外文翻譯---運用雙擺載荷動力學控制塔式起重機（節(jié)選）

1、<p>　　西 南 交 通 大 學</p><p>　　本科畢業(yè)設計（外文翻譯）</p><p>　　Control of Tower Cranes With Double-Pendulum Payload Dynamics</p><p>　　運用雙擺載荷動力學控制塔式起重機</p><p>　　年 級: 2

2、006 </p><p>　　學 號: 20061112 </p><p>　　姓 名: 陳東 </p><p>　　專 業(yè): 機械設計制造及其自動化</p><p>　　指導老師: 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>　　運用雙擺載荷動力學控制塔式起重機<

71、/p><p>　　Joshua Vaughan, Dooroo Kim, and William Singhose</p><p>　　摘要：起重機的作用之所以有限，是因為載荷由架空纜支撐著，而架空纜在起重機運行期間是允許振動的發(fā)生。在一定條件下，載荷動力可以產(chǎn)生一種能夠產(chǎn)生雙擺的額外的振動模式。這篇文章就是對在此作用下的塔式起重機進行了分析，同時也實驗結果證明了用推薦的方法可以提高起重機

72、操作者的技能去駕馭一臺雙擺塔式起重機，不管是現(xiàn)場控制還是遠程控制，操作性能都得到了改進。</p><p>　　關鍵字：起重機 輸入整形 塔式起重機振動 振幅</p><p><b>　　Ⅰ. 緒論</b></p><p>　　起重機的動力學和先進的控制方法的研究已經(jīng)獲得了極大關注。基于其動態(tài)特性以及能最自然描述出懸索連接點的位置坐標系統(tǒng)，

73、起重機可以大致分為3類。第一大類，橋式起重機，在笛卡兒空間進行運行操控，如圖1（a）。小車沿著主梁運動，而主梁的運動方向垂直于小車。其中能夠在一個移動的基座上運動的橋式起重機通常稱作龍門起重機。橋式起重機在工廠、倉庫和船廠是很常見的。</p><p>　　起重機的第二大類是懸臂起重機，其中一種結構如圖1（b）。懸臂起重機的最佳描述就是球面坐標，其中一對旋轉(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，懸起重機經(jīng)常放置在移動的基座上，這樣可使他們能夠改變他們的工作地點。</p><p>　　起重機的第三大類是塔式起重機，其結構如圖1（c）所示。用圓柱坐標描述塔式起重機是最合適的。水平臂架沿著垂直塔身旋轉(zhuǎn)。載荷懸掛在小車的纜繩上，其中下車是徑向沿著水平臂移動的。塔式起重機常用在多層樓房建筑中，并具有很小的進入工作區(qū)的比例。從控制設計的

75、角度看，懸臂起重機和塔式起重機主要缺點是由于起重機旋轉(zhuǎn)的性質(zhì)引起的非線性動力學，除此之外還缺乏較直觀的自然坐標系。</p><p>　　圖1 結構：（a）橋式起重機，（b）懸臂式起重機，（c）塔式起重機</p><p>　　圖2 輸入整形過程</p><p>　　所有起重機的共同特點是：負載是通過一架空懸索支持。雖然這提供了起重機的吊裝功能，但是它也提出了一些挑

76、戰(zhàn)，其中主要的是載荷振蕩。起重機動作通常會導致大量的載荷振蕩。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，增加任務的完成時間，并降低安全性。一個針對減少振蕩的大型研究工作已展開了。在起重機控制方面的成果的概述，主要集中在反饋方法上，見參考文獻[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，參考文獻見[2] - [5]。基于命令方式的起重機控制綜述可參考文獻[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上安裝更多附加的傳感器，也證明了人類操作員難以用此方法操控起重機。比如，一些岸邊起重機已配備了先進的反饋控制系統(tǒng)，以抑制有效載荷擺動。但是由電腦控制的運動讓一些人類操作員很惱火。因此，人類操作員無法使用反饋控制器。鑒于絕大多數(shù)起重機操作員以及無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[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[9]、[10]）。一種利用粗略估計的系統(tǒng)固有頻率、阻尼比以及一系列的脈沖的機器被設計出來了，稱為輸入成型機。輸入成型機的卷積和原始命令將用于驅(qū)動系統(tǒng)。這個過程可以用圖2中的兩個脈沖輸入和一步命令來展示。結果顯示了該命令信號的上升時間隨著脈沖輸入持續(xù)時間的增加而增加。這種上升時間略有增加，通常占主導的振動模態(tài)的0.5-1個周期命令。輸入整形法已經(jīng)成功的用在了許多震動系統(tǒng)中，

79、包括橋式起重機、塔式起重機、懸臂式起重機、坐標測量儀、機械臂、排雷機器人和微型銑床。</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)對關于輸入整形到非線性系統(tǒng)的擴展性進行了調(diào)查，盡管非線性系統(tǒng)已經(jīng)被開發(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)動力學，摩擦力學和動態(tài)非線性。其中一種非線性處理方法是自適應輸入整形器的使用或?qū)W習。盡管有了這些成果，但是最簡單最常見的用以解決系

