CHINESE JOURNAL OF MECHANICAL ·232· Vol. 26, No. 2, 2013

DOI: 10.3901/CJME.2013.02.232, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn

Metamorphic Manipulating Mechanism for MCCB Using Reduced Iteration

XU Jinghua*, ZHANG Shuyou, ZHAO Zhen, and LIN Xiaoxia State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China

Received May 5, 2012; revised November 25, 2012; accepted November 27, 2012

Abstract: The present research on moulded case circuit breaker(MCCB) focuses on the enhancement of current-limiting interrupting performance during short circuit, overload, under voltage and phase failure, involving electrics, magnetic, mechanics, thermal, material, friction, arc extinguishing, impact vibration, skin effect, etc. The rigid-flexible coupling of the parts and components of the metamorphic manipulating mechanism in multi-fields leads to the non-rigid, high frequency, high damping, singularity of the Euler-Lagrange equations which represents the multi-body dynamics. The small step iteration which is used for obtaining the instantaneous and short time critical interrupting performance of metamorphic mechanism appears inaccuracy. It is difficult to realize top-down design by existing CAD systems. Therefore, a metamorphic manipulating mechanism design method for MCCB using index reduced iteration(IRI) is put forward. The metamorphic manipulating mechanism of MCCB is decomposed into three mechanisms: main switch connector mechanism, electromagnet-drawbar-jump buckle mechanism, and bimetallic strip-drawbar mechanism, which is respectively described by electro-dynamic force, electromagnet force, and bimetallic strip force. The dummy part(virtual rigid) without moment of inertia and mass is employed as intermediate to join the flexible body and rigid body. The model of rigid-flexible coupling metamorphic mechanism multi-body dynamics is built. The differential algebraic equations(DAEs) of the multibody dynamics model are converted to pure ordinary differential equations(ODEs) by coordinate partition. Order reduced integration with multi-step and variable step-size is preceded based on IRI. The non-linear algebraic equations are solved in each integration step by Newton-Rapson iteration. There is no ill-condition and singularity of Jacobian matrix when step size reduces to zero. The independent prototype design system using ACIS R13, HOOPS V11.0 and Visual C++. 2003 has been developed, which verifies the effectiveness of the proposed method. The proposed method enhances the current-limiting interrupting performance of MCCB, and has reference significance for multi-body dynamics design for similar flexible metamorphic mechanisms in multi-fields.

Key words: index reduced iteration (IRI), moulded case circuit breaker (MCCB), metamorphic manipulating mechanism, Euler-Lagrange equations, rigid-flexible coupling, multi-body dynamics, current interrupting simulation

number of DOF changes during the movement of the ∗ 1 Introduction device under complicated working conditions. This means

the kinematic structural representation of a metamorphic Moulded case circuit breaker(MCCB), namely low linkage has different forms in which vertices and edges voltage(less than 1 kV) circuit breaker, is an automatically combine depending on the configuration of the device[1–2]. operated electrical switch designed to protect an electrical In recent years, many scholars have studied the circuit from damage caused by overload, short circuit, under voltage and phase failure. Its basic function is to mechanism or and proposed many design [3–9] [10] detect a fault condition and, by interrupting continuity, to methods . DAI, et al , analyzed the difference between immediately discontinue electrical flow. MCCB is widely metamorphic mechanism and deployable mechanism, used in power branches, such as electric power kinematotropic mechanism and discontinuity mobility [11–12] locomotive, CNC machine tools and complete sets of mechanism. ZHANG, et al , proposed kinematic electromechanical equipments. reverse design, product structure modeling method for The metamorphic manipulating mechanism is crucial to mechanism and product design. Ref. [13] uses the equation the current-limiting interrupting performance of MCCB. A of motion to analyze the dynamic response of the spring metamorphic mechanism has the property that the effective type manipulating mechanism. In Ref. [14], direct numerical simulation of the first three milliseconds * Corresponding author. E-mail: [email protected] This project is supported by National Basic Research Program of China following ignition of the arc in a low-voltage (973 Program, Grant No. 2011CB706506), National S&T Great Special of circuit-breaker is proposed using a computational- China(Grant Nos. 2012ZX04010011, 2011ZX04014-131), National Science Foundation for Young Scholars of China(Grant No. 51005204), fluid-dynamics code adapted for electric-arc modeling. In and Postdoctoral Fund of China(Grant No. 20100471000) Refs. [15–16], multiobjective constrained optimal synthesis © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2013

