运动副中摩擦力的确定资料
第5章机械的效率和自锁
P1 1
P’1
P2
Pk
2
k
P’2
P’k
Pr
总效率η不仅与各机器的效率ηi有关，而且与传递的功率 Pi有关。
设各机器中效率最高最低者分别为ηmax和ηmin 则有：
ηmin<η <ηmax
3.)混联 先分别计算，合成后按串联或并联计算。
P1 1
P2 P’d23‘ P’d3 4‘P’r 2
Pd
P”d23“ P”d3 P4“kP”r
无论F多大，滑块在F的作用下不可能运动
FR Ft F Fn
φβ 1
Ff
2
当驱动力的作用线落在摩擦角（锥）内时，则机械发生 自锁。
5.4.2转动副的自锁
a
对仅受单力F作用的回转运动副产 生的力矩为： Md=F·a
最大摩擦力矩为： Mf =FRρ
1F FR
2
当力F的作用线穿过摩擦圆(a<ρ)时，发生自锁。
Ff 21
简单平面移动副
2 FN21 G
Ff 21 fFN21 fG
v FN21
12
F 1
G
●槽面接触： fv= f / sinθ
G=(FN21 /2)sinθ+(FN21 /2)sinθ FN21 = G / sinθ Ff21= f FN21
= G (f / sinθ) =G fv
fv─当量摩擦系数。
第5章 机械的效率和自锁
本章教学内容
5.1运动副中摩擦力的确定 5.2考虑摩擦时机构的受力分析 5.3机构的效率 5.4机构的自锁
5.1 运动副中的摩擦力的确定
5.1.1移动副中摩擦力的确定
●水平面接触：
Ff 21 fFN21
4.3 计入运动副中摩擦的机构受力分析
4.3计入运动副中摩擦的机构受力分析当机械运转时，运动副中因存在摩擦而产生摩擦阻力。
在低副中，运动副两元素之间的相对运动为滑动，将产生滑动摩擦阻力；在高副中，运动副两元素之间的相对运动以滚动为主、兼有一定的相对滑动，将产生滚动摩擦阻力与滑动摩擦阻力。
4.3.1摩擦系数与摩擦角以及摩擦圆的半径若两个构件以单一的平面接触形成移动副，如图4.6(a)所示，其平面摩擦系数为f，摩擦角为φ，则 f 与φ存在以下关系=arctanfφ若两个构件以V形平面接触形成移动副，如图4.6(b)所示，则其当量摩擦系数f V＝f / sinθ，当量摩擦角φV=arctanf V由图4.6(b)得垂直方向的力平衡方程为φV为为此，水平向外的驱动力P为若两个构件形成转动副，转轴1作匀速转动，半径为r，其上作用有径向外力Q、驱动力矩M d，其余标注如图4.6(c)所示。
孔2对转轴1的摩擦阻力的合力为F21，支反力的合力为N21，F21与N21的合力R21＝－Q。
由图4.6(c)得出孔2对转轴1的摩擦阻力矩Mf为M f＝r×F21＝ρ×R21(4.11)摩擦阻力F21表达为定义ρ＝f V.r ，M f为M f＝r×(Qf V)＝(f V·r)×Q＝ρ×Q(4.13)4.3.2计入运动副中摩擦时曲柄滑块机构的受力分析在图4.7(a) 中，F21、F41、M f21、M f23、M f41、Md分别为F21产生的摩擦力矩M f21为M f21＝ρ×F21(4.14) M f21阻碍相对转动ω12，以及F23产生的摩擦力矩M f23为M f23＝ρ×F23(4.15) M f23阻碍相对转动ω32，由此确定连杆2上拉力作用线的位置。
F41形成的摩擦力矩M f41为M f41＝ρ×F41曲柄1上的驱动力矩Md为M d＝h1×F12(4.17)对于滑块3，由三力汇交原理得Fr与F23的交点即为F43应通过的点，由此确定F43的位置。
4移动副的摩擦
如果，P’为负值，成为驱动力的一部分，作用为促使滑块1沿斜面等速下滑。
F21 fN 21 tg f N 21 N 21
二、移动副中的摩擦（续）
2）总反力的方向
R21与移动副两元素接触面的公法线偏斜一摩擦角； R21与公法线偏斜的方向与构件1相对于构件2 的相对速度方向v12 的方向相反
3. 斜面滑块驱动力的确定 1）求使滑块1 沿斜面 2 等速上行时所需的水平驱动力P
2）两构件沿一槽形角为2q 的槽面接触
N21sinq = -Q
令 f fv sinq
F21 fN 21 f
Q f Q sinq sinq
F21 fN 21 f v Q
二、移动副中的摩擦（续）
3）两构件沿圆柱面接触 N21是沿整个接触面各处反力的总和。 整个接触面各处法向反力在铅垂方向的分力的总和等于外载荷Q。 取N21=kQ F21 （k ≈1～1.57） 令kf f v 4）标准式 不论两运动副元素的几何形状如何，两元素间产生的滑动摩擦力均可用通式：
（正行程）
根据力的平衡条件
P R Q 0 P Qtg( )
二、移动副中的摩擦（续）
2）求保持滑块1沿斜面2等速下滑所需的水平力 P’
（反行程）
根据力的平衡条件
P ' R Q 0
P Qtg( )
注意
当滑块1下滑时，Q为驱动力，P’为阻抗力，其作用为阻止滑块1 加速下滑。
fN 21 kfQ
F21 f v Q
F21 fN 21 f v Q
ƒv ------当量擦系数
来计算。
二、移动副中的摩擦（续）
5）槽面接触Biblioteka 应当运动副两元素为槽面或圆柱面接触时，均有ƒv＞ƒ 其它条件相同的情况下，沿槽面或圆柱面接触的运动副两元素之间所产生的 滑动摩擦力＞平面接触运动副元素之间所产生的摩擦力。 2. 移动副中总反力的确定 1）总反力和摩擦角 总反力R21 ：法向反力N21和摩擦力F21的合力。 摩擦角 ：总反力和法向反力之间的夹角。
机械原理-转动副中的摩擦分析
ρω12
21
l ,
P作用在摩擦圆外
M M f , 加速转动
l ， P与ρ 相切
R21
M M f ，临界平衡
l ,
P与ρ 相割
M M f ， 构件自锁
l
转动副的自锁条件：
r
F
ρω12
21
驱动力作用线与摩擦圆相割
此时，无论怎样增大驱动力， 其驱动力矩Md(=Pl )总小于由 它产生的摩擦阻力矩Mf 。
R21
小
结
转动副中的摩擦分析
总反力的确定：
总反力R21作用线切于摩擦圆 总反力R21对轴心的力矩Mf 的方向与ω 12的方向相反
总反力R21与驱动力P大小相等、方向相反
转动副的自锁条件：
驱动力作用线与摩擦圆相割
第 12 章 机械中的摩擦和机械效率
移动副中的摩擦
转动副中的摩擦 机械效率 机械的自锁
转动副中的摩擦
P ——作用在轴颈上的
l
M
P
r
P
驱动力 距离轴颈中心为 l
等效于
一对心力P和转矩M
Байду номын сангаас
转动副中的摩擦分析
P产生反力集： N21
P
