橡胶元件有限元技术-英文版-all
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 ⎟⎠⎞⎜⎝⎛∂∂<<⎟⎠⎞⎜⎝⎛∂∂.。
橡胶机械轮胎专业术语英文
橡胶机械轮胎专业术语英文Air Preparation Equipment- 空气清净化元件Air Line Equipment- 气动辅助元件Directional Control Equipment- 方向控制元件Actuator-执行元件Nytex810, Nytex820, Nytex840, Nytex8450, Nytex4700, Nyflex222B- 环保油polymer bound pre-dispersed rubber chemicals- 预分散橡胶化学品母胶粒processing promoters-力口工助剂bladder coating-胶囊隔离剂tyre paints-轮胎喷涂液specialty chemicals-特殊化学品anti-sun check wax- 微晶防护蜡steel cord-钢帘线Thermoplastics And Thermosetting Phenolic Resin- 热塑性和热固性酚醛树脂On-Line Profilometer-在线测厚仪pneumatic components and automatic systems -气动元件、组件和自动化系统radial tyre building machine- 全钢子午线轮胎成型机tyre curing press-双模轮胎定型硫化机系列plate vulcanizer-平板硫化机系列mixing mill-炼胶机系列tread extruding line- 胎面挤出线batch off-胶片冷却线Rubber Processing Analyzer-RPA 橡胶加工分析仪Tensile Strength Testers with High and Low Temp- 电脑系统拉力机试验机Batch Off Machine For Cabon Mixing Line- 胶片冷却联动生产线One Step Mix Sytem- 一次法低温炼胶系统Tread Extruder Line- 胎面压出联动生产线The internal mixer upstream equipment system -密炼机上辅机系统Tire Curing Press- 硫化机Tire Building Machines- 成型机Chemical Machinery-化工机械Extruder-胎胚内外喷涂机Cutters-磨白边机Tire Painting Machines- 带束层生产线White Sidewall Buffers- 钢丝帘布裁断机All-Steel Radial Tires- 全钢工程轮胎Bias OTR Tyres-斜交工程轮胎Light Truck Tire And Passenger Car Radial Tire- 卡车轮胎以及轿车轮胎Rubber （Plastics） Mixing Line- 橡胶（塑料）密炼生产线Rubber （Plastics） Calendaring Line- 橡胶（塑料）压延生产线Every Series Of Two Roll Mills- 橡胶（塑料）开炼机系列SLG series water jar —SLG 系列水缸SLQG series air cylinder —SLQG 系列气缸The SLYG series oil urn —SLYG 系列油缸Antioxidant TMQ、PAN、KL、SP —防老剂TMQ、PAN、KL、SPAccelerator M、DM、CZ、NS、DZ、D —促进剂M、DM、CZ、NS、DZ、DPlasticizer A —增塑剂AAntioxidant T-531 —抗氧剂T-531 中石油—三剂 ||Series of Semi-steel car / passenger radial tire molding machine —全钢载重子午胎系列成型机Series of all-steel light truck radial tire molding machine —全钢轻卡子午胎系列成型机Series of all-steel truck radial tire molding machine —半钢乘用子午胎系列成型机Series of all-steel engineering radial tire molding machine —全钢丝工程子午胎系列成型机Series of all-steel engineering giant fetal radial tire molding machine —全钢丝巨型工程子午胎系列成型机Meridian tyres single stage building machine for seml-steel tyre (two drums ) —半钢一次法成型机(两鼓) GK-Type Internal Mixer —密炼机系列XJY-ZS twin-Screw Roller Head Extruder —双锥双螺杆滚式机头挤出机Tyre Shaping and Curing Press —轮胎定型硫化机Multi-Unit Daylight Press —平板硫化机Rotocure —鼓式硫化机Tyre building machine (TBM) —轮胎成型机Rubber Antioxidant BLE —橡胶防老剂BLERubber Antioxidant AW —橡胶防老剂AWRubber antioxidant4010NA —防老剂4010NARubber antioxidant4020 —防老剂4020Tire High-speed / Endurance Testing Machine —轮胎高速/耐久试验机Five stiffness Index Tire Testing Machine —轮胎五刚性试验机Tire Contact Contour Analyzer —轮胎接地轮廓分析试验机Tire Strength & Static Loaded Testing Machine —轮胎强度/静负荷试验机Tire Bead Unseating Resistance T esting Machine —轮胎脱圈阻力试验机Hydraulic Burst Testing Machine —水压爆破试验机Electrical Resistance of Tire Under Load Testing Machine —轮胎电阻试验机Tire Run-out Testing Machine —轮胎偏心度试验机Standard Test Tire Rim —轮胎标准试验轮辋Mooney Viscosity —门尼粘度仪Instrument Configuration And After Serve Of Moving Die Rheometer —MDR-2000E 电脑型无转子硫化仪PNEUMATIC TWO-PLACE CUTTING VALVE —气动二位切断阀SELF-SUPPORTING ADJUSTMENT VALVE —力式压力调节阀PNEUMATIC TWO-PLACE AND FOUR-PASSAGE VALVE —气动二位四通滑阀PNEUMATIC FILM ADJUSTMENT VALVE —气动薄膜调节阀tyre-tubes —内胎rubberAdditives Series —橡胶助剂HA-8 N, N' -n -Phenylenedimaleimide(HA-8) —多功能橡胶硫化剂Environmental antioxidantTPS-2 —环保型防老剂TPS-2Reaction type is not taking antioxidant —反应型不抽出防老剂SF-98 ( MC) Mono pin type —销钉冷喂料挤出机cold feed pin type —普通冷喂料挤出机vent type extruder —排气式冷喂料挤出机new type hot feed extruder —新型热喂料挤出机rubber filtering extruder —滤胶挤出机NR inner tube —天然胶内胎Butyl rubber inner tube —丁基内胎Internal Release Agent —内脱模剂External Release Agent —外脱模剂Rubber product Release Agent —橡胶制品脱模剂Tire Capsule Partition —轮胎胶囊隔离剂Mould Care Agent —模具护理剂Rubber Additive —橡胶助剂steel cord —钢帘线Carbon Black Dispersing Agent- 白炭黑分散剂AR-205 —白烟力AR-205Motorcycle Tires —摩托车Small-Scale —电动车Agricultural V ehicle Tires —农用车V-Belt —轻卡车轮胎tyre cord fabric —橡胶骨架材料tyre cord canvas —帘帆布Wet Granule Carbon Black For Rubber —橡胶用炭黑Use And For Seal Use —密封条专用炭黑tire valve cores —轮胎气门芯air conditioner valve cores —空调气门芯valve cores for relief valves on LPG —液化气减压阀门气门芯tanks and valve stems on blowing out proof automobile tire —防爆胎气门芯Carbon Black —炭黑OTR Tyre-工程机械轮胎Big\Middle\And Small Agricultural Tyre- 大中小型农业轮胎Forklift Tyre-叉车轮胎Truck And Light Truck Tyre- 载重汽车轮胎Tire high speed / endurance testing machine- 轮胎高速/耐久性能试验机Tire section cutting machine- 轮胎强度/脱圈试验机Tire strength / bead unseating testing machine- 轮胎断面切割机Tire Integration performance testing machine- 轮胎综合性能试验机Testing rims-试验轮辋Tyres And Tubes Of Motorcycles- 摩托车内外胎Agricultural-Used V ehicles-农用车内外胎Light Trucks-轻型载重汽车内外胎Heavy-Duty Trucks- 中型载重汽车内外胎Industry Vehicles-工程工业车辆内外胎Rubber Machinery-橡胶机械The butyl rubber inner tube —丁基橡胶内胎motorcycle tyre —摩托车轮胎agriculture tyre —农业轮胎different truck tyre —各类轻卡及载重汽车轮胎Semi-Steel Radial Tire Building Machine (First Stage And Second Stage) —半钢成型机All-Steel Radial Tire Building Machine (Two Drum And Three Drum) —全钢成型机Tire Curing Press (45 ||48 ||55 ',63.5 1,65.5 I)—双模硫化机、大型平板硫化机组The Sealing Strip Of All Types Of Automobiles —密圭寸条Rubber Belt Track —橡胶履带Tire Molds —轮胎模具产品Rubber Machinery —橡胶机械all-steel radial truck tires- 全钢载重子午线轮胎semi-steel radial tires- 半钢子午线轮胎steel engineering tire- 全钢工程轮胎bias truck tire-斜交载重轮胎Tube-内胎Flap-垫带Rim-轮辋Sedan Radial Tires-轿车子午胎ultra high performance tire- 半钢高性能轮胎passenger car radial tire- 轿车子午胎light truck tire and all-steel radial tires- 半钢轻卡胎及全钢子午胎The tubeless wheels- 无内胎轮胎车轮Tube Steel Wheels-型钢汽车车轮Tubeless Steel Wheels- 无内胎汽车车轮Bias Heavy Truck Tyres- 汽车载重斜交轮胎Bias OTR Tyres-工程斜交轮胎All Steel Radial Tyres-全钢子午线载重轮胎Radial OTR Tyres —全钢子午线工程胎Giant OTR Tyres —工程巨胎Tyre Bladders —轮胎硫化胶囊Bearing —轴承cylinders-液压油缸power stations-液压系统hydraulic systems-液压泵站Tyre Molds-轮胎模具Motor Tube-摩托车系列轮胎Rubber Oxygen Pipe & Acetylene Pipe- 氧气管，乙炔管Rubber Products-塑制品，橡胶产品Brake pads —盘式和鼓式刹车片Motorcycles —摩托车轮胎small and medium sized agricultural vehicles —中小型农用车loaders —装载机ATV tires —沙滩车轮胎Bicycle V alves —自行车气门嘴Motorcycle V alves —摩托车气门嘴Bus And Heavy —Duty Truck V alves —客车卡车气门嘴等轮胎气门嘴Rubber Sundry Piece —橡胶杂件Bevel Gear & Drive Axle —锥齿轮及驱动桥rotor —汽车刹车盘brake drum —刹车鼓brake pads —杀U车片brake shoes —刹车蹄片Air hose-空气管water hose-水管suction hose-埋吸管hydraulic hose-液压胶管pvc hose-pvc 管twinline hose-连体管acetylene hose-乙炔管oxygen hose-氧气管rubber sheet-橡胶板Brake Rotors-盘式刹车片Brake Pads-鼓式刹车片。
橡胶密封制品的有限元模拟与结构优化
2、橡胶制品有限元分析具体案 例分析
