外文翻譯---運(yùn)用雙擺載荷動(dòng)力學(xué)控制塔式起重機(jī)（節(jié)選）

1、<p>　　西 南 交 通 大 學(xué)</p><p>　　本科畢業(yè)設(shè)計(jì)（外文翻譯）</p><p>　　Control of Tower Cranes With Double-Pendulum Payload Dynamics</p><p>　　運(yùn)用雙擺載荷動(dòng)力學(xué)控制塔式起重機(jī)</p><p>　　年 級(jí): 2

2、006 </p><p>　　學(xué) 號(hào): 20061112 </p><p>　　姓 名: 陳東 </p><p>　　專(zhuān) 業(yè): 機(jī)械設(shè)計(jì)制造及其自動(dòng)化</p><p>　　指導(dǎo)老師: 于蘭峰 </p&g

3、t;<p>　　2010 年 6 月 </p><p>　　Control of Tower Cranes With </p><p>　　Double-Pendulum Payload Dynamics</p><p>　　Joshua Vaughan, Dooroo Kim, and William Singhose</p>&l

4、t;p>　　Abstract：The usefulness of cranes is limited because the payload is supported by an overhead suspension cable that allows oscilation to occur during crane motion. Under certain conditions, the payload dynamics m

5、ay introduce an additional oscillatory mode that creates a double pendulum. This paper presents an analysis of this effect on tower cranes. This paper also reviews a command generation technique to suppress the oscillato

6、ry dynamics with robustness to frequency changes. Experimental result</p><p>　　Key words：Crane , input shaping , tower crane oscillation , vibration</p><p>　　I. INTRODUCTION</p><p>

7、;　　The study of crane dynamics and advanced control methods has received significant attention. Cranes can roughly be divided into three categories based upon their primary dynamic properties and the coordina

8、te system that most naturally describes the location of the suspension cable connection point. The first category, bridge cranes, operate in Cartesian space, as shown in Fig. 1(a). The trolley moves along a

9、 bridge, whose motion is perpendicular to that of the tr</p><p>　　The second major category of cranes is boom cranes, such as the one sketched in Fig. 1(b). Boom cranes are best described in spherical

10、 coordinates, where a boom rotates about axes both perpendicular and parallel to the ground. In Fig. 1(b), is the rotation about the vertical, Z-axis, and is the rotation about the horizontal, Y -axis. The payload is s

11、upported from a suspension cable at the end of the boom. Boom cranes are often placed on a mobile base that allows them to change their workspace.</p><p>　　The third major category of cranes is tower cranes,

12、 like the one sketched in Fig. 1(c). These are most naturally described by cylindrical coordinates. A horizontal jib arm rotates around a vertical tower. The payload is supported by a cable from the trolley, which moves

13、radially along the jib arm. Tower cranes are commonly used in the construction of multistory buildings and have the advantage of having a small footprint-to-workspace ratio. Primary disadvantages of tower and boom

14、 cranes, </p><p>　　A common characteristic among all cranes is that the pay- load is supported via an overhead suspension cable. While this provides the hoisting functionality of the crane, it also presents

15、several challenges, the primary of which is payload oscillation. Motion of the crane will often lead to large payload oscillations. These payload oscillations have many detrimental effects including degrading

16、 payload positioning accuracy, increasing task completion time, and decreasing safety. A </p><p>　　Many researchers have focused on feedback methods, which necessitate the addition necessitate the additio

17、n of sensors to the crane and can prove difficult to use in conjunction with human operators. For example, some quayside cranes have been equipped with sophisticated feedback control systems to dampen payload sway. Howev

18、er, the motions induced by the computer control annoyed some of the human operators. As a result, the human operators disabled the feedback controllers. Given that the vast ma</p><p>　　Input shaping [7], [8]

19、 is one control method that dramatically reduces payload oscillation by intelligently shaping the commands generated by human operators [9], [10]. Using rough estimates of system natural frequencies and damping ratios,

20、 a series of impulses, called the input shaper, is designed. The convolution of the input shaper and the original command is then used to drive the system. This process is demonstrated with atwo-impulse input shaper and

21、a step command in Fig. 2. Note that th</p><p>　　Fig. 1. Sketches of (a) bridge crane, (b) boom crane, (c) and tower crane.</p><p>　　Fig. 2. Input-shaping process.</p><p>　　Input sha

22、ping has been successfully implemented on many vibratory systems including bridge [11]–[13], tower [14]–[16], and boom [17], [18] cranes, coordinate measurement machines[19]–[21], robotic arms [8], [22], [23], demi

23、ning robots [24], and micro-milling machines [25].</p><p>　　Most input-shaping techniques are based upon linear system theory. However, some research efforts have examined the extension of input shapi