CHINESE JOURNAL OF MECHANICAL ENGINEERING ·233· of mechanisms is realized using a new evolutionary 100 A) use air alone to extinguish the arc. It avoids the . In Ref. [17], constraint analysis on mobility contact floating or bouncing at closure and during any change of a novel metamorphic parallel mechanism is inrush current under short circuit conditions. The MCCB is realized. In Ref. [18], the effect of different vent shown as Fig. 1–Fig. 3. configurations including middle vent and side vent on the interruption performance is investigated based on a simplified model of arc chamber with a single break, which can be opened by the electro-dynamics repulsion force automatically. Ref. [19] proposes a Bayesian methodology to assess the performance of circuit breaker utilizing its control circuit data. Ref. [20] uses improved empirical mode decomposition energy entropy and multi-class support vector machine to diagnose fault for high voltage circuit breaker. In Ref. [21], the evaluation of time measurements in case of different failures is carried out for two circuit-breaker solutions, i.e. with spring and hydraulic drive. Ref. [22] presents the results of the degree of irreversible changes of dielectric properties of vacuum Fig. 1. Geometric assembly model of metamorphic mechanism circuit breakers with CuCr and CuBi contacts before and after short-circuit breaking operations. Ref. [23] uses the constraint graph of computational geometry rather than the traditional topological graph to characterize a metamorphic linkage in order to simplify the representation of configuration changes. In Ref. [24], degree of motion constraint and its algorithm are put forward to add motion constraints among parts automatically for a MCCB multibody system. LIN, et al[25], focused on the application of asymmetric distributed loads on parts based on the eigenfunction distribution of FEM. ZHAO, et al[26], focused on the efficiency when the step small enough to guarantee the solution stability and precision of MCCB. The above methods promote the design quality of mechanism or product design. However, it is difficult to obtain the instantaneous and short time breaking Fig. 2. Metamorphic manipulating mechanism of MCCB characteristics of flexible metamorphic mechanism because 1—Rod for holding main switch connector K; 2—Contact spring for keeping pressure of K; of inaccuracy during small step iteration. Therefore, a 3—Shaft for restricting breaking displacement of main switch metamorphic manipulating mechanism design method for connector K; 4—Upper linkage BD; MCCB using IRI is put forward which mainly contains: 5—Hinge shaft D of upper linkage BD and jump buckle CD/CT; current-limiting interrupting process of metamorphic 6—Latch buckle HT; 7—Shaft C of jump buckle CD/CT; manipulating mechanism of MCCB; rigid-flexible coupling 8—Main spring for keeping or changing status of linkage multi-body dynamics resolution of metamorphic metamorphic mechanism; manipulating mechanism for MCCB based on IRI; 9—Drawbar spring for keeping or changing status of HT; 10—Shaft H of latch buckle HT; metamorphic manipulating mechanism design process via 11—Hinge shaft B of upper linkage BD and lower linkage AB; multiobjective optimization. The independent prototype 12—Lower linkage AB design system has been developed to verify the proposed method. Fig. 4 shows the interruption process of MCCB. If the actual current is more than or much larger than the rated 2 Current-Limiting Interrupting Process current, the jump buckle CD will execute action, Lock of Metamorphic Manipulating Mechanism catch HT will turn which results in the freedom of T, then, of MCCB Jump buckle CD will rotate around point C under the action of the spring JB, the point D becomes active point, the The DOF of main switch connector changes, therefore, upper linkage BD and the lower linkage AB have no the manipulating mechanism of MCCB can be considered constraints, the mechanism becomes five-bar linkage from as a kind of metamorphic mechanism. Miniature four-bar linkage. The DOF of MCCB is changing from one low-voltage circuit breakers (rated current not more than to two, so the circuit is protected.

