APPENDIX A
APPENDIX A SUPPORTING DATA FOR OPERATIONAL PROFILES This section contains data taken from the computer data generated in this study. These data are the maximum axial load in the strut, the maximum horizontal load, the maxi- mum stroke, and the minimum stability angle for each of the runs made. A table of these data is presented f o r each of the conditions studied. The operational profiles were plotted from these tables.
Table A-1. Summary Table of Critical Values for Series 1 Max. Horiz. Max. Stroke Max. Vert.
v V vH 3 (1) 1478 (1) 829 (1) ,860 1 4 1893 1186 ,943 1 5 23 24 1568 ,991 5 1601 1.154 2 2609 4 1889 1201 1.076 3 4 2278 1213 1.100 3 3 1865 837 .961 4 3 1979 880 1.014 4 4 2378 1229 1.124 5 4 2586 1283 1.152 5 3 209 1 924 1.069 3 964 1.111 6 2202 6 4 29 60 1334 Bottomed 7 4 3 285 1380 Bottomed 7 3 2307 1000 1.137 7 2 1800 644 1.037 8 1990 664 1.068 8 3 2460 1030 1.156 9 3 2647 1057 Bottomed 9 23 23 67 3 1.093 9 1 2101 329 .967 10 1 2383 337 .994 10 2261 0 .830 All values a r e for leg No. 1 as indicated by the (1) next to the f i r s t row o f numbers.
Table A-2. Summary Table of Critical Values for Series 2 Max. Vertical Max. Horizontal Max. Stroke Min. Beta
z-l--l-
The leg number for the vah ! given is in parentheses.
1110 (1) 1344 (1) 1 4 .788 (1) 1323 1730 1 5 .889 1329 1740 2 5 .9 27 4 ,827 1090 1647 1230 3 4 .834 1627 1675 3 5 .944 1.025 1753 2137 3 6 20 50 4 6 1.032 2300 2046 1568 4 5 .947 2269 (@ .084) 1513 5 5 .980 5 4 .946 1971 1180 6 4 1.039 .753 (1) 2081 1245 6 5 1.058 ,735 23 25 ,744 2446 1655 7 5 1.114 7 4 2221 1295 1.079 .758 (2) 8 4 1.106 .742 2334 1338 8 3 1.049 .752 (1) 2154 9 80 2478 1008 9 3 1.078 .752 9 2 1.014 ,755 2308 9 1 .923 .754 (2) 2136 3 23 10 1 .956 .7 56 2426 33 2 10 0 0 .84 .773 2337 Table A-3. Summary Table of Critical Values for Series 3 Max. Vertical Max. Horizontal
Min. Beta 7-- Max. Stroke
I
i
.eg nurnbdr for the value iven is in parentheses.
The 544 (1) 0 8 .320 (1) .7505 (1) 460 (1) 544 460 0 9 .320 .7502 .472 .7422 6 85 555 1 9 9 .582 ,7409 945 73 1 9 .663 .73 76 1401 1072 8 .670 .7368 1427 1120 1465 1173 3 7 ,678 .73 50 7 .7289 2642 1769 4 .794 4 6 .810 .7325 2189 1684 5 6 .899 .7371 2337 1878 1663 5 5 .913 .7378 5 .976 .7449 (1) 20 20 1799 6 4 .978 .7420 (2) 1935 1446 4 2178 1461 7 1.017 .7430 5 .7438 2253 190 5 7 1.027 8 5 2545 1965 1.064 .7408 8 4 1.051 .7475 (2) 2404 147.8 8 3 1.032 .7550 (1) 2216 1024 9 3 1.052 .7550 (2) 2479 9 2 .7532 (1) 1.014 23 16 664 9 1 .924 .7545 (2) 2137 3 26 10 1 .9 56 .7550 (2) 2486 334
-
10 0 .842 .7637 (2) 2406 Table A-4. Summary Table of Critical Values for Series 4 Max. Stroke Min. Beta Max. Vertical Max. Horizontal vV vH The leg numbc civen is in pare: :heses.
Toppled 1137 (1) 1527 (1) 0 11 .892 (1) 0 10 .870 .5366 (1) 1535 1440 0 8 .822 ,6323 (1) 1021 1282 0 9 .840 .6096 (1) 1041 1334 Toppled 1748 1 9 ,855 1366 8 Toppled 1282 1677 1 .943 1 7 .924 .524 (1) 1220 1596 2 7 .997 Toppled 1459 1909 6 1.009 .5481 (1) 1564 2 2192 2 5 .942 .6313 1380 1834 2 4 .844 .69 29 1136 1404 3 .872 .7126 1341 1389 5 .976 .6637 1882 3 1415 4 5 .998 .6897 1750 1883 4 4 .894 .7284 1695 1360 5 4 .929 .7421 1869 1355 5 1.018 ,7124 1989 1889 6 5 1.036 .7318 (1) 2134 1904 6 4 ,975 .7453 (2) 2037 1371 7 4 1.017 .7488 (2) 2227 1391 5 1.055 .7211 (2) 2365 1916 5 1.075 ,7152 (2) 2579 1927 8 4 1.053 .7526 (2) 1416 3 1.026 .7489 (1) 2216 995 3 1.057 .7526 (1) 2479 1020 9 2 1.014 .7531 (1) 664 9 1 .924 ,7545 (2) 3 26 1 .956 .7551 (2) 2486 334 10 0 .7637 (2)
. a42 2406
--
Table A-5. Summary Table of Critical Values for Series 6 Min. Beta Max. Vertical Max. Horizontal Max. Stroke for the value given i s in parentheses.
