Differential Equations in the New Millennium the Parachute Problem

有趣的英文谚语有趣的英文谚语篇一1、 Action speak louder than words.行动胜于言语！2、 Adversity reveals genius； fortune conceals it．苦难显才华，好运隐天资。

3、 All things are difficult before they are easy.凡事必先难后易。

4、 Believe in yourself.相信你自己！5、 Great hopes make great man.伟大的理想造就伟大的人。

6、 I can because i think i can.我行，因为我相信我行！7、While there is life， there is hope.有生命就有希望/留得青山在，不怕没柴烧。

8、 Keep on going never give up.勇往直前，决不放弃！9、 Knowledge is power.知识就是力量。

10、 Knowledge makes humble， ignorance makes proud.博学使人谦逊，无知使人骄傲。

11、 Learn and live.活着，为了学习。

12、 Never put off what you can do today until tomorrow.今日事今日毕！13、 Never say die.永不气馁！14、 Nothing is impossible for a willing heart.心之所愿，无所不成。

经典爱情语句。

15、 Whatever is worth doing is worth doing well.任何值得做的事就值得把它做好！有趣的英文谚语篇二1. Nothing is impossible if you try your best. 只要努力一切皆有可能。

2. Pride goes before a fall. 骄必败。

四元数分析中的幂函数与二项式定理鄢盛勇【摘要】研究了四元数分析中幂函数在算子D-q,Dq作用下的表示式, 得到了四元数分析中的二项式定理.【期刊名称】《成都师范学院学报》【年(卷),期】2017(033)007【总页数】5页(P120-124)【关键词】四元数分析;二项式定理;幂函数【作者】鄢盛勇【作者单位】成都师范学院数学系,成都 611130【正文语种】中文【中图分类】O175.27四元数分析是复分析在高维上最自然的推广, 是近代分析的重要分支, 它有非常重要的理论意义和应用价值, 如在Maxwell方程,Yang-Mill场理论以及量子力学等方面都应用它的一些结论。

近年来许多学者研究了四元数分析中的正则函数以及一些奇异积分算子, 并考虑了一些边值问题。

由于四元数代数的非交换性, 使得乘法运算, 特别是涉及到幂函数的运算变得复杂, 本文通过引入一些新记号, 考虑了四元数代数中二项式定理, 简化涉及到幂函数的运算。

所谓四元数分析, 是指在四元数代数H上建立的分析理论。

而H是一个四维的实向量空间, 以为基元, 且满足。

其中乘法运算满足结合律但不满足交换律。

从而H={q=x1+ix2+jx3+kx4|x1,x2,x3,x4∈R}。

H也可以看作二维的复向量空间, 从而也记为H={|z1,z2∈C},z1=x1+ix2,z2=x3+ix4。

在这种意义下也用C2表示四元数空间。

四元数的矩阵表示[4]为, 计算易得为了方便我们引入下列记号: 多重指标;当时, 记. 并且当都是奇数时, 称是奇的, 当都是偶数时, 称是偶的;下文中我们有时也把多重指标等同于四元数。

四元数函数是由H到H的函数, 定义微分算子:,其共轭算子分别为易计算得, 是四维空间中的Laplace算子。

若则称是(左)正则函数。

关于四元数分析以及正则函数详细内容见参考文献[4], 并且须要说明的是我们考虑的算子和文献[3,5]中的是不一样的, 这样做的目的是方便使用复分析方法和建立与多复变数之间的联系。

