影響反振蕩器的結(jié)構(gòu)響應(yīng)外文文獻(xiàn)翻譯、中英文翻譯、外文翻譯,影響,振蕩器,結(jié)構(gòu),響應(yīng),外文,文獻(xiàn),翻譯,中英文 XXXX 附錄1 影響反振蕩器的結(jié)構(gòu)響應(yīng) 摘要： 對(duì)靈活的結(jié)構(gòu)低速?zèng)_擊結(jié)構(gòu)響應(yīng)的預(yù)測(cè)，適當(dāng)?shù)倪x擇一個(gè)碰撞模型很重要。本文考慮基于所謂的反振蕩器的沖擊結(jié)構(gòu)的一個(gè)新的建模。通過其反振蕩器許可證的沖擊結(jié)構(gòu)的表示，以了解對(duì)結(jié)構(gòu)響應(yīng)的不同影響的參數(shù)的影響。結(jié)果表明，該網(wǎng)絡(luò)連接RST反振蕩器不僅允許預(yù)測(cè)預(yù)期結(jié)構(gòu)響應(yīng)，而且影響持續(xù)時(shí)間的良好估計(jì)。 1.簡(jiǎn)述 建模的影響在靈活的結(jié)構(gòu)應(yīng)該考慮到彈丸的議案、彈丸和結(jié)構(gòu)、并在動(dòng)態(tài)的互動(dòng)結(jié)構(gòu)。撞擊事件的完整建?？梢酝ㄟ^三維有限元方法來(lái)實(shí)現(xiàn)。相互作用的三維有限元的模型提高了建模工作，并要求進(jìn)行數(shù)值模擬的時(shí)間。這個(gè)問題可以主要通過其相互作用建模為來(lái)克服 彈簧（剛度K接點(diǎn)）描述的非線性接觸行為：赫茲定律是最廣泛使用的[1-3]。這樣的模擬可以在受影響的結(jié)構(gòu)來(lái)執(zhí)行通過接觸彈簧連接到所述射彈。 如果該結(jié)構(gòu)是復(fù)雜的或離散化以非常抽象，數(shù)值成本將非常高。為了減少的參與模擬自由的數(shù)量，可以使用的結(jié)構(gòu)的模態(tài)描述[1-3,5]。模態(tài)擴(kuò)張的截?cái)嗫梢宰龅剑簡(jiǎn)栴}在于怎樣去連接寧截?cái)嗟臄?shù)據(jù)信息。在某些情況下，結(jié)構(gòu)的行為可以被很好通過其基本振動(dòng)的模式描述的，在這種情況下，結(jié)構(gòu)被建模為單自由度的彈簧 - 質(zhì)量系統(tǒng)的連接于彈丸通過非線性接觸剛度[3-6]。 彈簧質(zhì)量模型是最簡(jiǎn)單的沖擊模型提供了合理的精度的結(jié)果，如果該結(jié)構(gòu)表現(xiàn)在準(zhǔn)靜態(tài)方式[3,4,6]。這就是為什么有些研究人員提出的方法來(lái)預(yù)測(cè)結(jié)構(gòu)響應(yīng)低速?zèng)_擊[3,5,6,9]。這樣的目的方法是選擇的建模方法模態(tài)描述和一個(gè)彈簧質(zhì)量系統(tǒng)。 Jacqueline et al. [7]開發(fā)了基于沖擊造型一些所謂的“反振蕩器”結(jié)構(gòu)參數(shù)。反振蕩器相關(guān)的位移影響區(qū)域這是基于約束模式和靜態(tài)受影響的結(jié)構(gòu)的模式。反振蕩器模式具有兩個(gè)彈簧質(zhì)量模型和模態(tài)特征描述：它可以被看作是模態(tài)描述基于一些'先驗(yàn)'有關(guān)的信息的影響 位置和方向。 本文的目的是利用反振蕩器模型對(duì)于涉及到各個(gè)方面的理解 影響建模以及制定更完備標(biāo)準(zhǔn)結(jié)構(gòu)響應(yīng)的預(yù)測(cè)。在下面，首先呈現(xiàn)是對(duì)防振子模型進(jìn)行檢討，其次是結(jié)構(gòu)性的解釋通過反振蕩器響應(yīng)，在的最后一部分，它表明這種建模方法可以檢測(cè)影響的持續(xù)時(shí)間。 2.反震蕩模型 2.1運(yùn)動(dòng)學(xué)描述 反振蕩器是基于一個(gè)特定的位移區(qū)域的定義，通過靜態(tài)模式和一些約束定義 受影響的結(jié)構(gòu)的模式中，德音響定義如下[7]（參見圖。 1）：靜態(tài)模式'FST''是結(jié)構(gòu)引起的形狀由靜態(tài)負(fù)載''FST'施加在撞擊中的點(diǎn)沖擊方向，使得在單位位移在這一點(diǎn)。用于與剛體模式的結(jié)構(gòu)中，該靜態(tài)模式替換為剛性體模式[7]。 所述約束其運(yùn)動(dòng)的沖擊方向點(diǎn)的影響，從而使在該點(diǎn)的位移是零，因此圓形特征頻率 約束模式將是圓形的反諧振對(duì)應(yīng)于原始頻率結(jié)構(gòu)點(diǎn)沖擊。 靜態(tài)模式不一定垂直于約束模式。所以，為了獲得所述位移區(qū)域?yàn)檎缓瘮?shù)一個(gè)無(wú)限總和，剩余模式為定義0，使得'F0'是垂直于約束形狀相對(duì)于操作者的質(zhì)量，使得{Φi（X）}i=0……∞，是一組正交函數(shù)： 其中Ci是由于正交性得到殘余模式Φo和約束之間屬性。 圖1 2.2 反振蕩器 那么位移區(qū)域被定義為無(wú)限的正交函數(shù)和： 其中的參數(shù)里是自由度、離散化可以通過分段來(lái)實(shí)現(xiàn)擴(kuò)展等級(jí)N： 在[7]中，已經(jīng)證明了位移音響場(chǎng)（式（2））導(dǎo)致由模型來(lái)表示的結(jié)構(gòu)在圖中描繪。自由系統(tǒng)2. N個(gè)單學(xué)位正在鋪設(shè)的自由程度單系統(tǒng)（M0，K0），使得所述N + 1個(gè)參數(shù)是結(jié)構(gòu)的自由度。重要的是要注意自由度l0etT給出的結(jié)構(gòu)位移在撞擊點(diǎn);因此，接觸力可以直接從l0etT來(lái)估計(jì)。這是第一個(gè)對(duì)于主要的區(qū)別的模式描述在那里接觸力取決于所有廣義坐標(biāo)。表達(dá)了人民群眾和剛度可能發(fā)現(xiàn)在[7]，并且回顧在附錄A. 2.3、評(píng)論 反振蕩器模型，如在圖2中描繪，可視為彈簧質(zhì)量模型的擴(kuò)展在很大程度上在文獻(xiàn)中使用的[3,4,6]（參照?qǐng)D3）其特征在于，所述參數(shù)。的確，如果所有的反振蕩器是消除，該模型將簡(jiǎn)化的成彈簧質(zhì)量模型特征的參數(shù)（K0，M0），而且附錄A中（A.4）和（A.6）將提供方程： 圖2 圖3 值得注意的是在等效的區(qū)別結(jié)構(gòu)質(zhì)量用在彈簧質(zhì)量模型和有效的結(jié)構(gòu)質(zhì)量的反用振子模型。取決于靜態(tài)模式通過施加一個(gè)靜力處的點(diǎn)獲得的結(jié)構(gòu)影響（見附錄A，方程（A.1））。 結(jié)構(gòu)等效質(zhì)量取決于受影響的結(jié)構(gòu)的基本頻率 [9]。為一個(gè)光束可以被估計(jì)為對(duì)于其中靜態(tài)模式變?yōu)轭愃频那闆r下結(jié)構(gòu)的基本模式形狀： 3.