基于ANSYS的汽車萬向傳動裝置有限元分析
基于ANSYS的汽車萬向傳動裝置有限元分析,基于,ansys,汽車,萬向,傳動,裝置,有限元分析
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譯文題目： A new analysis method of evaluation
to improve the vehicle vibration noise
評估改進(jìn)車輛振動噪音的全新分析法
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A new analysis method of evaluation to improve the vehicle vibration noise
Abstract
The design of an automotive powerplant mounting system is an essential part in vehicle safety and improving the vehicle noise, vibration and harshness (NVH) characteristics. One of the main problems encountered in the automotive design is isolating low frequency vibrations of the powerplant from the rest of the vehicle. The significant powerplant mass makes the choice of frequency and mode arrangements a critical design decision. Several powerplant mounting schemes have been developed to improve NVH properties concentrating on the positioning and design of resilient supports. However these methods are based on decoupling rigid body modes from a grounded powerplant model which ignores chassis and suspension system interactions.But it cannot be stated that decoupling the grounded rigid body modes of the powerplant will systematically reduce chassis vibrations. In this paper, a new analytical method is proposed to examine the mechanisms of coupling between the powerplant and the vehicle chassis and subsystems. The analytical procedure expands the equation of motion of the vehicle components to such that a domain of boundary conditions used in the 6 degrees-of-freedom powerplant mounting model can be defined. An example of this new procedure is given for improving NVH chassis response at idle speed using the torque roll axis decoupling strategy.
Keywords: Powerplant mounting system; Optimization; Dynamic isolation; Coupled systems
1. Introduction
In vehicles, the engine mounts play an essential role for the noise, vibration and harshness (NVH) comfort. The main functions of these mounts (rubber or hydraulic) are to provide static supports for the powerplant and to isolate the vibrations of the powerplant from the rest of the vehicle. To provide design characteristics necessary for the NVH improvement in terms of rigidity and damping it is essential to simulate the responses of the powerplant mounting system to low frequency vibrations. Is is essential that the model includes the primary interactions between the powerplant mounting system and each of the vehicle subsystems. In the early stages of the vehicle design most of the necessary data needed from the subsystems are not yet fully described. Thus, to begin a theoretical layout of the powerplant mounting system, some reasonable assumptions of the vehicle components must be made. Specifically, the model includes rigid body representations of the powerplant and the chassis; with appropriate values for the location of the centers of gravity, masses, and moments of inertia. This simulation model enables the assessment of the rigid body modes of the powerplant in the vehicle. As well, the motions of the powerplant and the chassis under various engine operating conditions (idle, full load speed sweep) and road/wheel inputs can be analyzed.
Equations of motion for the powerplant mounting system include parameters for a rigid chassis. On the contrary, the chassis flexibility may have a significant effect on powerplant vibrations and mounting forces transmitted from the powerplant to the structure, especially when flexible vibration modes of the chassis are excited. The dominant vibration modes of body structure at idle speed should be the first longitudinal bending mode and the first torsional mode, normally above 30–35 Hz. Experimental verification of the simulation model’s assumptions through measurements of the vibration modes of the chassis should be included in future work.
The current industrial strategies use a model approach to analyze the harmonic response of the powerplant on resilient supports attached to ground (Brach, 1997; Khajepour and Geisberger, 2002). The 6 degrees-of-freedom model used in the modal analysis is interesting insofar as the response to an excitation is calculated and interpreted according to the position in frequency and to the form of the modes.
Typical design strategies move input source frequencies away from the rigid body natural frequencies of the powerplant in order to avoid resonances (Gray et al., 1990; Kano and Hayashi, 1994). Vibrations are minimized within this design approach by manipulating the rigid body modes of the grounded powerplant and shaping the response through the torque roll axis decoupling and the elastic axis decoupling methods attempt. The background theory of these techniques is widely described in literature (Patton and Geck, 1984; Singh and Jeong, 2000; Brach, 1997). However, by considering the powerplant to be grounded these design strategies neglect the influences of the chassis, exhaust subsystem, drive-shaft, wheel suspension . . . .
