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1、3formulation公式dynamic behavior动态性能(xngnng)translational mechanical system线性机械系统element law元件定律、元素定律interconnection 连接2022-4-101第1页/共42页第一页，共43页。Chapter 3 Formulation and Dynamic Behavior of Translational Mechanical Systems线性机械系统的公式(gngsh)和动态性能Syllabus3.1 Introduction3.2 Variables3.3 Element Laws元件定律

2、元件定律(dngl)3.4 Interconnection Laws连接定律连接定律(dngl)3.5 Obtaining the System Model第2页/共42页第二页，共43页。3.1in advance事先describe描述(mio sh) motion patterns 运动模式quantitatively量化地 reveal揭示 parameters参数 performance功能derivation of the model 模型的推导 describe描述(mio sh) 2022-4-103第3页/共42页第三页，共43页。3.1 Introduction2022-4

3、-104p In order to analyze, synthesize and design a dynamic system, an accurate mathematical model must be determined in advance事先.p Mathematical models describe描述 the motion patterns 运动模式of physical systems.p Mathematical models quantitatively量化地 reveal揭示 the relationship between systems parameters参

4、数 and performance功能(gngnng), and the dynamic behavior of the system as well.p The derivation of the model 模型的推导is based on the fact that the dynamic system can be completely described完全描述 by known differential equations or experimental test data.p The ability to analyze the system and determine its

5、performance depends on how well the characteristics can be expressed mathematically.3.1.1 Concepts of Mathematical Models第4页/共42页第四页，共43页。2022-4-105Static models: reveal the system motion patterns 系统(xtng)运动模式independent of time.Dynamic models: reveal the system motion patterns dependent of time, in

6、cluding:External models: they only reveal relationship between the input and the corresponding output of a control system, e. g. differential equation, transfer function, etc.Internal models: they reveal relationship among the input, output and the internal variables, e. g. state space expression, e

7、tc.3.1.2 Types of Mathematical Models第5页/共42页第五页，共43页。3.2Displacement 位移(wiy)Velocity速度Force力Acceleration加速度2022-4-106第6页/共42页第六页，共43页。3.2 Variables2022-4-107The symbols for the basic variables used to describe the dynamic behavior of translational mechanical systems are:p x - Displacement 位移(wiy)in

8、 meter ( m )p v - Velocity速度 in meter per second ( m/s )p f - Force力 in Newton ( N )p a - Acceleration加速度 in meter per second square ( m/s2 )第7页/共42页第七页，共43页。3.3Mass质量friction摩擦damping阻尼(zn) stiffness刚性spring弹簧momentum动量oil film油膜laminar flow层流depict描绘 2022-4-108第8页/共42页第八页，共43页。viscous friction粘性(z

9、hn xn)摩擦friction coefficient 摩擦系数 unit单位proportional成正比的viscosity粘性(zhn xn) inversely proportional 成反比的dashpot阻尼器shock absorbers 吸振器2022-4-109第9页/共42页第九页，共43页。cylinder圆柱(yunzh)symbol符号elongation变形external forces外力equal in magnitude大小相等的opposite in direction 方向相反的equal magnitude大小相等opposite direction

10、方向相反2022-4-1010第10页/共42页第十页，共43页。3.3 Element Laws元件定律(dngl)/元素定律(dngl)p Physical devices are represented by one or more idealized elements that obey laws involving the variables associated with the elements. p Some degree of approximation is required in selecting the elements to represent a device,

11、and the behavior of the combined elements may not correspond exactly to the behavior of the device. p The major elements that we include in translational systems are mass 质量, friction(damping)摩擦（阻尼）, stiffness(spring)刚性(n xn)（弹簧）. 2022-4-1011第11页/共42页第十一页，共43页。2022-4-1012Newtons First LawWhen viewed

12、 in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force. Newtons Second LawThe vector sum of the external forces 外力(wil)F on an object is equal to the mass m of that object multiplied by the acceleration ve

13、ctor a of the object: F = ma.Newtons Third LawWhen one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction 大小相等方向(fngxing)相反on the first body.Review第12页/共42页第十二页，共43页。2022-4-10132.)()()(F(t)(B(t)(BF(t)()()(F(t)dttaKdttvKtKx

