Svetlitsky ~ Eng Vib Analysis,Vol.1

8/19/2019 Svetlitsky ~ Eng Vib Analysis,Vol.1

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Foundations of Engineering Mechanics

V. A. Svetlitsky


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Springer-VerlagBerlin HeidelbergGmbH

Engineeringspringeronline.com

ONLINELIBRARY


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v. A. Svetlitsky

Engineering

Vibration AnalysisWorked Problems 1

Translated by G.I. Merzon and V.A. Chechin

With 312 Figures

Springer


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Series Editors:V.1. BabitskyDepartmentofMechanical EngineeringLoughborough UniversityLoughboroughLEI13TU, LeicestershireGreat Britain

Author:V. A. SvetlitskyBauman Moscow StateTechnical UniversityDepartment of Applied Mechanics2nd Baumanskaya st.5105005 MoscowRussian Federation

Translators:0.1. MerzonV. A. ChechinLebedev Physics InstituteRussian Academy of SciencesLeninsky pro 53

119333 MoscowRussian Federation

J. WittenburgInstitutfUrTechnische MechanikUniversitiUKarlsruhe (TH)KaiserstraJ3e1276128 Karlsruhe / Germany

ISBN 978-3-642-05842-4 ISBN 978-3-540-40970-0 (eBook)DOl 10.1007/978-3-540-40970-0

Cataloging-in-Publication data applied forBibliographic information published by Die Deutsche Bibliothek.Die Deutsche Bibliothek lists this publication in the Deutsche N ationalbibliografie;detailed bibliographic data is available in the Internet at .

This work is subject to copyright. All rights are reserved, whether the whole or part of the materialis concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplicationof his publication or parts thereof is permitted only under the provisions of he German CopyrightLaw of September 9, 1965, in its current version, and permission for use must always be obtainedfrom Springer-Verlag. Violations are liable for prosecution act under German Copyright Law.

© Springer-Verlag Berlin Heidelberg 2004Originally published by Springer-Verlag Berlin Heidelberg in 2004Softcover reprint of the hardcover 1st edition 2004

The use of general descriptive names, registered names, trademarks, etc. in this publication doesnot imply, even in the absence of a specific statement, that such names are exempt from therelevant protective laws and regulations and therefore free for general use.

Cover-design: de'blik, BerlinPrinted on acid-free paper 62 I 3020 hu - 5 4 3 2 1 0


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Preface

Theory of vibrations belongs to principal subjects needed for training mechanical engineers in technological universities. Therefore, the basic goal of the monograph "Advanced Theory of Vibrations 1" is to help students studying vibrationtheory for gaining experience in application of this theory for solving particularproblems. Thus, while choosing the problems and methods to solve them, the closeattention was paid to the applied content of vibration theory.

The monograph is devoted to systems with a single degree of freedom and systems with a finite number of degrees of freedom. In particular, problems are formulated associated with determination of frequencies and forms of vibrations,study of forced vibrations, analysis of both stable and unstable vibrations (including those caused by periodic but anharmonic forces). The problems of nonlinearvibrations and of vibration stability, and those related to seeking probabilisticcharacteristics for solutions to these problems in the case of random forces are alsoconsidered. Problems related to parametric vibrations and statistical dynamics ofmechanical systems, as well as to determination of critical parameters and of dynamic stability are also analyzed.

As a rule, problems presented in the monograph are associated with particularmechanical systems and can be applied for current studies in vibration theory. Allowing for interests of students independently studying theory of vibrations, themajority of problems are supplied with either detailed solutions or algorithms ofthe solutions.

While preparing the manuscript of "Advanced Theory of Vibrations I", lecturesgiven by the author to students of the Applied Mechanics cathedra of BaumanMoscow State Ter.;hnological University, Moscow, Russia were partly used. Theselectures were published in Russian as a textbook (V. Svetlitsky, 1994) and formed

a basis of the present monograph.The monograph is intended to students, post-graduate students, and teachingstaff of technological universities. I t can also be useful for mechanical engineerswho apply theory of vibrations in their everyday practical work.

Valery SvetlitskyMoscow, September 2003


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Table of Contents

Problems and Examples in Vibration Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Vibrations of Systems with a Single Degree of Freedom ............................. 31.1 Free Vibrations . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . 31.2 Free Vibrations of Systems with Allowance for Resistance Forces ........ 141.3 Forced Vibrations... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 Critical States and Vibration Stability . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . 36

1.5 Parametric Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.6 Nonlinear Vibrations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 49

2 Vibrations of Systems with Several Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . 572.1 Free Vibrations... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.2 Forced Vibrations... . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . 722.3 Critical States and Vibration Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.4 Approximate Methods of Determining the Lowest Frequency . . . . . . . . . . . . . 882.5 Random Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Answers and Solutions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . 101

1 Vibrations of Systems with a Single Degree of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . 1031.1 Free Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031.2 Free Vibrations of Systems with Allowance for Resistance Forces ...... 1261.3 Forced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1361.4 Critical States and Vibration Stability ......................................... 1621.5 Parametric Vibrations . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . 1771.6 Nonlinear Vibrations.......................................... . . . . . . . . . . . . . ... 188

2 Vibrations of Systems with Several Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . 2092.1 Free Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2092.2 Forced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2482.3 Critical States and Vibration Stability . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . 2672.4 Approximate Methods of Determining the Lowest Frequency . . . . . . . . . . . . 2872.5 Random Vibrations . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . 291

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315


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BASIC NOTATION

AAllA22

c

ccDdEFFoF(t)GgHI, i10YoJJ x, JY' Jp , J kKkL

IMMj, M2, M3mP, P(t)Pj, P 2, P 3

vibration amplitude;bending stiffness of a rod;bending stiffness of a rod with respect to the y axis (orX2 axis);bending stiffness of a rod with respect to the z axis (orX3 axis);capacitance; capacitor; capacity;

coefficients;bending stiffness; torsional stiffness; spring rate:variance; diameter of a disk, cylinder, wheel;internal diameter; wire diameter;Young modulus;cross-sectional area;amplitude of a perturbing force;perturbing force;shear modulus;free fall acceleration;Heaviside functionelectric-current intensity;zero-order Bessel function of the first kind;zero-order Bessel function of the second kind;moment of inertia;geometric characteristics of the rod cross section;correlation function; Krylov function;rigidity of an elastic base;inductance;length;moment of force;torque and bending moments;mass;force;components of a concentrated force in the related coordinate system;components of a concentrated force in a Cartesian coordinate system;

eigenfrequency (natural frequency); free vibration frequency;force; generalized force;axial force and cutting forces, respectively;


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X Basic Notation

R

rSTtUvv.wwx;r;Z

I I

P0"

't


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Problems and Examples in Vibration Theory


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1 Vibrations of Systems with a Single Degree ofFreedom

1.1 Free Vibrations

1 Determine frequencies of natural bending vibrations for systems, shown inFig. 1.

2 Using the force method for systems with constant rigidity EJ. (Fig. 2), set up thedifferential equations for small free vibrations and determine their frequencies.The masses of the rods are assumed to be small compared to the load mass, thelatter being considered as a point mass.

3 Using the force method for systems shown in Fig. 3, set up the differential equations of small free vibrations in the case of point masses m and determine the eigenfrequencies.

4 For systems shown in Fig. 4, set up the differential equations of small torsionalvibrations of the flywheel with the moment of inertia J and determine the eigenfrequencies. The bearing shown in Fig. 4d is assumed to be long (i.e., the axial lineof the bent rod its tangent to the bearing axis) and GJ k =0.8EJ .

.. ' ~ , -- 11 Ie ..:4 Elxa b

~ .

E:J.

- ~cFig. I.


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4 1 Vibrations of Systems with a Single Degree of Freedom

a b

11

d

c

Fig. 2.

5 A sensor (pressure transducer, Fig. 5) is intended for controlling pressure of liquid or vapor in a certain volume. The variation in pressure causes a piston of massm1 to move and (via a rod) to power an actuator. The reduced mass of the rod withother mobile pieces of the device is m2• Determine the eigenfrequency of the pres

sure transducer (piston) in the case of a very small friction of seal rings on thecylinder.

a b

Fig. 3.


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1.1 Free Vibrations 5

-t-93«

- ~ ~a

1-zs:T.t

. ~3~ I tb

Fig. 4.

6 A load of mass m is fixed to an absolutely rigid weightless rod with length 31.The rod is connected to two springs each with stiffness c (Fig. 6). Determine frequencies of small free vibrations of the pendulum.

Fig. 5. Fig. 6.


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Fig. 7. Fig.S.

7 Solve Problem 6 for the case of the horizontal rod.

S A gear wheel] (Fig. 7) of mass m, with fixed carrier 2 of mass m2 lies on rack 3.Springs 4 each of stiffness c are fixed without tension at the free end of the carrier.Set up the differential equation describing small free vibrations of the system anddetermine its eigenfrequency.

9 A heavy cylinder of mass m =Qlg and of radius r (Fig. 8) lies on a concave surface with radius of curvature R. Rigid tension springs each with stiffness c areconnected to the upper point of the cylinder. Set up the differential equation forsmall free vibrations of the cylinder under the assumption that its slipping over thesurface is absent.

