Effect of large amplitude deflection profile on the dynamic behaviour of slender beams Suresh Gamini* and Kashi Nath Saha *

[email protected] , P. G. Student and [email protected], Faculty member Department Of Mechanical Engineering, Jadavpur University, Kolkata-700032. ABSTRACT A thorough study has been carried out to determine the frequency-amplitude relationship (backbone curves) of slender beams under different types of loading and for different boundary conditions. The complicated solution of dynamic problem is achieved by using the solution of static analysis as its kernel. The main objective of the present work is to investigate the effect of the large amplitude deflection profile on the non-linear frequency parameter. It is observed that the nonlinearity effect for a particular vibration mode is more predominant for a particular combination of boundary conditions and type of loading. As the assumed deflection profiles always correspond to some physical loading condition, the results may be useful to the practicing engineers for design purposes. 1. INTRODUCTION Vibration of beams with large displacement has come out as an emerging research area. This is due to the fact that the consequent geometric nonlinearity provides an additional stiffening to the beam structure. Thus in the actual application a substantial savings in weight and volume is achieved due to this stretching effect. However, when stretching displacement occurs due to the large amplitude of beam deflection a non-linear free vibration problem arises, the solution of which is quite complicated. In the present work the solution is obtained in two steps, the first one is a static analysis, which is used as a kernel in the subsequent step of dynamic problem solution. In this connection, naturally the question arises that whether the shape of the large amplitude deflection profile has any bearing on the non-linear frequency parameter. Seeking an answer to this query is the main objective of the present work. Research work on large amplitude vibration analysis of beam structures has gone through different phases and the evolution process can be traced back with the help of review works [7, 12, 13]. The geometrically non-linear beam vibration problem has been solved by different researchers through different techniques. Kreiger [5] reduced the governing partial differential equations to ordinary differential equations and the solution was obtained in terms of elliptic functions using a one-term approximation. Mei [8] presented the finite element formulations for large amplitude vibrations of beams whereas Nayfeh et al. [9] had adopted analytical approaches for obtaining non-linear normal modes of a cantilever beam. Feng and Bert [4] applied the differential quadrature (DQ) method to analyse non-linear vibration of clamped-clamped beams. The method has been further improved by Quan and Chang [11] and Chen et al. [1] through incorporation of non-uniformly spaced Chebyshev grid points and simplified Hadamard product method of matrices, respectively. These solution methods along with various other techniques are applied to solve beam vibration problems with various complicating effects. Evensen [3] extended the analytical perturbation method to study the

159

effect for various boundary conditions. Lou and Sikarske [6] employed form-function approximations to study the non-linear forced vibrations of buckled beam. Pohit et al. [10] investigated the problem of free transverse vibrations of rotating beams with non-linear elastomeric constraints. It is observed from the review work that significant amount of research work is carried out on large amplitude vibration analysis of slender beams with various complicating effects. However, the effect of large amplitude deflection profile on the dynamic behaviour of slender beams has not been addressed by any researchers. In the present work, the relation between the normalized amplitude of vibration and the resonance frequency parameter (usually known as the backbone curve) is used to represent the dynamic behaviour of the system. 2. MATHEMATICAL FORMULATION The mathematical model is framed on the assumptions that the beam material is homogeneous, isotropic and linear-elastic, the beam follows Euler-Bernoulli hypothesis and the damping effect of the system is negligible. Further it is assumed that the magnitude of the beam vibrations, about its static equilibrium position, is small and the beam is of uniform cross section. The beam problem is solved for various boundary conditions as shown in Fig. 1(a). Although this figure shows concentrated loading ' P' , the problem is solved for various other types of distributed loading p(x) as indicated in Fig. 1(b) for a clamped-clamped beam. In the case of large deflection analysis, the displacement of a beam structure due to external transverse loading is given by, u = u s + ub , where u s denotes displacement due to stretching and ub denotes displacement due to bending. For a planar system, the strain is the total derivative of the displacement, i.e., ε xx = du / dx = d (u s ) / dx + d (ub ) / dx . Substituting u s and ub in terms of the in-plane displacement and transverse deflection fields u (x) and w(x) , the expression of the strain becomes