The le{ number .6585 (2) 5.00 1807.9 (1) 1587.2 (1) .948 (1) .50 1.068 6.00 1731.8 2436.0 .6118 .55 1.077 1.50 6.00 1759.7 2512.2 .6010 1.020 ~ 5 8 6 1.40 5.00 1552.8 2128.7 1.30 1507.7 1591.4 .93 1 .65a4 4.00 4.00 1651.2 1595.7 .939 .65a5 2.20 5.00 1918.6 2073.0 1.020 ~ 5 8 7 2.40 .979 3.55 4.50 2705.6 1599.7 3.20 3.60 2427.7 1219.3 .922 .65a6 2.85 2.70 1952.3 855.0 .863 . m a 4
2.25 2121.1 57 2.7 . a47 .6584
3.50 .914 4.00 3 .OO 2779.3 787.6
588.3 . 886 m a 6
4.60 2.40 2728.4 3.95 1.80 2157.5 624.3 .a29 ~ 5 8 4 4.70 1.50 2349.0 670.7 .a33 m a 4 m a 2
3.90 1 .oo 1625.1 721.7 .770
4.60 1656.7 758.0 .772 .65a3 .70 5.50 1.05 2278.6 714.4 .a23 .e584 6.70 .45 2359.4 831.2 .ai7 .65a5 6.00 1849.4 .7a6 .30 842.9
1846. a -957.5 (2) .a08 (2) ,6583
6.40 0.
7.10 0. 2342.4 -997.9 .a52 .6584
6.70 - .45 1806.3 -1172.6 . a74 .65a2
7.40 - .60 2079.6 - 1280.1 m a 2
.927 8.10 -.75 2386.4 - 1388.3 .973 . m a 3
7.30 -1.75 1959.0 (2) - 1843.8 1.011 .6579
6.40 - 1.40 1706.6 -1597.7 .941 .6579
5.50 1474.7
- 2.00 - 1756.3 .943 .6576
4.70 -1.50 1214.6 - 1416.2 .a55
.6576
3.95 - 1.80 1141.1 - 1407.6 . a20 .6575
4.60 - 2.40 1349.8 (2) - 1762.7 ,926 .6573
4.00 -3.00 1433.2 (1) - 1885.9 .949 .6571
3.50
- 2.25 1197.5 (2) - 1496.3 .a45 .6573
2.85 - 2.70 1230.1 - 1534.7 .84a
3.20 -3.60 1447.9 - 1943.0 .950
.6351
2.20 - 4.00 1453.9 - 1854.9 .go8 .5a16
2.00
-3.00 1262.8 - 1465.5 .a09 ,6243
1.20 -3.00 1206.0
- 1310.8 .733 .5979
1.30 - 4.00 1423.2 - 1648.1
.832 .0014
.45 - 4.00 2051.4 - 1025.2 .809 .em8
.50 - 5.00 2090.6 -2119.2 .967 .6529 .
Table A-6. Summary Table of Critical Values for Series 7 Max. Vertical Min. Beta Max. Horizonta: vV vH I The g numb1 for the value .ven is in parentheses.
0 4 .013 (1) .755 (1) 322 (1) 388 (1) 0 .016 .751 480 390 0 6 .774 .202 (2) 1474 1800 1 6 ,887 .011 1436 20 16 1 5 .803 .372 1242 1636 2 5 .858 .437 1220 1648 2 6 .941 .118 1435 20 50 3 6 .97 1 .271 1523 3 .886 ,514 1487 1617 4 5 .902 .579 1980 1518 4 6 ,987 .399 2277 @ .090 1972 6 1.002 .482 2321 5 5 .939 .636 2082 1476 6 5 1.007 .654 2267 1542 6 4 .981 .732 200 4 7 4 1.044 .724 206 4 1258 7 5 1.069 .676 2341 1607 5 1.117 .652 2458 1665 8 4 1.076 .679 2264 @ .064 1302 3 1,001 ,688 2040 3 1.034 .686 2352 9 82 9 2 ,965 .729 2183 645 2 ,995 .728 2433 662 10 1 .908 .758 (1) 2337 3 25 10 0 ,788 .773 2297 , - ' .
Table A-7. Summary Table of Critical Values for Series 9 Max. Horizontal Max. Stroke Max. Vertical Min. Beta
vv I vH
' number for the value giy ,n is in parentheses.
The 11 5.00 1522.2 (1) 1691.8 1) ,802 (1) .3173 (2) .50 6.00 1483.0 2063.9 .886 .OO 15 .55 1546.5 2260.5 .953 .0005 1.50 6.00 1430.2 1856.6 ,872 .3439 1.40 5.00 1.30 4.00 1145.4 1467.3 .767 .5446 1147.4 1524.1 .5693 2.20 4.00 .822 1374.3 1926.0 .40 20 2.40 5.00 .925 4.50 1587 .O 1549.2 .888 ,5808 3.55 3.90 5.40 1877.7 1917.5 ,975 .463 7 5.00 4.50 1930.7 1342.9 .919 .6521 5.50 5.25 2202.4 1640.6 .988 .5903 2513.8 1946.6 1.047 .5059 6.00 6.00 4.80 2297.9 1579.3 1.085 .67 88 7.20 6.55 4.20 2057.7 1332.6 1.03 5 .7175 2104.3 .6836 7.90 3.50 1151.9 1.049 8.70 4.00 2368.9 1359.9 1.104 .6596 0.00 2.80 2720.9 964.8 1.062 .6792 9.10 2.45 2378.7 822.4 1.014 .6879 8.20 2.10 2044.9 685.8 .959 .7056 8.80 2262.3 .869 .7471 .90 299.4 8.10 .7 5 2001.3 247.3 .824 .7486 8.50 0. 21 22.2 -34.7 ,742 .7633 (1) 2399.7 9.20 0. -32.6 .772 .763 4 8.80 - .90 2248.3 -271.3 (2) ,839 (2) .7422 (1)
9.50 - 1.05 2501.5 -325.7 .883 .7562 (2)
9.10 - 2.45 2339.6 -783.6 .988 .7106 (1) 8.20 - 2.10 2011.1 (2) -647.3 .928 .7348 7.10 .947 - 3 .OO 1993.0 -905.2 .6966 7.90 -3.50 2246.5 -1101.9 1.019 .6900 8.70 - 4.00 2513.7 -1307.7 1.082 .6632