Fig. 9. Fig. 10.


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Fig.n.

Analyze the dependence of the cylinder eigenfrequency on the radius of curvature R, in particular, including the case of a plane surface (R = 0).10 Determine the frequency of natural vibrations of a vertically installed pendulum(Fig. 9) with mass m in the case of an absolutely rigid rod. The stiffness of eachspring equals c.

11 Determine the frequency of small free vibrations for a steel disk of an inductiongalvanometer (Fig. 10). The disk has the thickness h = mm and diameterd = 0 mm. The torsional stiffness of each spring is c =M/rp= .488 N m.12 A rod AB with torsional stiffness c has a cylindrical hinge at one end and a diskof radius R and of mass m at the other end (Fig. 11). The system is fixed to a planetilted at an angle a to the horizon. Set up the differential equation for small freevibrations of the disk under the assumption that it can perform vibratory motion ofrolling without slipping. Analyze the eigenfrequency of the system as a function ofthe tilt angle a and find the condition of stability.

Fig. 12.


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8 I Vibrations of Systems with a Single Degree of Freedom

Fig.l3.

13 A level-sensitive element (level sensor, Fig. 12) is composed of a float with di

ameter d and of mass m i ' The float is immersed in a liquid with density p.The sensor is also equipped with a system of levers, a spring of stiffness c, and abalance mass m2• With varying the level Ho of the liquid, the excess buoyancyforce displaces the float. The displacement is transferred via the lever system to arecording or actuating mechanism. Set up the differential equation of small freevibrations of the float and determine their frequency. The inertia of the liquid maybe ignored.

14 A disk of mass m and with moment of inertia 10 is set on a hub of radius r(Fig. 13). The disk hub is supported by a curvilinear circular guide of radius R. As

suming the disk motion to occur without slipping between the hub and the guide,set up the differential equation for small free vibrations of the disk. Derive the differential equation of motion by both the Lagrangian method and d' Alembertmethod.

15 A truck of mass m is installed on an inclined rough surface (Fig. 14). The truckis confined to a spring with stiffness c. Set up the differential equation for smallfree vibrations of the truck. The moment of inertia of each wheel pair is 1, the radius of the wheels is R, and their motion occurs without slipping.

16 A beam of mass m = Q/g lies on a cylinder of radius R (Fig. 15), the slippingfriction coefficient being f.1. Set up the differential equation for small free vibrations of the beam with respect to its equilibrium position.

,

Fig. 14. Fig. IS.


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Fig. 16. Fig. 17.

Determine the beam eigenfrequency and the maximum amplitude CPomax forwhich the vibrations are possible.

17 A ring of mass m with radius R of the inner surface and of the moment of inertia J with respect to the system center of mass is put on a shaft of diameter 2r(Fig. 16). Determine the frequency of ring natural vibrations.

18 In order to damp vibrations of a flywheel (Fig. 17) rotating at angular velocityOJ, a pendulum damper is often applied. This damper represents a load of massm =QIg, has a special shape, and is placed into the internal cavity of the flywheel.The flywheel surface and the load contact surface have radii of curvature R\ andR2 , respectively (R\ < R).

Find the frequency of free vibrations of the load in the case of the stopped(OJ =0) flywheel (the gravity force is directed along the yaxis) and in the caseOJ = onst (ignoring the gravity force). The difference R2 - R\ = JR can be considered as a small quantity, and the distance from the contact point to the center ofmass is a.

...

Fig. IS. Fig. 19.


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The moment of inertia of the load with respect to the axis passing through theload center of mass is J e , and the distance between the flywheel rotation axis andthe center of mass is r.

19 A massive disk of radius R uniformly rotates at an angular velocity OJ. On thedisk rim, a rod of length I is hinged. The rod carries a load of mass m at its freeend (Fig. 18). Assuming the rod mass to be small compared to the mass m, determine the load eigenfrequency in the field of centrifugal forces.

20 A load of mass m is fixed to a flexible steel pillar of a rectangular cross sectionand having bending rigidity c 1 =EJ x ' The pillar is supported by two springs each ofstiffness c 2 (Fig. 19). Determine the frequency of small natural vibrations of theload in the case that the masses of both the pillar and the springs are much smallerthan the load mass. An effect of the load gravity force on bending the pillar may beignored.

21 For damping torsional vibrations, additional massive loads are, sometimes, installed on flywheels, which represent pendulums in the field of centrifugal forces.Determine the eigenfrequency of a centrifugal pendulum (Fig. 20) at a given angular velocity OJ of the disk, for the distance r between its rotation axis and thependulum point of suspension, pendulum moment of inertia J B with respect to thispoint, and pendulum mass m.

22 A flexible rod of length I, which carries a mass m at its free end, is rigidly fixedto the rim of a massive disk with radius R. The disk uniformly rotates at an angularvelocity OJ (Fig. 21). Ignoring the rod mass, determine the eigenfrequency of theload in the field of centrifugal forces. The rigidity of the rod is EJ x'

23 Using the expression

Yo =yl(1 + NIP)for deflections caused by the transverse-longitudinal bending, derive an approximate formula for the vibration frequency of a load of mass m (see Problem 22).Here, N is the axial tensile force, Pc = EJ 4/2 is the critical force, and y is the deflection under the action of a transverse load.

Fig. 20. Fig. 21.


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1.1 Free Vibrations 1l

B.

Fig. 22. Fig. 23.

Plot the dimensionless vibration frequency p/Po as a function of the parameterat for the exact (see Problem 22) and approximate solutions in the case of R = .24 A rod AB (Fig. 22) having the bending rigidity EJ x is fixed at the end of a rigidlever BC uniformly rotating in the horizontal plane at an angular velocity OJ. Neglecting the rod mass, determine the eigenfrequency of a concentrated load ofmass m fixed at the point A of the rod. Consider the case that the vibration of theload occurs in the rotation plane.25 A rod of length 1 and of rigidity EJ x' which carries a load of mass m at its end, isfixed to the rim of a heavy disk with radius R (Fig. 23). Find the eigenfrequency ofthe load in the case of uniform disk rotation. Assuming R > 1 and ignoring the rodmass, determine the critical value of the disk angular velocity OJ for which the loadeigenfrequency is zero.

26 A rod of length 1 and of rigidity EJ x is fixed on a massive disk rotating at anangular velocity OJ (Fig. 23). There is a load of mass m fixed at the rod free end.The center of mass 0 of the load coincides with the disk rotation center. Ignoringthe rod mass, determine the disk eigenfrequency.

27 A load of mass m is fixed to a thin absolutely inextensible line of length 2/. Atension To in the line is produced by means of a spring (Fig. 24). Determine the eigenfrequency of small vibrations of the load for the case that the tensions in theboth line segments are identical and constant.

28 A point load of mass m is fixed to a string of length I, which has an initial tension To (Fig. 25). Set up the differential equation for small free vibrations of thepoint load. Plot its eigenfrequency as a function of the load position xo' The variation of the string tension in the vibration process and an effect of the gravity forcecan be ignored.


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...

x Y

m~ ~ - - - - ~ + 4 - - - - - - - - - - - - - - ~ ~ I

Fig. 24. Fig. 25.

29 A load of mass m is fixed at the end of an elastic extensible weightless ropewound on a drum (Fig. 26). Set up the differential equation for small free longitu

dinal vibrations of the load. The drum rotates uniformly at the angular velocity OJ.The drum diameter is 2r, the cross-sectional area of the rope and the Youngmodulus of its material are F and E, respectively. Assuming that initially (at themoment of time to) the length of the unwound portion of the rope is 1 , determinethe moments of time when the load passes by the equilibrium position.

Fig. 27.

Fig. 26. Fig. 28.


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The total velocity of motion is v = OJR+ L1X and Llxo =0 (Llxo is the dynamic elongation of the rope). Solve the problem in the special case OJ= onst form

=100 kg, E

=10 GPa, F

=10-5 m\ v

=OJR

=1 mis, to

=1 s, and 10

=1 m.

30 A load of mass m is mounted at the end of an absolutely flexible inextensibleweightless rope wound on a drum. Set up the differential equation for small freetransverse vibrations of the load in the case of a uniform rotation of the drum (seeFig. 26). Find the amplitude of the free vibrations as a function of both the length 1of the rope free segment and the drum angular velocity OJ. Analyze two cases: (a)ascending the load with an initial rope length 10and (b) descending the load. Theinitial conditions (at t = o) are CPo=0 and rp=0 (cp is the angle of rope deflectionfrom the vertical).31 A steel w e i g h t l e s ~beam of length 1 =2 m with rigidity EJ x lies on a weightlesselastic base having rigidity k =4 kN/m 2 (Fig. 27). A point load. of massm =100 kg is fixed in the middle of the beam. Determine the eigenfrequencies ofthe system provided that the beam cross section is a rectangle with the base of6 cm and height of 1 cm.

32 A point load of mass m =20 kg is fixed to a steel beam of length 1and has thecross-sectional area b x h = 6 x 1 cm 2 (Fig. 28). The beam with the restrained endlies on an elastic base having rigidity k =4 kN/m 2 • Ignoring the beam mass and inertia of the elastic base, determine the eigenfrequency of the entire system.