反振蕩器和彈簧質(zhì)量模型 反振蕩器在結(jié)構(gòu)反應(yīng)的影響，以簡(jiǎn)單的方式通過比較研究反振蕩器建模所獲得的結(jié)果通過傳統(tǒng)的彈簧質(zhì)量模型中得到的那些。如前所述，剩余質(zhì)量的位移具有物理顯著性，也就是說，它所對(duì)應(yīng)在點(diǎn)沖擊的結(jié)構(gòu)位移，這是由在給定的等效結(jié)構(gòu)質(zhì)量的情況下彈簧質(zhì)量模型的位移。這樣比較的結(jié)果通過兩個(gè)模型得出將在理解有幫助站在結(jié)構(gòu)響應(yīng)控制反振蕩器的作用。 3.1研究案例 研究的情況下，選擇相對(duì)應(yīng)的橫向在簡(jiǎn)支正方形截面鋼材沖擊梁，在其跨中通過球形彈丸鋼。該數(shù)據(jù)和材料特性列于表1。沖擊速度等于10米/秒。速度是在低速?zèng)_擊由給定的范圍內(nèi)[9]， 表1 其中m是射彈的質(zhì)量和是比率的平均接觸屈服壓力的單軸屈服壓力，即對(duì)于球形固體的接觸， 在赫茲接觸剛度，由下式給出: 其中和是等效的楊氏模量和相當(dāng)于曲率，由下式給出 和 Y是較軟的材料的屈服應(yīng)力，v是泊松比。除非另有說明，上述情況下，表示研究的標(biāo)稱情況。 對(duì)于研究的情況下的結(jié)構(gòu)響應(yīng)可以是從基本周期之間的比率的預(yù)測(cè)光束振動(dòng)和接觸期間，由下式給出[9] 結(jié)構(gòu)反應(yīng)變成準(zhǔn)靜態(tài)的一段比值小于或等于單位[9]。周期比等于23，這是比標(biāo)準(zhǔn)高得多，因此結(jié)構(gòu)響應(yīng)是動(dòng)態(tài)性的，是，光束的多個(gè)本征模式將參加動(dòng)態(tài)響應(yīng)和一個(gè)彈簧質(zhì)量模型為研究這種情況下，不好的效果。 3.2 結(jié)果的比較 三個(gè)影響模型將用于獲得研究的標(biāo)稱情況下的結(jié)果：一個(gè)有限元模型中，抗振蕩器模型和2度的自由彈簧質(zhì)量模型。有限元模型的結(jié)果將被視為用于評(píng)估的參考其他兩款車型的銜接。所使用的元素是Timoshenko梁元件和模擬，都用50個(gè)元素。從通過提取反振蕩器減少大量的有限元模型，剛度矩陣由杰奎林等作為解釋 [7]，該彈丸被建模為剛性塊附連到梁通過Hertz接觸剛度所有型號(hào)。 接觸力和位移的歷史，獲得通過有限元模型示于圖 4，而通過彈簧質(zhì)量所獲得的結(jié)果，反振蕩器模型示于圖圖5和6。 圖4 圖5 圖6 可以觀察到，與反振蕩器連接獲得的結(jié)果均符合的結(jié)果是一致的 有限元模型。 彈簧質(zhì)量模型沒有預(yù)測(cè)接觸虧損對(duì)于這種情況，最大接觸力高估也是如此。由于這樣的事實(shí)，即結(jié)構(gòu)響應(yīng)是動(dòng)態(tài)的性質(zhì)，不能通過一彈簧質(zhì)量模型很好地描述?？梢酝ㄟ^彈簧質(zhì)量模型的行為很好地理解，看著與彈簧質(zhì)量的模型和反振蕩器模型參數(shù)。彈丸質(zhì)量和接觸剛度是相同的兩個(gè)模型這是等于33.1g和而相關(guān)的模型結(jié)構(gòu)的參數(shù)列于表2和3。 表二 序號(hào) 質(zhì)量m（g） 力矩 頻率 1 32.3 45.4 6.0 2 10.3 128.2 17.8 3 5.1 225.7 33.6 4 3.0 317.3 51.9 5 1.9 390.7 71.9 （序號(hào)5） 由,即靜態(tài)模式類似于基本模式形狀因?yàn)闆_擊位置是在簡(jiǎn)支梁的中間跨度。為彈簧質(zhì)量模型，所述彈丸是連接到結(jié)構(gòu)的等效質(zhì)量通過接觸剛度，而對(duì)于反振蕩器建模它被連接到剩余質(zhì)量（參照?qǐng)D2）。殘余質(zhì)量為是有效的區(qū)別結(jié)構(gòu)質(zhì)量和多個(gè)反振蕩器（參見附錄A）。可以注意的是，等效結(jié)構(gòu)質(zhì)量（）事實(shí)上，比反振蕩器5對(duì)應(yīng)的模型剩余質(zhì)量多。等同結(jié)構(gòu)質(zhì)量是集中到了彈簧質(zhì)量單質(zhì)模型;相反的是，由于反振蕩器和剩余系統(tǒng)有效的結(jié)構(gòu)質(zhì)量m分布為反振蕩器模型。這就是為什么慣性效果成為彈簧質(zhì)量模型重要原因，盡管總的結(jié)構(gòu)質(zhì)量幾乎相同的兩款車型（即）。結(jié)構(gòu)對(duì)質(zhì)量的影響分布，可以通過觀察光束被觀察在初次接觸式位移，通過雙方獲得模型（圖7）?？梢灾赋龅氖牵瑥椈少|(zhì)量模型光束開始移動(dòng)相對(duì)緩慢的由于由集總慣性效應(yīng)結(jié)構(gòu)等效質(zhì)量。彈丸改變了在初次接觸，因?yàn)閯?dòng)能方向當(dāng)?shù)貕汉垡呀?jīng)使用（即，相對(duì)位移在接觸點(diǎn)波束和彈丸之間的聯(lián)系）而不是全球性的結(jié)構(gòu)變形（即光束位移）。 事實(shí)上，反振蕩器可被視為一組上的剩余質(zhì)量調(diào)諧質(zhì)量阻尼器，該控制殘余物質(zhì)的運(yùn)動(dòng)，或者換句話說，該結(jié)構(gòu)在沖擊點(diǎn)運(yùn)動(dòng)。這是很好理解，例如，假定沒有反振蕩器參與，并且模仿剩余質(zhì)量仍然等于6.3克（即剩余質(zhì)量相關(guān)的反振蕩器5）。結(jié)構(gòu)反應(yīng)將是如圖 8.相反的反振子模型（參見圖5），過多的結(jié)構(gòu)性位移一中由于低結(jié)構(gòu)慣性效應(yīng)被觀察到。 反振蕩器控制結(jié)構(gòu)位移通過施加一個(gè)力，由于該殘留的質(zhì)量殘留和反振蕩器之間的相對(duì)運(yùn)動(dòng)。這種現(xiàn)象在說明如下。 對(duì)于正在審議的情況下，位移反振蕩器和剩余質(zhì)量中呈現(xiàn)圖 9.清楚地了解，對(duì)于位移的目的，系統(tǒng)連接的兩個(gè)反振蕩器與比較位移圖4中的剩余質(zhì)量的。圖9（a），以及接下來(lái)的三防振蕩器的位移比較圖，圖9（b）。 反振蕩器之間的相對(duì)位移和剩余質(zhì)量引起的彈簧的延伸附連到抗振蕩器在沖擊早期實(shí)例。彈簧的延伸引起的阻力上的剩余質(zhì)量，將起到相反的沖擊方向上的第一個(gè)影響早期實(shí)例;然而，該阻力的方向不會(huì)一定是相反的沖擊方向。事實(shí)上是抵制力控制的運(yùn)動(dòng)剩余質(zhì)量;否則，殘留量會(huì)取代的方式，如圖所示。圖8（a）中，凈力施加通過在完成對(duì)剩余質(zhì)量反振蕩器沖擊持續(xù)時(shí)間示于圖 10。 另一個(gè)重要的發(fā)現(xiàn)是，在較高的抗振蕩器不顯示的太大的位移與剩余質(zhì)量不同（參照?qǐng)D9（b））。