1 Introduction
2 The class of three-point boundary value problems
2.1 Differential operator with boundary and continuity conditions
2.2 Self-adjointness
2.3 Consequences
2.4 Sign of eigenvalues
2.5 Determination of the eigenvalues
3 The Green function matrix
4 Calculation of the Green function matrix
4.1 Structure of the general solution
4.2 Calculation of the Green matrix function if \(\xi \in [a,b]\)
4.3 Calculation of the Green function if \(\xi \in [b,c]\)
5 Numerical examples
5.1 Governing equations of Timoshenko beams with cross-sectional heterogeneity
Boundary conditions | |
\(y_{1}(0)=0,\;\,y_{2}^{(1)}(0)=0\) | \(y_{1}(\ell )=0,\;\,y_{2}^{(1)} (\ell )=0\) |
Continuity conditions | |
\(y_{1}(b-0)=0,\quad y_{1}(b+0)=0\) | |
\(y_{2}(b-0)=y_{2}(b+0)\) | |
\(y_{2}^{(1)}(b-0)=y_{2}^{(1)}(b+0)\) |
5.2 Free vibration of the pinned–pinned beam with intermediate roller support
b
|
\(\sqrt{\lambda _{1}}/\pi ^{2}\)
|
\(\sqrt{\lambda _{2}}/\pi ^{2}\)
|
\(\sqrt{\lambda _{3}}/\pi ^{2}\)
|
\(\sqrt{\lambda _{4}}/\pi ^{2}\)
|
---|---|---|---|---|
0.000 | 1.5621 | 5.0624 | 10.5625 | 18.0625 |
0.025 | 1.6155 | 5.2367 | 10.9278 | 18.6884 |
0.050 | 1.6749 | 5.4359 | 11.3560 | 19.4398 |
0.075 | 1.7393 | 5.6557 | 11.8339 | 20.2840 |
0.100 | 1.8094 | 5.8983 | 12.3645 | 21.2200 |
0.125 | 1.8859 | 6.1657 | 12.9490 | 22.2396 |
0.150 | 1.9694 | 6.4595 | 13.5860 | 23.3168 |
0.175 | 2.0607 | 6.7814 | 14.2689 | 24.3626 |
0.200 | 2.1607 | 7.1322 | 14.9759 | 24.9737 |
0.225 | 2.2704 | 7.5113 | 15.6359 | 23.7298 |
0.250 | 2.3907 | 7.9145 | 15.9991 | 21.3157 |
0.275 | 2.5229 | 8.3287 | 15.4032 | 20.2353 |
0.300 | 2.6679 | 8.7164 | 14.0506 | 20.6523 |
0.325 | 2.8267 | 8.9743 | 12.8510 | 21.6829 |
0.350 | 3.0001 | 8.8963 | 12.2687 | 22.9544 |
0.375 | 3.1878 | 8.4188 | 12.3919 | 24.2493 |
0.400 | 3.3874 | 7.7859 | 12.9584 | 24.9736 |
0.425 | 3.5925 | 7.1860 | 13.7556 | 24.0686 |
0.450 | 3.7874 | 6.6965 | 14.6714 | 22.3577 |
0.475 | 3.9393 | 6.3659 | 15.5541 | 20.8734 |
0.500 | 3.9993 | 6.2462 | 15.9991 | 20.2276 |
b
|
\({\mathfrak {E}}=3.180\)
| |||||||
---|---|---|---|---|---|---|---|---|
\({\mathfrak {r}}=0.03\)
|
\({\mathfrak {r}}=0.05\)
| |||||||
\(\sqrt{\lambda _{i}}/\pi ^{2}\), \((i=1,\ldots ,4)\) | \(\sqrt{\lambda _{i}}/\pi ^{2}\), \((i=1,\ldots ,4)\) | |||||||
0.000 | 1.0595 | 3.8245 | 7.9307 | 12.9426 | 0.9840 | 3.4273 | 6.6691 | 10.274 |
0.025 | 1.3548 | 4.3119 | 8.5654 | 13.7075 | 1.1746 | 3.6959 | 7.0037 | 10.676 |
0.050 | 1.4863 | 4.6301 | 9.0789 | 14.4111 | 1.3038 | 3.9393 | 7.3477 | 11.115 |
0.075 | 1.5803 | 4.8902 | 9.5378 | 15.0777 | 1.4030 | 4.1586 | 7.6834 | 11.562 |
0.100 | 1.6648 | 5.1376 | 9.9903 | 15.7487 | 1.4907 | 4.3702 | 8.0213 | 12.021 |
