B6U1 WORDS2
英文有限元方法Finite element method讲义 (1)
MSc in Mechanical Engineering Design MSc in Structural Engineering LECTURER: Dr. K. DAVEY(P/C10)Week LectureThursday(11.00am)SB/C53LectureFriday(2.00pm)Mill/B19Tut/Example/Seminar/Lecture ClassFriday(3.00pm)Mill/B192nd Sem. Lab.Wed(9am)Friday(11am)GB/B7DeadlineforReports1 DiscreteSystems DiscreteSystems DiscreteSystems2 Discrete Systems. Discrete Systems. Tutorials/Example I.Meshing I.Deadline 3 Discrete Systems Discrete Systems Tutorials/Example IIStart4 Discrete Systems. Discrete Systems. DiscreteSystems.5 Continuous Systems Continuous Systems Tutorials/Example II. Mini Project6 Continuous Systems Continuous Systems Tutorials/Example7 Continuous Systems Continuous Systems Special elements8 Special elements Special elements Tutorials/Example III.Composite IIDeadline *9 Special elements Special elements Tutorials/Example10 Vibration Analysis Vibration Analysis Vibration Analysis III Deadline11 Vibration Analysis Vibration Analysis Tutorials/Example12 VibrationAnalysis Tutorials/Example Tutorials/Example13 Examination Period Examination Period14 Examination Period Examination Period15 Examination Period Examination Period*Week 9 is after the Easter vacation Assignment I submission (Box in GB by 3pm on the next workingday following the lab.) Assignment II and III submissions (Box in GB by 3pm on Wed.)CONTENTS OF LECTURE COURSEPrinciple of virtual work; minimum potential energy.Discrete spring systems, stiffness matrices, properties.Discretisation of a continuous system.Elements, shape functions; integration (Gauss-Legendre).Assembly of element equations and application of boundary conditions.Beams, rods and shafts.Variational calculus; Hamilton’s principleMass matrices (lumped and consistent)Modal shapes and time-steppingLarge deformation and special elements.ASSESSMENT: May examination (70%); Short Lab – Holed Plate (5%); Long Lab – Compositebeam (10%); Mini Project – Notched component (15%).COURSE BOOKSBuchanan, G R (1995), Schaum’s Outline Series: Finite Element Analysis, McGraw-Hill.Hughes, T J R (2000), The Finite Element Method, Dover.Astley, R. J., (1992), Finite Elements in Solids and Structures: An Introduction, Chapman &HallZienkiewicz, O.C. and Morgan, K., (2000), Finite Elements and Approximation, DoverZienkiewicz, O C and Taylor, R L, (2000), The Finite Element Method: Solid Mechanics,Butterworth-Heinemann.IntroductionThe finite element method (FEM) is a numerical technique that can be applied to solve a range of physical problems. The method involves the discretisation of the body (domain) of interest into subregions, which are known as elements. This enables a continuum problem to be described by a finite system of equations. In the field of solid mechanics the FEM is undoubtedly the solver of choice and its use has revolutionised design and analysis approaches. Many commercial FE codes are available for many types of analyses such as stress analysis, fluid flow, electromagnetism, etc. In fact if a physical phenomena can be described by differential or integral equations, then the FE approach can be used. Many universities, research centres and commercial software houses are involved in writing software. The differences between using and creating code are outlined below:(A) To create FE software1. Confirm nature of physical problem: solid mechanics; fluid dynamics; electromagnetic; heat transfer; 1-D, 2-D, 3-D; Linear; non-linear; etc.2. Describe mathematically: governing equations; loading conditions.3. Derive element equations: convert governing equations into algebraic form; select trial functions; prepare integrals for numerical evaluation.4. Assembly and solve: assemble system of equations; application of loads; solution of equations.5. Compute:6. Process output: select type of data; generate related data; display meaningfully and attractively.(B) To use FE software1. Define a specific problem: geometry; physical properties; loads.2. Input data to program: geometry of domain, mesh generation; physical properties; loads-interior and boundary.3. Compute:4. Process output: select type of data; generate related data; display meaningfully and attractively.DISCRETE SYSTEMSSTATICSThe finite element involves the transformation of a continuous system (infinite degrees of freedom) into a discrete system (finite degrees of freedom). It is instructive therefore to examine the behaviour of simple discrete systems and associated variational methods as this provides real insight and understanding into the more complicated systems arising from the finite element method.Work and Strain energyFLuxConsider a metal bar of uniform cross section, A , fixed at one end (unrestrained laterally) and subjected to an axial force, F , at the other.Small deflection theory is assumed to apply unless otherwise stated.The work done, W , by the applied force F is .a ()∫′′=uau d u F WIt is worth mentioning at this early stage that it is not always possible to express work in this manner for various reasons associated with reversibility and irreversibility. (To be discussed later)The work done, W , by the internal forces, denoted strain energy , is se22200se ku 21u L EA 2121EAL d EAL d AL W ==ε=ε′ε′=ε′σ=∫∫εεwhere ε=u L and stiffness k EA L=.The principle of virtual workThe principle of virtual work states that the variation in strain energy is equal to the variation in the work done by applied forces , i.e.()u F u u d u F du d W u ku u ku 21du d ku 21W u0a 22se δ=δ⎟⎟⎠⎞⎜⎜⎝⎛′′=δ=δ=δ⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛δ=δ∫()0u F ku =δ−⇒Note that use has been made of the relationship δf dfduu =δ where f is an arbitrary functional of u . In general displacement u is a function of position (x say) and it is understood that ()x u δ means a change in ()u x with xfixed. Appreciate that varies with from zero to ()'u F 'u ()u F F = in the above integral.Bearing in mind that δ is an arbitrary variation; then this equation is satisfied if and only if F , which is as expected. Before going on to apply the principle of virtual work to a continuous system it is worth investigating discrete systems further. This is because the finite element formulation involves the transformation of a continuous system into a discrete one. u ku =Spring systemsConsider a single spring with stiffness independent of deflection. Then, 2F21u1F1u2k()()⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−=−=2121212se u u k k k k u u 21u u k 21W()()()⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−δδ=δ−δ−=δ21211212se u u k k k k u u u u u u k W()⎟⎟⎠⎞⎜⎜⎝⎛δδ=δ+δ=δ21212211a F F u u u F u F W , where ()111u F F = and ()222u F F =.Note here that use has been made of the relationship δ∂∂δ∂∂δf f u u f u u =+1122, where f is an arbitrary functional of and . Observe that in this case is a functional of 1u u 2W se u u u 2121=−, so()()(121212*********se se u u u u k u u ku 21du d u du dW W δ−δ−=−δ⎟⎠⎞⎜⎝⎛=δ=δ).The principle of virtual work provides,()()()()0F u u k u F u u k u 0W W 21221121a se =−−δ+−−−δ⇒=δ−δand since δ and δ are arbitrary we have. u 1u 2F ku ku 11=−2u 2 F ku k 21=−+represented in matrix form,u F K u u k k k k F F 2121=⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−=⎟⎟⎠⎞⎜⎜⎝⎛=where K is known as the stiffness matrix . Note that this matrix is singular (det K k k =−=220) andsymmetric (K K T=). The symmetry is a result of the fact that a unit deflection at node 1 results in a force at node 2 which is the same in magnitude at node 1 if node 2 is moved by the same amount.Could also have arrived at equation above via()⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−=⎟⎟⎠⎞⎜⎜⎝⎛⇒=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−−⎟⎟⎠⎞⎜⎜⎝⎛δδ=δ−δ2121211121a se u u k k k k F F 0u u k k k k F F u u W WBoundary conditionsWith the finite element method the application of displacement constraint boundary conditions is performed after the equations are assembled. It is an interest to examine the implications of applying and not applying the displacement boundary constraints prior to applying the principle of virtual work. Consider then the single spring element above but fixed at node 1, i.e. 0u 1=. Ignoring the constraint initially gives()212se u u k 21W −=, ()()1212se u u u u k W δ−δ−=δ and 2211a u F u F W δ+δ=δ.The principle of virtual work gives 2211ku ku ku F −=−= and 2212ku ku ku F =+−=, on applicationof the constraint. Note that is the force required at node 1 to prevent the node moving and is the reaction force.21ku F −=21ku F =−Applying the constraint straightaway gives 22se ku 21W =, 22se u ku W δ=δ and 22a u F W δ=δ. The principle of virtual work gives with no information about the reaction force at node 1.22ku F =Exam Standard Question:The spring-mass system depicted in the Figure consists of three massless springs, which are attached to fixed boundaries by means of pin-joints at nodes 1, 3 and 5. The springs are connected to a rigid bar by means of pin-joints at nodes 2 and 4. The rigid bar is free to rotate about pivot A. Nodes 2 and 4 are distances and below pivot A, respectively. Each spring has the same stiffness k. Node 2 is subjected to an external horizontal force F 2/l 4/l 2. All deflections can be assumed to be small.