24、ng to nonlinear systems [26], [14]. Input shapers that are effective despite system nonlinearities have been developed. These include input shapers for nonlinear actuator dynamics, friction, and dynamic

25、nonlinearities [14], [27]–[31]. One method of dealing with nonlinearities is the use of adaptive or learning input shapers [32]–[34]</p><p>　　In Section II, the mobile tower crane used during exper

26、imental tests for this paper is presented. In Section III, planar and 3-D models of a tower crane are examined to highlight important dynamic effects. Section IV presents a method to design multimode input shapers with s

27、pecified levels of robustness. InSection V, these methods are implemented on a tower crane with double-pendulum payload dynamics. Finally, in Section VI, the effect of the robust shapers on human operator performance<

28、/p><p>　　II. MOBILE TOWER CRANE</p><p>　　The mobile tower crane, shown in Fig. 3, has teleoperation capabilities that allow it to be operated in real-time from anywhere in the world via the Intern

29、et [15]. The tower portion of the crane, shown in Fig. 3(a), is approximately 2 m tall with a 1 m jib arm. It is actuated by Siemens synchronous, AC servomotors. The jib is capable of 340° rotation about the tower.

30、The trolley moves radially along the jib via a lead screw, and a hoisting motor controls the suspension cable length. Motor encode</p><p>　　The measurement resolution of the camera depends on the suspension

31、cable length. For the cable lengths used in this research, the resolution is approximately 0.08°. This is equivalent to a 1.4 mm hook displacement at a cable length of 1 m. In this work, the camera is not used for f

32、eedback control of the payload oscillation. The experimental results presented in this paper utilize encoder data to describe jib and trolley position and camera data to measure the deflection angles of the hook.</p&g

33、t;<p>　　Base mobility is provided by DC motors with omnidirectional wheels attached to each support leg, as shown in Fig. 3(b). The base is under PD control using two HiBot SH2-based microcontrollers, with feedbac

34、k from motor-shaft-mounted encoders. The mobile base was kept stationary during all experiments presented in this paper. Therefore, the mobile tower crane operated as a standard tower crane. </p><p>　　Table

35、 I summarizes the performance characteristics of the tower crane. It should be noted that most of these limits are enforced via software and are not the physical limitations of the system. These limitations are enf

36、orced to more closely match the operational parameters of full-sized tower cranes.</p><p>　　Fig. 3. Mobile, portable tower crane, (a) mobile tower crane, (b) mobile crane base.</p><p>　　TABLE I

37、 MOBILE TOWER CRANE PERFORMANCE LIMITS</p><p>　　Fig. 4 Sketch of tower crane with a double-pendulum dynamics.</p><p>　　III. TOWER CRANE MODEL</p><p>　　Fig.4 shows a sketch of a

38、tower crane with a double-pendulum payload configuration. The jib rotates by an angle around the vertical axis Z parallel to the tower column. The trolley moves radially along the jib; its position along th

39、e jib is described by . The suspension cable length from the trolley to the hook is represented by an inflexible, massless cable of variable length . The payload is connected to the hook via an inflexible, massless cable

40、 of length . Both the hook and the</p><p>　　The angles describing the position of the hook are shown in Fig. 5(a). The angle represents a deflection in the radial direction, along the jib. The angle

41、 represents a tangential deflection, perpendicular to the jib. In Fig. 5(a), is in the plane of the page, and lies in a plane out of the page. The angles describing the payload position are shown in Fig. 5(b). Notice

42、that these angles are defined relative to a line from the trolley to the hook. If there is no deflection of the hook,</p><p>　　To give some insight into the double-pendulum model, the position of the hook an

43、d payload within the Newtonian frame XYZ are written as and , respectively</p><p>　　Where , and are unit vectors in the X , Y , and Z directions.</p><p>　　The Lagrangian may then be written as

44、</p><p>　　Fig. 5. (a) Angles describing hook motion. (b) Angles describing payload motion.</p><p>　　Fig. 6. Experimental and simulated responses of radial motion.</p><p>　　(a) Hook

45、responses () for ，(b) Hook responses for </p><p>　　The motion of the trolley can be represented in terms of the system inputs. The position of the trolley in the Newtonian frame is described by</p>

46、<p>　　This position, or its derivatives, can be used as the input to any number of models of a spherical double-pendulum. More detailed discussion of the dynamics of spherical double pendulums can be found in [39]–[

47、42].</p><p>　　The addition of the second mass and resulting double-pendulum dramatically increases the complexity of the equations of motion beyond the more commonly used single-pendulum tower model [1], [16

48、], [43]–[46]. This fact can been seen in the Lagrangian. In (3), the terms in the square brackets represent those that remain for the single-pendulum model; no terms appear. This significantly reduces the complexity of

