Self-Excited Vibrations. 293

29. Self-Excited Vibrations of Quasi-Linear Systems. 298

30. Self-Excited Relaxation Vibrations. 304

Index. 314