實(shí)際上， 高振蕩跟隨剩余質(zhì)量的移動(dòng)，由于它們的低質(zhì)量和高剛度值（參看表2）。這可以更清楚地從圖11中觀察到，即呈現(xiàn)質(zhì)的變化和剛度上升反振蕩器的數(shù)目。反振蕩器向下位移可以觀察到。該網(wǎng)絡(luò)第一個(gè)反振蕩器覆蓋結(jié)構(gòu)質(zhì)量的50％以上;此外，反振蕩器5，在所有中覆蓋超過90％的有效的結(jié)構(gòu)質(zhì)量。就是說為什么只有學(xué)習(xí)的名義下網(wǎng)絡(luò)反振蕩器提供了非常好的效果。該剛度在增加以相同的方式，這允許理解為什么高階反振蕩器跟隨的位移剩余質(zhì)量。 表三 圖7 圖8（a） 圖8（b） 圖9（a） 圖9（b） 附錄2 Structural response of impacted structure described through anti-oscillators S. Pashah, M. Massenzio, E. Jacquelin Laboratoire Biome ′canique et Me ′canique des Chocs (LBMC), Universite ′ Lyon-1 INRETS, IUT B, 17 Rue de France, 69627 Villeurbanne Cedex, France Received 19 February 2007; accepted 4 June 2007 Available online 28 June 2007 Abstract Prediction of structural response for low velocity impact on ?exible structures is important for the selection of an appropriate impactmodel. This paper considers a new modeling of an impacted structure based on so-called anti-oscillators. The representation of animpacted structure through its anti-oscillators permits to understand the effects of different impact parameters on the structural response. It is shown that the ?rst anti-oscillator permits not only to predict the expected structural response but also a good estimation of the impact duration. 2007 Elsevier Ltd. All rights reserved. Keywords: Anti-oscillators; Structural response; Impact duration; Low velocity impact 1. Introduction Modeling for impact on a ?exible structure should take into account projectile’s motion, interaction between the projectile and the structure, and the dynamics of the structure. A complete modeling of an impact event can be achieved through 3D Finite Element approach. The consideration of the interaction in a 3D Finite Element model increases the modeling efforts and the time required for numerical simulation. This problem can be overcome primarily by modeling the interaction as a non-linear spring (stiffness kcontact) describing the contact behavior: Hertz’s law is the most widely used [1–3]. So the simulations can be performed for the impacted structure connected to the projectile by a contact spring. If the structure is complex or discretized with a very ?ne mesh, the numerical cost will be very high. In order to decrease the number of degrees of freedom involved in the simulations, the modal description of the structures may be used [1–3,5]. A truncation of the modal expansion can be done: the problem rests in de?ning the right rank of truncation. In certain situations, the structural behaviorcan be well described through its fundamental mode of vibration, in such case the structure is modeled as the single degree of freedom spring-mass system attached to th projectile through a non-linear contact stiffness [3–6]. Spring-mass model is the simplest impact model that provides the results of reasonable accuracy if the structure behaves in a quasi-static manner [3,4,6]. That is why some researchers have proposed methods to predict structural response for low velocity impact [3,5,6,9]. The aim of such methods is to select a modeling approach between the modal description and a