49、the equations because is a function of the inputs and all four angles shown in Fig. 5.</p><p>　　It should be reiterated that such a complex dynamic model is not used to design the input-shaping controllers

50、presented in later sections. The model was developed as a vehicle to evaluate the proposed control method over a variety of operating conditions and demonstrate its effectiveness. The controller is designed using a much

51、simpler, planar model.</p><p>　　Experimental Verification of the Model </p><p>　　The full, nonlinear equations of motion were experimentally verified using several test cases. Fig.6 shows tw

52、o cases involving only radial motion. The trolley was driven at maximum velocity for a distance of 0.30 m, with =0.45m .The payload mass for both cases was 0.15 kg and the hook mass was approx

53、imately 0.105 kg. The two cases shown in Fig. 6 present extremes of suspension cable lengths . In Fig. 6(a), is 0.48 m , close to the minimum length that can be mea</p><p>　　Fig. 7. Hook responses to 20&#

54、176;jib rotation: </p><p>　　(a) (radial) response;(b) (tangential) response.</p><p>　　Fig. 8. Hook responses to 90°jib rotation: </p><p>　　(a) (radial) response;(b) (tangen

55、tial) response.</p><p>　　If the trolley position is held constant and the jib is rotated, then the rotational and centripetal accelerations cause oscillation in both the radial and tangential directio

56、ns. This can be seen in the simulation responses from the full nonlinear model in Figs. 7 and 8. In Fig. 7, the trolley is held at a fixed position of r = 0.75 m, while the jib is rotated 20°. This relatively sma

57、ll rotation only slightly excites oscillation in the radial direction, as shown in Fig. 7(a). The vibra</p><p>　　Fig.9.Planardouble-pendulummodel.</p><p>　　Dynamic Analysis</p><p>　

58、　If the motion of the tower crane is limited to trolley motion, like the responses shown in Fig. 6, then the model may be simplified to that shown in Fig. 9. This model simplifies the analysis of the system dynamics and

59、provides simple estimates of the two natural frequencies of the double pendulum. These estimates will be used to develop input shapers for the double-pendulum tower crane.</p><p>　　The crane is moved by appl

60、ying a force to the trolley. A cable of length hangs below the trolley and supports a hook, of mass , to which the payload is attached using rigging cables. The rigging and payload are modeled as a second cable,

61、 of length and point mass . Assuming that the cable and rigging lengths do not change during the motion, the linearized equations of motion, assuming zero initial conditions, are</p><p>　　where and descr

62、ibe the angles of the two pendulums, R is the ratio of the payload mass to the hook mass, and is the acceleration due to gravity.</p><p>　　The linearized frequencies of the double-pendulum dynamics modeled

63、 in (5) are [47]</p><p>　　Where </p><p>　　Note that the frequencies depend on the two cable lengths and the mass ratio.</p><p>　　Fig. 10. Variation of first and second mode f

64、requencies when .</p><p>　　Fig. 10 shows the two oscillation frequencies as a function of both the rigging length and the mass ratio when the total length from trolley to payload is held constant at 1.8 m.

65、The total length is set to this value because it corresponds to the maximum length of the tower crane that was shown in Fig. 3. This maximum length corresponds to the largest possible swing amplitudes, so Fig. 10 represe

66、nts the frequencies that are possible in this worst-case scenario. The low frequency is maximized wh</p><p>　?、? CONCLUSION</p><p>　　A dynamic analysis of a tower crane with a payload exhibitin

67、g double-pendulumdynamics was presented. A simplified model was used to estimate the frequency andcontribution to the total response of each of the vibratory modes. An input-shaping control method to limit the residual o

68、scillation, with robustness to errors in frequency, was then developed using the simple model.</p><p>　　This input shaper was experimentally tested for various cases, and its robustnessto changes in suspensi

69、on cable length and nonlinear effects during slewing werepresented. The influence of this input shaper on operator performance was then examined for two different obstacle courses, one simple and one difficult. The human

70、 operators negotiated the two obstacle courses both locally and remotely, teleoperating the crane via the Internet. Input shaping was shown to dramatically improve task completi</p><p>　　運(yùn)用雙擺載荷動(dòng)力學(xué)控制塔式起重機(jī)<

71、/p><p>　　Joshua Vaughan, Dooroo Kim, and William Singhose</p><p>　　摘要：起重機(jī)的作用之所以有限，是因?yàn)檩d荷由架空纜支撐著，而架空纜在起重機(jī)運(yùn)行期間是允許振動(dòng)的發(fā)生。在一定條件下，載荷動(dòng)力可以產(chǎn)生一種能夠產(chǎn)生雙擺的額外的振動(dòng)模式。這篇文章就是對(duì)在此作用下的塔式起重機(jī)進(jìn)行了分析，同時(shí)也實(shí)驗(yàn)結(jié)果證明了用推薦的方法可以提高起重機(jī)

