2. Benchmark solution#

2.1. Calculation method#

The problem can be in the following matrix form:

\(\left[\begin{array}{cc}{m}_{2}& 0\\ 0& {m}_{3}\end{array}\right]\left\{\begin{array}{c}{\dot{u}}_{2}\\ {\dot{u}}_{3}\end{array}\right\}+\left[\begin{array}{cc}{k}_{1}{+k}_{2}& -{k}_{2}\\ -{k}_{2}& {k}_{2}\end{array}\right]\left\{\begin{array}{c}{u}_{2}\\ {u}_{3}\end{array}\right\}=\left\{\begin{array}{c}{k}_{1}\mathrm{sin}({\mathrm{\omega }}_{p}t)\\ 0\end{array}\right\}\)

This problem requires an analytical solution [bib2]:

\(\{\begin{array}{c}{u}_{2}=\mathrm{sin}({\mathrm{\omega }}_{p}t)\\ {u}_{3}=\frac{1}{3}\left({\mathrm{\omega }}_{p}\mathrm{sin}\left(t\right)-\mathrm{sin}({\mathrm{\omega }}_{p}t)\right)\end{array}\)

2.2. Uncertainty about the solution#

None (analytical solution).

2.3. Bibliographical reference#

  1. « On A Composite Implicit Time Integration Procedure For Nonlinear Dynamics » K.J. BATHE, M.M.I. BAIG, Computers and Structures, 83 (2005) 2513—2524

  2. « An efficient backward Euler time-integration method for nonlinear dynamic analysis of structures » T. LIU, C. ZHAO, Q. LI, L. ZHANG, Computers and Structures 106—107 (2012) 20—28