0.125 | 1.7481 | 5.3890 | 10.4568 | 16.4456 | 1.5746 | 4.5831 | 8.3702 | 12.502 |
0.150 | 1.8340 | 5.6519 | 10.9456 | 17.1697 | 1.6586 | 4.8025 | 8.7336 | 13.004 |
0.175 | 1.9247 | 5.9311 | 11.4604 | 17.9037 | 1.7450 | 5.0322 | 9.1159 | 13.530 |
0.200 | 2.0216 | 6.2291 | 11.9953 | 18.4193 | 1.8357 | 5.2746 | 9.5161 | 13.979 |
0.225 | 2.1261 | 6.5473 | 12.5198 | 17.0047 | 1.9319 | 5.5316 | 9.9271 | 12.514 |
0.250 | 2.2393 | 6.8846 | 12.8418 | 15.1275 | 2.0349 | 5.8036 | 10.2282 | 11.113 |
0.275 | 2.3624 | 7.2343 | 12.2022 | 14.6668 | 2.1456 | 6.0890 | 9.5178 | 11.065 |
0.300 | 2.4966 | 7.5725 | 11.0197 | 15.1996 | 2.2653 | 6.3759 | 8.5537 | 11.552 |
0.325 | 2.6427 | 7.8149 | 10.0766 | 15.9733 | 2.3949 | 6.6050 | 7.8170 | 12.120 |
0.350 | 2.8016 | 7.7323 | 9.6797 | 16.8505 | 2.5352 | 6.5101 | 7.5654 | 12.739 |
0.375 | 2.9734 | 7.2676 | 9.8775 | 17.7789 | 2.6867 | 6.0616 | 7.8147 | 13.408 |
0.400 | 3.1567 | 6.7071 | 10.3662 | 18.4193 | 2.8490 | 5.5855 | 8.2233 | 13.979 |
0.425 | 3.3467 | 6.1954 | 10.9910 | 17.5490 | 3.0193 | 5.1649 | 8.7050 | 13.164 |
0.450 | 3.5305 | 5.7777 | 11.6979 | 16.2697 | 3.1884 | 4.8177 | 9.2406 | 12.230 |
0.475 | 3.6781 | 5.4898 | 12.4221 | 15.1810 | 3.3311 | 4.5695 | 9.8156 | 11.422 |
0.500 | 3.7382 | 5.3831 | 12.8418 | 14.6365 | 3.3925 | 4.4731 | 10.2282 | 10.928 |
b | \({\mathfrak {E}}=2.00\) | |||||||
---|---|---|---|---|---|---|---|---|
\({\mathfrak {r}}=0.03\) | \({\mathfrak {r}}=0.05\) | |||||||
\(\sqrt{\lambda _{i}}/\pi ^{2}\), \((i=1,\ldots ,4)\) | \(\sqrt{\lambda _{i}}/\pi ^{2}\), \((i=1,\ldots ,4)\) | |||||||
0.000 | 1.1003 | 3.9356 | 8.2553 | 13.6827 | 1.01268 | 3.5861 | 7.1615 | 11.2913 |
0.025 | 1.4225 | 4.5172 | 9.0457 | 14.6536 | 1.25312 | 3.9427 | 7.6057 | 11.8182 |
0.050 | 1.5420 | 4.8352 | 9.5946 | 15.4410 | 1.38851 | 4.2251 | 8.0226 | 12.3608 |
0.075 | 1.6291 | 5.0924 | 10.0735 | 16.1662 | 1.48736 | 4.4661 | 8.4121 | 12.8959 |
0.100 | 1.7097 | 5.3407 | 10.5480 | 16.8957 | 1.57434 | 4.6947 | 8.7977 | 13.4383 |
0.125 | 1.7910 | 5.5964 | 11.0408 | 17.6545 | 1.65819 | 4.9242 | 9.1934 | 14.0023 |
0.150 | 1.8761 | 5.8666 | 11.5602 | 18.4419 | 1.74302 | 5.1613 | 9.6049 | 14.5881 |
0.175 | 1.9668 | 6.1555 | 12.1080 | 19.2263 | 1.83124 | 5.4104 | 10.0367 | 15.1916 |
0.200 | 2.0645 | 6.4652 | 12.6748 | 19.7399 | 1.92451 | 5.6740 | 10.4859 | 15.6526 |
0.225 | 2.1703 | 6.7967 | 13.2197 | 18.4612 | 2.02416 | 5.9540 | 10.9345 | 14.2750 |
0.250 | 2.2853 | 7.1479 | 13.5362 | 16.5001 | 2.13138 | 6.2498 | 11.2288 | 12.7153 |
0.275 | 2.4108 | 7.5101 | 12.9486 | 15.8781 | 2.24728 | 6.5573 | 10.6001 | 12.4637 |
0.300 | 2.5478 | 7.8556 | 11.7534 | 16.3661 | 2.37289 | 6.8588 | 9.5679 | 12.9596 |