·234· XU Jinghua, et al: Metamorphic Manipulating Mechanism Design for MCCB Using Index Reduced Iteration

main spring JB is the same with upper linkage BD, the spring has the maximum mechanical energy. Once the main spring JB passes across the upper linkage BD, point B will move under the action of the main spring JB, then, the lower linkage AB pushes the main switch connector K to close. After the thorough close of K, the upper linkage BD and the lower linkage AB are all at the dead center position. The metamorphic manipulating mechanism of MCCB is decomposed into three mechanisms: main switch connector mechanism, electromagnet-drawbar-jump buckle mechanism, and bimetallic strip-drawbar mechanism which is respectively described by electro-dynamic force, electromagnet force, and bimetallic strip force which are shown as Fig. 5–Fig. 7. Fig. 3. Two dimensional full section view of MCCB 1—Main switch connector K; 2—Shaft for restricting breaking displacement of K; 3—C; 4—B; 5—D; 6—J; 7—Main spring; 8—T; 9—H; 10—Drawbar spring; 11—R(Main shaft); 12—Contact spring; 13—A. AB—Lower linkage; BD—Upper linkage; HT—Latch buckle; AR—Arm of moving contact

Fig. 5. Electromagnet trip break mechanism

Fig. 6. Bimetallic strip break mechanism

Fig. 4. Current-limiting interrupting process of metamorphic manipulating mechanism of MCCB Fig. 7. Main switch connector mechanism If we want to switch on manually, push the operating handle, let it rotate around point O. Then the main spring The attraction force of electromagnet trip mechanism Fem JB turns with JO and stores mechanical energy can be obtained by Maxwell formula and Ampere’s because of tensile strain. When the force direction of the circuital law:

CHINESE JOURNAL OF MECHANICAL ENGINEERING ·235·

INS22 µ II sinαα sin  0  12 1 2ddll , Fem 2 ,  2 12  4π r1  ll12 4µ0    µµ 12 where dB —Magnetic flux density due to Id,l

Idl —Current element, where I—Actual current, α —Vector angle of current element, S—Cross sectional area of electromagnetic r —Position vector from Idl to point P. iron-, µµ— 12, Magnetic permeability, 3 Rigid-flexible Coupling Multi-body µ —Permeability of vacuum, µ  2 0 0 0.4πμNA , Dynamics Resolution of Metamorphic — l Magnetic flux path, Manipulating Mechanism for MCCB — N Coil turn. Based on IRI The pushing force of bimetallic strip Fbi can be obtained by thermodynamic energy equation: The parts with large deformation should be considered as flexible body rather than rigid body, such as long linkage 23 d∆T 2 kbδδ E b E type, long shaft type. Index reduced iteration(IRI) means mc k S∆∆ T  I R  F T  f , d4tLT bi 4L3 0 converting the higher order differential equations into lower order differential equations to improve iteration stability in where m —Mass of bimetallic strip, each integration step. — kT Surface heat transfer coefficients, The differential order refers to derivation number, neither S —Surface heat transfer area, matrix size nor power exponent. The dynamics questions R —Electrical resistance of heating body, are expressed with differential algebraic equations(DAEs), c —Specific heat capacity of bimetallic strip, which are also called Euler-Lagrange Equations. For k —Specific thermal deflection, dynamics questions, the inputs are the constraints of the b —Width of bimetallic strip, kinematic pairs, the preset initial generalized coordinates of δ —Thickness of bimetallic strip, each rigid body, while the outputs are the generalized L —Length of bimetallic strip, coordinates (position, velocity and acceleration) of each — rigid body at any time in global coordinates system. The f0 Displacement of bimetallic strip, — DAEs are as follows: E Young’s modulus of , — ∆T Temperature rise.  T PF Φ TTλ H 0, The electro-dynamic repulsion force FR contains the  q  q Holm force FH and the Lorentz force FL.   T The contact area is far less than the contact surface area P  ,  q because the contact surface is always rough in microscopic  scales shown in Fig. 7. Then, the current elements shrink at Φ (q ,t ) 0,  the contact point which leads to repulsion force. The Holm Ff(uq , , t ), u q, force FH is caused by current constriction and skin effect which exists only during contact process of the main switch where q—Generalized coordinates of rigid body, connector K: qq(xyz ψθφ ),T Rn , (xyz )—Cartesian coordinates of mass center µµII22S ξ HS of rigid body, F 00ln lnB , H ψθφ —Euler angles of rigid body, 4π4π SF0 K ( ) — t Time coordinate, — where S—Cross sectional area of contact point K, T Kinetic energy of the system, P —Generalized momentum of the system, S0—Effective contact area, — ξ —Contact coefficient which is about 0.3–1, H Coordinates transformation matrix — 2 of external force, HB Brinell hardness of contact point (Nmm ), λ—Lagrange for constraint equations, FK —Contact force of contact point K. λ  Rm , The Lorentz force FL can be obtained by Biot-Savart law Φ —Constraints of the kinematic pairs, Φ  Rm , which exists till electric arc extinguishing. FL depends on T — current and contact angle of K: Φq Jacobian matrix of constraint equations,