(i) Write expressions for the extension of each spring in terms of the displacement of node 2.(ii) In terms of the degrees of freedom at node 2, write expressions for the total strain energy W of the spring-mass system. In addition, specify the variation in work done se a W δ resulting from the application of the force.2F (iii) Use Use the principle of virtual work to find a relationship between the magnitude of and the horizontal components of displacement at node 2.2F (iv) Use the principle of virtual work to show that the net vertical force imposed by the springs on the rigid-bar at node 2 is zero.Solution:(i) Directional vectors for springs are: 2112e 21e 23e +=, 2132e 21e 23e +−= and 145e e =. Extensions for bottom springs are: 221212u 23u e =⋅=δ, 223232u 23u e −=⋅=δ.Note that 2u u 24=, so 2u245−=δ.(ii)()2222222245232212se ku 87u 212323k 21k 21W =⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛−+⎟⎟⎠⎞⎜⎜⎝⎛=δ+δ+δ=, 222u F W δ=δ(iii) 2222a 22se ku 47F u F W u ku 47W =⇒δ=δ=δ=δ(iv) Need additional displacement degree of freedom at node 2. Let 22122e v e u u += and note that2221212v 21u 23u e +=⋅=δ and 2223232v 21u 23u e +−=⋅=δ.()⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛+−+⎟⎟⎠⎞⎜⎜⎝⎛+=δ+δ=222222232212se v 21u 23v 21u 23k 21k 21W ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛δ+δ−⎟⎟⎠⎞⎜⎜⎝⎛+−+⎟⎟⎠⎞⎜⎜⎝⎛δ+δ⎟⎟⎠⎞⎜⎜⎝⎛+=δ22222222se v 21u 23v 21u 23v 21u 23v 21u 23k W Setting and gives0v 2=0u 2=δ2vert 222222se v F v 0v 21u 23v 21u 23k W δ=δ=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛δ⎟⎟⎠⎞⎜⎜⎝⎛−+⎟⎠⎞⎜⎝⎛δ⎟⎟⎠⎞⎜⎜⎝⎛=δ hence . 0F vert 2=Method of Minimum PotentialConsider the expression,()()F u u u TT 21212121c se K 21F F u u u u k k k k u u 21W W P −=⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡−−=−=where W F and can be considered as a work term with independent of . u F u c =+1122F i u iThe approach of minimising P is known as the method of minimum potential .Note that,()()u F 0F -u u =F u u u +u u K K K K 21W W P T T T T c se =⇒=δδ−δδ=δ−δ=δwhere use has been made of the fact that δδu u =u u T TK K as a result of K 's symmetry.It is useful at this stage to consider the minimisation of an arbitrary functional ()u P where()()3T T O H 21P P u u u u u δ+δδ+∇δ=δand the gradient ∇=P P u i i ∂∂, and the Hessian matrix coefficients H P u u ij i j=∂∂∂2.A stationary point requires that ∇=, i.e.P 0∂∂Pu i=0.Moreover, a minimum point requires that δδu u TH >0 for all δu ≠0 and matrices that possess this property are known as positive definite .Setting P W W K se c T=−=−12u u u F T provides ∇=−=P K u F 0 and H K =.It is a simple matter to check that with u 10= (to prevent rigid body movement) that K is positive definite and this is a property commonly associated with FE stiffness matrices.Exam Standard Question:The spring system depicted in the Figure consists of four massless unstretched springs, which are attached to fixed boundaries by means of pin-joints at nodes 1 to 4. The springs are connected to a slider at node 5. Theslider is constrained to move in a frictionless channel whose axis is to the horizontal. Each spring has the same stiffness k. The slider is subjected to an external force F 0453 whose direction is along the axis of the frictionless channel.(i)The deflection of node 5 can be represented by the vector 25155v u e e u +=, where and areunit orthogonal vectors which are shown in the Figure. Write the components of deflection and in terms of , where is the magnitude of , i.e. e 1e 25u 5v 5U 5U 5u 25U 5u =. Show that the extensions of eachspring, in terms of , are: 5U ()22/31U 515+=δ, ()22/31U 525−=δ, and2/U 54535−=δ−=δ.(ii) In terms of k and write expressions for the total strain energy W of the spring-mass system. Inaddition, specify the variation in work done 5U se a W δ resulting from the application of the force . 5F (iii) Use the principal of virtual work to find a relationship between the magnitude of and thedisplacement at node 5.5F 5U (iv) Use the principal of virtual work to determine an expression for the force imposed by the frictionless channel on the slider.(v)Form a potential energy function for the spring system. Assume here that nodes 1, 3 and 4 are fixed and node 5 is restricted to move in the channel. Use this function to determine the reaction force at node 2.Solution:(i) Directional vectors for springs and channel are: ()2115e e 321e +=, ()2125e e 321e +−=, 135e e −=, 45e e = and (21c 5e e 21e +=). Deflection c 555e U u =, so 2U v u 555==. Extensions springs are: ()3122U u e 551515+=⋅=δ, ()3122U u e 552525−=⋅=δ, 2Uu e 553535−=⋅=δand 2Uu e 554545=⋅=δ(ii)()()()252522245235225215se kU U 83131k 8121k 21W =⎟⎠⎞⎜⎝⎛+−++=δ+δ+δ+δ=, 55a U F W δ=δ(iii)5555a 55se kU 2F U F W U kU 2W =⇒δ=δ=δ=δ(iv) Need additional displacement degree of freedom at node 3. A unit vector perpendicular to the channel is(21p 5e e 21e +−=) and let p 55c 555e V e U u += and note that()()3122V3122U u e 5551515−++=⋅=δ and ()()3122V3122U u e 5552525++−=⋅=δ, 2V 2U u e 5553535+−=⋅=δ and 2V 2U u e 5554545−=⋅=δ()()()()()()()()⎟⎠⎞⎜⎝⎛−+++−+−++=δ+δ+δ+δ=255255255245235225215se V U 831V 31U 31V 31U k 8121k 21W ()()()()()555se V 0V 831313131kU 81W δ=δ−+−+−+=δ, where variation is onlyconsidered and is set to zero. Principle of virtual work .5V δ3V 0F V F V 0W p 55p 55se =⇒δ=δ=δ(v)()3122U u e 551515+=⋅=δ, ()()5552525V 3122Uu u e −−=−⋅=δ, where 2522e V u =. ()()()223333223232233245235225215V F U F U V 3122U 3122U k 21V F U F k 21P −−⎟⎟⎠⎞⎜⎜⎝⎛+⎥⎦⎤⎢⎣⎡−−+⎥⎦⎤⎢⎣⎡+=−−δ+δ+δ+δ=and ()0F V 3122U k V P 2232=−⎥⎦⎤⎢⎣⎡−−−=∂∂, which on setting 0V 2= gives ()⎥⎦⎤⎢⎣⎡−−=3122U k F 32.The reaction is .2F −System AssemblyConsider the following three-spring system 2F 21u 1F 1u 2kF 3F 4u 3u 4k 1k2334()()()234322322121se u u k 21u u k 21u u k 21W −+−+−=,()()()()()()343432323212121se u u u u k u u u u k u u u u k W δ−δ−+δ−δ−+δ−δ−=δ,44332211a u F u F u F u F W δ+δ+δ+δ=δ,and δδ implies that,W W se a −=0u F K u u u u k k 0k k k k 00k k k k 00k k F FF F 43213333222211114321=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−+−−+−−=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=where again it is apparent that K is symmetric but also it is banded, i.e. the non-zero coefficients are located around the principal diagonal. This is a property commonly associated with assembled FE stiffness matrices and depends on node connectivity. Note also that the summation of coefficients in individual rows or columns gives zero. The matrix is singular and 0K det =.Note that element stiffness matrices are: , and where on examination of K it is apparent how these are assembled to form K .