Efficient Procedures of Sensitivity Analysis for Structural Vibration Systems with Repeated Frequencies

Table 1

Result of eigenvectors and the sensitivity analysis.

	Mode
	1	2	3	4	5
Generalized eigenvectors	−1.6940𝑒−4; −7.5774𝑒−2𝑖	−1.6940𝑒−4; +7.5774𝑒−2𝑖	−5.9292𝑒−4; −1.4832𝑒−1𝑖	−5.9292𝑒−4; +1.4832𝑒−1𝑖	−8.8391𝑒−3; −3.7710𝑒−1𝑖
	6.4403𝑒−5; +6.6628𝑒−2𝑖	6.4403𝑒−5; −6.6628𝑒−2𝑖	−7.6481𝑒−4; −3.8049𝑒−2𝑖	−7.6481𝑒−4; +3.8049𝑒−2𝑖	−9.1033𝑒−3; −3.6559𝑒−1𝑖
	2.0564𝑒−4; −1.8688𝑒−2𝑖	2.0564𝑒−4; +1.8688𝑒−2𝑖	−3.9167𝑒−4; +5.2447𝑒−2𝑖	−3.9167𝑒−4; −5.2447𝑒−2𝑖	−9.5377𝑒−3; −3.4587𝑒−1𝑖
	3.3666𝑒−5; +1.4966𝑒−6𝑖	3.3666𝑒−5; −1.4966𝑒−6𝑖	−1.7466𝑒−4; −1.1910𝑒−5𝑖	−1.7466𝑒−4; +1.1910𝑒−5𝑖	−1.0552𝑒−3; +5.7213𝑒−5𝑖
	3.3666𝑒−5; +1.4966𝑒−6𝑖	3.3666𝑒−5; −1.4966𝑒−6𝑖	−1.7466𝑒−4; −1.1910𝑒−5𝑖	−1.7466𝑒−4; +1.1910𝑒−5𝑖	−1.0552𝑒−3; +5.7213𝑒−5𝑖
	7.3452𝑒−1	7.3452𝑒−1	9.0427𝑒−1	9.0427𝑒−1	4.6611𝑒−1
	−6.4586𝑒−1; −8.1957𝑒−4𝑖	−6.4586𝑒−1; +8.1957𝑒−4𝑖	2.3200𝑒−1; −3.7356𝑒−3𝑖	2.3200𝑒−1; +3.7356𝑒−3𝑖	4.5189𝑒−1; −6.5990𝑒−4𝑖
	1.8114𝑒−1; +2.3984𝑒−3𝑖	1.8114𝑒−1; −2.3984𝑒−3𝑖	−3.1975𝑒−1; −3.6663𝑒−3𝑖	−3.1975𝑒−1; +3.6663𝑒−3𝑖	4.2755𝑒−1; −1.7674𝑒−3𝑖
	−1.5236𝑒−5; +3.2631𝑒−4𝑖	−1.5236𝑒−5; −3.2631𝑒−4𝑖	7.6868𝑒−5; −1.0646𝑒−3𝑖	7.6868𝑒−5; +1.0646𝑒−3𝑖	−4.0125𝑒−5; −1.3052𝑒−3𝑖
	−1.5236𝑒−5; +3.2631𝑒−4𝑖	−1.5236𝑒−5; −3.2631𝑒−4𝑖	7.6868𝑒−5; −1.0646𝑒−3𝑖	7.6868𝑒−5; +1.0646𝑒−3𝑖	−4.0125𝑒−5; −1.3052𝑒−3𝑖
First-order derivatives	4.9349𝑒−3; +4.1090𝑒−3𝑖	1.8620𝑒−4; +1.6593𝑒−6𝑖	−7.1896𝑒−6; −1.0523𝑒−6𝑖	−6.7317𝑒−6; −3.0627𝑒−7𝑖	−3.0855𝑒−6; +3.7924𝑒−5𝑖
	1.0543𝑒−2; +4.9187𝑒−3𝑖	3.1679𝑒−4; −8.3544𝑒−5𝑖	−7.4393𝑒−6; −1.9432𝑒−6𝑖	−4.9700𝑒−6; +5.0936𝑒−7𝑖	−3.9019𝑒−6; +3.1021𝑒−5𝑖