7.20 - 4.80 2421.2 - 1518.6 1.07 1 .6935
6.55 -4.20 2226.2 -1274.8 1.012 .7318
5.50 - 5.25 2428.1 - 1558.9 .968 .6273
5.00 -4.50 2166.2 - 1279.3 .902 .6858
3.90 - 5.40 2299.0 -1584.1 .906 ,5426
3.55 -4.50 1985.8
- 1217.0 .807 .6416
2.40 - 5.00 1494.9 - 1440.7 .819 .5218
2.60 - 6.00 1662.0 - 1851.3 .919 .3194
1.50 - 6.00
1656.0 - 1725.9 .873 .2963
1.40 - 5.00 1049.7 -1345.0 .773 .503 5
.50 - 5.00 1104.5 - 1423.7
.745 .4567
.55 - 6.00 1686.7 - 1770.0
.840 .229 5 Table A-8. Summary Table of Critical Values for Series 11 ~ ~~~ Max. Horizontal Max. Stroke Min. Beta Max. Vertical
vv vH
~~ The le1 number 'or the value gj 1869.0 (1) .969 (1) .6783 (1)
.50 5 .oo 1406.8 (1)
1153.8 1438.9 .874 .7106 (2)
.4 5 4 .oo
.7107 (2)
4 .oo 1285.8 1660.5 .go9
1.30 .6840 (1) 5 .OO 1503 . O 2052.7 .996 1.40 1.002 .7035 (1)
2.40 5 .oo 1566.4 2007.7
1498.8 .905 .7109 (2) 2.20 4 .OO 1389.0 2038.0 993.5 .844 .7109 3.20 3.60 .7111 3.55 4.50 2452.8 1338.1 .918 .830 .7111 4.50 3.75 2548.1 915.1 (2) 2223.2 700.2 ,806 .7109
4 .oo 3 .oo
4.60 2.40 2288.7 577.9 .859 .7109 3.95 1.80 1864.4 403.9 .782 .7107 343.1 (1) .797 .7108 4.70 1.50 2050.7
5.50 2 .oo 2477.8 '496.0 (2) .871 .7110
6.40 1.40 2501.1 349.2 (1) .854 .7110 .7108 5.50 1.05 2118.2 326.8 .789 6.70 .4 5 2449.7 331.7 .777 ,7109 6 .OO .30 2059.2 -361.1 (2) .734 .7108 .761 2) .7108 6.40 0. 2113.2 -463.9 7.10 0. 2413.7 -472.2 ,802 .7109
6.70 - .45 2130.6 - 634.9 ,7108
.829
7.40 - .60 2403.2 - 704.3 ,882 .7108
6.40 - 1.40 1769.4 - 1034.4 .912 .7106
7.30 - 1.75 -1212.1 .7106
2089.3 .987 8.20 -2.10 2461.6 -1389.8 1.049 .7107
7.10 - 3 .OO 2093.1 (2) -1811.0 1.060 .7104
6.30 -2.50 1867.0 -1545.4 .997 .7104
5.25 - 3 .OO 1822.2 - 1762.2
,973 .7102
4.60 -2.40 1578.3 - 1453 . O .894 .7102
4 .OO
- 3 .OO 1548.5 - 1712 . O .92 1 .7 100
3.50 -2.25
1330.9 - 1329.9 .802 .7101
2.85 -2.70 1175.9 -1428.7 .812 .7099 3.20 -3.60 1403.4 -1850.1 .929 .7060 2.20 -4.00 1414.0 -1814.0 .6631 ,903
2 .oo
-3.00 1216.8 -1398.3 .786 .6954 1.20
- 3 -00 1173.8 - 1267.7 .718 .6725
1.30 -4.00 1401.7 -1635.1 .834 .6300
1.40 - 5 .OO 1594.6 -2022.7 .922 .5763
.50 -5.00 1656.2 - 1961.5 .911 .6682
.4 5 -4 .OO 1701.9 -934.8 .733 ,7026 4 5 Table A-9. Summary Table of Critical Values for Series 12 Max. Vertical [ a x . Horizontal Max. Stroke Min. Beta vH vV The 1( 1606 (1) 2250 1) .5 5 1.032 (1) .591 (1) 4 .949 .631 1341 1753 .45 3 .834 .665 1262 .4 1334 1336 3 .847 .669 1.2 1424 1904 4 .964 .647 1.3 .658 1399 1861 2.2 4 .968 .857 .674 1309 1304 2 3 1330 948 ,817 .6a9 2.85 2.7 1753 1316 3.6 .899 .673 3.2 .651 2270 @ .lo8 17 27 3.55 4.5 .985 .796 .689 2583 1123 4.5 3.75 4 3 .760 .693 1943 878 (1) 2308 63 .lo2 696 (2) 2.4 .886 .706 4.6 .705 1788 5 19 3.95 1.8 .826 .847 .706 1776 4 80 4.7 1.5 2472 @ .092 644 2 ,914 .704 5.5 .704 2395 63.084 495 (2) 6.4 1.4 .go9 .850 .703 2081 414 (1) 5.5 1.05 .45 .847 .700 2551 63 .084 40 5 6.7
6 .3 .aoa .701 40 1
1816 4 10 6.4 0 .796 .700 .700 2463 406 7.1 0 .820 (1) - .45 6.7 .765 (2) 1751 46 5 .700
- .6
7.14 .819 2515 (1) 514 .700
- 1.4
6.4 .854 .699 1621 (2) 828
- 1.75
7.3 .928 2116 (1) 97 5 .699 - 2.1 8.2 .998 1128 .700 2454 63 .072 (1 -3 7.1 1.029 2005 (2) 1555 .680 -3.5 7.9 1.084 2234 1793 .648 -4.2 6.55 1.073 2205 2140 .609 -3.6 5.9 1.013 1839 .652 1964 - 3 5.25 .945 1540 .ea4 1729 -3.75 4.5 .965 1739 1899 .643 - 3 .868 1506 1524 .685 -3.6 3.2 .892 1329 1716 .672
- 2.7
2.85 .775 1300 ,693 1100 - 3 .746 1155 1296 .697 - 4 2.2 .875 1340 1716 .660 - 4 1.3 ,813 1315 1560 .670 - 5 1.4 .911 1483 1960 .616 - 5 .5 .841 .697 1494 1793 - 4 .45 .743 .704 1293 1434
~- - -
.
