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Tension effects on the natural frequencies of centre-loaded clamped beams
Journal of Sound and Vibration (1995) 181(4), 727–736
LETTER TO THE EDITOR TENSION EFFECTS ON THE NATURAL FREQUENCIES OF CENTRE-LOADED CLAMPED BEAMS G. B. C, K. H. L T. M. L School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 2263 (Received 13 June 1994, and in final form 13 September 1994)
1. In a recent work [1], both experimental and theoretical results were presented for clamped beams carrying a concentrated mass at mid-span. Tests were performed on several beams varying in material and geometrical properties. The theory presented in reference [1] was that for the linear elastic vibration of the loaded beams. It was found that for a beam of a specific length and width, experimental data for the thicker beams correlated very well with theory but not so well for the thinner beams. It was anticipated that the effect of fixed end constraints might be the main cause for these discrepancies. In what follows here, the fixed ends of the beam are completely restrained from any end movement and hence a tensile force is set up at the ends as a result. Due to this tensile force the natural frequencies of the beam are higher than those if the tensile force were absent, and the tensile force will be larger for a thinner beam carrying a concentrated mass than a corresponding thick beam, owing to the larger deflection in the former case. In the present analysis, correlations between theory and experimental data, with inclusion of the effect of the tensile reaction force, for all the cases are presented. 2. An illustration of a clamped beam carrying a concentrated mass is shown in Figure 1. The tensile reaction forces at the clamped supports due to both the concentrated mass and the mass of the beam can be obtained approximately, by assuming a cosine curve for the total deflection [2], as ymax +
0 1
0
1
A L3 wL 3 ymax = 4 W+ , 16EI 2p EI 2
(1)
where A is the cross-sectional area, I is the second moment of inertia, E is the Young’s modulus, ymax is the maximum deflection of the beam, W is the concentrated weight and w is the weight of the beam per unit length. Equation (1) can be rewritten as 3 ymax +
0
1
4t 2 8L 3g m M+ , y = 2 3 max p 4Ebt
(2)
where M is the concentrated mass and m is the beam’s mass. The solution of 3 the cubic equation (2), ymax + C1 ymax − C2 = 0, is ymax = Q 1/3 − C1 /(3Q 1/3 ), where 3 2 Q = C2 /2 + z12C1 + 81C2 /18. This value of ymax is then substituted into the following equation to calculate the tensile force P [2]: 2 P = (Ebt/4)(p/L)2 ymax .
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727 0022–460X/95/140727 + 10 $08.00/0
7 1995 Academic Press Limited
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Figure 1. A C–C beam carrying concentrated mass and a tension force.
The strain energy of bending for a simple beam with tension is [3]: Ub =
EI 2
g0 1 L
0
1 2y 1x 2
2
dx +
P 2
g0 1 L
0
1y 1x
2
dx.
(4)
The maximum kinetic energy for a beam carrying a concentrated mass is given by [4] Uke =
gv 2 2
g
L
y 2 dx +
0
Mv 2 y(a)2, 2
(5)
where g is the beam’s mass per unit length, and y(a) is the deflection of the beam at the position x = a. The total energy in the beam is thus the strain energy of bending less the kinetic energy: V = Ub − Uke . (6) The Ritz procedure [4] can then be used in conjunction with the principle of minimum potential energy for solving the free vibration problem. A series function that satisfies the geometric boundary condition is assumed in the form N
N
n=1
n=1
y = s An fn = s An sin
0 1 0 1
px npx sin . L L
(7)
The deflection function of equation (7) is substituted into equation (6) to give N
Q
6 g
L
V = 12 s s An Aq EI n=1 q=1
0
(f0n f0q ) dx + P
g
L
(f'n f'q ) dx
0
− v 2g
g
L
0
7
(fn fq ) dx − v 2Mfn (a)fq (a) ,
(8)
where a prime denotes differentiation with respect to x. According to the principle of minimum potential energy, the total potential energy of equation (8) is then minimized with respect to the unknown coefficients: i.e., 1V/1An = 0. As a result two matrices of size N × Q are formed, [K]{Aq } − v 2[S]{Aq } = 0, (9) where [K] is the stiffness matrix and [S] is the mass matrix. The non-trivial solution of equation (9) will give the eigensolutions of the beam. For a one-term solution, Rayleigh’s quotient is employed and the resulting expression for the natural frequencies is: vn = (ln /L)zEI/m, (10) where ln = z[Dn /Tn ], Dn = p 4(1 + 6n 2 + n 4) + (p 2PL/EI)(1 + n 2), Tn = fn L + (4M/m)fn2 (a), fn = 3/2 when n = 1 and fn = 1 when n q 1.
