A method for analyzing the dynamic response of a structural system with variable mass , damping and stiffness

The bridge bucket unloaders and container cranes are now the main facilities using for loading and unloading the goods in ports around the world. The high working efficiency requires bigger and faster cranes. In analyzing bridge bucket unloaders, conventional structural analysis methods produce some errors, sometimes causing serious problems because of lack of consideration of the complex dynamic behavior. The bridge crane is a machine with repeating movements including accelerating, hoisting, traversing and decelerating. There are also two combined movements such as hoisting with traversing and lowing with traversing. As the crane becomes heavier, the influence


Introduction
The bridge bucket unloaders and container cranes are now the main facilities using for loading and unloading the goods in ports around the world.The high working efficiency requires bigger and faster cranes.In analyzing bridge bucket unloaders, conventional structural analysis methods produce some errors, sometimes causing serious problems because of lack of consideration of the complex dynamic behavior.
The bridge crane is a machine with repeating movements including accelerating, hoisting, traversing and decelerating.There are also two combined movements such as hoisting with traversing and lowing with traversing.As the crane becomes heavier, the influence of vibration caused by start lifting, load moving and unloading will become significant and must be considered in the crane structure analysis.Based on above and our experience, we know that the crane steel structure bears the complex and strenuous vibrations during its operation.During the trolley and lifting load moving, the mass, damping and stiffness matrices of the crane structure will change simultaneously.So the complicated dynamic response should be taken into account in the actual structural design.Using the finite element method and the wide application of the computer, research and analysis on the vibration of the crane steel structure was conducted.However, the work including the dynamic response of the crane with the consideration of the aforementioned factors is quite rare.In this paper, the variable mass, damping and stiffness dynamic equations and the solution are presented with sufficient consideration of the above factors' influence, which comparatively really simulates and analyzes all these kinds of responses of the whole dynamic process during the unloader's actual operations.

Duty cycle description
One duty cycle for bridge bucket unloader consists of bucket loading goods from ship, lifting, trolley traveling back to the dock, unloading the goods, trolley traveling forth to the ship and bucket lowing for loading.There are six basic motions (i.e.bucket closing for loading, hoisting, trolley traveling back with loads, bucket opening for unloading, trolley traveling forth with empty bucket, bucket lowing) and their combinations, shown as Fig. 1.Based on the actual work process, The forces in the crane structure are relatively small for the three motions of bucket closing, trolley traveling with empty bucket and bucket lowing.So in this paper, we only simulate the rest three motions of hoisting with loads, trolley traveling back with loads and unloading.See the interval T 0 ∼ T 8 in Fig. 1 for reference.
Based on Fig. 1, there are seven moving combinations in the interval T 0 ∼ T 8 described as follows: 1) Hoisting Accelerating phase: The lifting weight leaves the ground and continues to be accelerated at the acceleration of Vow .The system stiffness matrix changes with different elevation of lifting weight at any moment.2) Hoisting at uniform speed: The lifting weight ascends at the rated hoist speed.3) Hoisting at uniform velocity and the trolley accelerating back move: When the lifting weight is ascending at the rated hoist speed, in the mean time, the trolley starts to move back (to dock side) at the acceleration of a x .The system mass matrix changes with the trolley's movement.4) Hoisting decelerating and the trolley accelerating back move: When the lifting weight is ascending at the deceleration of Vow , the trolley continues to move back at the acceleration of a x .5) Hoisting decelerating phase and the trolley moving back at the uniform speed: The lifting weight is ascending at the deceleration of Vow , the trolley moves back at the rated traversing speed of V x .6) Trolley moving back at the uniform speed and opening the bucket for unloading: The trolley continues to move back at uniform speed motion.The bucket's open-close rope begins to open the bucket for unloading at the same time.7) Trolley decelerating moving back and opening the bucket for unloading: When the trolley moves back at the deceleration of a x , the bucket continues unloading.

Establishment and solving method of variable mass, damping and stiffness dynamic equations
Based on above description, we know that the hoisting changes the length of hoisting ropes causing the variation of the system stiffness matrix and the movement of the trolley and goods causes the variation of system mass and stiffness matrixes.The dynamic equations are discussed and established in the following subdivisions.

