Chapter 2 Crane Mathematic Model

Chapter Crane Mathematic Model Abstract This chapter examines the dynamics of overhead cranes. Concerning single-pendulum-type overhead cranes, their equations of motion are first presented by means of its Euler Lagrange equations. Subsequently, the equations of motion are extended to double-pendulum-type overhead cranes. The two models are presented as references for examples throughout this book. Since the two models are established under some ideal assumptions, some uncertainties associated with real applications are discussed next. In addition, the chapter proceeds with the analysis of oscillations for pendulum-type motions on the basis of linearized models of the two types of overhead cranes. The analysis distills the essential properties of each. Keywords Overhead crane modeling Single-pendulum dynamics Double-pendulum dynamics Uncertainty.1 Modeling of Single-Pendulum-Type Cranes.1.1 Modeling Figure.1 shows the coordinate system of an overhead crane system with its payload. Apparently, the crane system consists of two subsystems, i.e., trolley and payload [1]. The former is driven by a force. The latter is suspended from the trolley by a rope. Other symbols in Fig..1 are described as the trolley mass M, the payload mass m, the rope length L, the swing angle of the payload with respect to the vertical line θ, the trolley position with respect to the origin x, and the driven force applied to the trolley f. Consider that the crane in Fig..1 is static and the payload is in its downward position. If the trolley moves toward the right direction by a positive driven force, then the payload will rotate clockwise. Apparently, the payload angle θ is inherently Springer-Verlag Berlin Heidelberg 016 D. Qian and J. Yi, Hierarchical Sliding Mode Control for Under-actuated Cranes, DOI 10.1007/978-3-66-48417-3_ 51

5 Crane Mathematic Model Fig..1 Structure of the single-pendulum-type overhead crane system a pendulum-type motion. For the purpose of simplification, the following assumptions [] are given. The payload is regarded as a material particle. The rope is considered as an inflexible rod. Compared with the payload mass, the rope mass is ignored. The trolley moves in the x-direction. The payload moves on the x y surface. No friction exists in the system. Using Lagrangian method, the Lagrangian equation with respective to the generalized coordinate q i [3] can be obtained as d dt @La @ _q i @La @q i ¼ T i ; ð:1þ where i =1,,La = K P (K means the system kinetic energy and P denotes the system potential energy.), q i is the generalized coordination (here, q 1 and q indicate x and θ, respectively), and T i is the external force. According to the assumption that the payload is regarded as a material particle, the system kinetic energy in Fig..1 can be depicted as K ¼ 1 M _x þ 1 mv ð:þ here, v is a vector and it denotes the payload velocity, defined as v ¼ v x þ v y ; ð:3þ where v x ¼ _x þ L _ h cos h and v y ¼ L _ h sin h. Note that the payload is assumed to be a particle such that its moment of inertia is not considered in (.). When it is failed

.1 Modeling of Single-Pendulum-Type Cranes 53 to assume that the payload is a particle, its moment of inertia has to be taken into considerations. From Fig..1, the potential energy of the trolley subsystem is kept unchanged. Owing to this fact, the system potential energy in Fig..1 is only exhibited by the potential energy of the payload subsystem, defined as P ¼ mglð1 cos hþ ð:4þ Here, g is the gravitational acceleration. From (.) and (.4), La has the form La ¼ K P ¼ 1 M _x þ 1 mv mglð1 cos hþ ð:5þ Consider the variable x. Differentiating La with respect to x in (.5) yields @La @x ¼ 0 ð:6þ Differentiating La with respect to _x in (.5) yields @La @ _x ¼ M _x þ mð_x þ L_ h cos hþ Further, differentiating (.7) with respect to time t can have d dt @La @ _x ¼ M x :: þm x :: þl h cos h L h _ sin h Finally, the Lagrangian equation with respective to x has the form d @La @La dt @ _x @x ¼ ðm þ MÞx:: þml h cos h h _ sin h ¼ f ð:7þ ð:8þ ð:9þ Consider the variable θ. Differentiating La with respect to θ in (.5) yields @La @h ¼ m½ð_x þ L_ h cos hþð L h _ sin hþ þðl h _ sin hþðl h _ cos hþš mgl sin h ¼ ml_x h _ sin h mgl sin h ð:10þ Differentiating La with respect to _ h in (.5) yields @La @ h _ ¼m½ð_x þ L_ h cos hþðl cos hþþð L h _ sin hþð L h _ sin hþš ¼mL_x cos h þ ml h _ ð:11þ