(1)轮胎分析：通过有限元分析，可以对轮胎的力学性能进行精确预测，从 而优化其结构设计，提高轮胎的使用性能和安全性。例如，对轮胎进行静力学分 析，可确定轮胎在不同工况下的变形和应力分布情况，避免轮胎在极端工况下出 现损坏或失稳现象。
(2)传动带分析：传动带是机械系统中重要的元件之一，其性能直接影响到 整个系统的稳定性和效率。通过有限元分析，可以对其传动力、传动效率、振动 等性能进行评估，进而优化传动带的设计和生产工艺。例如，对传动带进行动力 学分析，可确定其在不同转速和负载下的振动和疲劳寿命，提高传动系统的稳定 性和效率。
三、橡胶动态性能有限元分析的 未来展望
随着科技的不断发展，有限元分析在橡胶动态性能研究中的应用将更加广泛。 新的算法和计算能力的提升将使得对橡胶制品的动态性能进行更精细的模拟成为 可能。同时，随着材料科学和实验技术的发展，将为有限元模拟提供更准确的材 料模型和边界条件，从而进一步提高模拟的准确性。
橡胶密封制品的有限元模拟与 结构优化
基本内容
橡胶密封制品在工业和日常生活中应用广泛，如汽车、航空航天、电子产品 等领域。它们起着至关重要的作用，不仅可以防止气体和液体的泄漏，还可以防 止灰尘和污垢的侵入。为了提高橡胶密封制品的性能和可靠性，有限元模拟和结 构优化成为越来越重要的工具。
有限元模拟是利用数学方法模拟真实物理系统的过程，以获得系统在不同条 件下的行为和性能。在橡胶密封制品的有限元模拟中，通常选用橡胶材料属性， 如弹性模量、泊松比、剪切模量等，来描述其力学行为。同时，有限元模拟可以 预测和优化橡胶密封制品的结构，以获得最佳的性能。
结论：
有限元数值模拟已成为工程设计和科学研究的重要工具，对于橡胶制品的动 态性能分析具有特别的价值。通过对橡胶动态性能进行有限元模拟，我们可以预 测制品在实际工作环境下的行为，优化设计以改善其性能，并提高产品的可靠性 和使用寿命。
有限元在车辆橡胶元件中的应用
研究开发弹性体,20100825,20(4):34~38CH INAELASTOMERICS收稿日期63作者简介韩智慧(),女,山东省岛人,硕士,主要研究方向为减震系统及橡胶减震件的开发。
有限元在车辆橡胶元件中的应用韩智慧,万里翔,何宇林,曾力(西南交通大学机械学院,四川成都610031)摘要:分别利用闭型方程式与有限元对粘合圆柱橡胶块进行刚度分析对比;利用有限元与试验对3种不同汽车橡胶减震件进行分析对比,体现了有限元法在此设计领域中的可行性与优越性。
关键词:有限元;橡胶元件;刚度中图分类号:TQ 336.4+2文献标识码:A文章编号:10053174(2010)04003405橡胶减震件在汽车上应用非常广泛而且其品种繁多,例如各种衬套、发动机悬置,推力杆橡胶关节等等,但是由于橡胶材料的超强非线性及元件的复杂结构,若仅仅使用有限几个闭型分析方程式是满足不了设计要求的。
随着非线性有限元分析软件的不断发展与日臻完善,其已经可以在汽车橡胶减震元件中得到广泛的应用,成为工程技术人员解决设计分析工作的有效途径。
1粘合圆柱橡胶块的刚度分析对一种结构十分简单而且经典的结构粘合圆柱形橡胶块(结构如图1),通过闭型方程式与有限元的计算对其刚度进行分析并对结果进行比较。
图1粘合圆柱形橡胶块三维模型1.1利用闭型方程式求解1.1.1本构方程橡胶类各向同性不可压缩超弹性材料,文献[1]得其本构方程:e=-P +2W I 1B-W I 2B -1式中:I 1、I 2为Cauch Green 左张量B 的前2个基本不变量,e为高氏应力张量,角标e 表示弹性分析;W(I 1、I 2)是未变形物体单位体积的应变能密度。
本研究中W 的形式为:W =C 10(I 1-3)+C 01(I 2-3)+C 11(I 1-3)(I 2-3)+C 20(I 1-3)2+C 30(I 1-3)3式中由5个常量组成的集合{C 10,C 01,C 11,C 20,C 30}是材料的特性参数,这些特性参数的数值是从单轴和多轴应力松弛数据中得到的,本研究中C 10=100.8kPa,C 01=161.2kPa,C 11=1.338kPa,C 20=0.6206kPa,C 30= 6.206kPa 。
基于有限元方法的橡胶弹性件设计开发流程
基于有限元方法的橡胶弹性件设计开发流程史文库、陈志勇、姜莞 吉林大学汽车工程学院一、应用有限元方法进行橡胶减振隔振件设计的必要性和意义橡胶包括天然橡胶及合成橡胶，是无定形的高聚物[1]。
橡胶是一种超弹性材 料，具有良好的伸缩性和复原性。
橡胶的弹性与金属的弹性不同，若将金属棒和 橡胶棒各用力拉伸，橡胶的最大伸长通常在 500~1000%之间，处在小变形区域 外，没有固定的杨氏模量，小变形范围内的杨氏模量约为 1.0MPa，而弹性变形 仅为 1%或更小[2]。
对于橡胶材料来说，对一般工程材料适用的小变形理论已经 不再适用，即当载荷比较大时，应力应变不再保持为线性关系，但不会产生永久 变形，当载荷一旦消失，变形将完全恢复。
并且橡胶元件的形状对其弹性特性有 复杂的影响，因此，不能用传统的理论进行设计和计算[3]。
如果单纯靠试验的方 法设计，其成本较高。
随着计算机技术的发展以及有限元理论的完善，用有限元 方法对橡胶隔振件设计计算成为了较好的方法。
有限元方法的优势在于设计周期 短，成本低，精度高，能够对橡胶隔振件进行非线性计算。
二、目前的国内外橡胶件设计开发现状分析早在 20 世纪 70 年代，橡胶制品的有限元分析已经成为橡胶制品设计者的主 要设计手段，一些商业化的非线性有限元软件MARC、ABAQUS、ADINA等被 用来对橡胶制品进行辅助分析[4]。
世界上一些著名的科研机构和生产橡胶制品的 公司开始对橡胶制品进行有限元分析，以提高橡胶制品的质量。
其中包括英国的 MERL公司，欧洲的RAPRA公司，美国的PSP公司，HLA Engineers公司，英国 TUN ABDUL RAZAK研究中心的橡胶材料咨询中心， Akron橡胶研发实验室等以 及世界上一些著名的轮胎公司都在橡胶制品的有限元分析方面有非常成熟的研 究。
有限元分析技术已经广泛应用到生产中，并且在产品研制中发挥着重要作用 [5] 。
我国橡胶制品的有限元分析起步较晚， 但随着国外大型有限元软件引入我国， 国内的一些高校和科研机构已经开始对橡胶制品进行有限元分析。
英文有限元方法Finite element method讲义 (4)
(ii)
and u 3 are identical (i.e. u 3 (0 ) = u 2 (0 ) and u 3 (0 ) = u 2 (0 ) ), show that u 3 = u 2 . Also show by means of Lagrange’s equations of motion that the response of the system depicted in the Figure is governed by the equation M u + Ku = F , where
U1 U2
1
1 0 U1 0 1 U2
1 1
T
1
1 0 1 0 1 −1
The zero frequency is associated with the translation deformation mode. The zero occurs when the stiffness matrix K is singular, which happens when insufficient boundary conditions are specified to prevent bulk modes of movement.