	6.9776𝑒−3; −1.5213𝑒−3𝑖	1.2503𝑒−4; −1.5632𝑒−4𝑖	−9.1416𝑒−6; −3.4699𝑒−6𝑖	−6.7761𝑒−7; +3.0202𝑒−6𝑖	−5.3973𝑒−6; +2.1570𝑒−5𝑖
	1.2865𝑒−8; −5.3235𝑒−8𝑖	−6.7567𝑒−6; −1.3890𝑒−7𝑖	4.3822𝑒−4; −7.3751𝑒−3𝑖	4.3945𝑒−4; −7.3752𝑒−3𝑖	−4.1661𝑒−2; −2.4195𝑒−3𝑖
	−1.2865𝑒−8; +5.3235𝑒−8𝑖	6.7567𝑒−6; +1.3890𝑒−7𝑖	4.3822𝑒−4; −7.3751𝑒−3𝑖	4.3945𝑒−4; −7.3752𝑒−3𝑖	−4.1661𝑒−2; −2.4195𝑒−3𝑖
	−4.0427𝑒−2; −1.5051𝑒−2𝑖	−1.1609𝑒−3; +4.0699𝑒−4𝑖	4.9009𝑒−6; +2.4447𝑒−5𝑖	−2.3870𝑒−6; +1.5967𝑒−5𝑖	1.1140𝑒−4; +7.8994𝑒−6𝑖
	−2.3209𝑒−2; +3.7847𝑒−2𝑖	1.9018𝑒−4; +1.2372𝑒−3𝑖	1.7448𝑒−5; +2.6237𝑒−5𝑖	−1.1561𝑒−5; +4.3792𝑒−6𝑖	9.9583𝑒−5; +1.0880𝑒−5𝑖
	1.2020𝑒−2; +5.5792𝑒−3𝑖	3.5070𝑒−4; −9.4669𝑒−5𝑖	−1.0988𝑒−5; +2.4234𝑒−5𝑖	1.4771𝑒−5; +4.6761𝑒−6𝑖	5.8350𝑒−5; −1.5286𝑒−5𝑖
	1.4840𝑒−9; +4.4428𝑒−8𝑖	4.0257𝑒−7; −1.6972𝑒−7𝑖	−2.0897𝑒−2; −6.1056𝑒−5𝑖	−2.0897𝑒−2; −5.7389𝑒−5𝑖	−1.6748𝑒−4; +1.1803𝑒−1𝑖
	−1.4840𝑒−9; −4.4428𝑒−8𝑖	−4.0257𝑒−7; +1.6972𝑒−7𝑖	−2.0897𝑒−2; −6.1056𝑒−5𝑖	−2.0897𝑒−2; −5.7389𝑒−5𝑖	−1.6748𝑒−4; +1.1803𝑒−1𝑖
	6	7	8	9	10
Generalized eigenvectors	−8.8391𝑒−3; +3.7710𝑒−1𝑖	2.1782𝑒−4; −1.6872𝑒−5𝑖	2.1782𝑒−4; +1.6872𝑒−5𝑖	−2.1190𝑒−17; −2.9349𝑒−18𝑖	−2.1190𝑒−17; +2.9349𝑒−18𝑖
	−9.1033𝑒−3; +3.6559𝑒−1𝑖	1.8288𝑒−4; −1.8087𝑒−5𝑖	1.8288𝑒−4; +1.8087𝑒−5𝑖	−1.6163𝑒−17; −6.5559𝑒−19𝑖	−1.6163𝑒−17; +6.5559𝑒−19𝑖
	−9.5377𝑒−3; +3.4587𝑒−1𝑖	1.2866𝑒−4; −1.9205𝑒−5𝑖	1.2866𝑒−4; +1.9205𝑒−5𝑖	−1.3628𝑒−17; −1.3204𝑒−17𝑖	−1.3628𝑒−17; +1.3204𝑒−17𝑖
	−1.0552𝑒−3; −5.7213𝑒−5𝑖	−1.3333𝑒−2; −2.3533𝑒−1𝑖	−1.3333𝑒−2; +2.3533𝑒−1𝑖	1.2667𝑒−2; +2.3536𝑒−1𝑖	1.2667𝑒−2; −2.3536𝑒−1𝑖
	−1.0552𝑒−3; −5.7213𝑒−5𝑖	−1.3333𝑒−2; −2.3533𝑒−1𝑖	−1.3333𝑒−2; +2.3533𝑒−1𝑖	−1.2667𝑒−2; −2.3536𝑒−1𝑖	−1.2667𝑒−2; +2.3536𝑒−1𝑖
	4.6611𝑒−1	1.2794𝑒−5; +6.1779𝑒−4𝑖	1.2794𝑒−5; −6.1779𝑒−4𝑖	1.9126𝑒−17; −2.8812𝑒−17𝑖	1.9126𝑒−17; +2.8812𝑒−17𝑖