54 Crane Mathematic Model Further, differentiating (.11) with respect to time t can have d @La dt @ h _ ¼ ml x :: cos h ml_x h _ sin h þ ml h Finally, the Lagrangian equation with respective to θ has the form d @La dt @ h _ @La @h ¼ ml x:: cos h þ ml h þ mgl sin h ¼ 0 ð:1þ ð:13þ From (.9) and (.13), the dynamic model [4] of this overhead crane system with respect to x and θ can be obtained by means of the Lagrangian method. ðm þ MÞ x :: þmlð h cos h _ h sin hþ ¼f x :: cos h þ L h þ g sin h ¼ 0 ð:14þ ð:15þ Further, the above dynamic model composed of (.14) and (.15) can be transformed to the following state space model [5], formulated as _x 1 ¼ x _x ¼ f 1 ðxþþb 1 ðxþu _x 3 ¼ x 4 _x 4 ¼ f ðxþþb ðxþu ð:16þ Here, x ¼ ½x 1 ; x ; x 3 ; x 4 Š T ; x 1 ¼ x; x 3 ¼ h; x is the trolley velocity; x 4 is the angular velocity of the load; u is the control input; and f i and b i (i = 1, ) are described as f 1 ðxþ ¼ MLx 4 sin x 3 þ mg sin x 3 cos x 3 M þ m sin x 3 1 b 1 ðxþ ¼ M þ m sin x 3 f ðxþ ¼ ðm þ mþg sin x 3 þ mlx 4 sin x 3 cos x 4 ðm þ m sin x 3 ÞL b ðxþ ¼ cos x 3 ðm þ m sin x 3 ÞL Equation (.16) formulates the state space model of this single-pendulum-type overhead crane system. In (.16), four state variables can depict this dynamic system. As far as state-variable-based control methods are concerned, the four states can be employed and a diversity of control approached can be achieved.

.1 Modeling of Single-Pendulum-Type Cranes 55 Note that the model (.16) is ideal and it contains no uncertainties. Due to imperfect modeling and effects of environment, it is impossible to avoid uncertainties and external disturbance in real dynamical systems. In reality, overhead crane systems often are operated under uncertainty conditions such as parameter variations, unmodeled dynamics, skidding and slipping, etc. Considering the possible effects of these uncertainties, the dynamic model of the overhead crane in Fig..1 can have the form _x 1 ¼ x _x ¼ f 10 ðxþþb 10 ðxþu _x 3 ¼ x 4 _x 4 ¼ f 0 ðxþþb 0 ðxþu ð:17þ In (.17) f i0 ðxþ ¼ f i ðxþþdf i ðxþ; b i0 ðxþ ¼ b i ðxþþdb i ðxþði ¼ 1; Þ, where f i (x) and b i (x) are the nominal parts of f i0 ðþand x b i0 ðxþ, respectively. Both f i (x)andb i (x) are formulated in (.16). Without loss of generality, the terms depicting modeling errors and parameter variations, Df i0 ðxþ and Db i0 ðxþ are assumed to be differentiable with respect to time t..1. Model with Uncertainties Uncertainties can be categorized as matched uncertainties and unmatched uncertainties [6]. The uncertainties are matched if and only if the uncertainties enter a dynamical system from the control tunnel. In (.17), the so-called matched uncertainties mean Df i0 ðxþ and Db i0 ðxþspanfb i ðxþg ð:18þ Explicitly, (.18) can be written as Df i0 ðxþ ¼b i ðxþd~f i ðxþ Db i0 ðxþ ¼b i ðxþd~b i ðxþ Substituting (.18) into (.17) yields _x ¼ f 1 ðxþþb 1 ðxþ u þ D~b 1 ðxþu þ D~f 1 ðxþ ð:19þ _x 4 ¼ f ðxþþb ðxþ u þ D~b ðxþu þ D~f ðxþ Apparently, all the uncertainties in (.19) enter the dynamic model (.17) by the control tunnel, indicating that they are matched. Such an entering tunnel makes this kind of uncertainties resistible by suitable control methods. In the case that there are unmatched uncertainties, it is challenging to suppress them because it is hard to formulate these kinds of uncertainties. A common

