An exact frequency equation for an axially loaded beam-mass-spring system resting on a Winkler elastic foundation

Journal of Sound and Vibration (1995) 185(2), 357–363

LETTERS TO THE EDITOR AN EXACT FREQUENCY EQUATION FOR AN AXIALLY LOADED BEAM–MASS–SPRING SYSTEM RESTING ON A WINKLER ELASTIC FOUNDATION S. H. F  K. M. Z Department of Mechanical Design, Faculty of Engineering and Technology (Mataria) , P.O. Box 11718, Cairo, Egypt (Received 10 May 1994, and in final form 13 September 1994)

1.  Although much research has been done on beam–mass–spring systems in order to obtain their dynamic characteristics [1–13], no one, to the best of the authors’ knowledge, has completely treated the combination problem as presented in this work, in which parametric studies have been carried out for the effect of the 12 system design parameters: five for heavy tip body and an intermediate concentrated mass; five for end restraints and elastic foundation effects; and two for an axial load applied to the system. An exact frequency equation is derived based on the Laplace transform technique. The 12 system design parameters are considered in the numerical computation of all of the chosen cases for comparison purposes. Many of the previous studies can be regarded as special cases of the present study. Good agreement between the present results and those of previous investigators is observed. The shear deformation and rotary inertia effects of the beam are neglected. 2.        A uniform beam is considered of length l and cross-sectional area Ab resting on an elastic foundation of stiffness kf , and subjected to an axial compressive load F. The beam is carrying an intermediate concentrated mass m and a heavy tip body of mass me and is elastically supported at its ends, as shown in Figure 1. The offset, d, and rotary inertia effect J of the tip mass are considered in this work. The beam material has a density r and Young’s modulus of elasticity E. The equation of motion for a harmonically vibrated uniform beam with circular frequency v subjected to an axial force F with a mass placed at x = x1 and resting on a Winker type elastic foundation, when the transverse deformation due to shear and the rotary inertia are neglected, may be written in non-dimensional form as Y2(z) + P 2Y0(z) + {K* − l 4 − m¯l 4d(z − z1 )}Y(z) = 0,

(1)

where d(· · ·) is the Dirac delta function, K* is the elastic foundation modulus (=kf l 4/EI), m¯ is the intermediate mass parameter (=m/rAb l), P 2 is an axial load parameter, (=Fl 2/EI) and l is a frequency parameter, (=(rAb v 2l 4/EI)1/4). The boundary conditions [4, 6], non-dimensional form are Y0(0) − f0 Y(0) = 0,

Y1(0) − P 2Y'(0) + Z0 Y(0) = 0,

Y0(1) + aY'(1) + bY(1) = 0,

Y1(1) + gY'(1) + uY(1) = 0,

(2a, b) (2c, d)

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7 1995 Academic Press Limited

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Figure 1. The mathematical model for which the frequency equations is derived.

where a = f1 + Z1  d2 − l 4(m¯e  d2 +  J) − Pd2  d, g = b + P 2, f0 = F0 l/EI,

u = −Z1 + l 4m¯e ,

f1 = F1 l/EI,

m¯e = me /rAb l,

b = Z1  d − l 4m¯e  d,

Z0 = K0 l 3/EI,  J = J/rAb l 3,

Z1 = K1 l 3/EI,

 d = d/l.

Here Z0 and Z1 are the translational rigidity parameters of the left–hand support and right-hand support, respectively. f0 and f1 are the rotational rigidity parameters of the left–hand support and the right-hand support, respectively. 3.      Taking the Laplace transformation of equation (1), and using the boundary conditions (2a, b), and then taking the inverse Laplace transformation yields Y(g) = {1/(a 2 + b 2)}{[(a 2 cosh az + b 2 cos az) + P 2(cosh az − cosz bz) −Z0 ((1/a) sinh az − (1/b) sin bz)]A + [(a sinh az + b sin bz) + 2P 2((1/a) sinh az −(1/b) sin bz) + f0 (cosh az − cos bz)]B + l 4m¯Y(z1 )((1/a) sinh a(z − z1 ) − (1/b) sin b(z − z1 ))},

(3)

where a = {−(P 2/2) + [(P 2/2)2 + (l 4 − K*)]1/2}1/2,

b = {(P 2/2) + [(P 2/2)2 + (l 4 − K*)]1/2}1/2.

Substituting equation (3) and its derivatives with respect to z in equations (2c, d) results in two equations in three unknowns A, B and Y(z1 ). Another equation, called the consistency equation, is found by substituting z = z1 − e in equation (3) and then letting e:0, which results in Y(z1 ) = {[(a 2 cosh az1 + b 2 cos bz1 ) + P 2(cosh az1 − cos bz1 ) − Z0 ((1/a) sinh az1 −(1/b) sin bz1 )]A + (a sinh az1 + b sin bz1 ) + 2P 2((1/a) sinh az1 −(1/b) sin bz1 ) + f0 (cosh az1 − cos bz1 )]B}/(a 2 + b 2).