14、dttaxBtvtxmtvmtma第13页/共42页第十三页，共43页。3.3.1 Mass质量(zhling)p A mass M, which has units of kilograms (kg), subjected to a force f. p Newtons second law states that the sum of the forces acting on a body is equal to the time rate of the change of the momentum动量(dngling):()dMvfdt which, for a constant mas

15、s定常质量(zhling), can be written as:dvMfdt (3.4)2022-4-1014第14页/共42页第十四页，共43页。Mass M22( )( )( )MddftMv tMx tdtdt 2022-4-1015fM(t)(3.5)第15页/共42页第十五页，共43页。3.3.2 Friction A mass sliding on an oil film油膜that has laminar flow层流, as depicted描绘 in Fig. 3.3(a), is subject to viscous friction粘性(zhn xn)摩擦and obe

16、ys the linear relationship: fB vp The friction coefficient B摩擦系数 has units 单位(dnwi)of Newton-second per meter (Ns/m).p B is proportional成正比 to the contact area and to the viscosity粘性 of the oil, and inversely proportional 成反比to the thickness of the film. pv = v2-v1Figure 3.3 (a) Friction described b

17、y (3.6) with v = v2-v12022-4-1016(3.6)第16页/共42页第十六页，共43页。 Viscous friction also may be used to model a dashpot阻尼器, such as the shock absorbers 吸振器on an automobile. As indicated in Fig. 3.4(a), a piston moves through an oil-filled cylinder装满油的圆柱, and there are small holes in the face of piston throug

18、h which the oil passes as the parts零件(ln jin) move relative to each other. The symbol符号 often used for a dashpot is shown in Fig. 3.4(b). Figure 3.4 (a) A dashpot阻尼器. (b) Its representation表示(biosh).2022-4-1017(a)(b)第17页/共42页第十七页，共43页。Damping B 1212( )( )( )( )( )( )( )Bdd x tftBx tx tBdtdtB v tv tB

19、 v t 2022-4-1018(3.7)第18页/共42页第十八页，共43页。3.3.3 Stiffness刚性(n xn) The most common stiffness element刚性(n xn)元件 is the spring.Figure 3.6 (a)d0: the length of the spring when no force is applied;x: the elongation变形(bin xng) caused by the force f.d(t) = d0 + x: the total length at any instantFigure 3.6 (b

20、) The stiffness property refers to the algebraic relationship between x and fFor a linear spring, the curve in Figure 3.6(b) is a straight line and f = Kx, where K is a constant with units of Newtons per meter (N/m).Figure 3.6 Characteristic of a general spring2022-4-1019(a)(b)第19页/共42页第十九页，共43页。Spr

21、ing k1212( )( )( )( ) ( )( )( )kftk x tx tk x tkv tv t dtkv t dt2022-4-1020(3.8)第20页/共42页第二十页，共43页。3.4Interconnection 相互连接DAlemberts Law达朗贝尔定理momentum动量(dngling)Fictitious force虚力Inertial force内力Reaction Forces相互作用derivative导数2022-4-1021第21页/共42页第二十一页，共43页。3.4 Interconnection Laws相互连接定律(dngl)3.4.1 D

22、Alemberts Law达朗贝尔定理(dngl)DAlemberts law is just a restatement of Newtons Second Law governing the rate of the change of momentum动量(dngling). For a constant mass, we can write:ext()iidvfMdt (3.9) (fext)i: all the external forces acting on the body; Forces and velocity: vector quantities, can be treat

23、ed as scalars (the motion is constrained to be in a fixed direction). Rewriting (3.9) as:ext()0iidvfMdt (3.10)2022-4-1022第22页/共42页第二十二页，共43页。ext()0iidvfMdt Suggests from Eq.(3.10): the mass can be considered to be in equilibrium. That is the sum of the forces = 0 (Mdv/dt as an additional force).Mdv/

24、dt: the Fictitious force虚力, the Inertial force内力(nil).Including Inertial force along with the external forces, rewrite (3.10) as :(3.10)0iif (3.11)2022-4-1023fi: all the forces acting on the body, including the Inertial force.第23页/共42页第二十三页，共43页。3.4.2 The Law of Reaction Forces相互作用力定律(dngl)Newtons T

25、hird Law states( regarding reaction forces): Accompanying any force of one element on another, there is a reaction force on the first element of equal magnitude and opposite direction.fk : exerted by mass on the right end of spring, with positive sense defined to be to right.fk : acts on mass a reac