33 A steel beam of length 21 =200 cm has the cross section b x h =6 x 2 cm 2• Thebeam end sections are restrained (Fig. 29). The beam lies on an elastic base havingrigidity k =2.5 kN/m 2 and is loaded at the middle point by a mass m =50 kg. Determine the eigenfrequency of the load ignoring the beam mass and inertia of thebase.

34 Using the Rayleigh method, determine the eigenfrequency of a system shown inFig. 30 with allowance for the beam mass. The concentrated load may be considered as a point mass m, i.e., its moment of inertia can be neglected. The beam ri

gidity is EJ x' and the beam mass per unit length is mo.

35 Using the Rayleigh method, determine the eigenfrequency for a load of mass mfixed in the middle of a beam with rigidity EJ x (Fig. 31). The beam mass per unitlength is mo.

Fig. 29. Fig. 30.


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' ~ ' t

$m£1I. ~ I · w., t ~

Fig. 31.

ad

''f "" £.to\- mf JeIt• II. 6= 7tb ,

f te

t x £:Ixmt ,c ,

Fig. 32.

36 Using the Rayleigh method determine the eigenfrequencies of systems containing a point mass m, which are shown in Fig. 32. The beam mass per unit lengthis mo.

1.2 Free Vibrations of Systems with Allowance forResistance Forces

37 Determine the eigenfrequency for torsional vibrations of a mixer shaft as afunction of viscosity of a medium (Fig. 33). Find the decay time for which the vibration amplitude decreases by a factor of 10 after instantaneously stopping therotor whose rotation angular velocity before stopping was n. When solving theproblem, assume that the shaft mass is small compared to the mass of blades.


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1.2 Free Vibrations of Systems with Allowance for Resistance Forces 15

~ ~ ' - ' - ,

d~-.l

I ~ . : : : :t - t - - ~ - :t:t - ll, - -. - --~ , .

'

Fig. 33. Fig. 34.

The moment of inertia of the blades is J = .5 kg m 2, and the shaft length isI = 0.5 m. The coefficient of viscous resistance to the blade motion is a = 1.2Nms.

38 A setup shown in Fig. 34 can be used for measuring oil viscosity. In this case, a

massive steel cylinder of height H and of diameter D, is suspended by a thin wireof diameter d and of length I in a cup of diameter D 2• The oil is poured into thecup, and the cylinder is set in the vibratory-rotational motion. Determine the kinematic viscosity of the straw oil with the specific weight r= 9 kN/m 3• The momentof forces resisting to the motion can be calculated by the formulaM = (l/4)rcHD,3 OJpv I(D2 - D), where OJ= ip is the angular velocity and p is theoil density. It should be taken into account the fact that, while performing tests in asetup with D, =0.1 m, D2 =0.12 m, H = .2 m, d =0.001 m, and 1=0.5 m, theperiod of free vibrations turned out equal to To =6.5 s. The liquid may be considered as inertialess.39 For the system shown in Fig. 35, set up the differential equation for small freevibrations of the mass m under the assumption that the damper resistance force Fis directly proportional to the velocity of motion (F = aY).40 A shock-proof clutch (Fig. 36) serves for smoothing an effect of sharp variations in the driving-shaft velocity on an output shaft. The clutch represents steeldisk 1 freely fixed to shaft 3. A force is transferred from the shaft to the disk viaspring 2. Assuming the resistance between the shaft and the disk to be of viscous

nature, determine the coefficient a of the viscous resistance to the motion in thedisk-shaft pair. The numerical values of relevant parameters are the following: thelogarithmic decrement of disk free vibrations 8= 1.5; disk diameter D = 0.1 m;disk thickness H =20 mm; spring diameter D, =30 mm; spring wire diameterd, =3 mm; number of spring turns i =9; shear modulus of the spring materialG =80 GPa; and density of steel p =8 x 10 3 kglm3•


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y

Fig. 35.

41 A load of mass m is suspended on an absolutely rigid rod of length 21 (Fig. 37).Two elastic springs each having stiffness c are fixed in the middle of the rod. Theload is immersed in a vessel filled with a viscous liquid. In the process of smallfree vibrations of the load, the liquid produces the damping action on the system.

Determine the coefficient of viscous resistance to the motion of the system inthe case that the period of its damped vibrations is 1"0 = 1.0 s. Numerical values ofother parameters are the following: the load mass m =1 kg; the rod length1=0.3 m; the spring diameter D =0.2 m; the spring wire diameter d =2 X 10.3 m;the shear modulus of the spring material G =80 GPa; and the number of turns ineach spring i = .42 A hydraulic damper (dash-pot, Fig. 38) represents a piston of mass m movingin a liquid. Analyze the motion of the piston in the case that, at the initial momentof time (t = 0), the deviation of the piston from its equilibrium position isYo=5 mm.

Fig. 36.


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1.2 Free Vibrations of Systems with Allowance for Resistance Forces 17

Fig. 37. Fig. 38.

Determine the time at which the deviation of the piston from the equilibriumposition reduces by half. The stiffness of the spring is c =3 kN/m; the diameter ofthe cylinder is D = .1 m; the diameter of holes in the piston is d = 0 mm; thenumber of the holes is z=25; the piston mass is m =2.73 kg; the piston height isH = 50 mm; and the coefficient of dynamic viscosity of the liquid is f.1= 0.06 Pa s.

43 A device represents a load of mass m fixed by two springs each having stiffnessc. The load is placed into a tube filled with a liquid (Fig. 39). The resistance to themotion of the load can vary depending on the viscosity of the liquid, as well as onthe size of the gap between the load and the tube wall. In the case that the resistance force F damping the motion of the load is directly proportional to its velocity, i.e., F =a x set up the differential equation for small free vibrations of theload. Determine the time for which the amplitude of free vibrations decreases by afactor of 100 provided that at the initial moment of time (t =0), x = o and x =O.The values of relevant parameters are m =50 kg, c = 1 kN/m, anda =500 (N s)/m.44 A load of mass m is fixed by an absolutely rigid weightless rod of length I.(Fig. 40), which is kept in equilibrium by both a spring and a damper. The latterhas a linear friction characteristics Ff =ax. Determine the eigenfrequency ofthe system and the logarithmic decrement in the case of m = 1 kg, 1 =0.5 m,a =0.2 m, spring diameter D =50 mm, diameter of the spring wire d =5 mm,number of spring turns i =5, shear modulus G =80 GPa, and the coefficient ofviscous resistance to the damper motion a =300 (N s)/m.


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....

Fig. 39. Fig. 40.

45 For measuring low pressure differences in gases (or small pressure oscillations), U-shaped manometers supplied with an inclined tube are used (Fig. 41).Determine the frequency of small natural vibrations of a liquid poured into thetube. The density of liquid is p and the force of resistance to the motion of liquidin the tube is F = at where x is the displacement of the level of liquid from theequilibrium position.

46 At the initial moment of time, a load of mass m (Fig. 42a) is deflected from itsequilibrium position by the distance Xo and then is released at the zero initial velocity. When slipping over the surface, a friction force arises that may be considered as a Coulomb dry friction force. Determine the law of motion for the mass min the case of the dependence of the Coulomb friction force FI on the velocity ofmotion X, which is shown in Fig. 42b.

47 Disk 1 is put on shaft 2 and is kept against revolving by spring 3 (Fig. 43). Oneof the spring ends is attached to the disk, while the other end is fixed to a support.For the case that the moment of inertia of the disk-shaft-spring system is J and thespacing between the disk and the Coulomb friction surfaces A and B is rathersmall, set up the differential equation for small free vibrations of the disk. Analyzethe character of motion under the initial condition rp = po and ip =0 as t =0and find the vibration amplitude as a function of the number of vibration periods inthe case that the restoring elastic moment of the spring exceeds the moment offriction force. Determine also the vibration period.

I n s t r u c t i o n . Under rotation of the disk through an angle rp, the spring lengthchanges, and the disk is pressed to one of the friction surfaces (A or B). The Coulomb friction force is proportional to the pressing force.


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1.3 Forced Vibrations 19

Ii

b

Fig. 41. Fig. 42.

Fig. 43.

48 Using the phase-plane method, analyze free vibrations of the system shown inFig. 43 (see Problem 47).

49 Compose an analog electric circuit for a mechanical vibratory system consistingof a load of mass m suspended by a spring of stiffness c.

50 A piston with mass m of an oil damper is suspended by a spring with stiffness c(Fig. 38). Compose an analog electric circuit for small free vibrations of the system.

1.3 Forced Vibrations

51 A vibrograph schematically shown in Fig. 44 represents a device that recordsvibrations. On rigid frame 1 of the device, coil 2 is mounted whose core is magnet3 suspended by spring 4.


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Fig. 44. Fig. 45.

When moving the magnet inside the coil, electric current appears and then istransmitted to a recording device. Assuming the mass of magnet 3 and the stiffnessof spring 4 to be m and c, respectively, set up the differential equation of the magnet motion. Determine the condition corresponding to the minimal distortion of recorded vibrations. (The device registers vertical vibrations of the frame, whichproceed according to the harmonic law y = osinOJt.)52 A lever-free sensor intended for testing wobbles of a surface is supplied withmeasuring rod 1 of mass m mounted on two plane springs 2 (Fig. 45). Helical cylindrical spring 3 serves for pressing the measuring rod to piece 4 being tested. I fthis piece has wobbles of its surface with respect to the centerline (the wobble sizeis determined by eccentricity e) then, while rotating the piece at the angular veloc

ity OJ, the measuring rod is displaced in the vertical direction.