56 Crane Mathematic Model approach is to apply the available controllers as if there were no unmatched uncertainties. This method will inevitably result in a threshold on the size of the unmatched uncertainties. The unmatched uncertainties are then required to be smaller than the threshold value so that a stability result holds locally with respect to the size of the uncertainties. Since unmatched uncertainties are common in control practice, it is important to suppress them to guarantee the system stability in the presence of significant unmatched uncertainties. Consequently, the following dynamic model [7] with unmatched uncertainties can be directly formulated by (.0) without further simplification. _x 1 ¼ x _x ¼ f 1 ðxþþb 1 ðxþu þ n 1 ðx; uþ _x 3 ¼ x 4 _x 4 ¼ f ðxþþb ðxþu þ n ðx; uþ ð:0þ Here, n 1 ðx; uþ ¼Df 1 ðxþþdb 1 ðxþu and n ðx; uþ ¼¼Df ðxþþdb ðxþu..1.3 Linearized Model The nonlinear single-pendulum-type overhead crane model has been discussed and it is composed of (.14) and (.15). Because θ = 0 is the sole stable equilibrium of the overhead crane system, both of the equations can be linearized around the point. The linearized equations can be written as ðm þ MÞ x :: þ ml h :: ¼ f x :: þ L h :: þ gh ¼ 0 ð:1þ ð:þ According to Newton s second law, f =(m + M) x :: can be obtained. Substituting x :: ¼ f = ðm þ MÞ into (.1) yields f M þ m þ L h:: þ gh ¼ 0 ð:3þ Equation (.3) is a second order ordinary differential equation. The Laplace transform can be employed to solve Eq. (.3). Finally, the angular frequency describing the oscillation of the linearized Eq. (.3) can be formulated as r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x n ¼ 1 þ m gl M ð:4þ

.1 Modeling of Single-Pendulum-Type Cranes 57 Fig.. Variation of the system frequency Equation (.4) reveals the system frequency depending on the rope length and the mass ratio. It is of interest to investigate how the frequency changes as a function of the system parameters. Such information can be used for physical insights of the overhead crane system. Figure. demonstrates the function revealed in (.4). The MATLAB programs of the example are given in Appendix A. From Fig.., the frequency changes very little with respect to the mass ratio when the rope length is more than 4 m. On the other hand, the frequency value has a strong dependence on the mass ratio when the rope length is short..1.4 Modeling of Double-Pendulum-Type Cranes.1.4.1 Modeling Figure.3 illustrates the schematic representation of a double-pendulum-type crane. The crane is moved by a driven force F, applied to the trolley. This system consists of three subsystems, i.e., trolley, hook, and payload. That is, there exist three variables to describe the crane system. Each subsystem possesses one variable, described by trolley position with respect to the origin, x (m), hook angle with respect to the vertical line θ 1 (rad), and payload angle with respect to the vertical line θ (rad). Other symbols in Fig..3 are explained as trolley mass m 0 (kg), hook mass m 1 (kg), payload mass m (kg), cable length between trolley and hook l 1 (m), and cable length between hook and payload l (m). Consider the following ideal assumptions like no friction, massless cables, mass-point hook, and mass-point payload. To obtain the dynamic model of this crane system, the Lagrangian method is also adopted. The following Lagrangian equation with respect to the generalized coordinate q i can be obtained as