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Equation (10) shows that the term associated with the tensile force is in the numerator of the equation and thus the existence of a tensile force will increase the natural frequencies of the beam. The free vibration of a clamped beam carrying a concentrated mass with ends restrained from movement is analyzed by solving equation (9), and numerical results will next be presented to assess its validity. 3. The geometries and properties used in all numerical computations were extracted from the tested beams in reference [1] and they are shown in Table 1. The slenderness ratio shown in Table 1 is defined as a = z(AL 2/I).
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T 1 Properties and dimensions of the beam specimens Length (m) Width (mm) Thickness (mm) E (GPa) Density (Mg/m3) Slenderness ratio, a
Beam 1
Beam 2
Beam 3
Beam 4
1·0 12·7 3·175 71 2·71 1091
1·0 12·7 3·175 207 7·81 1091
1·0 12·7 4·7625 71 2·71 727
1·0 12·7 4·7625 207 7·81 727
T 2 Natural frequencies (Hz) for the unloaded beams (P = 0) Beam 1 Beam 2 Beam 3 Beam 4 ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV f 01† f 02 f 03 f 01 f 02 f 03 f 01 f 02 f 03 f 01 f 02 f 03 One-term‡ 10-term 20-term Experimental
17·62 16·71 16·70 16·00
48·02 46·09 46·06 45·00
86·85 90·49 90·30 82·75
17·12 16·81 16·80 16·25
47·46 46·36 46·32 44·00
86·44 91·02 90·83 83·75
25·53 25·07 25·06 23·75
70·78 69·14 69·08 65·50
128·9 135·7 135·5 126·8
25·68 25·21 25·20 24·00
71·19 69·54 69·48 66·50
129·7 136·5 136·2 130·5
† f 01 is the mode 1 frequency, f 02 is the mode 2 frequency and f 03 is the mode 3 frequency. ‡ Number of terms of solution used in the present theory; for the one-term solution Rayleigh’s quotient was used.
T 3 Natural frequencies (Hz) for the unloaded beams (with tension P) Beam 1 Beam 2 Beam 3 Beam 4 ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV f 01† f 02 f 03 f 01 f 02 f 03 f 01 f 02 f 03 f 01 f 02 f 03 One-term‡ 10-term 20-term Experimental
17·63 17·37 17·36 16·00
48·02 47·00 46·96 45·00
86·85 91·50 91·30 82·75
17·13 17·46 17·45 16·25
47·89 47·25 47·21 44·00
86·47 92·01 91·81 83·75
25·57 25·17 25·16 23·75
70·83 69·28 69·22 65·50
129·0 135·9 135·6 126·8
25·68 25·31 25·30 24·00
71·20 69·68 69·62 66·50
129·7 136·7 136·4 130·5
† f 01 is the mode 1 frequency, f02 is the mode 2 frequency and f03 is the mode 3 frequency. ‡ Number of terms of solution used in the present theory; for the one-term solution Rayleigh’s quotient was used.
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Figure 2. The effect of slenderness ratios and number of terms used in the solution (mode 1). Beam 1: – – – –, one-term solution; — —, four term solution; e, 10-term solution. Beam 3: ----, one-term solution; t, 10-term solution.
The results for the unloaded beam without and with the tensile force effect (due to the beam’s weight alone) are shown in Tables 2 and 3, respectively. It can be seen that when using a one-term solution the results are almost identical in both cases. However when using a 10-term or 20-term solution the effect of the tensile force can be clearly seen, in particular for beams 1 and 2 (thinner beams). The effect of the tensile force is not so obvious for beams 3 and 4 (thicker beams). Since beams 1 and 2 have a slenderness ratio of 1091 and beams 3 and 4 have a slenderness ratio of 727 only the theoretical results for beams 1 and 3 will be presented in graphical form, to draw some conclusions as to the effect of the tensile forces on the natural frequencies of the loaded beam with increasing mass ratio (concentrated mass divided by the beam’s mass). The effect of the number of terms used in the deflection function on the frequencies is shown in Figures 2–5 for modes 1–4, respectively. In these figures, f 1–f 4 are the first four natural frequencies of the loaded beam, f 01–f 04 are the corresponding frequencies of the
Figure 3. As Figure 2, but for mode 2. Key as Figure 2.
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Figure 4. As Figure 2, but for mode 3. Key as Figure 2.
unloaded beam, and Mc is the concentrated mass placed at the centre of the beam, i.e., at x = L/2. It can be seen that the difference between a one-term solution and a 10-term solution is very obvious in the mode 3 frequencies for both beams. There is also a noticeable difference between the results for beams 1 and 3 for all the four modes. It can also be concluded that a 10-term solution can be taken as the converged solution and hence comparisons from here on will be based on a 10-term solution. As obtained by using a 10-term solution, the effect of the tensile reaction forces on the beam vibration behaviour is shown in Figures 6–9 for modes 1–4, respectively. For all modes of vibration the curves for both beams are identical when there is no tensile force (P = 0) and the difference becomes more prominent when the tensile force is not zero. It is seen that the frequency ratio for beam 1 is always higher. Furthermore, the curves for P = 0 indicate that if the mass is placed at the nodes of vibration the frequencies for the loaded beam are identical to those of the unloaded beam (modes 2 and 4): i.e. f/f0 = 1. This does not hold when the tensile force is not zero. In fact, the results for such cases show that frequency ratio increase as the mass ratio is increased.