r
l
M
P
N21 N21
力平衡：
Mf N 21
F F21 21
P N 21
摩擦力： ∆F21 摩擦力矩： M f F21r N21 f r
N 21
总摩擦力矩：
M f M f N21 f r
kN21 fr kPfr
P
r
l
4-4 运动副中摩擦力
FR21
FN21
φ
V12
1
Ff21
F2
G
P’
FR21
FN21
φ
V12
1
Ff21
2
2. 楔形摩擦（楔形滑块）时的摩擦力
V 形槽导轨： 楔角为 2 θ；载荷为 G；驱动力为 F
G
F
摩擦力计算：
当楔角对称于载荷 G 时， 则两侧产生相等的
正压力(FN21/2) 和摩擦力（Ff 21/2）
显然： 2 (FN21 /2) sinθ = G →
FR21 φ
1
Gα
G
V1
2
F
2
（2）反行程：G----驱动力，F’----工作阻力 滑块平衡：F’ + G + FR21= 0
显然：tg（α - φ ）= F’ / G F’ = G tg（α - φ）
正行程形式：F’ = G tg（α +（-φ））
F’
FR21
G
α-φ
FR21
V1
α-φ
2
1
F’
α
2
对于跑合过的径向轴颈（有一定间隙----线接触）分析如下：
总反力 FR21： F2 R21 2= FN21 2+ Ff 21
而 Ff 21 = f FN21
摩擦力矩 M f ：
M f = Ff 21 r
若在B点轴颈平衡，有：
M= M f
G + FR21 = 0
大小： FR21 = G
→ FR21 2= FN21 2+ ( f FN21 ) 2 = G2
G ---- 驱动力 F’ ---- 阻力
G
4
运动副中的摩擦
§4-2 运动副中的摩擦滑动摩擦（低副）滚动摩擦（高副）—摩擦小一、移动副的摩擦根据接触面的几何形状不同，三种情况，即单一平面接触、槽面接触和圆柱面接触。
GF N -=21库仑定律)(2121大小N f fF F =滑块1的总反力212121f N R F F F +=f F fF F F N N N f ===21212121tan φfarctan =φ★总反力F R21的方向恒与相对速度V 12方向成（90°+ ϕ）1、单一平面接触总反力摩擦角ϕF R21F N21F f 212V 12P G1平面摩擦在斜面机构中的应用1）滑块沿斜面等速上升（正行程）G ——铅垂载荷；P ——水平驱动力；F R21——滑块1所受的总反力。
为使滑块沿斜面等速上升，求P 。
021=++G F P R )tan(ϕα+=G P F R21α+ϕP GF R212）滑块沿斜面等速下降（反行程）G ——驱动力；P ’——阻止滑块加速下滑的阻力；F R21——滑块1所受的总反力。
0'21=++G F P R )tan('ϕα-=G P α-ϕP’GF R21F R21α-ϕ> 0 ，说明P′>0，保持匀速状态的力；α-ϕ≤0 ，P′≤0, G 的分力不足以使滑块运动，只有工作阻力变为驱动力时,滑块才能运动。
)tan('ϕα-=G P F R212、槽面摩擦槽形角2θG F N =⋅⋅θsin 2221θsin 21G F N =V N f Gf fG fF F ===θsin 2121当量摩擦系数θsin f f V =当量摩擦角V V f arctan =φf V ≥f , 常用楔槽面增大摩擦力，如V 带传动、三角形螺纹联接等。
但注意并非实际摩擦系数增大，而是将增大的F N21折合到 f 变为f v 。
楔形滑块F N21/2F f21F N21/2不论两运动副元素的几何形状如何，两元素间产生的滑动摩擦力均可用通式：3、圆柱面摩擦F N21是沿整个接触面各处反力的总和。
1重点与难点1移动副中的摩擦力及总反力的确定
5.1重点与难点5.1.1移动副中的摩擦力及总反力的确定由库仑摩擦定律知，摩擦力=f。
式中为构件2对构件1的摩擦力大小;f为摩擦系数，与构成运动副的两构件的材料有关;为构件2对构件1的正压力大小。
摩擦力总是阻碍两构件之间的相对运动的。
如图5.1所示。
因此，的方向总和相对运动速度的方向相反(为构件1相对构件2的运动速度)。
为分析问题方便，我们总是把正压力和摩擦力合成运动副总反力,与的夹角称为摩擦角φ(tanφ=f)。
因此，与的夹角总为钝角π/2+φ。
综上所述，在移动副中确定运动副总反力的方法如下:(1)运动副总反力,和正压力的夹角为φ;(2)运动副总反反力和相对速度的夹角为钝角π/2+φ。
摩擦力总是成对出现的，和总是大小相等,方向相反，在同一条直线上分别作用在不同的构件上。
而运动副总反力也总是成对出现的，和。
也是大小相等，方向相反。
在同一条直线上分别作用在不同构件上。
它们是一对作用力与反作用力。
摩擦力与外载荷的关系可以用表示。
式中，为铅垂外载荷大小；称为当量摩擦系数。
当量摩擦系数除了与摩擦系数有关外，还与运动副的形状有关。
在图5.1所示的平面移动副中，=f;在图5.2所示的槽面移动副中，=f/sinβ；在图5.3所示的柱面移动副中，=kf，k为1~1.57，k 值与运动副的接触状态有关，即在相同的外载荷作用下，运动副形状不同，生产的摩擦力不同。
这是由于运动副的形状不同，所产生的正压力不同而引起的。
5.1.2转动副中的摩擦力及总反力的确定转动副中的摩擦力=P。
如图5.4所示，摩擦力对轴颈中心的力矩即为摩擦力矩，该摩擦力矩应阻碍构件1对构件2的相对运动，因此和角速度的方向相反，=，式中，r为轴颈半径。
而正压力对轴颈中心的力矩等于零。
因此，运动副总反力对轴颈中心的力矩即为对轴颈中心的摩擦力矩，即=r=。
由力的平衡条件，=-P ，所以有=r 。
以为半径作一圆.这个圆称为摩擦圆。
运动副总反力恒切于摩擦圆。
因此，在转动副中确定运动副总反力的方法图5.4是:(1)运动副总反力，与外载荷P等值反向，并恒切于摩擦圆。
运动副的摩擦和机械效率
以平滑块1为研究对象
摩擦力 F21=f N21
总反力R21：正反力N21与摩擦力F21的合力。
R21
N21
v12
1
2 F21 P
一、平滑块的摩擦 总反力R21：正反力N21与摩擦力F21的合力。
摩擦角:总反力R21与正反力N21之间的夹角，
tg =F21/N21= ｆ
总反力R21的方向：与滑块1相对平面2的相对速度v12的
• 力平衡： R21= Q
• 力矩平衡：
Md= R21 = Mf
• 即：
Mf = fvQr=fv R21 r= R21 • 可得： = fv r
• 对于具体的轴颈， 为定值.