Solution
2 2 1 1 2 2 m 1u 1 + m2u 2 k1 u1 − u 0 + k 2 u 2 − u1 + k 2 u 3 − u1 2 + m 3 u 3 , Wse = 2 2 2 2 2 1 1 2 2 L = T − Wse = m1 u 1 + m2u 2 k1 u1 − u 0 + k 2 u 2 − u1 + k 2 u 3 − u1 2 + m3u 3 − 2 2
有限元仿真的英语
有限元仿真的英语Finite Element SimulationThe field of engineering has seen a remarkable evolution in recent decades, with the advent of advanced computational tools and techniques that have revolutionized the way we approach design, analysis, and problem-solving. One such powerful tool is the finite element method (FEM), a numerical technique that has become an indispensable part of the modern engineer's toolkit.The finite element method is a powerful computational tool that allows for the simulation and analysis of complex physical systems, ranging from structural mechanics and fluid dynamics to heat transfer and electromagnetic phenomena. At its core, the finite element method involves discretizing a continuous domain into a finite number of smaller, interconnected elements, each with its own set of properties and governing equations. By solving these equations numerically, the finite element method can provide detailed insights into the behavior of the system, enabling engineers to make informed decisions and optimize their designs.One of the key advantages of the finite element method is its abilityto handle complex geometries and boundary conditions. Traditional analytical methods often struggle with intricate shapes and boundary conditions, but the finite element method can easily accommodate these complexities by breaking down the domain into smaller, manageable elements. This flexibility allows engineers to model real-world systems with a high degree of accuracy, leading to more reliable and efficient designs.Another important aspect of the finite element method is its versatility. The technique can be applied to a wide range of engineering disciplines, from structural analysis and fluid dynamics to heat transfer and electromagnetic field simulations. This versatility has made the finite element method an indispensable tool in the arsenal of modern engineers, allowing them to tackle a diverse array of problems with a single computational framework.The power of the finite element method lies in its ability to provide detailed, quantitative insights into the behavior of complex systems. By discretizing the domain and solving the governing equations numerically, the finite element method can generate comprehensive data on stresses, strains, temperatures, fluid flow patterns, and other critical parameters. This information is invaluable for engineers, as it allows them to identify potential failure points, optimize designs, and make informed decisions that lead to more reliable and efficient products.The implementation of the finite element method, however, is not without its challenges. The process of discretizing the domain, selecting appropriate element types, and defining boundary conditions can be complex and time-consuming. Additionally, the accuracy of the finite element analysis is heavily dependent on the quality of the input data, the selection of appropriate material models, and the proper interpretation of the results.To address these challenges, researchers and software developers have invested significant effort in improving the finite element method and developing user-friendly software tools. Modern finite element analysis (FEA) software packages, such as ANSYS, ABAQUS, and COMSOL, provide intuitive graphical user interfaces, advanced meshing algorithms, and powerful post-processing capabilities, making the finite element method more accessible to engineers of all levels of expertise.Furthermore, the ongoing advancements in computational power and parallel processing have enabled the finite element method to tackle increasingly complex problems, pushing the boundaries of what was previously possible. High-performance computing (HPC) clusters and cloud-based computing resources have made it possible to perform large-scale, multi-physics simulations, allowing engineers to gain deeper insights into the behavior of their designs.As the engineering field continues to evolve, the finite element method is poised to play an even more pivotal role in the design, analysis, and optimization of complex systems. With its ability to handle a wide range of physical phenomena, the finite element method has become an indispensable tool in the modern engineer's toolkit, enabling them to push the boundaries of innovation and create products that are more reliable, efficient, and sustainable.In conclusion, the finite element method is a powerful computational tool that has transformed the field of engineering. By discretizing complex domains and solving the governing equations numerically, the finite element method provides engineers with detailed insights into the behavior of their designs, allowing them to make informed decisions and optimize their products. As the field of engineering continues to evolve, the finite element method will undoubtedly remain a crucial component of the modern engineer's arsenal, driving innovation and shaping the future of technological advancement.。