Figure 5. As Figure 2, but for mode 4. Key as Figure 2.
Figure 6. The effect of tension and slenderness ratio (mode 1); 10-term solution. P $ 0: – – –, beam 1; — —, beam 3; P = 0; ----, beam 1; r, beam 3.
Figure 7. As Figure 6, but for mode 2. Key as Figure 6.
Figure 8. As Figure 6, but for mode 3. Key as Figure 6.
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Figure 9. As Figure 6, but for mode 4. Key as Figure 6.
Figure 10. A comparison of theoretical results with experimental data (mode 1). 10-term solution; ----, Beams 1 and 2; — —, Beams 3 and 4. Experimental: e, Beam 1; r, Beam 2; ×, Beam 3; t, Beam 4.
Figure 11. As Figure 10, but for mode 2. Key as Figure 10.
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Figure 12. As Figure 10, but for mode 3. Key as Figure 10.
To illustrate the validity of the present analytical model, the theoretical results from the 10-term series can be compared with the experimental data for beams 1–4 [1], and these comparisons are shown in Figures 10–12 for modes 1–3, respectively. As can be seen in Table 4 or Figure 10, the analytical results for mode 1 of beams 3 and 4 are in excellent agreement with the experimental results. In general, the trend is well predicted by equation (9), particularly for beams 3 and 4, and mode 3 for all four beam specimens. T 4 Comparison of theoretical frequency ratio (f 1/f 01) with experimental data Beam 1 Beam 3 ZXXXXXXXXXCXXXXXXXXXV ZXXXXXXXXXCXXXXXXXXXV Theory Theory ZXCXV ZXCXV Mc (g) Mc/m Experimental P = 0 P $ 0 Mc (g) Mc/m Experimental P = 0 P $ 0 0·00 30·41 50·68 70·27 90·51 123·77 143·88 163·47 183·71 203·98 224·29 244·32 263·91 284·02 304·44 324·81 344·40 375·07 395·10 425·23 475·62 525·90
0·00 0·28 0·46 0·64 0·83 1·13 1·32 1·50 1·68 1·87 2·05 2·24 2·42 2·60 2·79 2·97 3·15 3·43 3·62 3·89 4·35 4·81
1·00 0·80 0·73 0·67 0·67 0·63 0·63 0·61 0·59 0·59 0·58 0·58 0·58 0·58 0·56 0·56 0·56 0·55 0·55 0·55 0·53 0·53
1·00 0·76 0·68 0·61 0·56 0·50 0·48 0·45 0·43 0·41 0·40 0·38 0·37 0·36 0·35 0·34 0·33 0·32 0·31 0·30 0·28 0·27
1·00 0·79 0·72 0·67 0·63 0·59 0·57 0·55 0·53 0·52 0·51 0·50 0·49 0·48 0·47 0·47 0·46 0·45 0·45 0·44 0·43 0·42
0·00 30·18 50·34 70·58 90·87 113·84 134·13 164·07 194·30 224·29 244·53 274·58 304·71 331·72 350·06 382·07 400·41 425·36 745·52 525·90
0·00 0·18 0·31 0·43 0·55 0·69 0·82 1·00 1·19 1·37 1·49 1·68 1·86 2·02 2·14 2·33 2·44 2·60 2·90 3·21
1·00 0·82 0·76 0·69 0·65 0·61 0·59 0·55 0·51 0·48 0·47 0·45 0·44 0·43 0·42 0·41 0·40 0·39 0·38 0·37
1·00 0·82 0·75 0·69 0·64 0·60 0·57 0·53 0·50 0·47 0·45 0·43 0·41 0·40 0·39 0·38 0·37 0·36 0·34 0·33
1·00 0·83 0·75 0·70 0·65 0·61 0·58 0·54 0·51 0·49 0·48 0·46 0·44 0·43 0·42 0·41 0·40 0·39 0·38 0·37
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T 5 Frequency coefficients Vi = vi L zrA/EI of the centre-loaded clamped beam system (1/a = 0·00001 and a/L = 1/2) 2
Mc/m
V1
V2
V3
V4
V5
V6
V7
0·0† 0·0‡
22·3747 22·3733
61·6826 61·6728
120·946 120·903
199·962 199·859
298·818 298·556
417·436 416·991
556·072 —
0·2 0·2
18·2065 18·2054
61·6826 61·6728
107·130 107·095
199·962 199·859
270·284 269·968
417·436 416·991
511·103 510·509
0·5 0·5
14·8009 14·8000
61·6826 61·6728
100·034 99·9981
199·962 199·859
259·562 259·204
417·436 416·991
498·040 496·949
1·0 1·0
11·8233 11·8182
61·6826 61·6728
95·7988 95·7568
199·962 199·859
254·117 253·730
417·436 416·991
492·027 491·009
2·0 2·0
8·99706 8·99446
61·6826 61·6728
92·9934 92·9540
199·962 199·859
250·837 250·440
417·436 416·991
488·579 487·339
3·0 3·0
7·54661 7·54500
61·6826 61·6728
91·9224 91·8834
199·962 199·859
249·645 249·246
417·436 416·991
487·358 486·149
† Results in the first row were obtained by using the present method. ‡ Results in the second row were extracted from reference [8].