(3a)

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Substituting equation (3a) in the two equations resulting from the substitution in the boundary conditions yields two equations in the two unknown A and B: [(G1 e62 + L2 ) + P 2(G1 e61 + L3 ) − Z0 (G1 e51 + L1 )]A + [(G1 e52 + L4 ) + 2P 2(G1 e51 + L1 ) + f0 (G1 e61 + L3 )]B = 0,

(4a)

[(G2 e61 + L6 ) + P 2(G2 e61 + L7 ) − Z0 (G2 e51 + L5 )]A + [(G2 e52 + L8 ) + 2P 2(G2 e51 + L5 ) + f0 (G2 e61 + L7 )]B = 0.

(4b)

Equation (4a) and (4b) form a system of two homogeneous equations in two unknowns. For the problem to have non-trivial solution the two unknowns cannot all be zero. Hence the determinant of the coefficient matrix of the system of equations must vanish. This leads to the system frequency equation XV1 − X1 V = 0,

(5)

where X = e2 + P 2e3 − Z0 e1 ,

X1 = e6 + P 2e7 − Z0 e5 ,

V = e4 + 2P 2e1 + f0 e3 ,

V1 = e8 + 2P 2e5 + f0 e7 . Here e1 = L1 + G1 e51 ,

e5 = L5 + G2 e51 ,

e2 = L2 + G1 e62 ,

e6 = L6 + G2 e62 ,

e3 = L3 + G1 e61 ,

e7 = L7 + G2 e61 ,

e4 = L4 + G1 e52 ,

e8 = L8 + G2 e52 ,

and L1 = e12 + ae21 + be11 ,

L5 = e22 + ge21 + ue11 ,

L2 = e23 + ae13 + be22 ,

L6 = e14 + ge13 + ue22 , L3 = e22 + ae12 + be21 ,

L7 = e13 + ge12 + ue21 ,

L4 = e13 + ae22 + be12 ,

L8 = e23 + ge22 + ue12 , G1 = {l m¯ /(a + b )}(e32 + ae41 + be31 ), 4

2

G2 = {l 4m¯ /(a 2 + b 2)}(e42 + ge41 + ue31 ),

2

e11 = ((1/a) sinh a − (1/b) sin b),

e21 = (cosh a − cos b),

e12 = (a sinh a + b sin b),

e22 = (a 2 cosh a + b 2 cos b),

e13 = (a 3 sinh a − b 3 sin b),

e23 = (a 4 cosh a − b 4 cos b),

e14 = (a 5 sinh a + b 5 sin b), e31 = (1/a) sinh a(1 − z1 ) − (1/b) sin b(1 − z1 ), e32 = (a sinh a(1 − z1 ) + b sin b(1 − z1 )),

e42 = (a 2 cosh a(1 − z1 ) + b 2 cos b(1 − z1 )),

e51 = (1/a) sinh az1 − (1/b) sin bz1 ), e52 = (a sinh az1 + b sin bz1 ),

e41 = (cosh a(1 − z1 ) − cos b(1 − z1 )),

e61 = (cosh az1 − cos bz1 ),

e62 = (a 2 cosh az1 + b 2 cos bz1 ).

4.   The function obtained by solving the highly transcendental equation (5) shows rapid oscillations, attaining very large values between successive roots. The slope of the function

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T 1 Comparison of the present results with those of reference [1] for the first five modes, for cases with m¯e =m¯ =d=z1 =P 2 =Pd2 =VS and elastic foundation parameter K*=2p 4

End support parameters f0 Z0 f1 Z1 Frequency parameters l12 l22 l32 l42 l52

Reference [1]

Present

Reference [1]

Present

— — — —

VL† VL VS† VS

— — — —

VS VL VS VL

14·39 26·08 63·26 121·70 200·3

14·3932 26·0832 63·2562 121·7046 200·3450

17·10 41·87 89·92 158·50 247·10

17·0945 41·8731 89·9162 158·5285 247·1330

†VL=108 and VS=10−8.

at each root is, therefore, very close to vertical. The roots were first bracketed by means of a straight search process; then an iterative process was used to ensure convergence in the determination of the roots. The PC-MATLAB software version (4) has been used for all the computational processes in this work. The results and a brief discussion are presented as follows. Firstly, when P 2 =Pd2 =K*=Z1 =f1 =0, equation (5) reduces to that obtained recently by Chang [4]. The 12 system design parameters are considered in all of the recovered results of typical chosen examples of previous works. The first five frequency parameters are obtained for each case, and in Tables 1–4 is shown a comparison between present results and those obtained in references [1, 3, 4]. Good agreement is observed, which confirms the validity of the present general frequency equation. Secondly, the first five frequency parameters are obtained for the well known ten combinations of the four classical end T 2 Comparison of the present results with those of reference [3] for first five modes for cantilever beam with m¯e =0·6 and Z1 =−P 2 =10, with all other parameters zero