72、操作者的技能去駕馭一臺(tái)雙擺塔式起重機(jī)，不管是現(xiàn)場(chǎng)控制還是遠(yuǎn)程控制，操作性能都得到了改進(jìn)。</p><p>　　關(guān)鍵字：起重機(jī) 輸入整形 塔式起重機(jī)振動(dòng) 振幅</p><p><b>　?、? 緒論</b></p><p>　　起重機(jī)的動(dòng)力學(xué)和先進(jìn)的控制方法的研究已經(jīng)獲得了極大關(guān)注?；谄鋭?dòng)態(tài)特性以及能最自然描述出懸索連接點(diǎn)的位置坐標(biāo)系統(tǒng)，

73、起重機(jī)可以大致分為3類(lèi)。第一大類(lèi)，橋式起重機(jī)，在笛卡兒空間進(jìn)行運(yùn)行操控，如圖1（a）。小車(chē)沿著主梁運(yùn)動(dòng)，而主梁的運(yùn)動(dòng)方向垂直于小車(chē)。其中能夠在一個(gè)移動(dòng)的基座上運(yùn)動(dòng)的橋式起重機(jī)通常稱(chēng)作龍門(mén)起重機(jī)。橋式起重機(jī)在工廠、倉(cāng)庫(kù)和船廠是很常見(jiàn)的。</p><p>　　起重機(jī)的第二大類(lèi)是懸臂起重機(jī)，其中一種結(jié)構(gòu)如圖1（b）。懸臂起重機(jī)的最佳描述就是球面坐標(biāo)，其中一對(duì)旋轉(zhuǎn)軸的可以垂直或平行于地面。在圖1（b）中，角為豎直方向與Z

74、軸方向的夾角。角為Y軸與水平方向的夾角。懸掛在架空纜上的載荷是作用在懸臂的末端，懸起重機(jī)經(jīng)常放置在移動(dòng)的基座上，這樣可使他們能夠改變他們的工作地點(diǎn)。</p><p>　　起重機(jī)的第三大類(lèi)是塔式起重機(jī)，其結(jié)構(gòu)如圖1（c）所示。用圓柱坐標(biāo)描述塔式起重機(jī)是最合適的。水平臂架沿著垂直塔身旋轉(zhuǎn)。載荷懸掛在小車(chē)的纜繩上，其中下車(chē)是徑向沿著水平臂移動(dòng)的。塔式起重機(jī)常用在多層樓房建筑中，并具有很小的進(jìn)入工作區(qū)的比例。從控制設(shè)計(jì)的

75、角度看，懸臂起重機(jī)和塔式起重機(jī)主要缺點(diǎn)是由于起重機(jī)旋轉(zhuǎn)的性質(zhì)引起的非線性動(dòng)力學(xué)，除此之外還缺乏較直觀的自然坐標(biāo)系。</p><p>　　圖1 結(jié)構(gòu)：（a）橋式起重機(jī)，（b）懸臂式起重機(jī)，（c）塔式起重機(jī)</p><p>　　圖2 輸入整形過(guò)程</p><p>　　所有起重機(jī)的共同特點(diǎn)是：負(fù)載是通過(guò)一架空懸索支持。雖然這提供了起重機(jī)的吊裝功能，但是它也提出了一些挑

76、戰(zhàn)，其中主要的是載荷振蕩。起重機(jī)動(dòng)作通常會(huì)導(dǎo)致大量的載荷振蕩。這些載荷振蕩有許多不利的影響，包括降低載荷的定位精度，增加任務(wù)的完成時(shí)間，并降低安全性。一個(gè)針對(duì)減少振蕩的大型研究工作已展開(kāi)了。在起重機(jī)控制方面的成果的概述，主要集中在反饋方法上，見(jiàn)參考文獻(xiàn)[1]。一些研究者提出使用通順命令，以減少系統(tǒng)靈活模式的激勵(lì)，參考文獻(xiàn)見(jiàn)[2] - [5]?；诿罘绞降钠鹬貦C(jī)控制綜述可參考文獻(xiàn)[6]。 </p><p>　　許

77、多研究人員都集中于反饋方法的研究，而該方法必須在起重機(jī)上安裝更多附加的傳感器，也證明了人類(lèi)操作員難以用此方法操控起重機(jī)。比如，一些岸邊起重機(jī)已配備了先進(jìn)的反饋控制系統(tǒng)，以抑制有效載荷擺動(dòng)。但是由電腦控制的運(yùn)動(dòng)讓一些人類(lèi)操作員很惱火。因此，人類(lèi)操作員無(wú)法使用反饋控制器。鑒于絕大多數(shù)起重機(jī)操作員以及無(wú)法裝備基于電腦的反饋裝置的情況，在這篇論文中反饋方法將不予考慮。</p><p>　　輸入整形（參考文獻(xiàn)[7]、[8]