26、tion force of equal magnitude with its positive sense to left.fk : the fixed surface exerts a force on the spring with the positive sense to the left.fk : the spring exerts an force with the equal magnitude and opposite sense on the surface.2022-4-1024Figure 3.8 Example of reaction forces.第24页/共42页第

27、二十四页，共43页。3.5 intermediate variable中间(zhngjin)变量Translational system线性系统 horizontal force水平力respectively分别gravity重力2022-4-1025第25页/共42页第二十五页，共43页。Parallel Combinations并联Series Combinations串联identical相等的stretch拉伸 compress压缩(y su) substitute代替2022-4-1026第26页/共42页第二十六页，共43页。3.5 Obtaining the System Mod

28、elSome notes:p The system model must obey both the element laws(displacement x, velocity v, and acceleration a=dv/dt) and the interconnection laws.p Write all the element laws in terms of displacement x and its derivatives导数(do sh) (or in terms of x, v, dx/dt and dv/dt). p Choose the assumed positiv

29、e directions for a, v, and x to be the same, so it will not be necessary to indicate all three positive directions on the diagram. p Throughout this PPT, dots over the variables are used to denote derivatives with respect to time. For example,22 and dxd yxydtdt2022-4-1027第27页/共42页第二十七页，共43页。3.5.1 Fr

30、ee-Body DiagramsEXAMPLE 3.1Draw the free-body diagram and apply DAlemberts law to write a differetial equation for the system shown in Fig. 3.9(a). The mass is assumed to move horizontally on frictionless bearings, and the spring and dashpot are linear. fa(t) is the applied force, also is the input

31、variable. x(t) is the displacement of the mass M, also is the output variable.(v(t) is just an intermediate variable中间(zhngjin)变量 and should not exist in the final differential equation)Fig. 3.9 (a) Translational system 线性系统2022-4-1028第28页/共42页第二十八页，共43页。ANALYSISThe horizontal forces水平(shupng)力, whi

32、ch are included in the free-body diagram, are:p fK, the force exerted by the springp fB, the force exerted by the dashpot.p fI, the inertial force / the fictitious force.p fa(t), the applied force.2022-4-1029Fig. 3.9 (a) Translational system.(b) Free-body diagram.(c) Free-body diagram including elem

33、ent laws.p fK = Kxp fB = Bvp fI =Mdv/dt p fa(t), be given第29页/共42页第二十九页，共43页。Fig. 3.9 (a) Translational system.(b) Free-body diagram.(c) Free-body diagram including element laws.DAlemberts law can now be applied to the free-body diagram in Fig. 3.9(c), pay attention to the assumed arrow directions.

34、If forces acting to the right are regarded as positive, the law yields:( )()0af tMvBvKx Replacing v and by and by , respectively分别(fnbi), and rearranging the terms, we can rewrite this equation as:( )aMxBxKxf t2022-4-1030v SOLUTION第30页/共42页第三十页，共43页。EXAMPLE 3.2Draw the free-body diagrams for the two

35、-mass system shown in Fig. 3.10(a) and use DAlemberts law to write the two differential equations that describe mass 1 and mass 2, respectively. x1(t) and x2(t) are the displacements of the mass 1 and mass 2, respectively. fa (t) is the applied force. K1, K2 and B are coefficients for spring 1, spri

36、ng 2 and dashpot, respectively.Fig. 3.10 (a) Translational system.2022-4-1031第31页/共42页第三十一页，共43页。SOLUTIONBecause there are two masses that can move with different unknown velocities, a separate free-body diagram is drawn for each one:Fig. 3.10 (b) Free-body diagram for mass 1.Fig. 3.10 (c) Free-body

37、 diagram for mass 2.212211111()()0B xxKxxM xK x2221221( )()()0af tM xB xxKxx2022-4-1032第32页/共42页第三十二页，共43页。EXAMPLE 3.3Fig. 3.11 (a) Translational system with vertical motion.SOLUTIONFig. 3.11 (b) Free-body diagram.Draw free-body diagram, including the gravity重力(zhngl), and find the differential equa

38、tion describing the motion of the mass shown in Fig. 3.11(a). x is the displacement of the mass. fa(t) is the applied force. K and B are coefficients for spring and dashpot, respectively.2022-4-1033Assume that x is the displacement from the position corresponding to a spring that is neither stretche