spring-mass system. Jacquelin et al. [7] developed an impact modeling based on some structural parameters called ‘‘a(chǎn)nti-oscillators’’. The anti-oscillators are related to the displacement ?eld which is based on the constraint modes and the static mode of the impacted structure. The anti-oscillator model has the features of both the spring-mass model and modal description: it may be viewed as the modal description based on some ‘‘a(chǎn) priori’’ information regarding the impact location and direction. The aim of this paper is to use the anti-oscillator model for the comprehension of different aspects related to impact modeling as well as to develop an improved criterion for the prediction of structural response. In the following, a review of the anti-oscillator model is ?rstpresented, followed by the explanation of the structural response through anti-oscillators. In the last part of the paper, it is shown that this modeling approach permits the estimation of the impact duration. 2. Anti-oscillator model 2.1. Kinematic description The anti-oscillators are based on a speci?c displacement ?eld de?ned through a static mode and some constraint modes of the impacted structure, de?ned as follows [7] (see Fig. 1): The static mode ‘‘fst’’ is the shape of the structure caused byastaticload‘‘Fst’’ applied at the point of impact in the impact direction, such that there is unit displacement at that point. For the structures with rigid body modes, the static mode is replaced by a rigid body mode [7]. The constraint modes are the eigenmodesof the constrained impacted structure: an additional boundary condition is applied to the impacted structure for constraining its motion in the impact direction at the point of impact, so that the displacement is zero at that point. Hence the circular eigenfrequencies of theconstrained modes would be the circular anti-resonance frequencies of the original structure corresponding to the point of impact. The static mode is not necessarily orthogonal to the constraint modes. So, in order to obtain the displacement ?eld as an in?nite sum of orthogonal functions, a residual mode f0 is de?ned, such that ‘‘f0’’ is orthogonal to the constraint shapes with respect to the mass operator, so that is a set of orthogonal functions: where ci are obtained owing to the orthogonality property between residual mode f0 and the constraint where ci are obtained owing to the orthogonality property between residual mode f0 and the constraint 2.2. The anti-oscillators The displacement ?eld can then be de?ned as an in?nite sum of the orthogonal functions: In [7], it has been proved that the displacement ?eld (Eq. (2)) leads to represent the structure by the model depicted in Fig. 2. N single degree of freedom systemsare laying on the single degree of freedom system (m0, k0) such that the N+1 parameters are the degrees of freedom of the structure. It is important to note that the degree of freedom l0etT gives the structural displacement at the point of impact; hence the contact force can be estimated directly from. This is the ?rst major difference with respect to