Lately, researches have focused on the significance of the rigid body modes alignment for grounded powerplant to its invehicle behavior (Sirafi and Qatu, 2003; Hadi and Sachdeva, 2003). These studies deal with the accuracy of NVH vehicle models and raise the problem of interactions between the different subsystems. Various powertrain models have been studied and their accuracy was discussed through a full vehicle model. By the evaluation of actual cases, the existence of these interactions have been clearly demonstrated. Nevertheless, no general formalism have been introduced to evaluate the limits of the modeling assumptions made during the development of the classical 6 degrees-of-freedom powerplant mounting schemes.
The aim of the proposed method is to highlight and identify, through an analytical procedure, the relationships between the powerplant mounting schemes and the vehicle response characteristics. In the second section, the general equations of motion are reformulated using an original matrix, the coupling matrix introduced for coupled plates (Bessac and Guyader, 1996). With the characteristics of the coupling matrix, acceptable boundary conditions used in the traditional 6 degrees-of-freedom mounting strategies can be defined for different engine operating conditions. As an example in Section 3, these parameters are defined for engine models in the idle state. In the last section of the paper, the issue of the torque roll axis decoupling strategy is analyzed using the coupling parameters in terms of improvement of the dynamic chassis responses at idle speed.
2. Formulation of the coupling problem
2.1. Modelling of the vehicle system
Derivation of the equations of motion to simulate dynamic behaviors of powerplant mounting systems with supporting structures, a good modelling of the total vehicle system can consist of four subsystems: the powerplant which includes engine and transmission, the engine mounts, the chassis and the suspension. Since small displacements can be assumed, the powerplant is modelled as rigid body of time-invariant inertial matrix of 6 dimensions. The powerplant is supported by an arbitrary number of mounts on the vehicle chassis, that is modelled as an elastically suspended rigid body as shown in Fig. 1.
The mounts classically used in powerplant mounting application are bonded metal-rubber construction. It is possible to get better isolation effects than conventional rubber mount systems with hydraulic engine mount. Hydraulic engine mount control the damping characteristics by using the fluid viscosity. Elastomeric materials behave visco-elastically, thus engine mounts are represented by three sets of
.mutually perpendicular of linear springs and viscous dampers in parallel. No rotational stiffness of the mounts has been considered. The stiffness matrix Kmi and damping matrix Cmi of a mount i can be written in the local coordinate system as:
，and (1)
Fig. 1. Powerplant mounting model
Fig. 2. Translational u and rotational θ displacements of the powerplant center of gravity.
The subscript mi corresponds to the mount frame coordinates Rmi (Fig. 1) of the ith mount. The stiffness and damping matrices must be transformed from the local mount coordinate system Rmi to the global coordinate system R by the following linear transformation:
,and (2)
The element Πmi is the transformation matrix from the local coordinate system Rmi to the global one R. The elements of Пmi consist of directional cosines of the local frame with respect to R defined from Euler angles.
2.2. Equations of motion
Another transformation is necessary to express the equations of motion of the powerplant and chassis centers of gravity in terms of displacements and rotations. This transformation relates the displacements of each mount with respect to the displacements and rotations of the powerplant and chassis centers of gravity. The superscripts (e) and (c) stands for powerplant and chassis respectively. The superscript (b) may refer to either the powerplant or the chassis. A generalized vector q (Eq. (3)) is defined by combining translational u and rotational θ displacements of the centers of gravity of the powerplant (Fig. 2) and of the chassis.