摩擦圆：以轴颈中心 O为圆心， 为半径的圆。 为摩擦圆半径。
❖转动副中总反力R21的方位根据以下三点确定：
• 1)总反力R21始终切于摩擦圆， • 2）总反力R21对轴颈中心之力
• 自锁条件： v
蜗杆传动的效率
• 正正行程 ：蜗杆为主动件 • =tg /tg( + v) • —蜗杆的升角（导程角）
• 反行程 ：蜗轮为主动件 • = tg( - v)/ tg • 自锁条件： v
返回
本章教学要求
•了解：作用在机械中的力的分类；机械的 瞬时机械效率的计算和机械的自锁条件。 理解：机械效率和自锁的概念。 掌握：运动副中总反力的确定和考虑摩擦 时机构的静力分析。
运动副的摩擦和机械效率
• 运动副的摩擦 • 考虑摩擦时机构的静力分析 • 机械效率
❖ 运动副的摩擦
• 效率是衡量机械性能的重要指标。 • 研究运动副中摩擦的主要目的在于寻找提高 机械效率的途径,以及合理利用摩擦来工作。
• 移动副的摩擦 • 螺旋副的摩擦 • 转动副的摩擦
ADAMS中运动副以及其摩擦力参数理论原理
JointsIdealized JointsAbout Idealized JointsIdealized joints connect two parts. The parts can be rigid bodies, Flexible bodies, or Point mass es. You can place idealized joints anywhere in your model.Note:The joints you can attach to flexible bodies depend on the version of Adams/Solver you are using (C++ or FORTRAN). In addition, Adams/Solver (C++) does not support pointmasses.For a summary of which joints and forces are supported on flexible bodies, see Table ofSupported Forces and Joints in the Adams/Flex online help. Also refer to the Adams/Flexonline help for more information on attaching joints and forces to flexible bodies. Adams/View supports two types of idealized joints: simple and complex. Simple joints directly connect bodies and include the following:•Revolute Joints. See Revolute Joint Tool.•Translational Joints. See Translational Joint Tool.•Cylindrical Joints. See Cylindrical Joint Tool.•Spherical Joints. See Spherical Joint Tool.•Planar Joints. See Planar Joint Tool.•Constant-Velocity Joints. See Constant-Velocity Joint Tool.•Screw Joints. See Screw Joint Tool.•Fixed Joints. See Fixed Joint Tool.•Hooke/Universal Joint. See Hooke/Universal Joint Tool.Complex joints indirectly connect parts by coupling simple joints. They include:•Gears. See Gear Joint Tool.•Couplers. See Coupler Joint Tool.You access the joints through the Joint Palette and Joint and Motion Tool Stacks.Creating Idealized JointsThe following procedure explains how to create a simple idealized joint. You can select to attach the joint to parts or spline curves. If you select to attach the joint to a curve, Adams/View creates a curve marker,Adams/View2Jointsand the joint follows the line of the curve. Learn more about curve markers with Marker Modify dialogbox help. Attaching the joint to a spline curve is only available with Adams/Solver (C++). L earn aboutswitching solvers with Solver Settings - Executable dialog box help.Note that this procedure only sets the location and orientation of the joint. If you want to set the frictionof a joint, change the pitch of a screw joint, or set initial conditions for joints, modify the joint.To create a simple idealized joint:1.From the Joint palette or tool stack, select the joint tool representing the idealized joint that youwant to create.2.In the settings container, specify how you want to define the bodies the joint connects. You canselect:• 1 Location (Bodies Implicit)• 2 Bodies - 1 Location• 2 Bodies - 2 LocationsFor more on the effects of these options, see the help for the joint tool you are creating andConnecting Constraints to Parts.3.In the settings container, specify how you want the joint oriented. You can select:•Normal to Grid - Lets you orient the joint along the current Working grid, if it is displayed, or normal to the screen.•Pick Geometry Feature - Lets you orient the joint along a direction vector on a feature in your model, such as the face of a part.4.If you selected to explicitly define the bodies by selecting 2 Bodies - 1 Location or 2 Bodies - 2Locations in Step 2, in the settings container, set First Body and Second Body to how you wantto attach the joint: on the bodies of parts, between a part and a spline curve, or between two splinecurves.ing the left mouse button, select the first part or a spline curve (splines and data element curvesare all considered curves). If you selected to explicitly select the parts to be connected, select thesecond part or another curve using the left mouse button.6.Place the cursor where you want the joint to be located (for a curve this is referred to as its curvepoint), and click the left mouse button. If you selected to specify its location on each part or curve,place the cursor on the second location, and click the left mouse button.7.If you selected to orient the joint along a direction vector on a feature, move the cursor around inyour model to display an arrow representing the direction along a feature where you want the jointoriented. When the direction vector represents the correct orientation, click the left mouse button.Modifying Basic Properties of Idealized JointsYou can change several basic properties about an Idealized joints. These include:•Parts that the joint connects. You can also switch which part moves relative to another part.3Joints •What type of joint it is. For example, you can change a revolute joint to a translational joint. Thefollowing are exceptions to changing a joint's type:•You can only change a simple idealized joint to another type of simple idealized joint or to a joint primitive.•You cannot change a joint's type if motion is applied to the joint. In addition, if a joint has friction and you change the joint type, Adams/View displays an error.•Whether or not forces that are applied to the parts connected by the joint appear graphically on the screen during an animation. Learn about Setting Up Force Graphics.