ε xx = du / dx + 0.5(du / dx) 2 + 0.5(dw / dx) 2 − z (d 2 x / dx 2 ) . Strain energy stored in the structure is given by, U =

(1)

1 bE h / 2 L 2 (σε )dv = ∫ ∫ ∫ [ε xx ] dzdx . 2 Vol 2 −h / 2 0

(2)

Substitution of equation (1) in (2) and some mathematical manipulations yield, L

{

U = 0.5hbE ∫ (du / dx) 2 + 0.25(du / dx) 4 + 0.25(dw / dx) 4 + (du / dx)(du / dx) 2 + 0

2

2

2

}

L

(3) 2

2 2

(du / dx)(dw / dx) + 0.5(du / dx) (dw / dx) dx + 0.5 EI ∫ (d w / dx ) dx . 0

L

The potential energy of the external load is given by, V = − P w x − ∫ ( pw)dx .

(4)

Static displacements are obtained from the energy conservation principle δ (π ) = δ (U + V ) = 0 .

(5)

p

0

Substituting (3) and (4) in equation (5) and normalising the space co-ordinate ' x' through the transformation, ξ = x / l , the governing differential equation in non-dimensional form becomes,

160

3 3 2  1  EA ∫  2  du δ  du  + 1  du  δ  du  + 1  dw  δ  dw  + 3  du  δ  du   2 0  L  dξ   dξ  L3  dξ   dξ  L3  dξ   dξ  L2  dξ   dξ    2

2  du   dw   dw  1  dw   du  1  dw    δ +  2 δ  +  + 2  L  dξ   dξ   dξ  L2  dξ   dξ  L3  dξ  1  du   + 3  L  dξ 

2

2

 du   du   δ    dξ   dξ 

L   dw   dw  EI 1  d 2 w   d 2 w  dξ + 3 ∫  2 δ  2 dξ − Pδ ( w ξ ) − ∫ ( pw)dx  = 0 . δ   P L 0  dξ   dξ    dξ   dξ  0

(6)

The displacement functions u (ξ ) and w(ξ ) can be approximated by sets of coordinate functions 2n

n

i =1+ n

i =1

φi (ξ ) and ψ i (ξ ) , expressed as u(ξ ) ≅ ∑ ciφi − n and w(ξ ) ≅ ∑ ciψ i .

(7)

Substituting (7) in (6) and replacing the operator ' δ ' by ∂ / ∂c j , following Galerikin’s principle, yields n n

 EI

i =1 j =1

L

∑ ∑ ci 

3

1

′ ′j′dξ + ∫ψ i′ψ

0

  n   n 2 n  EA 1  n  n EA 1 ′ ′ ′ ′ ψ ψ ψ ψ c c    ∑ ∑ i2 i2  i1 i1  dξ  + ∑ ∑ ci  2 ∫ψ i′φ ′j −n  ∑ ci1ψ i′1 dξ + 3 ∫ i j 2L 0   i1=1   i =1 j =n+1  2 L 0  i 2=1  i1=1

1

  n  n   2n EA 1 EA ψ ′ψ ′  c φ ′ dξ + 3 ∫ φi′−nφ ′j −n  ∑ ci1ψ i′1  ∑ ci 2ψ i′2 dξ + 2 ∫ i j  ∑ i1 j −n  2L 0 L 0   i 2=1  i1=1   i1=1+n 2n 2 n  EA 1   2n    2 n  2n 3EA 1 EA 1 ψ ′ψ ′  c φ ′  φi′−nφ ′j −n dξ + 2 ∫ φi′−nφ ′j −n  ∑ ci1φi′1−n dξ c φ ′ dξ  + ∑ ∑ ci  ∫ 3 ∫ i j  ∑ i1 i1−n  ∑ i 2 i 2−n  L 2L 0 2 L 1 i n = + = + = + = + = + 1 1 i n j 1 n 2 1 i n i 1 n 1       0 0 