复试机械专业英语词汇必知金属切削metal cutting 机床machine tool金属工艺学technology of metals刀具cutter电路circuit半导体元件semiconductor element反馈feedback发生器generator直流电源DC electrical source门电路gate circuit逻辑代数logic algebra逻辑电路logic circuit触发器flip-flop脉冲波形pulse shape数模digital analogy液压传动机构fluid drive mechanism 机械零件mechanical parts摩擦friction联结link传动drive/transmission轴shaft弹性elasticity频率特性frequency characteristic误差error响应response定位allocation 机床夹具jig动力学dynamic运动学kinematic静力学static分析力学analyse mechanics拉伸pulling压缩hitting剪切shear扭转twist弯曲应力bending stress强度intensity三相交流电three-phase AC磁路magnetic circles 变压器transformer异步电动机asynchronous motor几何形状geometrical精度precision正弦形的sinusoid交流电路AC circuit机械加工余量machining allowance变形力deforming force变形deformation应力stress硬度rigidity热处理heat treatment退火anneal正火normalizing脱碳decarburization渗碳carburization淬火冷却quench淬火hardening回火tempering调质hardening and tempering磨粒abrasive grain结合剂bonding agent砂轮grinding wheel外圆磨削external grinding内圆磨削internal grinding平面磨削plane grinding变速箱gearbox离合器clutch绞孔fraising绞刀reamer螺纹加工thread processing螺钉screw铣削mill铣刀milling cutter功率power工件workpiece齿轮加工gear mechining齿轮gear主运动main movement主运动方向direction of main movement进给方向direction of feed进给运动feed movement合成进给运动resultant movement of feed合成切削运动resultant movement of cutting合成切削运动方向direction of resultant movement of cutting切削深度cutting depth前刀面rake face刀尖nose of tool前角rake angle后角clearance angle龙门刨削planing主轴spindle主轴箱headstock卡盘chuck加工中心machining center车刀lathe tool车床lathe钻削镗削bore车削turning磨床grinder基准benchmark钳工locksmith锻forge压模stamping焊weld拉床broaching machine拉孔broaching装配assembling铸造found流体动力学fluid dynamics流体力学fluid mechanics加工machining液压hydraulic pressure切线tangent机电一体化mechanotronics mechanical-electrical integration气压air pressure pneumatic pressure稳定性stability 介质medium液压驱动泵fluid clutch液压泵hydraulic pump阀门valve失效invalidation强度intensity载荷load应力stress安全系数safty factor可靠性reliability螺纹thread螺旋helix键spline销pin滚动轴承rolling bearing滑动轴承sliding bearing弹簧spring 制动器arrester brake十字结联轴节crosshead联轴器coupling链chain皮带strap精加工finish machining粗加工rough machining变速箱体gearbox casing腐蚀rust氧化oxidation磨损wear耐用度durability随机信号random signal离散信号discrete signal超声传感器ultrasonic sensor集成电路integrate circuit 挡板orifice plate金属切削metal cutting机床machine tool 金属工艺学technology of metals刀具cutter摩擦friction联结link传动drive/transmission轴shaft弹性elasticity频率特性frequency characteristic误差error响应response定位allocation机床夹具jig动力学dynamic运动学kinematic 静力学static分析力学analyse mechanics 拉伸pulling压缩hitting剪切shear扭转twist弯曲应力bending stress强度intensity三相交流电three-phase AC磁路magnetic circles变压器transformer异步电动机asynchronous motor几何形状geometrical精度precision正弦形的sinusoid交流电路AC circuit 机械加工余量machining allowance 变形力deforming force变形deformation应力stress 硬度rigidity热处理heat treatment退火anneal正火normalizing 脱碳decarburization渗碳carburization 电路circuit半导体元件semiconductor element反馈feedback发生器generator 直流电源DC electrical source门电路gate circuit 逻辑代数logic algebra外圆磨削external grinding内圆磨削internal grinding面磨削plane grinding变速箱gearbox离合器clutch 绞孔fraising绞刀reamer 螺纹加工thread processing螺钉screw 铣削mill铣刀milling cutter 功率power工件workpiece 齿轮加工gear mechining齿轮gear主运动main movement主运动方向direction of main movement 进给方向direction of feed 进给运动feed movement 合成进给运动resultant movement of feed 合成切削运动resultant movement of cutting合成切削运动方向direction of resultant movement of cutting切削深度cutting depth前刀面rake face刀尖nose of tool前角rake angle 后角clearance angle龙门刨削planing 主轴spindle主轴箱headstock 卡盘chuck加工中心machining center 车刀lathe tool车床lathe 钻削镗削bore车削turning 磨床grinder基准benchmark 钳工locksmith锻forge 压模stamping焊weld 拉床broaching machine拉孔broaching 装配assembling铸造found 流体动力学fluid dynamics流体力学fluid mechanics 加工machining液压hydraulic pressure 切线tangent气压air pressure pneumatic pressure机电一体化mechanotronics mechanical-electrical integration稳定性stability介质medium 液压驱动泵fluid clutch液压泵hydraulic pump 阀门valve失效invalidation 强度intensity载荷load 应力stress安全系数safty factor 可靠性reliability螺纹thread 螺旋helix键spline 销pin滚动轴承rolling bearing 滑动轴承sliding bearing弹簧spring 制动器arrester brake十字结联轴节crosshead 联轴器coupling链chain 皮带strap精加工finish machining 粗加工rough machining变速箱体gearbox casing 腐蚀rust氧化oxidation 磨损wear耐用度durability 随机信号random signal离散信号discrete signal 超声传感器ultrasonic sensor集成电路integrate circuit 挡板orifice plate残余应力residual stress 套筒sleeve扭力torsion 冷加工cold machining电动机electromotor 汽缸cylinder过盈配合interference fit 热加工hotwork摄像头CCD camera 倒角rounding chamfer优化设计optimal design 工业造型设计industrial moulding design 有限元finite element 滚齿hobbing插齿gear shaping 伺服电机actuating motor铣床milling machine 钻床drill machine镗床boring machine 步进电机stepper motor丝杠screw rod 导轨lead rail组件subassembly可编程序逻辑控制器Programmable Logic Controller PLC电火花加工electric spark machining电火花线切割加工electrical discharge wire - cutting相图phase diagram 热处理heat treatment固态相变solid state phase changes 有色金属nonferrous metal陶瓷ceramics 合成纤维synthetic fibre电化学腐蚀electrochemical corrosion 车架automotive chassis悬架suspension 转向器redirector变速器speed changer 板料冲压sheet metal parts孔加工spot facing machining 车间workshop工程技术人员engineer 气动夹紧pneuma lock数学模型mathematical model 画法几何descriptive geometry机械制图Mechanical drawing 投影projection视图view 剖视图profile chart标准件standard component 零件图part drawing装配图assembly drawing 尺寸标注size marking技术要求technical requirements 刚度rigidity内力internal force 位移displacement截面section 疲劳极限fatigue limit断裂fracture 塑性变形plastic distortion脆性材料brittleness material 刚度准则rigidity criterion垫圈washer 垫片spacer直齿圆柱齿轮straight toothed spur gear 斜齿圆柱齿轮helical-spur gear 直齿锥齿轮straight bevel gear 运动简图kinematic sketch齿轮齿条pinion and rack 蜗杆蜗轮worm and worm gear虚约束passive constraint 曲柄crank摇杆racker 凸轮cams共轭曲线conjugate curve 范成法generation method定义域definitional domain 值域range导数\\微分differential coefficient 求导derivation定积分definite integral 不定积分indefinite integral曲率curvature 偏微分partial differential毛坯rough 游标卡尺slide caliper千分尺micrometer calipers 攻丝tap二阶行列式second order determinant 逆矩阵inverse matrix线性方程组linear equations 概率probability随机变量random variable 排列组合permutation and combination气体状态方程equation of state of gas 动能kinetic energy势能potential energy 机械能守恒conservation of mechanical energyn 动量momentum 桁架truss轴线axes 余子式cofactor逻辑电路logic circuit 触发器flip-flop脉冲波形pulse shape 数模digital analogy液压传动机构fluid drive mechanism 机械零件mechanical parts淬火冷却quench 淬火hardening回火tempering 调质hardening and tempering磨粒abrasive grain 结合剂bonding agent砂轮grinding wheel阿基米德蜗杆Archimedes worm 安全系数safety factor; factor of safety安全载荷safe load 凹面、凹度concavity扳手wrench 板簧flat leaf spring半圆键woodruff key 变形deformation摆杆oscillating bar 摆动从动件oscillating follower摆动从动件凸轮机构cam with oscillating follower 摆动导杆机构oscillating guide-bar mechanism摆线齿轮cycloidal gear 摆线齿形cycloidal tooth profile摆线运动规律cycloidal motion 摆线针轮cycloidal-pin wheel包角angle of contact 保持架cage背对背安装back-to-back arrangement 背锥back cone ；normal cone背锥角back angle 背锥距back cone distance比例尺scale 比热容specific heat capacity闭式链closed kinematic chain 闭链机构closed chain mechanism臂部arm 变频器frequency converters变频调速frequency control of motor speed 变速speed change变速齿轮change gear ; change wheel 变位齿轮modified gear变位系数modification coefficient 标准齿轮standard gear标准直齿轮standard spur gear 表面质量系数superficial mass factor表面传热系数surface coefficient of heat transfer 表面粗糙度surface roughness并联式组合combination in parallel 并联机构parallel mechanism并联组合机构parallel combined mechanism 并行工程concurrent engineering并行设计concurred design, CD 不平衡相位phase angle of unbalance不平衡imbalance (or unbalance) 不平衡量amount of unbalance不完全齿轮机构intermittent gearing 波发生器wave generator波数number of waves 补偿compensation参数化设计parameterization design, PD 残余应力residual stress操纵及控制装置operation control device 槽轮Geneva wheel槽轮机构Geneva mechanism ；Maltese cross 槽数Geneva numerate槽凸轮groove cam 侧隙backlash差动轮系differential gear train 差动螺旋机构differential screw mechanism差速器differential 常用机构conventional mechanism; mechanism in common use 车床lathe 承载量系数bearing capacity factor承载能力bearing capacity 成对安装paired mounting尺寸系列dimension series 齿槽tooth space齿槽宽spacewidth 齿侧间隙backlash齿顶高addendum 齿顶圆addendum circle齿根高dedendum 齿根圆dedendum circle齿厚tooth thickness 齿距circular pitch齿宽face width 齿廓tooth profile齿廓曲线tooth curve 齿轮gear齿轮变速箱speed-changing gear boxes 齿轮齿条机构pinion and rack齿轮插刀pinion cutter; pinion-shaped shaper cutter 齿轮滚刀hob ,hobbing cutter 齿轮机构gear 齿轮轮坯blank齿轮传动系pinion unit 齿轮联轴器gear coupling齿条传动rack gear 齿数tooth number齿数比gear ratio 齿条rack齿条插刀rack cutter; rack-shaped shaper cutter 齿形链、无声链silent chain齿形系数form factor 齿式棘轮机构tooth ratchet mechanism插齿机gear shaper 重合点coincident points重合度contact ratio 冲床punch传动比transmission ratio, speed ratio 传动装置gearing; transmission gear传动系统driven system 传动角transmission angle传动轴transmission shaft 串联式组合combination in series串联式组合机构series combined mechanism 串级调速cascade speed control创新innovation ; creation 创新设计creation design垂直载荷、法向载荷normal load 唇形橡胶密封lip rubber seal磁流体轴承magnetic fluid bearing 从动带轮driven pulley从动件driven link, follower 从动件平底宽度width of flat-face从动件停歇follower dwell 从动件运动规律follower motion从动轮driven gear 粗线bold line粗牙螺纹coarse thread 大齿轮gear wheel打包机packer 打滑slipping带传动belt driving 带轮belt pulley带式制动器band brake 单列轴承single row bearing单向推力轴承single-direction thrust bearing 单万向联轴节single universal joint单位矢量unit vector 当量齿轮equivalent spur gear; virtual gear当量齿数equivalent teeth number; virtual number of teeth当量摩擦系数equivalent coefficient of friction当量载荷equivalent load 刀具cutter导数derivative 倒角chamfer导热性conduction of heat 导程lead导程角lead angle 等加等减速运动规律parabolic motion; constant acceleration and deceleration motion等速运动规律uniform motion; constant velocity motion 等径凸轮conjugate yoke radial cam等宽凸轮constant-breadth cam 等效构件equivalent link等效力equivalent force 等效力矩equivalent moment of force等效量equivalent 等效质量equivalent mass等效转动惯量equivalent moment of inertia 等效动力学模型dynamically equivalent model底座chassis 低副lower pair点划线chain dotted line （疲劳）点蚀pitting垫圈gasket 垫片密封gasket seal碟形弹簧belleville spring 动力学dynamics顶隙bottom clearance 定轴轮系ordinary gear train; gear train with fixed axes动密封kinematical seal 动能dynamic energy动力粘度dynamic viscosity 动力润滑dynamic lubrication动平衡dynamic balance 动平衡机dynamic balancing machine动态特性dynamic characteristics 动态分析设计dynamic analysis design动压力dynamic reaction 动载荷dynamic load端面transverse plane 端面参数transverse parameters端面齿距transverse circular pitch 端面齿廓transverse tooth profile端面重合度transverse contact ratio 端面模数transverse module端面压力角transverse pressure angle 锻造forge对称循环应力symmetry circulating stress 对心滚子从动件radial (or in-line ) roller follower对心直动从动件radial (or in-line ) translating follower对心移动从动件radial reciprocating follower对心曲柄滑块机构in-line slider-crank (or crank-slider) mechanism多列轴承multi-row bearing多楔带poly V-belt 多项式运动规律polynomial motion多质量转子rotor with several masses 惰轮idle gear额定寿命rating life 额定载荷load ratingII 级杆组dyad 发生线generating line发生面generating plane 法面normal plane法面参数normal parameters 法面齿距normal circular pitch法面模数normal module 法面压力角normal pressure angle法向齿距normal pitch 法向齿廓normal tooth profile法向直廓蜗杆straight sided normal worm 法向力normal force反馈式组合feedback combining 反向运动学inverse ( or backward) kinematics反转法kinematic inversion 反正切Arctan范成法generating cutting 仿形法form cutting方案设计、概念设计concept design, CD 防振装置shockproof device飞轮flywheel 飞轮矩moment of flywheel非标准齿轮nonstandard gear 非接触式密封non-contact seal非周期性速度波动aperiodic speed fluctuation 非圆齿轮non-circular gear粉末合金powder metallurgy 分度线reference line; standard pitch line分度圆reference circle; standard (cutting) pitch circle分度圆柱导程角lead angle at reference cylinder分度圆柱螺旋角helix angle at reference cylinder 分母denominator分子numerator 分度圆锥reference cone; standard pitch cone分析法analytical method 封闭差动轮系planetary differential复合铰链compound hinge 复合式组合compound combining复合轮系compound (or combined) gear train 复合平带compound flat belt复合应力combined stress 复式螺旋机构Compound screw mechanism复杂机构complex mechanism 杆组Assur group干涉interference 刚度系数stiffness coefficient刚轮rigid circular spline 钢丝软轴wire soft shaft刚体导引机构body guidance mechanism 刚性冲击rigid impulse (shock)刚性转子rigid rotor 刚性轴承rigid bearing刚性联轴器rigid coupling 高度系列height series高速带high speed belt 高副higher pair格拉晓夫定理Grashoff`s law 根切undercutting公称直径nominal diameter 高度系列height series功work 工况系数application factor工艺设计technological design 工作循环图working cycle diagram工作机构operation mechanism 工作载荷external loads工作空间working space 工作应力working stress工作阻力effective resistance 工作阻力矩effective resistance moment公法线common normal line 公共约束general constraint公制齿轮metric gears 功率power功能分析设计function analyses design 共轭齿廓conjugate profiles共轭凸轮conjugate cam 构件link鼓风机blower 固定构件fixed link; frame固体润滑剂solid lubricant 关节型操作器jointed manipulator惯性力inertia force 惯性力矩moment of inertia ,shaking moment惯性力平衡balance of shaking force 惯性力完全平衡full balance of shaking force 惯性力部分平衡partial balance of shaking force 惯性主矩resultant moment of inertia惯性主失resultant vector of inertia 冠轮crown gear广义机构generation mechanism 广义坐标generalized coordinate轨迹生成path generation 轨迹发生器path generator滚刀hob 滚道raceway滚动体rolling element 滚动轴承rolling bearing滚动轴承代号rolling bearing identification code 滚针needle roller滚针轴承needle roller bearing 滚子roller滚子轴承roller bearing 滚子半径radius of roller滚子从动件roller follower 滚子链roller chain滚子链联轴器double roller chain coupling 滚珠丝杆ball screw滚柱式单向超越离合器roller clutch 过度切割undercutting函数发生器function generator 函数生成function generation含油轴承oil bearing 耗油量oil consumption耗油量系数oil