•For a screw joint, you can also set the pitch of the threads of the screw (translational displacement for every full rotational cycle). Learn about screw joints.To change basic properties for a joint:1.Display the Modify Joint dialog box as explained in Accessing Modify Dialog Boxes.2.If desired, in the First Body and Second Body text boxes, change the parts that the joint connects.The part that you enter as the first body moves relative to the part you enter as the second body.3.Set Type to the type of joint to which you want to change the current joint.4.Select whether you want to display force graphics for one of the parts that the joint connects.5.For a screw joint, enter its pitch value (translational displacement for every full rotational cycle).6.Select OK.About Initial Conditions for JointsYou can specify initial conditions for revolute, translational, and cylindrical joints. Adams/View uses the initial conditions during an Initial conditions simulation, which it runs before it runs a simulation of your model.You can specify the following initial conditions for revolute, translational, and cylindrical joints:•Translational or rotational displacements that define the translation of the location of the joint on the first part (I marker) with respect to its location on the second part (J marker) in units oflength. You can set translational displacement on a translational and cylindrical joint and you can set rotational displacements on a revolute and cylindrical joint.Adams/View measures the translational displacement at the origin of the I marker along thecommon z-axis of the I and J markers and with respect to the J marker. It measures the rotational displacement of the x-axis of the I marker about the common z-axis of the I and J markers with respect to the x-axis of the J marker.•Translational or rotational velocity that define the velocity of the location of the joint on the first part (I marker) with respect to its location on the second part (J marker) in units of length per unit of time.Adams/View Joints 4Adams/View measures the translational velocity of the I marker along the common z-axis of I and J and with respect to the J marker. It measures the rotational velocity of the x-axis of the I marker about the common z-axis of the I and J markers with respect to the x-axis of the J marker.If you specify initial conditions, Adams/View uses them as the initial velocity of the joint during an assemble model operation regardless of any other forces acting on the joint. You can also leave some or all of the initial conditions unset. Leaving an initial condition unset lets Adams/View calculate the conditions of the part during an assemble model operation depending on the other forces acting on the joint. Note that it is not the same as setting an initial condition to zero. Setting an initial condition to zero means that the joint will not be moving in the specified direction or will not be displaced when the model is assembled, regardless of any forces acting on it.If you impose initial conditions on the joint that are inconsistent with those on a part that the joint connects, the initial conditions on the joint have precedence over those on the part. If, however, you impose initial conditions on the joint that are inconsistent with imparted motions on the joint, the initial conditions as specified by the motion generator take precedence over those on the joint.Setting Initial ConditionsTo modify initial conditions:1.Display the Modify Joint dialog box as explained in Accessing Modify Dialog Boxes .2.Select Initial Conditions .The Joint Initial Conditions dialog box appears. Some options in the Joint Initial Conditions dialog box are not available (ghosted) depending on the type of joint for which you are setting initial conditions.3.Set the translational or rotational displacement or velocity, and then select OK .Imposing Point Motion on a JointYou can impose a motion on any of the axes (DOF) of the idealized joint that are free to move. For example, for a translational joint , you can apply translational motion along the z-axis. Learn more About Point Motion .Note:If the initial rotational displacement of a revolute or cylindrical joint varies by anywherefrom 5 to 60 degrees from the actual location of the joint, Adams/Solver issues a warningmessage and continues execution. If the variation is greater than 60 degrees, Adams/Viewissues an error message and stops execution.Note:For translational, revolute , and cylindrical joints, you might find it easier to use the jointmotion tools to impose motion. Learn about Creating Point Motions Using the Motion Tools .5JointsTo impose motion on a joint:1.Display the Modify Joint dialog box as explained in Accessing Modify Dialog Boxes.2.Select Impose Motion.The Impose Motion(s) dialog box appears. Some options in the Impose Motion dialog box are not available (ghosted) depending on the type of joint on which you are imposing motion.3.Enter a name for the motion. Adams/View assigns a default name to the motion.4.Enter the values for the motion as explained in Options for Point Motion Dialog Box, and thenselect OK.Adding Friction to Idealized JointsYou can model both static (Coulomb) and dynamic (viscous) friction in revolute, translational, cylindrical, hooke/universal, and spherical joints.Note:Using Adams/Solver (C++), you can apply joint friction to joints if they are attached to flexible bodies; using Adams/Solver (FORTRAN), you cannot. In addition, Adams/Solver(C++) does not support point masses.For a summary of which joints and forces are supported on flexible bodies, see Table ofSupported Forces and Joints in the Adams/Flex online help. Also refer to the Adams/Flexonline help for more information on attaching joints and forces to flexible bodies.To add friction to a joint:1.Display the Modify Joint dialog box as explained in Accessing Modify Dialog Boxes.2.Select the Friction tool .The Create/Modify Friction dialog box appears. The options in the dialog box change depending on the type of joint for which you are adding friction.3.Enter the values in the dialog box for the type of joint as explained below, and then select OK.