78、）是一種可以大大減小因操作者命令產(chǎn)生載荷振蕩的控制方法（參考文獻(xiàn)[9]、[10]）。一種利用粗略估計(jì)的系統(tǒng)固有頻率、阻尼比以及一系列的脈沖的機(jī)器被設(shè)計(jì)出來(lái)了，稱(chēng)為輸入成型機(jī)。輸入成型機(jī)的卷積和原始命令將用于驅(qū)動(dòng)系統(tǒng)。這個(gè)過(guò)程可以用圖2中的兩個(gè)脈沖輸入和一步命令來(lái)展示。結(jié)果顯示了該命令信號(hào)的上升時(shí)間隨著脈沖輸入持續(xù)時(shí)間的增加而增加。這種上升時(shí)間略有增加，通常占主導(dǎo)的振動(dòng)模態(tài)的0.5-1個(gè)周期命令。輸入整形法已經(jīng)成功的用在了許多震動(dòng)系統(tǒng)中，

79、包括橋式起重機(jī)、塔式起重機(jī)、懸臂式起重機(jī)、坐標(biāo)測(cè)量?jī)x、機(jī)械臂、排雷機(jī)器人和微型銑床。</p><p>　　大多數(shù)輸入整形裝置是基于線性系統(tǒng)理論。但是一些研究成果已經(jīng)對(duì)關(guān)于輸入整形到非線性系統(tǒng)的擴(kuò)展性進(jìn)行了調(diào)查，盡管非線性系統(tǒng)已經(jīng)被開(kāi)發(fā)，輸入整形器還是有效地。這些包括輸入整形器的非線性驅(qū)動(dòng)力學(xué)，摩擦力學(xué)和動(dòng)態(tài)非線性。其中一種非線性處理方法是自適應(yīng)輸入整形器的使用或?qū)W習(xí)。盡管有了這些成果，但是最簡(jiǎn)單最常見(jiàn)的用以解決系

80、統(tǒng)的非線性的方式是利用一個(gè)穩(wěn)健的輸入成型機(jī)。具有更穩(wěn)健的抑制系統(tǒng)參數(shù)變化能力的輸入整形器一般也會(huì)具有更穩(wěn)健的抑制系統(tǒng)非線性化的能力，其通過(guò)改變線性頻率體現(xiàn)出來(lái)。輸入整形器除了可以設(shè)計(jì)出強(qiáng)固的整形外，還可同時(shí)用于抑制復(fù)合振動(dòng)模式。</p><p>　　在第二部分中，為此論文的論述，將對(duì)移動(dòng)塔式起重機(jī)進(jìn)行實(shí)驗(yàn)測(cè)試。在第三部分中，將用平面及的塔式起重機(jī)模型來(lái)檢測(cè)突出重要的動(dòng)態(tài)效果。第四部分會(huì)提出一種設(shè)計(jì)具備指定穩(wěn)健性等

81、級(jí)的多模輸入整形器的方法。第五部分，將把這些方法使用在一臺(tái)帶有雙擺負(fù)載動(dòng)力的塔式起重機(jī)上。在最后的第六部分將介紹對(duì)人類(lèi)操控者行為的強(qiáng)勁塑造的影響，包括本地和遠(yuǎn)距離操縱控制。</p><p><b>　　II.移動(dòng)式塔機(jī)</b></p><p>　　如圖3所示，移動(dòng)式塔機(jī)具有了遠(yuǎn)程操控的性能，允許在世界任何地方通過(guò)網(wǎng)絡(luò)實(shí)時(shí)操控它。如圖3（a）所示，該起重機(jī)的塔身部分是由

82、近2米的塔柱及1米長(zhǎng)的塔臂組成。它是由西門(mén)子同步交流伺服電機(jī)驅(qū)動(dòng)。塔臂可以繞塔柱進(jìn)行340度的旋轉(zhuǎn)。臺(tái)車(chē)沿吊臂通過(guò)絲桿徑向移動(dòng)，其上有個(gè)吊重電動(dòng)機(jī)控制著懸繩的長(zhǎng)度。馬達(dá)編碼器用于小車(chē)回轉(zhuǎn)和徑向運(yùn)動(dòng)的PD反饋控制。一臺(tái)西門(mén)子數(shù)碼相機(jī)被安裝在小車(chē)上，并記錄掛鉤在50赫茲下擺動(dòng)撓度的采樣率。相機(jī)的測(cè)量分辨率將根據(jù)懸繩的長(zhǎng)度而定。而此次測(cè)試中用到的懸繩長(zhǎng)度決定了分辨率在0.08度左右。這相當(dāng)于在1米的懸繩上吊鉤移動(dòng)了1.4毫米的距離，相機(jī)將不被

