龍凱,谷先廣,王選
1.華北電力大學(xué) 能源的安全與清潔利用北京市重點(diǎn)實(shí)驗(yàn)室,北京 102206 2.華北電力大學(xué) 新能源電力系統(tǒng)國(guó)家重點(diǎn)實(shí)驗(yàn)室,北京 102206 3.合肥工業(yè)大學(xué) 汽車(chē)與交通工程學(xué)院,合肥 230009 4.大連理工大學(xué) 工業(yè)裝備結(jié)構(gòu)分析國(guó)家重點(diǎn)實(shí)驗(yàn)室,大連 116024
基于多相材料的連續(xù)體結(jié)構(gòu)動(dòng)態(tài)輕量化設(shè)計(jì)方法
龍凱1,2,*,谷先廣3,王選4
1.華北電力大學(xué) 能源的安全與清潔利用北京市重點(diǎn)實(shí)驗(yàn)室,北京 102206 2.華北電力大學(xué) 新能源電力系統(tǒng)國(guó)家重點(diǎn)實(shí)驗(yàn)室,北京 102206 3.合肥工業(yè)大學(xué) 汽車(chē)與交通工程學(xué)院,合肥 230009 4.大連理工大學(xué) 工業(yè)裝備結(jié)構(gòu)分析國(guó)家重點(diǎn)實(shí)驗(yàn)室,大連 116024
為了實(shí)現(xiàn)基于多相材料的結(jié)構(gòu)輕量化設(shè)計(jì),遵循獨(dú)立連續(xù)映射法,提出了以結(jié)構(gòu)總重最小化為目標(biāo)和給定特征值為約束的拓?fù)鋬?yōu)化模型。方法采用2類(lèi)獨(dú)立拓?fù)渥兞繉?shí)現(xiàn)了單元?jiǎng)偠染仃?、質(zhì)量矩陣和重量插值。推導(dǎo)了固有特征值和總重量的敏度表達(dá)式,通過(guò)1階和2階泰勒展開(kāi)得到其近似表達(dá)式。對(duì)約束函數(shù)一次項(xiàng)過(guò)濾轉(zhuǎn)換為偏微分方程的求解,消除了棋盤(pán)格現(xiàn)象和網(wǎng)格依賴性等數(shù)值不穩(wěn)定性。通過(guò)二維數(shù)值算例驗(yàn)證了提出方法的可行性和優(yōu)越性。結(jié)果表明,與單相組分材料拓?fù)鋬?yōu)化結(jié)構(gòu)相比,多相材料拓?fù)鋬?yōu)化結(jié)構(gòu)具有更輕的重量。通過(guò)附加相鄰頻率間隔約束或增加高階頻率約束,避免了模態(tài)交換現(xiàn)象。
拓?fù)鋬?yōu)化;獨(dú)立連續(xù)映射法;連續(xù)體結(jié)構(gòu);固有頻率;模態(tài)分析
自Bendsoe和Kikuchi[1]提出連續(xù)體結(jié)構(gòu)拓?fù)鋬?yōu)化概念以來(lái),迄今為止涌現(xiàn)出各類(lèi)拓?fù)鋬?yōu)化方法,包括均勻化法、實(shí)體各向同性懲罰微結(jié)構(gòu)(Solid Isotropic Microstructures with Penalization,SIMP) 的變密度法、雙向結(jié)構(gòu)進(jìn)化式(Bi-directional Evolutionary Structural Optimization,BESO)法和水平集法等。文獻(xiàn)[2-6]給出了在不同發(fā)展階段各類(lèi)方法的理論和應(yīng)用綜述。早期的動(dòng)態(tài)優(yōu)化研究大多以結(jié)構(gòu)固有頻率為目標(biāo)或約束,體現(xiàn)在各類(lèi)拓?fù)鋬?yōu)化方法中[7-11]。在動(dòng)態(tài)響應(yīng)方面,以降低結(jié)構(gòu)振動(dòng)幅度、減小噪聲的拓?fù)鋬?yōu)化方法也被相繼提出[12-19]。文獻(xiàn)[20]綜述了現(xiàn)有的動(dòng)態(tài)拓?fù)鋬?yōu)化研究。