the modal description where contact force depends on all generalized coordinates. The expressions for the masses and the stiffnesses may be found in [7] and are recalled in Appendix A. 2.3. Comments The anti-oscillator model, as depicted in Fig. 2, can beregarded as the extension of the spring-mass model largely used in the literature [3,4,6] (cf. Fig. 3) characterized by the parameters (kst, ms). Indeed, if all of the anti-oscillators are eliminated, the model will be simpli?ed into a spring-mass model characterized by the parameters (k0, m0). Moreover, Eqs. (A.4) and (A.6) of Appendix A would provide: It is worth noting the difference between the equivalent structural mass ‘‘ms’’ used in the spring-mass model and the effective structural mass ‘‘mst’’ used in the anti-oscillator model. mst depends on the static mode of the structure obtained by applying a static force at the point of impact (see Appendix A, Eq. (A.1)). The structural equivalent mass ms depends on the fundamental frequency o1 of the impacted structure [9]. For a beam it can be estimated as For the case where static mode becomes similar to thefundamental mode shape of the structure, 3. Anti-oscillator and spring-mass model The in?uence of anti-oscillators on the structural response can be studied in a simple manner by comparing the results obtained through anti-oscillator modeling with the ones obtained through traditional spring-mass model. As stated earlier, the displacement of residual mass has a physical signi?cance, i.e., it corresponds to the structural displacement at the point of impact, which is given by the displacement of the equivalent structural mass in case of spring-mass model. So the comparison of the results obtained through two models will be helpful in under- standing the role of anti-oscillators in controlling the structural response. 3.1. Case of study The selected case of study corresponded to the transverse impact on a simply supported square cross-section steel beam at its mid span by a spherical steel projectile. The geometrical and material properties are given in Table 1. The impact velocity was equal to 10m/s. The velocity was in the range of low velocity impact given by [9]: where m is the mass of the projectile and is the ratio of mean contact yield pressure to the uniaxial yield stress,. For the contact of spherical solids, Y is the yield stress of the softer material, n is the Poisson’sratio. Unless otherwise stated, the above case represents the nominal case of study. The structural response for the case of study can be predicted from the ratio between the fundamental period of beam vibration and the contact period, given by [9] The structural response becomes quasi-static for a period ratio less than or equal to unity [9]. The period ratio is equal to 23, which is much higher than unity, hence the structural response would be dynamic in nature, i.e., several eigenmodes of the beam would participate in the dynamic response and a spring-mass model would not provide good results for this case of study. 