Table 2 Comparison between theoretical solution and numerical LCP solution.
From: An Improved Linear Complementarity Solver for the Dynamic Analysis of Blast Loaded Structures
Theoretical solution | Numerical Lumped mass | Error Lumped mass (%) | Numerical Continuous mass | Error Continuous mass (%) | |
|---|---|---|---|---|---|
\({P}_{0}=2.5{P}_{c}\) | |||||
Hinge position \(\overline{t }=0\) (Fig. 12) \({\xi }_{1}^{*}/L\) | 0.54265 | 0.54000 | 0.49 | 0.54500 | − 0.4331 |
Hinge position \(\overline{t }=0\) (Fig. 12) \({\xi }_{2}^{*}/L\) | 0.61629 | 0.62000 | − 0.60 | 0.61500 | 0.2093 |
Hinge position at the end of Phase 1 \({\xi }_{*}/L\) | 0.58579 | 0.59000 | − 0.72 | 0.59000 | − 0.7187 |
Displacement at the end of Phase 1 \({\bar{W}}_{1}=({W}_{1}/L).(mL){M}_{p}/{I}^{2}\) | 0.05224 | 0.05205 | 0.35 | 0.05292 | − 1.3017 |
Time when hinges coalesce \({\bar{t}}_{1}={M}_{p}{T}_{1}/IL\) | 0.05300 | 0.05308 | − 0.14 | 0.05339 | − 0.7358 |
Displacement at the end of motion \({\bar{W}}_{3}=({W}_{3}/L).(mL){M}_{p}/{I}^{2}\) | 0.63318 | 0.62575 | 1.17 | 0.63128 | 0.3001 |
Cessation time \(\overline{t }={M}_{p}T/IL\) | 0.34272 | 0.34005 | 0.78 | 0.34394 | − 0.3560 |
\({P}_{0}=12.5{P}_{c}\) | |||||
Hinge position \(\overline{t }=0\) (Fig. 12) \({\xi }_{1}^{*}/L\) | 0.28706 | 0.28500 | 0.72 | 0.28500 | 0.7176 |
Hinge position \(\overline{t }=0\) (Fig. 12) \({\xi }_{2}^{*}/L\) | 0.79702 | 0.79500 | 0.25 | 0.79500 | 0.2534 |
Hinge position at the end of Phase 1 \({\xi }_{*}/L\) | 0.58579 | 0.59000 | − 0.72 | 0.59000 | − 0.7187 |
Displacement when the pulse terminates \({\bar{W}}_{1}=({W}_{1}/L).(mL){M}_{p}/{I}^{2}\) | 0.14686 | 0.14689 | − 0.02 | 0.14666 | 0.1362 |
Displacement at the end of Phase 1 \({\bar{W}}_{2}=({W}_{2}/L).(mL){M}_{p}/{I}^{2}\) | 0.38234 | 0.38247 | − 0.03 | 0.38145 | 0.2328 |
Time when hinges coalesce \({\bar{t}}_{1}={M}_{p}{T}_{1}/IL\) | 0.11372 | 0.11214 | 1.39 | 0.11339 | 0.2902 |
Displacement at the end of motion \({\bar{W}}_{3}=({W}_{3}/L).(mL){M}_{p}/{I}^{2}\) | 0.83904 | 0.84450 | − 0.65 | 0.83042 | 1.0274 |
Cessation time \(\overline{t }={M}_{p}T/IL\) | 0.34248 | 0.34439 | − 0.56 | 0.34323 | − 0.2190 |