Table A-10. Summary of Critical Values for Series 13 Max. Stroke Min. Beta Max. Vertical Max. Horizontal vH - 1344 (1) 1882 (1) .414 (2) 0 5 ,897 (1) 1153 1525 ,613 0 4 ,804 1157 1521 ,622 1 4 ,787 1351 1876 1 5 .E84 .404 1329 5 .906 .452 (2) 1125 4 .E18 .614 (1) 1393 1523 3 4 .a45 .646 (1) 1525 1880 3 5 .935 .533 (2) 2346 (3 .lo4 1857 4 5 .954 .594 (2) 2304 (3 .lo6 4 4 .a37 .655 (1) 1836 4 828 (1) 3 ,700 .681 2352 (3 .098 5 773 (2) 3 .757 .699 2200 5 524 .769 .707 2451 (3 .090 585 (2) .704 6 2 .828 1921 344 (1) .703 6 1 .747 2626 (3 .082 3 67 .703 7 1 .801 2297 (3 .084 7 335 0 .694 .701 2665 (3.074 338 8 0 .744 .703 2616 9 340 (1) 0 ,787 (1) .704 2574 (3 .066 9 661 (2) -1 .885 (2) .703 243 1 8 668 -1 .E42 .704 1570 7 670 -1 .789 .689 1649 (2) 1078 .665 7 - 2 .a97 1085 .668 2339 (1) @ .074 8 - 2 .943 1077 (2) 0 1294 - 5 .766 .481 1211 0 1635 -6 .854 .307 0 -7 .947 1440 207 2 Toppled 1770 1 2500 -7 .935 Toppled 1618 1 2139 -6 .a73 Toppled 1424 1 1780 - 5 .797 .303 (1) 1220 1 1435 -4 .697 .482 1265 2 1584 -4 .784 .518 1445 2 - 5 .877 .351 1510 3 - 5 .439 2143 .943 3 -4 .E56 .548 1300 1742 1096 3 1357 - 3 ,739 .620 1308 1489 -3 .813 .581 1390 1912 -4 .914 .417 5 200 1 -4 .964 .396 1555 5 -3 .a59 .624 1713 6 -3 .921 1534 .637 6 -4 ,988 1926 2017 .415 7 -4 2103 20 19 1.033 ,595 7 -3 1858 1541 .975 .639 2056 (1) 1538 -3 1.018 .638 8 -4 2215 (2) 203 1 1.071 .592 2359 (2) -4 1.103 .593 9 -3 1.054 .637 2605 (1) @ .068 1538 9 - 2 .679 2307 (1) I 1079 .982 Table A-11. Summary Table of Critical Values for Series 14 Beta(Min.)
FS (Max.) FH(Max.) S(Max.)
vH vV ~~ 1 I I lumber for the value given is in parentheses.
The lec .50 5.00 1953.4 (1) 1623.8 (1) .974 (1) .6530 (1) 1.086 .6096 a55 6.00 2215.5 2370.3 6.00 2014.7 2873.2 1.119 .5711 1.50 5.00 1777.1 2502.0 1.082 .6405 (1) 1.40 1.30 4.00 1716.4 1892.3 1.008 ,6584 (2) 1605.1 .6582 1.20 3 .OO 1196.7 .933 1660.3 1417.9 .943 .6583 2.00 3 .OO 2.20 4.00 1882.6 2033.7 1.021 .6585 3.20 3.60 2520.5 1673.2 .992 .6586 2153.3 1209.1 .942 .6584 2.85 2.70 3.50 2.25 2332.8 885.9 .932 .6584 3.00 1.50 1879.0 653.0 .871 .6582 3.30 1.20 1615.2 645.5 .869 .6582 2187.7 .919 .6584 3.95 ' 1.80 736.3 4.60 2712.5 ,6586 2.40 832 . 6 ,966 5.50 2.00 2824.0 1054.8 .969 .6586 4.70 1.50 2150.6 1092.2 .930 .6584 5.50 1.05 2103.2 .6584 1177.2 ,929 6.40 1.40 2916.1 1173.4 .964 .6586 7.40 .60 2573.2 1345.8 .957 .6586 6.70 .45 2033.2 1322.6 .936 .6585 7.10 1873.2 1461.1 0 . ,939 .6.584 7.80 0. 2195.7 1521.3 .953 .6585 8.50 0 . 2635.4 (1) 1570.6 .963 (1) .6585
- .7 5
8.10 1906.5 (2) 1760.3 .979 (2) .6583
7.40 - .60 1694.4 (2) 1645.8 .942 (1)
.6582
- 1.40
6.40 1453.9 (1) -1669.0 (2) .936 (2) .6579 2 )
- 1.75 1607.9 (2) - 1970.0
7.30 1.012 .6468 1 )
6.30 - 2.50 1635.3 (1) .6091 (1)
- 2137.6 1.012
5.50 - 2.00 1456.2 - 1773.6 .6576 (2)
.93 1
4.70 - 1 . 5 0 1327.0 - 1432.7 .831 .6576
- 1.80 1319.6
3.95 - 1414.6 .822 .6575
1736.4
4.60 - 2.40 - 1766.8 .926 .6574
2 )
4.00 -3.00 1572.3 - 1874.9
.946 .6483 1 )
3.50 - 2.25 1587.4 (1) ,844
- 1493.3 .6573
2)
2.85 - 2.70 1223.6 (2) - 1519.3 .842 .6568
3.20 -3.60 1431.8 (2) - 1914.1 ,942 .63 54
- 4.00 1594.7 (1)
2.20 - 1811.2 .895 .5827
2 . 0 0 - 3 .OO 1248.9 (2)
- 1440.0 .799 (2) .6246
1.20 -3.00 1427.0 (1) -1285.5 .775 (1) .5985
1.30 - 4.00 1400.1 (2) - 1602.7
.814 (2) .0067
.45 - 4.00
1877.4 -922.6 .810 .6529
.50 - 5.00 1551.3
- 1802.9 .838 .0074
.