consumption factor 赫兹公式H. Hertz equation合成弯矩resultant bending moment 合力resultant force合力矩resultant moment of force 黑箱black box横坐标abscissa 互换性齿轮interchangeable gears花键spline 滑键、导键feather key滑动轴承sliding bearing 滑动率sliding ratio滑块slider 环面蜗杆toroid helicoids worm环形弹簧annular spring 缓冲装置shocks; shock-absorber灰铸铁grey cast iron 回程return回转体平衡balance of rotors 混合轮系compound gear train积分integrate 机电一体化系统设计mechanical-electrical integration system design 机构mechanism 机构分析analysis of mechanism机构平衡balance of mechanism 机构学mechanism机构运动设计kinematic design of mechanism 机构运动简图kinematic sketch of mechanism机构综合synthesis of mechanism 机构组成constitution of mechanism机架frame, fixed link 机架变换kinematic inversion机器machine 机器人robot机器人操作器manipulator 机器人学robotics技术过程technique process 技术经济评价technical and economic evaluation技术系统technique system 机械machinery机械创新设计mechanical creation design, MCD 机械系统设计mechanical system design, MSD机械动力分析dynamic analysis of machinery 机械动力设计dynamic design of machinery机械动力学dynamics of machinery 机械的现代设计modern machine design机械系统mechanical system 机械利益mechanical advantage机械平衡balance of machinery 机械手manipulator机械设计machine design; mechanical design 机械特性mechanical behavior机械调速mechanical speed governors 机械效率mechanical efficiency机械原理theory of machines and mechanisms 机械运转不均匀系数coefficient of speed fluctuation机械无级变速mechanical stepless speed changes 基础机构fundamental mechanism基本额定寿命basic rating life 基于实例设计case-based design,CBD基圆base circle 基圆半径radius of base circle基圆齿距base pitch 基圆压力角pressure angle of base circle基圆柱base cylinder 基圆锥base cone急回机构quick-return mechanism 急回特性quick-return characteristics急回系数advance-to return-time ratio 急回运动quick-return motion棘轮ratchet 棘轮机构ratchet mechanism棘爪pawl 极限位置extreme (or limiting) position极位夹角crank angle between extreme (or limiting) positions计算机辅助设计computer aided design, CAD计算机辅助制造computer aided manufacturing, CAM计算机集成制造系统computer integrated manufacturing system, CIMS计算力矩factored moment; calculation moment 计算弯矩calculated bending moment加权系数weighting efficient 加速度acceleration加速度分析acceleration analysis 加速度曲线acceleration diagram尖点pointing; cusp 尖底从动件knife-edge follower间隙backlash 间歇运动机构intermittent motion mechanism减速比reduction ratio 减速齿轮、减速装置reduction gear减速器speed reducer 减摩性anti-friction quality渐开螺旋面involute helicoids 渐开线involute渐开线齿廓involute profile 渐开线齿轮involute gear渐开线发生线generating line of involute 渐开线方程involute equation渐开线函数involute function 渐开线蜗杆involute worm渐开线压力角pressure angle of involute 渐开线花键involute spline简谐运动simple harmonic motion 键key键槽keyway 交变应力repeated stress交变载荷repeated fluctuating load 交叉带传动cross-belt drive交错轴斜齿轮crossed helical gears 胶合scoring角加速度angular acceleration 角速度angular velocity角速比angular velocity ratio 角接触球轴承angular contact ball bearing角接触推力轴承angular contact thrust bearing 角接触向心轴承angular contact radial bearing角接触轴承angular contact bearing 铰链、枢纽hinge校正平面correcting plane 接触应力contact stress接触式密封contact seal 阶梯轴multi-diameter shaft结构structure 结构设计structural design截面section 节点pitch point节距circular pitch; pitch of teeth 节线pitch line节圆pitch circle 节圆齿厚thickness on pitch circle节圆直径pitch diameter 节圆锥pitch cone节圆锥角pitch cone angle 解析设计analytical design紧边tight-side 紧固件fastener径节diametral pitch 径向radial direction径向当量动载荷dynamic equivalent radial load 径向当量静载荷static equivalent radial load径向基本额定动载荷basic dynamic radial load rating径向基本额定静载荷basic static radial load tating径向接触轴承radial contact bearing 径向平面radial plane径向游隙radial internal clearance 径向载荷radial load径向载荷系数radial load factor 径向间隙clearance静力static force 静平衡static balance静载荷static load 静密封static seal局部自由度passive degree of freedom 矩形螺纹square threaded form锯齿形螺纹buttress thread form 矩形牙嵌式离合器square-jaw positive-contact clutch绝对尺寸系数absolute dimensional factor 绝对运动absolute motion绝对速度absolute velocity 均衡装置load balancing mechanism抗压强度compression strength 开口传动open-belt drive开式链open kinematic chain 开链机构open chain mechanism可靠度degree of reliability 可靠性reliability可靠性设计reliability design, RD 空气弹簧air spring空间机构spatial mechanism 空间连杆机构spatial linkage空间凸轮机构spatial cam 空间运动副spatial kinematic pair空间运动链spatial kinematic chain 框图block diagram空转idle 宽度系列width series雷诺方程Reynolds‘s equation 离心力centrifugal force离心应力centrifugal stress 理论廓线pitch curve离合器clutch 离心密封centrifugal seal理论啮合线theoretical line of action 隶属度membership 力force力多边形force polygon 力封闭型凸轮机构force-drive (or force-closed) cam mechanism力矩moment 力平衡equilibrium力偶couple 力偶矩moment of couple连杆connecting rod, coupler 连杆机构linkage连杆曲线coupler-curve 连心线line of centers链chain 链传动装置chain gearing链轮sprocket ; sprocket-wheel ; sprocket gear ; chain wheel 联组V 带tight-up V belt联轴器coupling ; shaft coupling 两维凸轮two-dimensional cam临界转速critical speed 六杆机构six-bar linkage龙门刨床double Haas planer 轮坯blank轮系gear train 螺杆screw螺距thread pitch 螺母screw nut螺旋锥齿轮helical bevel gear 螺钉screws螺栓bolts 螺纹导程lead螺纹效率screw efficiency 螺旋传动power screw螺旋密封spiral seal 螺纹thread (of a screw)螺旋副helical pair 螺旋机构screw mechanism螺旋角helix angle 螺旋线helix ,helical line绿色设计green design ; design for environment 马耳他机构Geneva wheel ; Geneva gear马耳他十字Maltese cross 脉动无级变速pulsating stepless speed changes脉动循环应力fluctuating circulating stress 脉动载荷fluctuating load铆钉rivet 迷宫密封labyrinth seal密封seal 密封带seal belt密封胶seal gum 密封元件potted component密封装置sealing arrangement 面对面安装face-to-face arrangement面向产品生命周期设计design for product`s life cycle, DPLC名义应力、公称应力nominal stress模块化设计modular design, MD 模块式传动系统modular system模幅箱morphology box 模糊集fuzzy set模糊评价fuzzy evaluation 模数module摩擦friction 摩擦角friction angle摩擦力friction force 摩擦学设计tribology design, TD摩擦阻力frictional resistance 摩擦力矩friction moment摩擦系数coefficient of friction 摩擦圆friction circle磨损abrasion ;wear; scratching 末端执行器end-effector目标函数objective function 耐腐蚀性corrosion resistance耐磨性wear resistance 内齿轮internal gear挠性机构mechanism with flexible elements 挠性转子flexible rotor内齿圈ring gear内力internal force 内圈inner ring能量energy 能量指示图viscosity逆时针counterclockwise (or anticlockwise) 啮出engaging-out啮合engagement, mesh, gearing 啮合点contact points啮合角working pressure angle 啮合线line of action啮合线长度length of line of action 啮入engaging-in牛头刨床shaper 凝固点freezing point; solidifying point扭转应力torsion stress 扭矩moment of torque扭簧helical torsion spring 诺模图NomogramO 形密封圈密封O ring seal 盘形凸轮disk cam盘形转子disk-like rotor 抛物线运动parabolic motion疲劳极限fatigue limit 疲劳强度fatigue strength偏置式offset 偏( 心) 距offset distance偏心率eccentricity ratio 偏心质量eccentric mass偏距圆offset circle 偏心盘eccentric偏置滚子从动件offset roller follower 偏置尖底从动件offset knife-edge follower 偏置曲柄滑块机构offset slider-crank mechanism 拼接matching评价与决策evaluation and decision 平底宽度face width频率frequency 平带flat belt平带传动flat belt driving 平底从动件flat-face follower平分线bisector 平均应力average stress平均中径mean screw diameter 平均速度average velocity平衡balance 平衡机balancing machine平衡品质balancing quality 平衡平面correcting plane平衡质量balancing mass 平衡重counterweight平衡转速balancing speed 平面副planar pair, flat pair平面机构planar mechanism 平面运动副planar kinematic pair平面连杆机构planar linkage 平面凸轮planar cam平面凸轮机构planar cam mechanism 平面轴斜齿轮parallel helical gears普通平键parallel key 其他常用机构other mechanism in common use起动阶段starting period 启动力矩starting torque气动机构pneumatic mechanism 奇异位置singular position起始啮合点initial contact , beginning of contact 气体轴承gas bearing千斤顶jack 嵌入键sunk key强迫振动forced vibration 切齿深度depth of cut曲柄crank 曲柄存在条件Grashoff`s law曲柄导杆机构crank shaper (guide-bar) mechanism 曲柄滑块机构slider-crank (or crank-slider) mechanism曲柄摇杆机构crank-rocker mechanism 曲齿锥齿轮spiral bevel gear曲率curvature 曲率半径radius of curvature曲面从动件curved-shoe follower 曲线拼接curve matching曲线运动curvilinear motion 曲轴crank shaft驱动力driving force 驱动力矩driving moment (torque)全齿高whole depth 权重集weight sets球ball 球面滚子convex roller球轴承ball bearing 球面副spheric pair球面渐开线spherical involute 球面运动spherical motion球销副sphere-pin pair 球坐标操作器polar coordinate manipulator燃点spontaneous ignition 热平衡heat balance; thermal equilibrium人字齿轮herringbone gear 冗余自由度redundant degree of freedom柔轮flexspline 柔性冲击flexible impulse; soft shock柔性制造系统flexible manufacturing system; FMS 柔性自动化flexible automation 润滑油膜lubricant film 润滑装置lubrication device润滑lubrication 润滑剂lubricant三角形花键serration spline 三角形螺纹V thread screw三维凸轮three-dimensional cam 三心定理Kennedy`s theorem砂轮越程槽grinding wheel groove 砂漏hour-glass少齿差行星传动planetary drive with small teeth difference设计方法学design methodology设计变量design variable 设计约束design constraints深沟球轴承deep groove ball bearing 生产阻力productive resistance升程rise 升距lift实际廓线cam profile 十字滑块联轴器double slider coupling; Oldham‘s coupling矢量vector 输出功output work输出构件output link 输出机构output mechanism输出力矩output torque 输出轴output shaft输入构件input link 数学模型mathematic model实际啮合线actual line of action 双滑块机构double-slider mechanism, ellipsograph 双曲柄机构double crank mechanism 双曲面齿轮hyperboloid gear双头螺柱studs 双万向联轴节constant-velocity (or double) universal joint双摇杆机构double rocker mechanism 双转块机构Oldham coupling双列轴承double row bearing 双向推力轴承double-direction thrust bearing松边slack-side 顺时针clockwise瞬心instantaneous center 死点dead point四杆机构four-bar linkage 速度velocity速度不均匀( 波动) 系数coefficient of speed fluctuation速度波动speed fluctuation 速度曲线velocity diagram 速度瞬心instantaneous center of velocity塔轮step pulley 踏板pedal台钳、虎钳vice 太阳轮sun gear弹性滑动elasticity sliding motion 弹性联轴器elastic coupling ; flexible coupling弹性套柱销联轴器rubber-cushioned sleeve bearing coupling 套筒sleeve梯形螺纹acme thread form 特殊运动链special kinematic chain特性characteristics 替代机构equivalent mechanism调节modulation, regulation 调心滚子轴承self-aligning roller bearing调心球轴承self-aligning ball bearing 调心轴承self-aligning bearing调速speed governing 调速电动机adjustable speed motors调速系统speed control system 调压调速variable voltage control调速器regulator, governor 铁磁流体密封ferrofluid seal停车阶段stopping phase 停歇dwell同步带synchronous belt 同步带传动synchronous belt drive凸的，凸面体convex 凸轮cam凸轮倒置机构inverse cam mechanism 凸轮机构cam , cam mechanism凸轮廓线cam profile 凸轮廓线绘制layout of cam profile凸轮理论廓线pitch curve 凸缘联轴器flange coupling图册、图谱atlas 图解法graphical method推程rise 推力球轴承thrust ball bearing推力轴承thrust bearing 退刀槽tool withdrawal groove退火anneal 陀螺仪gyroscopeV 带V belt 外力external force外圈outer ring 外形尺寸boundary dimension万向联轴器Hooks coupling ; universal coupling 外齿轮external gear弯曲应力beading stress 弯矩bending moment腕部wrist 往复移动reciprocating motion往复式密封reciprocating seal 网上设计on-net design, OND微动螺旋机构differential screw mechanism 位移displacement位移曲线displacement diagram 位姿pose , position and orientation稳定运转阶段steady motion period 稳健设计robust design蜗杆worm 蜗杆传动机构worm gearing蜗杆头数number of threads 蜗杆直径系数diametral quotient蜗杆蜗轮机构worm and worm gear 蜗杆形凸轮步进机构worm cam interval mechanism蜗杆旋向hands of worm 蜗轮worm gear涡圈形盘簧power spring 无级变速装置stepless speed changes devices无穷大infinite 系杆crank arm, planet carrier现场平衡field balancing 向心轴承radial bearing向心力centrifugal force 相对速度relative velocity相对运动relative motion 相对间隙relative gap象限quadrant 橡皮泥plasticine细牙螺纹fine threads 销pin消耗consumption 小齿轮pinion小径minor diameter 橡胶弹簧balata spring修正梯形加速度运动规律modified trapezoidal acceleration motion修正正弦加速度运动规律modified sine acceleration motion斜齿圆柱齿轮helical gear 斜键、钩头楔键taper key泄漏leakage 谐波齿轮harmonic gear谐波传动harmonic driving 谐波发生器harmonic generator斜齿轮的当量直齿轮equivalent spur gear of the helical gear心轴spindle 行程速度变化系数coefficient of travel speed variation行程速比系数advance-to return-time ratio 行星齿轮装置planetary transmission行星轮planet gear 行星轮变速装置planetary speed changing devices行星轮系planetary gear train 形封闭凸轮机构positive-drive (or form-closed) cam mechanism虚拟现实virtual reality 虚拟现实技术virtual reality technology, VRT虚拟现实设计virtual reality design, VRD 虚约束redundant (or passive) constraint 许用不平衡量allowable amount of unbalance许用压力角allowable pressure angle 许用应力allowable stress; permissible stress 悬臂结构cantilever structure 悬臂梁cantilever beam循环功率流circulating power load 旋转力矩running torque旋转式密封rotating seal 旋转运动rotary motion选型type selection 压力pressure压力中心center of pressure 压缩机compressor压应力compressive stress 压力角pressure angle牙嵌式联轴器jaw (teeth) positive-contact coupling雅可比矩阵Jacobi matrix 摇杆rocker液力传动hydrodynamic drive 液力耦合器hydraulic couplers液体弹簧liquid spring 液压无级变速hydraulic stepless speed changes液压机构hydraulic mechanism 一般化运动链generalized kinematic chain移动从动件reciprocating follower 移动副prismatic pair, sliding pair移动关节prismatic joint 移动凸轮wedge cam盈亏功increment or decrement work 应力幅stress amplitude应力集中stress concentration 应力集中系数factor of stress concentration 应力图stress diagram 应力—应变图stress-strain diagram优化设计optimal design 油杯oil bottle油壶oil can 油沟密封oily ditch seal有害阻力useless resistance 有益阻力useful resistance有效拉力effective tension 有效圆周力effective circle force有害阻力detrimental resistance余弦加速度运动cosine acceleration (or simple harmonic) motion预紧力preload 原动机primer mover圆带round belt 圆带传动round belt drive圆弧齿厚circular thickness 圆弧圆柱蜗杆hollow flank worm圆角半径fillet radius 圆盘摩擦离合器disc friction clutch圆盘制动器disc brake 原动机prime mover原始机构original mechanism 圆形齿轮circular gear圆柱滚子cylindrical roller 圆柱滚子轴承cylindrical roller bearing圆柱副cylindric pair 圆柱式凸轮步进运动机构barrel (cylindric) cam圆柱螺旋拉伸弹簧cylindroid helical-coil extension spring圆柱螺旋扭转弹簧cylindroid helical-coil torsion spring圆柱螺旋压缩弹簧cylindroid helical-coil compression spring圆柱凸轮cylindrical cam 圆柱蜗杆cylindrical worm圆柱坐标操作器cylindrical coordinate manipulator圆锥螺旋扭转弹簧conoid helical-coil compression spring圆锥滚子tapered roller 圆锥滚子轴承tapered roller bearing圆锥齿轮机构bevel gears 圆锥角cone angle。