Several researchers have studied the vibration problems of beams and plates carrying elastically mounted concentrated masses [5–8]. Rossi et al. [8] provided an exact solution of the vibration problems of loaded beams for the well-known Timoshenko vibrating beam model. The centre-loaded clamped beam system considered in this work is equivalent to the system shown in their Figure 1(c) [8], where K = a (spring constant) and a = L/2 (centre-loaded). Results obtained by using the present method with a 20-term series can be compared with those of reference [8], the comparison of the system frequencies Vi = vi zrA/EI being summarized in Table 5. To further illustrate the effect of tension, frequency coefficients of beam 1 are shown in Table 6. It is again shown that the frequencies increase by including the tension. T 6 Frequency coefficients for beam 1 with and without tension (1/a = 0·00092 and a/L = 1/2) Mc/m
V1
V2
V3
V4
V5
V6
V7
0·0† 0·0‡
22·3747 22·6255
61·6826 62·0243
120·946 121·321
199·962 200·356
298·818 299·224
417·436 417·850
556·072 556·493
0·2 0·2
18·2065 18·5935
61·6826 62·3247
107·130 107·727
199·962 200·703
270·284 270·970
417·436 418·216
511·103 511·832
0·5 0·5
14·8009 15·3942
61·6826 62·8938
100·034 101·061
199·962 201·365
259·562 260·819
417·436 418·914
498·040 499·399
1·0 1·0
11·8190 12·7661
61·6826 64·0023
95·7936 97·7137
199·962 202·666
254·111 256·565
417·436 420·290
492·021 494·696
2·0 2·0
8·99516 10·3890
61·6826 66·4841
92·9918 96·7561
199·962 205·636
250·835 255·830
417·436 423·446
488·578 494·065
3·0 3·0
7·54558 9·26816
61·6826 68·8665
91·9216 97·5272
199·962 208·558
249·645 257·211
417·436 426·574
487·356 495·698
† Results in the first row were obtained with P = 0 (no tension). ‡ Results in the second row were obtained with P $ 0.
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4.
An approximate model, which includes the ends’ tension, has been presented for clamped beams carrying a concentrated mass at the centre. It has been shown that the effect of the tension force is appreciably greater for a thinner beam than for a thicker beam. This seems logical, since a thinner beam carrying a concentrated mass will deflect more than a thicker beam carrying the same concentrated mass. Thus, with larger deflection, the tensile force will also increase proportionately and this will in turn increase the natural frequencies of the beam. Comparisons between experimental data and theoretical results indicate good correlations and thus give the required confidence in the theory. The authors express their gratitude to the reviewer for his valuable comments and information provided. 1. K. H. L, T. M. L and G. B. C 1993 Computers and Structures 48, 1157–1162. Experimental and analytical investigations of vibration frequencies for centre-loaded beams. 2. R. J. R and W. C. Y 1976 Formulas for Stress and Strain. New York: McGraw-Hill. 3. L. H. Donnell 1976 Beams, Plates and Shells. New York: McGraw-Hill. 4. W. T. T 1988 Theory of Vibrations with Applications. Englewood Cliffs, New Jersey: Prentice-Hall; third edition. 5. L. E and P. A. A. L 1987 Journal of Sound and Vibration 114, 519–533. Analytical and experimental investigation on continuous beams carrying elastically mounted masses. 6. P. A. A. L, C. F and V. H. C´ 1987 Journal of Sound and Vibration 117, 459–465. Vibrations of beams and plates carrying concentrated masses. 7. M. J. M and P. M. B 1991 Journal of Sound and Vibration 150, 330–334. Natural frequencies of the beam–mass system: comparison of the two fundamental theories of beam vibrations. 8. R. E. R, P. A. A. L, D. R. A and H. L 1993 Journal of Sound and Vibration 165, 209–223. Free vibrations of Timoshenko beams carrying elastically mounted, concentrated masses.