3.2. Comparison of results Three impact models will be used for obtaining the results of the nominal case of study: a Finite Element model, an anti-oscillator model and a two degree of freedom spring-mass model. The results of Finite Element model will be considered as a reference for evaluating the convergence of other two models. The used element is a Timoshenko beam element and the simulations are carried out with 50 elements. The anti-oscillators are extracted from the Finite Element model through reduced mass and stiffness matrices as explained by Jacquelin et al. [7]. The projectile is modeled as a rigid mass attached to the beam through Hertz contact stiffness for all models. The contact force and displacement histories, obtained through the Finite Element model are presented in Fig. 4, whereas the results obtained through the spring-mass and the anti-oscillator models are presented in Figs. 5 and 6. It can be observed that the results obtained with only ?ve anti-oscillators are in good agreement with the results of Finite Element model. The spring-mass model does not predict the contact loss for this case; moreover, the maximum contact force is overestimated as well. This is due to the fact that thestructural response is dynamic in nature which cannot be well described through a spring-mass model. The behaviorof the spring-mass model can be well understood by looking at the modeling parameters related to spring-mass and anti-oscillator models. The projectile mass and the contact stiffness are the same for both models which are equal torespectively, whereas the structural parameters related to the models are given in Tables 2 and 3. It can be noted that. This is due to the fact that the static mode is similar to the fundamental mode shape because the impact location is at the mid span of the simplysupported beam. For spring-mass model, the projectile i on nected to the equivalent mass of the structure ‘‘ms’’ through contact stiffness, whereas for the anti-oscillator model it is connected to the residual mass ‘‘m0’’ (cf. Fig. 2). The residual mass m0 is the difference between the effective structural mass ‘‘mst’’ and the masses of the anti-oscillators(cf. Appendix A). It can be noted that the equivalent structural mass (ms ? 58.8 g) is much more than the residual mass corresponding to the model with ?ve anti- oscillators (m0 ? 6.3 g). In fact, the equivalent structural mass is lumped into a single mass ms for spring-mass model; contrary to that, the effective structural mass mst is distributed for the anti-oscillator model thanks to the anti-oscillators and the residual system. That is why the inertial effects become more important for the spring-mass model despite the total structural mass is almost the same for the two models. The effects of structural mass distribution can be observed by looking at the beam displacement during ?rst contact, obtained through both models (Fig. 7). It can be noted that the beam started moving relatively slowly for the spring-mass model due to important inertial effects caused by the lumped structural equivalent mass ms. The projectile changed its direction during ?rst contact because the