4. NUMERICAL SOLUTION PROCEDURES AND VIRTUAL PROTOTYPE
188.8.131.52 Baumgarte’s stabilization technique (BTS)
shown a higher numerical stability towards the integration over the digging time, it is not sufficient to accurately estimate the trajectories of the ropes and their corresponding initial values. The precedent analyses performed in Section (184.108.40.206) were based on the assumption that the constraint equations (4.1) are smooth and differentiable twice. It was found that the Jacobian matrix is not symmetric and that resulted in singularity at different time steps. Equations (4.1), with the embedded linear displacements of the drag and hoist ropes, is defined by equation (4.11).
F𝑗𝑗(qi(𝑡𝑡), t) = 0 (i, j) = (1, . . ,3) (4.11)
Differentiating equation (4.11) with respect to time results in the constraint equations at the velocity level as shown in equation (4.12). h𝑖𝑖(q, t) is called the hidden constraint and is given by the equation (4.13).
𝑑𝑑𝑖𝑖�F𝑗𝑗(qi(𝑡𝑡), t)� = 𝐽𝐽(q(t), t). q̇(t) + h𝑖𝑖(q(t), t) = 0 (4.12)
Figure 4.6. Singularity of dragline kinematics models: (a) digging phase (b) full-bucket hoisting phase
h𝑖𝑖(q(t), t) =
The hidden constraints are just the right-hand side of equation (4.5) and are nonlinear algebraic equations. Differentiating equation (4.12) with respect to time yields the second order differential equation (4.14) at the acceleration level.
𝑑𝑑𝑖𝑖2�F𝑗𝑗(qi(𝑡𝑡), t)� = 𝐽𝐽(q(t), t). q̈(t) + J𝛿𝛿(q(t), t). (𝑞𝑞.̇ 𝑞𝑞̇) + h𝑖𝑖𝑖𝑖(q(t), t) + 2h𝑖𝑖𝛿𝛿(q(t), t) = 0 (4.14)
J𝛿𝛿(q(t), t) = 𝜕𝜕J(q(𝑖𝑖),t))𝜕𝜕𝛿𝛿
,h𝑖𝑖𝑖𝑖(q(t), t) = 𝜕𝜕(ℎ𝑖𝑖(q(𝑖𝑖),t))𝜕𝜕𝑖𝑖
,h𝑖𝑖𝛿𝛿(q(t), t) =𝜕𝜕(ℎ𝑡𝑡(q(𝑡𝑡),t))𝜕𝜕𝑞𝑞
The last two terms in the equation (4.14) are also hidden constraints and their representation is often undesirable due to the complexity of the equations in a constraint multibody system. Equation (4.14) can be rewritten in a simplified form without augmented hidden constraint and is given in equation (4.16).
𝑑𝑑𝑖𝑖2�F𝑗𝑗(qi(𝑡𝑡), t)� = 𝐽𝐽(q(t), t). q̈(t) + 𝒜𝒜(𝑞𝑞, 𝑞𝑞̇, 𝑡𝑡) (4.16)
𝒜𝒜(𝑞𝑞, 𝑞𝑞̇, 𝑡𝑡) = J𝛿𝛿(q(t), t). (𝑞𝑞.̇ 𝑞𝑞̇) + h𝑖𝑖𝑖𝑖(q(t), t) + 2h𝑖𝑖𝛿𝛿(q(t), t) (4.17)
A central process that follows these derivations is to combine the differential equations (4.12) and (4.16) with the constraint equation (4.11). This combination leads to the Baumgarte’s formalism (Baumgarte, 1972), which is given by equation (4.18).
F̈ + 𝛼𝛼𝐵𝐵. Ḟ + 𝛽𝛽𝐵𝐵. F = 0 (4.18)
𝛼𝛼𝐵𝐵and 𝛽𝛽𝐵𝐵 are parameters defined by the user and are more likely selected as stated in
equation (4.19) (Baumgarte, 1972). The benefit of using equation (4.18) is that the numerical violations resulted from embedded constraints of velocity and constraint equations are minimized, if not eliminated. The system of equations becomes more stable during integration and the drift error is reduced to a minimum. The drift error can be regarded as the perturbations in the acceleration when the constraint equations and constraint velocity equations are differentiated. Thus, the drift error in equation (4.20) is a quadratic function of time and is related to the constraint equations violation (Simeon, 2010).