橡胶工业中有限元计算问题过盈配合
橡胶工业中有限元计算问题过盈配合作者：清华大学工程力学系范成业摘要本文分析了过盈配合的有限元计算时用到超弹性本构时可压缩性对计算结果的影响情况，得到在过盈配合中必须考虑这种可压缩性的结论并分析考虑可压缩性的原因。
1、引言过盈配合是橡胶工业中的一种常见的配合方式。
橡胶为超弹性材料，有限元计算中通常假定为不可压或者几乎不可压。
本文首先给出一种不可压橡胶模型过盈配合的理论解，并与ABAQUS计算解进行比较。
进一步本文探讨过盈配合中假定橡胶不可压时遇到的问题，提出处理过盈配合中橡胶计算的方法。
2、可压模型理论解与ABAQUS数值解的比较2.1、理论解理论解模型如图1，内层为钢，中间不可压橡胶，最外层为钢给出橡胶和橡胶之间的过盈量求整个结构的应力应变状态假设平面应变状态。
图1 理论解模型示意图本构方程：对于钢：对于橡胶：2.1材料性质：钢：E=210000v=0.3橡胶：C10=0.461312, C20=0.01752, C30=8.8e-05，其余为0，（三次多项式模型，材料不可压缩）2.2.2几何特性如图2所示，R59.50为内层钢的半径和中间层橡胶的内径，R73.00为中间层橡胶的外径，R71.10为外层钢的内径，R80.00为外层钢的外径。
图2 不可压模型算例几何特征理论解与计算解的比较（理论解由Maple计算得出）表1 理论解与ABAQUS 解的比较半径（mm ） 理论解 ABAQUS 计算解 误差 位移59.5 -9.2984E-02 -9.73152E-2 4.6% 径向应力S1159.5（钢）-660.51 -631.60 -4.38% 59.5（橡胶） -660.51 -631.60 -4.38% 73.0（橡胶） -660.51 -631.60 -4.38% 71.1（钢） -660.51 -664.30 0.57% 80.0（钢） 0 28.15 － 环向应力S2259.5（钢）-660.51 -631.20 -4.44% 59.5（橡胶） -660.51 -631.20 -4.38% 73.0（橡胶） -660.51 -631.40 -4.38% 71.1（钢） 5626.36 5541.00 -1.52% 80.0（钢）4956.854957.000.00%3、可压缩模型橡胶的应变能采用多项式模型时，在静水压力荷载下p 与J 的关系如下：用ABAQUS 对这1-4组系数进行评估：图3 不同系数对应的橡胶静水压力下的应力应变关系将这六种橡胶本构代入第二部分中的算例中进行计算结果如下：图4 第6组系数对应的位移图图5 第1组系数对应的位移图由图4和图5容易看到这两组系数对应的位移差异非常大。
有限元外文翻译
DEVELOPMENT OF A METHODOLOGY FOR DETERMINATION AND ANALYSIS OF THERMAL DISPLACEMENTS OF MACHINE TOOLS USING ELEMENTS METHOD FINITE AND ARTIFICIAL NEURAL NETWORKSRomualdo Figueiredo de Sousa, Instituto Federal da Paraíba, Av. Primeiro de maio, 720, Bairro: Jaguaribe. João Pessoa, CEP 58015-430, Campus Cajazeiras-PB, Brasil. engfsousa@.brFrancisco Augusto Vieira da Silva, Universidade Federal da Paraíba, Centro de Tecnologia, Bloco “F” Campus I – CEP 58.059-900 – João Pessoa – PB, Brasil. francisco.avs@João Bosco de Aquino Silva, Universidade Federal da Paraíba, Centro de Tecnologia, Bloco “F” Campus I –CEP 58.059-900 – João Pessoa – PB, Brasil. jbosco@ct.ufpb.brJosé Carlos de Lima Júnior, Universid ade Federal da Paraíba, Centro de Tecnologia, Bloco “F” Campus I –CEP 58.059-900 – João Pessoa – PB, Brasil. limajr@ct.ufpb.br利用有限元法和人工神经网络技术测定和分析机床热位移的方法一種方法的測定的和機床熱利用移動分析單元法有限和神經網絡分析發展菲格雷多羅穆阿爾德索薩，聯邦研究所DA帕拉伊巴，大道。
橡胶有限元分析基础
-20 -50
30
实验
20
Ogden 3
Ogden 5
10
Ogden 7
M-R 2 Gauss
实验 M-R 1 SVD M-R 2 SVD M-R 3 SVD
M-R 3 Gauss
0 50 100 150 200
应变 (%)
Mooney-Rivlin
应力(pa)
0
-10
Ogden
-20
-50 0 50 100 150 200
0 0.0
轴向 径向
0.5 1.0 1.5 2.0
位移(mm)
• 复杂形状悬架衬套弹性静力学特性分析的有限元方法(2/2)
计算结果和实验结果的比较！！
284
275
291
• 单元特性对橡胶主簧静态特性有限元分析结果的影响
－对垂直刚度的影响
单元类型
单元最 大尺寸
(mm) 结点数
单元数
垂直刚度 (N/mm)
相对误差(%)
8/1 （六面体） 4
9763 16171
305
7
10/4
6 14304 8517
265
1.6
（四面体） 8
8868 5186
265
弹簧手册k rΒιβλιοθήκη 1L(Ea B
G)
ln( )
A
轴向刚度
ka
2GL
B
ln( )
A
• 复杂形状悬架衬套弹性静力学特性分析的有限元方法(1/2)
应力(Mpa) 力 (N)
6
4
实验
Ogden 3
2
0
-2
-4 -0.4 0.0 0.4 0.8 1.2 1.6 2.0
橡胶减振器参数化有限元法优化设计
34橡胶减振器参数化有限元法优化设计黄祖宇焦作市高级技工学校实习工厂(焦作机床厂) (454100)摘 要 为获得与理想减振特性相吻合的橡胶减振器，建立了1/8立体参数化有限元模型。
通过试验确定有限元仿真的材料模型参数并对分析结果进行试验验证。
基于灵敏度分析选择橡胶减振器的外倾角α、半径r 2、高度h 2作为优化变量，以理想载荷-变形特性曲线为优化目标，运用多约束非线性二次规划算法优化得到相应的尺寸值并对橡胶减振器的强度进行了校核。
借助于理想载荷-变形特性曲线并结合有限元法优化得到橡胶减振器的最优外形尺寸，为橡胶元件性能最优化设计提供了新的思路和方法。
关键词 橡胶减振器 参数化建模 有限元 优化设计橡胶减振器是汽车上常见而重要的减振元件，由于橡胶减振器外形结构的复杂性以及它的弹性模量随形状系数而变化的时变特性，使形状不规则的橡胶减振器很难通过解析公式预测其刚度、强度及疲劳寿命，因此早期的橡胶产品开发中大多采用反复试验修正的方法。
20世纪70年代中期以来，随着计算机技术的发展，有限元仿真已成为橡胶件各种性能研究的有力工具：Morman [1]等人用有限元法分析了发动机橡胶减振件的静态和动态特性，Seong Beom Lee [2]借助有限元法预测了汽车橡胶衬套扭转特性，J Pelc [3]研究了充气橡胶轮胎在外力作用下的变形和分层应力特性，Lee [4]用有限元法对发动机橡胶支撑件进行了形状参数优化设计，但未见由理想载荷-变形特性曲线优化得到橡胶减振器外形参数的相关文献。
本文以某工程自卸车橡胶减振器为研究对象，对该减振器进行参数化建模、灵敏度研究以及试验验证，优化得到与理想载荷—变形特性曲线相吻合的工程自卸车橡胶减振器。
1 橡胶大变形数值模拟基本方程橡胶材料在外载作用下的大变形，同时具有材料非线性、几何大变形非线性以及接触非线性的特点，解析求解困难，目前一般借助有限元方法对其研究，其基本求解过程如下：根据虚功原理，外力在虚位移上所作的虚功等于内力在虚应变上所作的虚功，将虚位移原理应用于橡胶元件初始构形的每一个单元： 0()()ee TT e e T e V a B SdV W a F δδ≡=∫(1)式中 B ——应变-位移矩阵S ——克希荷夫应力张量V 0e ——单元初始构形下的体积 F e ——作用在单元上的外力。
有限元、力学专业词汇汇总--中英文对照
有限元、力学专业词汇汇总拉力tensile force 正应力normal stress 切应力shear stress静水压力hydrostatic pressure 集中力concentrated force 分布力distributed force 线性应力应变关系linear relationship between stress andstrain弹性模量modulus of elasticity 横向力lateral force transverse force轴向力axial force 拉应力tensile stress 压应力compressive stress平衡方程equilibrium equation 静力学方程equations of static比例极限proportional limit 应力应变曲线stress-strain curve 拉伸实验tensile test‘屈服应力yield stress 极限应力ultimate stress 轴shaft 梁beam纯剪切pure shear 横截面积cross-sectional area 挠度曲线deflection curve 曲率半径radius of curvature 曲率半径的倒数reciprocal of radius of curvature纵轴longitudinal axis 悬臂梁cantilever beam 简支梁simply supported beam微分方程differential equation 惯性矩moment of inertia 静矩static moment扭矩torque moment 弯矩bending moment弯矩对x的导数 derivative of bending moment with respect to x弯矩对x的二阶导数the second derivative of bending moment with respect to