kinetic energy had been used in local indentation (i.e., relative displace- ment between beam and projectile at the point of contact) rather than global structural deformation (i.e., beam displacement). In fact, the anti-oscillators may be viewed as a set of tuned mass dampers on the residual mass, which control the motion of the residual mass, or in other words, the motion of the structure at the point of impact. This can be well understood if, for instance, it is assumed that no anti- oscillators were present, and as if the residual mass were still equal to 6.3 g (i.e., the residual mass related to the anti-oscillator model with ?ve anti-oscillators). The structural response would be as shown in Fig. 8. Contrary to the anti- oscillator model (cf. Fig. 5), excessive structural displace- ment can be observed due to low structural inertia effects. The anti-oscillators control the structural displacement by exerting a force on the residual mass owing to the relative motion between the residual and anti-oscillator masses. This phenomenon is explained in the following. For the case under consideration, the displacements of the anti-oscillators and the residual mass are presented in Fig. 9. For the purpose of clear visibility, the displacements of the ?rst two anti-oscillators are compared with the displacement of the residual mass in Fig. 9(a), and the displacements of the next three anti-oscillators are com-pared in Fig. 9(b). The relative displacement between the anti-oscillators and the residual mass causes the extension of the springsattached to the anti-oscillators at the early instance of theimpact. The extension of the springs causes a resisting force on the residual mass that would act opposite to the impact direction at the early instance of the ?rst impact; never- theless, the direction of the resisting force would not necessarily be opposite to the impact direction afterwards. Indeed, the resisting force controls the motion of the residual mass; otherwise, the residual mass would displace in the manner as shown in Fig. 8(a). The net force exerted by the anti-oscillators on the residual mass during complete impact duration is shown in Fig. 10. Another important observation is that the higher anti-oscillators do not show the displacement much differentfrom the residual mass (cf. Fig. 9(b)). In fact, the higher anti-oscillators follow the movement of the residual mass due to their low mass and high stiffness values (cf. Table 2). This can be observed more clearly from Fig. 11, that presents the variation of mass and stiffness with ascending number of anti-oscillators. An important decrease in the masses of anti-oscillators can be observed. The ?rst anti- oscillator covers more than 50% of the structural mass; moreover, the masses of ?rst ?ve anti-oscillators cover more than 90% of the effective structural mass. That is why only ?ve anti-oscillators provide very good results for the nominal case of study. The stiffnesses increase in the same manner, which permits to understand why thehigher-order anti-oscillators follow the displacement of theresidual mass. 25