83、用于載荷振蕩的反饋控制。本論文提供的實(shí)驗(yàn)結(jié)果，是利用編碼器的數(shù)據(jù)來(lái)描述吊臂和小車(chē)位置并且利用相機(jī)數(shù)據(jù)來(lái)衡量的掛鉤偏轉(zhuǎn)角。</p><p>　　圖3 移動(dòng)便捷式塔機(jī)，（a）移動(dòng)式塔機(jī)（b）移動(dòng)的基座</p><p>　　如圖3（b）所示，移動(dòng)基座部分是由帶直流電機(jī)的萬(wàn)向輪安裝于各個(gè)支撐腿上而組成。這個(gè)基座是由PD通過(guò)兩個(gè)基于HiBot SH2的微型控制器控制的，并能得到電機(jī)軸式編碼器的反饋

84、。移動(dòng)基座在本論文中提及的所有試驗(yàn)過(guò)程中都是保持固定的。因此，可按操控標(biāo)準(zhǔn)塔機(jī)一樣操控該移動(dòng)式塔機(jī)。</p><p>　　表I 塔式起重機(jī)的性能限制</p><p>　　表I總結(jié)了塔機(jī)的性能特點(diǎn)。應(yīng)當(dāng)指出，這些限制大多是通過(guò)軟件進(jìn)行強(qiáng)行規(guī)定的，不屬于該系統(tǒng)的物理限制。被規(guī)定的這些限制可以更加緊密地配合全尺寸塔式起重機(jī)的操作參數(shù)。</p><p>　　III.塔式起

85、重機(jī)的模型</p><p>　　如圖4所示，一個(gè)具有雙擺載荷配置的塔式起重機(jī)結(jié)構(gòu)簡(jiǎn)圖。吊臂平行于塔柱面繞垂直Z軸旋轉(zhuǎn)角，小車(chē)沿吊臂徑向移動(dòng)，它移動(dòng)的距離可用r來(lái)表示。從小車(chē)到吊鉤之間長(zhǎng)度為的架空纜可視為剛性無(wú)質(zhì)量并且可變。負(fù)載到吊鉤之間長(zhǎng)度為的懸繩也是剛性無(wú)質(zhì)量。吊鉤和負(fù)載可以分別看作為重和的兩個(gè)質(zhì)點(diǎn)。描述吊鉤位置的角度如圖5(a) 所示。角表示在徑向方向的偏轉(zhuǎn)。角表示在垂直于吊臂的切向方向的偏轉(zhuǎn)。在圖5(a)中

86、，角是在頁(yè)面所在平面內(nèi)，角則是在頁(yè)面所在平面外。描述負(fù)載位置的角度如圖5(a)所示。</p><p>　　圖4 帶雙擺載荷動(dòng)力的塔機(jī)結(jié)構(gòu)簡(jiǎn)圖</p><p>　　值得注意的是，這些角度的定義是相對(duì)于吊鉤和負(fù)載之間的直線的。如果吊鉤沒(méi)有偏轉(zhuǎn)，則角描述了沿吊臂的徑向撓度，角表示在切線方向垂直于吊臂的繞度。使用商業(yè)性動(dòng)力學(xué)組件可推導(dǎo)出包這個(gè)模型的運(yùn)動(dòng)方程，但是由于每一個(gè)方程復(fù)雜到都超過(guò)一頁(yè)紙，

87、在此論文中就不予全部給出。</p><p>　　圖5 （a）吊鉤運(yùn)動(dòng)中的角度描述（b）負(fù)載運(yùn)動(dòng)中的角度描述</p><p>　　具體分析這個(gè)雙擺動(dòng)力學(xué)的起重機(jī)模型，在牛頓參考系中可以將吊鉤和負(fù)載的位置分別表示為和。</p><p>　　其中、、是X、Y、Z軸方向的單位向量。</p><p>　　用拉格朗日法可表示為</p>&l

88、t;p>　　小車(chē)的運(yùn)動(dòng)可看作系統(tǒng)輸入。用牛頓參考系小車(chē)的位置可用下列式子表示</p><p>　　這個(gè)位置方程或者它的衍生方程都可以被看做任何一個(gè)球型雙擺模型的輸入表達(dá)。更多關(guān)于球型雙擺模型動(dòng)力學(xué)的分析可參考文獻(xiàn)[39]–[42]。</p><p>　　增加了第二個(gè)質(zhì)量以及由此產(chǎn)生的雙擺大大提高了運(yùn)動(dòng)方程的復(fù)雜性，遠(yuǎn)遠(yuǎn)超過(guò)了較常用的單塔模型。這一點(diǎn)在式子（3）中可以體現(xiàn)出來(lái)。其中方括