58 Crane Mathematic Model Fig..3 Schematic of the double-pendulum-type overhead crane system d dt @ _q i @q i ¼ T i ð:5þ In (.5), i =1,,3.La d = K d P d (K d means the system kinetic energy and P d denotes the system potential energy.), q i is the generalized coordination (here, q 1, q, and q 3 indicates x, θ 1, and θ, respectively), and T i is the external force. According to the aforementioned assumptions that the payload and hook are regarded as mass-points, the system kinetic energy in Fig..3 can be written as K d ¼ 1 m 0 _x þ 1 m 1v 1 þ 1 m v ð:6þ here, the vectors v 1 and v denote the hook and payload velocities, respectively. They are defined as v 1 ¼ v x1 þ v y1 v ¼ v x þ v y ; ð:7þ where v x1 ¼ _x þ l 1 _ h1 cos h 1 ; v y1 ¼ l 1 _ h1 sin h 1 ; v x ¼ _x þ l 1 _ h1 cos h 1 þ l _ h cos h and v y ¼ l 1 _ h1 sin h 1 l _ h sin h. From Fig..3, the potential energy of the trolley subsystem is kept unchanged. Owing to this fact, the system potential energy in Fig..3 is only exhibited by the potential energies of the hook and payload subsystems, defined as P d ¼ m 1 gl 1 ð1 cos h 1 Þþm gl ½ 1 ð1 cos h 1 Þþl ð1 cos h ÞŠ ð:8þ Here, g is the gravitational acceleration. Then, La d has the form

.1 Modeling of Single-Pendulum-Type Cranes 59 La d ¼ K d P d ¼ 1 m 0 _x þ 1 m 1v 1 þ 1 m v ð:9þ m 1 gl 1 ð1 cos h 1 Þ m gl ½ 1 ð1 cos h 1 Þþl ð1 cos h ÞŠ Consider the variable x. Differentiating La d with respect to x in (.9) yields Differentiating La d with respect to _x in (.9) yields @x ¼ 0 ð:30þ @ _x ¼ðm 0 þ m 1 þ m Þ_x þðm 1 þ m Þl 1 _ h1 cos h 1 þ m l _ h cos h ð:31þ Further, differentiating (.31) with respect to time t can have d dt @ _x @x ¼ðm 0 þ m 1 þ m Þ x þðm 1 þ m Þl 1 h1 cos h 1 ðm 1 þ m Þl 1 _ h 1 sin h 1 þ m l h cos h m l _ h sin h Finally, the Lagrangian equation with respective to x has the form d dt @ _x @x ¼ðm 0 þ m 1 þ m Þ x þðm 1 þ m Þl 1 h1 cos h 1 ðm 1 þ m Þl 1 _ h 1 sin h 1 þ m l h cos h m l _ h sin h = F ð:3þ ð:33þ Consider the variable θ 1. Differentiating La d with respect to θ 1 in (.9) yields @h 1 ¼ ðm 1 þ m Þl 1 _x _ h 1 sin h 1 ð:34þ m l 1 l _ h1 _ h sinðh 1 h Þ ðm 1 þ m Þgl 1 sin h 1 Differentiating La d with respect to _ h 1 in (.9) yields @ _ h 1 ¼ðm 1 þ m Þðl 1 _x cos h 1 þ l 1 _ h 1 Þþm l 1 l _ h cosðh 1 h Þ ð:35þ