80、統(tǒng)的非線性的方式是利用一個穩(wěn)健的輸入成型機。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具有更穩(wěn)健的抑制系統(tǒng)非線性化的能力，其通過改變線性頻率體現(xiàn)出來。輸入整形器除了可以設計出強固的整形外，還可同時用于抑制復合振動模式。</p><p>　　在第二部分中，為此論文的論述，將對移動塔式起重機進行實驗測試。在第三部分中，將用平面及的塔式起重機模型來檢測突出重要的動態(tài)效果。第四部分會提出一種設計具備指定穩(wěn)健性等

81、級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帶有雙擺負載動力的塔式起重機上。在最后的第六部分將介紹對人類操控者行為的強勁塑造的影響，包括本地和遠距離操縱控制。</p><p><b>　　II.移動式塔機</b></p><p>　　如圖3所示，移動式塔機具有了遠程操控的性能，允許在世界任何地方通過網(wǎng)絡實時操控它。如圖3（a）所示，該起重機的塔身部分是由

82、近2米的塔柱及1米長的塔臂組成。它是由西門子同步交流伺服電機驅(qū)動。塔臂可以繞塔柱進行340度的旋轉(zhuǎn)。臺車沿吊臂通過絲桿徑向移動，其上有個吊重電動機控制著懸繩的長度。馬達編碼器用于小車回轉(zhuǎn)和徑向運動的PD反饋控制。一臺西門子數(shù)碼相機被安裝在小車上，并記錄掛鉤在50赫茲下擺動撓度的采樣率。相機的測量分辨率將根據(jù)懸繩的長度而定。而此次測試中用到的懸繩長度決定了分辨率在0.08度左右。這相當于在1米的懸繩上吊鉤移動了1.4毫米的距離，相機將不被

83、用于載荷振蕩的反饋控制。本論文提供的實驗結果，是利用編碼器的數(shù)據(jù)來描述吊臂和小車位置并且利用相機數(shù)據(jù)來衡量的掛鉤偏轉(zhuǎn)角。</p><p>　　圖3 移動便捷式塔機，（a）移動式塔機（b）移動的基座</p><p>　　如圖3（b）所示，移動基座部分是由帶直流電機的萬向輪安裝于各個支撐腿上而組成。這個基座是由PD通過兩個基于HiBot SH2的微型控制器控制的，并能得到電機軸式編碼器的反饋

84、。移動基座在本論文中提及的所有試驗過程中都是保持固定的。因此，可按操控標準塔機一樣操控該移動式塔機。</p><p>　　表I 塔式起重機的性能限制</p><p>　　表I總結了塔機的性能特點。應當指出，這些限制大多是通過軟件進行強行規(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的操作參數(shù)。</p><p>　　III.塔式起

85、重機的模型</p><p>　　如圖4所示，一個具有雙擺載荷配置的塔式起重機結構簡圖。吊臂平行于塔柱面繞垂直Z軸旋轉(zhuǎn)角，小車沿吊臂徑向移動，它移動的距離可用r來表示。從小車到吊鉤之間長度為的架空纜可視為剛性無質(zhì)量并且可變。負載到吊鉤之間長度為的懸繩也是剛性無質(zhì)量。吊鉤和負載可以分別看作為重和的兩個質(zhì)點。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、，角是在頁面所在平面內(nèi)，角則是在頁面所在平面外。描述負載位置的角度如圖5(a)所示。</p><p>　　圖4 帶雙擺載荷動力的塔機結構簡圖</p><p>　　值得注意的是，這些角度的定義是相對于吊鉤和負載之間的直線的。如果吊鉤沒有偏轉(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)性動力學組件可推導出包這個模型的運動方程，但是由于每一個方程復雜到都超過一頁紙，

87、在此論文中就不予全部給出。</p><p>　　圖5 （a）吊鉤運動中的角度描述（b）負載運動中的角度描述</p><p>　　具體分析這個雙擺動力學的起重機模型，在牛頓參考系中可以將吊鉤和負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>　　小車的運動可看作系統(tǒng)輸入。用牛頓參考系小車的位置可用下列式子表示</p><p>　　這個位置方程或者它的衍生方程都可以被看做任何一個球型雙擺模型的輸入表達。更多關于球型雙擺模型動力學的分析可參考文獻[39]–[42]。</p><p>　　增加了第二個質(zhì)量以及由此產(chǎn)生的雙擺大大提高了運動方程的復雜性，遠遠超過了較常用的單塔模型。這一點在式子（3）中可以體現(xiàn)出來。其中方括

89、號內(nèi)的表達式仍表示單擺模型，沒有項出現(xiàn)。這大大降低了方程的復雜性，因為是一個關于輸入和圖5中所有的四個角度的函數(shù)。應當重申的是這樣一個復雜的動力學模型，將不被用于后面章節(jié)中提出的輸入整形控制器的設計中。該模型是作為一種手段，在評估了各種工作條件下提出的控制方法，并證明其有效性。該控制器設計采用了一個更簡單，平面模型。</p><p>　　圖6. 掛鉤徑向運動的實驗和模擬仿真的響應</p><p