𝜅𝜅= 12 (𝑡𝑡 − 𝑡𝑡0)2𝜁𝜁𝑎𝑎+ (𝑡𝑡 − 𝑡𝑡0)𝜁𝜁𝑎𝑎+ 𝜁𝜁𝑝𝑝 (4.20)
𝜁𝜁𝑎𝑎, 𝜁𝜁𝑎𝑎 and 𝜁𝜁𝑝𝑝 are constants associated with the error at acceleration, velocity and position
levels, respectively. The use of Baumgarte’s method has been found to solve the singularity problem and improve the accuracy of resulting trajectories. However, the choice of parameters bigger than 3 did not dampen the errors, but resulted in a stiff DAE system. Correct trajectories are generated after integration for the selected parameters 𝛼𝛼𝐵𝐵= 1 and 𝛽𝛽𝐵𝐵= 0.25 and are shown in Figure 4.7 (a). Other trajectories that are not within
machine limits are not a part of the solution of the equations of motions. In addition, the applicability of BTS was also evaluated against the errors, which are calculated from the invariants (constraints algebraic equations). Caution must to be taken when choosing the values of Baumgarte’s parameters. From Figure 4.7 (b), the selection of 𝛼𝛼𝐵𝐵 = 6 and 𝛽𝛽𝐵𝐵 = 9 meets the conditions of equation (4.19). However, it has changed the structure of the
constraints and their derivatives. Consequently, the numerical analysis has produced inaccurate results.
It can be concluded from Figure 4.7 (a) that the angular displacements of the hoist and drag ropes follow the same behavior and their trajectories vary within the machine limits for the parameters 𝛼𝛼𝐵𝐵= 1 and 𝛽𝛽𝐵𝐵 = 0.25. The same behavior can be seen for values 𝛼𝛼𝐵𝐵 = 6 and 𝛽𝛽𝐵𝐵= 9, but the values of the angular displacements of the hoist and drag ropes
are around 70 ° and 90 ° and are not within the machine limits. The initial conditions of the angular displacements of the rope angles and their initial angular velocities are listed in Table 4.1.
Figure 4.7. Trajectories of ropes using BTS during digging phase: (a) α =1, β =0.25 (b), α =6, β =9
Table 4.1 Initial angular displacements and angular velocities of ropes Rope Initial Angle (rad) Initial Velocity (rad/s) Hoist rope 𝑞𝑞4 = −0.0437 𝑞𝑞̇4 = 0.0 Dump rope 𝑞𝑞5 = 0.2831 𝑞𝑞̇5 = 0.0 drag rope 𝑞𝑞6 = −0.4692 𝑞𝑞̇6 = 0.0
Dynamics Solution Procedures. The dynamic model of the dragline front-
end assembly was formulated based on Kane’s method. The mathematical model, in Section 3, has all the relevant information to perform the inverse dynamic analysis. Figure 4.8 shows the flowchart for developing and implementing the simultaneous kinematics and dynamic analyses. The solution of the dynamic model during the digging phase is two- fold: (i) feedforward displacement calculations based on Newton-Raphson method in Mathematica, and (ii) inverse dynamics based on the calculated feedforward displacements from step (i). These procedures are also used for calculating the drag force and hoist and swing torques for the loaded bucket swing motion. The inverse kinematics analysis must be integrated in the solution of the inverse dynamic procedures.The BST used in Section 220.127.116.11 is integrated into the dynamics solution procedures to enforce the constraints of equation (4.4).