+

(8)

1 n      2 n  2n EA 1 c φ ′ dξ  = ∑  Pψ i ξp + L ∫ ( pψ i )dξ  . φ ′ φ ′  c φ ′  3 ∫ i − n j −n  ∑ i1 i1−n  ∑ i 2 i 2−n  2L 0   i =1   i 2=n+1  i1=n+1 0 

[K ] {c} = {P }

Equation (8) can be expressed in matrix form as, where,

[K ] [K ]12  [K ] =  11  [K ]21 [K ]22 

and

{P } = Pψ i ξ

(9)

1

p

+ L ∫ ( pψ i )dξ

(10)

0

The elements of the stiffness matrix [K ] are given below. [ K11 ] =

  n  n EI 1 EA 1 ′ ′ ′ ′ ψ ψ ξ ψ ′ψ ′  c ψ ′  c ψ ′ dξ , d + 3 ∫ i j  ∑ i1 i1  ∑ i 2 i 2  3 ∫ i j 2L 0 L 0   i 2=1  i1=1

[ K12 ] =

  2n   n EA 1 EA 1 ′ ′ ′ c d ψ φ ψ ξ ψ ′ψ ′  c φ ′ dξ +   ∑ ∫ i j n i i 1 1 −   2 ∫ i j  ∑ i1 j −n  2 2L 0 L 0   i1=1+n   i1=1

+

[ K 21 ] = 0

  2 n  2n   n  n EA 1 EA 1 ′ ′ ′ ′ ′ ′ ′ φ φ ψ ψ ξ ψ ψ φ c c c d +      ∑ ∑ ∑ ∫ ∫ i n j n i i i i 1 1 2 2 i j i 1 i n 1 − − −  ∑ ci 2φi′2−n dξ      3 3 2L 0 2L 0   i 2=n+1  i1=n+1   i 2=1  i1=1

[ K 22 ] =

  2 n  2n   2n 3EA 1 EA 1 EA 1 φi′−nφ ′j −n dξ + 2 ∫ φi′−nφ ′j −n  ∑ ci1φi′1−n dξ + 3 ∫ φi′−nφ ′j −n  ∑ ci1φi′1−n  ∑ ci 2φi′2−n dξ . ∫ L 0 2L 0 2L 0   i 2=n+1  i1=n+1   i1=n+1

It should be pointed out that the coefficient matrix [K ] is a function of unknown coefficients {c} . These unknown coefficients can be obtained from

{c} = [K ]−1{P }

through an iterative scheme using

a relaxation technique. The details of the procedure may be obtained in Ref. [2] and hence is not

{}

elaborated here. In each step of the iteration, the error vector {ε } = [K ]−1 P − {c} is computed and if it is not within the permitted value of tolerance, the process is repeated with new values of {c} until

{ε } becomes less than the specified tolerance. 161

2.1 Generation of characteristic orthogonal polynomials: Appropriate flexural and in-plane starting functions ψ 0 (ξ ) and φ0 (ξ ) have been selected for different boundary conditions. These starting functions φ0 and ψ 0 satisfy all the boundary conditions of the beam and are used to generate the sets of higher order functions φi and ψ i . Following GramSchimidt scheme, when a polynomial φ0 is given, an orthogonal set of polynomials in the interval a ≤ x ≤ b can be generated by following the scheme, φ1 ( x) = (x − B1 )φo ( x) and φ k ( x) = ( x − Bk )

φ k −1 ( x) − ck φ k −2 ( x) b

where,

b

b

b

a

a

a

Bk = ∫ xw( x)φ k2−1 ( x)dx / ∫ w( x)φ k2−1 ( x)dx and Ck = ∫ xw( x)φ k −1 ( x)φ k −2 ( x)dx /

2

∫ w( x)φ k − 2 ( x)dx . In the present application, the weight function is chosen as unity and the interval is

a

from 0 to 1. It should also be noted that the generated sets of functions are orthogonal in nature. 2.2 Analysis for the dynamic problem In the previous sections the solution of static problem is obtained through minimum potential energy principle. The dynamic problem is formulated by using Hamilton’s principle, according to which, t δ ( ∫ 2 L dt ) = 0 , where, L (= T − U ) is known as the Lagrangian, and 'T ' is the total kinetic energy of t1 the system, expressed as, T = 0.5 m ∫

