An iterative \(GL(n,\mathbb{R})\) method for solving non-linear inverse vibration problems. (English) Zbl 1279.22033
Summary: For the inverse vibration problem, a differential-algebraic equation (DAE) method is proposed to simultaneously estimate the time-dependent damping and stiffness coefficients by using two sets of displacement and velocity as input data. We combine the equations of motion and the supplemental data into a set of DAEs. We develop an implicit \(GL(n,\mathbb{R})\) scheme and a Newton iterative algorithm to stably solve the DAEs to find the unknown structural coefficients. The unknown force is also recovered by the present method. A linear oscillator and a non-linear Duffing oscillator are used as testing examples. The estimated results are rather accurate and robust against random noise; hence, the new method can be used in the solutions of non-linear inverse vibration problems.
MSC:
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 65L80 | Numerical methods for differential-algebraic equations |
| 65L07 | Numerical investigation of stability of solutions to ordinary differential equations |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
Keywords:
inverse vibration problem; time-dependent damping and stiffness coefficients; recovery of external force; Duffing non-linear oscillatorReferences:
| [1] | Adhikari, S., Woodhouse, J.: Identification of damping: part 1, viscous damping. J. Sound Vib. 243, 43-61 (2001) · doi:10.1006/jsvi.2000.3391 |
| [2] | Adhikari, S., Woodhouse, J.: Identification of damping: part 2, non-viscous damping. J. Sound Vib. 243, 63-88 (2001) · doi:10.1006/jsvi.2000.3392 |
| [3] | Feldman, M.: Consider high harmonics for identification of non-linear systems by Hilbert transform. Mech. Syst. Signal Process. 21, 943-958 (2007) · doi:10.1016/j.ymssp.2006.01.004 |
| [4] | Gladwell, G.M.L.: Inverse Problem in Vibration. Kluwer Academic, Dordrecht (1986) · Zbl 0646.73013 · doi:10.1007/978-94-015-1178-0 |
| [5] | Gladwell, G.M.L., Movahhedy, M.: Reconstruction of a mass-spring system from spectral data I: theory. Inverse Probl. Eng. 1, 179-189 (1995) · doi:10.1080/174159795088027578 |
| [6] | Hosseini, M.M.: Adomian decomposition method for solution of nonlinear differential algebraic equations. Appl. Math. Comput. 181, 1737-1744 (2006) · Zbl 1106.65071 · doi:10.1016/j.amc.2006.03.027 |
| [7] | Huang, C.H.: A non-linear inverse vibration problem of estimating the time-dependent stiffness coefficients by conjugate gradient method. Int. J. Numer. Methods Eng. 50, 1545-1558 (2001) · Zbl 0983.70017 · doi:10.1002/nme.83 |
| [8] | Huang, C.H.: A generalized inverse force vibration problem for simultaneously estimating the time-dependent external forces. Appl. Math. Model. 29, 1022-1039 (2005) · Zbl 1163.74507 · doi:10.1016/j.apm.2005.02.006 |
| [9] | Ingman, D., Suzdalnitsky, J.: Iteration method for equation of viscoelastic motion with fractional differential operator of damping. Comput. Methods Appl. Mech. Eng. 190, 5027-5036 (2001) · Zbl 1020.74049 · doi:10.1016/S0045-7825(00)00361-3 |
| [10] | Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution. Eur. Math. Soc., Zurich (2006) · Zbl 1095.34004 · doi:10.4171/017 |
| [11] | Lancaster, P., Maroulas, J.: Inverse eigenvalue problems for damped vibrating systems. J. Math. Anal. Appl. 123, 238-261 (1987) · Zbl 0637.34017 · doi:10.1016/0022-247X(87)90306-4 |
| [12] | Liang, J.W., Feeny, B.F.: Balancing energy to estimate damping parameters in forced oscillator. J. Sound Vib. 295, 988-998 (2005) · doi:10.1016/j.jsv.2006.01.060 |