多相材料的布局優(yōu)化是拓?fù)鋬?yōu)化研究中另一類(lèi)值得關(guān)注的問(wèn)題。Thomsen[21]較早開(kāi)展了多相材料的拓?fù)鋬?yōu)化研究。Sigmund和Torquato[22]采用常規(guī)材料,實(shí)現(xiàn)了負(fù)熱膨脹系數(shù)的材料微結(jié)構(gòu)優(yōu)化設(shè)計(jì)。Gibiansky和Sigmund[23]提出了達(dá)到體積模量上限的多相材料組成的微結(jié)構(gòu)。除了SIMP法外,多相材料布局優(yōu)化在水平集法[24]、相場(chǎng)法[25]和BESO法[26]中也有所體現(xiàn)。Yin和Ananthasuresh[27]提出的峰值函數(shù)法用單變量描述多相材料以減少密度變量數(shù)目。Gao等[28-29]提出了線性對(duì)等混合材料插值和質(zhì)量約束拓?fù)鋬?yōu)化模型,指出質(zhì)量約束在尋找全局最優(yōu)解等方面的優(yōu)越性。Takakoli和Mohseni[30]將多相布局優(yōu)化問(wèn)題轉(zhuǎn)化為序列兩相材料優(yōu)化問(wèn)題,采用優(yōu)化準(zhǔn)則法求解,實(shí)現(xiàn)了各相材料體積比約束下的布局優(yōu)化。Zuo和Saitou[31]歸一化處理密度變量,減少了密度變量數(shù)目,提出了序列SIMP方法。
不同于常見(jiàn)拓?fù)鋬?yōu)化方法,獨(dú)立連續(xù)映射(Independent Continuous Mapping,ICM)方法以獨(dú)立于單元具體物理參數(shù)的變量表征單元的有無(wú)[32]。其獨(dú)立性體現(xiàn)在拓?fù)渥兞繑[脫了物理量的約束,提出了磨光、過(guò)濾函數(shù)等新概念。自1996年被提出以來(lái),ICM方法實(shí)現(xiàn)了重量最小化目標(biāo)下的應(yīng)力、位移、頻率和屈曲因子等約束條件的拓?fù)鋬?yōu)化結(jié)構(gòu)設(shè)計(jì)[33],研究成果集中體現(xiàn)在文獻(xiàn)[34]中。
目前多相材料布局優(yōu)化較多地以各相材料體積比或總質(zhì)量為約束條件建立優(yōu)化模型。與此不同,本文將針對(duì)結(jié)構(gòu)動(dòng)態(tài)拓?fù)鋬?yōu)化問(wèn)題,基于ICM方法建立多相材料布局的拓?fù)鋬?yōu)化模型。優(yōu)化目標(biāo)為總重最小化,這種以輕量化設(shè)計(jì)為目的的優(yōu)化模型具有更為直觀的工程意義。方法定義了2類(lèi)獨(dú)立拓?fù)渥兞恳员碚鲉卧挠袩o(wú)和不同材料相。采用1階、2階泰勒近似表達(dá)固有特征值和重量函數(shù),建立的二次優(yōu)化近似模型采用序列二次規(guī)劃方法求解。最后通過(guò)數(shù)值算例來(lái)證明本文提出的方法在多相材料布局優(yōu)化的可行性。
ICM方法定義2類(lèi)獨(dú)立拓?fù)渥兞縮i(0≤si≤1)和ti(0≤ti≤1)用于表征單元的有無(wú)和不同相材料,其與單元?jiǎng)偠染仃噆i、質(zhì)量矩陣mi和重量wi的關(guān)系為
(1)
式中:N為設(shè)計(jì)區(qū)域單元總數(shù);上標(biāo)Ⅰ和Ⅱ表示可選用材料Ⅰ和Ⅱ;si=1或0表示單元i由材料填充或空洞;ti=1或0表示單元i選用材料Ⅰ或材料Ⅱ;α和β分別為單元?jiǎng)偠染仃嚭唾|(zhì)量矩陣懲罰因子。
以重量最小化為目標(biāo),給定特征值約束限值的拓?fù)鋬?yōu)化模型為
(2)
在動(dòng)力學(xué)優(yōu)化問(wèn)題中,普遍存在著局部模態(tài)的現(xiàn)象,這種現(xiàn)象是由低密度區(qū)的剛度懲罰不當(dāng)引起的[35]。在SIMP法和BESO法中,通常修正低密度區(qū)的彈性模量來(lái)避免局部模態(tài)現(xiàn)象[35-36]。這里通過(guò)對(duì)質(zhì)量矩陣予以剛度矩陣相同的懲罰來(lái)消除局部模態(tài)現(xiàn)象,即α=β=4。數(shù)值實(shí)驗(yàn)結(jié)果表明,這種參數(shù)設(shè)置能有效地避免局部模態(tài)現(xiàn)象,并得到清晰的拓?fù)鋬?yōu)化構(gòu)型。