x静定梁statically determinate beam 静不定梁statically indeterminate beam相容方程compatibility equation 补充方程complementary equation中性轴neutral axis 圆截面circular cross section两端作用扭矩twisted by couples at two ends 刚体rigid body 扭转角twist angle 静力等效statically equivalent 相互垂直平面mutually perpendicular planes通过截面形心through the centroid of the cross section 一端铰支pin support at one end 一端固定fixed at one end 弯矩图bending moment diagram剪力图shear force diagram 剪力突变abrupt change in shear force、旋转和平移rotation and translation虎克定律hook’s law边界条件boundary condition 初始位置initial position、力矩面积法moment-area method 绕纵轴转动rotate about a longitudinal axis 横坐标abscissa 扭转刚度torsional rigidity 拉伸刚度tensile rigidity剪应力的合力resultant of shear stress 正应力的大小magnitude of normal stress 脆性破坏brittle fail 对称平面symmetry plane 刚体的平衡equilibrium of rigid body 约束力constraint force 重力gravitational force实际作用力actual force 三维力系three-dimentional force system合力矩resultant moment 标量方程scalar equation、矢量方程vector equation 张量方程tensor equation 汇交力系cocurrent system of forces任意一点an arbitrary point 合矢量resultant vector 反作用力reaction force反作用力偶reaction couple 转动约束restriction against rotation平动约束restriction against translation 运动的趋势tendency of motion绕给定轴转动rotate about a specific axis 沿一个方向运动move in a direction控制方程control equation 共线力collinear forces 平面力系planar force system 一束光a beam of light 未知反力unknown reaction forces 参考框架frame of reference大小和方向magnitude and direction 几何约束geometric restriction刚性连接rigidly connected 运动学关系kinematical relations运动的合成superposition of movement 固定点fixed point平动的叠加superposition of translation 刚体的角速度angular speed of a rigid body 质点动力学particle dynamics 运动微分方程differential equation of motion 工程实际问题practical engineering problems 变化率rate of change 动量守恒conservation of linear momentum 定性的描述qualitative description点线dotted line 划线dashed line 实线solid line 矢量积vector product点积dot product 极惯性矩polar moment of inertia 角速度angular velocity角加速度angular acceleration infinitesimal amount 无穷小量definite integral 定积分a certain interval of time 某一时间段kinetic energy 动能conservative force 保守力damping force 阻尼力coefficient of damping 阻尼系数free vibration 自由振动periodic disturbance 周期性扰动viscous force 粘性力forced vibration 强迫震动general solution 通解particular solution 特解transient solution 瞬态解steady state solution 稳态解second order partial differential equation 二阶偏微分方程external force 外力internal force 内力stress component 应力分量state of stress 应力状态coordinate axes 坐标系conditions of equilibrium 平衡条件body force 体力continuum mechanics 连续介质力学displacement component位移分量additional restrictions 附加约束compatibility conditions 相容条件mathematical formulations 数学公式isotropic material 各向同性材料sufficient small 充分小state of strain 应变状态unit matrix 单位矩阵dilatation strain 膨胀应变the first strain invariant 第一应变不变量deviator stress components 应力偏量分量resonance frequency 谐振频率the first invariant of stress tensor 应力张量的第一不变量bulk modulus 体积模量constitutive relations 本构关系linear elastic material 线弹性材料mathematical derivation 数学推导 a state of static equilibrium 静力平衡状态Newton‘s first law of motion 牛顿第一运动定律directly proportional to 与……成正比stress concentration factor 应力集中系数state of loading 载荷状态st venant’ principle 圣维南原理uniaxial tension 单轴拉伸cylindrical coordinates 柱坐标buckling of columns 柱的屈曲critical value 临界值stable equilibrium 稳态平衡unstable equilibrium condition 不稳定平衡条件critical load 临界载荷a slender column细长杆fixed at the lower end下端固定free at the upper end上端自由critical buckling load 临界屈曲载荷potential energy 势能fixed at both ends 两端固定hinged at both ends 两端铰支tubular member 管型杆件transverse dimention 横向尺寸stability of column 柱的稳定axial force 轴向力elliptical hole 椭圆孔plane stress 平面应力nominal stress 名义应为principal stress directions 主应力方向axial compression 轴向压缩dynamic loading 动载荷dynamic problem 动力学问题inertia force 惯性力resonance vibration 谐振static states of stress 静态应力dynamic response 动力响应time of contact 接触时间length of wave 波长。
橡胶工业中的有限元计算问题
中文摘要摘要作为一种工程材料硫化橡胶早在19世纪就被广泛的应用。
由于它良好弹性的特性被用于承载结构轴承，密封圈，吸收震动的衬垫，连接器，轮胎等。
然而，不同于金属材料仅需要几个参数描述其材料特性，橡胶的行为复杂，材料本构关系是非线性的。
它的力学行为对温度，环境，应变历史，加载的速率都非常敏感，这样使得描述橡胶的行为变得更为复杂。
橡胶的制造工艺和成分也对橡胶力学性能有显著的影响。
这也意味着橡胶作为工程材料的研究是一段不断的尝试和改进的过程，而不是完全彻底的理解。
幸运的是，由于计算机以及有限元分析的飞速发展，我们可以借助计算机来对超弹性材料工程应用进行深入研究以及优化设计。
本文给出如何用有限元方法来分析工业中的橡胶元件的力学性能的完整的方法，包括选取橡胶的本构模型，拟合本构模型，有限元建模，处理计算结果。
有限元分析的精度是直接与输入的材料数据相关的。
理想情况下，数据应该来自一系列的独立的实验。
本文给出了常用的用于拟合橡胶本构关系的实验方案。
另外本文详细讨论了一种橡胶元件中常用的超弹性材料轴对称过盈配合问题。
分别用解析的方法和有限元计算方法详细研究了此问题，包括平面应变大变形和小变形的解析解，有限元解，平面应力的小变形理论解，平面应力情况大变形和小变形的有限元解，橡胶体积模量对过盈配合的影响，接触面的摩擦系数对过盈配合的影响。