53 The end cross section A of a rod shown in Fig. 46 is forced to tum according tothe harmonic law O(t) =00 cos OJt . This kind of exciting vibrations is calledkinematic. Assuming the vibration regime to be stable, determine the maximumnormal stress in the restraint. Assuming that in the absence of eccentricity (e = 0),the force pressing the measuring rod to the piece is Ro, determine the ultimate angular velocity OJ,. In doing so, use the condition of providing the continuous contact in the process of testing a piece with eccentricity e. Masses of elastic elements

of the device may be neglected. What is the measuring force of the device (i.e., theforce of pressing the measuring rod to the piece) in the case of OJ = 0.1 OJ.?


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h- $V'tt) tg tVJt}

$- ~ ~ -t f;}JjJ. tFig. 46. Fig. 47.

When calculating, take m =0.05 kg, ro =5 N, d t =50 mm, and e =0.1 mm. Thelength and the cross-sectional area of the springs are t =30 mm andb x h =5 x 0.5 mm\ respectively. The diameter of the helical spring is D =5 mm,the spring wire diameter is d =1 mm, and the number of spring turns is i =20. Thespring is made of steel having E =200 GPa and G =80 GPa.54 The cross section k of the rod (Fig. 47) experiences the vertical displacementYk(t) = Yo cos mt . Assuming rod vibrations to be stable, find the reaction in thehinge.

55 A lever electrocontact sensor (roughness indicator, Fig. 48) is intended forautomatically testing the waviness of a surface. The indicator consists of meas

urement rod 1 contacting the surface to be measured and lever 3 enhancing thedisplacement of the rod by a factor of l i t . Measuring rod 6 is pressed to a surfacebeing tested by spring 2 having stiffness Ct. Lever 3 is mounted on spring hinge 5possessing stiffness c2 • I f a piece to be tested does not adhere to assigned tolerances, one of contacts 4 is being closed, i.e., a rejection signal is generated. The efficiency of testing is proportional to the displacement velocity of the piece with respect to the sensor. However, the velocity cannot be arbitrarily high becausebreaking the contact at the point B t becomes possible at high velocities.

56 In the roughness indicator shown in Fig. 48, violation of the contact between

measuring rod 1 and lever 3 at the point B2 is possible at a sufficiently high velocity of motion of a surface being tested with respect to the device. Assuming thesurface profile to be a sinusoid x =aosin(2nz/A), determine the condition for thebreaking contact at the point Br The lever moment of inertia is J, and the stiffnessof the spring in hinge 5 is c2• The springs have the initial angular twist


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4

Fig. 48. Fig. 49.

To do this, additional springs 2 and 3 are attached to the trimming tab in a special experimental setup.

Find the elevator trimming-tab eigenfrequency if the resonance frequency lV, ofthe system is known. The mass of measuring rod 1 is m, and the moment of inertiaof lever 3 with respect to hinge 0 is J. Ignoring friction forces, determine the ultimate velocity v. for the motion of the piece in the case that the equation of its surface has the form x = osin(21tz/A), where A is the irregularity wavelength. Allowfor the fact that when installing the measuring rod on the surface to be tested,spring 2 and the springs of hinge 5 have the initial tension Xo and twist ll'o'

58 In order to reduce an effect of base vibrations on the operation of a device, apassive vibration isolation is used, e.g., the suspension (installation) of the device

on soft compliant shock absorbers. For schematic diagrams shown in Fig. 50, determine the dynamic coefficient (ratio of the vibration amplitude of mass m to theamplitude Xo of base vibrations) of the system.

f,

a b

Fig. SO.


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Fig. 51.

59 When operating a single-phase motor, an alternating torque M = Mo + MlsinllJtarises. In this case, a special support is applied as a vibration isolation (Fig. 51).The support is arranged in such a manner that inclined segments of steel strips areintersected at the center of a bearing. This support turns out to be rather rigid inboth the vertical and horizontal directions, which is necessary for the normal operation of a gear wheel fixed to the motor shaft. At the same time, such a hinge issufficiently compliant in the case of turning the motor in the picture plane.

Determine necessary dimensions b x 2b of the supporting steel strips under thecondition that the dynamic coefficient of the torque transfer to the base must notexceed 1115.

The dimensions of the system are 13= 100 mm; 12= 1600 mm; and II = 50 mm.The moment of inertia of the motor is J = 20 kg m 2 • Determine also the safety factor for endurance if Mo = 50 N m; MI = 30 N m; the normal ultimate strength is

0;, = 1 GPa; OJ = 105 S-I; and the fatigue limit for the symmetric cycle isq=O.4GPa.

60 To reduce the action of inertial forces arising in an engine (motor), which arecaused by unbalanced rotating masses, various systems of vibration isolation areemployed, for instance, elastic suspensions or supports. For systems whose schematic diagrams are shown in Fig. 52, determine the dynamic coefficient of theforce transfer to the base at the fixation points of elastic supports with stiffness c.Assume the beams on which the motors are mounted to be absolutely rigid.

61 A load of mass m = 9.1 kg (Fig. 53) is suspended by a spring with stiffnessc = 7.2 kN/m. The upper end of the spring is connected to a slider-crank mechanism and executes reciprocate motion described by the equation x = xosinllJt withXo = 15.5 mm.


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Fig. 52.

Fig. 53. Fig. 54. Fig. 55.

The crank angular velocity OJ is equal to the frequency of free undamped vibrations of the load, the coefficient of the viscous resistance of the damper beinga= 27 (N s)/m. Determine the maximum force stretching the spring and the safetyfactor for the spring fatigue strength. The diameters of the spring and of the springwire are D = 50 mm and d = 5 mm. The spring is made of steel with the followingmechanical characteristics for a torsional load: the ultimate strength, yield


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strength, and fatigue limit for a symmetric cycle are '" = 850 MPa, 't"y = 600 MPa,and 1f= 300 MPa, respectively.

62 A load of mass m (Fig. 54) supported by two springs each of stiffness c is attached to a slider-crank mechanism via a damper whose coefficient of viscous resistance is a. Set up the differential equation of motion for the load. Determinestresses arising in the springs in the case that the crank angular velocity OJ is equalto the frequency of undamped vibrations of the load mounted on the springs. Thediameters of the springs and of the spring wire are D and d, respectively.

63 A periodic perturbing force F =Fosinwt acts on a load of mass m (Fig. 55)linked to a rigid wall by both a spring with stiffness c and a damper whose coefficient of viscous friction is a. Determine the frequency of natural vibrations of theload and the frequencies OJ of the perturbing force for which the maximum forceacting on the spring is equal to the perturbing-force amplitude.

64 A motor of mass ml' which has a rotor of mass mo' is rigidly fixed to a frame ofmass m2 (Fig. 56). The frame is kept on a base by means of two springs with thetotal stiffness c. There exists a lubrication layer between the frame and the base.The center of mass mo is shifted by the distance e with respect to the rotation axis.Determine the coefficient of viscous friction of the lubricant in the case that thedissipated energy (i.e., the work of friction forces) attains the highest value, andthe rotor uniformly rotates at an angular velocity co.

65 For the system shown in Fig. 56, plot the amplitude of the perturbing forceacting on the springs as a function of its frequency in the case of steady-state vibrations.

66 A motor having an unbalanced mass is mounted on two steel beams (Fig. 57).In order to reduce the amplitude of vibrations induced by the motor, a viscousfriction damper is introduced into the system. The motor mass is m =50 kg, theangular velocity is OJ =63 s-', and the amplitude of the perturbing force isPo = 2000 N.

p= p"Stll41t

Fig. 56. Fig. 57.


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Find the characteristic a of the damper such that the safety factor nu for thebeam fatigue strength would be equal to 2. The beam has a square cross sectionb x h = 500 x 500 mm 2 and is made of steel with the following mechanical parameters: 0;. = 220 MPa, OJ= 180 MPa, and cr. = 400 MPa. The beam length isI m. Effects of both the absolute size of the beam and its strengthening treatment,as well as of the stress concentration in the restraint cross section may be ignored.

67 In order to provide vibration isolation, dissipative elements, e.g., viscousfriction dampers are introduced into elastic suspensions of various engines and devices. Determine the dynamic coefficient for the system schematically shown inFig. 58 for the case of the following numerical parameters: m = 100 kg,m = 100 s-', a = 6000 (N s)/m, I = I m, and EJ x = 20 kN/m.68 For registration of vibratory processes in the presence of various random perturbations (pushes, shocks), low-frequency vibrographs are employed. These vibrographs are supplied with viscous-friction dampers suppressing induced vibrations. A schematic diagram of such a device is shown in Fig. 59. The motion of aload of mass m suspended on a spring with stiffness c is damped by the viscousforce F = a y , where y is the load displacement with respect to the base. Find thedisplacement recorded by the device as a function of time. Analyze the case thatthe base motion is described by the periodic law y, = O


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8J ....3

%

84Fig. 59. Fig. 60.

