I am working on the structural dynamic response with Daubechies wavelet. The Equation of Motion of dynamic system with single degree of freedom can be described as $$ m\ddot{x}(t) +c\dot{x}(t)+k{x}(t)=f(t) $$ The dynamic displacement response $x(t)$ can be projected into the DbN (N is posivtive even integer) wavelet domain given by $$ x(t) = \sum\limits_{k}\hat{x}_{j,k}\phi_{j,k}(t) $$ $$ \phi_{j,k}(t) = 2^{j/2}\phi(2^{j}t-k) $$ then, the velocity and acceleration of system are given by $$ \dot{x}(t) = \sum\limits_{k}\hat{x}_{j,k}\dot{\phi}_{j,k}(t) $$ $$ \ddot{x}(t) = \sum\limits_{k}\hat{x}_{j,k}\ddot{\phi}_{j,k}(t) $$ where $ k\in{Z}$. $j$ is the decomposition scale. Similarly, the external force can also expanded in DbN wavelet domain as $$ f(t) = \sum\limits_{k}\hat{f}_{j,k}\phi_{j,k}(t) $$ Then, the Equation of Motion is rewritten as $$ m\sum\limits_{k}\hat{x}_{j,k}\ddot{\phi}_{j,k}(t)+c\sum\limits_{k}\hat{x}_{j,k}\dot{\phi}_{j,k}(t) +k\sum\limits_{k}\hat{x}_{j,k}\phi_{j,k}(t) =\sum\limits_{k}\hat{f}_{j,k}\phi_{j,k}(t) $$ Now define $\tau=2^{j}t, \tau=0,1,2,..,n-1$. Then signal are discretized into $n=2^{j}$ points.
Both sides are taking inner product operation and one can do some arrangemenet based on the orthogonality of DbN wavelet $$ \frac{1}{2^{2j}}m\sum\limits_{k=l-N+2}^{k=l+N-2}\hat{x}_{j,k}\Omega_{k-l}^{2}(t)+\frac{1}{2^{j}}c\sum\limits_{k=l-N+2}^{k=l+N-2}\hat{x}_{j,k}\Omega_{k-l}^{1} +k\hat{x}_{j,l}=\hat{f}_{j,l} $$ $$ \Omega_{k-l}^{1}=\int{\dot{\phi}(\tau-k)\phi(\tau-l)d\tau} $$ $$ \Omega_{k-l}^{2}=\int{\ddot{\phi}(\tau-k)\phi(\tau-l)d\tau} $$ where $l=0,1,...,n-1$.
The Equation of Motion is transformed into wavelet domain. The key problem is how to solve scaling funciton coefficients $\hat{x}_{j,l}$, $\hat{f}_{j,l}$? I have referred to many references[1-3], which seems to avoid this problem. Is there any suggestion and recommendation? I am really hurry!
References
[1] D.M. Joglekar, M. Mitra, A wavelet-based method for the forced vibration analysis of piecewise linear single- and multi-DOF systems with application to cracked beam dynamics, J Sound Vib, 358 (2015) 217-235.
[2] R. Ghanem, F. Romeo, A wavelet-based approach for the identification of linear time-varying dynamical systems, J Sound Vib, 234 (2000) 555-576.
[3] X. Xu, Z. Shi, Q. You, Identification of linear time-varying systems using a wavelet-based state-space method, Mech Syst Signal Pr, 26 (2012) 91-103.