APPROXIMATE DETERMINATION OF THE FUNDAMENTAL FREQUENCY OF A CANTILEVERED BEAM WITH POINT MASSES AND RESTRAINING SPRINGS

Journal of Sound and Vibration (1996) 195(2), 229–240

APPROXIMATE DETERMINATION OF THE FUNDAMENTAL FREQUENCY OF A CANTILEVERED BEAM WITH POINT MASSES AND RESTRAINING SPRINGS P. K. S Directorate of ENTEST Laboratories, Research Center Imarat, Hyderabad 500 069, India (Received 28 July 1994, and in final form 3 January 1996) This paper deals with the approximate determination of the fundamental vibration frequency of a uniform cantilever beam with any number of mounted masses and restraining springs at various locations along the length of the beam. Numerical results are obtained by using a finite element technique. The beam has a uniform rectangular cross-section and the constraint is a translational spring. A method known as the equivalent center weight method forms the basis of this investigation. Design charts have been established for equivalent factors for both the mounted masses and springs. These charts are useful for quick determination of the natural frequency of a cantilever beam with masses and springs at any arbitrary locations. 7 1996 Academic Press Limited

1. INTRODUCTION

The free vibration natural frequencies of restrained uniform cantilever beams have received considerable attention (see references [1, 2] and references therein). Gu¨rgo¨ze [3] has studied the fundamental frequency of a restrained cantilever beam carrying a heavy tip mass by the combined use of Dunkerley’s and Southwell’s method. Wu and Lin [4] have investigated the case of a uniform cantilever beam with any number of concentrated masses using an analytical and numerical combined method in which first the natural frequency equation is developed analytically and then is solved numerically to obtain the frequency and mode shapes. In this paper an alternate approximate method of determining the fundamental natural frequency of free vibration of a uniform cantilever beam with any number of mounted masses and restraining springs at any arbitrary locations along the length of the beam is presented. Low [5, 6] has studied the natural frequencies of flat rectangular plates with a mounted mass at any location by using the equivalent center weight method analytically as well as from experimental test data. In the present technique, instead of determining the equivalent weight factor with reference to the center of the beam, the reference point is chosen to be the tip of the cantilever. By following the same logic as in reference [5], one finds a mass or a spring located at the tip of the beam that produces the same effect on the natural frequency as another mass or spring located away from the tip. This is called equivalent tip mass or equivalent tip spring. By adding up all equivalent tip masses and tip springs, one can find a total equivalent tip mass or total equivalent tip spring whose effect on the resonance frequency should be the same as that of all the individual masses 229 0022–460X/96/320229 + 12 $18.00/0

7 1996 Academic Press Limited

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. . 

and springs located at various places. An equivalent tip factor design chart for both masses and springs has been established. Numerical results are obtained by using a finite element technique. To validate the design charts data obtained by the present method are compared with those available in the existing literature, wherever possible.

2. METHOD OF ANALYSIS

2.1.         The sole aim of this study is to establish design charts for quick and easy determination of the fundamental frequency of the title problem. As already mentioned, the equivalent center weight method suggested in reference [5] forms the basis of the logic for generating design charts for the equivalent tip factor for both the mounted mass and the restraining spring in the present investigation. The procedure for establishing equivalent tip factor design charts is as follows (a list of nomenclature is given in appendix D): (1) the fundamental frequency of the unloaded and unrestrained beam, f01 is obtained; (2) the mass, mt , ranging from 0·05mb to 2mb in steps of 0·05mb is placed at the tip and the fundamental frequency of the loaded beam ( f1 ) is calculated; (3) a graph of ( f1 /f01 ) vs (mt /mb ) is then plotted; (4) the effect of the location of the mass on the beam’s fundamental frequency is then studied by placing a dummy mass of 0·25mb at 19 different locations along the length of the beam, and the fundamental frequency of the beam with the dummy mass at each location, f1 , is obtained; (5) the frequency ratio ( f1 /f01 ) at each location is computed and, by using the graph generated in step (3), (mt /mb ) is calculated corresponding to this ( f1 /f01 ); (6) the equivalent tip mass factor for any location is then calculated by using the relationship Em = (mt /mb ) × (mb /dummy mass at that location), and likewise for all other locations; (7) the effect of the dummy mass on Em is studied for different dummy masses and steps (4) to (6) are then repeated. In a similar fashion equivalent tip spring factors for the same 19 locations are obtained.