Vibration equation of the lifting weight 1) Y -direction
Shown in Fig. 2, the wire rope tension is F at a certain time, the mass of the lifting weight is M w , W = M w •g, the dynamic equation in Y-direction is: Where V w is the elastic displacement of the lifting weight's center of gravity and F is the elastic restoring force of the wire rope.C w is damping.
Assuming the initial vertical length of the wire rope is l 0 , l t is the vertical length at the time t, the elastic extension of the rope is δ 1 and the vertical displacement at the trolley position is V w .
Where V ow is the shortening of the vertical length of the wire rope only caused by hoisting the lifting weight.From Fig. 3, we can establish the equation: From Eqs ( 2) and (3), we get: The tension F of the wire rope at the time t is given by: Where k L (t) is the stiffness of the wire rope at the t-time: Where E r is the elastic modulus of the wire rope and A r is the cross -section area of the wire rope.
Replace the F in Eq. ( 1) by the F in Eq. ( 5), the dynamic equation in Y -direction is got: Where Vow varies with the different moving phases as follows: Due to the unloading T 5 − T 8 , the M w in Eq. ( 7) is defined as: Where M 1 is the mass of the bucket, M 2 is the mass of goods.

2) X -direction
The vibration of the lifting weight at the horizontal (X) direction.The lifting weight exerts different horizontal force F x on the crane structure in different time interval described as follows: A) In T 0 − T 2 , the lifting weight has no movement in X -direction.
, following the horizontal acceleration of the trolley, the lifting weight gradually acquires the same horizontal speed as the trolley, which exerts the horizontal force F x on the structure.The F x is: Where T (t) is the swing period of the lifting weight at the time t, and its swing angle is relatively small (θ < 5 • ), we can calculate the T (t) using the following formula: C) In T 4 − T 6 , The lifting weight moves horizontally at the same speed as the trolley, In T 6 −T 7 , the trolley has a horizontal decelerated motion, according to the analysis in B): (14')

Structural dynamic equation
In order to establish the dynamic equation of the structural system, we first study the displacement relationship among the lifting weight's center of gravity, the hanging point on the trolley and the related structure nodes, shown in Fig. 4.
Assuming at a certain moment t, the left and right wheels of the trolley have been respectively on the spaces among the nodes of i, j, k and l of the boom discretized by the finite element.The V w in Fig. 4 is same as the V w in Fig. 3.According to the Fig. 4, the displacement relationship between the V w and the  structure nodes of i, j, k and l can be obtained, Where V a is the displacement of the trolley's left wheel, V b is the displacement of the right wheel.
s 1 is the distance from the trolley's left wheel to the node i, shown in Fig. 4. s 2 is the distance from the trolley's right wheel to the node k, shown in Fig. 4. l 1 is the distance between node i and node j, shown in Fig. 4. l 2 is the distance between node k and node l, shown in Fig. 4. s 1 and s 2 are based on Eqs (18) and (19): (18 Where, x 0 is the initial x coordinates of the trolley's left wheel, x i is the x coordinates of node i in Fig. 4. L ab is the distance between the trolley's left wheel a and the right wheel b, shown in Fig. 4. Substituting Eqs ( 16) and (17) into Eq.(15), we can obtain: Now, we know the displacement relationship between the structure and the hanging point on the trolley.
The structural dynamic equation is: Where N is the total node number of the structure.Following discusses the effect of the moving trolley and the lifting weight on the M , K and f (t) in the structural dynamic equation.

The effect of the trolley's horizontal movement on M and f (t) (1) For the mass matrix M
When the trolley is moving, its mass is distributed to the nodes of i, j, k and l as follows: Where, m is the half mass of the trolley.α = s 1 /l 1 and α 2 = s 2 /l 2 , see Eqs ( 16) and (17).
(2) For the load f (t) A) X -direction (horizontally) The trolley gives the structure horizontal forces in the phase of acceleration and deceleration described as follow: a.In T 0 − T 2 , there is no horizontal force.b.In T 2 − T 4 , there is no horizontal force.d.In T 6 − T 7 , the trolley moves at an acceleration of −a x .
The trolley's reaction on the structure in Y -direction is always existing.These are constant forces, which are considered in the static analysis.We simply offer the formula herein: The effect of the lifting weight and wire rope on the K and f (t) (1) For the f (t) A) In X -direction, reference to section 2) of 3.1, we have already known the lifting weight's reaction on the structure in this direction.It only needs to be distributed to the nodes of i, j, k and l based on following laws: (2) For the stiffness matrix K and damping matrix C From Eq. ( 27), we know that the force in Y -direction caused by the lifting weight is related to the displacements of the hanging point and the lifting weight's center of gravity (i.e.V w and V w ).In the actual calcu- lation, first the V w in Eqs (7) and ( 21) is canceled by Eq. ( 20), and then substitute Eq. ( 27) into Eq.( 21), and the items of node displacements in Eqs ( 7) and ( 21) are moved to their left side respectively.Finally we obtain the following global dynamic equation: Where Where C is the structure damping matrix and C w is the damping of the lifting weight. Where Where We can get the dynamic response of the structure by solving the Eq. ( 28).The explicit  difference method is used to solve the Eq. ( 28) in this paper.The unique condition for solving is that the time step length should be small enough, i.e. ∆t ∆t cr (∆t cr = T n /π), ∆t cr is the critical step length and T n is the minimal period of the structure system.In the following section, we give an example to show the actual application of this method.