LETTER TO THE EDITOR TENSION EFFECTS ON THE NATURAL FREQUENCIES OF CENTRE-LOADED CLAMPED BEAMS G. B. C, K. H. L T. M. L School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 2263 (Received 13 June 1994, and in final form 13 September 1994)
1. In a recent work [1], both experimental and theoretical results were presented for clamped beams carrying a concentrated mass at mid-span. Tests were performed on several beams varying in material and geometrical properties. The theory presented in reference [1] was that for the linear elastic vibration of the loaded beams. It was found that for a beam of a specific length and width, experimental data for the thicker beams correlated very well with theory but not so well for the thinner beams. It was anticipated that the effect of fixed end constraints might be the main cause for these discrepancies. In what follows here, the fixed ends of the beam are completely restrained from any end movement and hence a tensile force is set up at the ends as a result. Due to this tensile force the natural frequencies of the beam are higher than those if the tensile force were absent, and the tensile force will be larger for a thinner beam carrying a concentrated mass than a corresponding thick beam, owing to the larger deflection in the former case. In the present analysis, correlations between theory and experimental data, with inclusion of the effect of the tensile reaction force, for all the cases are presented. 2. An illustration of a clamped beam carrying a concentrated mass is shown in Figure 1. The tensile reaction forces at the clamped supports due to both the concentrated mass and the mass of the beam can be obtained approximately, by assuming a cosine curve for the total deflection [2], as ymax +
0 1
0
1
A L3 wL 3 ymax = 4 W+ , 16EI 2p EI 2
(1)
where A is the cross-sectional area, I is the second moment of inertia, E is the Young’s modulus, ymax is the maximum deflection of the beam, W is the concentrated weight and w is the weight of the beam per unit length. Equation (1) can be rewritten as 3 ymax +
0
1
4t 2 8L 3g m M+ , y = 2 3 max p 4Ebt
(2)
where M is the concentrated mass and m is the beam’s mass. The solution of 3 the cubic equation (2), ymax + C1 ymax − C2 = 0, is ymax = Q 1/3 − C1 /(3Q 1/3 ), where 3 2 Q = C2 /2 + z12C1 + 81C2 /18. This value of ymax is then substituted into the following equation to calculate the tensile force P [2]: 2 P = (Ebt/4)(p/L)2 ymax .
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727 0022–460X/95/140727 + 10 $08.00/0
7 1995 Academic Press Limited
728
Figure 1. A C–C beam carrying concentrated mass and a tension force.
The strain energy of bending for a simple beam with tension is [3]: Ub =
EI 2
g0 1 L
0
1 2y 1x 2
2
dx +
P 2
g0 1 L
0
1y 1x
2
dx.
(4)
The maximum kinetic energy for a beam carrying a concentrated mass is given by [4] Uke =
gv 2 2
g
L
y 2 dx +
0
Mv 2 y(a)2, 2
(5)
where g is the beam’s mass per unit length, and y(a) is the deflection of the beam at the position x = a. The total energy in the beam is thus the strain energy of bending less the kinetic energy: V = Ub − Uke . (6) The Ritz procedure [4] can then be used in conjunction with the principle of minimum potential energy for solving the free vibration problem. A series function that satisfies the geometric boundary condition is assumed in the form N
N
n=1
n=1
y = s An fn = s An sin
0 1 0 1
px npx sin . L L
(7)
The deflection function of equation (7) is substituted into equation (6) to give N
Q
6 g
L
V = 12 s s An Aq EI n=1 q=1
0
(f0n f0q ) dx + P
g
L
(f'n f'q ) dx
0
− v 2g
g
L
0
7
(fn fq ) dx − v 2Mfn (a)fq (a) ,
(8)
where a prime denotes differentiation with respect to x. According to the principle of minimum potential energy, the total potential energy of equation (8) is then minimized with respect to the unknown coefficients: i.e., 1V/1An = 0. As a result two matrices of size N × Q are formed, [K]{Aq } − v 2[S]{Aq } = 0, (9) where [K] is the stiffness matrix and [S] is the mass matrix. The non-trivial solution of equation (9) will give the eigensolutions of the beam. For a one-term solution, Rayleigh’s quotient is employed and the resulting expression for the natural frequencies is: vn = (ln /L)zEI/m, (10) where ln = z[Dn /Tn ], Dn = p 4(1 + 6n 2 + n 4) + (p 2PL/EI)(1 + n 2), Tn = fn L + (4M/m)fn2 (a), fn = 3/2 when n = 1 and fn = 1 when n q 1.