It can be seen from Figure 4.8 that the dynamic analysis of a dragline is similar to any multibody dynamic analysis. It starts with a mathematical formulation of the constraint equations, followed by a full kinematics analysis for defining the independent generalized speeds, and finally the formulation of the EOM using Kane’s method. The latter are solved simultaneously by enforcing the acceleration constraints using BST to minimize the drift errors. Finally, the outputs of the numerical analysis are plotted and verified based on the machine operational limits. The outputs of the dynamics model during the digging phase include the drag force and hoist torque. In the case of loaded bucket swinging phase, the outputs include the drag force and hoist and swing torques that are used as inputs in the advanced finite element analysis on wire ropes. The errors due to constraint violations are
also plotted and visualized to verify the accuracy of the solution approach and the numerical algorithm.
Figure 4.8. Flowchart of the dynamics solution algorithm
The equations of motion in equation (3.87) combined with the acceleration constraint equations in equation (4.4) form a stiff system of highly nonlinear differential equations. The initial conditions found in Section (18.104.22.168) must be consistent and produce minimal errors over the whole integration domain. The derivation of the constraint algebraic equations with respect to time produces a system of equations with a reduced index and it changes the structure of the original equations. As was seen in equations set
(4.5), the derived equations contain additional expressions of nonlinear trajectory functions, which result in drift errors. The DAE solver in Mathematica contains the necessary algorithms to handle a stiff system for a twice-differentiable drift. The reduction of the drift error, in the solution algorithm, is based on using a simplified method that minimizes the residual at each iteration step. In other words, the right-hand side of the complete system, in equation (4.18), is subtracted from the left-hand side to create a residual function. A general type of this residual is given by equation (4.21).
F𝑗𝑗(qi(𝑡𝑡), q̇i(𝑡𝑡), t) = 0 (4.21)
The initial conditions proposed earlier satisfy this residual, as well as its derivative, which is given in the equation (4.22).
𝑑𝑑F𝑗𝑗(qi(𝑡𝑡), q̇i(𝑡𝑡), t) = 0 (4.22)
Inconsistent initial conditions are more likely to violate the residual equations and their derivatives resulting in accumulated errors. Solving DAE with higher index is very challenging because of the requirement to satisfy several equations along with second-to- third degrees of their derivatives. The procedures of the DAE solvers are described in the integrated solution algorithm in Figure 4.9. The algorithm contains a solver for the first order kinematics DAE, a solver for the second order kinematics DAE, and a dynamic solver that integrates the equations of motion. The algorithm starts evaluating the index of the constraints algebraic equations and determines the order of differentiations that is required to relate the variables together. If the index of the DAE1, DAE2, and DAE3 is 1, the equations are solved by integration and their results are passed onto the equations of motion solver.
Figure 4.9. Scheme of numerical implementation of the DAE solvers No No Outputs Differentiate Rebuild DAEs Inputs (Geometry, Inertia, Masses)
(Motion Variables 𝒒𝒒𝒊𝒊) Index Reduction Differentiate Yes Index Reduction
∫DAE3 Yes DAE2 DAE1 DAE3 DAE5 DAE4 DAE6
∫DAE6 Constraints Algebraic
DAE4 DAE5 DAE6
If the index of the DAE1, DAE2, and DAE3 is 1, the equations are solved by integration and their results are passed onto the equations of motion solver. When the DAE index is 2, a second derivation is performed on the constraints equations and the DAE 4, DAE 5, and DAE6 are then integrated and plotted. In addition, their integration results are passed onto the dynamics solver to calculate the required forces and torques.
The constraint algebraic equations may have an index of three that requires an additional differential order to be carried out to relate the hidden variables. This case is encountered in the vibration and jerky motion of links in the mechanical system. It can be concluded that the formulation of the constraint equations has a profound impact on the numerical solution process. In general, the dynamics of a closed kinematics mechanism that possesses a number of links less than 3 can be done without any difficulty. However, for a multilink mechanism, such as the dragline front-end assembly, it requires a substantial amount of work and fine tuning of the model at all stages of its development. Therefore, it is recommended to start the kinematic and dynamic analyses using a simplified vector loop equation and then expanding it to incorporate additional links. It should be pointed out that the derivations of the kinematic and dynamic models are done in Mathematica and are provided in details in Appendix A.