89、號(hào)內(nèi)的表達(dá)式仍表示單擺模型，沒(méi)有項(xiàng)出現(xiàn)。這大大降低了方程的復(fù)雜性，因?yàn)槭且粋€(gè)關(guān)于輸入和圖5中所有的四個(gè)角度的函數(shù)。應(yīng)當(dāng)重申的是這樣一個(gè)復(fù)雜的動(dòng)力學(xué)模型，將不被用于后面章節(jié)中提出的輸入整形控制器的設(shè)計(jì)中。該模型是作為一種手段，在評(píng)估了各種工作條件下提出的控制方法，并證明其有效性。該控制器設(shè)計(jì)采用了一個(gè)更簡(jiǎn)單，平面模型。</p><p>　　圖6. 掛鉤徑向運(yùn)動(dòng)的實(shí)驗(yàn)和模擬仿真的響應(yīng)</p><p

90、>　?。╝）架空纜長(zhǎng)0.48米；（b）架空纜長(zhǎng)1.28米</p><p><b>　　模型的實(shí)驗(yàn)驗(yàn)證</b></p><p>　　完全非線性方程組已經(jīng)在很多測(cè)試實(shí)驗(yàn)中被驗(yàn)證了。圖6顯示了兩種只包括徑向運(yùn)動(dòng)的分析。臺(tái)車(chē)的最大驅(qū)動(dòng)速度為0.30m/s，附帶一根長(zhǎng)度為為0.45米的懸繩。負(fù)載的質(zhì)量為0.15千克，掛鉤的質(zhì)量接近0.105千克。圖6顯示了架空纜的極限長(zhǎng)度

91、。在圖6（1）中，=0.48 m，這是接近架空的相機(jī)能夠測(cè)量到得最小長(zhǎng)度。在這個(gè)長(zhǎng)度，雙擺的效果可立即觀察到。圖中可以看出實(shí)驗(yàn)曲線和模擬仿真曲線十分近似。在圖6(b)中，長(zhǎng)1.28米，這個(gè)長(zhǎng)度是保持負(fù)載不接觸地面的最長(zhǎng)長(zhǎng)度。這個(gè)長(zhǎng)度上，可以大大減小第二次振動(dòng)模式對(duì)響應(yīng)的影響。仿真模型的響應(yīng)十分近似于實(shí)驗(yàn)的響應(yīng)。這個(gè)線性平面模型的響應(yīng)同樣也在圖6中黑線表示出來(lái)。第三部分B中也將用到的這個(gè)模型。這個(gè)平面模型的響應(yīng)同時(shí)與實(shí)驗(yàn)響應(yīng)以及架空纜在兩

92、種長(zhǎng)度下的全非線性模型的響應(yīng)十分近似。</p><p>　　圖7.吊臂水平轉(zhuǎn)過(guò)20°掛鉤的響應(yīng)</p><p>　?。╝）角的響應(yīng)（徑向）；（b）角的響應(yīng)（法向）</p><p>　　圖8. 吊臂水平轉(zhuǎn)過(guò)90°掛鉤的響應(yīng)</p><p>　?。╝）角的響應(yīng)（徑向）；（b）角的響應(yīng)（切向）</p><p&

93、gt;　　如果保持小車(chē)的位置不變，吊臂旋轉(zhuǎn)。然后利用旋轉(zhuǎn)和向心力加速度引發(fā)的徑向和切向兩種方向的振蕩。從圖7和圖8可以看到全非線性的仿真響應(yīng)。圖7中，小車(chē)固定在離塔柱0.75米的位置，同時(shí)吊臂轉(zhuǎn)過(guò)20°。這種相對(duì)小的旋轉(zhuǎn)只會(huì)輕微地激發(fā)徑向振蕩，如圖7（a）所示。振動(dòng)動(dòng)力學(xué)主要由切線方向的振蕩決定，如圖7（b）所示。但是，如果吊臂發(fā)生了一個(gè)很大的角位移，那么在徑向和切向方向都會(huì)引發(fā)強(qiáng)烈的振動(dòng)，如圖8所示。在這個(gè)分析中，小車(chē)被固定