Table A-12. Summary Table of Critical Values for Series 15 I I I Beta (Min.)
S(Max.)
I I I ;es.
The leg number for the value given is in parenthe 1956.6 (1) .917 (1) ,4363 (2) 5.00 1396.3 (1) .50 1616.7 .836 .6240 4.00 1211.6 .45 1511.8 .820 .5907 1.30 4.00 1137.5 .4293 5.00 1326.0 1843.8 .908 1.40 1838.6 .926 .4879 2.40 5.00 1485.7 1498.9 .832 .6197 2.20 4.00 1293.8 1873.4 1388.2 .804 .6586 3.20 3.60 .6316 4.50 2422.7 1683.9 .902 3.55 .852 .6588 4.50 3.75 3000.0 1176.9 .783 .6586 4.00 3.00 2304.3 953.4 1807.0 739.0 .708 .6584 3.50 2.25 .6584 1785.8 504.1 .712 3.95 1.80 .6586 2340.0 581.6 .782 4.60 2.40 597.7 ,791 .6586 5.50 2.00 2508.2 1927.1 613.7 .715 ,6584 4.70 1.50 ,6584 1.05 2029.3 690.6 .717 5 . 5 0 ,785 .6586 6.40 1.40 2597.5 673.0 791.7 .761 .6586 7.40 .60 2472.5 2072.0 799.3 .723 (1) ,6585 6.70 .45 7.10 1965.0 906.3 (1) .747 (2) .6584 0 .
.800 ,6585 7.80 0 . 23 1 6 . 0 -939.5 (2)
1778.8 - 1168.0 .833 .6582
7.40 - .60
8.10 -.75 2141.7 - 1281.6 .890 .6583
8.80 -.go 2505.3 (1) - 1391.0 ,941 .6500
- 1939.0 1.001 .4791
8.20 - 2.10 1951.0 (2)
7.30 - 1685.2 .926 .5630
- 1.75 1677.4
- 1876.4 ,5179
6.30 - 2.50 1363.7 ,936
5.50 - 2.00 1157.8 - 1526.0 .825 .6481
4.60 - 2 . 4 0 1160.5 (2) - 1520.6 .814 .6573
.6031
5.25 -3.00 1420.7 (1) - 1883.0 .918
.938 .4347
4.50 -3.75 1444.0 (2) - 2024.0
4.00 -3.00 1229.2 - 1631.3 .837 .6071
3.20 -3.60 1288.5 - 1685.0 ,833 .5646
3.55 .4464
- 4.50 1477.5 - 2090.2 .93 1
2 . 4 0 - 5.00 1519.6 -1993.2 .875 ,3253
2.20 - 4.00 1326.2 -1614.1 .780 ,5085
1.30
- 4.00 1243.5 - 1446.5 .689 .5385
1.40 - 1771.9
- 5.00 1473.2 .782 ,3949
.50 - 5.00 1530.1 - 1200.8 ,763 ,5769
.55 - 6.00 1638.6 - 1659.9 .890 .3899
.60 -7.00 1664.0 - 2092.9 .967 .0012
, Table A-13. Summary Table o f Critical Values for Series A Min. Beta Max. Vertical Max. Horizontal
- 1 Max. Stroke
I I I 1 I I I The leg numb6r for the value given i s in parentheses.
7 1.120 (1) Toppled 2076 (1) 3249 (1) 0.6 6 1.068 .61 ( 2 ) 1743 243 2 .55 2 . 6 6 1.081 .61 ( 2 ) 2090 2469 7 Toppled 2 . 8 1.122 2300 2857 1.099 .60 ( 2 ) 3824 2496 4.25 6 . 3 .66 3 268 2031 ( 1 ) 3.9 5 . 4 1.044 5.25 3 .66 3292 630 (2) .94 5 . 9 3 . 6 .992 .66 3972 814 ( 2 ) 7.3 1.75 .922 .66 3584 684 (1) 6.4 1 . 4 .873 .66 2838 687 (1) 0 ,892 .66 2718 -1033 ( 2 ) 7 . 8
8.5 0 .928 .66 3129 - 1062
9.2 0 .960 .66 3572 - 1092
-9 1.014 .66 2763 - 1499
8.8 10.2 -1.2 1.081 .66 3677 -1717
9.5 - 1 . 0 5 1.051 .66 3136 ( 1 ) - 1606
- 2 . 4 5 1.115 2 ) 9 . 1 .66 2456 ( 2 ) - 2349
10 -2.8 1.151 .62 ( 2 ) 2753 (1) - 2605
6.55 -4.2 1.122 .53 ( 1 ) 2070 ( 2 ) -3082 -3.6 5 . 9 1.075 .65 tl) 1770 - 2544 -5 2 . 4 .984 .52 ( 2 ) 1618 -2257
-6 1.042 3487 - 2664
2 . 6 .47 ( 2 ) Table A-14. Summary Table o f Critical Values for Series B Min. Beta Max. Vertical Max. Horizontal
-Vv 1 VH I Max. Stroke
I I The leg number for the value iven is in parentheses.
7 1.156 (1) Toppled 2561 1) 4174 (1) . 6 6 1.104 .61 (1) 2126 2600 .55 1946 1646 .5 5 .976 .65 .63 2088 2414 2 . 4 5 1.077 6 1.116 .57 (1) 2290 2770 2 . 6 3 . 6 4 . 5 1.052 .66 (2) 3092 2173 1.099 (1) 3482 2538 3 . 9 5 . 4 .64 .968 .66 (2) 2824 836 4 . 6 2 . 4 5.25 1.011 .66 (2) 3551 951 3 .O 7 . 3 1 . 8 .998 .66 3767 1169 .963 .66 2918 1172 6.4 1 . 4 0 .964 .66 270 5 1569 8 . 5 9.2 0 .972 (1) .66 3148 1610 9.9 0 .996 (2) .66 3621 1647
9.5 - 1.05 1.056 .66 (2) 2586
10.2 -1.2 1.085 .66 (1) 2976 2038 (1) 10.9 -1.35 1,110 .64 (1) 3432 (1) -2162 (2) 10 -2.8 1.143 .54 (2) 2559 (2) -2912
9 . 1 - 2.45 1.110 .60 (1) 2259 - 2608
-2.1 1.069 .62 1948 (2) - 2290
8 . 2 5.25 -3 1.011 .60 2125 (1) -2152 5 . 9 -3 1,011 Toppled 2216 (1) -2551 2 . 6 -6 1.030 .47 -2595 1761 (2)
2 . 4 -5 .972 .51 1667 (2) - 2206
Table A-15. Summary Table of Critical Values for Series C I Max. Stroke Min. Beta Max. Vertical Max. Horizontal vV vH The leg number for the value given is in parentheses.