Figure 1. Beam-mass-spring sytem: (a) the whole system; (b) separated into mass only and spring only systems; (c) system with total equivalent tip mass and total equivalent tip spring.

-   

231

2.2.         --  The whole system can be considered as consisting of two sub systems, namely a mass only system and a spring only system, as shown in Figure 1, with the coupling that exists between subsystems neglected. After having established the equivalent tip factor for both the mass and spring, Em and Ek respectively, associated with each dimensionless location

T 1 Specifications of the beam-system Length (L) Moment of inertia (I) Modulus of elasticity (E) Beam mass (mb ) Equivalent stiffness of beam at the tip, kb ,(3EI/L 3)

0·6m 6·4 × 10−10 m4 2·07 × 1011 N/m2 0·0576kg 1840N/m

T 2 Study of convergence of finite element results No. of elements

f1 (rad/s)

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

356·1491 361·0350 361·9561 362·2800 362·4304 362·5121 362·5617 362·5944 362·6164 362·6315 362·6434 362·6528 362·6533 362·6536 362·6538 362·6538

T 3 Comparison of Rayleigh’s solution with those of other methods Condition Fixed-free beam with k=0 m=0

f1 (Rayleigh-equation (3)) (rad/s)

f1 (FEM) (rad/s)

368·1330

362·6538

368·1330 Fixed-free beam with k = 0 m = 0·0288 at tip Fixed-free-beam with k = 1840, m = 0 at tip

208·3738

362·6538 208·0057

520·6187

505·3902

f1 (exact) (rad/s)

Difference (%) 1·50

362·8155 362·8155

1·46 0·05 0·18 3.00

. . 

232

parameter h1 , h2 , h3 , etc. (see figure 1(b)) one can arrive at a total equivalent tip mass (me ) and total equivalent tip spring stiffness (ke ) (see figure 1(c)) as follows: n

me = s mi × (Em )i , i=1

n

ke = s ki × (Ek )i ,

i = 1, 2, 3, . . . , n, locations. (1, 2)

i=1

Figure 2. Variation of first natural frequency with (a) tip mass and (b) tip spring.

T 4 Design chart for equivalent tip-mass factor h 0·10 0·15 0.20 0·25 0·30 0·35 0·40 0·45 0·50

Mounted dummy mass

f1 (FEM)

f1 /f01

mt /mb (from Figure 2(a))

0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144

362·4997 362·5504 362·6016 361·9134 362·1604 362·4069 360·4287 361·1721 361·9129 357·5184 359·2274 360·9397 352·6821 355·9708 359·2953 345·5784 351·1114 356·8041 336·1347 344·4965 353·3301 324·5837 336·1473 348·7963 311·3966 326·2565 343·1974

0·9991 0·9992 0·9995 0·9975 0·9981 0·9989 0·9934 0·9956 0·9975 0·9853 0·9902 0·9950 0·9720 0·9811 0·9903 0·9524 0·9677 0·9835 0·9264 0·9495 0·9739 0·8946 0·9265 0·9614 0·8582 0·8993 0·9461

0·0002 0·0001 0·0000 0·0011 0·0008 0·0004 0·0035 0·0023 0·0011 0·0081 0·0054 0·0027 0·0157 0·0105 0·0053 0·0269 0·0181 0·0091 0·0417 0·0285 0·0146 0·0673 0·0417 0·0218 0·0983 0·0593 0·0306

Em 0·0002 0·0002 0·0000 0·0014 0·0016 0·0016 0·0047 0·0046 0·0044 0·0108 0·0108 0·0108 0·0209 0·0210 0·0212 0·0359 0·0362 0·0364 0·0556 0·0570 0·0584 0·0836 0·0834 0·0872 0·1191 0·1186 0·1224 continued

-   

233

T 4—continued h 0·55 0·60 0·65 0·70 0·75 0·80 0·85 0·90 0·95 1·00

Mounted dummy mass

f1 (FEM)

f1 /f01

mt /mb (from Figure 2(a))

