Experimental damping determination in beam bifurcation process.

Dolecek, Vlatko ; Isic, Safet ; Voloder, Avdo 等

1. INTRODUCTION

A beam is the most used element in many engineering constructions. Because of its specific dimensions, stability loss most likely occurs than overcome of allowable stress. So, beam bifurcation of stability is still the problem of interests of many authors (Isic et al.,2007). Bifurcation of beam equilibrium position is mostly analyzed by using static approach, where postcritical equilibrium path is the subject to be determined (Thompson & Hunt,1973; Zyczkowski, 2005). Because of that, dynamics of bifurcation process is not widely analyzed.

Load causing bifurcation is almost assumed as the constant, while in many practical problems load changes its magnitude according to deformation of buckled beam. Bifurcation of equilibrium of beam straight-line position under axial load, which changes magnitude with beam deformation, may produce moderate displacement and beam still remains in elastic range (Isic et al.,2007). As the result of the bifurcation, after departure from primary position, vibration around new secondary equilibrium position appears. While stiffness and stress stiffness change due to large displacement may be taken into account by using higher order terms of cross-section curvature and beam shortening (Isic et al.,2007; Thompson & Hunt,1973), damping remains unknown. Experimental modeling may only be used to its determination (Beards, 1996; Virgin, 2000).

In this paper experimental determination of dumping in beam vibration is presented, caused by bifurcation buckling under axial load, which is produced by compression of elastic body with known stiffness. The load magnitude depends on deformation of buckled beam. Equipment for experimental measurement is also presented, which consists of supports for beam-like specimens, mechanism for force adjustment with strain-gauge dynamometer, piezo-accelerometer and digital amplifier. It is assumed viscous damping scheme, and damping coefficient is determined in function of beam length and second moment area of cross-section. For specific specimen, viscous damping coefficient is calculated from logarithmic decrement of amplitude and vibration frequency, which are determined from time-domain analysis of acquiesced accelerograms of bifurcation process. Acceleration in imposed vibrations is measured by mounted piezo-accelerometer, connected to PC over digital amplifier, while applied force is controlled by mechanism with dynamometer.

2. EXPERIMENT SETUP AND MEASUREMENT

On the Fig. 1 scheme of the experiment setup is presented. Beam-like specimen 4 is connected to the testing platform 7 using two supports 3 and 5 (pinned or clamped). Axial force is applied over mechanism in the axially movable support 3 and measured with strain gage dynamometer 2. The dynamometer is hollow steel tube with bonded strain gages LY11 6/120, connected in the full bridge. The dynamometer is also main guide bush of axially movable support. An acceleration is measured on the middle point of the specimen by connected piezo-accelerometer KD37V. The specimen is restrained in straight-line initial position by auxiliary support 6, until force is regulated to the desired value, greater than calculated critical buckling force. After that, specimen is allowed to buckle and record of axial force and acceleration is acquired using digital amplifier SPIDER 8 and PC with CATMAN 5.0 software.

Using presented experiment setup the acceleration for given axial force and stiffness of dynamometer are measured. Characteristic accelerogram is given on the Fig. 2.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Measured accelerograms show two different rates of amplitude decay. In the beginning of the process large amplitudes appear, which are non-symmetric around time axes, what shows presence of departure of beam from initial straight-line position. After that process is finished in the vibration around new buckled configuration. The accelerograms of the process enable measurement of acceleration amplitudes, vibration period and frequency of vibration.

3. CALCULATION OF DAMPING COEFFICIENT

In case of a single d.o.f. system, logarithmic decrement of the amplitude may be calculated from the following expression

[LAMBDA] = C[pi]/M[omega] (1)

where C is viscous damping, M is mass of the system, [omega] is vibration frequency and [LAMBDA] is logarithmic decrement.

Previous expression could be easily derived from corresponding differential equation of damped vibration. It is shown in [2] that beam deform according to buckling eigenmode. The vibration in the bifurcation process could then be described by using single nonlinear differential equation, which is derived by using finite elements method. Mass M and damping C in (1) is then replaced by

M = [D.sup.T.sub.0][MD.sub.0]

C = [D.sup.T.sub.0][CD.sub.0] (2)

where M is mass matrix, C is viscous damping matrix for damping coefficient [kappa] = 1, and [D.sub.0] is normalized buckling eigenvector in which characteristic displacement (in experiment--displacement of the specimen middle point) has unit value.