Φi mn µ   ()Φq(, ij )R ,i  1, 2, , mj ,  1, 2, , n , 0 Idlr  dB   q j 4π r3   F( I  BV )d  µ  L contact 0 I d lr  where m —Number of constraints, number B  3  4π l r  of the kinematic pairs,

·236· XU Jinghua, et al: Metamorphic Manipulating Mechanism Design for MCCB Using Index Reduced Iteration

p n —Number of rigid and flexible body, m 1 p ΦΦ  6n —Number of generalized coordinates. q  (q )(, ij )  ,p  [1, ). p   The generalized coordinates q of rigid body can be i1 decomposed into independent coordinates and non- independent by full rank factorization of matrix of After obtaining generalized coordinates q of rigid body, constraints Φ of the kinematic pairs. The independent the n order interpolation polynomial Pxn ()is constructed to coordinates are solved by integration iteration while the obtain the derivative of the function gx(): non-independent coordinates are solved by non-linear  algebraic equations resolution. nn n   xx j  T P() x l () xy  y, Jacobian matrix of constraint equations Φq can be n kk    k kk00 j0 xxkj obtained by derivation of ()uqλ through m order  jk   backward differentiation formula (m 1): n1 ξ g () Rx() gx () Pnn () x  ω 1(),x  (n  1)! ykk fx( ), k 0,1  m1 n    xk x0 kh  ωni1(x ) ( xx ).   my ()  m1 y  mm  11 yy   i0 k k kk1 ΦΦ  Φ Rx() represents truncation error. When k  5 , we can  11 1     xx12 xn  get the numeric differential expression with five discrete    ΦΦ  Φ points based on Lagrange interpolation polynomial. Thus,  22 2 T   if we get the resultant displacement of main switch Φq  xx12  xn  (,xxx123 , ) (u,q ,λ ).   connector, we can use Lagrange interpolation polynomial      rules s to obtain the velocity and acceleration[27]:   ΦΦmm  Φ m       xx12 xn      25y1 48 y 2 36 y 3 16 yy 45 3 gx()1  , 12h The nonlinear algebraic equationsΦ (,)q t  0are solved  3y 10 y  18 y  6 yy  by improved Newton-Raphson iteration algorithm. When gx() 1 2 3 45, 2 12h the integration step approaches zero h  0 , Jacobian  T y88 y yy matrix Φ becomes into ill-conditioned matrix, therefore,  i2 i  1 ii  12 q gx()i  , an IRI method is applied to analyze metamorphic 12h  manipulating mechanism: si3, p  2  & i Z ,  y6 y  18 y  10 yy  3  T  p43 p p  2 pp  1   TT gx()p1  , PFΦq λ H 0, 12h  q     3yyyyypp43 16 36 p  21 48 pp  25  T gx()p  , P  , 12h  q   p 5,&,  p Z Φ (q ,t ) 0,     s :,qq  Φ (uq , ,t ) 0,    ss :,qq  Ff (uq , , t ),     Φ Tµµ where  stands for composite mapping. uq q , 0.