⎥⎦⎤⎢⎣⎡−−1111k k k k ⎥⎦⎤⎢⎣⎡−−2222k k k k ⎥⎦⎤⎢⎣⎡−−3333k k k kIf a boundary constraint is imposed then row one is removed to give:0u 1=u F K u u u k k 0k k k k 0k k k F F F 432333322221432=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−+=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=. If however a boundary constraint (say) is imposed then row one is again removed but a somewhatdifferent answer is obtained: 1u 1=u F K u u u k k 0k k k k 0k k k F F k F 4323333222214312=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−+=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+=)Direct FormulationIt is possible to formulate the stiffness matrix directly by moving one node and keeping the others fixed and noting the reactions.The above system can be solved for u , once possible rigid body motion is prevented, by setting u (say) to give 10=⇒=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−+=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=u F K u u u k k 0k k k k 0k k k F F F 432333322221432⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−+−−+=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−4321333322221432F F F k k 0k k k k 0k k k u u uThe inverse stiffness matrix, K −1, is known as the flexibility matrix and, for this example at least, can be assembled directly by noting the system response to prescribed forces.In practice K −1is never calculated and the system K u F = is solved using a modern numerical linear system solver.It is a simple matter to confirm thatu u K 21u u u u k k 0k k k k 00k k k k 00k k u u u u 21W T 4321333322221111T4321se =⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−+−−+−−⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛= with F u T4321T4321a F F F F u u u u W δ=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛δδδδ=δThus,()u F F u u K 0K W W Ta se =⇒=−δ=δ−δExample:k1F 2u23k2u3F321With use a direct method to find the assembled stiffness and flexibility matrices.0u 1=Solution:The equations of interest are of the form: 3232222u k u k F += and 3332323u k u k F +=.Consider and equilibrium at nodes 2 and 3. At node 2, 0u 3=()2212u k k F += and at node 3,.223u k F −=Consider and equilibrium at nodes 2 and 3. At node 2, 0u 2=322u k F −= and at node 3, . 323u k F =Thus: , , 2122k k k +=223k k −=232k k −= and 233k k =.For flexibility the equations of interest are of the form: 3232222F c F c u += and . 3332323F c F c u +=Consider and equilibrium at nodes 2 and 3. At node 2, 0F 3=122k F u = and at node 3,1223k F u u ==.Consider and equilibrium at nodes 2 and 3. At node 2, 0F 2=122k F u = and at node 3,()2133k 1k 1F u +=.Thus: 122k 1c =, 123k 1c =, 132k 1c = and 2133k 1k 1c +=.Can check that ⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡−−+1001k 1k 1k 1k 1k 1k k k k k 2111122221 as required,It should be noted that the direct determination requires boundary constraints to be applied to ensure that the flexibility matrix exists, which requires the stiffness to be non-singular. However, the stiffness matrix always exists, so boundary conditions need not be applied prior to constructing the stiffness matrix with the direct approach.Large deformation theory for spring elementsThus far small deflection theory has been applied where the strains are measured using the Cauchy strainxu11∂∂=ε. A conjugate stress can be obtained by differentiating with respect the expression for strain energy density (energy per unit volume) 11ε211E 21ε=ω, i.e. 111111E ε=ε∂ω∂=σ, where E is Young’s Modulusand is the Cauchy stress (sometimes referred to as the Euler stress). 11σIn the case of large deformation theory we will restrict our attention to hyperelastic materials which are materials that possess an expression for strain energy density Ω (say) that is analytical in strain.The strain used in large deformation theory is Green’s strain (see Appendix II) which for a uniformly loadeduniaxial bar is 211x u 21x u E ⎟⎠⎞⎜⎝⎛∂∂+∂∂=.An expression for strain energy density (energy per unit volume) 211EE 21=Ω and the derived stress is 111111EE E S =∂Ω∂=, where E is Young’s Modulus and is known as the 211S nd Piola-Kirchoff stress . 2F21u1F1u2kBar subject to longitudinal deformationConsider a bar of length L and cross sectional area A represented by a spring element and subject to nodal forces and . 1F 2FThe strain energy is∫∫∫∫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂+∂∂==Ω=Ω=212121x x 22x x 211x x V se dx x u 21x u EA 21dx E EA 21dx A dV WConsider further a linear displacement field of the form ()21u L x u L x L x u ⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−= and note thatL u u xu 12−=∂∂. ()()221212x x 221212se u u L 21u u L EA 21dx L u u 21L u u EA 21W 21⎥⎦⎤⎢⎣⎡−+−=⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−+−=∫ ()()()⎥⎦⎤⎢⎣⎡−+−+−=4122312212se u u L 41u u L 1u u k 21W()()()(12312221212se u u u u L 21u u L 23u u k W δ−δ⎥⎦⎤⎢⎣⎡−+−+−=δ) and 2211a u F u F W δ+δ=δ.The principle of virtual work gives()()()⎥⎦⎤⎢⎣⎡−+−+−−=3122212121u u L 21u u L 23u u k F and()()(⎥⎦⎤⎢⎣⎡−+−+−=3122212122u u L 21u u L 23u u k F ), represented in matrix form as()()()()()⎟⎟⎠⎞⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−+−−−−−−−+−+⎥⎦⎤⎢⎣⎡−−=⎟⎟⎠⎞⎜⎜⎝⎛21121212121221u u L 3u u 1L 3u u 1L 3u u 1L 3u u 1L 2u u k 3k k k k F Fwhich is of the form[]u F G L K K += where is called the geometrical stiffness matrix and is the usual linear stiffnessmatrix. G K L KA common approximation used, depending on the magnitude of L /u u 12−, is⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡−−+⎥⎦⎤⎢⎣⎡−−=⎟⎟⎠⎞⎜⎜⎝⎛2121u u 1111L 2P 3k k k k F F where ()12u u k P −=.The fact that is non-linear (even in its approximate form) means that iterative solution procedures are required to be employed to determine the unknown displacements. G KNote that the approximate form is arrived at using the following strain energy expression()()⎥⎦⎤⎢⎣⎡−+−=312212se u u L 1u u k 21WExample:The strain energies for the springs in the above system (fixed at node 1) are k 1 F 2u 23k 2u3F321⎥⎦⎤⎢⎣⎡+=1322211seL u u k 21W and ()()⎥⎦⎤⎢⎣⎡−+−=323222322se u u L 1u u k 21WUse the principle of virtual work to obtain the assembled linear and geometrical stiffness matrices.()()()3322a 2322322322122212se1sese u F u F W u u u u L 23u u k u L 2u 3u k W W W δ+δ=δ=δ−δ⎥⎦⎤⎢⎣⎡−+−+δ⎥⎦⎤⎢⎣⎡+=δ+δ=δThus ()(⎥⎦⎤⎢⎣⎡−+−−⎥⎦⎤⎢⎣⎡+=2232232122212u u L 23u u k L 2u 3u k F ) and ()()⎥⎦⎤⎢⎣⎡−+−=22322323u u L 23u u k F⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡αα−α−α+α+⎥⎦⎤⎢⎣⎡−−+=⎟⎟⎠⎞⎜⎜⎝⎛32222212222132u u k k k k k F F where 1211L 2u k 3=α and ()23222u u L 2k 3−=α.Note that the element stiffness matrices are[][]111k K α+= and ⎥⎦⎤⎢⎣⎡αα−α−α+⎥⎦⎤⎢⎣⎡−−=222222222k k k k Kand it is evident how these should be assembled to form the assembled linear and geometrical stiffness matrices.2v21u 1v1u 2kxBar subject to longitudinal and lateral deflectionConsider a bar of length L and cross sectional area A represented by a spring element and subject to longitudinal and lateral displacements u and v, respectively.The normal strain is 2211x v 21x u 21x u E ⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+∂∂= and the associated strain energy∫∫∫∫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+∂∂==Ω=Ω=212121x x 22x x 211x x V se dx x v 21x u 21x u EA 21dx E EA 21dx A dV W ∫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂≈21x x 232se dx x v x u x u x u EA 21WConsider further a linear displacement field of the form ()21u L x u L x L x u ⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−= and()21v L x v L x L x v ⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−=, and note thatL u u x u 12−=∂∂ and L v v x v 12−=∂∂. ()()()()⎥⎦⎤⎢⎣⎡−−+−+−=L v v u u L u u u u L EA 21W 21212312212se()()()()()()()1212121221221212se v v L v v u u k u u L 2v v L 2u u 3u u k W δ−δ⎦⎤⎢⎣⎡−−+δ−δ⎥⎦⎤⎢⎣⎡−+−+−=δ2v 22h 21v 11h 1a v F u F v F u F W δ+δ+δ+δ=δ and the principle of virtual work gives()()()⎥⎦⎤⎢⎣⎡−+−+−−=L 2v v L 2u u 3u u k F 21221212h1and ()()⎥⎦⎤⎢⎣⎡−−−=L v v u u k F 1212v1 ()()()⎥⎦⎤⎢⎣⎡−+−+−=L 2v v L 2u u 3u u k F 21221212h2and ()()⎥⎦⎤⎢⎣⎡−−=L v v u u k F 1212v2()()⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−−−+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛22111212v 2h 2v 1h 1v u v u 101005.105.1101005.105.1Lu u k 1010000010100000L2v v k 0000010100000101k F F F FExam Standard Question:The spring system depicted in the Figure consists of two massless springs of equal length , which are attached to fixed boundaries by means of pin-joints at nodes 1 and 2. The springs are connected to a slider atnode 3. The slider is constrained to move in a frictionless channel whose axis is 45 to the horizontal. Each spring has the same stiffness . The slider is subjected to an external force F 1L =0L /EA k =3 whose direction is along the axis of the frictionless channel.FigureAssume the springs have strain density ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂=Ω232x v x u x u x u E 21.(i) Write expressions for the longitudinal and lateral displacements for each spring at node 3 in terms of thedisplacement along the channel at node 3.