Differential Equations ( Assignment )Solve the following differential equation : Problem 1: Solve the following O.D.E :(a) x d y – yd x = d x – dy . Ans : y+1=c(x +1) (b) (1-ycoty)dy y0dx x+= Ans : x y=csiny (c)dy xy 3x y 3dx xy 2x 4y 8+--=-+-Ans : [y-x +5lnx 4y 3++=c] (d) ln (dydx)=3x+4y , given y(0)=0 . Ans : 4e 3x + 3e -4y = 7 (e) 2x 3 dy = 3x 2y dx – y 3 dy Ans : x 3-y 3 = cy 2(f )dy 8x 4y 1dx 4x 2y 1++=-++ Ans : (y+2x)2+(y+2x)=x+C (g) (x+1)dydx+ y = lnx , given y(1) = 10 . Ans : (x+1)y=x(lnx-1)+21 (h)2dy 1dx y x=- Ans : 222y x y y Ce -=-++ ( i) 3cos sin cos n dyxny x x dx++= , where y=0 when x=0 . Ans : y 2=tan 2x + 4tanx -1 (j) 1tan()dy x y x dx =+- Ans : y = x + sin -1( 22x c e +)(k) sin()cos()dy x y x y dx =+++ Ans : ln(1tan )2x y x c ++=+ (l ) 2229d y dyx x dx dx+= Ans : y = x 3 + Aln x + B Problem 2: . Using the substitution u= y(1+x 2) , solve 22221(1)421d y dy x x y dx dx x+++=+. [ Ans : y ( 1 + x 2 ) = xtan -1x - 21ln(1)2x Ax B +++]Problem 3: (i) Solve (i) (2x + e y ) dx + xe y dy = 0 Ans : x 2 + xe y = c(ii) e x+y dx dy + 1 = e 2xAns : ln 121x y x e e c e --=-++(iii)y x dy x y xe dx =+ , given that y(1) = 1 . Ans : -1ln y x e x e-=-Problem 4: .The gradient at a variable point (x,y) of a curve y=f(x) is given by21yx -+. If the curve passes through the point (1,1) , find the equation of the curve . [ ans : 24(1)y x =+] Problem 5:Solve the following differential equations :(a) 222d y dy y dx dx=(b) y 4y 3y 0'''-+= , given y(0)=6 , y (0)10'=. Ans : y = 4e x +2e 3x(c) 2y 2d y 2e dx= , given that y = 0 , dy dxAns : -2e y/2 = or -2e y/2 = 4-(d) ln lny dyx y x y dx+= Ans : y = xe cx + 1(e) ( y 4 – 3x 2 ) dy + xy dx = 0 Ans : x 2 = y 4 + Cy 6(f) 22323dy xy dx x y y -=+ Ans : y C =(g) xy y ''+=tan )C -= (h) (y 2 – ycosx ) dx + (2xy-sinx)dy = 0 Ans : xy 2 – ysinx = C(i) (2x-4y+6)dx + (x+y-3)dy = 0 Ans : (y-2x) 3 = C(y-x-1)2 , y = x + 1Problem 6:Solve the following Bernoulli DE :(a) 420y xy xy '++= Ans : 23312x y Ce --=+ ; y = 0 (b)22sin cos dyy y x y x dx++= Ans : 1sin x y Ce x -=- ; y = 0 Problem 7 :An object is released from rest . Apart from being acted on by the gravitational pull g, the resistance that the object encounters varies with its velocity v m/s , thus giving its acceleration a as (g-kv) , where k is a positive constant .Show that v = (1)kt ge k-- , where t represents the time when the velocity of the object is v .。

