Office of Naval Research. Department of the Navy.Contract N6ori-71, Task Order VI, Project NR-064-18

Newmark, N.M.

Veletsos, A.S.

English

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CIVIL ENGINEERING STUDIES .~. ~ .~~ tTeT~L RESEARCH SERIES NO. 25 ~ .r .. ~.~ /:5 ~~~2 fl~' e ~ ~ A SIMPLE APPROXIMATION FOR THE NATURAL FREQUENCIES OF PARTLY RESTRAINED BARS .... .-' [-: By N. M. NEWMARK and A. S. VELETSOS Technical Report to OFFICE OF NAVAL RESEARCH Contract N6onr-71, Task Order VI Project NR-064-1 83 UNIVERSITY OF ILLINOIS URBANAr ILLINOIS .. ' -", ! SIMPLE.APPROXIMATION FOR THE NATURAL FREQUENCIES OF PARTLY. RESTRAINED BARS by N. M. Newmark ·and A. S. Veletsos A Technic·al Report of a Research Project Sponsored by TEE OFFICE OF NAVAL RESEARCH DEP ARTMENT OF THE NAVY in Cooperation with TEE DEP AR.T'MmNT OF CIVIL ENGINEERING UNivERSITY OF ILLINOIS Contract N6onr--7l, Task Order VI Project Designation No. Nh-064-183 Urbana, Illinois 15 April 1952 ~ SIMPLE APP~OXIMATION FOR THE NATURAL .FREQUENC.IES OF PARTLY RESTRAINED BARS' by No M. Newmark and A~ S. Veletsos University of Illinois This paper presents a simple approximate formula. for the natural frequencies of flexural v.ibration of a bar which rests at each end on non-deflecting supports but which is elastically restrained against rotation at the ends 0 The bar is of consta.nt cross·-sectionand· density. The end restraints are assumed to be proportional to the end rotations, and the restraint stiffness may have any value in the range between perfectly hinged and completely fixed conditions. The restraints may be due to elements such as coil springs, or they may result from the continuity of the bar 'with adjoining members. Consider the bar tD be oscillating in 'oneof its natural modes of free vibration. Let BL and GR be the angular rotations of the bar at the left and .right ends respectively, and let ML and MR be the moments at the corresponding ends. The relation between moments and rotations at the ends is expressed by the equat~ons: and where KL and KR represent the stiffnesses of the restraints at the left and the right ends. The stiffness K is de~ined as the moment necessary to rotate the spring restraint, or the restraining member -which it may symbolize, by a unit amount 0 A restraint is referred to as positive when the moment which it exerts on the bar "acts in a direction opposite to the direction of rotation of the bar; when the rea.ctive moment acts so as to augment this rotation, the restraint is r~ferred to as a negative restr~int. A positive restr~int~ there~ore, resists the tendency of the bar to rotateo It is convenient to express thestiffnesses of " the restraints by dimensionless coefficients which are related tn the characteristics of the bar ~s follows: and (2) where L - the length of the span; E = the modulus 01: elasticity of the material in the bar; and I = the moment of inertia of the cross-section of the ba.r about "its centroidal axis. In terms of the coefficients f3L a.nd f3 R it is possible to derive lI exact" v~lues of the natural frequencies of flexural vibrationo These appear as the roots of transcend"ental equations involving trigo'-nometric and hyperbolic functions 0 In gener~l the roots are tedious to evaluate, and the formulas do not give a simple picture of the mechanism of the action of the bar. Gr"aphs or tables are required to obtain useful values of the frequencies for a range iIi the parameters, The frequencies can also be expressed by approximate formulas 0 It ~s convenient to state the frequencies of the elastically restra.ined bar as the product of a dimensionless coefficient Cn multi-plied by the fundamental frequency f of the same bar with hinged ends: o 3. where E, .!, and!: are as previously defined, ~ is the masS ,per unit of length oithe bar, and in is the n-th frequency of 'the ba.r in its elastically restrained condition, expressed in cycles per S~conda From the results of numeric'al calculations based on the exact, solutions the following 'simple approximatioTh was developedf'or the coefficient