0.325 | 2.6972 | 8.0956 | 10.7653 | 17.1809 | 2.50919 | 7.0831 | 8.7652 | 13.6014 |
0.350 | 2.8597 | 8.0177 | 10.3249 | 18.1259 | 2.65691 | 7.0014 | 8.4502 | 14.3137 |
0.375 | 3.0353 | 7.5631 | 10.4916 | 19.1120 | 2.81629 | 6.5666 | 8.6593 | 15.0762 |
0.400 | 3.2224 | 6.9924 | 10.9879 | 19.7399 | 2.98630 | 6.0655 | 9.0896 | 15.6526 |
0.425 | 3.4154 | 6.4634 | 11.6431 | 18.9095 | 3.16304 | 5.6137 | 9.6202 | 14.8497 |
0.450 | 3.6006 | 6.0313 | 12.3874 | 17.5642 | 3.33542 | 5.2430 | 10.2156 | 13.7924 |
0.475 | 3.7474 | 5.7357 | 13.1324 | 16.4155 | 3.47609 | 4.9839 | 10.8381 | 12.8887 |
0.500 | 3.8064 | 5.6271 | 13.5362 | 15.8728 | 3.53435 | 4.8862 | 11.2288 | 12.4014 |
b | \({\mathfrak {E}}=1.00\) | |||||||
---|---|---|---|---|---|---|---|---|
\({\mathfrak {r}}=0.03\) | \({\mathfrak {r}}=0.05\) | |||||||
\(\sqrt{\lambda _{i}}/\pi ^{2}\), \((i=1,\ldots ,4)\) | \(\sqrt{\lambda _{i}}/\pi ^{2}\), \((i=1,\ldots ,4)\) | |||||||
0.000 | 1.1792 | 4.0958 | 8.6240 | 14.4666 | 1.0626 | 3.7628 | 7.6813 | 12.3840 |
0.025 | 1.4999 | 4.7620 | 9.6049 | 15.7288 | 1.3648 | 4.2644 | 8.3275 | 13.1495 |
0.050 | 1.5984 | 5.0520 | 10.1478 | 16.5578 | 1.4917 | 4.5727 | 8.8234 | 13.8258 |
0.075 | 1.6761 | 5.2956 | 10.6261 | 17.3146 | 1.5822 | 4.8213 | 9.2574 | 14.4519 |
0.100 | 1.7520 | 5.5395 | 11.1119 | 18.0879 | 1.6641 | 5.0573 | 9.6825 | 15.0770 |
0.125 | 1.8308 | 5.7964 | 11.6243 | 18.8980 | 1.7453 | 5.2969 | 10.1183 | 15.7197 |
0.150 | 1.9148 | 6.0718 | 12.1694 | 19.7392 | 1.8293 | 5.5474 | 10.5732 | 16.3823 |
0.175 | 2.0053 | 6.3686 | 12.7459 | 20.5598 | 1.9181 | 5.8129 | 11.0490 | 17.0387 |
0.200 | 2.1034 | 6.6886 | 13.3398 | 21.0607 | 2.0130 | 6.0955 | 11.5386 | 17.4702 |
0.225 | 2.2103 | 7.0319 | 13.8987 | 19.9450 | 2.1154 | 6.3961 | 12.0073 | 16.3884 |
0.250 | 2.3269 | 7.3952 | 14.2072 | 17.9641 | 2.2263 | 6.7129 | 12.2797 | 14.7490 |
0.275 | 2.4544 | 7.7681 | 13.6708 | 17.1863 | 2.3468 | 7.0383 | 11.7719 | 14.2354 |
0.300 | 2.5938 | 8.1190 | 12.4862 | 17.6009 | 2.4779 | 7.3477 | 10.7415 | 14.6531 |
0.325 | 2.7461 | 8.3560 | 11.4622 | 18.4380 | 2.6204 | 7.5624 | 9.8855 | 15.3367 |
0.350 | 2.9118 | 8.2823 | 10.9807 | 19.4367 | 2.7750 | 7.4926 | 9.5043 | 16.1260 |
0.375 | 3.0909 | 7.8388 | 11.1145 | 20.4593 | 2.9415 | 7.0851 | 9.6525 | 16.9460 |
0.400 | 3.2812 | 7.2618 | 11.6129 | 21.0607 | 3.1183 | 6.5727 | 10.0855 | 17.4702 |
0.425 | 3.4768 | 6.7183 | 12.2926 | 20.2897 | 3.3003 | 6.0957 | 10.6527 | 16.7754 |
0.450 | 3.6631 | 6.2734 | 13.0667 | 18.9147 | 3.4746 | 5.7041 | 11.2938 | 15.6547 |
0.475 | 3.8091 | 5.9707 | 13.8218 | 17.7249 | 3.6126 | 5.4348 | 11.9330 | 14.6918 |
0.500 | 3.8672 | 5.8604 | 14.2072 | 17.1890 | 3.6680 | 5.3356 | 12.2797 | 14.2351 |