Em

0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144 0·0432 0·0288 0·0144

297·1494 315·1408 336·5961 282·3991 303·1688 329·1130 267·6047 290·7017 320·9019 253·1043 278·0498 312·1326 239·1223 265·4596 302·9659 225·7945 253·1113 293·5489 213·5485 241·1292 284·0053 201·3366 229·5939 274·4352 190·2297 218·5520 264·9170 179·8341 208·0057 255·4664

0·8190 0·8686 0·9278 0·7783 0·8356 0·9072 0·7375 0·8013 0·8845 0·6976 0·7664 0·8605 0·6590 0·7316 0·8351 0·6223 0·6976 0·8091 0·5885 0·6646 0·7828 0·5550 0·6328 0·7565 0·5243 0·6024 0·7303 0·4956 0·5734 0·7042

0·1223 0·0818 0·0409 0·1611 0·1072 0·0592 0·2067 0·1384 0·0738 0·2598 0·1742 0·0877 0·3204 0·2144 0·1078 0·3884 0·2598 0·1313 0·4612 0·3107 0·1562 0·5498 0·3678 0·1850 0·6448 0·4310 0·2163 0·7500 0·5000 0·2500

0·1631 0·1636 0·1636 0·2148 0·2144 0·2368 0·2756 0·2786 0·2952 0·3464 0·3484 0·3508 0·4272 0·4288 0·4312 0·5179 0·5196 0·5252 0·6149 0·6214 0·6248 0·7329 0·7356 0·7400 0·8597 0·8620 0·8652 1·0000 1·0000 1·0000

Em and Ek have been established for 19 locations. For any other intermediate locations Em and Ek can be obtained by interpolation. By employing Rayleigh’s energy formulation it can be established that the first natural frequency for a cantilever having a restraining spring and a mounted mass, both at the tip, is given by f1 = z(3EI + ke L 3 )/L 3(0·2357142mb + me ).

(3)

To arrive at equation (3) the static deflection curve is used as the beam shape function, as it produces a better result than a polynomial or trigonometric shape function for this case. A brief derivation of this expression is given in Appendix (A). By putting the values of ke and me in equation (3), the fundamental frequency of the whole system can be obtained. 3. RESULTS

3.1.        A beam of cross-section 0·015 m × 0·008 m is considered for analysis, the details of which are listed in Table 1. A study of the convergence properties of the finite element method was made by calculating the first natural frequency of the unloaded and

. . 

234

unrestrained beam for different numbers of elements, and the results are shown in Table 2. As the first natural frequency is the one of concern in this study, only the convergence of this frequency is considered. It is clear from Table 2, that with 75 beam elements or more convergence is achieved. However a further analysis was carried out with 80 beam elements. Table 3 presents a comparison of the first natural frequencies of the cantilever beam for different conditions, as obtained by the finite element method, Rayleigh’s method (equation (3)) and the exact method. It is shown in Table 3 that f01 obtained by the FEM with 80 elements is very close to the exact value. As the final output of the present method, the first natural frequency f1 , was obtained by using equation (3) based on Rayleigh’s energy formulation; the accuracy of the prediction to a great extent depends on how accurate equation (3) predicts the value of f1 . From Table 3 it is seen that equation (3) predicts a slightly higher value of f1 than does the exact method or the FEM for a beam without a point mass and restraining spring, or with a point mass or spring at the tip. 3.2.      The variations of the fundamental frequency, with a varying tip mass and tip spring, are presented in Figures 2(a) and 2(b), respectively. T 5 Design chart for equivalent tip-spring factor h

Dummy spring

f1 (FEM)

f1 /f01

kt /kb (Figure 2(b))

Ek

0·1

920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840

362·6773 362·7010 362·7022 362·7720 362·8314 362·8914 363·0106 363·1890 363·3669 363·4816 363·8933 364·3038 364·2802 365·0864 365·8867 365·5035 366·9086 368·3012 367·2433 369·4953 371·7192 369·5826 372·9644 376·2933 372·5854 377·4146 382·1455

0·9996 0·9996 0·9997 0·9999 1·0000 1·0003 1·0005 1·0011 1·0017 1·0018 1·0030 1·0042 1·0040 1·0063 1·0086 1·0074 1·0114 1·0153 1·0122 1·0184 1·0246 1·0187 1·0280 1·0372 1·0269 1·0402 1·0534