(ii) In terms of displacement along the channel at node 3, write expressions for the total strain energy W of thespring-mass system. In addition, specify the variation in work done se a W δ resulting from the application of the force .3F (iii) Use the principle of virtual work to find a relationship between the magnitude of and the displacementalong the channel at node 3. 3FSolution:(i) Directional vectors for springs and channel are: ()2113e e 321e +=and ()2123e e 321e +−= and (21c 3e e 21e +=). Perpendicular vectors are: ()2113e 3e 21e +−=⊥and ()2123e 3e 21e +=⊥Deflection c 333e U u =, so 2U v u 333==.Longitudinal displacement: ()3122U u e 331313+=⋅=δ, ()3122U u e 332323−=⋅=δ.Lateral displacement: ()3122U u e 331313+−=⋅=δ⊥⊥, ()3122U u e 332323+=⋅=δ⊥⊥(ii) The strain energy density for element 1 is ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛δ⎟⎠⎞⎜⎝⎛δ+⎟⎠⎞⎜⎝⎛δ+⎟⎠⎞⎜⎝⎛δ=Ω⊥21313313213L L L L E 21 The strain energy density for element 2 is ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛δ⎟⎠⎞⎜⎝⎛δ+⎟⎠⎞⎜⎝⎛δ+⎟⎠⎞⎜⎝⎛δ=Ω⊥22323323223L L L L E 21 The total strain energy with substitution of 1L = gives()()()()[]()()()()[][]3322312232332322321313313213se U Uk 21k 21k 21W α+α=δδ+δ+δ+δδ+δ+δ=⊥⊥where and are constants determined on collecting up terms on substitution of and .1α2α231313,,δδδ⊥⊥δ2333a U F W δ=δ.(iii) The principle of virtual work gives⎥⎦⎤⎢⎣⎡α+α=⇒δ=δ=δ⎥⎦⎤⎢⎣⎡α+α=δ32133332323231se U 23kU F U F W U U 23U k WPin-jointed structuresThe example above is a pin-jointed structure. A reasonable good approximation reported in the literature for strain energy density, commonly used with pin-jointed structures, is⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂=Ω22x v x u x u E 21This arises from strain-energy approximation 211x v 21x u E ⎟⎠⎞⎜⎝⎛∂∂+∂∂=. Can be used when 22x v x u ⎟⎠⎞⎜⎝⎛∂∂<<⎟⎠⎞⎜⎝⎛∂∂.。
W25Q64中文资料精编版
W25Q64BV出版日期：2010年7月8日- 1 - 版本E64M位与串行闪存双路和四路SPIW25Q64BV- 2 -目录1，一般DESCRIPTION (5)2。
FEATURES (5)3引脚配置SOIC208-MIL.......................................... .. (6)4，焊垫配置WSON8X6-MM.......................................... . (6)5，焊垫配置PDIP300-MIL.......................................... . (7)6引脚说明SOIC208密耳，PDIP300密耳和WSON8X6-MM................................ 7......7引脚配置SOIC300mil的.......................................... .. (8)8引脚SOIC封装说明300-MIL (8)8.1包装Types (9)8.2片选(/CS) (9)8.3串行数据输入，输出和IO（DI，DO和IO0，IO1，IO2，IO3）............................. 9.......8.4写保护(/WP) (9)8.5控股(/HOLD) (9)8.6串行时钟(CLK) (9)9座DIAGRAM (10)10功能DESCRIPTION (11)10.1 SPI OPERATIONS (11)10.1.1标准SPI Instructions (11)10.1.2双SPI Instructions (11)10.1.3四路SPI Instructions (11)10.1.4保持功能 (11)10.2写保护 (12)10.2.1写保护Features (12)11，控制和状态寄存器............................................ .. (13)11.1状态REGISTER (13)11.1.1 BUSY (13)11.1.2写使能锁存(WEL) (13)11.1.3块保护位（BP2，BP1，BP0）..................................... .. (13)11.1.4顶/底块保护（TB）....................................... .................................................. ..1311.1.5部门/块保护(SEC) (13)11.1.6状态寄存器保护（SRP，SRP0）....................................... . (14)11.1.7四路启用(QE) (14)11.1.8状态寄存器内存保护........................................... .. (16)11.2 INSTRUCTIONS (17)11.2.1制造商和设备标识........................................... .. (17)11.2.2指令集表1 (18)W25Q64BV11.2.3指令表2（阅读说明书）....................................... (19)出版日期：2010年7月8日- 3 - 修订版E11.2.4写使能(06h) (20)11.2.5写禁止(04h) (20)11.2.6读状态寄存器1（05H）和读状态寄存器2（35H）.............................. (21)11.2.7写状态寄存器（01H）......................................... .................................................. .. (22)11.2.8读取数据(03h) (23)11.2.9快速阅读(0Bh) (24)11.2.10快速读双输出（3BH）........................................ .................................................. 0.25 11.2.11快速读四路输出（6BH）........................................ .. (26)11.2.12快速读双I / O (BBh) (27)11.2.13快速读取四I/ O (EBh) (29)11.2.14八进制字读取四I/ O（E3H）..................................... (31)11.2.15页编程(02h) (33)11.2.16四路输入页编程（32H）........................................ . (34)11.2.17扇区擦除（20H） (35)11.2.1832KB的块擦除（52H） (36)11.2.1964KB的块擦除(D8h) (37)20年2月11日芯片擦除（C7H/ 60h) (38)21年2月11日擦除挂起(75h) (39)22年2月11日擦除恢复(7Ah) (40)23年11月2日掉电(B9h) (41)24年2月11日高性能模式（A3H）......................................... (42)25年2月11日发布掉电或高性能模式/设备ID（ABH） (42)26年2月11日读制造商/设备ID（90H）....................................... . (44)27年2月11日阅读唯一的ID号（4BH）........................................ . (45)28年2月11日读JEDEC的ID (9Fh) (46)29年2月11日连续读取模式复位（FFH或FFFFH）...................................... .. (47)12，电气特性.............................................. (48)12.1绝对最大Ratings (48)12.2操作范围 (48)12.3上电时序和写抑制阈值......................................... (49)12.4直流电气Characteristics (50)12.5 AC测量条件.............................................. .. (51)12.6 AC电气Characteristics (52)12.7 AC电气特性（续）......................................... . (53)12.8串行输出Timing (54)12.9输入Timing (54)12.10持有Timing (54)13包装SPECIFICATION (55)W25Q64BV13.18引脚SOIC208密耳（包装代号SS）..................................... .. (55)- 4 -13.28引脚PDIP300密耳（封装代码DA）..................................... (56)13.38触点WSON8x6毫米（封装代码ZE）....................................... (57)13.416引脚SOIC300密耳（封装代码SF）..................................... . (58)14订货INFORMA TION (59)14.1有效的部件号和顶端标记.......................................... (60)15版本HISTORY (61)W25Q64BV出版日期：2010年7月8日- 5 - 修订版E1概述该W25Q64BV（64M位）串行Flash存储器提供了有限的系统存储解决方案空间，引脚和电源。
实验三 过滤实验
一、实验原理
恒压过滤过程中，过滤速率由过滤压差及过滤阻力决 定，过滤阻力由二部分组成，一为滤布，一为滤渣。 在恒压过滤情况下，滤液量qe与过滤时间e的关系为： (V+Ve)2= K· 2· A (e）
若令q=
V
A
A
单位：m3/m2；
qe= V e
单位：m3/m2。
则上式改写为： （q+qe)2=K（e） 将K，e ，qe统称为过滤常数
16
三、操作要点：
1.验证流体静力学原理 打开上水阀8，关闭 调节阀5(此时流体不流动，没有循环，满足 p 静力学方程， gz 常数 ，在水平管路中，z=0， 同一流体，所以压强相等)，因此，这时各 测压管液面高度相同，且与活动测压头位置 无关。这说明当流体静止时，其内部各点的 压强只与深度和流体的密度有关。 请思考：此时测压管中液柱的高度决定于什 么？（流体的垂直位置，与基准水平面的距 离）
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五、注意事项
1.滤框装反，滤框上的开孔没对准进料口。 2.滤布未开口，或滤布安装时，有线头外漏， 致使滤框安装的不紧。 3.压力太小或仅为常压。 4.滤框出料阀或进料阀未打开。
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1 50 0
q
1 0 00
50 0
截距 2qe/K
0 0 0.0 25 q 0.0 5
３．接上步，继续开大调节阀5，观察测压 管液位高度的变化，此时测压管中液位的 高度为H，，。 请思考：测压孔正对水流方向，开大调节 阀5，流速增大，动压头增大，为什么测压 管的液位反而下降？(能量损失增加)
2012-6-18
(完整word版)图书目录分类表
图书目录分类表A1 马克思、恩格斯著作A2 列宁著作A3 斯大林著作A4 毛泽东著作A49 邓小平著作A5 马克思、恩格斯、列宁、斯大林、毛泽东、邓小平著作汇编A7 马克思、恩格斯、列宁、斯大林、毛泽东、邓小平生平和传记A8 马克思、恩格斯、列宁、斯大林、毛泽东、邓小平理论的学习和研究B0 哲学理论B1 世界哲学B2 中国哲学B3 亚洲哲学B4 非洲哲学B5 欧洲哲学B6 大洋州哲学B7 美洲哲学B80 思维科学B81 逻辑学（论理学）B82 伦理学（道德哲学)B83 美学B84 心理学B9 宗教C0 社会科学理论与方法论C1 社会科学现状及发展C2 社会科学机构、团体、会议C3 社会科学研究方法C4 社会科学教育与普及C5 社会科学丛书、文集、连续性出版物C6 社会科学参考工具书C[7］社会科学文献检索工具书C8 统计学C91 社会学C92 人口学C93 管理学C［94］系统科学C95 民族学C96 人才学C97 劳动科学D0 政治理论D1 国际共产主义运动D2 中国共产党D33/37各国共产党D4 工人、农民、青年、妇女运动与组织D5 世界政治D6 中国政治D73/77各国政治(完整word版)图书目录分类表D8 外交、国际关系D9 法律E0 军事理论E1 世界军事E2 中国军事E3/7 各国军事E8 战略学、战役学、战术学E9 军事技术E99 军事地形学、军事地理学F0 经济学F1 世界各国经济概况、经济史、经济地理F2 经济计划与管理F3 农业经济F4 工业经济F49 信息产业经济(总论）F5 交通运输经济F59 旅游经济F6 邮电经济F7 贸易经济F8 财政、金融G0 文化理论G1 世界各国文化与文化事业G2 信息与知识传播G3 科学、科学研究G4 教育G8 体育H0 语言学H1 汉语H2 中国少数民族语言H3 常用外国语H4 汉藏语系H5 阿尔泰语系（突厥—蒙古—通古斯语系) H61 南亚语系（澳斯特罗–亚细亚语系)H62 南印语系（达罗毗荼语系、德拉维达语系）H63 南岛语系（马来亚–玻里尼西亚语系)H64 东北亚诸语言H65 高加索语系（伊比利亚—高加索语系）H66 乌拉尔语系（芬兰–乌戈尔语系）H67 闪–含语系（阿非罗–亚西亚语系）H7 印欧语系H81 非洲诸语言H83 美洲诸语言H84 大洋州诸语言H9 国际辅助语I0 文学理论I1 世界文学I2 中国文学I3/7 各国文学J0 艺术理论J1 世界各国艺术概况J2 绘画J29 书法、篆刻J3 雕塑J4 摄影艺术J5 工艺美术J[59] 建筑艺术J6 音乐J7 舞蹈J8 戏剧艺术J9 电影、电视艺术K0 史学理论K1 世界史K2 中国史K3 亚洲史K4 非洲史K5 欧洲史K6 大洋州史K7 美洲史K81 传记K85 文物考古K89 风俗习惯K9 地理N0 自然科学理论与方法论N1 自然科学现状及发展N2 自然科学机构、团体、会议N3 自然科学研究方法N4 自然科学教育与普及N5 自然科学丛书、文集、连续性出版物N6 自然科学参考工具书N［7] 自然科学文献检索工具N8 自然科学调查、考察N91 自然科学研究、自然历史N93 非线性科学N94 系统科学N［99] 情报学、情报工作O1 数学O3 力学O4 物理学O6 化学O7 晶体学P1 天文学P2 测绘学P3 地球物理学P4 大气科学（气象学)P5 地质学P7 海洋学P9 自然地理学Q1 普通生物学Q2 细胞生物学Q3 遗传学Q4 生理学Q5 生物化学Q6 生物物理学Q7 分子生物学Q81 生物工程学（生物技术）Q［89] 环境生物学Q91 古生物学Q93 微生物学Q94 植物学Q95 动物学Q96 昆虫学Q98 人类学R1 预防医学、卫生学R2 中国医学R3 基础医学R4 临床医学R5 内科学R6 外科学R71 妇产科学R72 儿科学R73 肿瘤学R74 神经病学与精神病学R75 皮肤病学与性病学R76 耳鼻咽喉科学R77 眼科学R78 口腔科学R79 外国民族医学R8 特种医学(完整word版)图书目录分类表R9 药学S1 农业基础科学S2 农业工程S3 农学（农艺学)S4 植物保护S5 农作物S6 园艺S7 林业S8 畜牧、动物医学、狩猎、蚕、蜂S9 水产、渔业TB 一般工业技术TD 矿产工程TE 石油、天然气工业TF 冶金工业TG 金属学与金属工艺TH 机械、仪表工业TJ 武器工业TK 能源与动力工程TL 原子能技术TM 电工技术TN 无线电电子学、电信技术TP 自动化技术、计算机技术TQ 化学工业TS 轻工业、手工业TU 建筑科学TV 水利工程(完整word版)图书目录分类表U1 综合运输U2 铁路运输U4 公路运输U6 水路运输U［8］航空运输V1 航空、航天技术的研究与探索V2 航空V4 航天（宇宙航行）V［7］航空、航天医学X1 环境科学基础理论X2 社会与环境X3 环境保护管理X4 灾害及其防治X5 环境污染及其防治X7 废物处理与综合利用X8 环境质量评比与环境监测X9 安全科学Z1 丛书Z2 百科全书、类书Z3 辞典Z4 论文集、全集、选集、杂著Z5 年鉴、年刊Z6 期刊、连续性出版物Z8 图书目录、文摘、索引。
Maxwell_v16_2D_WS02_BasicEddyCurrentAnalysis
Note: On symmetry axis, “Balloon Boundary” assignment is automatically skipped, This can also be achieved by selecting the edges of region which are not on symmetry axis.