| [13] | Liu, C.-S.: Exact solutions and dynamic responses of SDOF bilinear elastoplastic structures. J. Chin. Inst. Eng. 20, 511-525 (1997) · doi:10.1080/02533839.1997.9741858 |
| [14] | Liu, C.-S.: Cone of non-linear dynamical system and group preserving schemes. Int. J. Non-Linear Mech. 36, 1047-1068 (2001) · Zbl 1243.65084 · doi:10.1016/S0020-7462(00)00069-X |
| [15] | Liu, C.-S.: Two-dimensional bilinear oscillator: group-preserving scheme and steady state motion under harmonic loading. Int. J. Non-Linear Mech. 38, 1581-1602 (2003) · Zbl 1348.70060 · doi:10.1016/S0020-7462(02)00123-3 |
| [16] | Liu, C.-S.: Reid’s passive and semi-active hysteretic oscillators with friction force dependence on displacement. Int. J. Non-Linear Mech. 41, 775-786 (2006) · Zbl 1160.70342 · doi:10.1016/j.ijnonlinmec.2006.04.006 |
| [17] | Liu, C.-S.: New integrating methods for time-varying linear systems and lie-group computations. Comput. Model. Eng. Sci. 20, 157-175 (2007) · Zbl 1152.93340 |
| [18] | Liu, C.-S.: Identifying time-dependent damping and stiffness functions by a simple and yet accurate method. J. Sound Vib. 318, 148-165 (2008) · doi:10.1016/j.jsv.2008.04.003 |
| [19] | Liu, C.-S.: A Lie-group shooting method for simultaneously estimating the time-dependent damping and stiffness coefficients. Comput. Model. Eng. Sci. 27, 137-149 (2008) · Zbl 1232.74027 |
| [20] | Liu, C.-S.: A method of Lie-symmetry \(\mathit{GL}(n,{\mathbb{R}})\) for solving non-linear dynamical systems. Int. J. Non-Linear Mech. 52, 85-95 (2013) · doi:10.1016/j.ijnonlinmec.2013.01.015 |
| [21] | Liu, C.-S., Huang, Z.M.: The steady-state responses of SDOF viscous-elastoplastic oscillator under sinusoidal loadings. J. Sound Vib. 273, 149-173 (2004) · doi:10.1016/S0022-460X(03)00423-1 |
| [22] | Liu, C.-S., Chang, J.R., Chang, K.H., Chen, Y.W.: Simultaneously estimating the time-dependent damping and stiffness coefficients with the aid of vibrational data. Comput. Mater. Continua 7, 97-107 (2008) |
| [23] | Ma, C.K., Tuan, P.C., Lin, D.C.: A study of inverse method for the estimation of impulse loads. Int. J. Syst. Sci. 29(6), 663-672 (1998) · doi:10.1080/00207729808929559 |
| [24] | Ma, C.K., Tuan, P.C., Chang, J.M., Lin, D.C.: Adaptive weighting inverse method for the estimation of input loads. Int. J. Syst. Sci. 34(3), 181-194 (2003) · Zbl 1142.93436 · doi:10.1080/0020772031000135432 |
| [25] | Sand, J.: On implicit Euler for high-order high-index DAEs. Appl. Numer. Math. 42, 411-424 (2002) · Zbl 1005.65081 · doi:10.1016/S0168-9274(01)00164-7 |
| [26] | Soltanian, F., Karbassi, S.M., Hosseini, M.M.: Application of He’s variational iterative method for solution of differential-algebraic equations. Chaos Solitons Fractals 41, 436-445 (2009) · Zbl 1198.65154 · doi:10.1016/j.chaos.2008.02.004 |
| [27] | Starek, L., Inman, D.J.: On the inverse vibration problem with rigid-body modes. ASME J. Appl. Mech. 58, 1101-1104 (1991) · doi:10.1115/1.2897693 |
| [28] | Starek, L., Inman, D.J.: A symmetric inverse vibration problem with overdamped modes. J. Sound Vib. 181, 893-903 (1995) · doi:10.1006/jsvi.1995.0176 |
| [29] | Starek, L., Inman, D.J.: A symmetric inverse vibration problem for nonproportional underdamped systems. ASME J. Appl. Mech. 64, 601-605 (1997) · Zbl 0900.70262 · doi:10.1115/1.2788935 |
| [30] | Starek, L., Inman, D.J., Kress, A.: A symmetric inverse vibration problem. ASME J. Vib. Acoust. 114, 565-568 (1992) · doi:10.1115/1.2930299 |
| [31] | Wu, A.L., Loh, C.H., Yang, J.N.: Input force identification: application to soil-pile interaction. J. Struct. Control Health Monit. 16, 223-240 (2009) · doi:10.1002/stc.308 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