90、>　　（a）架空纜長0.48米；（b）架空纜長1.28米</p><p><b>　　模型的實驗驗證</b></p><p>　　完全非線性方程組已經(jīng)在很多測試實驗中被驗證了。圖6顯示了兩種只包括徑向運動的分析。臺車的最大驅(qū)動速度為0.30m/s，附帶一根長度為為0.45米的懸繩。負載的質(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度

91、。在圖6（1）中，=0.48 m，這是接近架空的相機能夠測量到得最小長度。在這個長度，雙擺的效果可立即觀察到。圖中可以看出實驗曲線和模擬仿真曲線十分近似。在圖6(b)中，長1.28米，這個長度是保持負載不接觸地面的最長長度。這個長度上，可以大大減小第二次振動模式對響應的影響。仿真模型的響應十分近似于實驗的響應。這個線性平面模型的響應同樣也在圖6中黑線表示出來。第三部分B中也將用到的這個模型。這個平面模型的響應同時與實驗響應以及架空纜在兩

92、種長度下的全非線性模型的響應十分近似。</p><p>　　圖7.吊臂水平轉(zhuǎn)過20°掛鉤的響應</p><p>　　（a）角的響應（徑向）；（b）角的響應（法向）</p><p>　　圖8. 吊臂水平轉(zhuǎn)過90°掛鉤的響應</p><p>　　（a）角的響應（徑向）；（b）角的響應（切向）</p><p&

93、gt;　　如果保持小車的位置不變，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。圖7中，小車固定在離塔柱0.75米的位置，同時吊臂轉(zhuǎn)過20°。這種相對小的旋轉(zhuǎn)只會輕微地激發(fā)徑向振蕩，如圖7（a）所示。振動動力學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)生了一個很大的角位移，那么在徑向和切向方向都會引發(fā)強烈的振動，如圖8所示。在這個分析中，小車被固定

94、在離塔柱0.75米的位置，同時吊臂轉(zhuǎn)過90°。圖7和圖8顯示了實驗響應與為這些旋轉(zhuǎn)運動而作的模型的那些預測十分近似。圖8（b）中部分偏差是由于在起重機所處的地板不平衡所致。經(jīng)過90°的旋轉(zhuǎn)，根據(jù)架空相機所測，吊臂和有效載荷振蕩的平衡點略有不同。</p><p><b>　　動力學分析</b></p><p>　　如果塔式起重機的運動僅限于小車運動，

95、如圖6所示的響應，這樣模型可簡化為圖9所示的模型。該模型簡化了系統(tǒng)的動力學分析，并提供了兩個雙擺的固有頻率簡單的估量。這些估量將用于開發(fā)雙擺塔式起重機的輸入整形儀。</p><p>　　通過提供一個作用在小車上的力使起重機移動。在小車下懸掛一根長的纜繩，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。索具和有效載荷被第二纜繩套在一起，其長度為，質(zhì)點質(zhì)量為。假設纜繩和索具長度在運動時不改變，在零初始條件下

96、，線性運動方程組如下：</p><p>　　其中，角和描述了兩個擺的角度，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度以及質(zhì)量比有關。</p><p

97、>　　圖10.當時，兩個模型的頻率區(qū)別。</p><p>　　圖10顯示了當從小車到有效載荷總長度恒為1.8米時，兩個振動頻率同時作為了索具長度和質(zhì)量比的一個函數(shù)。總長度之所以被設定為這個值，是因為它對應了圖3所示的塔式起重機的最大長度。這個最大長度是由最大可能的擺動幅度而定的。所以圖10顯示了在這種最壞的情況下的頻率。當兩根纜繩長度相等時，最低頻率可取到最大值，值得注意的是當超過圖10所示的參數(shù)值得范圍時

98、，最低頻率將在它的中間值0.42Hz上下10%浮動變化。相比之下，第二個模型在同樣的參數(shù)范圍下，浮動變化為34%。</p><p><b>　?、? 結論</b></p><p>　　上述就是用關于一臺帶有雙擺動力載荷的塔式起重機的動力分析。利用簡化的模型估測頻率并且在每一個振動模型的整體響應中起了很大的作用。該模型中運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。這種輸入整型器已在各類例子中被實驗性測試過，而且它也提供了關于架空纜長度變化以及在扭轉(zhuǎn)側(cè)移中非線性的影響因素的魯棒性分析。這種輸入整型器對于操作者技能的影響也用兩種不同的障礙課程考核過，一種簡單，一種困難。人類操作員現(xiàn)場或遠程順利通過這兩個障礙課程后，可以通過網(wǎng)絡遠程操控起重機。輸入整形結果表明不僅可以大大提高任務完成時間，同時也減少了障礙碰撞。ANOVA分析顯示這種改進在統(tǒng)計學上幾乎對所有的測試都具有重大的意義。