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 ⎟⎠⎞⎜⎝⎛∂∂<<⎟⎠⎞⎜⎝⎛∂∂.。

.LECTURE NOTESFeedback Control Engineering and Dynamic Systems:Control4,Control5H.GolleeK.J.HuntUniversity of GlasgowCopyright c K.J.Hunt,H.Gollee2003–2012These lecture notes are in two parts:1.Differential Equations.Contents/concepts:•Equations of motion of simple biomechanical systems(linear ODEs)•Time-domain solution(free and forced responses)•Transient and steady-state solutions•Standard2nd-order equation;natural frequency and damping•Stability of standing•Frequency-domain solution(via Laplace)•The transfer-function•The frequency response•Direct links between time-and frequency-domain solutions2.Feedback Control.Contents/concepts:•Open-loop and closed-loop control:–plant model–static open-loop control–static closed-loop control–control and plant inversion•Dynamic closed-loop control:sensitivity functions•Feedback design using pole assignment•Stability and stability robustness•Design goals and loop shaping•Limitations of performance2Part1:DIFFERENTIAL EQUATIONS(dynamic systems) The purpose of this chapter is to develop mathematical representations of some simple biomechanical systems.The systems are described as lin-ear differential equations.We develop the time-domain solution to these equations,and then contrast this with the(equivalent)frequency-domain solution.The concept of frequency response is reviewed.More general the-ories and concepts are assumed to be covered elsewhere:[1,2,3].Summary(linear ODEs)The general solution of an n-th order linear,homogeneous,constant-coefficient differential equation depends on the rootsλ1...λn of the characteristic equa-tion.For a second-order equation we have the following cases:•Real and distinct rootsλ1andλ2give the solutionx(t)=Aeλ1t+Beλ2t•Repeated real rootsλ1=λ2=λgive the solutionx(t)=(At+B)eλt•Complex conjugate rootsλ1,2=a±jb give the solutionx(t)=(A cos bt+B sin bt)e atAlternatively,this solution may be expressed in the equivalent formx(t)=C sin(bt+φ)e atwhere the unknown coefficients C andφare related to A and B by √C=Figure1:Seated person with shank rotation about the knee joint.1Example:response of the lower leg1.1System descriptionWe will investigate the dynamic response of the lower leg.Consider a person seated on a bench with the shank free to move around the knee joint,as shown infigure1.We approximate this system using the uniform single link depicted infigure2.The link has a length l and is free to rotate about O. The mass is m,and the centre of mass is located at the point G,situated a distance l/2from each end of the link.The moment of inertia of the link around the point O is denoted by I,and this is related to the mass and length through the relation I=m(l/2)2=ml2/4(see[3]).Wefirst consider a situation in which the knee-extensor muscles(prin-cipally the quadriceps group of muscles)contract in such a way that the system is held in equilibrium at an angleθe by application of a constant equilibrium torqueτe acting around O.We denote an arbitrary angular displacement from equilibrium asθ,as shown infigure3.With the system in static equilibrium the moment arm a for the gravitational force mg is a=(l/2)sinθe.Thus the equilibrium moment isτe=(mgl/2)sinθeWhen the system is in motion(i.e.˙θ=0)we consider it to be subject to a viscous damping moment c˙θ,where c is the damping coefficient shown in figure2.In reality,this damping arises from the physical properties of the contracting muscle and its tendon-bone connection.4Figure2:Idealised model of lower leg.1.2Equations of motionWe now derive the differential equation which describes motion of the lower leg model when it is in motion around the equilibrium.It is instructive to derive the system equation in two ways:first,by considering all forces acting on the system;second,by ignoring the constant forces associated with equilibrium,and considering only the forces developed in displacement away from equilibrium.We will see that both approaches lead to the same result.1.All forces:The inertial moment of the limb is given by I¨θ,where Iis the moment of inertia.Balancing all moments we obtainI¨θ=τe−c˙θ−mg l(sinθe+(cosθe)θ)2Using the expression derived above for the equilibrium moment,τe= (mgl/2)sinθe,this simplifies tolI¨θ=−c˙θ−mg˙θ+mgl cosθeIFigure3:Displacementθfrom equilibrium positionθe.which we recognise as a2nd-order linear constant-coefficient differen-tial equation.2.Displacement forces:We now re-derive the equation of motion byneglecting the equilibrium moments.A positive displacement ofθcauses the gravitational moment arm to increase by a′,as shown in figure4.We see from the geometric arrangement that(d/2)/(l/2)= sin(θ/2),or d=l sin(θ/2).Assuming again thatθis small we have d≈l2θ.Thus,the additional gravitational momentτd generated by a displacementθisτd=mga′=mgl cosθe2θ6Figure4:Displacementθfrom equilibrium positionθe.which on rearrangement gives¨θ+cθ=0(3)2IWe see that this is identical to expression(2).Thus we see that by defining the displacement variableθto be zero at equilibrium we may ignore the equal and opposite forces associated with equilibrium.We noted above that the system can be described as a2nd order linear constant-coefficient differential equation.However,it is important to note that the coefficients of the equation of motion depend not only on the phys-ical properties of the system through the parameters c,I,m and l,but also on the chosen equilibrium point.This is because the equilibrium parameter θe appears within one of the coefficients.This means that solutions of the equation of motion obtained at a given equilibrium pointθe are valid only for“small”angular deviations around that point.71.3Solution of equation of motionIt is common practice to rewrite linear2nd order differential equations of the form(3)in a standard format.To do this we define the variablesωn andζ(zeta)asωn= 2I andζ=cζ2−1)(6) Clearly the value ofζdetermines the character of the solution:when0≤ζ<1there is a pair of complex conjugate roots;whenζ=1there is a pair of equal real roots;and whenζ>1the roots are real and distinct.We now consider these cases individually,beginning by considering the special case ofζ=0.1.Undamped system,ζ=0.If no viscous damping is present then c=0andζ=0(equation(4)).In this case the roots of the characteristic equation areλ1,2=±jωn.This leads to the general solutionθ(t)=A cosωn t+B sinωn t(7) or,equivalently,θ(t)=C sin(ωn t+φ)(8) Particular solutions are obtained by using the initial conditionsθ0=θ(0)and˙θ0=˙θ(0)to evaluate the constants A and B(or C andφ).8Evaluating(7)at t=0we obtain A=θ0.Differentiating(7)and evaluating at t=0results in B=˙θ0/ωn.The particular solution isthereforeθ(t)=θ0cosωn t+˙θ0θ20+ ˙θ0˙θ0 The particular solution is thereforeθ(t)= ωn 2sin(ωn t+tan−1 ωnθ01−ζ2.It is convenient to introduce a new variableωd=ωnandφ=tan−1 θ0ωd sinφωdThe solution(10)shows that the system in this case oscillates with a frequencyωd=ωn1−ζ2).Figure6shows a graphical representation of the damped motion for a system having the parameters m=5kg,l=0.4m,c=0.5Nm·rad−1·s,thus giving ωn=7rad·s−1andζ=0.1785(see equation(4)).The equilibrium position is againθe=0deg and the initial conditions areθ0=10deg and˙θ0=0.The damping factorζcan be experimentally determined for under-damped systems.This experimental approach can be useful when the viscous damping coefficient c cannot be easily determined from physi-cal considerations.Assume that the system is set in motion with some initial conditions and two successive amplitudesθ1andθ2are mea-sured at times t1and t2,respectively.The ratio of the two amplitudes10θ2=Ce−ζωn t1θ2 =ζωn T d=2πζ1−ζ2Solving this equation forζwe obtainζ=δ4π2+δ23.Critically damped system,ζ=1.In this case the roots of thecharacteristic equation areλ1=λ2=−ωn,which leads to the general solutionθ(t)=(At+B)e−ωn t(11)A andB are given in terms of the initial conditions asA=˙θ0+ωnθ0,B=θ011ζ2−1). The general solution isθ(t)=Ae(−ζ+√ζ2−1)ωn t(12)Again,the motion decays in a non-oscillatory fashion towardsθ(t)=0.The rate of decay becomes slower as the damping increases.Figure7 shows solutions for various values ofζ≥1.for a system having the parameters m=5kg,l=0.4m,c=0.5Nm·rad−1·s.1.4Forced response of the lower legWe now consider the situation in which motion of the lower leg is con-tinuously excited by a disturbing moment.The moment will typically be generated by contraction of the muscles which extend andflex the knee joint (principally the quadriceps and hamstring muscle groups).Various forms of forcing functions might be applied.However,we consider the special case in12Figure8:Forced motion of the lower leg.which the forcing function is harmonic(sinusoidal).This type of function gives a useful introduction to the analysis of forced motion,but is also very important because,as we shall see later,the response to harmonic excitation defines the frequency response of the system.Consider the system offigure8where the lower leg is subject to the applied harmonic momentτ=τe+τ0sinωt.Here,τe,as before,is the static moment corresponding to the equilibrium positionθe.τ0is the am-plitude of the driving moment andωis the driving frequency(in radians per second).It is important to distinguish betweenω,which is a property of the moment applied to the system,andωn=2θwhich on rearrangement gives¨θ+c2I θ=τ0standard form of the equation of motion is¨θ+2ζωn ˙θ+ω2nθ=τ0ω2n[(1−(ω/ωn)2)2+(2ζω/ωn)2]1/2(16)ψ=tan−1 2ζω/ωnthe physical system or of the forcing function,except for the constants C and φin the transient solution,which depend only on the initial conditions.The relationship between the components of the complete solution is summarised as follows:θ(t)=θc(t)+θp(t)=Ce−ζωn t sin(ωd t+φ)+D sin(ωt−ψ)Complementary function,i.e.Particular integral,i.e.general solution of solution of non-homogeneous equation.homogeneous equation.→Transient Solution→Steady-state Solution Since the transient solution diminishes with time(forζ>0),the steady-state solution is of particular interest.We see from equation(16)that the magnitude D of the steady-state solution depends strongly on the damping factorζand the non-dimensional frequency ratioω/ωn.The amplification (or attenuation)of the forcing function is determined by the ratio of the steady-state solution’s amplitude and the forcing function’s amplitude,i.e. D/(τ0/I).This quantity,denoted by M,is called the amplitude ratio or magnification factor,and from(16)is given by1M=sinωt(20)Iwhere(τ0/I)sinωt is the forcing function.The variablesζandωn are given by(4)for the lower leg.Thus the solution derived below is valid for both systems,or for any other system which can be represented in the standard format.17We proceed by taking Laplace transforms throughout equation(20). This gives(s2Θ(s)−sθ0−˙θ0)+2ζωn(sΘ(s)−θ0)+ω2nΘ(s)=τ0ω/Is2+2ζωn s+ω2n +τ0ω/I(s2+ω2)(s2+2ζωn s+ω2n)(22) The next step in the procedure is to decompose the above expression forΘinto partial fractions.The form of the partial fractions will depend upon the nature of the roots of the characteristic equation s2+2ζωn s+ω2n.The roots are given by equation(6),i.e.s1,2=ωn(−ζ±1−ζ2=−ζωn±jωd sinceωd=ωn(s+ζωn)2+ω2d +Cs+D(s+ζωn)2+ω2d+Cs+DThe numerical values of the constants A,B,C and D are obtained in the usual way by multiplying equations(22)and(23)by the denominator and then equating the coefficients of corresponding powers of s(this procedure is left as an exercise).The solution forθis then obtained by inverse trans-formation.In order to apply standard tables of Laplace transforms it is convenient to rearrange the terms in equation(23)asΘ(s)=A(s+ζωn)(s+ζωn)2+ω2d+Css2+ω2(24)where B′=(B−Aζωn)/ωd,D′=D/ω.The inverse transform can then be obtained from standard tables asθ(t)=(A cosωd t+B′sinωd t)e−ζωn t+C cosωt+D′sinωtor,equivalently,θ(t)=Ee−ζωn t sin(ωd t+φ)+F sin(ωt−ψ)(25) where the constants E,F,φandψare related straightforwardly to A,B′,C and D′:E= C2+D′2,ψ=tan−1(−D′/C)Thus,we see that the solution(25)obtained by the Laplace transform pro-cedure is identical to the result obtained previously in the time domain (equation(18)).2.2Transfer function and amplitude ratioFrom equation(21)we see that the system response can be split into two parts:one containing the initial conditions,and one including the forcing function.This holds for general forcing functions u(t)and not only the sinusoidal functions considered above.Denoting the transform of a general forcing function u as U(s)=L{u(t)}we see from(21)that,in general,Θ(s)=1The transfer function is usually defined as the ratio of the Laplace transform of the system output(here,Θ(s))and the transform of the system input(or forcing function,U(s))under the assumption that the initial conditions are zero,i.e.G(s) U−1(s)Θ(s)whenθ0=˙θ0=0.We shall see below(section2.3)that the frequency response of the sys-tem is given by G(s),but with the complex variable s replaced by jω.It is clear from the relationshipΘ(s)=G(s)U(s),and from the analysis to follow in section2.3,that the magnitude|G(jω)|determines the amount of amplification(or attenuation)of the forcing signal u at the frequencyω. This observation allows us to make an explicit link between|G(jω)|and the amplitude ratio(or magnification factor)M which was derived previously from time-domain considerations(see equation(19)).We will show that these quantities are identical.Since G(s)=1/(s2+2ζωn s+ω2n)we have1G(jω)=(ω2n−ω2)2+4ζ2ω2nω2The magnitude of G is then found to be|G(jω)|=[(ω2n−ω2)2+4ζ2ω2nω2]1/2[(ω2n−ω2)2+4ζ2ω2nω2]1/2 which can be rearranged as|G(jω)|=1We consider an input signal u(t)=(τ0/I)sinωt.From equation(27)the system response,assuming zero initial conditions,isΘ(s)=G(s)U(s)=G(s)τ0ω/Is−jω+bs−p1+ds2+ω2=as+jω+cs−p2which we multiply throughout by s2+ω2=(s+jω)(s−jω)to giveτ0ωs−p1+d(s+jω)(s−jω)s+jω s=jω=τ0 s−jω s=−jω=−τ0 2IjG(jω)e jωt−τ0The complex number G(jω)can be expressed in polar form as G(jω)= |G(jω)|e j arg G(jω),where|G(jω)|denotes the magnitude of G(jω)and the term arg G(jω)is its phase.Since G(−jω)=|G(jω)|e−j arg G(jω)the steady-state response in(35)becomesτ0θss(t)=I|G(jω)|sin(ωt+arg G(jω))(37) This expression defines the system’s frequency response,i.e.the response to a sinusoidal input of frequencyω.The frequency response is seen to be characterised by the magnitude,|G(jω)|,and phase,arg G(jω),of G(s) when evaluated at s=jω.We have seen in section2.2that|G(jω)|is equivalent to the amplitude ratio(or magnification factor)M.It is often referred to simply as the gain of G.The quantity arg G(jω)is known as the phase shift.The response to a sinusoidal input signal u(t)=(τ0/I)sinωt of frequency ωis also a sinusoid of the same frequency,but its amplitude is scaled by |G(jω)|and its phase is shifted by the angle arg G(jω).It may atfirst sight appear restrictive to consider only sinusoidal sig-nals.However,it should be recalled that all periodic signals can be exactly represented as the sum of sinusoids of different frequencies(using Fourier’s theorem).Moreover,other signals will have a frequency spectrum which allows them to be represented arbitrarily accurately by a sum of sinusoids over a range of frequencies.Superposition then allows us to represent the response of a system as the sum of the responses to the individual sinu-soidal inputs.The frequency response therefore provides a very powerful description of the system.The information characterising the frequency response can be conve-niently displayed in a variety of graphical formats.One common format is the Bode diagram which consists of two graphs:a plot of|G(jω)|against frequencyω;and a plot of arg G(jω)against frequency.22References[1]G.James,Modern Engineering Mathematics,3rd ed.Prentice Hall,2001.[2]——,Advanced Modern Engineering Mathematics,2nd ed.Addison-Wesley,1999.[3]J.L.Meriam and L.G.Kraige,Engineering Mechanics,Volume2:Dy-namics,4th ed.Wiley,1998.23。