0·0000 0·0000 0·0002 0·0004 0·0008 0·0012 0·0019 0·0029 0·0040 0·0047 0·0070 0·0095 0·0093 0·0142 0·0188 0·0166 0·0248 0·0329 0·0267 0·0399 0·0529 0·0405 0·0601 0·0796 0·0579 0·0862 0·1148

0·0000 0·0000 0·0002 0·0008 0·0010 0·0012 0·0038 0·0038 0·0040 0·0094 0·0093 0·0095 0·0186 0·0189 0·0188 0·0332 0·0331 0·0329 0·0534 0·0532 0·0529 0·0810 0·0801 0·0796 0·1158 0·1149 0·1148

0·15 0·20 0·25 0·30 0·35 0·40 0·45 0·50

continued

-   

235

T 5—continued h

Dummy spring

f1 (FEM)

f1 /f01

kt /kb (Figure 2(b))

Ek

0·55

920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840 920 1380 1840

376·3254 382·9187 389·3623 380·8175 389·5185 397·9885 386·0841 397·2238 408·0216 392·1156 406·0026 419·4029 398·8793 415·7863 432·0166 406·3172 426·4592 445·6822 414·3484 437·8695 460·1611 422·8716 449·8247 475·1593 431·7692 462·1084 490·3469 440·9145 474·5001 505·3902

1·0373 1·0554 1·0733 1·0496 1·0736 1·0970 1·0641 1·0949 1·1248 1·0810 1·1191 1·1560 1·0994 1·1461 1·1910 1·1199 1·1754 1·2284 1·1420 1·2069 1·2683 1·1655 1·2399 1·3097 1·1901 1·2738 1·3517 1·2154 1·3080 1·3930

0·0799 0·1191 0·1584 0·1063 0·1592 0·2120 0·1381 0·2071 0·2765 0·1754 0·2632 0·3513 0·2177 0·3273 0·4377 0·2653 0·3993 0·5343 0·3178 0·4787 0·6408 0·3749 0·5643 0·7551 0·4358 0·6555 0·8753 0·5000 0·7500 1·0000

0·1598 0·1588 0·1584 0·2126 0·2123 0·2120 0·2762 0·2762 0·2765 0·3708 0·3509 0·3513 0·4354 0·4361 0·4377 0·5306 0·5324 0·5343 0·6356 0·6382 0·6408 0·7498 0·7524 0·7551 0·8716 0·8740 0·8753 1·0000 1·0000 1·0000

0·60 0·65 0·70 0·75 0·80 0·85 0·90 0·95 1·00

Figure 3. Three different configurations for validation of design charts.

. . 

236

The design chart for the equivalent tip mass factor (Em ) and the equivalent tip spring factor (Ek ) are presented in Tables 4 and 5, respectively. To establish Em a dummy mass of 0·0144 kg is placed at different locations along the length of the beam. The corresponding frequency calculated by the FEM is shown in the third column. The frequency ratio ( f1 /f01 ) for each position is shown in the fourth column. The (mt /mb ) ratio obtained from Figure 2(a), corresponding to ( f1 /f01 ), and is shown in the fifth column. Finally Em is calculated from Em = (mt /mb ) × (mb /dummy mass); Em is unity at the tip location, as expected. It is also expected that Em will be invariant to the dummy mass magnitude. To prove this two other dummy masses of different magnitudes (0·0288 kg and 0·0432 kg) were considered and Em was calculated for each case. In Table 4, for each position, the first, second and third rows correspond to dummy masses of 0·0432 kg, 0·0288 kg and 0·0144 kg, respectively. It is seen from Table 4 that the variation in Em for each case is very slight. In an exactly similar manner, the design chart for Ek was prepared, and it is presented in Table 5. 3.3.      Verification of the Em and Ek design charts was carried out by computing the first natural frequency for three different beam-mass-spring systems as shown in Figure 3. A sample calculation is given in appendix B. Though any one of the Em and Ek values is suitable to T 6 Verification of the design charts Type

Location

Mass

Config. a

0·8L

0·0576

Config. b

0·4L 0·6L L

Freq. (FEM)

Diff. (%)

212·1299

212·5664

0·21

220·3981 253·6467

220·8508 257·3467

0·21 1·43

275·0080

1·38

0·576 0·0864 0·0576 0·0288 2500 3000 2500 2000 2500 2500

0·55L 0·9L 0·15L 0·2L 0·45L 0·6L L

Freq.*

2500

0·2L

Config. c

Spring-Stiff.