关键词：橡胶过盈配合超弹性大变形- I -目录摘要 (II)Abstract（英文摘要） (III)目录 (V)第一章超弹性材料本构关系 (1)引言： (1)1.1 超弹性模型概况 (1)1.2 橡胶模型的特征 (3)1.3 常用的橡胶本构模型介绍 (3)1.3.1 多项式形式及其特殊情况 (3)1.3.1.1 Mooney-Rivlin模型和Neo-Hookean模型 (4)1.3.1.2 Yeoh形式（Yeoh, 1993） (5)1.3.2 Ogden形式 (6)1.3.3 Arruda-Boyce形式 (6)1.3.4 Van der Waals模型 (7)1.4 本文的主要内容 (8)第二章超弹性材料过盈配合的解析解和数值解 (10)引言： (10)2.1 橡胶大变形和小变形本构关系 (11)2.1.1 大变形 (11)2.1.2 小变形 (12)2.2 平面应变情况下的解析解和有限元解 (14)2.2.1解析解 (14)2.2.1.1 线弹性小变形解析解 (14)2.2.1.2 大变形超弹性本构关系解析解 (15)2.2.1.3 线弹性与超弹性解析解的比较 (17)- II -2.2.2解析解与ABAQUS数值解的比较 (20)2.3 平面应力情况下解析解和有限元解 (22)2.3.1 解析解（小变形线弹性） (22)2.3.2有限元解 (23)2.3.2.1解析解与有限元解（线弹性橡胶本构关系）的比较 (23)2.3.2.2 两种本构关系的有限元解的比较（线弹性和超弹性） (25)2.4 可压缩性对过盈配合的影响 (26)2.5 摩擦系数对过盈配合的影响 (27)2.5.1 ABAQUS中接触的定义 (28)2.5.2 ABAQUS模拟过盈配合 (28)2.6 本章总结 (32)第三章实验拟和超弹性本构模型系数 (33)引言： (33)3.1 超弹性材料试验简介 (33)3.1.1 多种应变状态测试 (34)3.2 超弹性材料基本试验 (35)3.2.1单轴拉伸实验 (35)3.2.2 纯剪（平面拉伸）实验 (36)3.2.3等轴拉伸实验 (37)3.2.4压缩实验 (38)3.2.5体积压缩实验 (39)3.3 弹性本构模型中的系数 (39)3.3.1 最小二乘法用于多项式形式 (40)3.3.2 非线性最小二乘法 (40)3.3.3 非线性最小二乘法用于Ogden模型 (41)第四章橡胶定位器的有限元计算 (43)4.1 定位器建模 (43)4.1.1数值方法的选择 (44)4.1.2 有限元建模 (44)- III-4.2 静力学分析 (45)4.2.1 垂向刚度 (45)4.2.2 横纵向刚度 (46)4.2.3 静态分析结果对比 (48)4.3 动态分析 (49)4.3.1 模态分析基本方程 (49)4.3.2 定位器振型有限元分析结果 (49)4.4 本章总结 (52)第五章球铰的有限元计算 (53)5.1 球铰建模 (53)5.1.1数值方法的选择 (54)5.2 静力学分析 (54)5.2.1有限元计算扭转刚度 (55)5.2.2 偏转刚度 (56)5.2.3 有限元计算与实验的比较 (58)5.3 本章总结 (59)第六章结论 (60)参考文献 (61)致谢 (62)附录A 纯剪实验方法 (63)附录B 体积模量实验方法 (65)个人简历和在学期间的研究成果及发表的学术论文 (67)- IV -目录第一章超弹性材料本构关系引言作为一种工程材料硫化橡胶早在19世纪就被广泛的应用。
橡胶有限元分析之研究_魏泳涛
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收稿日期 1996 - 09 - 17
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有限元技术在橡胶悬置正向开发中的应用
有限元技术在橡胶悬置正向开发中的应用康一坡;霍福祥;魏德永;叶绍仲【摘要】通过与经验公式法对比，总结有限元技术辅助进行橡胶悬置正向开发的优势；以Abaqus有限元软件为基础，阐述如何应用有限元技术进行橡胶悬置正向开发流程；依据流程，介绍了两例橡胶悬置正向开发案例。
结果表明，有限元技术辅助设计的橡胶悬置不仅达到了刚度目标要求，而且强度、寿命大幅提高，同时产品开发周期有效缩短。
%Comparing with empirical formula, the advantages of FEM aiding forward design of rubber mounting is summarized in the paper. And based on the FEM software Abaqus, the process is presented in details on how to conduct the rubber mounting forward design. Then, the two cases of the forward design of rubber mounting are introduced. The results indicate that the designed rubber mounting using FEM can not only meet the stiffness requirement, but also improve the strength and fatigue life significantly, and the development cycle is shortened greatly.【期刊名称】《汽车技术》【年(卷),期】2014(000)006【总页数】6页(P6-10,47)【关键词】橡胶悬置;有限元;刚度;正向开发【作者】康一坡;霍福祥;魏德永;叶绍仲【作者单位】中国第一汽车股份有限公司技术中心汽车振动噪声与安全控制综合技术国家重点实验室;中国第一汽车股份有限公司技术中心汽车振动噪声与安全控制综合技术国家重点实验室;中国第一汽车股份有限公司技术中心汽车振动噪声与安全控制综合技术国家重点实验室;中国第一汽车股份有限公司技术中心汽车振动噪声与安全控制综合技术国家重点实验室【正文语种】中文【中图分类】U463.21 前言有限元技术不仅可以分析评价橡胶悬置结构是否满足设计要求，而且还能够根据具体要求设计出满足一定条件的橡胶结构，即橡胶悬置结构的正向开发。
有限元常用命令英文汉译
有限元常用英文汉译File（文件）New----新建Save----保存Close----关闭Open----打开Save as----另存为（当前模型也更名为）Save to----另存为（当前模型不更名为）Browse----浏览Import/Export----输入/输出（支持MSC-Nastran, .dat .nas文件,AutoCAD DXF, .dxf文件, ACIS, .sat文件, Stero-listography, .stl文件, Bitmap，可直接编辑文本文件.txt）Save sub model----存储子模型Make beam section----建立梁截面特性文件Make library----建立材料特性库或截面特性库Print setup----打印设置Preference----预置：Scratch files----临时文件路径Clean----清理临时文件Property libraries----特性库Default units----缺省单位制Auto save (minutes)----自动存储时间间隔Backup files on open----后备文件打开名Inspector font----实体检查字体Show full group----显示整个组Toggle select: Additive/Exclusive----锁定选择：复选/单选Exit----退出Edit （编辑）:Node----节点Element----单元Geometry----几何Select----选择：select all----全选，by region----按区域选，by property----按特性选by group----按组选clear all select----清除全部选择Cut----剪切Copy----复制Paste----粘贴Delete----删除Find----查找Undo----取消Redo----重做Online editor----在线编辑View:（显示）Redraw----重画Dynamic----动态显示Fresh----刷新Clear----清理Zoom----缩放：In----将指定区域图象放大到整个屏幕；Out----将整个图象缩至指定区域内；Last----上一次Draw----画Pan----平移Scale----按比例显示Angles----视角MultiView----多个模型同时显示Snap grid----捕捉栅格Show/Hide----显示/隐藏Display----显示选择node----节点, beam----梁, plate-----板, brick----块, links----连接, node attributes----节点属性, element attributes----单元属性,vertices----角节点, geometry----几何Show by property----按特性显示Beam free ends----梁自由端Plate free edges----板自由边Brick free edges----块自由面Element display----单元显示Attribute display----属性显示Options----选项Axes & screen Tab----关于轴和屏幕,Dynamic Rotation Tab----动态显示,Drawing Tab----绘图,Numbers Tab----数字显示,Free Edge Tab----自由边显示,Selecting Tab----关于选择Toolbars----工具条Entity toggles----实体锁定Summary:（摘要）Information----模型信息Total----总数,Units----单位,Load cases----载荷工况,Freedom cases----自由度工况,Properties----材料特性,Table----表,Comments----评述Property----特性信息Beam 梁，Plate 板，Brick 块Model----模型Bill of material 材料清单（具有这种材料的单元实体等数量等情况）Mass distribution质量分布(质心)Local and Global Mass Moments of Inertia----局部坐标系下\总体坐标系下的惯性矩等截面特性。
有限元分析软件Marc及在橡胶材料分析中的应用
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有 限 元 分 析 软 件 Ｍａ ｃ及 在 ｒ 橡 胶 材 料 分 析 中 的应 用
张 宝 生 ， 力 培 ， 灿 丰 ， 嘉 庆 蒋 周 陈
（ 京 石 油 化 工 学 院机 械 工程 系 ， 京 １ ２ ０ ） 北 北 ０ ６０
4结束语橡胶材?的分析由于涉及材?的大变形?可压缩性?稳定性以及多解情况的发生因而是比较困难的采用marc这种优秀的非线性分析软件将为橡胶材?的?为研究橡胶元件的设计提供可靠有效的手段但是由于计算时设置参数较多?设置?合?还得?到计算建立了帘线分布角度密度轮胎的下沉位移与轮胎内部的应?分布关系为提高轮胎的设计质?提供了依据成晓明陈兆静蔡显新
有限元分析英文文献
The Basics of FEA Procedure有限元分析程序的基本知识2.1IntroductionThis chapter discusses the spring element,especially for the purpose of introducing various concepts involved in use of the FEA technique.本章讨论了弹簧元件，特别是用于引入使用的有限元分析技术的各种概念的目的A spring element is not very useful in the analysis of real engineering structures;however,it represents a structure in an ideal form for an FEA analysis.Spring element doesn’t require discretization(division into smaller elements)and follows the basic equation F=ku.在分析实际工程结构时弹簧元件不是很有用的;然而,它代表了一个有限元分析结构在一个理想的形式分析。