Unit 2 Poems强化训练AAmerican poet, one of the finest of rural New England's 20-century poets, Frost published his first book in Great Britain in the 1910s, but he soon became the most popular poet in his own country, whose works were made familiar in classrooms and lectures. Frost was awarded the Pulitzer Prize four times. Nature and Frost's rural surroundings were for him a source for insights "from delight to wisdom", or as he also said, "Literature begins with geography."Robert Frost (1874-1963) was born in San Francisco, California. His father, a journalist, died when Frost was about eleven years old. His Scottish mother worked as a schoolteacher to support the family. The family lived in Lawrence, Massachusetts, with Frost's grandfather, William Prescott Frost, who gave his grandson a good schooling. In 1892, Frost graduated from a high school and attended Darthmouth College for a few months. Over the next ten years, he held a number of jobs. Frost worked among others in a textile mill (纺织厂) and taught Latin at his mother's school. In 1894, the New York Independent published Frost's poem My Butterfly.From 1897 to 1899, Frost studied at Harvard, but left without receiving a degree. When he sent his poems to The Atlantic Monthly, they were returned with this note, "We regret that The Atlantic has no place for your wonderful poems."In 1912, Frost went to England. There he published his first collection of poems A Boy’s Will, at the age of 39. It was followed by North Boston (1914), which gained international reputation. The collection contains one of Frost's best- known poems Mending Wall.1. Which of the following is TRUE of Frost?A. His works were used in textbooks in Britain.B. His first book was not published in the U.S.C. He learned poem writing in his mother's school.D. He was more popular in Britain than in the U.S.2. By saying "Literature begins with geography", Frost meant that _______A. poets should have a good knowledge of geographyB. writers from different areas had different writing stylesC. literature can help readers learn more about geographyD. environment played an important role in a writer's development3. The Atlantic Monthly refused Frost's poems because ________.A. they only published famous poets' worksB. they didn't think the poems were goodC. the poems' writing style was out of dateD. the poems were too long to be used in the magazine4. Before studying at Harvard, Frost published his ________.A. A Boy's WillB. My ButterflyC. Mending WallD. North BostonBJohn Milton, the most learned poet, is the greatest writer of the seventeenth century and one of the giants of English literature as a whole. He was greatly influenced by two historical movements of Renaissance (文艺复兴) and Reformation(宗教改革). Like Spenser and Shakespeare, he was also one of the Renaissance giants not only in England but also in the whole world. Almost all later poets in English literature respected Milton highly. Milton towers over his age as Shakespeare towers over the Elizabethan age, and as Chaucer towers over the medieval period.John Milton was born in London in 1608. His education began at St. Paul's School, where he was very hard-working, where he showed wonderful gifts as a student of languages mastering Greek, Latin, Hebrew and many modern European languages. He also received very good home education under the influence of his father who was a Puritan and a lover of music and literature. He attended Christ's College, Cambridge University, where he was very popular because of his handsomeness and talent, where he explained the true aim of knowledge as making the spirit of man "reach out far and wide, until it fills the whole world and the space far beyond with the expansion of its greatness", where he graduated with B. A. in 1629 and M. A. in 1632.Milton wrote in Latin, Italian and English, and had an international reputation during his lifetime. Milton's poetry was slow to see the light of day, at least under his name. His first published poem was On Shakespeare (1630), anonymously included in the Second Folio edition of Shakespeare. In the midst of the excitement attending the possibility of establishing a new English government, Milton collected his work in 1645 Poems. John Milton is best known for his epic poem Paradise Lost. John Milton died in November 16745. According to the passage all of the following are true except that ________.A. John Milton was one of the giants of English literatureB. John Milton was a most learned poetC. Spenser and Shakespeare were the Renaissance giants in EnglandD. Spenser, Shakespeare and John Milton had the friendly relationship between them6. The underlined word "towers" in the first paragraph most probably means ________.A. is better than others in ability and qualityB. looks like a tall buildingC. builds a tall building like a towerD. likes to visit towers7. Which of the following statements is NOT true about John Milton's education?A. He was a clever boy so he didn't take pains, but got good results.B. He was very good at learning languages and he mastered several foreign languages at school.C. He was interested in literature.D. He graduated from Cambridge University.8. What was Milton's opinion to the true aim of knowledge?A. To think about what we don't know.B. To make the character of man strong and great.C. To research the whole world and space.D. To make the spirit of man reach out far and wide.CThomas Sterns Eliot was born in St. Louis, Missouri, on September 26, 1888. He was raised in a family that had distinguished(区别) Americans since colonial days. At 18 years old, T.S. Eliot entered Harvard, and after graduating he went abroad. Then he taught in a boys school briefly before spending eight years in Lloyds Bank in London.T. S. Eliot inspired the musical Cats in the 1980s from his well-known book Old Possum's Book of Cats. He also wrote The Love Song of J. Alfred Prufock in 1915. After 1915, T. S. Eliot wrote such poems as Portrait of a Lady.The Waste Land appeared in 1922. It was considered by many to be his most challenging work. In 1927 Thomas Sterns Eliot became a British subject and was confirmed in the Church of England.His essays, For Lancelot Andrew(1928) and his poetry, Four Quarters (1943) increasingly reflected this association with a traditional culture. The Rock (1934), his first drama, was a pageant (露天表演) play. This was followed by Murder in the Cathedral (1935), a play dealing with the assassination of Archbishop Thomas a Becket, who was later praised.In 1948, King George VI bestowed the order of Merit on T. S. Eliot, and in that same year he was awarded the Nobel Prize in Literature. On January 4, 1965 Thomas Sterns Eliot died at the age of 76. Another one of his poems I enjoy is Aunt Helen.9. Where did the inspiration of Cats come from?A. Thomas Sterns Eliot.B. The animal cat.C. Old Possum's Book of Cats.D. King George VI.10. How many works are mentioned in the passage?A. 7.B. 8.C. 9.D. 10.11. Which of the following is TRUE according to the passage?A. Thomas Sterns Eliot was once a teacher in a school.B. Eliot won the Nobel Prize because King George VI bestowed the order of Merit on him.C. Thomas Sterns Eliot spent eight years in London.D. Thomas Sterns Eliot only wrote poems in his life.12. What does the underlined word "assassination" mean?A. Murder.B. Assignment.C. Deeds.D. Hero.DNo poem should ever be discussed or "analyzed", until it has been read aloud by someone, teacher or student. Better still, perhaps, is the practice of reading it twice, once at the beginning of the discussion and once at the end, so the sound of the poem is the last thing one hears of it.All discussions of poetry are, in fact, preparations for reading it aloud, and the reading of the poem is, finally, the most telling "interpretation" of it, suggesting tone, rhythm, and meaning all at once. Hearing a poet read the work in his or her own voice, on records or on film, is obviously a special reward. But even those aids to teaching can not replace the student and teacher reading it or, best of all, reciting(背诵) it.I have come to think, in fact, that time spent reading a poem aloud is much more important than "analyzing" it, if there isn't time for both. I think one of our goals as teachers of English is to have students love poetry. Poetry is "a criticism of life", and "a heightening(提升) of life". It is "an approach to the truth of feeling", and it "can save your life". It also deserves a place in the teaching of language and literature more central than it presently occupies.I am not saying that every English teacher must teach poetry. Those who don't like it should not be forced to put that dislike on anyone else. But those who do teach poetry must keep in mind a few things about its essential nature, about its sound as well as its sense, and they must make room in the classroom for hearing poetry as well as thinking about it.13. According to the text, to have a better understanding of a poem, the best way is ________.A. to hear it read outB. to analyze it by oneself.C. to discuss it with othersD. practise reading it aloud.14. The best explanation for the underlined sentence in Paragraph 1 should be ________.A. students should read the poem at the end of poem classB. we should end the analyzing of the poem by reading it aloudC. sometimes the sound of the poem represents the meaning of itD. we feel the original meaning of the author by hearing the poem read out after discussion15. What does the last sentence in the third paragraph imply?A. The teaching of poetry should have been much more stressedB. One cannot enjoy life fully without an understanding of poetry.C. Poetry is the foundation of all language and literature courses.D. The teaching of poetry is more important than the teaching of any other subject.16. The phrase "make room" in the last paragraph could be best replaced by _________.A. "build a booth".B. "provide equipment".C. "leave a certain amount of time".D. "set aside enough space".EDo you still remember your favorite poem from high school or mine other important period in your life? Why is it that decades later it still stands out in your mind? Probably the main reason is that some aspect of that poem resonates(引起共鸣) with you. In the same way, you too as a school leader can touch the hearts of your staff and students.Poetry allows us to experience strong spiritual connections to things around us and to the past. Thus, it can inspire whatever and whomever it touches. The power that poetry has displayed over time and across cultures actually satisfies this common need of the human heart and soul.As one of the oldest art forms, poetry has successfully connected various strands of humanity(人性) from one generation to another. Referring to poetry, Hillyer makes a simple yet meaningful statement, "With this key mankind unlocked his heart."School leaders can find and make use of the value of poetry for themselves, their students and their staff members. Beyond the simple use of poetry, techniques of poetry such as metaphors, repetitions and imagery can be used to take advantage of the power of language to transform communication, create meaning and a culture of care and attention.Since schools are mainly about people and relationships, school leaders, like poets, are required to inspire and encourage the human heart. The use of poetry--or even of some techniques of poetry--in school leadership not only helps to improve communication, but also serves to meet the human need for inspiration.17. Who is the passage mainly for?A. Students.B. School teachers.C. School leaders.D. Poets.18. According to the passage, some poems are unforgettable mainly because ________.A. they are easy to understandB. they usually tell true storiesC. they can inspire people's imaginationD. they can touch people's hearts19. The underlined part "this common need" in the second paragraph refers to ________.A. the need to read poetryB. the need to be inspiredC. the need to learn about the pastD. the need to be connected with other people20. The use of poetry or techniques of poetry in school leadership is in fact to make use of ______.A. the power of languageB. the power of school leadersC. people's preference for poetryD. people's desire for communication/。