模態(tài)分析有限元方程為
Kuj=λjMuj
(3)
式中:K和M分別為總體剛度矩陣和質(zhì)量矩陣;uj為第j階特征向量。由伴隨法易推導(dǎo)得
(4)
式中:K和M對(duì)si和ti的偏導(dǎo)數(shù)計(jì)算可歸結(jié)為單元?jiǎng)偠染仃嚭唾|(zhì)量矩陣的偏導(dǎo)數(shù)計(jì)算上,由式(1)可得
(5)
與SIMP方法不同,ICM方法引入si和ti的倒數(shù)函數(shù)作為設(shè)計(jì)變量,定義設(shè)計(jì)變量xi和yi為
(6)
由此可得
(7)
固有特征值采用2類(lèi)設(shè)計(jì)變量的1階泰勒展開(kāi)得到其近似表達(dá)式為
(8)
(9)
將式(4)、式(5)和式(7)代入式(9),可求得敏度值。
基于設(shè)計(jì)變量表達(dá)的重量函數(shù)為
(10)
易推導(dǎo)重量函數(shù)的1階偏導(dǎo)數(shù)為
(11)
同理可得重量函數(shù)的2階偏導(dǎo)數(shù)為
(12)
根據(jù)式(11)和式(12),在當(dāng)前設(shè)計(jì)變量下,采用2階泰勒展開(kāi)得到重量函數(shù)的近似表達(dá)式為
(13)
經(jīng)過(guò)上述推導(dǎo),優(yōu)化模型式(2)可轉(zhuǎn)換為
(14)
式(14)中的約束函數(shù)和目標(biāo)函數(shù)分別采用式(8)和式(13)的近似表達(dá)式。
動(dòng)態(tài)拓?fù)鋬?yōu)化中存在著模態(tài)交換問(wèn)題,除了關(guān)心特征值大于等于指定值外,ICM方法采用增加約束的處理方式來(lái)避免模態(tài)交換。例如,2階特征值始終不小于1階特征值[37-38];或2階特征值大于指定值。上述2種方法的目的在于相鄰特征值間保持間隔,約束方程分別表達(dá)為
λ2≥bλ1
(15a)
(15b)
注意到模型式(14)是標(biāo)準(zhǔn)的二次規(guī)劃形式,這里采用序列二次規(guī)劃法求解,具體過(guò)程可參考文獻(xiàn)[39-40]。在優(yōu)化求解后,更新結(jié)構(gòu)直至滿足
|W(a+1)-W(a)|/W(a+1)≤ε
(16)
式中:ε為收斂率。
連續(xù)體拓?fù)鋬?yōu)化問(wèn)題中普遍存在著棋盤(pán)格現(xiàn)象和網(wǎng)格依賴性問(wèn)題[41-42]。基于數(shù)字圖像原理的過(guò)濾方法被廣泛應(yīng)用解決上述問(wèn)題。定義固有特征值對(duì)設(shè)計(jì)變量的偏導(dǎo)數(shù)與設(shè)計(jì)變量乘積為
(17)
(18a)
(18b)
容易證明,過(guò)濾場(chǎng)函數(shù)滿足
(19)
即過(guò)濾前后,式(8)中的一次項(xiàng)總和不變,這對(duì)穩(wěn)定優(yōu)化求解非常有利。
過(guò)濾措施帶來(lái)的平均效果不可避免地導(dǎo)致優(yōu)化構(gòu)型邊界不清晰。為了獲得清晰的拓?fù)鋬?yōu)化構(gòu)型,采用2階段優(yōu)化策略[39-40],即在第1階段采用過(guò)濾措施消除棋盤(pán)格現(xiàn)象和網(wǎng)格依賴性問(wèn)題。第2階段不采用過(guò)濾措施直至優(yōu)化收斂。這2個(gè)階段采用的收斂率ε不同,分別取值為0.5‰和0.1‰。