Assuming that the mirror is subjected to the force F =Bli, set up the differential equation of small free vibrations of the mirror. Here, i is the electric-current

intensity, B is the magnetic-field strength, and I is the tape length within the magnetic field. Determine the eigenfrequency of the system. The tension Q in the loopbranches is assumed to be constant, the mirror mass is m, and the distance betweenthe tape supports is II.

70 An oscilloscope coil vibrator (coil) is schematically shown in Fig. 61. In contrast to a loop vibrator, the coil has a sensitive element (sensor) that represents aplane coil 1 with several wire turns, which is placed into the magnetic field of apermanent magnet.

11

,1 1-'1

~ N.,IL UJ (

01::: $( ..../1 )/

2 d

Fig. 61.


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The sensitivity of such a device is higher than that of a loop vibrator because,in this case, there exist a larger number of lines of flow in the magnetic field. Vibrations of the coil (which rotates through angle rp by electromagnetic forces) arerecorded with the help of a light beam being reflected from mirror 2 attached to thecoil. Tension members holding the coil and the mirror are made in the form ofsolid tapes or rods of circular cross section.

For the case that the moment of inertia J of the mirror and coil with respect tothe vertical axis is J = 1.02 X 10-2 kg m 2, determine the eigenfrequency of the device. The coil is suspended by a bronze string with a diameter d = 0.2 mm, shearmodulus G =40 GPa, and length I =30 mm. Analyze the mirror motion as a function of the coefficient a of viscous friction (i.e., taking the moment of resistanceforces as alp) after suddenly switching on the electric current in the case that the

torque moment M =Mo = i is proportional to the current intensity i. Determine thevibration frequencies of the coil for the cases of a =0 and a =0.6 (c/J)lf2, where cis the torsional stiffness of the rod.

71 In order to study torsional vibrations of rotary shafts, special devices called torsiographs are used (Fig. 62). In a Geiger-type torsiograph, massive disk (flywheelmass) 2 is mounted with the help of spiral spring 1 inside light pulley 3. The pulleyis rigidly attached to shaft 4 that rotates in bearing 5. The pulley 3 is connected tothe shaft whose motion is being studied. Thus, the rotation of the pulley 3 repeatswith a sufficient accuracy the rotation of the shaft under investigation. In the caseof a uniform rotation of the shaft, flywheel 2 also rotates uniformly. As the angularvelocity of the shaft varies, pulley 3 also changes its velocity, whereas flywheel 2possessing a noticeable inertia continues its uniform rotation. The difference in themotions of the flywheel and pulley is recorded by a special mechanical device (notshown in the figure). The steel flywheel has the shape of a ring with the dimensions d =50 mm, D =100 mm, and b =50 mm and is mounted on a spring withstiffness c = 146 N m.

Fig. 62.


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Set up the differential equation for relative vibrations of the flywheel in thecase of rotation of the pulley at the angular velocity OJ = OJo + OJ,coskt. Allow fordamping the system, with the damping moment being proportional to the relativeangular velocity of the flywheel and the disk. The coefficient of the viscous resistance is a =0.73 N m s. Determine the phase shift between vibrations of the flywheel and of the disk, as well as the logarithmic decrement for vibrations of theentire device in the case of k = 50 s".

72 For compacting concrete poured into a base of structure foundations, specialdevices called vibrators are applied. A vibrator shown in Fig. 63 consists of aheavy frame of mass m with two disks each of mass mI. The disks rotate in a vertical plane in mutually opposite directions at an angular velocity OJ. Loads each ofmass mo are fixed to the disks with eccentricity e with respect to the rotation axis.At the final stage of the compaction process, properties of the concrete base can beapproximately described by a rheological model presented in Fig. 63b.

Set up the differential equation for steady-state vibrations of the vibrator case.Determine the amplitude of the vibrations under the assumption that the vibratorcase does not lose a contact with the mass being compressed.

73 A contactless pickup of the membrane type (Fig. 64) is employed for measuringpressure variations in pipelines. The measurement results depend on the length of aspacing between tip 2 and piece 1. The pressure is registered according to the flexure of membrane 3 by means of special indicator 4 supplied with mobile rod 5.

The pressure P on the membrane is assumed to vary proportionally to a changein the spacing LI, i.e.,

LIp/Po = (e/LI)Asinmt.

(Here, Po is the nominal pressure in the chamber of the device; e is the amplitudevariation of the spacing LI in the process of testing the piece; OJ is the angular velocity of the piece, and A is a dimensionless coefficient.) Plot the dependence ofthe angular velocity OJ of the piece on pickup readings, i.e., the amplitude-frequency characteristic of the device.

a

"Fig. 63.


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30 1 Vibrations of Systems witha Single Degree of Freedom

The mass of the rod 5 is m, the stiffness of the indicator spring attached to therod is c, and the coefficient of the viscous friction arising as a result of the rodmotion is a.

The numerical values of the parameters are the following: m = 0.05 kg,c = 2 X 103 N/m, P = 2 X 105 Pa, L1= 1 mm, and e = 0.1 mm. The membrane area isF = 300 mm\ A = 0.5, and P = 1. The damping coefficient obtained as a result ofthe analysis of device free vibrations is a =10 (N s)/m. The membrane mass canbe ignored.

I n s t r u c t i on . As the pressure changes by LIp, the rod is subjected to the force,1P = flF LIp from the membrane. Here, fl is the proportionality coefficient depending onelastic properties of the membrane side and F is its area.

74. Corrugated cylinders (bellows) are widely used in contactless measuringdevices as mechanical enlargers. A negligible variation of the internal pressure inbellows causes relatively noticeable changes in their lengths. Figure 65 demonstrates a schematic diagram of a bellows-based sensor intended for automaticallycontrolling ovality of pieces. In case 1 of the device, cylinder 4 of mass m is suspended by plane springs 2 having total stiffness cJ" Cylinder 4 is rigidly connectedto bellows 5 having stiffness c2 • The pressure in the bellows depends on pressure Poin pipeline 6. The preliminary elongation t>oand tuning of the device are realizedby spring 7 having stiffness c3• When escaping the piece size beyond the toleranceregion, the closure of one of contacts 8 occurs, and a rejection signal is formed.

In order to reduce the damping time of device vibrations, when testing a piece,viscous-friction damper 3 is introduced into the system. The damping force of the

damper is proportional to the displacement velocity Ff = ax.

Fig. 64.


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2 7

:J 8

4- .t5 a

8~

Fig. 65.

Plot the amplitude--frequency characteristic of the device under the assumption

that the pressure vibration in the bellows is proportional to the variation of thespacing ~ between the measuring tip and the piece, i.e., L1p/Po = (e/LI)Asinmt.(Here, e is the amplitude deviation of the piece from the cylindrical shape, OJ is theangular velocity of the piece, and A is a dimensionless coefficient.) Determine therange of angular velocities OJ in which the dynamic measurement error does notexceed 10% of the quantity being measured. (In other words, an interval of OJ inwhich the vibration amplitude of mass 4 differs from the static deviation by nomore than 10%.)

Use the following numerical values: m = 0.1 kg; c J =20 N/cm; c2 =5 N/cm;c) =10 N/cm; Po =3 X 10

5Pa; LI =1 mm; and A =1.0. The effective area of thebellows is F = 400 mm 2• The coefficient of viscous resistance determined accord

ing to an oscillogram of free damped vibrations is a =75 (N s)/m.75 A trailer of mass m moves at a constant velocity v along an irregular road(Fig. 66a). Assuming that the point of fixation of the trailer to a car has no verticaldisplacements, determine the velocity of steady-state motion of the system forwhich the vibration amplitude becomes maximal. Admit that the stiffness of tiresis much larger than spring rate C of the leaf springs. The road irregularity profilecan be described by the equation h =ho[1 - cos(2JZX l)]. The trailer moment of inertia with respect to the point 0 is lx' Assume also that the viscous friction (withthe coefficient a) arises in the leaf springs. The mass of the trailer wheels is considered to be small compared to the mass m.


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,m

V 0

~ .c:;~

a

11'

b

Fig. 66.

I n s t r u c t ion. Solving the problem, see the trailer schematic diagram presented inFig. 66b and suppose that the irregularity height h is considerably smaller than I (i.e., therotation through angle qJ results in only vertical displacement of the point A).

76 An external perturbing torque that varies with time according to the periodiclaw M = Mosinwt is applied to a mechanical system containing a disk (see Fig. 43).Derive the set of differential equations determining the steady-state motion of thedisk.

In s t r u c t ion. Consider a half-period T/2 = niL of disk vibrations, which consists oftwo segments of motion (see the solution to Problem 47). Matching these segments, derivethe equation for determination of parameters of stable vibrations.

77 Assuming that a periodically varying torque M =Mosinwt acts on the systemshown in Fig. 43, determine the amplitude of steady-state vibrations and of theperturbing force.

I n s t r u c t ion. Change the Coulomb friction by viscous friction using the conditionof equality of the incoming and dissipated energy over a period of stable vibrations.