94、在離塔柱0.75米的位置，同時(shí)吊臂轉(zhuǎn)過(guò)90°。圖7和圖8顯示了實(shí)驗(yàn)響應(yīng)與為這些旋轉(zhuǎn)運(yùn)動(dòng)而作的模型的那些預(yù)測(cè)十分近似。圖8（b）中部分偏差是由于在起重機(jī)所處的地板不平衡所致。經(jīng)過(guò)90°的旋轉(zhuǎn)，根據(jù)架空相機(jī)所測(cè)，吊臂和有效載荷振蕩的平衡點(diǎn)略有不同。</p><p><b>　　動(dòng)力學(xué)分析</b></p><p>　　如果塔式起重機(jī)的運(yùn)動(dòng)僅限于小車(chē)運(yùn)動(dòng)，

95、如圖6所示的響應(yīng)，這樣模型可簡(jiǎn)化為圖9所示的模型。該模型簡(jiǎn)化了系統(tǒng)的動(dòng)力學(xué)分析，并提供了兩個(gè)雙擺的固有頻率簡(jiǎn)單的估量。這些估量將用于開(kāi)發(fā)雙擺塔式起重機(jī)的輸入整形儀。</p><p>　　通過(guò)提供一個(gè)作用在小車(chē)上的力使起重機(jī)移動(dòng)。在小車(chē)下懸掛一根長(zhǎng)的纜繩，用以支撐質(zhì)量為的掛鉤，使用索具纜繩將有效載荷與掛鉤連起來(lái)。索具和有效載荷被第二纜繩套在一起，其長(zhǎng)度為，質(zhì)點(diǎn)質(zhì)量為。假設(shè)纜繩和索具長(zhǎng)度在運(yùn)動(dòng)時(shí)不改變，在零初始條件下

96、，線性運(yùn)動(dòng)方程組如下：</p><p>　　其中，角和描述了兩個(gè)擺的角度，R為有效載荷與掛鉤的質(zhì)量比，為重力加速度。</p><p>　　在式子（5）中的雙擺動(dòng)力模型的線性頻率是</p><p><b>　　其中</b></p><p>　　要注意的是頻率跟兩根纜繩的長(zhǎng)度以及質(zhì)量比有關(guān)。</p><p

97、>　　圖10.當(dāng)時(shí)，兩個(gè)模型的頻率區(qū)別。</p><p>　　圖10顯示了當(dāng)從小車(chē)到有效載荷總長(zhǎng)度恒為1.8米時(shí)，兩個(gè)振動(dòng)頻率同時(shí)作為了索具長(zhǎng)度和質(zhì)量比的一個(gè)函數(shù)?？傞L(zhǎng)度之所以被設(shè)定為這個(gè)值，是因?yàn)樗鼘?duì)應(yīng)了圖3所示的塔式起重機(jī)的最大長(zhǎng)度。這個(gè)最大長(zhǎng)度是由最大可能的擺動(dòng)幅度而定的。所以圖10顯示了在這種最壞的情況下的頻率。當(dāng)兩根纜繩長(zhǎng)度相等時(shí)，最低頻率可取到最大值，值得注意的是當(dāng)超過(guò)圖10所示的參數(shù)值得范圍時(shí)

98、，最低頻率將在它的中間值0.42Hz上下10%浮動(dòng)變化。相比之下，第二個(gè)模型在同樣的參數(shù)范圍下，浮動(dòng)變化為34%。</p><p><b>　　Ⅳ. 結(jié)論</b></p><p>　　上述就是用關(guān)于一臺(tái)帶有雙擺動(dòng)力載荷的塔式起重機(jī)的動(dòng)力分析。利用簡(jiǎn)化的模型估測(cè)頻率并且在每一個(gè)振動(dòng)模型的整體響應(yīng)中起了很大的作用。該模型中運(yùn)用一種輸入整型控制方法，此方法可限制多余的且與魯

99、棒性的頻率誤差的振動(dòng)。這種輸入整型器已在各類(lèi)例子中被實(shí)驗(yàn)性測(cè)試過(guò)，而且它也提供了關(guān)于架空纜長(zhǎng)度變化以及在扭轉(zhuǎn)側(cè)移中非線性的影響因素的魯棒性分析。這種輸入整型器對(duì)于操作者技能的影響也用兩種不同的障礙課程考核過(guò)，一種簡(jiǎn)單，一種困難。人類(lèi)操作員現(xiàn)場(chǎng)或遠(yuǎn)程順利通過(guò)這兩個(gè)障礙課程后，可以通過(guò)網(wǎng)絡(luò)遠(yuǎn)程操控起重機(jī)。輸入整形結(jié)果表明不僅可以大大提高任務(wù)完成時(shí)間，同時(shí)也減少了障礙碰撞。ANOVA分析顯示這種改進(jìn)在統(tǒng)計(jì)學(xué)上幾乎對(duì)所有的測(cè)試都具有重大的意義。