采用數(shù)值算例來(lái)說(shuō)明提出方法的可行性和優(yōu)越性。可選用的材料及屬性如表1所示。每個(gè)算例均為平面結(jié)構(gòu),厚度為1 mm。結(jié)構(gòu)采用四節(jié)點(diǎn)方形單元離散,單元尺寸為1 mm×1 mm。當(dāng)結(jié)構(gòu)全部由碳鋼材料組成,其質(zhì)量和重量分別為m0和W0,結(jié)構(gòu)拓?fù)鋬?yōu)化后的重量為W,以重量比W/W0來(lái)說(shuō)明減重效果。在拓?fù)鋬?yōu)化構(gòu)型中,碳鋼、鋁和鎂分別采用紅色、藍(lán)色和綠色表示,空洞部分不顯示。
表1 可選用材料的屬性Table 1 Properties of candidate material
算例1圖1所示的長(zhǎng)梁結(jié)構(gòu),結(jié)構(gòu)尺寸為280 mm×40 mm。梁兩端全固定,平板中心附加一個(gè)質(zhì)量為m0的質(zhì)量點(diǎn)。可選材料為碳鋼和鋁,當(dāng)結(jié)構(gòu)分別采用碳鋼和鋁時(shí),對(duì)應(yīng)的1階頻率分別為1 237和813 Hz。對(duì)于多相材料布局優(yōu)化問(wèn)題,設(shè)結(jié)構(gòu)1階頻率下限值為750 Hz。本算例首先考察懲罰參數(shù)對(duì)優(yōu)化結(jié)果的影響。選取參數(shù)α=4、β=1,拓?fù)鋬?yōu)化過(guò)程迭代振蕩無(wú)法收斂。結(jié)構(gòu)固有頻率值在優(yōu)化迭代過(guò)程中會(huì)突然變小,該現(xiàn)象由低密度區(qū)的虛假模態(tài)引起。當(dāng)參數(shù)α=β=4時(shí),優(yōu)化迭代78步收斂,優(yōu)化結(jié)果重量比為0.271,拓?fù)鋬?yōu)化結(jié)果如圖2(a)所示。為了驗(yàn)證提出方法的可行性和有效性,優(yōu)化模型采用式(2)形式,采用移動(dòng)漸近線(MMA)算法更新變量,在指定的最大優(yōu)化迭代步(≥300)收斂,優(yōu)化重量比為0.278,拓?fù)鋬?yōu)化構(gòu)型如圖2(b)所示。
由圖2可知,2種不同方法得到的拓?fù)鋬?yōu)化構(gòu)型類(lèi)似,說(shuō)明本文提出方法具有可行性。長(zhǎng)梁結(jié)構(gòu)1階振型為彎曲振型,圖2(a)中更多的強(qiáng)相材料布置在上下表面以提高結(jié)構(gòu)抗彎性,優(yōu)化后的結(jié)構(gòu)重量更輕。由此可知,ICM方法得到的優(yōu)化結(jié)構(gòu)合理,且優(yōu)化迭代效率高。
圖1 長(zhǎng)梁結(jié)構(gòu)Fig.1 A long beam
圖2 不同方法下的優(yōu)化構(gòu)型Fig.2 Optimal configurations with different methods
算例2本算例是算例1的擴(kuò)展,用于考察不同頻率約束下多相材料拓?fù)鋬?yōu)化結(jié)果與單相拓?fù)鋬?yōu)化結(jié)果的對(duì)比。設(shè)結(jié)構(gòu)1階頻率下限在500~900 Hz之間變化,不同頻率約束下的多相材料拓?fù)鋬?yōu)化結(jié)果如表2所示。為了便于比較,在相同的頻率約束條件下,單相組分材料下的拓?fù)鋬?yōu)化結(jié)果如表3和表4所示。
由表2可知,在不同的頻率約束條件下,多相材料拓?fù)鋬?yōu)化結(jié)構(gòu)包含碳鋼、鋁和空洞,強(qiáng)相材料碳鋼始終布置在約束位置、質(zhì)量點(diǎn)位置等區(qū)域。對(duì)比表2~表4的結(jié)果可知,在相同的1階頻率約束條件下,多相材料拓?fù)鋬?yōu)化結(jié)構(gòu)比2種組分材料下的優(yōu)化結(jié)構(gòu)更輕。當(dāng)1階頻率要求不低于900 Hz時(shí),整個(gè)結(jié)構(gòu)全部采用鋁材料無(wú)法滿足設(shè)計(jì)要求。綜合上述2點(diǎn),多相材料不僅能更好地滿足設(shè)計(jì)要求,同時(shí)達(dá)到了結(jié)構(gòu)輕量化設(shè)計(jì)的目的。優(yōu)化結(jié)果說(shuō)明所提方法具有協(xié)調(diào)多相材料布局優(yōu)化的能力。