机械专业机械工具扳子spanner (美作:wrench)双头扳子double-ended spanner活扳子,活络扳手adjustable spanner, monkey wrench半自动滚刀磨床semi-automatic hob grinder半自动化semi-automation; semi-automatic 扳手wrench备件spare parts 边刨床side planer 变速箱transmission gear 柄轴arbor 部件units; assembly parts 插床slotting machine 拆卸to disassemble超高速内圆磨床ultra-high-speed internal grinder车床lathe; turning lathe 车刀lathe tool车轮车床car wheel lathe 车削turning 车管钳子box spanner (美作:socket wrench) 卡规calipers夹钳pincers, tongs 剪子shears 钢锯hacksaw剪线钳wire cutters万能手钳multipurpose pliers, universal pliers可调手钳adjustable pliers 冲子punch 钻drill 卡盘chuck 三角刮刀scraper扩孔钻reamer 孔径规calliper gauge铆钉rivet 螺母nut 自锁螺母,防松螺母locknut 螺栓bolt销钉pin, peg, dowel 垫圈washer U形钉staple油壶oil can 工作服jack 注油枪grease gun机械加工抛光polishing 安装to assemble 衬套bushing半机械化semi-mechanization; semi-mechanized 轴axle金属切削metal cutting机床machine tool金属工艺学technology of metals刀具cutter摩擦friction联结link传动drive/transmission轴shaft弹性elasticity频率特性frequency characteristic误差error响应response定位allocation机床夹具jig动力学dynamic运动学kinematic静力学static分析力学analyse mechanics拉伸pulling压缩hitting剪切shear扭转twist弯曲应力bending stress强度intensity三相交流电three-phase AC磁路magnetic circles变压器transformer异步电动机asynchronous motor 几何形状geometrical精度precision正弦形的sinusoid交流电路AC circuit机械加工余量machining allowance 变形力deforming force变形deformation应力stress硬度rigidity热处理heat treatment退火anneal正火normalizing脱碳decarburization渗碳carburization电路circuit半导体元件semiconductor element 反馈feedback发生器generator直流电源DC electrical source门电路gate circuit逻辑代数logic algebra外圆磨削external grinding内圆磨削internal grinding平面磨削plane grinding变速箱gearbox离合器clutch绞孔fraising绞刀reamer螺纹加工thread processing螺钉screw铣削mill铣刀milling cutter功率power工件workpiece齿轮加工gear mechining齿轮gear主运动main movement主运动方向direction of main movement进给方向direction of feed进给运动feed movement合成进给运动resultant movement of feed合成切削运动resultant movement of cutting合成切削运动方向direction of resultant movement of cutting 切削深度cutting depth前刀面rake face刀尖nose of tool前角rake angle后角clearance angle龙门刨削planing主轴spindle主轴箱headstock卡盘chuck加工中心machining center车刀lathe tool车床lathe钻削镗削bore车削turning磨床grinder基准benchmark钳工locksmith锻forge压模stamping焊weld拉床broaching machine拉孔broaching装配assembling铸造found流体动力学fluid dynamics流体力学fluid mechanics加工machining液压hydraulic pressure切线tangent机电一体化mechanotronics mechanical-electrical integration 气压air pressure pneumatic pressure稳定性stability介质medium液压驱动泵fluid clutch液压泵hydraulic pump阀门valve失效invalidation强度intensity载荷load应力stress安全系数safty factor可靠性reliability螺纹thread螺旋helix键spline销pin滚动轴承rolling bearing 滑动轴承sliding bearing 弹簧spring制动器arrester brake十字结联轴节crosshead 联轴器coupling链chain皮带strap精加工finish machining 粗加工rough machining 变速箱体gearbox casing 腐蚀rust氧化oxidation磨损wear耐用度durability随机信号random signal离散信号discrete signal超声传感器ultrasonic sensor集成电路integrate circuit挡板orifice plate残余应力residual stress套筒sleeve扭力torsion冷加工cold machining电动机electromotor汽缸cylinder过盈配合interference fit热加工hotwork摄像头CCD camera倒角rounding chamfer优化设计optimal design工业造型设计industrial moulding design 有限元finite element滚齿hobbing插齿gear shaping伺服电机actuating motor铣床milling machine钻床drill machine镗床boring machine步进电机stepper motor丝杠screw rod导轨lead rail组件subassembly可编程序逻辑控制器Programmable Logic Controller PLC 电火花加工electric spark machining电火花线切割加工electrical discharge wire - cutting相图phase diagram热处理heat treatment固态相变solid state phase changes有色金属nonferrous metal陶瓷ceramics合成纤维synthetic fibre电化学腐蚀electrochemical corrosion车架automotive chassis悬架suspension转向器redirector变速器speed changer板料冲压sheet metal parts孔加工spot facing machining 车间workshop工程技术人员engineer气动夹紧pneuma lock数学模型mathematical model 画法几何descriptive geometry 机械制图Mechanical drawing 投影projection视图view剖视图profile chart标准件standard component零件图part drawing装配图assembly drawing尺寸标注size marking技术要求technical requirements 刚度rigidity内力internal force位移displacement截面section疲劳极限fatigue limit断裂fracture塑性变形plastic distortion脆性材料brittleness material刚度准则rigidity criterion垫圈washer垫片spacer直齿圆柱齿轮straight toothed spur gear 斜齿圆柱齿轮helical-spur gear直齿锥齿轮straight bevel gear运动简图kinematic sketch齿轮齿条pinion and rack蜗杆蜗轮worm and worm gear虚约束passive constraint曲柄crank摇杆racker凸轮cams共轭曲线conjugate curve范成法generation method定义域definitional domain值域range导数\\微分differential coefficient求导derivation定积分definite integral不定积分indefinite integral曲率curvature偏微分partial differential毛坯rough游标卡尺slide caliper千分尺micrometer calipers攻丝tap二阶行列式second order determinant逆矩阵inverse matrix线性方程组linear equations概率probability随机变量random variable排列组合permutation and combination气体状态方程equation of state of gas动能kinetic energy势能potential energy机械能守恒conservation of mechanical energy 动量momentum桁架truss轴线axes余子式cofactor逻辑电路logic circuit触发器flip-flop脉冲波形pulse shape数模digital analogy液压传动机构fluid drive mechanism 机械零件mechanical parts淬火冷却quench淬火hardening回火tempering调质hardening and tempering磨粒abrasive grain结合剂bonding agent砂轮grinding wheel2D Solid 二维实体2D Wireframe 二维线框3D Array 三维阵列3D Dynamic View 三维动态观察3d objects 三维物体3D Orbit 三维轨道3D Orbit 三维动态观察3D Studio 3D Studio3D Viewpoint 三维视点3dpoly 三维多段线3dsin 3DS 输入3DSolid 三维实体3dsout 3DS 输出abort 放弃abort 中断absolute coordinates 绝对坐标abut 邻接accelerator key 加速键access 获取acisin ACIS 输入acisout ACIS 输出action 操作active 活动（的）adaptive sampling 自适应采样add 添加Add a Printer 添加打印机Add mode 添加模式Add Plot Style Table 添加打印样式表Add Plot Style Table 添加打印样式表Add Plotter 添加打印机Add Plotter 添加打印机Add to Favorites 添加到收藏夹ADI ADI(Autodesk 设备接口)adjacent 相邻Adjust 调整Adjust Area fill 调整区域填充AdLM (Autodesk License Manager) AdLM（Autodesk 许可管理器）Administration dialog box 管理对话框Advanced Setup Wizard 高级设置向导Aerial View 鸟瞰视图affine calibration 仿射校准alert 警告alias 别名aliasing 走样align 对齐aligned dimension 对齐标注alignment 对齐(方式) allocate 分配Altitude 标高ambient color 环境色ambient light 环境光angular dimension 角度标注angular unit 角度单位annotation 注释anonymous block 无名块anti-aliasing 反走样aperture 靶框apparent intersections 外观交点append 附加Application key 授权申请号appload 加载应用程序Apply 应用/申请approximation points 近似点arc 圆弧Architectual Ticks 建筑标记area 区域,面积Argument 参数Arrange icons 排列图标array 阵列arrowhead 箭头ASCII ASCII （美国标准信息交换码）aseadmin ASE 管理aseexport ASE 输出aselinks ASE 链接aserows ASE 行aseselect ASE 选择asesqled SQL 编辑Aspect 纵横向间距aspect ratio 宽高比assign 指定Assist 助理associative dimension 关联标注associative hatches 关联填充attach v.附着attdef 属性定义attdisp 属性显示attedit 属性编辑attenuation 衰减attenuation of light 灯光衰减attext 属性提取attredef 属性重定义attribute definition 属性定义Attribute Display 属性显示attribute extraction file 属性提取文件attribute extraction template file 属性提取样板文件attribute prompt 属性提示attribute tag 属性标签attribute value 属性值audit 核查authorization code 授权码AutoCAD library search path AutoCAD 库搜索路径autocommit 自动提交AutoTrack 自动追踪axis tripod 三轴架azimuth 方位角Back Clipping On 后向剪裁打开back view 后视图background color 背景色backup 备份Backward 反向bad 不正确的base 基点base dimension 基准标注base grips 基夹点base point 基点baseline 基线baseline dimension 基线标注Basic color 基本色batch plotting 批处理打印beam angles of spotlights 聚光灯光束角度Beep on Error 出错报警bevel 倒角bevel 倒角beveling objects 斜角对象Bezier curve Bezier 曲线Big Font 大字体bind 绑定bitmap 位图blend 合成blipmode 点标记模式block 块block definition 块定义block reference 块参照block table 块表bmpout BMP 输出body 体Boolean operation 布尔运算borders 边框bottom view 仰视图boundary 边界boundary sets 边界集bounding 边（框）break (v.) 打断Bring Above Object 置于对象之上Bring to Top 顶置brower 浏览器built-in 内置的bulge 凸度bump map 凹凸贴图button menu 按钮菜单BYBLOCK 随块BYLAYER 随层byte 字节cabling 电缆布线cal 计算器calibrate 校准call 调用callback 回调(for LISP) callback 回叫camera 相机camera angle 相机角度Cancel 取消cap 封口cascade 层叠（的）case （大小）写cast 投射catalog 目录cell 单元Center 圆心center mark 圆心标记centerline 中心线centroid 形心,质心chamfer 倒角change 修改character 字符Check 检查Check Box 复选框Check Spelling 拼写检查child dimension style 下级标注样式chord 弦chprop 修改特性circle 圆circular external reference 循环外部参照Circumference 圆周class 类clause 子句Clean 清除Clean 清除clear 清除client 客户机clip 剪裁Clipboard 剪贴板clipping boundaries 剪裁边界clipping planes 剪裁平面Close 闭合cluster 组code pages 代码页color 颜色color depth 颜色深度color map 色表Color Wheel 颜色轮盘color-dependent 颜色相关Color-Dependent Plot Style Table 颜色相关打印样式表column 列combine 结合comma 逗号command history 命令历史command line 命令行command window 命令窗口comment 注释commit 提交Compatibility 兼容性compile 编译complex solids 复杂实体component 分量Component 部件composite regions 组合面域composite tolerances 混合公差compound documents 合成文档cone 圆锥面config 配置configuration 配置conic 圆锥曲线Constant 常量Construction Line 构造线containers 容器Content Explorer 内容管理器Content Explorer 内容管理器contents 内容context 环境continuation 顺序Continue 继续Continue 继续continue dimension 连续标注contour lines 轮廓线contrast 对比度contrast 对比度control (n.)控件coordinate filters 坐标过滤器coordinate system 坐标系coplanar 同一平面的copy 副本Copy Hyperlink 复制超级链接copyclip 复制剪贴copyhist 复制历史copylink 复制链接Corner 角点corner joint 角点correct v. 