60 Crane Mathematic Model Further, differentiating (.35) with respect to time t can have d dt @ h _ 1 ¼ ðm 1 þ m Þ l 1 x :: cos h 1 l 1 _x h _ 1 cos h 1 þ l 1 h 1 þ m l 1 l h cosðh 1 h Þ m l 1 l h1 _ h _ sinðh 1 h Þ þ m l 1 l h _ sin ð h 1 h Þ ð:36þ Finally, the Lagrangian equation with respective to θ 1 has the form d dt ð @ _ h 1 Þ @h 1 ¼ðm 1 þ m Þl 1 x cos h 1 þðm 1 þ m Þl 1 h 1 þ m l 1 l h cosðh 1 h Þþm l 1 l _ h sinðh 1 h Þ þðm 1 þ m Þgl 1 sin h 1 =0 ð:37þ Consider the variable θ. Differentiating La d with respect to θ in (.9) yields ¼ m l 1 _x h @h _ sin h þ m l 1 l h1 _ h _ sinðh 1 h Þ m gl sin h ð:38þ Differentiating La d with respect to _ h in (.9) yields @ h _ ¼ m l 1 _x cos h 1 þ m l h _ þ m l 1 l h1 _ cosðh 1 h Þ ð:39þ Further, differentiating (.39) with respect to time t can have d @L dt @ h _ ¼ m l x cos h m l _x h _ cos h þ m l h þ m l 1 l h1 cosðh 1 h Þ m l 1 l h _ 1 sinðh 1 h Þþm l 1 l h1 _ h _ sinðh 1 h Þ ð:40þ Finally, the Lagrangian equation with respective to θ has the form d dt @ h _ ¼ m l x cos h þ m l @h h þ m l 1 l h1 cosðh 1 h Þ m l 1 l h _ 1 sin ð h 1 h Þþm gl sin h =0 ð:41þ

.1 Modeling of Single-Pendulum-Type Cranes 61 From (.33), (.37), and (.41), the dynamic model of the double-pendulum-type overhead crane system with respect to x, θ 1, and θ [8] can be obtained by means of the Lagrangian method. ðm 0 þ m 1 þ m Þ x þ ðm 1 þ m Þl 1 h1 cos h 1 þ m l h cos h ðm 1 þ m Þl 1 h _ 1 sin h 1 m l h _ sin h ¼ F ð:4þ ðm 1 þ m Þl 1 x cos h 1 þ ðm 1 þ m Þl 1 h 1 þ m l 1 l h cosðh 1 h Þ þ m l 1 l h _ sin ð h 1 h Þþðm 1 þ m Þgl 1 sin h 1 ¼ 0 m l x cos h þ m l 1 l h1 cosðh 1 h Þþm l h m l 1 l h _ 1 sin ð h 1 h Þþ m gl sin h =0 ð:43þ ð:44þ Rearrange (.4), (.43), and (.44) in the form of a matrix. The three equations can be rewritten as Mq ð Þq :: þ Cq; ð _q Þ_q þ Gq ð Þ ¼ s ð:45þ Here, q ¼½x; h 1 ; h Š T is a vector of the three generalized coordinates, s ¼ ½F; 0; 0Š T is a vector of the generalized force, g is the gravitational acceleration, M (q)isa3 3 inertia matrix, Cðq; _qþ _q is a vector of Coriolis and centripetal torques, and G(q) is a vector of the gravitational term. Mq ð Þ; Cðq; _qþ and G(q) are defined as Mq ð Þ ¼ 3 m 0 þ m 1 þ m ðm 1 þ m Þl 1 cos h 1 m l cos h 4 ðm 1 þ m Þl 1 cos h 1 ðm 1 þ m Þl 1 m l 1 l cosðh 1 h Þ5 m l cos h m l 1 l cosðh 1 h Þ m l 3 0 ðm 1 þ m Þl 1 h1 _ sin h 1 m l h _ sin h 4 0 0 m l 1 l h _ sinðh 1 h Þ5 0 m l 1 l h1 _ sinðh 1 h Þ 0 Cq; ð _q Þ ¼ Gq ð Þ ¼ ½0 ðm 1 þ m Þgl 1 sin h 1 m gl sin h Further, (.45) can be transformed to its state space expression. The expression [9, 10] has the form Š T