Example
Figure 5 is the elevation sketch of a typical bucket ship unloader.Finding the maximal displacement of the key nodes from the phases of the hoisting of the lifting weight and the moving of the trolley to the phase of unloading (i.e. the dynamic response).
Given: Mass of lifting weight = 20,000 kg, mass of bucket = 10,000 kg, mass of trolley = 34,000 kg, initial length of wire rope = 22 m, total metallic section area of wire rope = 0.0012 m 2 , elastic modulus = 1011 N/m 2 , hoist speed = 2.8 m/s, hoist acceleration = 0.78 m/s 2 , trolley's speed V x = 2.8 m/s, trolley's acceleration = 0.74 m/s 2 .Sizes and geometrical properties for members in the structure.We neglect the influence of damping.We use Super SAP 91 and the central difference algorithm above mentioned respectively to analyze this example.The results are summarized in Table 1.(Table 1 only includes the maximal displacement of concerned nodes.) As shown in Table 1, the analysis model in Super SAP 91 is in complete agreement with the one in DOOF (DOOF is a program using this method with no changing mass matrix M and stiffness matrix K).That is, at a certain moment we only take into account the influence of the weight's force in the structural dynamic response, neglecting both the variable stiffness of the wire rope and the vibration of the trolley and lifting weight.The analysis time is the hoisting phase (T0-T2).In Table 1 the results of Super SAP 91 and DOOF is comparatively close to each other when the conditions stay the same.The errors come from the different algorithm for solving the dynamic equation, because the method of Wilson-θ is used in Super SAP 91, we choose the explicit center difference method.In DKMF(1) and DKMF(2) (DKMF is a program using this method with changing mass matrix M and stiffness matrix K), the variations of mass matrix and stiffness matrix are taken into account.The analysis time interval in DKMF(1) is  T0-T2 (the hoisting phase).The analysis time interval in DKMF( 2) is T0-T8 (the whole process, from ascending to travelling and unloading).One can clearly find out that the difference of the maximal displacement between considering the variations of M, K and not from Table 1.We can see from the results, the maximal displacement occurs after the moment of T > 6.24 s (the hoisting phase).These results are also distinctly different from those of the traditional dynamic response analysis in which the maximal displacement occurred in the hoisting.Also the maximal values between the two methods are also distinctly different.For the sake of intuitive observation, we draw the variation of displacement in x and y directions of the concerned nodes in T0-T8 shown from Fig. 6 to Fig. 10.There are sig-nificant difference between considering the changes of (M, K) and not.

Conclusions
1) By comparing the analysis results of this method with that of Super SAP 91, we find: (1) This method is correct and reliable.
(2) The distinct difference of the results shows the necessity of this method.
2) The traditional design only takes into account the ascent phase.This phase is thought of the most dangerous and governing the design.However, results of this method shows that the maximal dynamic displacement after the ascent phase is generally greater than that of in the ascent phase (approximately 60%-80%).3) If the static displacement is considered, the dynamic displacement is 20%-30% in the total displacement (static displacement plus dynamic displacement).So the total displacement solved by this method is about 15% greater than that solved by the traditional method.4) Most of the bridge bucket unloaders and the container cranes on the shore that are traditionally designed are safe in their operations.There are also accidents of collapsing.The reasons for the accidents consist of many factors, but neglect of the influence of variable mass and variable stiffness can not be ignored.