4 Metamorphic Manipulating Mechanism After using IRI method, condition number cond(Φq ) T Design Process via Multiobjective calculated by norm Φq of Jacobian matrix Φq decreases from +∞ to zero. There is no violation during calculating Optimization

Φ and Φ . Metamorphic manipulating mechanism design for MCCB can be converted into a multiobjective optimization model: cond(ΦΦqq ) cond( )  0,

1 cond(ΦΦqq )   Φ q, minFx ( ) xRn m s.t.Gxie ( ) 0, i 1, 2, , m , ΦΦq  max (q )(, ij ) , 1 1jn i1 Gx( ) 0, i  m 1, , m , ie T Φ λ ( ΦΦ ), xlu xx, q2 max qq n n ΩΩ{xxR },  , ΦΦq  max (q )(, ij ) , m  1im Λ  j1 y Fx( ), { y R },

CHINESE JOURNAL OF MECHANICAL ENGINEERING ·237· where Gxi () 0 stands for equality constraints, the maximum break angular velocity as the main desired

Gxi () 0 stands for inequality constraints. The goal design object during the process from the dead center attainment method is employed to solve the problem. It position to extreme position. The status when MCCB involves expressing a set of design goals, which is completely closed is taken as initial condition. associated with a set of objectives. The problem The key known parameters of metamorphic mechanism formulation allows the objectives to be under- or are shown in Table 1. θK stands for opening angle of overachieved, enabling the to be relatively main switch connector K, means the angle between moving imprecise about initial design goals. An element of contact and fixed contact across the main shaft. If θK is slackness is introduces into the problem, which otherwise too small, it will cause electric arc reignition. LK stands imposes that the goals be rigidly met. Hard constraints can for opening resultant displacement of main switch be incorporated into the design by setting a particular connector K, means the motion resultant displacement of weighting factor to zero. The goal attainment method the centroid of main switch connector in global coordinate provides a convenient intuitive interpretation of the design system. vK stands for linear velocity of main switch problem, which can be solved using standard optimization connector K, means the resultant velocity of the moving connector in global coordinate system. a stands for procedures. K linear acceleration of main switch connector K, means the The flowchart of the metamorphic mechanism synthesis resultant acceleration of the moving connector in global is shown in Fig. 8. coordinate system. F means the minimum trip force trip which is enough to trigger tripping automatically of latch buckle.

Table 1. Key parameters of metamorphic mechanism when MCCB completely closed

Parameter –1 Stiffness of main spring kmain(N • mm ) 14.468

Preload of main spring FmainN 102.126 –1 Stiffness of contact spring kcontact(N • mm ) 16.556

Preload of contact spring FcontactN 25.346 –1 Stiffness of drawbar spring kdrawbar(N • mm ) 0.943

Preload of drawbar spring FdrawbarN 1.236 x coordinate of fixed axial center C of jump buckle 6.119 xCmm y coordinate of fixed axial center C of jump buckle 26.133 yCmm x coordinate of connection point J of handle 12.022 and main spring xJmm y coordinate of connection point J of handle 33.545 and main spring yJmm

x coordinate of axial center B of linkage shaft xBmm 11.014

y coordinate of axial center B of linkage shaft yBmm 25.389

x coordinate of axial center A of connecting shaft xAmm 23.845

y coordinate of axial center A of connecting shaft yAmm 31.334

The breaking performance obtained by IRI is shown in Fig. 9 and Table 2. From Fig. 9(a), we can know that main Fig. 8. Metamorphic manipulating mechanism design process switch connector K starts breaking due to electro-dynamic via multiobjective optimization repulsion force FR or pushing force of bimetallic strip Fbi at t1. The metamorphic mechanism occurs just when the latch buckle HT disconnects at t2, concurrently, the jump 5 Example of Metamorphic Manipulating buckle CDCT starts breaking due to electromagnetic force Mechanism Design for MCCB Fem, manipulating mechanism becomes five-bar linkage from four-bar linkage. The metamorphic manipulating The independent prototype design system using ACIS mechanism endures impact vibration at t3. The jump buckle R13, HOOPS V11.0 and Visual C++.NET 2003 has been terminates breaking and K breaks thoroughly at t4. developed to verify the proposed method. The break If we want to determine the precise minimum tripping velocity of the manipulating mechanism of the MCCB force, we can first divide the rough interval 0.5–0.8 N into means the average velocity during the process from six trial force and calculate the opening angle, opening tripping to thoroughly breaking off. The more quickly the resultant displacement, linear resultant velocity, linear break velocity is, the shorter the breaking time is. We make resultant acceleration, based on these values, we can further

·238· XU Jinghua, et al: Metamorphic Manipulating Mechanism Design for MCCB Using Index Reduced Iteration determine that the minimum tripping force is 0.55–0.60 N. metamorphic mechanism analysis can’t be precededwithout If we want to know the more precise minimum tripping IRI method, because the step size h of Newton-Raphson force, we use Index Reduced Iteration method and repeat iteration reduces to zero and condition number of Jacobian T the process whose results are shown in Table 2. The matrix approaches plus infinity Φq  .