6 Crane Mathematic Model _x 1 ¼ x _x ¼ f 1 ðxþþb 1 ðxþu _x 3 ¼ x 4 _x 4 ¼ f ðxþþb ðxþu ð:46þ _x 5 ¼ x 6 _x 6 ¼ f 3 ðxþþb 3 ðxþu In (.46), the vector x is defined by [x 1, x, x 3, x 4, x 5, x 6 ] T ; x 1 = x; x 3 = θ 1 ; x 5 = θ ; x is the trolley velocity; x 4 is the angular velocity of the hook; x 6 is the angular velocity of the payload; u = F is the control input; and f i (x) and b i (x)(i = 1,, 3) are nonlinear functions of the vector x, formulated by f i (x) =Γ i /Δ and b i (x) =Τ i /Δ. Here Γ i, Τ i, and Δ are determined by D = ðm 1 þ m Þm l 1 l ðm 0 þ m 1 þ m Þ ðm 1 þ m Þcos ðx 3 Þ m l 1 l ðm 1 þ m Þcos ðx 5 Þþðm 0 þ m 1 þ m Þcos ðx 3 x 5 Þ ðm 1 þ m Þcosðx 3 Þcosðx 5 Þcosðx 3 x 5 ÞŠ C 1 ¼ ðm 1 þ m Þm l 1 l m l 1 l cos ðx 3 x 5 Þ ð m1 þ m Þl 1 x 4 sin ð x 3Þ þ m l x 6 sin ð x 5Þ þ ðm 1 þ m Þm l 1 l cos ð x 3Þ m l 1l cos ð x 5Þcosðx 3 x 5 Þ m l 1 l x 6 sin ð x 3 x 5 Þ þðm 1 þ m Þgl 1 sinðx 3 ÞŠþ ðm 1 þ m Þm l 1 l cosðx 5 Þ m l 1 l cosðx 3 Þcosðx 3 x 5 Þ m l 1 l x 4 sin x ð 3 x 5 Þþm gl sinðx 5 Þ T 1 ¼ ðm 1 þ m Þm l 1 l m l 1 l cos ðx 3 x 5 Þ C ¼ m l 1l cos ð x 5Þcosðx 3 x 5 Þ ðm 1 þ m Þm l 1 l cos ð x 3Þ ð m1 þ m Þl 1 x 4 sin ðx 3 Þ þm l x 6 sin ð x 5Þ þ m l cos ðx 5 Þ ðm 0 þ m 1 þ m Þm l m l 1 l x 6 sin ð x 3 x 5 Þ þðm 1 þ m Þgl 1 sinðx 3 ÞŠþ½ðm 0 þ m 1 þ m Þm l 1 l cosðx 3 x 5 Þ ðm 1 þ m Þm l 1 l cosðx 3 Þcosðx 5 ÞŠ m l 1 l x 4 sin x ð 3 x 5 Þþm gl sinðx 5 Þ T ¼ m l 1 l cos ð x 5Þcosðx 3 x 5 Þ ðm 1 þ m Þm l 1 l cos ð x 3Þ C 3 ¼ ðm 1 þ m Þm l 1 l ½cosðx 3 Þcosðx 3 x 5 Þ cosðx 5 ÞŠ ½ð m1 þ m Þl 1 x 4 sin ð x 3Þ þm l x 6 sin ð x 5Þ þ ½ðm 0 þ m 1 þ m Þm l 1 l cosðx 3 x 5 Þ ðm 1 þ m Þm l 1 l cosðx 3 Þcosðx 5 ÞŠ m l 1 l x 6 sin x ð 3 x 5 Þþðm 1 þ m Þgl 1 sinðx 3 Þ h i þ ðm 1 þ m Þ l 1 cos ðx 3 Þ ðm 0 þ m 1 þ m Þðm 1 þ m Þl 1 m l 1 l x 4 sin x ð 3 x 5 Þþm gl sinðx 5 Þ