(3)
The position vector of the ith mount’s center of elasticity with respect to the center of gravity of the powerplant and the chassis are given in terms of global coordinate system as:
(4)
and each has a corresponding skew asymmetric matrix defined as:
(5)
with a generalized form:
(6)
Let ui(b) be the translational displacement vector at the mounting point i for the rigid body (b) side. The relative translational displacement vector δi for the ith mount for small motions is related to the rigid body center of gravity motions and the translational displacements at the mounting point according to Eq. (7).
(7)
The translational reaction force fi(e) and fi(c) and moment reactionτi(e) andτi(c) resulting from the application of the elastic forces of mounting i on the powerplant and the chassis centers of gravity can be expressed in the R frame as:
(8)
At idle speed, the connection to the ground is simply represented by four systems of linear spring and viscous damper in parallel at each wheel, characterized by their stiffness and damping coefficients following the three directions of the vehicle frame coordinates R. The translational reaction force fk(c) and moment reactionτk(c) from the kth suspension applied to the chassis can be expressed in the frame R with the displacement of the chassis u k(c) at the supporting point as:
(9)
Similarly, for the road/wheel inputs, a simple model can be used for the wheel-suspension system, with a single degree of freedom. This can be represented by a mass and a spring accounting for the wheel mass and the tire’s stiffness in parallel with a spring and a damper accounting for the suspension system. The dynamic interaction between the vehicle suspension and the powerplant mounting system should be included in future work.
Assuming all elastic loadings from all mounts and suspension, the total elastic loadings on the powerplant and chassis centers of gravity can be expressed through a generalized square stiffness matrix K of 12 dimensions (10), resulting from the assembly of the elementary stiffness matrices (mounts and suspensions).
(10)
The matrix K(e→c) is the powerplant’s matrix of influence on the chassis and the reciprocal, K(c→e) is the chassis’s matrix of influence on the powerplant. Using a similar assembly procedure to the elastic loadings, the total damping loadings on the powerplant and chassis centers of gravity can be expressed by a generalized square stiffness damping matrix C of 12 dimensions (11).
(11)
Since all component reactive forces are derived in terms of the generalized coordinates, and assuming small oscillations, the equations of motion of the powerplant and the chassis can be written as the matrix form in the frequency domain:
(12)
The vector F =t{F(e) F(c)} is the generalized external load vector. The external excitations are harmonics with known frequencies,amplitudes and phases. Engine excitation forces are applied to the powerplant at the center of the crankshaft location.
The response to road inputs can be studied by applying forces or displacements at the suspensions location of ground contact.The matrix M is the generalized mass matrix of the system (13).
(13)
With
and
m(b) is the mass of the rigid body (b) and Mτ(b) its inertia matrix. C is the generalized viscous damping matrix assuming a proportionally damped system. If a structural damping matrix H is considered, viscous damping term jωC may be replaced by the structural damping term jH. For the following developments, a complex stiffness is used to model the dynamic behavior of the isolators. The bar indicates that the stiffness term is complex ().
2.3. Introduction of the coupling matrix
The response of the powerplant and chassis centers of gravity can be calculated through the solving of Eq. (12). Then, the complex matrix inversion of Eq. (14) is classically used.
(14)
The inversion of the impedance matrix can be numerically resolved. Nevertheless, this method hinders the understanding of the coupling phenomena between the powerplant and the chassis. From the traditional equation of motion (14), one can isolate a matrix presenting only terms related to the coupling from the two bodies (15).
(15)
For the sake of physical meaning of the coupling mechanism, the term (?ω2M(e)) ?1F(e) in Eq. (15) represents the displacement of the powerplant subjected to its own excitation when the chassis is blocked (suspensions with infinite stiffnesses).This configuration represents the typical industrial model of the grounded behavior of the powerplant (Fig. 3(a)). The term (?ω2M(c)) ?1F(c) represents the displacement of the chassis subjected to his own excitation when the powerplant is blocked (null displacements) (Fig. 3(b)). This configuration, however, does not represent a realistic behavior. We can express the two configurations by the generalized vector displacement of the uncoupled blocked bodies t{q0(e) q0(c) } (16).