Using equation (2), coefficient of viscous damping [kappa], in case of considered vibration of buckled beam, could be calculated as

[kappa] = M/C[pi] [omega][LAMBDA] (3)

Equation (3) implies that damping factor for specific case could be calculated from measured vibration frequency and logarithmic decrement of a acceleration amplitude.

4. MATHEMATICAL MODEL OF DAMPING

It is assumed that viscous damping coefficient k may be expressed in shape of polynomial function of length and second moment area of the beam-like specimen as

[kappa] = [b.sub.0] + [b.sub.1] L + [b.sub.2][I.sub.z] + [b.sub.12] [LI.sub.z] (4)

where [b.sub.0], [b.sub.1], [b.sub.2], [b.sub.12] are constants to be determined, L is specimen length and [I.sub.z] is second moment area.

Unknown constants in the polynomial expression (4) are determined from [2.sup.2] factor experiment with repetition in every point of experimental plan. Experimental measurements are done with four specimens with two different lengths and two different second moment areas, from which is possible to construct four points in plan of the experiment. Obtained results in eight provided measurements are given in the Table 1.

[FIGURE 3 OMITTED]

The analysis of given results shows that all coefficients [b.sub.i], i = 1,2,12 are significant if Student t-test is used with confidence level of 95%. From presented experimental results following mathematical model for damping coefficient could be derived

[kappa] = 159.98 - 515.70L - 17.77 x [10.sup.12][I.sub.z] + 59.08 x [10.sup.12][LI.sub.z] (5)

which provides regression coefficient R [greater than or equal to] 0.95.

Fig. 3 shows dependency of damping coefficient on used factors of experiment: length and second moment area.

5. CONCLUSION

Experimental damping determination of vibration induced by bifurcation buckling under displacement dependent axial force is presented. Damping is assumed as viscous linear damping and damping coefficient is experimentally determined. Polynomial expression is given for damping coefficient, where appropriate coefficients are given. It is shown that damping could be modelled in function of beam geometry (length and second moment area). Given mathematical model for viscous damping calculation is chosen to give average amplitude decay in buckling process, while experimental results show different amplitude decay in the short beginning part than in the other part of measured accelerograms.

6. REFERENCES

Beards, C.F. (1996). Structural Vibration--Analysis and Damping, Arnold, ISBN 0-340-64580-6, London.

Isic S.; Dolecek, V.; Karabegovic, I. (2007). Numerical and Experimental Analysis of Postbuckling Behaviour of Prismatic Beam Under Displacement Dependent Loading, Proceedings of First Serbian Congress on Theoretical and Applied Mechanics, pp. 331-338, ISBN 978-86-909973-05, Sumarac, D., Kuzmanovic, D. (Ed.), Kopaonik, Serbia, April 2007.

Thompson, J.M.T.; Hunt, G.W. (1973). A General Theory of Elastic Stability, John Wiley & Sons, ISBN 0 471 85991 5, London.

Virgin, L.N. (2000). Introduction to Experimental Nonlinear Dynamics: A Case Study in Mechanical Vibration, Cambridge University Press, ISBN 0-521-66286-9.

Zyczkowski, M. (2005). Post-buckling analysis of nonprismatic columns under general behaviour of loading, International Journal of Non-Linear Mechanics, Vol. 40, Issue 4, pp. 445-463, May 2005, ISSN 0020-7462.

Tab. 1. Results of experiment.

 Value of experimental Viscous Damping Factor
 factors [kappa]

Point of plan L [m] [I.sub.z] Outcome Outcome
of experiment [[m.sup.4]] [().sub.(1)] [().sub.(2)]

 1 0.300 0.7840-10-11 4.911 4.875
 2 0.339 0.7840-10-11 2.921 2.764
 3 0.300 0.1065-10-10 4.355 5.158
 4 0.339 0.1065-10-10 9.033 9.328