.1 Modeling of Single-Pendulum-Type Cranes 63 T 3 ¼ ðm 1 þ m Þm l 1 l cosðx 3 Þcosðx 3 x 5 Þ ðm 1 þ m Þm l 1 l cosðx 5 Þ Note that the model (.45) has two important assumptions that are mass-point hook and payload. Usually, the mass-point assumption can be satisfied for the hook subsystem. However, the mass-point assumption for the payload subsystem can be satisfied under some operating conditions. Concerning these extreme operating conditions, the moment of inertia of the payload subsystem cannot be ignored and it has to be taken into consideration..1.5 Model with Uncertainties Equation (.46) is an ideal model. It can be treated as the nominal model of double-pendulum-type overhead cranes. Considering the system uncertainties, the uncertain equations can be derived from (.46). The analysis is very similar to the process in Sect..1.. Briefly, the uncertain model of the double-pendulum-type overhead crane in Fig..1 can be described as _x 1 ¼ x _x ¼ f 1 ðxþþb 1 ðxþu þ n 1 ðx; uþ _x 3 ¼ x 4 _x 4 ¼ f ðxþþb ðxþu þ n ðx; uþ ð:47þ _x 5 ¼ x 6 _x 6 ¼ f 3 ðxþþb 3 ðxþu þ n 3 ðx; uþ The uncertain part n i ðx; uþ in (.47) is matched, if it can be written as n i ðx; uþ ¼ b i ðxþdn i ðx; uþ where i = 1,, 3. Otherwise, the three uncertain terms are unmatched because they cannot enter the crane model by the control channel. Note that the three uncertain terms have to be treated as a whole. The uncertainties are still unmatched, if only a part of the three terms can enter the crane model by the control channel..1.6 Linearized Model The nonlinear double-pendulum-type overhead crane model is shown in (.13). Because θ 1 = θ = 0 is the sole stable equilibrium of the double-pendulum-type crane system, (.13) can be linearized around θ 1 = 0 and θ = 0. The linearized crane model can be written as

64 Crane Mathematic Model M q :: þkq ¼ 0 ð:48þ In the linearized model (.48), the matrixes M and K are determined as m 0 þ m 1 þ m ðm 1 þ m Þl 1 m l 3 M ¼ 4 ðm 1 þ m Þl 1 ðm 1 þ m Þl 1 m l 1 l 5 m l m l 1 l m l 3 0 0 0 K ¼ 4 0 ðm 1 þ m Þgl 1 0 5 0 0 m gl The two natural frequencies of the double-pendulum-type overhead crane system can be obtained by the nonzero eigenvalues of the matrix M K. Their expressions [11] are determined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g pffiffiffi x 1 ¼ a b rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g x ¼ a þ p ffiffiffi b ð:49þ ð:50þ Here, α has a form a ¼ m 1 þ m 1 þ 1 m 1 l 1 l Another parameter β is formulated as b ¼ m 1 þ m 1 þ 1 4 m 1 þ m 1 m 1 l 1 l m 1 l 1 l From (.49) and (.50), the two natural frequencies not only depend on the length of the cables but also depend on the masses of payload and hook. It is interesting to investigate how the frequencies change as a function of the system physical parameters. To simplify this problem, R = m /m 1 is defined as the payload-to-hook mass ratio, the cable length between hook and payload l is considered as a variable when the total length l determined by l 1 plus l is held constant at 6 m. Figure.4 illustrates the two oscillation frequencies as a function of R and l. In Fig..4, ω 1 changes very little for a constant l 1 + l. It corresponds closely to the frequency of a single pendulum with the length of l 1 + l. On the other hand, the value of ω 1 is maximized for a constant l 1 + l when the two cable lengths are equal to l 1þl ; but it can be dramatically changed by the hoisting operation [1].