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Equation (10) shows that the term associated with the tensile force is in the numerator of the equation and thus the existence of a tensile force will increase the natural frequencies of the beam. The free vibration of a clamped beam carrying a concentrated mass with ends restrained from movement is analyzed by solving equation (9), and numerical results will next be presented to assess its validity. 3. The geometries and properties used in all numerical computations were extracted from the tested beams in reference [1] and they are shown in Table 1. The slenderness ratio shown in Table 1 is defined as a = z(AL 2/I).
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T 1 Properties and dimensions of the beam specimens Length (m) Width (mm) Thickness (mm) E (GPa) Density (Mg/m3) Slenderness ratio, a
Beam 1
Beam 2
Beam 3
Beam 4
1·0 12·7 3·175 71 2·71 1091
1·0 12·7 3·175 207 7·81 1091
1·0 12·7 4·7625 71 2·71 727
1·0 12·7 4·7625 207 7·81 727
T 2 Natural frequencies (Hz) for the unloaded beams (P = 0) Beam 1 Beam 2 Beam 3 Beam 4 ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV f 01† f 02 f 03 f 01 f 02 f 03 f 01 f 02 f 03 f 01 f 02 f 03 One-term‡ 10-term 20-term Experimental
17·62 16·71 16·70 16·00
48·02 46·09 46·06 45·00
86·85 90·49 90·30 82·75
17·12 16·81 16·80 16·25
47·46 46·36 46·32 44·00
86·44 91·02 90·83 83·75
25·53 25·07 25·06 23·75
70·78 69·14 69·08 65·50
128·9 135·7 135·5 126·8
25·68 25·21 25·20 24·00
71·19 69·54 69·48 66·50
129·7 136·5 136·2 130·5
† f 01 is the mode 1 frequency, f 02 is the mode 2 frequency and f 03 is the mode 3 frequency. ‡ Number of terms of solution used in the present theory; for the one-term solution Rayleigh’s quotient was used.
T 3 Natural frequencies (Hz) for the unloaded beams (with tension P) Beam 1 Beam 2 Beam 3 Beam 4 ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV ZXXXCXXXV f 01† f 02 f 03 f 01 f 02 f 03 f 01 f 02 f 03 f 01 f 02 f 03 One-term‡ 10-term 20-term Experimental
17·63 17·37 17·36 16·00
48·02 47·00 46·96 45·00
86·85 91·50 91·30 82·75
17·13 17·46 17·45 16·25
47·89 47·25 47·21 44·00
86·47 92·01 91·81 83·75
25·57 25·17 25·16 23·75
70·83 69·28 69·22 65·50
129·0 135·9 135·6 126·8
25·68 25·31 25·30 24·00
71·20 69·68 69·62 66·50
129·7 136·7 136·4 130·5
† f 01 is the mode 1 frequency, f02 is the mode 2 frequency and f03 is the mode 3 frequency. ‡ Number of terms of solution used in the present theory; for the one-term solution Rayleigh’s quotient was used.
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Figure 2. The effect of slenderness ratios and number of terms used in the solution (mode 1). Beam 1: – – – –, one-term solution; — —, four term solution; e, 10-term solution. Beam 3: ----, one-term solution; t, 10-term solution.
The results for the unloaded beam without and with the tensile force effect (due to the beam’s weight alone) are shown in Tables 2 and 3, respectively. It can be seen that when using a one-term solution the results are almost identical in both cases. However when using a 10-term or 20-term solution the effect of the tensile force can be clearly seen, in particular for beams 1 and 2 (thinner beams). The effect of the tensile force is not so obvious for beams 3 and 4 (thicker beams). Since beams 1 and 2 have a slenderness ratio of 1091 and beams 3 and 4 have a slenderness ratio of 727 only the theoretical results for beams 1 and 3 will be presented in graphical form, to draw some conclusions as to the effect of the tensile forces on the natural frequencies of the loaded beam with increasing mass ratio (concentrated mass divided by the beam’s mass). The effect of the number of terms used in the deflection function on the frequencies is shown in Figures 2–5 for modes 1–4, respectively. In these figures, f 1–f 4 are the first four natural frequencies of the loaded beam, f 01–f 04 are the corresponding frequencies of the
Figure 3. As Figure 2, but for mode 2. Key as Figure 2.
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Figure 4. As Figure 2, but for mode 3. Key as Figure 2.
unloaded beam, and Mc is the concentrated mass placed at the centre of the beam, i.e., at x = L/2. It can be seen that the difference between a one-term solution and a 10-term solution is very obvious in the mode 3 frequencies for both beams. There is also a noticeable difference between the results for beams 1 and 3 for all the four modes. It can also be concluded that a 10-term solution can be taken as the converged solution and hence comparisons from here on will be based on a 10-term solution. As obtained by using a 10-term solution, the effect of the tensile reaction forces on the beam vibration behaviour is shown in Figures 6–9 for modes 1–4, respectively. For all modes of vibration the curves for both beams are identical when there is no tensile force (P = 0) and the difference becomes more prominent when the tensile force is not zero. It is seen that the frequency ratio for beam 1 is always higher. Furthermore, the curves for P = 0 indicate that if the mass is placed at the nodes of vibration the frequencies for the loaded beam are identical to those of the unloaded beam (modes 2 and 4): i.e. f/f0 = 1. This does not hold when the tensile force is not zero. In fact, the results for such cases show that frequency ratio increase as the mass ratio is increased.