78. Solve Problem 47 in the case that the perturbing moment M =Mo + M1sinwtacts on the system. Allow for the fact that, in addition to Coulomb friction with afriction factor /1, a resistance-force moment acts on the disk. This moment is proportional to the angular velocity of the disk and is equal to a ip .


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a

P(t.J

t

Fig. 67. Fig. 68.

Using instruction to Problem 77, estimate the amplitude of stable vibrations ofthe system and find the phase angle.

79 A perturbing force varying according to the periodic law P(t) =P)sinwtl acts ona load of mass m (Fig. 67). Derive the equation describing stimulated vibrations ofthe load and find the solution in the case of the stable vibration mode. Ignoring resistance forces, determine the frequency (() of a perturbing force at which a resonance of stable vibrations takes place. Find the solution to the problem by both theFourier method and Duffing method.

80 A load of mass m, which is suspended by a spring with stiffness c, is subjectedto the attraction force P(t) produced by an electromagnet (Fig. 68a). This force canbe approximately represented in the form (see Fig. 68b)

{Posin wt for 2Jl'n~ wt (2n + 1 Jl';

P(t)= o for (2n+ 1)Jl' ~ wt 2(n+ 1 Jl'.Find the solution to the problem by both the Fourier method and Duffing

method in the case of stable vibrations and ignoring of resistance forces.

81 Analyze motion of a load of mass m (i.e., determine the vertical coordinate y(t)and vertical velocity yet) as a function of time t). The load is fixed at the end of acantilever beam of length I and is subjected to the action of a unit impulse of forcer (t)dt = p (see Fig. 69). The beam mass and resistance force may be ignored.


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11,ty

t

peta Ift}

t

b t

Fig. 69. Fig. 70.

82 Analyze motion of a load of mass m (see Fig. 69) in the case of a suddenly applied force P, i.e., taking P(t) =Po =const (Fig. 70).83 Analyze motion of mass m fixed at the end of a cantilever rod (Fig. 69a) under

the action of a force P(t). The force varies according to the laws shown inFigs. 7la and 71b.

84 Analyze motion of mass m (Fig. 69) under the action of a linearly increasingforce P(t) = at in the case y = y = 0 as t = O.85 A load of mass m (Fig. 72a) is subjected to periodic unit pulses (of the equalsign and with period n of the force pet) (Fig. 72b). Analyze stimulated stable vibrations of the load. Using the Duffing method, determine the coordinate x(t) andthe velocity x(t) as a function of time t within the interval 0 :s; t :s; T. The frictionbetween mass m and the guide can be ignored.

pet) F t)

a b

Fig. 71.


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1.4 Critical States and Vibration Stability 35

a

ot

Fig. 72.

p(t)

8tr n D.tFig. 73. Fig. 74.86 Analyze stimulated stable vibrations of a load of mass m (see Problem 85) under the action of a periodic pulsed alternating-sign force (Fig. 73).

87 Analyze stimulated stable vibrations of a load of mass m (Fig. 72a) under theaction of a periodic force with period T (Fig. 74). Determine the coordinate x(t)within the time range 0::;; t::;; T.

88 Solve Problem 85 with allowance for a viscous resistance force Ff =ax.Find the steady-state regime for the motion of a load having mass m (see Fig. 72a).

1.4 Critical States and Vibration Stability

89 The velocity of motion of fluid in pipelines affects their eigenfrequency. Thetransportation velocity for which the eigenfrequency equals zero is called critical.For the tube shown in Fig. 75 set up the differential equation describing small free


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vibrations in the case of the constant fluid flow velocity va' Determine the flowvelocity v. for which the motion of the tube becomes aperiodic. The tube mass andlength are m and I, respectively. The orifice cross-sectional area is Fa' and the den

sity of the fluid is p. The tube can freely rotate with respect to the hinge A.

90 A piece of mass m is fixed to a thin-walled horizontal pivoted massless tube(Fig. 76). A perfect incompressible fluid flows through the tube at a velocity v.The average tube diameter, wall thickness, Young modulus of the tube material,and moment of inertia of the tube cross section are, respectively, d, 0, E, and J x

Determine the velocity v. of the fluid flow through the tube in the case of thezero frequency of system free vibrations. When analyzing the tube vibrations, boththe initial flexure of the system, which is associated with the action of the gravityforce, and inertial forces of the translational motion of the fluid (including theCoriolis force) may be ignored.

91 A sprayer disk having n radial holes (Fig. 77) is fixed to a hollow shaft havingtorsional stiffness c. The velocity of the fluid efflux from the disk holes is va' andthe disk moment of inertia is J.

Set up the differential equation describing small free vibrations of the disk withrespect to steady-state uniform rotation proceeding at an angular velocity OJ (withallowance for an effect of the fluid flow). The fluid is considered as perfect, i.e.,incompressible.

Determine the eigenfrequency of the disk and the fluid flow velocity v. forwhich the disk vibration frequency becomes zero.

Fig. 75. Fig. 76.


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Fig. 77. Fig. 78.

92 Solve Problem 90 (see Fig. 76) in the case that the mass m is additionally supported by a spring having stiffness c.

93 A disk of diameter D =100 mm and of thickness h =10 mm is built up on aflexible steel shaft (Fig. 78) with diameter d = 5 mm and length 21 = 0.5 m. Thecenter of gravity 0 1 of the disk is located at a distance e = 1 mm from the shaftsymmetry axis O.

Set up the differential equation (in the vector form) for small free vibrations of

the disk at a constant angular velocity aJ of the shaft and in the case of only translational disk motion. Assume the shaft mass to be small compared to the diskmass. Determine the critical angular velocity aJ. of the shaft and its maximal stressat aJ =100 S-I.

Ins t r u c t ion. The critical angular velocity is implied as a velocity for which the vibration amplitude can be infinitely large.

Fig. 79. Fig. 80.


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a

~ f = - 4 , ; ~ - %\. , ~

Fig. 81. Fig. 82.

94 A rotating cantilever shaft (Fig. 79) of length 1 and having bending rigidity EJ xcarries a point load of mass m at its free end. Ignoring resistance forces and theshaft mass, determine the critical angular velocity of the shaft.

95 A flexible shaft of a constant cross section is fixed at its ends in short hingesand can be considered as a pivoted beam of length 1 and of bending rigidity EJ x(Fig. 80). Determine the critical angular velocity of the shaft for which its rectilinear shape becomes unstable.

96 Determine the critical angular velocity of flexible shafts with a constant crosssection for which their rectilinear shape becomes unstable. Consider the cases offixed shaft ends, which are presented in Fig. 81.

97 Determine the critical angular velocity of a tubular hollow shaft with allowancefor an ideal incompressible fluid flowing inside it (Fig. 82). The mass of the fluidper unit length of the tubular shaft is m J ; the mass of the tube unit length is mo; thefluid flow velocity is v = onst; and the bending rigidity of the shaft is EJ x• TheCoriolis inertial forces of the fluid may be ignored.

98 A sensitive component of the device shown in Fig. 83 consists of spring 1 having stiffness c, levers 2 each of mass m J , which are hinged at the points A, andfreely moving rod 3 of mass m2 • In the steady-state operation mode ( 0 ) = const),the moment of inertia of the levers is balanced by moments of both the rod gravityforce and spring compression force. Determine the eigenfrequency of the rod and

the critical angular velocity.99 Determine the eigenfrequency of loads (each of mass mJ ) installed in a centrifugal regulator (Fig. 84) for the case of its rotation occurring at a constant angular velocity 0) . Ignoring an effect of the vertical displacement of the loads on boththe potential and kinetic energies of the device, derive the differential equation for


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Fig. 83. Fig. 84.

small vibrations of the system and determine its eigenfrequency. The rod mass isnegligibly small compared to both mass m l and sleeve mass m2 • The stiffness of thecompression springs is c. The sleeve transferring the rotation to the loads can shiftonly in the vertical direction.

100 Solve Problem 8 under the assumption that in the equilibrium position, thecarrier is directed vertically upward, and the springs attached to the carrier have acertain initial elongation 8 (see Fig. 7). Determine the critical length t. of the car

rier, i.e., the length for which the natural frequency of small free vibrations is zero.101 A homogeneous rod of mass m, length t, and with a constant cross section ishinged to an axis rotating uniformly at an angular velocity ( j ) (Fig. 85). Analyzesmall free vibrations of the rod with respect to the positions of static and dynamicequilibria (in Fig. 85, the angle a characterizes the state of dynamic equilibrium).Determine the angular velocity ( j ) at which the equilibrium state becomes unstable.

102 A load of mass m is fixed at the end of a weightless rigid rod of length I(Fig. 86). The rod is hinged to a sleeve rotating at a constant angular velocity (j).

Determine the critical angular velocity ( j ) . at which the vertical position of therod becomes unstable. Find the frequency of small free vibrations of the load withrespect to the vertical position of equilibrium. (at ( j ) < ( j ) , ) and the load vibrationfrequency with respect to the position of the dynamic equilibrium (at ( j ) > ( j) . ) .


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40 1 Vibrations of Systems with a Single Degreeof Freedom

Fig. 85. Fig. 86.

103 A massive disk uniformly rotates at an angular velocityOJ

(Fig. 87). A load ofmass m connected to two identical springs is installed in a disk slot. Each spring ischaracterized by stiffness c and has initial elongation bIT Set up the differentialequation for small free vibrations of the load. Determine the critical angular velocity OJ, of the disk for which the load position at the disk center becomes unstable.