•
2D Geometry: Iron Disk above a Spiral Coil
– A sinusoidal 500 Hz current will be assigned to an eight turn spiral coil underneath of a cast iron disk. The coil induces eddy currents and losses in plate. The 2D model will be setup as shown below using the 2D RZ axisymmetric solver.
iManager U2000 V200R016C50SPC200 LCT 版本说明书(传送域) 01
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1.1 产品版本信息 ............................................................................................................................................................. 1 1.2 与操作系统和数据库配套说明.................................................................................................................................. 1 1.3 与主机软件的配套说明 ............................................................................................................................................. 2 1.3.1 配套 MSTP 系列产品 .............................................................................................................................................. 2 1.3.2 配套 WDM 系列产品 ............................................................................................................................................ 11 1.3.3 配套 PTN 系列产品............................................................................................................................................... 21
浩瀚电器重型电锅替换零部件手册说明书
VTMB22VB2S VB21VTMB2VB SERIES1 THRU45 - 678 - 9101112 - 13 - 14ELECTRICAL COMPONENTSVB SERIESELECTRICAL COMPONENTSILLUS.PART OF PART AMT.1417934-G1Block Assy. - Porcelain (208-240 V. Machine)(Incls.items 2, 3 & 4)..........................................AR 2SC-117-3Mach. Screw 10-24 x 11⁄4 Rd. Hd................................................................................................AR 3WS-23-20Washer.......................................................................................................................................AR 4NS-44-9Nut Assy 10-24 Hex “KEPS”......................................................................................................AR 5411497-B1Contactor (208-240 V. Machines)..............................................................................................AR 6417856-1Ground Lug (VB21, VB221) (1)7406708-G1Door - Breaker Panel (1)8412578-1Closure - Breaker (208-240 V.)(Shown) (1)9412578-2Closure - Breaker (480 V.) (1)10411501-14Circuit Breaker (50 Amp.)(208-240 V. Machines)......................................................................AR 11412577-1Box - Breaker. (1)12410472-9Block - Terminal (208/240 V.) (1)13410472-11Terminal Block (208-240 V.) (1)14410472-8Terminal Block (480 V.) (1)VB SERIESVB SERIES 5VB SERIESDOOR MECHANISM — OVENPL-40408-11011 THRU141516171821 - 2219 - 20VB SERIESDOOR MECHANISM - OVENILLUS.PART OF PART AMT.1410110-1Anchor - Spring (1)2417865-1Turnbuckle (36") (1)3405618-1Hook - Oven Door (1)4SC-90-1Bolt - Shoulder 1⁄2-13 x 2 Hex Hd (1)5408895-G1Crank Support Assy (1)6409240-2Spacer - 3⁄4 Pipe x 7⁄16 Lg (1)7404629-1Bearing - Hinge Pin (1)8408896-1Support - Crank (2)9410111-1Crank - Bell (1)10406363-1Spring - Door (Long) (1)11408895-G1Crank Support Assy (1)12409240-1Spacer - 3⁄4 x 5⁄16 Lg (1)13404629-1Bearing - Hinge Pin (1)14408896-1Support - Crank (1)15410090-G1Radius Bar Assy (1)16410089-1Arm - Door Control (1)17SC-90-1Bolt - Shoulder 1⁄2-13 x 2 Hex Hd (1)18PC-3-36Cotter Pin 3⁄32 x 1 Lg (1)19417860-1Bolt - Shoulder 3⁄8-16 x 3⁄4 Hex Hd (1)20417862-1Pin - Door Hinge (1)21410114-1Cover - Bottom Terminal (1)22410113-1Cover - Terminals (Top Oven Heat Elements) (1)412887-G1Door Frame Assy (1)VB SERIESBROILER GRID & DRIP SHIELDNAME OF PARTGrid Assy (1)Handle (1)Retainer - Drip Shield (1)Mach. Screw 10-24 x 1 Truss Hd (2)Spacer (2)Lock Nut 10-24 Light Flexloc (2)Post - Door (R.H.) (1)Post - Door (L.H.) (1)Drip Shield (1)Ball Bearing (Grid Carriage).........................................................................................................VB SERIESPL-40410-1915 - 16 CARRIAGE & LIFT UNIT (BROILER)PART OF PART406966-G1Carriage & Lift Mechanism (Incls.items 2 thru 7)......................................................................... 417866Spring - Carriage.. (1)405655-1Ball Bearing (Grid Carriage)......................................................................................................... 407488-1Pin - Bearing (2)417869-1Spacer (2)WS-3-3Washer (4)PC-3-36Cotter Pin 3⁄32 x 1 Lg (2)407619-1Ratchet Stop (1)406046-1Sleeve - Handle (1)414819-1Red Plastic Ball Handle................................................................................................................ 406056-2Adapter - Ball Handle. (1)407636-G1Push Rod Assy (1)417810-2Spring - Compression (1)NS-13-22Full Nut 3⁄8-16 Hex (2)406990-G1Grease Collector (1)407006-1Handle - Grease Collector (1)VB SERIESPL-40411-1ELECTRIC HEAT ELEMENTS(BROILER)PARTNAME OF PART407012-4Electric Heat Element 208 V........................................................................................................407012-5Electric Heat Element 240 V........................................................................................................407012-6Electric Heat Element 480 V........................................................................................................410465-1Retainer - Heating Element (3)407930-G1Heat Shield Assy (1)VB SERIES9 THRU141715 - 16PL-40412-1ELECTRIC HEAT ELEMENTS (OVEN)NAME OF PARTFrame Assy. - Top Elements (1)Frame Assy. - Bottom Elements................................................................................................... Element Assy. - Outer Top (208 V.)............................................................................................. Element Assy. - Outer Top (240 V.)............................................................................................. Element Assy. - Outer Top (480 V.)............................................................................................. Element - Outer Bottom (208 V.).. (1)Element - Outer Bottom (240 V.) (1)Element - Outer Bottom (480 V.) (1)VB SERIES2PL-40413-1MISCELLANEOUSNAME OF PARTFlue Riser (1)Strap - Back Attaching................................................................................................................Leg (SST) (4)Caster W/O Brake (2)Caster W/Brake (2)。
Nonlinear in-plane free oscillations of the suspended cable investigated by the HA method
(Received June 26, 2012, Revised March 12, 2014, Accepted March 20, 2014)
An analytical solution for the nonlinear in-plane free oscillations of the suspended cable which contains the quadratic and cubic nonlinearities is investigated via the homotopy analysis method (HAM). Different from the existing analytical technique, the HAM is indeed independent of the small parameter assumption in the nonlinear vibration equation. The nonlinear equation is established by using the extended Hamilton's principle, which takes into account the effects of the geometric nonlinearity and quasi-static stretching. A non-zero equilibrium position term is introduced due to the quadratic nonlinearity in order to guarantee the rule of the solution expression. Therefore, the mth-order analytic solutions of the corresponding equation are explicitly obtained via the HAM. Numerical results show that the approximate solutions obtained by using the HAM are in good agreement with the numerical integrations (i.e., RungeKutta method). Moreover, the HAM provides a simple way to adjust and control the convergent regions of the series solutions by means of an auxiliary parameter. Finally, the effects of initial conditions on the linear and nonlinear frequency ratio are investigated.