表2 不同頻率約束下的多相材料優(yōu)化結(jié)果Table 2 Optimal results for multiple phases materials under different frequency constraints
表3 相材料1下的優(yōu)化結(jié)果Table 3 Optimal results for Phase 1
表4 相材料2下的優(yōu)化結(jié)果Table 4 Optimal results for Phase 2
算例3如圖3所示的短懸臂梁,結(jié)構(gòu)尺寸為160 mm×100 mm。梁左端全約束,右端中心附加一個(gè)質(zhì)量為0.8m0的質(zhì)量點(diǎn)。分別選用碳鋼、鋁和鎂來(lái)填充結(jié)構(gòu),1階頻率分別為1 284.9、834和669 Hz。設(shè)結(jié)構(gòu)1階頻率不低于700 Hz。顯然,整個(gè)結(jié)構(gòu)全部采用鎂材料無(wú)法滿足設(shè)計(jì)要求。對(duì)于多相材料布局優(yōu)化問(wèn)題,考慮不同的多相材料組合:碳鋼和鋁、碳鋼和鎂,拓?fù)鋬?yōu)化結(jié)果如表5所示。為便于比較,碳鋼和鋁單相材料拓?fù)鋬?yōu)化結(jié)果也列于表5中。
由表5可知,2類(lèi)多相材料組合和2類(lèi)單相材料拓?fù)鋬?yōu)化結(jié)果均能滿足動(dòng)態(tài)設(shè)計(jì)要求,多相材料拓?fù)鋬?yōu)化結(jié)構(gòu)均比組分單相材料拓?fù)鋬?yōu)化結(jié)構(gòu)輕。鋁和鎂材料比剛度較為接近,在多相材料拓?fù)鋬?yōu)化結(jié)果中,這2種弱相材料所占的體積不同,最優(yōu)結(jié)構(gòu)重量接近。這說(shuō)明多相材料的最優(yōu)拓?fù)浣Y(jié)構(gòu)隨著組分材料的不同而不同。
圖3 短懸臂梁Fig.3 A short cantilever beam
表5 不同相材料組合下的優(yōu)化結(jié)果Table 5 Optimal results for different combinations of phases materials
算例4如圖4所示的平面結(jié)構(gòu),結(jié)構(gòu)尺寸為120 mm×40 mm,在下端面1/4、1/2和3/4位置處分別附加質(zhì)量為0.8m0、0.6m0和0.8m0的質(zhì)量點(diǎn)。可選材料為碳鋼和鋁。設(shè)結(jié)構(gòu)1階頻率不低于2 000 Hz。本算例用來(lái)考察附加約束對(duì)拓?fù)鋬?yōu)化結(jié)果的影響。計(jì)算3種情況:① 2階頻率無(wú)任何約束(Case 1);② 根據(jù)式(15a),設(shè)置λ2≥bλ1,強(qiáng)迫2階頻率始終大于1階頻率(Case 2);③ 根據(jù)式(15b)增加2階頻率約束λ2≥2 500 Hz(Case 3)。3種情況下,優(yōu)化迭代過(guò)程中的前2階頻率變化如圖5所示。
由圖5(a)可知,當(dāng)沒(méi)有2階頻率約束時(shí),結(jié)構(gòu)1、2階頻率在優(yōu)化迭代過(guò)程中較為接近,容易產(chǎn)生模態(tài)交換現(xiàn)象,無(wú)法得到理想的拓?fù)鋬?yōu)化結(jié)果。由圖5(b)可知,λ2≥bλ1導(dǎo)致2階頻率與1階頻率保持一定的距離。由圖5(c)可知,前2階頻率均滿足設(shè)計(jì)要求,說(shuō)明所提方法的有效性。