•Cylindrical Joint Options•Revolute Joint Options•Spherical Joint Options•Translational Joint Options•Universal/Hooke Joint OptionsAdams/View6JointsFriction Regime Determination (FRD)Three friction regimes are allowed in Adams/View:diagram of the friction regimes available in Adams/Solver.7JointsConventions in Friction Block DiagramsThe following tables identify conventions used in the block diagrams:•Legend for Block Diagrams identifies symbols in the diagrams.•Relationship Between the Inputs Option and Switches Used in the Block Diagrams describes the relationship between the Input Forces to Friction option in the Create/Modify Friction dialog box and the switches used in the block diagrams.Legend for Block DiagramsRelationship Between the Inputs Option and Switches Used in the Block DiagramsCylindrical Joint frictionJoint reaction (F) and reaction torque (Tm) combined with force preload (Fprfrc) and torque preload (Tprfrc) yield the frictional force and torque in a cylindrical joint. As the block diagram indicates, you can turn off one or more of these force effects using switches SW1 through SW3. The frictional force inSymbol:Description: Scalar quantityVector quantitySumming junction:c=a+bMultiplication junction:c=axbMAGMagnitude of a vector quantity ABSAbsolute value of a scalar quantity FRD Friction regime determinationSwitch:Inputs are: Symbol: Acceptable values:SW1Preload Fprfrc or Tprfc On or offSW2Reaction force f or F On or off SW3Bending moment Tr On or off SW4 Torsional moment Tn On or off All or None sets all applicable switches On or off, respectivelyAdams/View8Jointsa cylindrical joint acts at the mating surfaces of the joint. The FRD block determines the direction of thefrictional force. Based on the frictional coefficient direction, the surface frictional force is broken downinto an equivalent frictional torque and frictional force acting along the common axis of translation androtation.9JointsCylindrical Joint OptionsFor the option: Do the following:Mu Static Define the coefficient of static friction in the joint. The magnitude of thefrictional force is the product of Mu Static and the magnitude of the normalforce in the joint, for example:Friction Force Magnitude, F = µNwhere µ = Mu Static and N = normal forceThe static frictional force acts to oppose the net force or torque along theDegrees of freedom of the joint.The range is > 0.Mu Dynamic Define the coefficient of dynamic friction. The magnitude of the frictionalforce is the product of Mu Dynamic and the magnitude of the normal forcein the joint, for example:Friction force magnitude, F = µNwhere µ = Mu Dynamic and N = normal forceThe dynamic frictional force acts in the opposite direction of the velocityof the joint.The range is > 0.Initial Overlap Defines the initial overlap of the sliding parts in either a translational orcylindrical joint. The joint's bending moment is divided by the overlap tocompute the bending moment's contribution to frictional forces.The default is 1000.0, and the range is Initial Overlap > 0.Adams/View Joints 10Overlap To define friction in a cylindrical joint, Adams/Solver computes the overlapof the joint. As the joint slides, the overlap can increase, decrease, or remainconstant. You can set:•Increase indicates that overlap increases as the I marker translates in the positive direction along the J marker; the slider moves to be within thejoint.•Decrease indicates that the overlap decreases with positive translationof the joint; the slider moves outside of the joint.•Remain Constant indicates that the amount of overlap does not changeas the joint slides; all of the slider remains within the joint.The default is Remain Constant.Pin Radius Defines the radius of the pin for a cylindrical joint.The default is 1.0, and the range is > 0.Stiction TransitionVelocity Define the absolute velocity threshold for the transition from dynamic friction to static friction. If the absolute relative velocity of the joint markeris below the value, then static friction or stiction acts to make the joint stick.The default is 0.1 length units/unit time on the surface of contact in thejoint, and the range is > 0.Max StictionDeformation Define the maximum displacement that can occur in a joint once the frictional force in the joint enters the stiction regime. The slightdeformation allows Adams/Solver to easily impose the Coulombconditions for stiction or static friction, for example:Friction force magnitude < static * normal forceTherefore, even at zero velocity, you can apply a finite stiction force if yoursystem dynamics require it.The default is 0.01 length units, and the range is > 0.Friction Force Preload Define the joint's preload frictional force, which is usually caused bymechanical interference in the assembly of the joint.Default is 0.0, and the range is > 0.Friction Torque Preload Define the preload friction torque in the joint, which is usually caused bymechanical interference in the assembly of the joint.The default is 0.0, and the Range is > 0.For the option:Do the following: 搭接接头丠丠JointsFor the option: Do the following:Effect Define the frictional effects included in the friction model, either Stictionand Sliding, Stiction, or Sliding. Stiction is static-friction effect, whileSliding is dynamic-friction effect. Excluding stiction in simulations thatdon't require it can greatly improve simulation speed. The default isStiction and Sliding.Input Forces to Friction Define the input forces to the friction model. By default, all user-definedpreloads and joint-reaction force and moments are included. You cancustomize the friction-force model by limiting the input forces you specify.The inputs for a translational joint are:•Preload•Reaction Force•Bending MomentFriction Inactive During Specify whether or not the frictional forces are to be calculated during aStatic equilibrium or Quasi-static simulation.Revolute Joint FrictionJoint reactions (Fa and Fr), bending moment (Tr), and torque preload (Tprfrc) determine the frictional torque in a revolute joint. You can turn off one or more of these force effects using switches SW1 through SW3. The joint reactions (Fa and Fr) are converted into equivalent torques using the respective friction arm (Rn) and pin radius (Rp). The joint bending moment (Tr) is converted into an equivalent torque usingpin radius (Rp) divided by bending reaction arm (Rb). The frictional torque (Tfrict) is applied along the axis of rotation in the direction that the FRD block computes.Joints Revolute Joint OptionsFor the option: Do the following:Mu Static Define the coefficient of static friction in the joint. The magnitude of thefrictional force is the product of Mu Static and the magnitude of the normalforce in the joint, for example:Friction Force Magnitude, F = µNwhere µ = Mu Static and N = normal forceThe static frictional force acts to oppose the net force or torque along theDegrees of freedom of the joint.The range is > 0.Mu Dynamic Define the coefficient of dynamic friction. The magnitude of the frictionalforce is the product of Mu Dynamic and the magnitude of the normal forcein the joint, for example:Friction force magnitude, F = µNwhere µ = Mu Dynamic and N = normal forceThe dynamic frictional force acts in the opposite direction of the velocityof the joint.The range is > 0.Friction Arm Define the effective moment arm used to compute the axial component ofthe friction torque. The default is 1.0, and the range is > 0.Bending Reaction Arm Define the effective moment arm use to compute the contribution of thebending moment on the net friction torque in the revolute joint. The defaultis 1.0, and the range is > 0.Pin Radius Defines the radius of the pin.The default is 1.0, and the range is > 0.Stiction Transition Velocity Define the absolute velocity threshold for the transition from dynamic friction to static friction. If the absolute relative velocity of the joint marker is below the value, then static friction or stiction acts to make the joint stick. The default is 0.1 length units/unit time on the surface of contact in the joint, and the range is > 0.Max Stiction Deformation Define the maximum displacement that can occur in a joint once the frictional force in the joint enters the stiction regime. The slight deformation allows Adams/Solver to easily impose the Coulomb conditions for stiction or static friction, for example:Friction force magnitude < static * normal forceTherefore, even at zero velocity, you can apply a finite stiction force if your system dynamics require it.The default is 0.01 length units, and the range is > 0.Friction Torque Preload Define the preload friction torque in the joint, which is usually caused bymechanical interference in the assembly of the joint.The default is 0.0, and the Range is > 0.Effect Define the frictional effects included in the friction model, either Stictionand Sliding, Stiction, or Sliding. Stiction is static-friction effect, whileSliding is dynamic-friction effect. Excluding stiction in simulations thatdon't require it can greatly improve simulation speed. The default isStiction and Sliding.Input Forces to Friction Define the input forces to the friction model. By default, all user-definedpreloads and joint-reaction force and moments are included. You cancustomize the friction-force model by limiting the input forces you specify.The inputs for a translational joint are:•Preload•Reaction Force•Bending MomentFriction Inactive During Specify whether or not the frictional forces are to be calculated during aStatic equilibrium or Quasi-static simulation.For the option: Do the following:JointsSpherical Joint FrictionThe reaction force (F) and the preload frictional torque (Tprfrc) are the two forcing effects used in computing the frictional torque on a Spherical joint. The ball radius is used to compute an equivalent frictional torque. The FRD block determines the direction of the frictional torque.Spherical Joint OptionsFor the option: Do the following:Mu Static Define the coefficient of static friction in the joint. The magnitude of thefrictional force is the product of Mu Static and the magnitude of the normalforce in the joint, for example:Friction Force Magnitude, F = µNwhere µ = Mu Static and N = normal forceThe static frictional force acts to oppose the net force or torque along theDegrees of freedom of the joint.The range is > 0.Mu Dynamic Define the coefficient of dynamic friction. The magnitude of the frictionalforce is the product of Mu Dynamic and the magnitude of the normal forcein the