烟雾报警器用户手册说明书
User ManualCOMBINED STROBE & VIBRATING PADW2-SVP-630LINTRODUCTION IMPORTANT: If you are installing this product for use by others, you must leave this manual (or a copy of it) for the end user.Wi-Safe 2 technology enables wireless interlinking between Wi-Safe 2 products.The Strobe and Vibrating Pad, when interlinked with Wi-Safe 2 smoke, heat and/or carbon monoxide (CO) alarms will provide warning of danger from smoke and/or carbon monoxide.In addition to the audible sound from your smoke, heat or CO alarm, the flashing Strobe and Vibrating Pad will be triggered to alert those who may not be able to hear the audible alarm. The flashing strobe provides a visual warning for waking hours, while the vibrating pad is designed to be placed under a pillow or a mattress and is suitable for waking an individual to alert them when an alarm sounds. If placed under a mattress, ensure the thickness of the mattress does not cushion the pad to the extent it cannot be felt adequately.KEY FEATURES• Wi-Safe 2 products can be interlinked to createa network, meaning if smoke or CO triggers any alarm in the network, all other networked alarms and ancillary devices activate.• When used with Wi-Safe 2 enabled smoke, heat and/or CO alarms, the kit provides additional protection when sound alone may not be enough.• Provides a simple two button “learn in” process to wirelessly interlink with Wi-Safe 2 smoke, heat or CO alarms.• The Strobe provides a remote low level testing facility for testing interlinked smoke and CO4alarms, avoiding the need to reach up to ceiling mounted alarms to test.• Mains powered with rechargable battery back-up (must be replaced after 5 years).• The Strobe will detect and demonstrate a fault or low battery warning with any smoke alarm or CO alarm within the interlinked network.• The Strobe is suitable for wall mounting with 2 screw fixings or free standing.• Designed for use with smoke, heat and CO alarms from the Wi-Safe 2 range.Wi-Safe 2 Strobe and Vibrating Pad pack contains:1. CP-LED - Strobe light with integrated control unit (referred to as “strobe”)2. CP-VPAD - Vibrating Pad to be placed under pillow or chair (referred to as ”pad”)3. AD-DC12V05A - AC mains adaptor, 12V, 0.5A rated output4. B-450L - Replaceable, rechargeable 5 yearlife battery5. Wall fixing screwsNOTE: The Strobe and Vibrating Pad is only compatible with Wi-Safe 2 alarms from FireAngel Safety Technology Limited.I t cannot be wirelessly interlinked with other manufacturers’ products. I t is not compatible with products from FireAngel Safety Technology Limited original Wi-Safe range.I NSTALLI NG THE STROBEAND VIBRATING PADPREPARATION: Please ensure you have read and understood this manual before installing your Strobe and Vibrating Pad.It is recommended that for optimum protection from smoke and/or carbon monoxide, one Wi-Safe 2 smoke alarm is fitted in each living room of your home, including bedrooms, and oneWi-Safe 2 CO alarm is fitted in every room containinga fossil-fuel appliance, connected to your Strobe and Vibrating Pad.510TESTI NG THE WI -SAFE 2 STROBE AND VIBRATINGPAD WI TH SMOKE /CO ALARMSTesting the network from the smoke alarm or CO alarmBriefly press the test button on the smoke alarm or CO alarm and release (fig5). The alarm will give an audible sound (consisting of 2 cycles of 3 loud beeps on smoke alarms, 2 cycles of 4 rapid beeps on CO alarms), then stop automatically. The red (alarm) LED on the alarm will flash rapidly during the audible signal. Any other wirelessly interlinked alarms which are also part of the network will also give an audible alarm, and then stop automatically.3. T he red LEDs will then perform a sequence of two long and three short flashes. T he amber fault LED will illuminate continuously to confirm that the Strobe is no longer interlinked to the wireless network.4. Press the test button on the Strobe. It should not cause other Wi-Safe 2 alarms to sound. If the Strobe is still interlinked, repeat the above procedure.IMPORTANT : If you wish to remove the Strobe and Vibrating Pad from the wireless network, it is important to “unlearn” the unit from the network. Failure to do so means that the unit continues to try and communicate with the removed unit, and will result in a system fault.12Position the Vibrating Pad:• Under your pillow.• Under a cushion on a chair where you are likely to fall asleep.Test and check that the Vibrating Pad can be felt in all circumstances where you may fall asleep. Ensure it is placed securely and cannot fall out – ideally within the pillowcase or cushion cover.Position the Strobe:• Close to a power socket where it can be easily plugged in and the cable does not create a tripping hazard.• Where you can see the Strobe flashing from anywhere in the room.• Where you can see the LEDs on the front of the unit.NOTE: The Strobe can be fitted to a wall or it can be placed on a table.POSITIONINGWhere should the Strobe and Vibrating Pad be installed?If you have one Strobe and Vibrating Pad, it should be installed in the main room where you sleep. You may wirelessly interlink additional Strobes and Vibrating Pads to the network and install them in any room where you may need to be alerted in the event of an alarm. It is important that if you install additional Strobes and Vibrating Pads you test each of these upon installation to check they are “learned in” to the rest of the network.Wireless Range: The wireless range of Wi-Safe 2 products is over 200 metres in clear air/clear line of sight.However it is recommended not to exceed 35m as the maximum distance between any interlinked Wi-Safe 2 smoke alarm or carbon monoxide alarm and the Strobe and Vibrating Pad. This is because the range can be reduced by walls, and other obstructions in the building.all other alarms in the network, for a period of 2minutes, except the alarm that has sensed smoke or carbon monoxide. This will enable theinstigating alarm to be located. UNPLUGGI NG FROM THEMAI NS / POWER CUTThe Strobe and Vibrating Pad is designed to bemains powered. The internal rechargeable batterywill still provide power if the mains is disconnected.A fully charged battery will allow the Strobe and Padto continue to operate for three days. Only use thebattery supplied in the pack or a genuine FireAngelreplacement. Other rechargeable batteries maydamage the equipment or create a safety hazard. WARN I NG: When the mains power is disconnected, the green power LED will no longer illuminate and the amber (Fault) LED will double flash every 5 seconds. Plug the Strobe back into the mains as soon as possible to recharge the battery and maintain operation.f the amber LED is flashing once every 5 seconds when the green power LED is off then URGENTL Y restore power to the unit as battery status is very low, and may fail to operate in an alarm situation.NOTE: Once the Strobe is plugged back into the mains, the amber (Fault) LED will continue to flash once every 5 seconds until the battery has fully recharged.16PRODUCT INDICATORS (LEDs)STATUS STROBE/PADACTIVITYPOWERGREENSMOKEREDFAULTAMBERCOREDACTIONNEEDEDStandby mode Illuminated None requiredMains not connectedor on battery power Double flashevery fivesecondsReconnectStrobe to mainspowerVibrating Pad not connected to Strobe Illuminated(providing Strobeis plugged intomains)Flashing once persecondReconnectVibrating Pad tostrobeMains notconnected and battery capacity low Strobe will chirpFlashing onceevery 5 secsURGENTL Yrestore powerto unit asbattery status isvery lowAlarm mode Smoke Strobe flashing/Pad vibratingIlluminated(providing Strobeis plugged intomains)FlashingAlarm – fire- evacuatepropertyAlarm mode CO Strobe flashing/Pad vibratingIlluminated(providing Strobeis plugged intomains)FlashingAlarm – CO –Open windowsand evacuateproperty17If after proceeding with extreme caution you have been able to confirm that your Strobe and Vibrating Pad System was triggered by a nuisance alarm (which may occur from time to time from cooking or other non-emergency situations) you should silence the alarm that has been triggered. T his will stop the Vibrating Pad from vibrating and the Strobe from flashing. It is important to practice escape plans and show everyone in the house, including children, what to do in the event of an alarm.WHAT TO DO IN THE EVENT OF A CARBON MONOXIDE ALARM TRI GGERI NG THESTROBE AND PADAssume carbon monoxide has been detected in your home!WHAT TO DO IN THE EVENT OF A CO ALARM WARNING: A loud alarm is a warning that unusually high and potentially lethal levels of carbon monoxide are present. Never ignore this alarm; further exposure could be fatal. Immediately check residents for symptoms of carbon monoxide poisoning, (see product manual) and contact the proper authorities to resolve all CO problems. NEVER IGNORE ANY ALARM.What to do during an alarm• Keep calm and open the doors and windows to ventilate the property.• Stop using all fuel-burning appliances and ensure, if possible, that they are turned off.• Evacuate the property leaving the doors and windows open.• Ring your gas or other fuel supplier on their emergency number; keep the number in a prominent place.• Do not re-enter the property until the alarmhas stopped. When exposed to fresh air it can take up to 10 minutes for the sensor to clear and the alarm to stop depending on the level of carbon monoxide detected.20• Get medical help immediately for anyone suffering the effects of carbon monoxide poisoning (headache, nausea, drowsiness), and advise that carbon monoxide poisoningis suspected.• Do not use the appliances again until they have been checked by an expert. Contact a Gas Safe registered engineer in the case of a gas appliance.REPAIRWARNING: DO NOT attempt to repair your Strobe and Vibrating Pad System. Y our Wi-Safe 2 Strobe and Vibrating Pad is a sealed AC electrical device, and no attempt should be made to open the casing on any part of the system. Attempting to open any case will:• Damage your system.• Possibly result in exposure to a potentially lethal electric shock.• Impair its operation.