0·0576 0·0288 0·0150 0·0144 0·4000 278·8523

0·2L 0·4L 0·45L 0·75L 0·9L

02500 02200 02300 02000 01800

*Frequency calculated by using design charts. Italicized values correspond to the case of non-identical mass and spring for config.b.

-   

237

T 7 Comparison of f1 obtained by using different methods (see Figure C1 of Appendix C) Station nos. at which point masses located 10 10 9 10 9 8 10 9 8 7 10 9 8 7 6 10 9 8 7 6 5 10 9 8 7 6 5 4 10 9 8 7 6 5 4 3 10 9 8 7 6 5 4 3 2 10 9 8 7 6 5 4 3 2 1

Method

f1 (rad/s)

A* C* P* A C P A C P A C P A C P A C P A C P A C P A C P A C P

5·0110 5·0110 5·0131 5·3087 5·3027 5·2996 5·6186 5·6120 5·5986 5·9360 5·9407 5·9081 6·2547 6·2472 6·2063 6·5671 6·5629 6·5154 6·8646 6·8528 6·8131 7·1399 7·1264 7·0914 7·3886 7·3739 7·3451 7·6098 7·5938 7·5674

*A & C, analytical-and-numerical combined method and transfer matrix method, respectively, available in reference 4, Table 6; P, present method using design chart.

produce accurate results, as they are almost the same, however, for the purpose of the sample calculation, the greatest Em among the three is chosen as it gives a large me which in turn produces a lower natural frequency. This is desirable from the design point of view. By following the same logic, the lowest of the Ek values is chosen for calculation as it also gives a lower natural frequency. A comparison of the frequency values thus obtained with the finite element values is presented in Table 6. For generality, different mass and spring values are used at different locations. A particular case, with an identical mass and spring, is also presented to show the differences between the results for the two cases (case b). It is found that the frequency predicted by the present method is quite close to that obtained by the FEM. This comparison further reveals that (i) though f1 given by equation (3) for the beam with a mass or spring at the tip is higher than the exact or FEM values, it may be lower than that given by the FEM with a mass and spring at locations other than the tip, and (ii) the deviations in the f1 values obtained from the design charts for identical masses and

238

. . 

springs and from the FEM are less than those for different values of the mass and spring (case b, Table 6). The title problem of reference [4] can be considered as a special case of the present study (a mass-only beam system). Table 7 shows a comparison of the first natural frequency value obtained using the design chart in Table 4 of the present study with that obtained by the TMM (transfer matrix method) and the ANC (analytical numerical combined) method available in reference [4]. The comparison clearly shows that the latter two results agree very well with the results obtained by the present method, thus establishing the reliability of the design chart. 4. CONCLUDING REMARKS

An approximate method for calculating the fundamental natural frequency of a cantilever beam with any number of mounted masses and restraining springs at any locations has been presented in this paper. Equivalent tip mass factor (Em ) and equivalent tip spring factor (Ek ) design charts have been established. Equivalent factors have been determined at nineteen different locations along the length of the beam. For any other intermediate location Em and Ek can be obtained by interpolation. The primary aim in this paper has been to present design charts for the equivalent factors, which can be of use in estimating the fundamental frequency quickly and easily, with little calculation but with reasonable accuracy, in order to provide a preliminary idea of the dynamic characteristics of a cantilever with a combination of masses and springs. ACKNOWLEDGMENTS