39、d nor compressed. The gravity on the mass is Mg. By summing the forces on the free-body diagram, we obtain:( )aMxBxKxf tMg第33页/共42页第三十三页，共43页。Two springs or dashpots are said to be in parallel if the first end of each is attached to the same body and if the remaining ends are also attached to a comm

40、on body.3.5.2 Parallel Combinations并联(bnglin)EXAMPLE 3.4The system shown in Fig. 3.12(a) includes two linear springs between wall and mass M.Write the differential equation describing the motion of the mass. (Assume that the springs have the same unstretched length相同(xin tn)未拉伸长度/自然长度)Find the sprin

41、g constant Keq for a single spring that could replace K1 and K2. (Assume that the springs have the same unstretched length)Fig. 3.12(a) Translational system2022-4-1034第34页/共42页第三十四页，共43页。SOLUTION 1.If the unstretched lengths of the two springs are identical相等(xingdng)的, then they will have the same

42、elongation, denoted by x, when the mass is in motion. Summing the forces gives：Fig. 3.12(b) Free-body diagrams when the springs have the same unstretched lengths.12()( )aMxBxKKxf tFig. 3.12(a) Translational system.2022-4-1035第35页/共42页第三十五页，共43页。Fig. 3.12(a) Translational system.Fig. 3.12(c) Free-bod

43、y diagram when the combination of K1 and K2 is replaced by a single equivalent spring.If the combination of K1 and K2 is replaced by a single equivalent spring, then the system reduces to that shown in Fig. 3.12(d):( )eqaMxBxK xf tComparing the above two equations reveals that：12( )()( )eqaaMxBxK xf

44、 tMxBxKKxf t 12eqKKKSOLUTION 2.2022-4-1036第36页/共42页第三十六页，共43页。SUMMARYTwo parallel springs or dashpots have their respective ends joined，as shown in Fig. 3.13. From the last example, we see that for the parallel combination of two springs,12eqKKKSimilarly, it can be shown that for two dashpots in par

45、allel, as in part (b) of Fig. 3.13,12eqBBBFigure 3.13 Parallel combinations. (a) Keq = K1+K2 (b) Beq= B1+B22022-4-1037第37页/共42页第三十七页，共43页。3.5.3 Series Combinations串联(chunlin)Two springs or dashpots are said to be in series if they are joined at only one end of each element and if there is no other e

46、lement connected to their common junction.EXAMPLE 3.5When x1 = x2 = 0, the two springs shown in Fig. 3.14(a) are neither stretched拉伸(l shn) nor compressed压缩 . Draw free-body diagrams for the mass M and for the massless junction A.Write the equations describing the system with the displacement x1 not

47、 x2. (Show that the motion of point A is not independent of that of the mass M and that x1and x2 are proportional to one another)Find Keq for a single spring that could replace the combination of K1 and K2.Fig. 3.14 (a) Translational system.2022-4-1038第38页/共42页第三十八页，共43页。SOLUTION 1.Fig. 3.14 (b) Fre

48、e-body diagram of mass M.Fig. 3.14 (c) Free-body diagram of massless junction A.11112()( )aMxBxK xxf t22112()K xK xxEquation for M:Equation for junction A:2022-4-1039第39页/共42页第三十九页，共43页。12112KxxKK 11111121( )aKMxBxKxf tKK 1211112( )aK KMxBxxf tKK SOLUTION 2.11112()( )aMxBxK xxf t22112()K xK xxEquati

49、on for junction A :Equation for M:This equation describes the system formed when the two springs in Fig. 3.14(a) are replaced by a single spring for which,SOLUTION 3.1212eqK KKKK substitutingsimplifying2022-4-1040第40页/共42页第四十页，共43页。SUMMARYFor the two springs in Fig. 3.15 (a), the equivalent spring c

50、onstant is given,Similarly, For two dashpots in series, as in Fig. 3.15 (b), it can be shown that,Figure 3.15 Series combinations. (a) Keq = K1K2/(K1 +K2) (b) Beq = B1B2/(B1 +B2).1212eqK KKKK 1212eqB BBBB 2022-4-1041(a) (b) 第41页/共42页第四十一页，共43页。2022-4-1042感谢您的观赏(gunshng)！第42页/共42页第四十二页，共43页。NoImage内容(nirng)总结3。element law元件定律(dngl)、元素定律(dngl)。第1页/共42页。第2页/共42页。第41页/共42页。感谢您的观赏第四十三页，共43页。