joint, for example:Friction force magnitude, F = µNwhere µ = Mu Dynamic and N = normal forceThe dynamic frictional force acts in the opposite direction of the velocityof the joint.The range is > 0.Ball Radius Defines the radius of the ball in a spherical joint for use in friction-force andtorque calculations.The default is 1.0, and the range is > 0.Stiction Transition Velocity Define the absolute velocity threshold for the transition from dynamic friction to static friction. If the absolute relative velocity of the joint marker is below the value, then static friction or stiction acts to make the joint stick. The default is 0.1 length units/unit time on the surface of contact in the joint, and the range is > 0.JointsTranslational Joint FrictionJoint reaction force (F), bending moment (Tm), torsional moment (Tn), and force preload (Fprfrc) are used to compute the frictional force in a translational joint. You can individually turn off the force effects using switches SW1 through SW4.Max StictionDeformation Define the maximum displacement that can occur in a joint once the frictional force in the joint enters the stiction regime. The slightdeformation allows Adams/Solver to easily impose the Coulombconditions for stiction or static friction, for example:Friction force magnitude < static * normal forceTherefore, even at zero velocity, you can apply a finite stiction force if yoursystem dynamics require it.The default is 0.01 length units, and the range is > 0.Friction Torque Preload Define the preload friction torque in the joint, which is usually caused bymechanical interference in the assembly of the joint.The default is 0.0, and the Range is > 0.Effect Define the frictional effects included in the friction model, either Stictionand Sliding, Stiction, or Sliding. Stiction is static-friction effect, whileSliding is dynamic-friction effect. Excluding stiction in simulations thatdon't require it can greatly improve simulation speed. The default isStiction and Sliding.Input Forces to FrictionDefine the input forces to the friction model. By default, all user-definedpreloads and joint-reaction force and moments are included. You cancustomize the friction-force model by limiting the input forces you specify.The inputs for a translational joint are:•Preload•Reaction Force Friction Inactive During Specify whether or not the frictional forces are to be calculated during aStatic equilibrium or Quasi-static simulation .For the option:Do the following:The bending moment (Tm) is converted into an equivalent force using the Xs block. Similarly, torsional moment is converted into an equivalent joint force using the friction arm (Rn). Frictional force (Ffrict) is applied along the axis of translation in the direction that the FRD block computes.Joints Translational Joint OptionsFor the option: Do the following:Mu Static Define the coefficient of static friction in the joint. The magnitude of thefrictional force is the product of Mu Static and the magnitude of the normalforce in the joint, for example:Friction Force Magnitude, F = µNwhere µ = Mu Static and N = normal forceThe static frictional force acts to oppose the net force or torque along theDegrees of freedom of the joint.The range is > 0.Mu Dynamic Define the coefficient of dynamic friction. The magnitude of the frictionalforce is the product of Mu Dynamic and the magnitude of the normal forcein the joint, for example:Friction force magnitude, F = µNwhere µ = Mu Dynamic and N = normal forceThe dynamic frictional force acts in the opposite direction of the velocityof the joint.The range is > 0.Reaction Arm Define the effective moment arm of the joint-reaction torque about thetranslational joint's axial axis (the z-direction of the joint's J marker). Thisvalue is used to compute the contribution of the torsional moment to the netfrictional force.The default is 1.0, and the range is > 0.Initial Overlap Defines the initial overlap of the sliding parts in either a translational orcylindrical joint. The joint's bending moment is divided by the overlap tocompute the bending moment's contribution to frictional forces.The default is 1000.0, and the range is Initial Overlap > 0.Overlap To define friction in a cylindrical joint, Adams/Solver computes the overlapof the joint. As the joint slides, the overlap can increase, decrease, or remainconstant. You can set:•Increase indicates that overlap increases as the I marker translates in thepositive direction along the J marker; the slider moves to be within thejoint.•Decrease indicates that the overlap decreases with positive translationof the joint; the slider moves outside of the joint.•Remain Constant indicates that the amount of overlap does not changeas the joint slides; all of the slider remains within the joint.The default is Remain Constant.Stiction Transition Velocity Define the absolute velocity threshold for the transition from dynamic friction to static friction. If the absolute relative velocity of the joint marker is below the value, then static friction or stiction acts to make the joint stick. The default is 0.1 length units/unit time on the surface of contact in the joint, and the range is > 0.Max Stiction Deformation Define the maximum displacement that can occur in a joint once the frictional force in the joint enters the stiction regime. The slight deformation allows Adams/Solver to easily impose the Coulomb conditions for stiction or static friction, for example:Friction force magnitude < static * normal forceTherefore, even at zero velocity, you can apply a finite stiction force if your system dynamics require it.The default is 0.01 length units, and the range is > 0.Friction Force Preload Define the joint's preload frictional force, which is usually caused bymechanical interference in the assembly of the joint.Default is 0.0, and the range is > 0.Effect Define the frictional effects included in the friction model, either Stictionand Sliding, Stiction, or Sliding. Stiction is static-friction effect, whileSliding is dynamic-friction effect. Excluding stiction in simulations thatdon't require it can greatly improve simulation speed. The default isStiction and Sliding.For the option: Do the following:。