104 A rigid frame rotates in the horizontal plane at a constant angular velocity OJ(Fig. 88). A load of mass m, which is supported by a spring with stiffness c, canslip along the radial slot made in the frame.

Fig. 87. Fig. 88.


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Fig. 89. Fig. 90.

U sing the Lagrangian method, set up the differential equation describing smallfree vibrations of the load with respect to its dynamic equilibrium position at adistance Xo from the rotation axis. Determine the eigenfrequency of the load andthe critical angular velocity of the frame.

105 A thin absolutely rigid plate of thickness 0 is fixed to a rod of the circularcross section having diameter d and angular stiffness c. The plate is placed into airflow whose velocity is v (Fig. 89).

A lifting force arising as a result of deflection of the plate through a small angle a is considered to be applied at a distance hl4 from the plate free end and isdetermined by the expression

Y =7rpv 2Sa.

(Here, p is the air density, S is the area of the plate surface, and a is the angle ofattack.) Analyze the plate eigenfrequency as a function of the air flow velocity. Ig

noring the drag force, determine also the critical velocity of plate divergence.I n s t r u c t i on . The critical velocity of plate divergence is implied as a velocity at

which one more form of equilibrium appears with a. *-O.

106 A massive lever is fixed on an elastic roller of length I and having the torsional stiffness c (Fig. 90). The lever moment of inertia with respect to the rolleraxis is 1. To the roller, a force Po is applied whose line of action passes through theroller axis. Set up the differential equation describing small free vibrations of thesystem in the case that under rotation of the lever, the line of force shifts in parallelto its initial position. Determine the system eigenfrequency and the critical forceF" that corresponds to the zero vibration frequency.


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N

Fig. 91. Fig. 92.

107 A load of mass m is fixed with the help of elastic tension members each oflength l. The load is placed into a constant homogeneous magnetic field producedby a magnet (Fig. 91).

Determine the eigenfrequency of the load for the case that the tension To in the

tension members does not vary in the course of the vibratory process. The magnetic attraction force F/.2 = [ (/J02/(a/ + X)2] is inversely proportional to the distancefrom the magnet. Find also the critical value of the magnetic flux (/Jo and the tension To' (The critical values correspond to the zero frequency of the load naturalvibrations).

108 A load of mass m (Fig. 92) fixed at the end of a long elastic rod is placed intoa constant homogeneous magnetic field having the magnetic flux (/JO" In the case ofsmall deviations of the load from its equilibrium position, the magnetic-field forcecan be considered as being proportional to the displacement x (see Problem 107).

Determine the load eigenfrequency and the critical value of the magnetic-field intensity. An effect of the rod mass may be ignored.

Fig. 93.


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1.5 Parametric Vibrations 43

109 A point load of mass m is fixed to a hinged weightless beam (Fig. 93). A concentrated force Ro moves along the beam at a constant velocity v. Find an approximate law for the vertical displacement of the load and determine whetherthere is a critical value of the velocity v for the motion of the force Ro when theamplitude of load vibrations unboundedly increases.

1.5 Parametric Vibrations

110 A pendulum (Fig. 94) represents a load of mass m fixed to a weightless rod oflength I. The upper point of the pendulum suspension executes vertical periodicmotion that obeys the harmonic law y = osinax. Analyze stability of small vibrations of the pendulum in the case of OJ =10 s-', 1=0.5 m, m =1 kg, andYo= 10 mm.

111 In Fig. 95, the schematic diagram of an astatic pendulum is shown. As thefixation point 0 of the pendulum vibrates at a certain frequency OJ exceeding theminimal frequency OJrnin , the vertical position of the load remains stable. Determinethe value of OJrnin in the case that the motion of the point 0 is described by the periodic law y/ = osinOJt with 1=0.2 m and Yo =5 mm.112 A steel absolutely rigid rod of a constant cross section, which has the length1= 0.12 m and mass m = 0.2 kg, is hinged at its lower end to a certain base. Thebase vibrates with an amplitude Yo = 10 mm (Fig. 96). Assuming motion of thebase to be harmonic (Y1= yosinOJt), determine the minimal frequency OJmin forwhich the vertical position of the rod becomes stable.

Fig. 94. Fig. 95.


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Fig. 96. Fig. 97.

113 A rectangular steel plate ofthe size b x 1 x Ii (Fig. 97) is placed into air flow.One of the plate ends is pivoted, whereas the other end has an elastic fixation withthe total stiffness c = 2.5 kN/m. The air flow velocity varies according to the periodic law v = Va + v Isinmt, where 0 ) = 100 S-I, Va = 30 mis, andVI = 5;6;7;8;9;lO mls.

Set up the differential equation for small vibrations of the plate and analyzetheir stability in the case ofv a » VI. The application point of the lifting force corresponds to the distance 1/4 from the leading edge of the plate. Numerical values ofthe relevant parameters are the following: 1= b = 0.2 m and Ii = 5 mm. The airdensity is p = 0.125 kg/m 3•

I n s t r u c t i on . When deriving the differential equation for small vibrations, the lifting force can be written out as

Y =0.5c yF pv\where cy = 21(a is the lifting factor, F = hI is the plate area, and a is the angle of attack.

114 An elastic rod of length 21 with pivoted ends is subject to the axial periodicforce P(t) = Po + PlsinO)t (Fig. 98). Set up the differential equation for the motionof a load of mass m fixed to the weightless rod undergoing small transverse vibrations. Analyze stability of small vibrations of the system in the case of the rodlength 21 = 2 m, the moment of inertia of the rod cross-sectional area J x = lO-9 m4,Young modulus E = 200 GPa, m = 5 kg, Po = 50 N, P I= 20 N, and (J) = lO S-I.

115 Set up the differential equation for small vibrations of a load of mass m fixedat a pillar of length 21 (Fig. 99). The pillar is subjected to the axial periodic forceP(t) = Po + Plsinrot. Analyze stability of small vibrations of the load for the following values of parameters: 1 = 1 m, J = lO-9 m4, E = 200 GPa, m = 5.2 kg,Po = 50 N, P I= 20 N, and ro = 30 S-I.


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-11

Fig. 98. Fig. 99.

116 A load of mass m is suspended by two tension strings each of length 1(Fig. 100). Ignoring masses of the strings and their tensility, analyze small verticalvibrations of the load in the case of a tensile force varying according to the periodic law T(t) =To + TJsinmt. The values of relevant parameters are To =20 N,TJ= 0 N, OJ = 5 s-\ m =0.25 kg, and 1=0.2 m.117 A steel beam of length 1carries a concentrated load of mass m at its free end(Fig. 101). Analyze stability of small free vertical vibrations of the load in the case

of the beam length varying according to the periodic law I(t) = 10+ 11sinOJt. Thebeam mass is negligibly small compared to the load mass. The values of relatedquantities are 10= 1 m, 11= 0.2 m, m = 2 kg, OJ = 20 S-l, J x = 8.3 x 10-10 m4, andE=200GPa.

T

Fig. 100. Fig.lO .


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Fig. 102. Fig. 103.

118 A load of mass m is suspended by an absolutely flexible inextensible line insuch a manner that the length of the free line segment is 10 (Fig. 102). Set up thedifferential equation describing small vibrations of the load in the case that thepoint ° the position of the hole from which the line emerges) shifts with respectto its mean position according to the harmonic law Yj = yosinmt. Derive the differential equation by two methods: (i) using the theorem for the variation of angularmomentum and (ii) using the d'Alembert method.

119 Set up the differential equation for small vibrations of a pendulum having theline length that varies according to the periodic law 1(1)= 10+ IjsinOJl (Fig. 103).120 In a heavy frame of mass mj, which has a pivoted suspension at the point 0, aslider-crank mechanism driving a load of mass m is set up (Fig. 104). While oper

ating the system, the position of the system center of mass is shifted because of avariation in the load position

Derive the differential equation for small vibrations of the system in the casethat the distance from the point of suspension to the system center of mass equalsIj • In the horizontal position of the crank, the distance between the point 0 1 and thecenter of the load is 10 • The moment of inertia of the frame with respect to the pointof suspension is J j ; the crank angular velocity is w; and the crank shoulder lengthis Xo. The masses of both the crank and the crosshead are small compared to theload mass.

121 A load of mass m attached to a pendulum is forced to shift along its axis according to a periodic law (Fig. 105). Set up the differential equation describingsmall vibrations of the pendulum. The masses of both the rod and the frame are assumed to be small compared to the load mass.


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Fig. 104. Fig. lOS.

122 An extensible rope is fixed at a tub of mass m (Fig. 106). At the point A, arope is rigidly connected to a pulley of radius R. In the equilibrium state, (x = 0),the length of the rope is 1 0, and its tension is To. The distance between the tub railsand the pulley is 101~ 10. Set up the differential equation describing small horizon

tal vibrations of the tub in the case that the pulley performs a periodic motion according to the harmonic law cp= CPosinrot, the tub having the displacement Xo at theinitial moment of time. Analyze stability of small vibrations of the tub in the caseof To = 103 N; m = 500 kg; 10= 1 m; CPo= 0.1 rad; and 101= 0.7 m. The rope crosssectional area is F = 10-4 m2, R = 0.1 m; and ( j ) = 10 S-I. The Young modulus of therope material equals E = 200 GPa. The friction in the system may be ignored.