更正corrupt 毁损create 创建Create Layout 创建布局criteria 条件criteria 条件Crop 修剪crosshair 十字光标crossing 窗交crossing polygon 交叉多边形crossing window 交叉窗口cube 立方体Cubic 三次Current UCS 当前UCS Current UCS 当前UCS cursor 游标cursor menu 光标菜单curve 曲线custom 自定义Custom installation 自定义安装Customize Menus 自定义菜单Customize Menus 自定义菜单Cut 剪切cutclip 剪切cylinder 圆柱体cylindrical coordinate 柱坐标dangle 不固定的Dark Color 暗色dash 虚线data integrity 数据完整性database 数据库datum 基准datum axis 基准轴datum dimension 基准标注datum identifier 基准标识datum reference frames 基准参考框架datum reference letters 基准参考字母dbConnect 数据库连接dbConnect Manager 数据库连接管理器dblist 数据库列表DBMS drivers DBMS 驱动ddattdef 属性定义对话框ddatte 属性编辑对话框ddattext 属性提取对话框ddcolor 颜色对话框ddedit 文字编辑对话框ddgrips 夹点对话框ddim 标注设置对话框ddinsert 插入对话框ddmodify 图元编辑对话框ddptype 点类型对话框ddrename 重命名对话框ddrmodes 绘图模式对话框ddselect 对象选择对话框dducs UCS 对话框dducsp UCS 方向对话框ddunits 单位对话框ddview 视图对话框ddvpoint 视点对话框deactivate 释放dealer 经销商decal effect 修剪效果decimal dimensions 十进制标注decurve 非曲线化default 缺省default drawing 缺省图形default drawing 缺省图形definition point 定义点Degenerate 退化delay 延迟delete 删除DELta 增量demand loading 按需加载dependent symbols 依赖符号deployment 展开Depth Map 深度贴图derive 导出description 说明Design Center 设计中心detach 拆离Detection 检测deviation 极限偏差deviation tolerances 极限公差device 设备device 设备Device and Default Selection 设备和默认选择Dia 直径diameter 直径（标注）dictionary 词典diffuse color 漫射色digitizer 数字化仪digitizing puck 数字化仪游标digitizing puck 数字化仪游标dim 标注dimaligned 对齐标注dimangular 角度标注dimbaseline 标注基线dimcenter 圆心标注dimcontinue 连续标注dimdiameter 直径标注dimedit 标注编辑dimension 标注dimension definition points 标注定义点dimension format 标注格式dimension geometry 构成要素dimension line arc 尺寸线圆弧dimension properties 标注特性dimension scale 标注比例dimension style 标注样式dimension style families 标注样式族dimension style name 标注样式名dimension style overrides 标注样式替代dimension text 标注文字dimension units 标注单位dimension variables 标注变量dimlinear 线性标注dimordinate 坐标标注dimoverride 标注替代dimradius 半径标注dimstyle 标注样式dimtedit 标注文字编辑Direct Hatch 直接填充Direction Control 方向控制Directory 目录Disable 禁用discard 放弃Discontinued 停止使用的dish 下半球面disk space 磁盘空间displacement point 位移点display 显示Display Order 显示次序dist 距离distant light 平行光distributing 分布dithering 抖动diverge 分散的Divide 等分divide 等分Division 等分dock(undock) 固定（浮动）document 文档dome 上半球面donut 园环Draft 草图drafting standards 绘图标准drafting techniques 绘图技术drag and drop 拖放draw 绘制/绘图(如果后面未接宾语) drawing 图形Drawing Aids 绘图辅助工具drawing area 绘图区域drawing boundaries 图形边界drawing browser 图形浏览器drawing database 图形数据库drawing environment 图形环境drawing extents 图形范围drawing file 图形文件drawing limits 图形界限drawing order 图形次序drawing project 图形项目drawing scale 图形比例drawing standard 图形标准drawing status 图形状态drawing time 绘图时间drawing units 图形单位driver 驱动程序Dropdown List 下拉列表dsviewer 鸟瞰视图dtext 动态文本dump 转储duplicate 重复duplicating 复制dview 动态观察dxbin DXB输入dxfin DXF输入dxfout DXF输出Dynamic 动态Dynamic Dragging 动态拖动Dynamic Update 动态更新dynamic viewing 动态观察dynamic zooming 动态缩放edge 边Edge Surface 边界曲面edgesurf 边界曲面editor 编辑器EDUCATION VERSION 教学版effect 效果Element 元素elev 标高elevation 标高ellipse 椭圆embed 内嵌，嵌入Encapsulated 封装See also "EPS" end 端点end angle 端点角度end tangent 端点切向end width 端点宽度Ending 终止English units 英制单位ENTER ENTER（输入）entity 图元entry 条目environment 环境environment variable 环境变量equation 方程式erase 删除existing 现有的Exit 退出export 输出expression 表达式extend 扩展extend 延伸，超出量（用于标注）extension line 尺寸界线Extent(s) 范围external data 外部数据external database 外部数据库External Reference 外部参照Extract 选集extrude 拉伸face 面facet 镶嵌面factor 因子(see Scale Factor)fade 褪色度falloff angle 收缩角Fast Zoom mode 快速缩放模式fatal 致命错误Favorites 收藏夹Favorites 收藏夹feature 功能/（几何）特征fence 栏选(See also Selection fence field 字段file 文件fill 填充Filled Text 填充文字Filmroll Filmrollfilter 过滤器find 查找finish 完成finish 修饰（for render only)fit 自适应设置fit points 拟合点flag 标志Flat Shaded, Edges on 带边框平淡着色flat-shaded （平淡）着色floating viewports 浮动视口flood 布满Flyout Properties 弹出特性对话框fog 雾fold 折叠Follow 跟随font 字体font map file 字体映射文件form tolerance 形状公差formatting text 设置文字格式frame 框架frame 帧frame 边框free-form 自由形式（的）freehand line 徒手画线Freeplotting 自由绘图freeze 冻结freezing layers 冻结图层From 自Front Clipping On 前向剪裁打开front view 主视图Full Preview 全视口预览General 基本generate 生成geometric characteristic symbols 几何特性Geometric Tolerance 形位公差geometry 几何图形global 全局（的）Gouraud renderings Gouraud Gouraud 着色Gradient 百分度Grads 百分度graphic area 图形区graphics cursor 图形光标graphics screen 图形屏graphics window 图形窗口graphscr 图形屏gray 灰度grid 网格grid 网格,栅格grid mode 栅格模式grip 夹点group 编组group code 组码Haltftoning 半色调handle 句柄handshaking 握手信号Hardcopy 硬拷贝hardware linetype 硬件线型hardware lock 保密锁hardware requirement 硬件需求hatch 图案填充hatch areas 填充区域hatch boundaries 填充边界hatch pattern 填充图案hatch styles 填充样式hatchedit 填充编辑Heads-up Design 轻松设计Height 高度help 帮助Hidden Line 隐藏线hidden-line image 消隐图像hide 隐藏(adj.)hide 消隐(v.)hideplot 消隐出图highlight 突出显示highlight 亮光(for the color of 3D objeHIGHLIGHT 亮显home page 主页home position 起始位置hook line 钩线hotspot 聚光角Hyperlink 超级链接Icon 图标Identifier 标识符IGES (International Graphics Exchange Sp 初始图形交换标准Ignore 忽略image 图像implement 实现import 输入Imprint 压印Imprint 压印included angle 包含角infinite lines 无限长线information 信息Inherit Properties 继承特性initial environment 初始环境Initialize 初始化in-place 在位inquiry 查询inscribed polygons 内接正多边形insert 插入Insertion 插入点insertobj 插入对象instance 引用Instruction 指示integer 整数IntelliMouse 智能鼠标IntelliMouse 智能鼠标Intensity 强度interactive 交互的Interchange 互换interface 介面interfere 干涉Interference 干涉Interference 干涉interlace 隔行internal 内部的Internet Utilities Internet应用程序interpolation points 插值点intersect 交集interval 间距/间隔Invalid 无效（的）inverse linear attenuation of light 线性衰减inverse square attenuation of light 平方衰减Invisible 不可见invoke 调用island 孤岛Island detection 孤岛检测ISO (International Standards Organizatio ISO(国际标准化组织) Isolation levels 隔离级别isoline 素线isometric 等轴测isometric snap style 等轴测捕捉样式isometric view 等轴测视图isoplane 等轴测平面ISOPLANE 等轴测平面Italic 斜体iterator 枚举器join 合并joint 连接Justification 对正justify 对正key 主键key n.名称knot vector 节点矢量label 标签landscape 横向Landscape 配景，风景,if used for "landsc Landscape Edit 编辑配景Landscape Library 配景库lateral tolerance symbol 尺寸公差符号layer index 图层索引Layout 布局Layout from Template 来自样板的布局Layout from Template 来自样板的布局Layout Wizard 布局向导leader 引线Leading 前导Learning Assistance 学习助手Left View 左视图legacy 传统length 长度lengthen 拉长Lens Length 镜头长度License Agreement 许可协议license key 许可证号License Manager 许可管理器License Server 许可服务器licenses 授权light 光源light color 浅色(for RAMT-wood ) Light Color 暖色lighting 照明效果Lightness 亮度limits 图形界限line 行line 直线line font 线型line object 线性对象line segment 线段line width 线宽Linear Dimension 线性标注linetype 线型Lineweight 线宽link 链接list 表list 列表list box 列表框List Files 文件列表load 加载locale 局部location 网址(for Internet) location 位置(for file and directory) locked 锁定locked layer 锁定图层logfileoff 关闭日志文件logfileon 打开日志文件logical 逻辑long file name 长文件名loop 环ltscale 线型比例LTSCALE 线型比例Ltype 线型lump 块magnet 磁吸magnification （缩放）比例magnifying glass 放大镜main window 主窗口major axis 长轴Make 新建Make 建立malformed 有缺陷Manipulate 操作Manipulate 操作Manufacture 制造商mapper 贴图mapping 贴图marble 大理石mark 标记massprop 质量特性match 匹配material condition 包容条件material condition symbols 包容条件符号Materials 材质对话框Materials Library 材质库matlib 材质库measure 测量measure 等距等分（菜单内容）Measurement 测量单位MEASUREMENT 在图形中设置测量值member 成员memory 内存menu 菜单menu bar 菜单栏menuload 加载菜单Merge 合并message 信息metafile 图元文件Method 方法Middle 中央点Middle Center 正中点Midpoint 中点minsert 多重插入mirror 镜像mirror line 镜像射线mirror3d 三维镜像Miscellaneous File Names 其他文件名Miter 斜接mledit 多线编辑mline 多线mlstyle 多线样式mode 模式model 型号model 型号model (v)建模，(n)模型model space 模型空间Modification 修改monitor resolution 显示器分辨率monochrome 单色（的）mouse 鼠标move 移动mslide 制作幻灯mspace 模型空间mtext 多行文字Multiline 多线比例multiline scale 多线比例Multiline Style 多线样式mview 生成视口mvsetup 设置图纸规格named object 命名对象named plot style table 命名的打印样式表Named UCS 命名UCSnamed view 命名视图NE Isometric 东北等轴测NE Isometric 东北等轴测Nearest 最近点nested 嵌套nested blocks 嵌套块new 新建new 新的New Layout 新建布局New Layout 新建布局NURB surfaces NURBS（非一致有理B 样条曲线）NW Isometric 西北等轴测NW Isometric 西北等轴测object 对象Object Properties 对象特性Object Properties Manager 对象特性管理器Object Snap 对象捕捉object snap override 对象捕捉覆盖Object Snap Setting 对象捕捉设置objects 对象Oblique 斜尺寸界线obsolete 废弃ODBC database ODBC数据库offset 偏移OLE OLE（对象链接和嵌入）olelinks OLE链接online help 联机帮助online manuals 联机手册oops 恢复opacity map 不透明贴图open 打开open 打开(的)OPM(Object Property Manager) 对象特性管理器Optimization 优化option 选项Orbit 轨道Ordinate 坐标标注orientation 方向origin 原点（对于坐标系）origin 原始位置(used for external Dat origin 起点（对于尺寸界线和标注）ortho 正交Ortho mode 正交模式orthogonal 正交orthographic 正交osnap 对象捕捉outside 外部的overall dimension scale 全局比例overflow 溢出overlay 覆盖overline 上划线override 忽略- For common sentences override 替代- For dimension overwrite 覆盖Palette 控制板。