运动副的摩擦和机械效率讲解
2. 连杆2为示力体，判定相对角速度23、21的方向
V等速 A Q 3 23 K 4
2 21 P
1
B
21
返回
3. 杆2受压 ,并为二力杆,其两端总反力方向相反,在同 一条直线上。判定出两端总反力R32、R12方向如图。
V等速
Q
A
3
23
2
21
R12
1 B
返回
R32
4
P
22
3.4 考虑摩擦时机构的静力分析
Qv Q P0 以力的形式表达 Pv P P
以力矩的形式表达
30
（2）同样的驱动力
以力的形式表达
Q = = Pv P Q0
Qv Q
以力矩的形式表达
Mr = M r0
31
二 、机组的效率
1.串联
Nd N1
1 2
N2
Nk-1
K
Nk
系统的总效率：
Nk N1 N 2 N 3 Nk = = . . = 1.2 .3 k Nd Nd N1 N 2 Nk -1
二、止推轴颈转动副
• 自学
18
例1 :图示为一偏心夹具。已知：轴颈rA、fv， 偏心距e，圆盘r1 及其与工件之间f。
求：撤去力P，仍能夹的楔角。
o r e 1 2
19
rA
1
P
o1
•P去除后，R21为主动力，当其与摩擦圆相切或相割时， 自锁。 •即：OC-CB
•e Sin()-r1Sin •arcSin [(r1Sin+)/e] +
V3
1
⒊分析力已知的构件1，
⒋分析力未知的构件3，
机械原理(第5章 机械中的摩擦、机械效率及自锁)
二、转动副中摩擦力：
轴 轴承
轴径
Northwest A&F University
第五章 机械中的摩擦、机械效率及自锁
二、转动副中摩擦力：
1.轴径摩擦： 轴用于承受径向力放在轴承中的部分称为轴径。 1）摩擦力矩的确定： 设有径向载荷G作用的轴径1，在驱 动力矩Md的作用下，在轴承2中等速运动。 此时转动副两元素必将产生摩擦力以阻 止轴径向对于轴承的滑动。则：
Northwest A&F University
第五章 机械中的摩擦、机械效率及自锁
一、移动副中摩擦力的确定：
2）三角形螺纹螺旋中的摩擦：
β
β △N β △N
Q
△N
△N
β
Q
β－牙形半角
Northwest A&F University
第五章 机械中的摩擦、机械效率及自锁
一、移动副中摩擦力的确定：
2）三角形螺纹螺旋中的摩擦： 螺母和螺纹的相对运动完全相同两者受力分析的方法一致。 运动副元素的几何形状不同在轴向载荷完全相同的情况下， 两者在运动副元素间的法向反力不同接触面间产生的摩擦力不 同。 引入当量摩擦系数: 当量摩擦角: fv = f / cosβ
第五章 机械中的摩擦、机械效率及自锁
二、研究机械中摩擦的内容：
1.几种常见的运动副中摩擦的分析； 2.考虑摩擦时机构的受力分析； 3.机械效率的计算； 4.由于摩擦的存在而可能发生的所谓机械的“自锁” 现 象，以及自锁现象发生的条件。
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第五章 机械中的摩擦、机械效率及自锁
Northwest A&F University
第五章 机械中的摩擦、机械效率及自锁
第5章运动副中的摩擦和机械效率
5.4.2摩擦圆
Mf R21 f Qr f r Q
以轴颈O为圆心，为半 径所做的圆为摩擦圆
结论：R21与摩擦圆相切,所形成的Mf与w12方向相反,与Q等值反向。 Q和 M
Q’=Q
h M Q
h M Mf h M Mf h M Mf
轴颈等速转动或静止不动 轴颈加速转动 轴颈减速转动至静止不动 或保持静止不动状态
1 1
P 理想驱动力 0 P 实际驱动力
Q 实际阻力 Q0 理想阻力
力矩表示法：
理想驱动力矩 实际阻力矩 实际驱动力矩 理想阻力矩
5.5.2效率的计算
1.机器或机组的效率的计算
（1）串联
Wk Wd
Wd
Wk 123 Wk 1
1
W1
2Leabharlann W23W3
Wk-1
2
2. 滑块等速下降
Q F R 0
-
F Qtg ( )
结论：等速下降时的自锁条件：
5.2.3 楔形滑块的摩擦
Rn sin a Q 0
Ff fRn
f f sin a
Ff
f Q f Q sin a
构件1和2间的摩擦角:
artgf 758
f rA 3mm
R1A
机构自锁条件: e sin( ) r 1 sin
sin( 758) 0.2829
rA
A
F RA1 e -
1 R21
2
r1
最大楔角为: 2424
5.2 移动副中的摩擦
07-04 运动副中摩擦力的确定解析
7.4 运动副中摩擦力的确定
如图所示，滑块1是置于一升角为a的斜面2上，Q为作用在 滑块1上的铅垂载荷（包括滑块的自重），欲使滑块1沿斜面2等
速上行时，所需的水平驱使力为 F，总反力R21的方向与所作用
的构件的运动方向υ12成90°+ φ，于是有
F R21 Q 0
从而能够作出力的多边形，如图 b 所示，由力的平衡条件得
7.4 运动副中摩擦力的确定
【例7-3】 如图所示的铰链四杆 机构中，已知机构的位置、各构件
的尺寸和驱动力F，各转动副的半径
和当量摩擦系数均为 r 和 f0 。若不计 各构件的重力、惯性力，求各转动
副中反作用力的作用线和作用在从
动件3上的阻力偶矩M3的方向。 【解】 （1）计算摩擦圆半径 按ρ＝r f0算出各转动副的摩擦圆半 径，并将这些摩擦圆以虚线画在图 上；
用在滑块上的铅垂载荷(包括滑块自重)，F为推动滑块1沿着槽面
于是每一侧面的摩擦力F21的大小为
2等速向右运动的水平力，N21为槽的每一侧面给滑块的法向反力，
F21 fN 21 F f 2F21 2 fN 21
a）
b）
7.4 运动副中摩擦力的确定
根据滑块1在铅垂线上的受力平衡条件得
Q 2 N 21 sin θ
7.4 运动副中摩擦力的确定
【例 7-2】 如图所示的偏心夹具 中，已知轴颈O的半径r0，当量摩 擦系数 f0 ，偏心圆盘 1 的半径 r1 以 及它与工件 2 之间的摩擦系数 f ，
求不加力 F 仍能夹紧工件的楔紧
角β。 【解】摩擦圆半径ρ＝r0 f0。偏心
圆盘与工件之间的摩擦角为arctan
f。
7.4 运动副中摩擦力的确定
运动副中摩擦力的确定详述
ψv= arctan fv 其中：90°-β为三角形螺纹的楔形 半角，β为螺纹工作面的牙形斜角。
则拧紧螺母所需的力矩为：
M = G d2 tan(α+ψv) /2 放松螺母所需的力矩为：
图4-6
图4-5a） 图4-5 b）
设螺母1上受有轴向载荷G，在 螺母上加一力矩M，使螺母旋转并 逆着G力等速向上运动（对螺纹联接 来说，这时为拧紧螺母），则在图b 中，就相当于在滑块2上加一水平力 F，使滑块2沿着斜面等速上升。 则： F =G·tan(α+ψ)
α为螺杆在中径d2上的螺纹导 程角，即：
tanα= l /πd2= zp /πd2。
它是一阻抗力矩，其作用是阻止螺母的加速松退。
当α<ψ时，M′为负值，其方向与预先假定的方向相 反，即与螺母运动方向相同，这时，它是放松螺母
时所需外力的驱动力矩。
2、三角形（普通）螺纹螺旋副中的摩擦
如图4-6所示为三角形（普通）螺纹，其螺旋副中的 摩擦可简化为一槽形滑块沿槽形斜面滑动的摩擦问题。
在研究三角形（普通）螺纹螺旋副中的摩擦时，只要
图4-5 a） 图4-5 b）
F =G·tan(α+ψ) F相当于拧紧螺母时必须在螺纹中 径处施加的圆周力，故拧紧螺母时
所需的力矩M为： M= F d2 /2= Gd2 tan(α+ψ) /2
图4-5 b）
同理，放松螺母（相当于滑块等速下滑）时所需的
力矩M ′为：
M ′= F ′d2/2 = G d2 tan(α-ψ) /2 注意：当α>ψ时，M′为正值，其方向与螺母运动方向相反，
机械原理——5.摩擦与效率
90 0 + ϕ
ϕ
α +ϕ Q
注意 : R12 = R21
90 −ϕ
0
R21
900 −α −ϕ
P = Q tan(α + 2ϕ )
P0 = Q tan α
P0 tan α η= = P tan(α + 2ϕ )
P
α + 2ϕ ≥ 900 , 发生自锁 发生自锁. 当
机械原理第四章 16
2.反行程 2.反行程
2
Q e Q
1
dF
Md
r
R21
ρ
机械原理第四章
4
3）确定机构中运动副总反力方向的步骤 （1）从二力杆开始； ）从二力杆开始； （2）在不考虑摩擦力的情况下，初步确定总反 ）在不考虑摩擦力的情况下， 力的方向； 力的方向； （3）再考虑摩擦，确定出移动副或转动副中总 ）再考虑摩擦， 反力的真实作用线的方位。 反力的真实作用线的方位。 **注意： **注意： 注意
22
η′ =
机械原理第四章
的取值范围； （3）正行程不自锁而反行程自锁时α、β的取值范围； 正行程不自锁而反行程自锁时α
①正行程不自锁条件
tgα tg ( β − 2ϕ ) η= ⋅ tg (α + 2ϕ ) tgβ
须满足: 须满足:
两个机构均不能自锁! 两个机构均不能自锁!
900 > α > 0 0 α + 2ϕ < 900 0 β − 2ϕ > 0 β < 900
第五章
运动副的摩擦和机械效率
一．运动副中的摩擦； 运动副中的摩擦； 二．考虑摩擦时机构的受力分析； 考虑摩擦时机构的受力分析； 三．机械的效率计算； 机械的效率计算； 四．机械的自锁条件分析。 机械的自锁条件分析。 重点: 考虑摩擦时机构的受力分析; 重点: 考虑摩擦时机构的受力分析; 机械的自锁。 机械的自锁。 难点: 难点: 运动副中总反力作用线的确定 机械自锁条件的确定。 机械自锁条件的确定。
对轴颈形成的摩擦力矩
在确定运动副（包括转动副、高副）的反力时，常 注意： 用总反力R21来表示，而不分法向反力N21与摩擦力 F21。
例：如图4-3、4-4所示的斜面机构中，将滑块1置于升角 为α的斜面2上，G为作用在滑块1上的铅垂载荷（包括 滑块自重）。试求： 1）使滑块1沿斜面2等速上升（通常称此行程为正行程） 时所需的水平驱动力F； 2）保持滑块1沿斜面2等速下滑（称此行程为反行程）时
如图所示的平面高副两元素在K 点接触，如构件1相对于构件2 的相 对速度V12 的方向如箭头所示。 则： R21与构件1相对于构件2的相对 速度V12的方向成90°+ψ，其中ψ为摩 擦角。
图45b图45a设螺母1上受有轴向载荷g在螺母上加一力矩m使螺母旋转并逆着g力等速向上运动对螺纹联接来说这时为拧紧螺母则在图b中就相当于在滑块2上加一水平力f使滑块2沿着斜面等速上升
§4—2 运动副中摩擦力的确定
在机械运动时运动副两元素间将产生摩擦力。下面分 析移动副、螺旋副、转动副和平面高副中的摩擦。 一、移动副中的摩擦 如图4-2，a所示为滑块1与水 平平台2构成的移动副，G为作用 在滑块1上的铅垂载荷，设滑块1 在水平力F的作用下等速向右移动。
所需的水平力F ′。
图4-3
图4-4Biblioteka 解： 1）滑块等速上升：如图4-3a) 斜面2对滑块1的总反力为R21。 根据力的平衡条件：
G + F + R21=0 方向：√ √ √v12 大小： √ ？ ？
作力多边形，如图4-3 b)。 ∴ F = G· tan(α+ψ)
图4-3 a）
图4-3 b）
2）滑块等速下滑：如图4-4a) 斜面2对滑块1的总反力为R21′。 根据力的平衡条件： G + F ′ + R21′=0 方向：√ √ √v12 大小：√ ？ ？ 作力多边形，如图4-4 b)。 ∴ F ′= G·tan(α-ψ)