Gn : c ~ [~ + ! ( ~L )] In + ! ( ~R· )] n 2 ,5n + 13L " 2 5n + ~R (4 ) , I In this expression,. n is the order of the desired natural frequency and' ~L and ~R are the restraint coefficients as defin-ed' by equation (Z). For the fundamental frequency, n = 1. In,genera.l, for the n-th mode there' are E. waves, alternately upward and downward, in the deflection curve along the length of the bar. The foregoing formula is a.pplicable to bars 'having: positive end restra.ints. It is valid f'orthe f'undamental as well as the higher -n.atu:r:a.l frequencies and it is accurate within a maximum error Df"4 per cent, with the maximum error occurring in the lowest or fundamental tr-equency. For the first and second natural frequencies, 'tne percentage error resulting from the us,e of 'this approximate formula is indicated in Fig. 1 in the form of a contour map. Since the error curves are sym-metrical with respect to a diagonal line, only one-half of the contour map is plotted for each frequency. The curves above the diagonal correspond to the first natural frequency while those below the diagonal pertain to the second natural frequency. Bec'aus e of symmetry, 1 t is possible to specify that the subscript ~ always refers to the larger 4. end restraint 0 The scale o£ the coordinates is chosen so chat the entire renge of positive restraints, from zero to infinity, is represented. A positive error indicates tha.t the £requency given by equa. tion (4) is too high. It can be observed that the maximum err'or in the second frequency is negligible for all practi-cal purposes.,.· The errors become prDgressivaly smaller for the higher modes. The greatest error is obtained for a bar fixed at one end 8J:J.d hinged at the other, where the fundamental frequency given by equ~.tion (4) is 1.50 f~, and the true value is 1.56 fo~ The large errors in the first mode are con-centr'ated over 'an extremely ,small region, which corresponds to the cases of bars with one end practically unrestrained and'the other subj ect to 'a very stiff restr'ainto With the exception of this localized region, the maximum error over the remaining domain of end fixities is consistently less than 2 per cent~ When the end restr'aints are very small, i 0 e o 'for -values of 13 approaching' zero, the value of C is express ed more- cOINe-nie-ntly- by n the relation: (5) Equation (5) is in error by less than 1·5 per cent whenf3L + f3 R is less than 1.0, and the error is always positive. The approximate formulas are based on exact 'Solutions derived from the ordinary theory of flexure of beams J which-neglects the effects of shearing deformation and of rotatory inertia, Because the influence of these neglected factors may become significant in the higher modes, the present formulas should not be used without regard for the limitations imposed by the ordinary theory of flexure. -I I I This paper is based upon an observation of' the senior author. \ The approxim'ate equatIons are similar to those given in a previous paper* relating to buckling loads for partly restrained bars. The calculations and d-etailed study were made as a part of a program sponsored by the Office of Nav'al Research (Mechanics Branch) in the structural Research Laboratory, Department of Civil Engineering, of the University of Illinois 0 *N. M. Newmark, "A Simple Approximate Fermula for Effective End'-Fixi ty of Columns," Journal of the Aeronautical Sciences, Vol. 16, No.2, February 1949, p~ 116. i ~ ~ ~ i ~ I i 1 I 1 I I >. U C (l) ::l 0-(l) ~ I.J.... ...... V> Q) ~ 0 --I 'I... 0 I.J.... " ~ A 5+{3R' For Lowest Frequency 0.2 0.8 0.6 0 .. 4 For n =2 -0 .. 2 0.?,':J ". ",. ",. ... 0:':> /' /' /' /' / o 0 0.2 0 .. 4 0.6 0.8 {3 .' IO+L/BL' For Second Fr e que ncy FIG.I. CONTOUR PLOTS OF PERCENTAGE ERROR IN NATURAL FREQUENCY AS DETERMINED BY EQUATION (4) ', .. '-: .. '.'- ., 0.8 ~ (,) c:: (l) ::l 0-(l) ~ 0.6 i.J... " "'0 c:: 0 U (l) (j) 0.4 'I... 0 l.J... ------- 1t 0.2 LO O 

A Simple Approximation for the Natural Frequencies of Partly Restrained Bars

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