Mode

Reference [3]

Present

1 2 3 4 5

2·304 4·549 7·467 10·502 13·581

2·3044 4·5494 7·4669 10·5017 13·5813

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T 3 Comparison of the present results with those of reference [4], for the first five modes for case with m¯ = d = 0, m¯e = J = Z0 = 1, f1 = Z1 = VS and f0 = 0·01

Mode

Reference [4]

Present

1 2 3 4 5

0·794228 1·024830 2·692257 5·628385 8·702749

0·79422 1·0245 2·6922 5·6283 8·7027

conditions. Four values of the pair (K*, m¯ ) are chosen : (0, 0); (0, 2); (10, 0) and (10, 2) and z1 = 0·5. The results are shown in Table 5. One can conclude the following from these results. (1) There is no variation in the modal frequency parameter if the intermediate mass is located at the associated nodal point in which the vibrations are diminished. These situations are distinguished by an asterisk with the mode number. In these situations, a slight increase in the frequency parameters occurs when the Winkler elastic foundation is considered, as shown in the third and fourth columns of Table 5. (2) The effect of a Winkler elastic foundation on the natural frequencies are dominant for the first mode and diminished for higher modes even for high values of K*. This is consistent with findings of Yokoyama [1]. Finally, it is hoped that the present analysis may be of interest for designers and analysts concerned with vibration and stability in structural and mechanical design applications.-

T 4 Comparison of the results with those of reference [4], for the first five modes for case with f0 = Z0 = 0·01, m¯ = 0·1, m¯e = 0·2, d= 0·025, f1 = Z1 = K* = VS, J = 0·00005 and z1 = 0·55

Mode

Reference [4]

Present

1 2 3 4 5

1·554336 3·421981 5·231131 7·743512 10·233140

1·5664 3·4562 5·2852 7·8275 10·1898

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T 5 Five frequency parameters for beams with central mass and Winkler elastic foundation

Case C–F

Mode I II III* IV V*

C–C

(K*, m¯ ) ZXXXXXXXXXXCXXXXXXXXXXV (2p 4, 2) (0, 0) (0, 2) (2p 4, 0) 1·8751 4·6940 7·8547 10·9955 14·1371

1·5745 3·5341 7·8505 9·6126 14·1371

3·7935 5.1071 7·9533 11·0320 14·1543

— 4·1293 7·9522 9·6633 14·1543

I II* III IV* V

4·7300 7·8530 10·9956 14·1371 17·2786

2·9991 7·8532 9·6412 14·1371 15·8552

5·1352 7·9518 11·0320 14·1543 17·2881

— 7·9519 9·6916 14·1543 15·8371

F–F

I II* III IV* V

4·7300 7·8530 10·9956 14·1371 17·2787

3·9702 7·8532 9·5832 14·1371 15·8280

5·1353 7·9519 11·0320 14·1541 17·2882

4·5327 7·9519 9·6342 14·1541 15·8400

C–P

I II III IV V

3·9266 7·0686 10·2102 13·3517 16·4933

2·5873 6·8395 9·0798 9·0798 13·1197

4·5604 7·2026 10·2556 13·3721 16·5041

— 6·9835 9·1404 9·1404 13·1411

P–F

I II III IV V

3·9266 7·06858 10·2102 13·3517 16·4933

3·2517 6·8745 9·0379 9·0379 13·1181

4·5604 7·2026 10·2556 13·3721 16·5041

4·1197 7·0168 9·0991 9·0991 13·1395

P–P

I II* III IV* V

p 2p 3p 12·5663 15·7079

2·0959 6·2830 8·0730 12·5663 14·2679

4·1345 6·4709 9·4824 12·5908 15·7205

— 6·4709 8·1559 12·5908 14·2841

S–S

I* II III* IV V*

3·1415 6·2830 9·4245 12·5664 15·7080

3·1415 5·0487 9·4245 11·1597 15·7080

4·1344 6·4709 9·4824 12·5908 15·7205

4·1344 5·3372 9·4824 11·1927 15·7205

C–S

I II III IV V

2·3650 5·4978 8·6394 11·7809 14·9225

1·1915 4·1197 7·1901 10·6144 14·6862

3·7641 5·7604 8·7107 11·8106 14·9371

— 4·8500 7·3109 10·6531 14·7014

S–F

I II III IV V

2·3650 5·4978 8·6394 11·7810 14·9225

2·3491 4·5908 8·4058 10·6032 14·6876

3·8774 5·7700 8·7139 11·8106 14·9372

3·8725 4·9771 8·4852 10·6420 14·7028

P–S

I II III IV V

p/2 3p/2 5p/2 10·9956 14·1372

1·1915 4·1197 7·1901 10·2984 13·4210

3·7641 5·1214 11·0320 11·0320 14·1532

— 4·6417 7·3109 10·3414 13·4407

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