(16)
Fig. 3. Uncoupled blocked bodies.
While revealing the displacement vector of the coupled systems, Eq. (15) takes the form of a coupling matrix D (Bessac and Guyader, 1996) (Eq. 17).
(17)
With
Each coupling matrix term represents the action of the powerplant mass displacement (respectively chassis) on the chassis mass displacement (respectively powerplant). The matrix of coupling describes the exchange between the masses independent of the external excitation. The coupling matrix, studied in more details in Section 3, is a practical solution to predict the global behavior of a system starting from the behavior of the isolated subsystem.
評估改進(jìn)車輛振動噪音中的全新分析法
摘要
汽車發(fā)動機(jī)裝備系統(tǒng)的設(shè)計(jì)是車輛安全以及汽車振動噪音改善不可或缺的重要部分。汽車設(shè)計(jì)中遇到的主要問題之一是將發(fā)動機(jī)的低頻振動與其它交通工具隔離開來。頻率和模式安排的選擇成為了質(zhì)量顯著的發(fā)動機(jī)的一個(gè)重要的設(shè)計(jì)決策。幾個(gè)發(fā)動機(jī)裝備方案已被開發(fā)出來，并且應(yīng)用于改善有關(guān)定位以及彈力支撐設(shè)計(jì)的噪音性能中。將剛體模式從接地發(fā)動機(jī)模式中解耦，這些方法是以此基礎(chǔ)的，然而，接地發(fā)動機(jī)模式忽略了底盤與懸架系統(tǒng)之間相互作用。但是我們不能說發(fā)動機(jī)接地剛體模式去耦將會減弱底盤系統(tǒng)的振動。本論文提出了一個(gè)全新的分析法來檢測發(fā)動機(jī)、車輛底盤以及子系統(tǒng)的耦合機(jī)制。分析程序?qū)ζ嚵悴考倪\(yùn)動方程進(jìn)行了擴(kuò)展，如此一來，就能對應(yīng)用在自由度為6度的發(fā)動機(jī)裝備模式中的邊界條件域進(jìn)行定義了。本論文通過扭矩輥軸耦戰(zhàn)略，給出了一個(gè)全新程序?qū)嵗?，本?shí)例詣在改善低速底盤噪音反應(yīng)。
關(guān)鍵詞：發(fā)動機(jī)裝備系統(tǒng)，最優(yōu)化，動態(tài)隔離，耦合系統(tǒng)
1.簡介
發(fā)動機(jī)架在緩和交通工具的振動噪音方面起著重要的作用。這些裝備（橡膠或液壓）的主要作用是為發(fā)動機(jī)提供靜態(tài)支撐，以及將其振動與其它交通工具隔離。如果要想從剛性和阻尼方面得到改善振動噪音所需的設(shè)計(jì)特征，就很有必要將發(fā)動機(jī)裝備系統(tǒng)的反應(yīng)刺激至低頻振動，這一點(diǎn)非常重要。這個(gè)模式包括發(fā)動機(jī)裝備系統(tǒng)與每個(gè)車輛子系統(tǒng)之間的基本相互作用。