1.141 (1) .69 (1) 2171 1) 2663 (1)
I :E5 I 1878
1.102 .71 2222 5 1.104 .73 237 1 1861 2.4 2.6 6 1.143 .72 2624 2433 2.8 7 Bottomed .70 27 40 2967 7 1.164 3141 4.2 .71 2569 3.9 1.122 .73 2830 2041 6 Bottomed .74 3 593 2144 5.9 5.25 5 Bottomed .75 293 2 1713 4 1.155 .76 2457 4.6 1311 6.4 4 Bottomed .76 2875 1404 1.135 .76 2262 1006 1.164 .76 2529 1056 Bottomed .76 3139 1.144 .76 2610 746 Bottomed .76 2918 89 5 1.148 (1) .76 3104 5.8 5 1.064 (2) .76 (2) 2870 (2) 1.076 .76 3 188 0 1.087 .76 3541 (2) 0 Table A-16. Summary Table o f Critical Values for Series D Min. Beta Max. Vertical .93 2590 1090 .93 1865 696 .93 2204 745 .93 3047 1204 5 .93 3 Bottomed 3224 5 .93 2 Bottomed 2354 80 7 .93 1 1.082 1691 3 87 .93 1 1.119 2098 409 7 2 Bottomed .93 261 2 847 9 2 Bottomed .93 2842 884 .93 1 1.147 2668 4 27 .93 1 1.159 303 2 43 5 10 .93 2 Bottomed 3165 .93 1 Bottomed 3417 .93 0 1.079 3278 0
. . ’
Table A-17. Summary Table o f Critical Values for Series E Max. Stroke Min. Beta Max. Vertical Max. Horizontal vV vH number for the value g The 11 ‘en is in parentheses.
5 . 0 Bottomed (1) .83 (3) 3011 1734 (1) .5 .83 45 4 . 0 1.151 2177 1278 Bottomed .83 2427 1445 2.2 4.0 2 1.675 .83 1972 1017 2.85 2 . 7 1 . 0 3 4 .83 2386 8 28 2.5 1.8 .907 .83 2212 540 3.3 1 . 2 .792 .83 2451 426 .83 2.65 0.6 .642 2237 305 (1) 0 .578 .83 2. a 2272 -347 (2) 2.65 -0.6 ,677 .83 2216 -491 3.3 -1.2 .826 .83 2307 -766
-3 .83 - 1355
3.2 1.063 (3) 2230 -2 .83 3.2 .927 (3) 2231 -1016
-3 1.115 (3) .83 2268 - 1499
-3.3 .83 2 1.019 (3) 2514 -1178 (2) -4 1.088 (3) .83 -1361 (1) 1.4 2779
---
I
* . .
Table A-18. Summary Table o f Critical Values for Series F Min. Beta Max. Horizontal Max. Stroke Max. Vertical vV vH
I
I I
The 11 ; numbt I for the value given i s in parentheses.
4 Bottomed (1) .83 (3) 1610 (1) .4 2782 (1) . 3 3 1.130 .83 2086 1143 1 . 7 3 1.159 .83 2242 1334 2 . 2 1.161 .83 3 108 1903 2 1.065 .83 2.0 2351 909 1 . 6 2 1.054 .83 2115 869 . 8 1 .889 .83 2149 449 1 . 6 1.1 .934 .83 2133 514 ,832 1 .o .7 .83 236 1 4 16 1.0 0 .669 (1) .83 23 10
1 . 0 - .7 .643 (3) .83 2294 -343 (1)
-2 .843 .83
2 . 0 2563 - 698
.7 - 1 .722 .83 -417
1 . 2 -2 .875 .83 1973 - 680
-3 .964 1 . 7 .83 2413 (1) - 981 . 3 -3 .992 .83 2291 (3) - 999 .2 - 2 .855 (3) .83 2123 (1) -729
. - ’
Table A-19. Summary Table o f Critical Values for Series G Max. Vertical Max. Stroke Min. Beta Max. Horizontal vH vV The 1 .4 4 .974 (1) .32 (1) 1473 (1) 2119 (1) . 3 3 ,928 .42 1352 1876 ,806 1345 1227 .2 2 .46 ,821 1390 1235 1 . 2 2 .47 1 . 7 3 .914 .39 1308 1806 3 .917 .28 1874 1762 3 . 2 2 . 6 2 . 5 .872 .41 1493 1609 .867 2013 1565 3 . 8 1 . 9 .46 ,915 .46 2413 1718 4 . 5 2 . 2 5 1716 1 .889 .45 1421 203 5 1414 5.8 1.2 ,915 .44 1 . 3 5 ,918 .44 2351 1380 6.4 7 0 .933 .41 1936 1270 8 0 1.025 .43 2223 1508 9 . 4 - 1 1.077 .40 2074 1806
- 1 . 7 1.069 2166
9.8 .34 1892 9 -2 .981 2059 -1838 2 ) .36 8 . 6 -1.3 .999 .38 1966 -1480 7 -1.4
.805 .29 1912 - 1345
- 2 . 2
7 . 3 .881 (2) .32 2364 -1715 -2.8 .888 2080 -1755 5 . 9 .24 5 . 2 -2.5 .817 2042 (1) -1525 .26
3 . 8 -3.5 ,834 .17 1230 (2) - 1644
-4
4 . 4 ,917 Toppled 1395 - 1943
-4 .751 2 . 2 .29 1242 -1510
- 5 .875 .ll 1425 - 1929
2 . 7 -5 .717 1284 -1527 .5 .45
. 6 -6 .749 1518 - 1813
.48 Table A-20. S u m m a r y Table of Critical V a l u e s for Series H Max. H o r i z o n t a l Max. Stroke Min. Beta Max. V e r t i c a l vV vH 1567 (1) 2258 (1) .6 6 .930 1) .22 (2) 5 .944 .26 (1) 1400 .5 4 .32 1591 .4 .731 1462 1478 4 .773 .33 2.2 1404 1990 2.7 5 .959 .29 4 ,906 .3a 1774 4.4 2141 1909 .969 .37 5.0 4.5 2014 1260 2.8 .go8 .46 5.9 2185 1454 6.6 3.1 .972 .45 1.026 .44 2328 7.3 3.4 2167 1275 1.063 .49
a. 2 1.8
2375 1450 9 2 1.108 .49 249 2 90 1 10 0 1.037 .4a .gag .48 2158 797 9 0 -1.8 .793 .4a 1796
a. 2
2073 4 17 9 -2 .a28 .4a 2386 (1) 417
9. a -2.2 . a60 .4a
-3 .49 2392 (2) 9.6 .a27 (2) - 4 .935 .47 2252 642 8.7 -3.4 .856 .48 1754 470 7.3 .30 2322 -1282 (2) 6.8 -6 1.070 -5.5 1.023 2114 -1138 6.2 .37 3.7 -7 .993 .15 2386 - 1746 -1410 3.2 -6 .907 .28 2090 -1841 .6 -6 ,901 .03 1729 (1) 1354 (1) - 1460 .5 - 5 .795 .21 .