{

}

L [∂w( x, t ) / ∂t ]2 + [∂u ( x, t ) / ∂t ]2 dx , 0

(11)

where ' m' is mass per unit length of the beam. The expression for strain energy of the system is already developed in equation (5), but here it is to be calculated corresponding to the static equilibrium position and hence the solution of the static problem is a prerequisite for this analysis. Upon substitution for the Lagrangian terms the governing differential equation of the dynamic system can be expressed as, 2 4 4 2  1 L  ∂w  2  ∂u  2  hbE L  du  1  du  1  dw   du  du  δ  ∫ m  +   dx + + + + dx           ∫ 2 0  dx  4  dx  4  dx   dx  dx   2 0  ∂t   ∂t    2 2 2 2  2  du  dw  1  du   dw   EI L  d w   = 0. dx +    +  +    ∫   dx  dx  2  dx   dx   2 0  dx 2  

(12)

Equation (12) is expressed in normalised form following the same procedure as in the case of static formulation, and is not elaborated here to maintain brevity. 2.3 Solution through assumed mode method The solution of the governing differential equation (12) is obtained through assumed mode method, where, the displacement functions are assumed to be separable in space and time co-ordinates, i.e., w(ξ , t ) = w(ξ ) f (t ) and u (ξ , t ) = u (ξ ) g (t ) . The static equilibrium positions w(ξ ) and u (ξ ) of the beams due to the externally applied loads is already known and it is assumed that the temporal functions f (t ) and g (t ) are constituted through a linear combination of harmonic functions. Thus the dynamic

162

displacements may be approximated as,

n

w(ξ , t ) = ∑ wi (ξ )di f i (t ) i =1

and

2n

u (ξ , t ) = ∑ ui (ξ )d i g i − n (t ) , i =1+ n

where, di ’s are the contribution or participation of the respective modes. The harmonic nature of the temporal functions, f (t ) and g (t ) are expressed as, f i (t ) = eiωt and g i (t ) = eiωt . Here again, as in the case of static formulation, ' δ ' is replaced by ∂ / ∂d j , j = 1,2...n , and we obtain the governing differential equation in the matrix form as, − ω 2 [M ]{d } + [K ]{d }= 0 , which is a standard eigen value problem of the form,

[[M ]

−1

[K ]]{d } = ω 2 {d } . The standard eigen value problem is solved numerically

for the natural frequencies ω i by using IMSL routines. The stiffness matrix is already mentioned in connection with the static problem and the elements of the mass matrix are given below,

[M ] = 

1 M 11 M 12  , [M ]11 = Lm ∫ wi w j dξ ,   M 21 M 22  0

1

[M ]22 = Lm ∫

u i u j dξ ,

[M ]12 = [M ]21 = 0 .

0

3. RESULTS AND DISCUSSIONS A thorough study has been carried out to determine the backbone curves of slender beams under different types of loadings and boundary conditions. The results are generated for a rectangular cross section beam, b x h x L = 20 mm x 5 mm x 1000 mm, having E = 2.1 x 1011 Pa and mass density ρ = 7850 Kg/m3. The two displacement functions for describing transverse deflection and in-plane displacement are constructed by using eight coordinate functions and the problem is solved for an error limit of 0.1%. The number of functions and the error limit are determined after carrying out the necessary convergence test but is not reported here to maintain brevity. The dimensionless free vibration frequency parameter corresponding to linear case is compared with other researchers and a fairly good agreement is observed for all the four boundary conditions. Comparison for large amplitude vibration frequencies is also done and it is observed that for lower amplitude of vibration the matching is quite good but for increasing amplitude the present method shows more nonlinearity effect. The large amplitude dynamic behaviour of the beam is shown graphically for first three vibration modes in dimensionless amplitude-frequency plane in figures 2 and 3. In Fig. 2, the results for different boundary conditions are obtained corresponding to the deflection profile of uniform loading and are shown in cases (i)-(iii) for the first three vibration modes. The ratio of the maximum beam deflection to beam thickness is taken as the dimensionless amplitude w* (= wmax / h) while the non-linear frequency ω nl is normalized by the frequencies of the respective linear vibration modes. In fig. 3, the ordinate remains same but in case of abscissa, normalization is carried out by the corresponding fundamental linear frequency ω1 . It is observed from Fig. 2 that the degree of nonlinearity is more predominant for some boundary condition for a particular vibration mode, whereas for another vibration mode some other boundary condition gets upper hand. For example in the first vibration mode, the degrees of