弹簧元件不需要离散化(分裂成更小的元素)只遵循的基本方程F=ku We will use it solely for the purpose of developing an understanding of FEA concepts and procedure.我们将使用它的目的仅仅是为了对开发有限元分析的概念和过程的理解。
2.2Overview概述Finite Element Analysis(FEA),also known as finite element method(FEM)is based on the concept that a structure can be simulated by the mechanical behavior of a spring in which the applied force is proportional to the displacement of the spring and the relationship F=ku is satisfied.有限元分析(FEA),也称为有限元法(FEM)，是基于一个结构可以由一个弹簧的力学行为模拟的应用力弹簧的位移成正比,F=ku切合的关系。
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Fig stiffness curve
Fig mullins effects curve
有限元技术创新性
Improvement of Analysis Precision: during the past 3 years, the analysis precision of FEA has improved dramatically, which help us short our development period .
有限元技术创新性 Analysis objective ： including pure rubber parts, elastic component，and even highly elastic system. The analysis is serving our daily development.
Understand mesh arrangement technology According to deformation trend characteristic of rubber component
Grasp establishment technology of hexahedral element’s by Hypermesh software
Multiple Strain
Following latest technology
succefully apply Manual and automatic remesh technology to rubber analysis！ Good meshing arrangement and element quality!
Fig strain contours
Fig composite stiffness
Fig stability predict
有限元技术平台
Rubber Meshing Technology: Hexahedral Element is Essential to Rubber FEA for Good Result
Rubber Part
Elastic Component
Elastic System
有限元技术创新性
Material constitutive type ： including elastic mental material, non-linear rubber material, engineering plastic, compressive foam material
பைடு நூலகம்
Building good rubber material constitutive model for large deformation！
Resolve by explicited technology on rubber simulation！
Fig nonlinear stiffness curve
ABAQUS
有限元技术平台
Material Database：Including Rubber Material, Metal Material and Patent Technology of Rubber Basic Test
Fig1 Simple Tension
Multiple Strain
The suitable pre-compression of rubber bush：the suitbale precompression is determined by the FEA and bush fatigue test。
Scientific design the suitable precompression by compare the radial stress, axial stress and mises stress of different pre-compression
Multiple Strain The area with high compression strain will have equal high tensile strain.
Fig radial stress of pre-compression
Fig Strain contour of rubber
Foam mateial Engineering plastic Rubber material Metal Material
有限元技术创新性
Material constitutive type ： including elastic mental material, non-linear rubber material, engineering plastic, compressive foam material and cord material
有限元技术探讨
Content of Rubber FEA
Introduction to FEA Platform Innovation of FEA Fatigue Prediction New FEA Development
有限元技术平台 Professional FEA Platform：ABAQUS, FE-SAFE and Hypermesh Software System
Fig simulating large deformation of rubber component
有限元技术创新性
Rubber Constitutive Model：Hyperelastic Model with Mullins Effect Model are Adopted in our Rubber Model. Good Fitting Result can Obtain on Condition of Multi-axis Pre-loaded
Scientific design the suitable precompression by compare the radial stress, axial stress and mises stress of different pre-compression The area with high compression Multiple Strain strain will have equal high tensile strain.
Nonlinear analysis
Stability prediction
Fatigue prediction
Design optimization
有限元技术与疲劳预测 Fatigue analysis of mental：successfully predict the fatigue life of metal parts by masting fatigue prediction technology.
We are keeping on：
Building up a FEA team with highly execution and research capability.
Establishing rubber material database for FEA！
Gaining good experience by developing basic research！ Closely combining analysis, test and design！ Learn advanced technology from abroad！
Fig2 Planar Tension
Multiple Strain Test
Fig3 Biaxial Extension
有限元技术平台
FEA Platform：Linear / nonlinear Stiffness, Strength ,Stability Fatigue Life Prediction
Load condition include heating vulcanization,precompression and radial fatigue load Quickly obtain comparable fatigue result
Multiple Strain
有限元技术与疲劳预测
Fig Mises stress of pre-compression
有限元技术与疲劳预测
The suitable pre-compression of rubber bush：the suitbale precompression is determined by the FEA and bush fatigue test。
Fig radial stress of pre-compression
Fig axial stress of pre-compression
Fig Strain contour of rubber
有限元技术与疲劳预测
Fatigue design guidelines ： with mass analysis and fatigue test ,we have sum up the basic design information for rubber material fatigue life。
空气弹簧系统
有限元技术创新性
Analysis Field: including linear stress analysis, nonlinear stiffness calculation, stability simulation, fatigue prediction, design optimization