Table A-21. Summary Table o f Critical Values for Series J _ _ _ _ _ _ _ ~ ~ Min. Beta Max. Vertical Max. Horizontal Max. Stroke vH vV The 1c 1.166 (1) .8821 (3) 5 . 0 0 3142.4 (1) 1856.4 (1) .50 1377.6 1.150 .8820 4.00 2243.0 .40 1263.8 1.166 .8823 2.20 4.00 2336.9 1624.1 1022.1 1.054 .8821 1.70 3 .OO ,8823 3 .OO 1771.5 829.8 (2) 1.055 3.20 1014.5 1.121 .8824 3.80 3.50 2003.9 1.165 .8825 4.40 4.00 2389.7 1209.3 600.0 .9 57 .8824 4.50 2.20 1954.5
2183.1 716.9 1 .o 10 .8825
5.20 2.50 2.80 2447.9 840.7 (2) 1.062 .8826 5.90 393.2 (1) .8826 6.60 1.40 2498.2 .891 2233.6 351.1 (1) .841 .8824 5.80 1.20
5.80 - 1.20 1978.0 -766.2 (2) ,980 (3) .8820
.8821
6.60 - 1.40 2262.6 - 866.1 1.041
- 1372.7 1.150 .8817
5.90 - 2.80 2335.0 (3)
2015.6 - 1227.1 1.098 .8817
5.20 - 2.50
5.00 -4.50 3677.1 - 1927.9 1.163 .8812
.&I813
4.40 - 4.00 2954.7 - 1720.2 1.164
- 1493.9 1.130 .a813
3.80 -3.50 2267.6
1967.5 - 1273.3 1.044 .8813
3.20 -3.00
1 . 7 0 -3.00 2064.2 - 1018.9 .896 .8811
2.20 - 4.00 2377.6 - 1445.8 1.066 .8810
1.034 .8808
.40 - 4.00 2444.9 - 1222.0
-940.8 ,903 .8809
.30 - 3 .OO 2011.0
.
' .
Table A-22. Summary Table o f Critical Values for Series K Max. Vertical Max. Horizontal Max. Stroke Min. Beta vV
vH I I I
~~ I ~~ I I number for the value given is in parentheses.
The leg 1.163 (1) .8560 (1) .50 5.00 3763.0 (1) 2161.6 (1) 2950.5 1704.4 1.162 .8653 .40 4.00 3 .OO 2112.4 1251.4 1.144 .8645 .30 1.70 3.00 2303.9 1418.8 1.164 .8623 888.3 1.037 ,8776 1.20 2 . 0 0 1549.9 2.50 2251.6 1105.1 1.118 .8668 2 . 6 0 3 .OO 2439.3 1244.1 1.163 .863 5 3.20 .8799 3.10 1.60 2381.4 751.4 .997 2.40 1.30 2175.0 667.4 .961 .8812 2244.8 458.2 .807 .8750 2 . 6 0 .40 3.40 2441.6 534.6 .839 (1) .8752 .60
3.40 - .60 2406.1 476.2 .609 (3) .8751
- .40 371.6 .8756
2.60 2205.9 .610 (1)
2.40 - 1.30 2166.7 -470.8 (2) .644 (3) .8763
3.10 - 1.60 2288.6 -661.3 .760 .8720
3.20 -3.00 2225.3 -1141.1 .971 .8510
3.80 -3.50 2253.2 - 1419.8 1.085 .8384
4.40 - 4.00 2641.2 (3) - 1688.2 1.164 .8217 ,
- 1736.8
2 . 7 0 - 5.00 2777.4 1.147 .8398
2.20 - 4.00 2460.9 -1260.9 (2) 1.021 .8489
1.70 -880.7 (1) .887 .8666 -3.00 2049.9 -3.00 2009.2 -853.8 (2) .904 .863 1 .30
.40 - 4.00 2396.1 -1151.2 (2) 1 . O 26 .8526
APPENDIX B
APPENDIX B EQUATIONS OF MOTION - . .
X - Y Coordinate system with the ground a s reference (Y normal to ground) R - Vehicle radius from centerline to footpads i - Subscript used to denote ith leg 8i - Angle of leg i from the X-axis - Angular orientation o f vehicle with respect to X-Y axis
+
ai - Angular deflection of leg i.
- Ground slope H - Vertical distance from c.g. to hardpoint of strut L - Length of shock absorber fully extended Si - Amount of stroke (Compression) of the ith strut AH - Cross-sectional area of inner cylinder (I.D.)