• Invalidate your warranty.If your Wi-Safe 2 Strobe and Vibrating Pad is not working correctly and you are unable to resolve the problem after consulting the “Troubleshooting” section, please contact T echnical Support.The battery provided has been chosen specifically for this application but will need to be replaced every 5 years under normal operating conditions. If during normal operation (i.e. connected to the mains supply for at least 72 hours) the amber LED flashes once every 5 seconds while the green LED is illuminated, contact T echnical Support for advice. IMPORTANT: Do not replace with any other type of battery.21To remove the Vibrating Pad from anestablished network:Switch the power off at the socket and disconnectthe battery. Disconnect the Vibrating Pad from thestobe unit (Fig.3). Press and hold the test button ofthe Strobe. Whilst doing so, switch the power onand continue to hold the test button. After about 5seconds, the Strobe will beep and both red LEDswill flash. You can now release the test button andconnect the battery. No fault conditions, indicated bya flashing amber LED, should be present even withthe Vibrating Pad disconnected.PLEASE NOTE:The V ibrating Pad can be reconnectedat any point and will operate correctly. If the VibratingPad is then removed the system will show a fault,indicated by a flashing amber LED once per second.If this occurs and you require the Strobe to operateindependently, please follow the steps above.23TROUBLESHOOTINGThe Vibrating Pad does not vibrate when testing Check the Vibrating Pad is correctly connected to the Strobe. If it is correctly connected and the Fault (amber) LED is flashing after 2 minutes, contact T echnical Support on ***********.The Strobe is connected to the power supply but the Power (green) LED is not illuminated Check the power supply is plugged in and switched on at a working power socket.The fault (amber) LED flashes once every 5 seconds continuously Urgently restore mains power to the strobe unit. If the strobe unit is plugged in to the mains supply and has been switched on for at least 72 hours, the battery may be faulty. Call T echnical Support on ***********.The Power (green) LED is illuminated, and the Fault (amber) LED is also continuously lit The strobe unit is not interlinked (“learned in”) to the rest of the network. If you have learned in the unit, and this is happening regularly, call T echnical Support on ***********.The Strobe and Vibrating Pad does not respond to a smoke or CO alarm that is being tested Check that the alarm and the Strobe and Vibrating Pad have been interlinked (“learned in”) correctly. Check that the units are in range, see “Positioning”. If the Strobe and Vibrating Pad still fails to respond, call T echnical Support on ***********.The Fault (amber) LED double flashes, rapidly One or more alarms in the network are missing, faulty or disabled (removed from its base).Unit chirps once per minute Refer to product indicators for LED flash pattern. If in doubt contact Technical Support on ***********.24DISPOSALWaste electrical products should not be disposed of with regular household waste. Your Strobe and Vibrating Pad should be disposed of in line with the waste electronic and electrical equipment (WEEE) regulations. Please recycle where facilities exist. Check with your local authority, retailer or contact our technical support team for recycling/disposal advice as regional variations apply. Please “unlearn” the Strobe from its wireless network and disconnect the battery within the Strobe before disposal. Failure to do so could result in a system fault. WARNING: DO NOT ATTEMPT TO OPEN. DO NOT BURN.WARRANTYFireAngel Safety Technology Limited warrants to the original purchaser that its enclosed W i-Safe 2 Strobe and V ibrating Pad alert (W2-SVP-630L) be free from defects in materials and workmanship under normal residential use and service for a period of 5 (five) years from the date of purchase. Provided it is returned with postage prepaid and proof of purchase date, FireAngel Safety Technology Limited hereby warrants that during the 5 (five) year period commencing from the date of purchase FireAngel Safety Technology Limited, at its discretion, agrees to replace the unit free of charge. The warranty on any replacement W2-SVP-630L will last for the remainder of the period of the original warranty in respect of the product originally purchased – that is from the date of original purchase and not from the date of receipt of the replacement product. FireAngel Safety Technology Limited reserves the right to offer an alternative product similar to that being replaced if the original model is no longer available or in stock. This warranty applies to the original retail purchaser from the date of original retail purchase and is not transferable. Proof of purchase is required. This warranty does not cover damage resulting from accident, misuse, disassembly, abuse or lack of reasonable care of the product, or applications not in accordance with the user manual. It does not cover events and conditions outside of FireAngel Safety Technology Limited’s control, such as Acts of God (fire, severe weather etc.). It does not apply to retail stores, service centres or any distributors or agents. FireAngel Safety Technology Limited will not recognise any changes to this warranty by third parties. FireAngel Safety Technology Limited shall not be liable for any incidental or consequential damages caused by the breach of any expressed or implied warranty. Except to the extent prohibited by applicable law, any implied warranty of merchantability or fitness for a particular purpose is limited in duration for 5 (five) years. This warranty does not affect your statutory rights. Except for death or personal injury, FireAngel Safety Technology Limited shall not be liable for any loss of use, damage, cost or expense relating to this product or for any indirect or consequential loss, damages or costs incurred by you or any other user of this product.25(UK)********************************** (DE)*************************(FR)********************************** (NL)******************************** (INT)**************************************。
高中物理人教版2019选修二习题答案第2章电磁感应习题课电磁感应中的电路、电荷量和图像问题
习题课电磁感应中的电路、电荷量和图像问题必备知识基础练1.如图所示，导体轨道OPQS固定，其中PQS是半圆弧，Q为半圆弧的中点，O为圆心。
轨道的电阻忽略不计，OM是有一定电阻、可绕O转动的金属杆，M端位于PQS 上，OM与轨道接触良好。
空间存在与半圆所在平面垂直的匀强磁场，磁感应强度的大小为B，现使OM从OQ位置以恒定的角速度逆时针转到OS位置并固定(过程Ⅰ)；再使磁感应强度的大小以一定的变化率从B增加到B'(过程Ⅱ)。
在过程Ⅰ、Ⅱ中，流过OM的电荷量相等，则B'等于()BA.54B.32C.74D.22.(2021山西太原模拟)如图所示，一电阻为R的导线弯成边长为L的等边三角形闭合回路。
虚线MN右侧有磁感应强度大小为B的匀强磁场，方向垂直于闭合回路所在的平面向里。
下列对三角形导线框以速度v向右匀速进入磁场过程中的说法正确的是()A.回路中感应电流方向为顺时针方向B.回路中感应电动势的最大值E=√3BLv2C.回路中感应电流的最大值I=√3RBLv2D.导线所受安培力的大小可能不变3.在竖直向上的匀强磁场中，水平放置一个不变形的单匝金属圆线圈，规定线圈中感应电流的正方向如图甲所示，当磁场的磁感应强度B随时间t如图乙变化时，图中正确表示线圈中感应电动势E的变化的是()4.如图所示，用粗细相同的铜丝做成边长分别为l和2l的两个闭合正方形线框a和b，以相同的速度从磁感应强度为B的匀强磁场区域中匀速地拉到磁场外，不考虑线框的重力，若闭合线框的电流分别为I a、I b，则I a∶I b为()A.1∶4B.1∶2C.1∶1D.1∶35.在匀强磁场中，有一个接有电容器的单匝导线回路，如图所示，已知C=30μF，l1=5cm，l2=8cm，磁感应强度以5×10-2T/s的变化率增加，则()A.电容器上极板带正电，带电荷量为6×10-5CB.电容器上极板带负电，带电荷量为6×10-5CC.电容器上极板带正电，带电荷量为6×10-9CD.电容器上极板带负电，带电荷量为6×10-9C6.如图所示，半径为R的左端开有小口的圆形导轨处在垂直于圆平面的匀强磁场中，磁感应强度为B，方向垂直于纸面向里。
高中英语选择性必修第三册教学设计 B6U1
Unit 1 ARTReading and Thinking: A Short History of WesternPainting文本简析本单元阅读文本的主题是西方绘画简史。
通过介绍不同时期的西方绘画以及因各时代代表画家引领而发展出的独特风格，激发读者品鉴艺术历史，探究西方绘画未来可能的发展方向和促进艺术发展的源动力，思考艺术的功能和价值，提高审美素养，崇扬人文价值。
该文本采用说明性文体，以第三者视角客观简述西方绘画发展的四个关键时期，内容紧凑，语言平实，文风平易近人。
标题“A Short History of Western Painting”指向文本主题，标示语篇类型。
全文以“总—分”建构，分述部分以副标题切分。
第一段为总起段，阐述为什么要了解绘画发展史。
其余段落以时间为主线，选取代表人物，依次介绍四个不同时期的西方绘画，并在最后留下思考空间，引发对于艺术本质的思考。
在进行文本教学设计时，要帮助学生梳理关于绘画艺术和历史发展相关的话题类语言，包括绘画特点，以及变化发展的相关表达。
此外，逻辑功能性语言，例如while…still…, less…more…, …but instead等，也是值得关注的。
本文四个时期虽由副标题统领独立成文，但全文在内容过渡、行文逻辑和语言衔接上是串联的整体，在文本梳理中也需特别注意。
此外，本课需要关注西方艺术区别于中国艺术的独特魅力，同时要培养发展的眼光看待文化，感受跨越文化的艺术之美。
教学设计（共2课时）第1课时一、教学内容梳理全文信息，了解西方绘画四个时期的代表人物及绘画特色和风格，理清信息的整合方式。
二、课时目标1. 围绕西方绘画进行主题讨论，根据文本布局和语篇要素预测文本内容，激活背景知识和主题词汇。
2. 运用略读、细读、归纳、整合等阅读策略，梳理文本信息，根据原文逻辑进行加工输出，理清代表人物在西方绘画发展史的作用，并完善对于西方绘画史的认知和理解。
3. 通过文本分析和语篇解读整理西方绘画各时期代表人物的绘画风格，并能够使用相关信息表达观点和解决实际问题。
Han-Tarantello-2014-Cal.Var.PDE
Received: 25 September 2012 / Accepted: 2 March 2013 / Published online: 31 March 2013 © Springer-Verlag Berlin Heidelberg 2013
Calc. Var. (2014) 49:1149–1176 DOI 10.1007/s00526-013-0615-7
Calculus of Variations
Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model
123
1150
X. Han, G. Tarantello
nonlinearity and Dirac source terms. Within this framework we mention for example the (2 + 1)-dimensional Abelian Chern–Simons model of Hong et al. [23] and Jackiw–Weinberg [27], for which Taubes’ approach has led to the existence of topological multivortices (as described in [42,52]), non-topological multivortics (as constructed in [8,9,11,13,43]) and doubly periodic vortices (as given in [7,14,15,31,38,47,49,53]). In the non-Abelian context, rigorous existence results are established in [4,10,44,45], while a series of sharp existence results have been obtained in [12,30,32,33,48] for non-Abelian models proposed in connection with the quark confinement phenomenon [24,25,34,35]. For more results about self-dual vortices, we refer the readers to the monographs [46,55]. Here, we are going to analyze a relativistic (self-dual) non-Abelian Chern–Simons model proposed by Dunne in [16,17]. For this model, Yang [54] first established the existence of topological solutions in a very general situation. Subsequently, for the gauge group SU (3), Nolasco and Tarantello [37] proved a multiplicity result about the existence of doubly periodic vortices. The purpose of this paper is to establish analogous multiplicity results for theories that involve more general gauge groups. More precisely, we focus on gauge groups with a semi-simple Lie algebra of rank 2. From the technical point of view, we need to handle a 2 × 2 nonlinear elliptic system on the flat 2–torus, with coupling matrix given by the Cartan matrix associated to the gauge group. Clearly, this more general situation poses new analytical difficulties compared to the (already nontrivial) case analyzed in [37], where the authors handle a (specific) symmetric 2 × 2 system. Actually, we manage to resolve such difficulties for a larger class of 2 × 2 systems, where our vortex problem is included as a particular case. 2 Derivation of a general 2 × 2 nonlinear elliptic system and statement of the main results The non-Abelian Chern–Simons model introduced by Dunne in [16,17], is formulated over the Minkowski space R1+2 with metric tensor: diag(1, −1, −1), that will be used in the usual way to raise and lower indices. Using the summation convention over repeated lower and upper indices ( ranging over 0, 1, 2), we consider the Lagrangian density: κ 2 L = − Tr μνα ∂μ Aν Aα + Aμ Aν Aα 2 3 + Tr [ Dμ φ ]† [ D μ φ ] − V municated by P. Rabinowitw. X. Han Institute of Contemporary Mathematics, School of Mathematics, Henan University, Kaifeng 475004, Henan, People’s Republic of China G. Tarantello (B) Dipartimento di Matematica, Unversità di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy e-mail: tarantel@axp.mat.uniroma2.it
非齐次方程
令 t t , 混合问题（2.5）就化为 2 2 w w 2 , (t 0) 2 a 2 t x w(0, t , ) w( L, t , ) 0, w( x, t , ) t 0 0, wt ( x, t , ) t 0 f ( x, )
数学的思维功能
加里宁：“数学是锻炼思维的体操。”
解出一道数学题的收获究竟是什么 ?