163

nonlinearity of the two symmetric boundary conditions are the bounds for the other two asymmetric boundary conditions whereas in the third vibration mode, an opposite effect is observed. In Fig. 3 the primary observation is that, for clamped-clamped boundary condition the nonlinearity effect is minimum whereas it is maximum for clamped-free boundary condition. These figures also reveal that with increase in the vibration mode number, the nonlinearity effect becomes more dominant. 4. CONCLUSIONS A novel method is established where static analysis serves as the basis for the subsequent dynamic study, which is quite robust, stable and realistic. Large amplitude free vibration analysis of beam structures with different boundary conditions is investigated. The results for first vibration mode are compared for two symmetric boundary conditions. The backbone curves documented for different boundary conditions, may be used by the practicing engineers for design purposes. The extent of nonlinearity of the system can be visualized from the magnitude of dimensionless load and amplitude parameter. In the present study, the assumed deflection profiles always correspond to some physical loading condition, which is in general absent for other research works on large amplitude vibration. 5. REFERENCES 1. Chen, W., Zhong, T.X., and Liang, S.P., 1997, “On the DQ analysis of geometrically non-linear vibration of immovably simply-supported beams”, Journal of Sound and Vibration, 206, 745-748. 2. Crandall, S.H., 1956, “Engineering Analysis: A survey of numerical procedures”, McGraw Hill,. 3. Evensen, D.A., 1968, “Nonlinear vibrations of beams with various boundary conditions”, AIAA Journal, 6, 370-372. 4. Feng, Y. and Bert, C.W., 1992, “Application of the quadrature method to flexural vibration analysis of a geometrically non-linear beam”, Nonlinear Dynamics, 3, 13-18. 5. Kreiger, S.W., 1950, “The effect of an axial force on the vibration of hinged bars”, Journal of Applied Mechanics, ASME, 17, 35-36. 6. Lou, C.L. and Sikarske, D.L., 1975, “Nonlinear vibration of beams using a form-function approximation”, Journal of Applied Mechanics, ASME, 42, 209-214. 7. Marur, S.R., 2001, “Advances in nonlinear vibration analysis of structures. Part-I. Beams”, Sadhana, Journal of Indian Acacemy of Science, 26, 243-249. 8. Mei, C., 1973, “Finite element displacement method for large amplitude free flexural vibrations of beams and plates”, Computers and Structures, 3,163-174. 9. Nayfeh, A.H., Chin, C. and Nayfeh, S.A., 1995, “Nonlinear Normal Modes of a Cantilever Beam”, Journal of Vibration and Acoustics, 117, 477-481. 10. Pohit, G, Mallik, A.K. and Venkatesan. C., 1999, “Free out-of plane vibrations of a rotating beam with non-linear elastomeric constraints”, Journal of Sound and Vibration, 220, 1-25. 11. Quan, J.R. and Chang, C.T., 1989, “New insights in solving distributed systems equations by the quadrature methods - II”, Computer and Chemical Engineering, 13, 1017-1024. 12. Reddy, J.N., 1979, “Finite element modelling of structural vibrations: A review of recent advances”, Shock Vibration Digest, 11, 25-39.

13. Sathyamoorthy, M, 1982, “Nonlinear analysis of beams, Part-1: A survey of recent advances”, Shock Vibration Digest, 14, 19-35.

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