在車輛設(shè)計(jì)的初級階段，大多數(shù)子系統(tǒng)所需的必要數(shù)據(jù)尚未被完全描述出來。因此，想要開始發(fā)動機(jī)裝備系統(tǒng)的理論設(shè)計(jì)，就必須對車輛零部件做出一些合理設(shè)想。具體來講，該模式包含發(fā)動機(jī)和底盤的剛體再現(xiàn)，以及具有重心位置，質(zhì)量和慣性矩的恰當(dāng)值。該仿真模型有助于發(fā)動機(jī)剛體模式的評估。同樣，不同發(fā)動機(jī)條件下（閑置或滿載負(fù)載速度下）的發(fā)動機(jī)和底盤的運(yùn)動都能被分析出來。
發(fā)動機(jī)裝備系統(tǒng)的運(yùn)動方程包含一個(gè)剛性底盤的參數(shù)。與之對應(yīng)的，底盤靈活性可能會對發(fā)動機(jī)震動和由發(fā)動機(jī)傳輸至結(jié)構(gòu)的裝備力量，尤其是在底盤的彈性震動模式被激活的時(shí)候，產(chǎn)生強(qiáng)烈作用。閥體結(jié)構(gòu)在空轉(zhuǎn)速度時(shí)的顯性運(yùn)動模式應(yīng)該是第一縱彎曲模式和第一扭轉(zhuǎn)模式，通常是在30—35赫茲之上。通過測量底盤振動模式，得出經(jīng)實(shí)驗(yàn)鑒定的仿真模型假設(shè)，這一點(diǎn)需被列入今后的工作中。
現(xiàn)有的工業(yè)戰(zhàn)略用模型方法來分析發(fā)動機(jī)在接地彈性支撐上的和聲反應(yīng)（布拉齊，1997；Khajepour and Geisberger, 2002）。模式分析中所采用的6度自由模型是有趣的，它對刺激的反應(yīng)可以被計(jì)算出來，也能根據(jù)頻率定位和模式形式被解釋出來。
典型的設(shè)計(jì)戰(zhàn)略將輸入端電源頻率從發(fā)動機(jī)剛性主體自然頻率中移除以避免共振（Gray 等，1990；卡諾和林文夫，1994）。該設(shè)計(jì)方法能通過兩種方式將振動最小化，一種是手動操作接地發(fā)動機(jī)的剛體模式，另一種是令轉(zhuǎn)矩軋輥軸線解耦和彈性軸解耦形成反應(yīng)的方式。很多文獻(xiàn)中都討論了這些技術(shù)的背景理論（Patton and Geck, 1984; Singh and Jeong, 2000; Brach, 1997）。但是，考慮到將要接地的發(fā)動機(jī)，這些設(shè)計(jì)策略忽略了底盤、排氣子系統(tǒng)、驅(qū)動器軸和車輪懸浮等的影響。
最近，很多研究都集中在接地發(fā)動機(jī)的剛體模式校準(zhǔn)對其車輛行為的影響上（Sirafi and Qatu, 2003; Hadi and Sachdeva, 2003）。這些研究論述了振動噪音車輛模型的準(zhǔn)確性，并引發(fā)了不同子系統(tǒng)之間的相互作用的問題。目前已有人研究出了各種動力系統(tǒng)模型，并使用完整的車輛模型討論其準(zhǔn)確性。對實(shí)際案例進(jìn)行評估發(fā)現(xiàn)：這些相互作用是存在的。然而，目前還沒有引入任何形式體系來評估在6度自由發(fā)動機(jī)裝備方案的發(fā)展中制定的關(guān)于模型假設(shè)的限制性。