Figure 5. As Figure 2, but for mode 4. Key as Figure 2.
Figure 6. The effect of tension and slenderness ratio (mode 1); 10-term solution. P $ 0: – – –, beam 1; — —, beam 3; P = 0; ----, beam 1; r, beam 3.
Figure 7. As Figure 6, but for mode 2. Key as Figure 6.
Figure 8. As Figure 6, but for mode 3. Key as Figure 6.
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Figure 9. As Figure 6, but for mode 4. Key as Figure 6.
Figure 10. A comparison of theoretical results with experimental data (mode 1). 10-term solution; ----, Beams 1 and 2; — —, Beams 3 and 4. Experimental: e, Beam 1; r, Beam 2; ×, Beam 3; t, Beam 4.
Figure 11. As Figure 10, but for mode 2. Key as Figure 10.
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Figure 12. As Figure 10, but for mode 3. Key as Figure 10.
To illustrate the validity of the present analytical model, the theoretical results from the 10-term series can be compared with the experimental data for beams 1–4 [1], and these comparisons are shown in Figures 10–12 for modes 1–3, respectively. As can be seen in Table 4 or Figure 10, the analytical results for mode 1 of beams 3 and 4 are in excellent agreement with the experimental results. In general, the trend is well predicted by equation (9), particularly for beams 3 and 4, and mode 3 for all four beam specimens. T 4 Comparison of theoretical frequency ratio (f 1/f 01) with experimental data Beam 1 Beam 3 ZXXXXXXXXXCXXXXXXXXXV ZXXXXXXXXXCXXXXXXXXXV Theory Theory ZXCXV ZXCXV Mc (g) Mc/m Experimental P = 0 P $ 0 Mc (g) Mc/m Experimental P = 0 P $ 0 0·00 30·41 50·68 70·27 90·51 123·77 143·88 163·47 183·71 203·98 224·29 244·32 263·91 284·02 304·44 324·81 344·40 375·07 395·10 425·23 475·62 525·90
0·00 0·28 0·46 0·64 0·83 1·13 1·32 1·50 1·68 1·87 2·05 2·24 2·42 2·60 2·79 2·97 3·15 3·43 3·62 3·89 4·35 4·81
1·00 0·80 0·73 0·67 0·67 0·63 0·63 0·61 0·59 0·59 0·58 0·58 0·58 0·58 0·56 0·56 0·56 0·55 0·55 0·55 0·53 0·53
1·00 0·76 0·68 0·61 0·56 0·50 0·48 0·45 0·43 0·41 0·40 0·38 0·37 0·36 0·35 0·34 0·33 0·32 0·31 0·30 0·28 0·27
1·00 0·79 0·72 0·67 0·63 0·59 0·57 0·55 0·53 0·52 0·51 0·50 0·49 0·48 0·47 0·47 0·46 0·45 0·45 0·44 0·43 0·42
0·00 30·18 50·34 70·58 90·87 113·84 134·13 164·07 194·30 224·29 244·53 274·58 304·71 331·72 350·06 382·07 400·41 425·36 745·52 525·90
0·00 0·18 0·31 0·43 0·55 0·69 0·82 1·00 1·19 1·37 1·49 1·68 1·86 2·02 2·14 2·33 2·44 2·60 2·90 3·21
1·00 0·82 0·76 0·69 0·65 0·61 0·59 0·55 0·51 0·48 0·47 0·45 0·44 0·43 0·42 0·41 0·40 0·39 0·38 0·37
1·00 0·82 0·75 0·69 0·64 0·60 0·57 0·53 0·50 0·47 0·45 0·43 0·41 0·40 0·39 0·38 0·37 0·36 0·34 0·33
1·00 0·83 0·75 0·70 0·65 0·61 0·58 0·54 0·51 0·49 0·48 0·46 0·44 0·43 0·42 0·41 0·40 0·39 0·38 0·37
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T 5 Frequency coefficients Vi = vi L zrA/EI of the centre-loaded clamped beam system (1/a = 0·00001 and a/L = 1/2) 2
Mc/m
V1
V2
V3
V4
V5
V6
V7
0·0† 0·0‡
22·3747 22·3733
61·6826 61·6728
120·946 120·903
199·962 199·859
298·818 298·556
417·436 416·991
556·072 —
0·2 0·2
18·2065 18·2054
61·6826 61·6728
107·130 107·095
199·962 199·859
270·284 269·968
417·436 416·991
511·103 510·509
0·5 0·5
14·8009 14·8000
61·6826 61·6728
100·034 99·9981
199·962 199·859
259·562 259·204
417·436 416·991
498·040 496·949
1·0 1·0
11·8233 11·8182
61·6826 61·6728
95·7988 95·7568
199·962 199·859
254·117 253·730
417·436 416·991
492·027 491·009
2·0 2·0
8·99706 8·99446
61·6826 61·6728
92·9934 92·9540
199·962 199·859
250·837 250·440
417·436 416·991
488·579 487·339
3·0 3·0
7·54661 7·54500
61·6826 61·6728
91·9224 91·8834
199·962 199·859
249·645 249·246
417·436 416·991
487·358 486·149
† Results in the first row were obtained by using the present method. ‡ Results in the second row were extracted from reference [8].