Case 2和Case 3下的拓?fù)鋬?yōu)化結(jié)果如表6所示。由表6可知,拓?fù)鋬?yōu)化結(jié)果均滿足設(shè)定的約束條件,拓?fù)鋬?yōu)化構(gòu)型略有區(qū)別。Case 3的約束條件比Case 2的約束條件更為苛刻,故而優(yōu)化結(jié)構(gòu)稍重。
圖4 平面結(jié)構(gòu)Fig.4 A plane structure
圖5 固有頻率迭代歷程Fig.5 Iteration history of natural frequency
表6 不同約束條件下的優(yōu)化結(jié)果Table 6 Optimal results for different constraints
1) 在結(jié)構(gòu)輕量化設(shè)計(jì)要求下,提出基于多相材料的動(dòng)態(tài)拓?fù)鋬?yōu)化的ICM方法。與目前的多相材料拓?fù)鋬?yōu)化方法不同,建立了指定固有特征值約束下的重量最小化優(yōu)化模型。遵循ICM建模方式,將結(jié)構(gòu)固有特征值和重量函數(shù)分別采用1階、2階泰勒展開(kāi)得到其近似表達(dá)式,從而將優(yōu)化模型轉(zhuǎn)換為標(biāo)準(zhǔn)二次規(guī)劃形式,采用序列二次規(guī)劃法求解,方法穩(wěn)健高效。
2) 在相同的固有特征值約束條件下,多相材料拓?fù)鋬?yōu)化結(jié)構(gòu)更輕。在動(dòng)態(tài)設(shè)計(jì)要求下,提出方法實(shí)現(xiàn)了各相材料的合理分配和結(jié)構(gòu)輕量化設(shè)計(jì)。
3) 根據(jù)優(yōu)化設(shè)計(jì)要求,優(yōu)化模型可拓展為多約束拓?fù)鋬?yōu)化模型并實(shí)現(xiàn)優(yōu)化求解,通過(guò)強(qiáng)迫相鄰頻率間距或增加高階頻率約束,能避免模態(tài)交換現(xiàn)象,并得到理想的拓?fù)鋬?yōu)化結(jié)果。
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Lightweightdesignmethodforcontinuumstructureundervibrationusingmultiphasematerials
LONGKai1,2,*,GUXianguang3,WANGXuan4
1.BeijingKeyLaboratoryofEnergySafetyandCleanUtilization,NorthChinaElectricPowerUniversity,Beijing102206,China2.StateKeyLaboratoryforAlternateElectricalPowerSystemwithRenewableEnergySources,NorthChinaElectricPowerUniversity,Beijing102206,China3.SchoolofAutomobileandTrafficEngineering,HefeiUniversityofTechnology,Hefei230009,China4.StateKeyLaboratoryofStructuralAnalysisforIndustrialEquipment,DalianUniversityofTechnology,Dalian116024,China
Toachievelightdesignofcontinuumstructurecontainingmultiphasematerials,atopologicaloptimizationmodelforweightminimizationwiththegiveneigenvalueconstraintisproposedusingtheindependentcontinuousmappingmethod.Twosetsofindependenttopologicalvariablesareemployedtointerpolatetheelementalstiffnessmatrix,themassmatrixandweight.Thesensitivityexpressionsfortheeigenvalueandtotalweightarederived.Theapproximationsoftheeigenvalueandtotalweightcanbeobtainedviathefirst-orderandsecond-orderTaylorexpansion.Thefilteringtechniqueforthefirsttermoftheconstraintfunctionisadoptedasasolutiontothepartialdifferentialequation.Thenumericalinstabilitiesincludingcheckerboardpatternsandmeshdependenceareremoved.Thefeasibilityandsuperiorityoftheproposedmethodarevalidatedbytwo-dimensionalnumericalexamples.Theresultsshowthattheweightoftheoptimalstructureconstructedbymultiphasematerialsislighterthanthatcomposedofconstituentphase.Themodeswitchcanbepreventedbyimposingtheconstraintonthegapoftheadjacentfrequencyoradditionalhighorderfrequencyconstraint.
topologyoptimization;independentcontinuousmappingmethod;continuumstructure;naturalfrequency;modeanalysis
2016-12-05;Revised2017-02-16;Accepted2017-04-25;Publishedonline2017-05-310942
URL:http://hkxb.buaa.edu.cn/CN/html/20171013.html
s:NationalNaturalScienceFoundationofChina(51405123);FundamentalResearchFundsfortheCentralUniversity(2017MS077)
.E-maillongkai1978@163.com
http://hkxb.buaa.edu.cnhkxb@buaa.edu.cn
10.7527/S1000-6893.2017.221022
V414.5;O343.1
A
1000-6893(2017)10-221022-10
2016-12-05;退修日期2017-02-16;錄用日期2017-04-25;< class="emphasis_bold">網(wǎng)絡(luò)出版時(shí)間
時(shí)間:2017-05-310942
http://hkxb.buaa.edu.cn/CN/html/20171013.html
國(guó)家自然科學(xué)基金(51405123); 中央高?;究蒲袠I(yè)務(wù)費(fèi)專項(xiàng)資金(2017MS077)
*
.E-maillongkai1978@163.com
龍凱, 谷先廣, 王選. 基于多相材料的連續(xù)體結(jié)構(gòu)動(dòng)態(tài)輕量化設(shè)計(jì)方法J. 航空學(xué)報(bào),2017,38(10):221022.LONGK,GUXG,WANGX.LightweightdesignmethodforcontinuumstructureundervibrationusingmultiphasematerialsJ.ActaAeronau-ticaetAstronauticaSinica,2017,38(10):221022.
(責(zé)任編輯:徐曉)