Fig. 9. Breaking performance verified by ADAMS simulation

Table 2. Parameters of MCCB during interruption process 1 222 σe[( σσ 12  )  ( σσ 23 )  ( σ 31 σ ) ]. at a given time using IRI 2

Opening Linear Trip Opening Linear resultant The strength analysis of improved metamorphic resultant resultant force angle acceleration mechanism is shown in Fig. 10 and Table 3. From Fig. 10 displacement velocity -2 FtripN θKrad –1 aK(km • s ) LKmm vK(m • s ) and Table 3, we can know that the jump buckle and upper 0.55 0 0 0 0 linkage have larger deformation under a certain working 0.60 0.074 2 26.164 39 4.253 937 0.521 620 condition. Therefore, defining the jump buckle and upper 0.65 0.101 5 29.316 19 3.943 733 2.036 455 linkage as rigid body is suitable. The rigid-flexible 0.70 0.146 4 32.416 08 3.914 446 2.942 640 coupling metamorphic mechanism of MCCB under a 0.75 0.203 3 39.823 44 4.894 420 2.755 787 certain working condition which is taken jump buckle and 0.80 0.269 3 37.189 73 5.899 657 2.641 224 upper linkage as flexible body is shown as Fig. 11.

The key parameters of metamorphic mechanism are The parts of the metamorphic mechanism have more optimized by rigid-flexible coupling multi-body dynamics than one functional direction of strains and stress. The three design using IRI shown as Table 4. principal stresses are labeled σ1, σ2, and σ3. The principal By using the proposed method, the maximum opening stresses are ordered so that σ1 is the most positive (tensile) angle and maximum opening resultant displacement of and σ3 is the most negative (compressive). The Von Mises main switch connector K are increased by shape parameters or equivalent stress σ e is employed to determine the optimization to eliminate electric arc reignition. The combined strength analysis: average absolute opening resultant velocity is increased to

CHINESE JOURNAL OF MECHANICAL ENGINEERING ·239· enhance current-limiting interrupting performance. The average absolute linear resultant acceleration is decreased to weaken the impact vibration to arm of the moving contact and frame of MCCB. The maximum equivalent stress of jump buckle and upper linkage are decreased to improve the reliability of MCCB.

Fig. 10. Strength analysis of improved metamorphic mechanism

Table 3. Von Mises stress of improved metamorphic mechanism under a certain working condition

Minimum Maximum Decreasing Maximum Von Mises Von Mises ratio of Part displacement stress stress max(σ ) max ε mm V min(σV)Pa max(σV)GPa ∆ %

Jump buckle 509 117 0.344 0.017 5 12.89 CDCT Upper linkage 563 801 0.347 0.004 4 16.34 BD Lower linkage 55 648 0.089 0.014 5 10.49 AB Latch buckle 61 718 0.267 0.050 3 11.53 HT Drawbar 18.216 0.301 0.086 0 8.85 Main shaft R 9 098 0.225 0.246 0 6.74 Cheek 25.542 0.080 0.004 5 4.18 Shaft H 218 102 0.047 0.426 8 9.43 Shaft for restricting 137 555 0.016 0.071 8 7.89 displacement of K

Rod for 7 0.389×10 0.035 0.006 2 9.36 holding K Hinge shaft D 11 281 0.005 0.002 7 5.54 Hinge shaft B 5 377 0.006 0.003 6 3.83 Shaft C 3 410 0.001 0.021 5 4.65

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inertia and mass is employed as intermediate to join the flexible body and rigid body. The precision of nonlinear dynamics calculation of MCCB is improved by getting micro element deformation of metamorphic manipulating mechanism. (3) The independent prototype design system has been developed to design the geometric structure of MCCB. The proposed method enhances the current-limiting interrupting performance of MCCB.

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