Several researchers have studied the vibration problems of beams and plates carrying elastically mounted concentrated masses [5–8]. Rossi et al. [8] provided an exact solution of the vibration problems of loaded beams for the well-known Timoshenko vibrating beam model. The centre-loaded clamped beam system considered in this work is equivalent to the system shown in their Figure 1(c) [8], where K = a (spring constant) and a = L/2 (centre-loaded). Results obtained by using the present method with a 20-term series can be compared with those of reference [8], the comparison of the system frequencies Vi = vi zrA/EI being summarized in Table 5. To further illustrate the effect of tension, frequency coefficients of beam 1 are shown in Table 6. It is again shown that the frequencies increase by including the tension. T 6 Frequency coefficients for beam 1 with and without tension (1/a = 0·00092 and a/L = 1/2) Mc/m
V1
V2
V3
V4
V5
V6
V7
0·0† 0·0‡
22·3747 22·6255
61·6826 62·0243
120·946 121·321
199·962 200·356
298·818 299·224
417·436 417·850
556·072 556·493
0·2 0·2
18·2065 18·5935
61·6826 62·3247
107·130 107·727
199·962 200·703
270·284 270·970
417·436 418·216
511·103 511·832
0·5 0·5
14·8009 15·3942
61·6826 62·8938
100·034 101·061
199·962 201·365
259·562 260·819
417·436 418·914
498·040 499·399
1·0 1·0
11·8190 12·7661
61·6826 64·0023
95·7936 97·7137
199·962 202·666
254·111 256·565
417·436 420·290
492·021 494·696
2·0 2·0
8·99516 10·3890
61·6826 66·4841
92·9918 96·7561
199·962 205·636
250·835 255·830
417·436 423·446
488·578 494·065
3·0 3·0
7·54558 9·26816
61·6826 68·8665
91·9216 97·5272
199·962 208·558
249·645 257·211
417·436 426·574
487·356 495·698
† Results in the first row were obtained with P = 0 (no tension). ‡ Results in the second row were obtained with P $ 0.
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4.
An approximate model, which includes the ends’ tension, has been presented for clamped beams carrying a concentrated mass at the centre. It has been shown that the effect of the tension force is appreciably greater for a thinner beam than for a thicker beam. This seems logical, since a thinner beam carrying a concentrated mass will deflect more than a thicker beam carrying the same concentrated mass. Thus, with larger deflection, the tensile force will also increase proportionately and this will in turn increase the natural frequencies of the beam. Comparisons between experimental data and theoretical results indicate good correlations and thus give the required confidence in the theory. The authors express their gratitude to the reviewer for his valuable comments and information provided. 1. K. H. L, T. M. L and G. B. C 1993 Computers and Structures 48, 1157–1162. Experimental and analytical investigations of vibration frequencies for centre-loaded beams. 2. R. J. R and W. C. Y 1976 Formulas for Stress and Strain. New York: McGraw-Hill. 3. L. H. Donnell 1976 Beams, Plates and Shells. New York: McGraw-Hill. 4. W. T. T 1988 Theory of Vibrations with Applications. Englewood Cliffs, New Jersey: Prentice-Hall; third edition. 5. L. E and P. A. A. L 1987 Journal of Sound and Vibration 114, 519–533. Analytical and experimental investigation on continuous beams carrying elastically mounted masses. 6. P. A. A. L, C. F and V. H. C´ 1987 Journal of Sound and Vibration 117, 459–465. Vibrations of beams and plates carrying concentrated masses. 7. M. J. M and P. M. B 1991 Journal of Sound and Vibration 150, 330–334. Natural frequencies of the beam–mass system: comparison of the two fundamental theories of beam vibrations. 8. R. E. R, P. A. A. L, D. R. A and H. L 1993 Journal of Sound and Vibration 165, 209–223. Free vibrations of Timoshenko beams carrying elastically mounted, concentrated masses.