A, - Net orifice area g - Gravity T - Thrust of engine FA - Force due to air spring of shock absorber FF - Friction of internal parts of shock absorber FS - Total shock absorber axial force F H - Force due to bending of the shock absorber - Moment about hardpoint of leg i due to FH(i) Mi m - Mass of vehicle p - Mass density of fluid in shock absorber I( - Coefficient of friction between footpad and surface v - coefficient of friction between footpad and obstacle C - Discharge coefficient of orifice FGV - Force normal to ground surface FGH - Force along ground surface FLG - Force along-ledge (obstacle) .
Line of Action of i th Shock Strut Direction o f Gravitional Field
t X
Figure B-1. Schematic Diagram of Vehicle - . .
A diagram o f the vehicle with appropriate symbols is shown in Figure B-1. The directions indicated in this diagram a r e positive. A s is indicated theX-Y coordinate system chosen is parallel ( X ) and normal (Y) to the ground surface.
The vehicle motion is assumed confined to translation and rotation in the plane of i t s approach path which is the X-Y plane.
The only force acting on the vehicle during landing are gravity, thrust of the engine, the shock absorber forces, and the force due to bending o f the shock absorber.
The axial load in the strut consists of three parts: 1) hydraulic load, 2) air load, and 3) friction o f the internal parts. The formula for the total axial load in the shock absor- 3 .2 2
ber is: FS = P ( A H ) s /2(cAo) + F A + FF
The f i r s t expression is the formula f o r hydraulic load. The air load, FA, a t any stroke, s, is dependent on the extended air pressure and the stroke s. For this study the ex- tended air pressure was not changed so that the air load becomes a direct function of s and was put into the computer in the form of a table. The friction load FF was assumed equal to zero. The horizontal load (FH) due to bending of the strut was cal- culated from the graph shown in Figure B-2. The formula for this graph is given by: For -ABS a I A B FH = - KMT/(L - s) and for a > AB FH = [- KMT * A B - KBT ( a - AB) ] / ( L - s)
and for a <- AB F H = [KMT * AB - KBT ( a + A B ) ]
/ ( L - s) Figure B-2. Shock Mount Loading The sum of the strut forces resolved into the X-Y direction and resulting moments about the center of gravity are given by , .
FX = Z FS (i) sin (a(i) +&) + FH (i) cos (a(i) +4) i= 1 F H (i) sin (a(i) +@)-FS (i) cos (a(i) +@) FY = Z i= 1 Mo = Z M (i) + (FH (i) c o s a ( i ) + FS (i) s i n a ( i ) ) H i= 1 + ( F H (i) s i n a ( i ) - FS (i) c o s a ( i ) ) R cos 8 (i) where M (i) = FH ( L - s (i)) Now adding in the gravity and thrust forces resolved in the X-Y direction gives
FXX = mg sin r- T s i n 4 - FX
F Y Y = - F Y + T C O S + - ~ ~ C ~ S I Gravity and thrust do not add to the moment about the center of gravity since they act through the center of gravity.
The accelerations are: X = FXX/m Y = FYY/m ;p' = -Mo/MX These equations were programmed on a CDC G-20 digital computer.
The process of solving these equations can best be illustrated by outlining the steps involved in the computer program.
Read in initial data.
1.
2. Calculate position of each footpad in the X-Y coordinate system.
Determine which footpads are on the surface and which are off the surface.
3.
Calculate s, i, and Q f o r each strut whose footpad is on the surface. (These 4.
t e r m s a r e zero i f the footpad is off the surface.)
Calculate FS, FH individually and sum them (torque can be calculated from 5.
kr.own geometry).
6. Sum all forces and moments acting.
7. Using these forces and moments, the center of gravity motion and angular . .
motion about the center of gravity can be determined at the end o f the time interval A t . Rectangular integration is used for the vehicle motion.
8. Set all parameters for next time interval.
9. Check for vehicle stopped (within epsilon quantities). I f not, return to step 2 for another iteration.
A
-. .-f$ 1 . .
Figure B-3. Footpad Force Diagram The time interval of t = .002 seconds proved to be sufficiently small to adequately describe the vehicle motion for all cases.
The method of calculating a and FH were changed to handle those runs where the footpads were allowed to slide. In order to check for sliding, the f o r c e s normal to the ground, FGT, and along the ground, FGH, had to be calculated. The condition of sliding is FGH = p FGV. For FGH less than pFGV the footpads would not slide.
The computations were made on the assumption that the footpads did not slide and FGH was compared to pFGV. I f FGH was l e s s than pFGV, the assumption was correct and the calculations for that time interval completed and the program continued f o r the next time interval.
If FGH was g r e a t e r than pFGV, then sliding would have occurred, which means that of the footpad has to be found.
the position This can be done by finding the c o r r e c t angle that the strut would be bent to fulfill the condition of sliding. The measure of bending of the strut,a, had been calculated on the assumption that the footpad did not slide.
Now ais calculated so that the condition of sliding is fulfilled.
The average of these two values of a is used f o r the recalculations made f o r this time interval. Because of the interdependency of a, FGH, and FGV, it was considered that just using a based on the condition of sliding might be a n over-correction of strut bend- ing. The values recalculated f o r this time interval are &, 8, FS, FH, FGV, and FGH.
These new values of FGV and FGH are then tested f o r the condition of sliding and the process o f revaluing a repeated until FGH = a F G V within an epsilon value. When this of FS happens the program continues with the other computations and the last values and FH a r e used in the equations of motion.
There were a few runs made to study the dynamic effect on the vehicle of the footpads striking an obstacle. This obstacle was defined as a six-inch ledge as shown in Figure B-4. For these runs there w a s an additional force, FLG, along the ledge when the foot- pad w a s moving along and in contact with the ledge. This f o r c e would be opposite the direction of movement and equal to the coefficient of friction,
Y , times the force normal
to the ledge, FGH. When this force, FLG, w a s acting it was added into the equations of motion
Surface Ground / H \
Figure B-4. Obstacle Impact NASA-Langley, 1966 CR-428 G G