1 3 2 3 解决了一个问题 得到了一种愉悦 找到了解决问题的一般门路 获得了一种成功感 增强了今后克服困难的信心与勇气
4 5 320若源自w( x, t ; ) 2
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英语选修Unit 同步系列教案(Word power)
英语选修6译林牛津版Unit 1同步系列教案（3）（Word power）●Word powerStep 1：BrainstormingT：Have you heard of Shakespeare? Can you say out some of his works? S: “Romeo and Julie”，“Merchants of Venice” and so on.T：Have you seen them performed on the stage？S: Yes。
/No.T：A piece of writing to be performed by actors in the theatre is called drama. Are you interested in drama？Do you want to learn something more about drama? Today we’ll learn some words used in a drama on the stage.Step 2: Brainstorming and vocabulary learning1 First，let’s check how much you have known about drama.T: What do we call the person who acts in a play？S：An actor or actress。
T：What do we call the person who is in charge of a play?S：A director.T：what are the words that the actors say called？S: linesT: Where can the lines can be found?S：In a script。
T：what is a script made up of?S: Acts and scene.T: What are the words called that tell the actors how to act?S：They are called stage directions。
牛津译林版选修六Unit2 Word power教案
Unit 2 What is happiness to you?Word powerStep 1: BrainstormingIn this section we’ll learn the words and idioms used to describe emotions and feelings. Now I’d like you to answer the following questions:What words do you know can express happiness?Apart from happiness, what other types of emotions do you know?Can you think of some words that can express these kinds of feelings?(love joy excitement hate fear jealousy delight surprise astonishment frustration depression contentment satisfaction concern worry fury curiosity ) Step 2: Vocabulary learning1. Read the instructions on page 22 and study the examples listed in the table.2. Write the adjective forms of other nouns related to emotion.ReferenceNouns Adjectivesastonishment astonisheddelight delightedfury furiouscuriosity curioussatisfaction satisfieddepression depressedamazement amazeddisappointmemt disappointedStep 3: Practice1. Let’s focus on Part A. Circle the right word according to each different situation.Step 4: CompetitionHow many emotional words do you know? Now I’ll divide you into several groups. You’ll have a competition to see which group has the most words. In the end you need to group emotional words into three different categories: happiness, sadness and anger.·Words describing happiness:Noun forms: joy, happiness, delightAdjective forms: joyful, happy, delighted·Words describing sadness:Noun forms: sadness, depressionAdjective forms: sad, depressed·Words describing anger:Noun forms: anger, furyAdjective forms: angry, furiousStep 5: Vocabulary extension1. Choose the correct word to complete each sentence.①I'm not_____(satisfied, satisfaction) with what I've done. I can't get_____ (satisfied, satisfaction) from it. (satisfied; satisfaction)②I was _____ (amazed, amazement) by the change in his appearance. All of us looked at him in_____ (amazed, amazement). (amazed; amazement)③The boy is _____ (curious, curiosity) about everything. His burning_____ (curious, curiosity) inspires him to learn more. (curious; curiosity)2. Do part C. In the English language, there are some idioms about emotions. Guess theirThere are some very good things about open education. This way of teaching allows the students to grow as people, and to develop their own interests in many subjects. Open education allows students to be responsible for their own education, as they are responsible for what they do in life. Some students do badly in a traditional classroom. The open classroom may allow them to enjoy learning. Some students will be happier in an open education school. They will not have to worry about grades or rules. For students who worry about these things a lot, it is a good idea to be in an open classroom.But many students will not do well in an open classroom. For some students, there are too few rules. These students will do little in school. They will not make good use of open education. Because open education is so different from traditional education, these students may have aproblem getting used to making so many choices. For many students it is important to have some rules in the classroom. They worry about the rules even when there are no rules. Even a few rules will help this kind of students. The last point about open education is that some traditional teachers do not like it. Many teachers do not believe in open education. Teachers who want to have an open classroom may have many problems at their schools.You now know what open education is. Some of its good points and bad points have been explained. You may have your own opinion about open education. The writer thinks that open education is a good idea, but only in theory. In actual fact, it may not work very well in a real class or school. The writer believes that most students, but of course not all students, want some structure in their classes. They want and need to have rules. In some cases, they must be made to study some subjects. Many students are pleased to find subjects they have to study interesting. They would not study those subjects if they did not have to.1. Open education allows the students to ___________ .A. grow as the educated B．be responsible for their futureC. develop their own interests D．discover subjects outside class2. Open education may be a good idea for the students who__________.A．enjoy learning B．worry about gradesC．do well in a traditional classroom D．are responsible for what they do in life.3. Some students will do little in an open classroom because_________ .A．there are too few rulesB．they hate activitiesC．open education is similar to the traditional educationD．they worry about the rules4. Which of the following is NOT mentioned in the passage?A．Some traditional teachers do not like it.B．Many teachers do not believe in open education.C．Teachers may have problems in open classrooms.D．The teacher’s feelings and attitudes are important to the students.5. Which of the following best summarizes the passage?A．Open education is a really complex idea.B．Open education is better than traditional education.C．Teachers dislike open education.D．The writer thinks that open education is a good idea in practice.Step 6: Homework1.Part C on page 114 in the Workbook.Prepare Grammar and usage.。
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分项工程质量检验评定表
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混凝土试模检验表
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混凝土试模检验表
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混凝土坍落度筒检验表
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a great deal attempt
appeal
use the new words to make a speech
I have a dream
我有一个梦想：有一天, 这个世界能 力争(aim)停止释放(release)大量的 二氧化碳。 我有一个梦想：有一天, 在地球上人 们会试图种植更多的树木。 我有一个梦想：有一天, 在非洲的孩 子们将拥有更多的淡水和更少的 饥饿(starvation)。 这是我们的希望。这也是我要向全 人类呼吁的信仰(faith)。有了这个 信念, 我们就能够打破所有的预言。
urgent request
appeal
The advertisement aims at appealing to people not to smoke. The government makes an appeal for people’s health. Does the idea of it appeal to you?
Jack is late again. It is____ of him to keep others waiting . （2008 江西） A.normal B. ordinary C. common D. typical
aim
On the other hand faith
adopt
possess
______ soldiers were sent to the area where the earthquake had happened. A. A great deal of B. A large number of C. A great amount of D. A plenty of
This kind of desk can be_____ to the height you need. A. adopted B. adjusted C. addicted D. adapted
---How did you like Nick’s performance last night? ---To be honest, his singing didn’t ____ to me much. （2010 北京） A.appeal B. belong C. refer D. occur
5 minutes
Show time
I have a dream that one day this world will _______ aim at stopping releasing _____________ a great deal of carbon dioxide. I have a dream that one day on the earth attempt to plant more trees. people will __________ I have a dream that one day the children in possess more fresh water and Africa will ________ less starvation. This is our hope. This is the faith that I will appeal tothe whole human. With this faith ________ we will be able to break all the _________. prediction
Have a try
lead a low-carbon life 低碳生活
Completion (介词填空)
1.He aimed the gun__ at the criminal' s head.
2.I passed my driving test ___ at the first attempt. 3. The ball was ___ in the possession of Kobe. 4. They needed to appeal ___ to his sense of humor.
22、appeal vi. 呼吁，要求，有感染力，求助于， 诉诸（常与to连用） appeal to sb. for sth. 为某事向某人呼吁 appeal to sb. to do sth. 呼吁某人做某事 对某人有吸引力；使某人感兴 appeal to sb. 将…上诉至… appeal…to… 那些领导要求群众支持。 Those leaders appeal _______________________support . to the masses for 政府要求大家节约用水。 The government is appealing ________________save water . to everyone to 这则广告对公众很有吸引力。 to the public. The advertisementappeals _____ very much__
19. aggressive adj. 敢作敢为的，有冲劲的，有 进取心的；好斗的，侵略的，挑衅性的 →aggressively adv.好争斗地；进取地 →aggression n.侵略，侵犯；挑衅 20. in the flesh 本人；活生生的（人），亲自 他们想去看他们的英雄本人。 They want to see their heroes ____________. in the flesh 21. preference n. 喜爱；偏爱（不可数名词）； 更喜欢的人或物（可数名词） have a preference for 更喜欢…
15. specific →specifically to be specific be specific to
adj. 确切的；特定的 adv.确切地；具体地；特别地 说得明确些（具体地说） 特有的；独特的
我给过你明确的提示。 I gave you specific instructions . 具体地说，他已经34了。 ______________, he is 34 years old . To be specific 这种动物是这个地区特有的。 The animal ______________ this area . is specific to