所提出的方法詣在通過分析程序強(qiáng)調(diào)和確認(rèn)發(fā)動機(jī)裝備方案和車輛反應(yīng)特點(diǎn)之間的關(guān)系。在第二部分中，作者采用原始矩陣模型以及為耦合平板引進(jìn)的耦合矩陣模型，再次形成了運(yùn)動的一般方程（Bessac and Guyader, 1996）。通過耦合矩陣模型的特點(diǎn)，就能為不同的引擎操作條件定義傳統(tǒng)6度自由裝備戰(zhàn)略所采用的可接受的邊界條件。正如第三部分的例子所示：這些參數(shù)是為處于靜止?fàn)顟B(tài)的引擎模型定義的。該論文的最后一部分通過使用改善空轉(zhuǎn)速度下動態(tài)底盤反應(yīng)的耦合參數(shù)，分析了轉(zhuǎn)矩軋輥軸線去耦戰(zhàn)略。
2. 耦合問題構(gòu)想
2.1.車輛系統(tǒng)模型化
為了引出運(yùn)動方程來刺激具有支撐結(jié)構(gòu)的發(fā)動機(jī)裝備系統(tǒng)的動態(tài)行為，總車輛系統(tǒng)的好模型由4個(gè)子系統(tǒng)構(gòu)成：包含引擎和傳動裝置的發(fā)動機(jī)、發(fā)動機(jī)架、底盤和懸架。由于可對小排量做出假設(shè)，發(fā)動機(jī)被設(shè)計(jì)成了6維不變時(shí)的慣性矩陣剛體模型。發(fā)動機(jī)被車輛底盤上任意數(shù)量的裝備支撐，那就形成了如圖1所示的彈性懸浮剛體模型。
發(fā)動機(jī)裝備應(yīng)用系統(tǒng)通常使用的掛架是由金屬和橡膠制作而成的。有了液壓發(fā)動機(jī)架，就有可能帶來比傳統(tǒng)的橡膠裝備系統(tǒng)更好的隔離效果。液壓發(fā)動機(jī)架通過采用流體粘度來控制減幅特征。彈性材料的運(yùn)動也是有彈性的，因此，發(fā)動機(jī)架是由三組相互垂直的線性彈簧以及相互平行的粘性減震器為代表的。沒有考慮掛架的旋轉(zhuǎn)勁度。在局部坐標(biāo)系中，可通過如下方式表達(dá)掛架i的剛性矩陣Kmi和阻尼矩陣Cmi：
,and (1)
圖1：彈性懸浮剛性模型
圖2：發(fā)動機(jī)重心平移u和旋轉(zhuǎn)θ位移
,and (2)
相當(dāng)于框架的下標(biāo)mi能與ith掛架的Rmi（圖1）相協(xié)調(diào)。剛體和阻尼矩陣必須按照以下線性轉(zhuǎn)化，從局部掛架坐標(biāo)系Rmi轉(zhuǎn)換至總坐標(biāo)系R。Πmi是從局部坐標(biāo)系Rmi轉(zhuǎn)換至總坐標(biāo)系R的轉(zhuǎn)換矩陣。Πmi的構(gòu)成元素包含與由歐拉角定義的與R相關(guān)的局部框架的方向余弦。
2.2. 運(yùn)動方程
很有必要用另外一個(gè)變形來表達(dá)發(fā)動機(jī)和底盤重心在移位和旋轉(zhuǎn)方面的運(yùn)動方程。該變形描述了有關(guān)發(fā)動機(jī)和底架重心的移位和旋轉(zhuǎn)。下標(biāo)(e)和(c)分別代表發(fā)動機(jī)和底盤。下標(biāo)(b)可能指的是發(fā)動機(jī)和底盤二者之一。通過結(jié)合發(fā)動機(jī)（圖2）和底盤重心平移的u和旋轉(zhuǎn)的θ位移來定義一個(gè)廣義矢量q （方程3）。
(3)
依照總坐標(biāo)系，這里給出了有關(guān)發(fā)動機(jī)和底盤重心的ith掛架彈性中心的位矢：
(4)
每一個(gè)都具有對應(yīng)的斜交非對稱矩陣，定義如下：
, (5)
廣義形式為：
(6)
將ui(b)看成剛體(b)側(cè)裝備點(diǎn)i的平行位移矢量。根據(jù)方程7，ith掛架的小幅度運(yùn)動相對平行位移矢量δi是與剛體重心運(yùn)動以及裝備點(diǎn)的平行位移相關(guān)的。
(7)