The author wishes to express his sincere thanks to Mr. K. Jayathirtha Rao, Director, ENTEST Laboratories for his co-operation and suggestions. Useful and important comments made by the referees are gratefully acknowledged. REFERENCES 1. A. R 1978 Journal of Applied Mechanics 45, 422–423. Vibration frequencies for a uniform cantilever with a rotational constraint at a point. 2. J. H. L 1984 Journal of Applied Mechanics 51, 182–187. Vibration frequencies and mode shapes for a constrained cantilever. 3. M. G¨¨ 1986 Journal of Sound and Vibration 105, 443–449. On the approximate determination of a restrained cantilever beam carrying a tip body. 4. J. S. W and T. L. L 1990 Journal of Sound and Vibration 136, 201–213. Free vibration analysis of a cantilever beam with point masses by an analytical and numerical combined method. 5. K. H. L 1993 Journal of Sound and Vibration 160, 111–121. Analytical and experimental investigation on a vibrating rectangular plate with mounted weights. 6. K. H. L 1993 Journal of Sound and Vibration 168(1), 123–139. An equivalent center-weight-factor method for predicting fundamental frequencies of plates carrying multiple masses from experimental test data. 7. M. L. J, G. M. S, J. C. W and P. W. W 1989 Vibration of Mechanical and Structural Systems. Singapore: Harper & Row.

APPENDIX A: NATURAL FREQUENCY FOR A CANTILEVERED BEAM WITH MASS AND SPRING BOTH AT THE TIP

A brief derivation of equation (3) by using Rayleigh’s energy method, is as follows. The static deflection curve is taken as the beam shape function because it is generally supposed to produce better results than trigonometric or polynomial shape functions.

-   

239

The static deflection curve for a cantilever of length L under a load P at the tip is given by [6] y = (P/6EI)(3Lx 2 − x 3 ),

x E L.

(A1)

Equation (A1) can be expressed in terms of a static deflection of the tip (D) given by D = (PL 3/3EI).

(A2)

By using equation (A2), equation (A1) can be written as y = (D/2L 3 )(3Lx 2 − x 3 ).

(A3)

The maximum kinetic energy (Tmax ) of the system includes that of the beam and the point mass (m) at the tip: i.e., Tmax = (g/2)

g

L

(yvn )2 dx + (m/2)(Dvn )2.

(A4)

0

Substituting equation (A3) in equation (A4) gives Tmax = 0·5(Dvn )2(0·2357142mb + m).

(A5)

The maximum potential energy of the system can be expressed as Umax = (EI/2)

g

L

(d2y/dx 2 )2 dx + 0·5kD 2,

(A6)

0

where k is the stiffness of the restraining spring at the tip. Evaluation of equation (A6) gives Umax = 0·5D 2L 3(3EI + kL 3 ). From Rayleigh’s principle, Tmax = Umax , one obtains f1 = z(3EI + kL 3 )/L 3(0·2357142mb + m): By expressing k and m in terms of ke and me , this can be rewritten as f1 = z(3EI + ke L 3 )/L 3(0·2357142mb + me ). APPENDIX B: SAMPLE CALCULATION FOR FIGURE 3, CONFIGURATION (b), (TABLE 6)

Step 1: The total equivalent tip mass me = Em × mounted mass = 0·2368 × 0·0576 + 1 × 0·0576 = 0·0712396 kg. The total equivalent tip spring stiffness ke = 0·038 × 2500 + 0·1584 × 2500 + 0·7498 × 2500 = 2280 N/m. Step 2: Using equation (3) gives f1 = 220·3981 rad/s.

Figure C1. Beam-mass system of reference [4].

240

. .  APPENDIX C: BEAM-MASS SYSTEM OF REFERENCE [4]

Figure C1 showing the beam-mass system of reference [4] is reproduced here for the sake of comparison of results (see Table 7). Numerical values: g = 0·8891 lbm/in, d = 2 in, E = 30 × 106 psi, l = 40 in, mc = 35·564 lbm. APPENDIX D: LIST OF NOMENCLAUTRE f1 f01 f1 /f01 E I m k mt kt ke me Em Ek mb kb h g

1st fundamental frequency of the beam 1st fundamental frequency of the beam without mass and spring frequency ratio modulus of elasticity moment of inertia mounted mass restraining spring stiffness mass at tip spring stiffness at the tip total equivalent spring stiffness at the tip total equivalent mass at the tip equivalent tip mass factor equivalent tip spring factor mass of the beam equivalent stiffness of the beam at the tip (3EI/L 3 ) dimensionless position parameter mass per unit length of the beam