.1 Modeling of Single-Pendulum-Type Cranes 65 Fig..4 Variation of the two frequencies Furthermore, the high-frequency - has a strong dependence on the cable length l ; the value of ω varies substantially more than the value of ω 1, the contribution of ω to problematic swing amplitude is particularly large for a constant l 1 + l when the two cable lengths are approximately equal. Concerning the mass ratio, R has a relatively small effect on ω 1, but the high-frequency ω becomes more important to the double-pendulum motions for low payload-to-hook mass ratios. In brief, low payload-to-hook mass ratios and equal cable lengths are more representative to depict the double-pendulum motions of the crane for the point-to-point transport control with a constant l 1 + l. Appendices A Matlab Codes to Plot Fig.. k=0.01:0.01:; % Mass ratio from 0.01 to, every each 0.01. l=0.1:0.1:10; % Rope length from 0.1 to 10, every each 0.1 [K,L]=meshgrid(k,l); % K and L arrays for 3-D plots. w=sqrt((1+k)*9.8./l); % Array of angular frequency surf(k,l,w) % 3-D colored surface.

66 Crane Mathematic Model B Matlab Codes to Plot Fig..4 r=0.1:0.1:; %mass ratio l=0.1:0.1:6; %cable length between hook and payload l1=6-l; %l1 plus l is held constant at 6m g=9.8; % gravitational acceleration [R,L]=meshgrid(r,l); % R and L arrays for 3-D plots p=sqrt((1+r).^.*(1./(6-l)+1./l).^-4.*((1+r)./(l.*(6-l)))); w1=sqrt(g/).*sqrt((1+r).*(1./(6-l)+1./l)-p); % Array of w1 w=sqrt(g/).*sqrt((1+r).*(1./(6-l)+1./l)+p); % Array of w surf(r,l,w1) %3-D colored surface hold; % Another surface surf(r,l,w) %3-D colored surface References 1. Abdel-Rahman EM, Nayfeh AH, Masoud ZN (003) Dynamics and control of cranes: a review. J Vib Control 9(7):863 908. Lee HH (1998) Modeling and control of a three-dimensional overhead crane. J Dyn Syst Meas Control Trans ASME 10(4):471 476 3. Spong MW, Hutchinson S, Vidyasagar M (006) Robot modeling and control. Wiley, New York 4. Oguamanam D, Hansen JS, Heppler G (001) Dynamics of a three-dimensional overhead crane system. J Sound Vib 4(3):411 46 5. Liu D, Yi J, Zhao D, Wang W (005) Adaptive sliding mode fuzzy control for a two-dimensional overhead crane. Mechatronics 15(5):505 5 6. Cheng C, Chen CY (1996) Controller design for an overhead crane system with uncretainty. Control Eng Pract 4(5):645 653 7. Park MS, Chwa D, Hong SK (008) Antisway tracking control of overhead cranes with system uncertainty and actuator nonlinearity using an adaptive fuzzy sliding-mode control. IEEE Trans Ind Electron 55(11):397 3984 8. Tuan LA, Lee SG (013) Sliding mode controls of double-pendulum crane systems. J Mech Sci Technol 7(6):1863 1873 9. Liu D, Guo W, Yi J (008) Dynamics and GA-based stable control for a class of underactuated mechanical systems. Int J Control Autom Syst 6(1):35 43 10. O Connor W, Habibi H (013) Gantry crane control of a double-pendulum, distributed-mass load using mechanical wave concepts. Mech Sci 4:51 61 11. Vaughan J, Kim D, Singhose W (010) Control of tower cranes with double-pendulum payload dynamics. IEEE Trans Control Syst Technol 18(6):1345 1358 1. Singhose W, Kim D, Kenison M (008) Input shaping control of double-pendulum bridge crane oscillations. J Dyn Syst Meas Control Trans ASME 130(3): doi:10.1115/1.907363

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