由發(fā)動機(jī)和底盤重心i裝備的彈力所產(chǎn)生的平移反作用力fi(e) 和fi(c)以及瞬間作用力τi(e) 和τi(c)可在R框架內(nèi)按如下方程表達(dá)：
(8)
在空轉(zhuǎn)速度，接地僅由每個(gè)車輪4組線性彈簧和平行粘性減震器代表，其特點(diǎn)為：它們的剛性和阻尼系數(shù)遵循車架坐標(biāo)R的3個(gè)方向。底盤Kth懸架的平行反作用力fk(c)和瞬間作用力τk(c)可在總坐標(biāo)系R（在該坐標(biāo)系中，底盤在支撐點(diǎn)有位移：u k(c)中表達(dá)為：
(9)
同樣，路段/車輪輸入可在車輪懸浮系統(tǒng)中使用一個(gè)自由度單一的簡單模型。這是以能夠解釋車輪質(zhì)量的彈簧和質(zhì)量與能夠解釋懸浮系統(tǒng)的彈簧和減震器相互平行為代表的。車輛懸浮和發(fā)動機(jī)裝備系統(tǒng)間的動態(tài)相互作用需被列入到未來工作中。
假設(shè)所有彈性負(fù)荷均來自掛架和懸浮，發(fā)動機(jī)和底盤重心的彈性負(fù)荷可通過12維一般方形剛性矩陣K來表達(dá)，該矩陣由基礎(chǔ)剛性矩陣（掛架和懸?。┙M裝而成。
(10)
K(e→c)矩陣是發(fā)動機(jī)對底盤的影響力矩陣，它的求逆K(e→c)是底盤對發(fā)動機(jī)的影響力矩陣。對彈性負(fù)荷使用類似的組裝程序，發(fā)動機(jī)和底盤重心的總阻尼負(fù)荷可通過12維一般方形剛性阻尼矩陣C來表示。
(11)
由于所有組件反作用力是由總坐標(biāo)系派生出來的，假設(shè)振幅很小，發(fā)動機(jī)和底盤運(yùn)動方程可在頻域內(nèi)寫成矩陣形式：
(12)
F =t{F(e) F(c)}這一矢量是廣義的外加載矢量。外部刺激和已知的頻率、振幅以及階段都是和聲。發(fā)動機(jī)刺激力被運(yùn)用到了發(fā)動機(jī)機(jī)軸中心。對路段輸入的反應(yīng)可通過在接地懸浮位置施加外力或者位移來研究。M矩陣是該系統(tǒng)（13）的廣義質(zhì)量矩陣。
(13)
與
,and
在該廣義質(zhì)量矩陣中，m(b)是剛體（b）質(zhì)量，Mτ(b)是其慣性矩陣。C是一個(gè)廣義的粘滯阻尼矩陣，它假設(shè)的是成比例的阻尼系統(tǒng)。若考慮到結(jié)構(gòu)阻尼矩陣H，粘滯阻尼術(shù)語jωC就可能被結(jié)構(gòu)阻尼術(shù)語jH所取代。在隨后的發(fā)展中，我們采用復(fù)雜的剛度來模擬隔離器的動態(tài)行為。該欄顯示：剛度術(shù)語是合成的()。
2.3. 耦合矩陣簡介
通過求解方程12，就能計(jì)算出發(fā)動機(jī)和底盤重心的反應(yīng)。然后，再次應(yīng)用方程14的復(fù)雜矩陣求逆。
(14)
阻抗矩陣求逆可通過數(shù)值求解。然而，這種方法阻礙了對發(fā)動機(jī)和底盤之間的耦合現(xiàn)象的理解。從傳統(tǒng)的運(yùn)動方程14，我們可以將只顯示耦合術(shù)語的矩陣與這兩個(gè)主體（15）隔離。
(15)
為了探求耦合機(jī)制的物理含義，方程15中的(?ω2M(e)) ?1F(e)表示的是底盤堵塞時(shí)（以無窮大的剛度懸?。碳ぐl(fā)動機(jī)所產(chǎn)生的位移。該結(jié)構(gòu)代表的是發(fā)動機(jī)接地行為的典型工業(yè)模型（圖3(a)）。(?ω2M(c)) ?1
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