	4.5189𝑒−1; +6.5990𝑒−4𝑖	2.1815𝑒−5; +5.1933𝑒−4𝑖	2.1815𝑒−5; −5.1933𝑒−4𝑖	2.7275𝑒−17; +1.0443𝑒−16𝑖	2.7275𝑒−17; −1.0443𝑒−16𝑖
	4.2755𝑒−1; +1.7674𝑒−3𝑖	3.3649𝑒−5; +3.6639𝑒−4𝑖	3.3649𝑒−5; −3.6639𝑒−4𝑖	−5.0624𝑒−17; +6.1754𝑒−17𝑖	−5.0624𝑒−17; −6.1754𝑒−17𝑖
	−4.0125𝑒−5; +1.3052𝑒−3𝑖	6.6667𝑒−1; −6.1319𝑒−15𝑖	6.6667𝑒−1; +6.1319𝑒−15𝑖	−6.6667𝑒−1; +1.9880𝑒−14𝑖	−6.6667𝑒−1; −1.9880𝑒−14𝑖
	−4.0125𝑒−5; +1.3052𝑒−3𝑖	6.6667𝑒−1	6.6667𝑒−1	6.6667𝑒−1	6.6667𝑒−1
First-order derivatives	1.7349𝑒−4; −1.6163𝑒−4𝑖	3.9518𝑒−4; −3.7717𝑒−4𝑖	−6.6110𝑒−6; −6.2260𝑒−7𝑖	−1.3023𝑒−4; +4.9260𝑒−5𝑖	−3.0598𝑒−6; +3.8496𝑒−5𝑖
	2.0752𝑒−4; −3.4405𝑒−4𝑖	4.7933𝑒−4; −8.0841𝑒−4𝑖	−4.6504𝑒−6; −1.2688𝑒−6𝑖	−3.0853𝑒−4; +1.7329𝑒−4𝑖	−3.4365𝑒−6; +3.2083𝑒−5𝑖
	−1.9104𝑒−5; −2.4330𝑒−4𝑖	−5.3099𝑒−5; −5.6976𝑒−4𝑖	−1.2556𝑒−7; −4.0572𝑒−6𝑖	−6.5242𝑒−5; +2.1995𝑒−4𝑖	−4.4476𝑒−6; +2.2222𝑒−5𝑖
	−1.7809𝑒−7; −6.5745𝑒−6𝑖	−4.1770𝑒+7; −1.4953𝑒+7𝑖	4.3894𝑒−4; −7.3760𝑒−3𝑖	1.1347𝑒−7; +1.1973𝑒−7𝑖	−4.1661𝑒−2; −2.4196𝑒−3𝑖
	1.7809𝑒−7; +6.5745𝑒−6𝑖	4.1770𝑒−7; +1.4953𝑒−7𝑖	4.3894𝑒−4; −7.3760𝑒−3𝑖	−1.1347𝑒−7; −1.1973𝑒−7𝑖	−4.1661𝑒−2; −2.4196𝑒−3𝑖
	−6.8458𝑒−4; +1.3706𝑒−3𝑖	−1.5683𝑒−3; +3.1981𝑒−3𝑖	−3.6769𝑒−6; +2.1007𝑒−5𝑖	8.6146𝑒−4; −6.5660𝑒−4𝑖	1.0981𝑒−4; +3.3384𝑒−6𝑖
	1.2345𝑒−3; +9.1495𝑒−4𝑖	2.8941𝑒−3; +2.1296𝑒−3𝑖	−1.3319𝑒−5; +2.6782𝑒−5𝑖	−3.5926𝑒−4; −1.2558𝑒−3𝑖	9.4929𝑒−5; +6.8591𝑒−6𝑖
	2.1805𝑒−4; −3.7015𝑒−4𝑖	5.1275𝑒−4; −8.8146𝑒−4𝑖	−5.2960𝑒−6; −1.9547𝑒−6𝑖	−4.6276𝑒−4; +2.3522𝑒−4𝑖	6.2620𝑒−5; −7.5296𝑒−6𝑖
	2.1276𝑒−7; −4.9301𝑒−7𝑖	4.8581𝑒−7; −1.1570𝑒−8𝑖	−2.0895𝑒−2; −5.8814𝑒−5𝑖	−3.5540𝑒−7; +3.0228𝑒−7𝑖	−1.6732𝑒−4; +1.1803𝑒−1𝑖
	−2.1276𝑒−7; +4.9301𝑒−7𝑖	−4.8581𝑒−7; +1.1570𝑒−8𝑖	−2.0895𝑒−2; −5.8814𝑒−5𝑖	3.5540𝑒−7; −3.0228𝑒−7𝑖	−1.6732𝑒−4; +1.1803𝑒−1𝑖