123 Set up the differential equation describing small free vibrations of a load ofmass m, which is fixed at the end of an elastic rod (see Fig. 92). The rod is placedin the magnetic field with the alternating magnetic flux l/X...t)= (fJo+ (fJlsinmt. Themagnitude (fJI of the magnetic flux is low, i.e., the term containing (fJI can be neglected. Analyze small vibrations of the load in the case of m = 500 kg and1= 0.21 m. The bending rigidity of the rod is EJ x = 2 x 10 3 N m 2; ( j ) = 200 S-I;al = 10 mm; (kia/)(fJo = 10 N; and (kia/)(fJI = 1 N.

124 In a magnetic field formed by a permanent electric current of intensity hwhich flows in infinitely long rigid conductor 1, bus 2 of mass m is placed(Fig. 107). In the bus, electric current of intensity 1] = ho + hlsinmt also flows. Theinteraction of the two magnetic fields results in attraction with a force

F = 2pllhlal between conductor 1 and bus 2, where f.l is the magnetic permeabilityand a 1 is the distance between the conductor and the bus at an arbitrary moment oftime.

Set up the differential equations for translational motion of bus 2 in the casethat the stiffness of each of the two springs supporting the bus equals c and thedistance between the conductor and the bus in their rest state (i.e., at h = ho) is ao.


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II

,

..

Fig. 106. Fig. 107.

125 Analyze stability of small vibrations of the system in the case that a periodicforce P applied to the lever varies according to the periodic law P = Po +PIsinOJl(see Problem 106, Fig. 90). The numerical values of relevant variables arec = 80 N m; Po = 100 N; PI = 40 N; R = 0.2 m; LI = 10 S-I; andJ= 0.4 kg m2.

126 Derive the differential equation describing small vibrations of a load of massm (see Fig. 25), which moves at a constant velocity v along an absolutely flexiblestring. The tension To ofthe string is assumed to be constant.

127 A pendulum is suspended by an absolutely flexible line of length I (Fig. 108).Analyze stability of small vibrations of the pendulum in the case of constant linetension T = To. The static flexure of the line at the point of suspension is Yst. Thependulum length and mass are II and m. At the initial moment of time t = 0, whenthe pendulum is deflected through an arbitrary angle rp, the pendulum point of

suspension has a vertical displacement Yoo and velocity Yoo =o. Two casesshould be considered:(1) The coordinate of the point of suspension is SI = 0.3/; the initial tension isTo = 40 N, the pendulum length is II = 0.3/; the pendulum mass is m = 0.1 kg; andYoo= O.l/.(2) To = 40 N; II = 0.31 [other parameters are the same as in variant (1)].

128 A load of mass m = 0.2 kg is fixed to a weightless rod of length 1= 80 mm(Fig. 109). The upper end of the rod (the pendulum point of suspension 0) is connected to a spring having stiffness c = 1 kN/m.


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1.6 Nonlinear Vibrations 49

Fig. lOS. Fig. 109.

For the case that the point 0 can perform only vertical vibrations and assumingthe crosshead mass m' to be small compared to the load mass, analyze stability ofsmall vibrations of the pendulum for two variants of initial conditions:(1) at the moment of time t = 0, the displacement of the pendulum point of suspen

sion with respect to the equilibrium point is Yo = 40 mm and the velocity isYo=O;(2) at the moment of time t = 0, the displacement of the pendulum point of sus

pension with respect to the equilibrium point is Yo = 0 and the velocity is Yo = 0.1ms- I .

1.6 Nonlinear Vibrations

129 Set up the differential equation of free vibrations of a system consisting oftwo disks of diameters Dl and D2 with masses m 1 and m2 respectively (Fig. 110).The disks are connected to each other by a massless rod. The upper disk can rollon the plane without slipping.

130 A rigid rod of mass m and length 21 with a constant cross section is hinged atits middle point A. Two springs each with stiffness c are attached to the rod ends(Fig. 111). Set up the differential equation of free vibrations of the rod and determine its natural frequency. The elastic forces of the springs are assumed to beproportional to variations in their lengths.

131 A rigid rod of mass m and length 21 with a constant cross section is hinged atits upper end. Two springs each with stiffness c are attached to its middle point


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I

Fig. 810. Fig. 111.

(Fig. 112). Set up the differential equation of free vibrations of the rod and determine its natural frequency.

132 A load of mass m can freely move along a guide without friction (Fig. 113).

Two free springs each with stiffness c are fastened to the load sides. There areclearances L1between the walls and the spring ends. Find the dependence of thevibration frequency on the vibration amplitude xO'

133 Solve Problem 132 in the case of different springs (c, *-c2 ).

134 A load of mass m can freely move on a plane without friction, compressingtwo springs in tum (Fig. 114). The springs with stiffnesses c, and C2 are not fas

I I

I I

Fig. 112.


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Fig. 113. Fig. 114.

tened to the load and have no preloads. Determine the frequency of free vibrationsof the system.

135 In order to suppress the influence of vibrations on devices, they are usuallyinstalled on a shock absorber whose eigenfrequency should be significantly lowerthan the vibration frequency. It is desirable that the ratio of the vibration frequencyto eigenfrequencies of all the devices to be cushioned would be independent oftheir masses. Determine the characteristic of such a shock absorber (i.e., the dependence of its contraction on the force applied to it) in the case when its contraction caused by a force Po is equal to xo'

136 The right end of the line considered in Problem 116 is fixed, and the preliminary tension To is given (Fig. 100). Derive the equation of vertical vibrations of themass m with allowance for tension variations due to displacements of the mass m.Plot the solution on the phase plane under the initial conditions x(O) = o anddx(O)/dt = O.

137 Two springs are attached (without preliminary tensions) to a body of mass m(Fig. 115). Each of the springs has length 10and stiffness c. The body can move ona horizontal plane without friction. Determine the amplitude-frequency relationfor free vibrations of the body under the initial conditions x(O) = Xo anddx(O)/dt =O.138 Determine the frequency of free vibrations of the body considered in Problem 137 as a function of the initial velocity Vo at t =0 when x(O) =O.139 The springs shown in Fig. 115 have preliminary tensions To (at the equilibrium position). Determine the trajectory of the load on the phase plane. The fric

tion force may be ignored. Use the following numerical values: x(O) =10 mm,dx(O)/dt =0,1 = 50 mm, c = 500 N/m, m = 0.1 kg, and To = 1 N.140 The springs shown in Fig. 115 have preliminary tensions To (at the equilibrium position). The Coulomb friction force acting upon the load is F =2.5 10-2 N.


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y

mx

Fig. US. Fig.U6.

Detenmne the trajectory of the load on the phase plane in the case of To =1 N,m = .1 kg, e =500 N/m, and 10=50 mm. At t =0, x(O) =10 mm and dx(O)/dt =O.141 A body of mass m is attached to a spring, with its nonlinear elasticity characteristic having the form F = x + e / (Fig. 116). Set up the differential equation offree vibrations of the body and find the amplitude-frequency relation. The frictionforce may be ignored. The initial conditions are x(O) =A and dx(O)/dt =O.Take that A =1 cm, e =100 N/m, e) =50 N/m, and m =100 kg.

Instruction. The problem should be solved in the first approximation using the Lyapunov-Lindsteadt method.

142 Solve Problem 141 by the linearization method, i.e., by substituting the nonlinear elastic force by a linear one found by the method of least squares.

143 Using the Galerkin method, find the amplitude-frequency relation for thesystem considered in Problem 141.

144 Using the linearization method (and the method of least squares), determinethe vibration frequency for the load of mass m considered in Problem 132. The vibration amplitude is equal to A.

145 A body of mass m is fastened to two springs having preliminary tensions To(Fig. 115). Ignoring the friction force, find the initial displacement Xo such that thevibration frequency be equal to 5 1/s. Use the method of small parameter (Lyapunov-Lindsteadt method) in the first approximation. Take that m =10 kg,10=0.5 m, To =50 N, and e =2 kN/m.146 Analyze the motion of the mass described in Problem 108 (Fig. 92). The initial displacement of the mass is x(O) = xo' Allowing for only the first nonlinearterm of the magnetic force, derive the differential equation of motion of the mass.


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Determine the critical value Xo' of the initial displacement Xo and plot the dependence xo(


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Use both the method of small parameter and the Galerkin method and compare theresults obtained.

152 A lever of mass m is fixed on a shaft of length I with torsional stiffness e(Fig. 117). One end of the shaft is fixed. Another end is supported in a bearing,with the modulus of the frictional torque Mr proportional to the angular velocitysquared: IMrl= (dqtdt( The moment of inertia of the lever about the shaft axis isJ, and the distance from the center of gravity to the shaft axis is 1 , Determine thedependence of the vibration frequency of the lever on its initial deflections in thefirst and second half-periods and find the vibration amplitudes at the ends of thehalf-periods. Use the method of small parameter

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Svetlitsky ~ Eng Vib Analysis,Vol.1