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Theoretical Mechanics
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Introduction
Statics is the branch of mechanics which studies the laws of
equilibria of bodies under the action of force systems,
It mainly studies simplifications and equilibrium conditions
of force systems and applications of the corresponding results,
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Theoretical mechanics
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§ 1–1 Fundamental concepts in statics
§ 1–2 Principles of statics
§ 1–3 Constraints and their reaction forces
§ 1–4 Force analysis and force diagrams
Chapter 1,Fundamental
principles of statics and force analysis
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§ 1–1 静力学的基本概念
§ 1–2 静力学公理
§ 1–3 约束与约束反力
§ 1–4 物体的受力分析与受力图

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Chapter 1,Fundamental
principles of statics and force analysis
§ 1-1 Fundamental concepts in statics
1,Force
1) Concept,the mechanical interaction of bodies is caused by
forces,This interaction can change the state of motion of a
body,
2) Effects of forces,① motion (external effect)
② deformation (internal effect),
3) The three elements,
magnitude,direction and
point of application,
A F
4) Unit of force,newton (N) (International System of Units)
kilonewton (kN)
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§ 1-1 静力学基本概念

1．定义, 力是物体间的相互机械作用,这种作用可以改变物

2,力的效应,① 运动效应 (外效应 ) ② 变形效应 (内效应 )。
3,力的三要素,大小，方向，作用点 A F
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Force system,a group of forces acting on a body,
Balanced force system,if a body subjected to a
force system is in equilibrium state,
we call the system to be a balanced,
A body is called to be in the state of equilibrium if it does not move
or if it moves with an uniform velocity,
2.Rigid body
A body which does not change its shape and dimension under
applied forces is called a rigid body
3.Equilibrium state
A
F
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A
F
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§ 1-2 Principles of statics
Principles,conclusions obtained from longtime practice and from
many experiments,They are repeatedly proved by practice and now
are recognized without prove,
1st Principle,two force equilibrium
A rigid body subjected to the action of two forces can be in
equilibrium,the necessary and sufficient conditions are,
the two forces are equal in magnitude,| F1 | = | F2 |
the two forces are opposite in direction,F1 = –F2
the two forces are collinear,
the two forces act on the same body,
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§ 1-2 静力学基本公理

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Explanation,
① As far as rigid bodies are concerned,the conditions above are
necessary and sufficient,
② As far as deformable bodies are concerned,the conditions
above are only necessary,
bodies (deformable body) string

Double force
equivalent body
Double force
equivalent rod
③ double force equivalent body,is a rigid body subjected to
two forces must be in equilibrium,
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③ 二力体,只在两个力作用下平衡的刚体叫二力体。
② 对变形体来说，上面的条件只是必要条件 (或多体中 )

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The action of a given force system on a rigid body remains unchanged
if another balanced force system is added to or subtracted from the
original system,
Corollary 1,transmissibility of a force
The point of application of a force acting on a rigid body can be
transferred to any other point on the line of action of the force without
altering its effect,
Therefore,as far as a rigid body is concerned,a force can be
characterized by three elements,magnitude,direction and line of
application,
2nd Principle,addition and subtraction of balanced force systems
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If a free rigid body remains in equilibrium
under the action of three nonparallel coplanar
forces,the lines of action of those forces
intersect at one point,Moreover,three forces are
coplanar,(under special circumstances,forces
intersect at infinite and become a
parallel force system),
3rd principle,the law of parallelogram
Corollary 2,three equivalent forces intersect
at one point
21 FFR ??
Two forces applied at one point of a body
have as their resultant a force applied at the
same point and represented by the diagonal
of a parallelogram constructed with two
given forces as its sides,
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21 FFR ??
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4th principle,the law of action force and reaction force
To any action of one material body on
another there is always an equal and
oppositely directed reaction,They are
equal in magnitude,opposite in
direction,collinear and exist together,
but act on different body,
Prove,∵ is a balanced force system,
∴ is a balanced force system too,
again∵ If two forces are in equilibrium,
they must be equal in magnitude,
opposite in direction and collinear,
∴ intersect at a point and are coplanar,
321,,FFF
321,,FFF
3,FR
Example
droplet
The force the droplet act
on the string
The force the string act
on the droplet
Gravitation
The reaction force of p
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[证 ] ∵ 为平衡力系,
∴ 也为平衡力系。

∴ 三力 必汇交，且共面。
321,,FFF
321,,FFF
3,FR
[例 ] 吊灯
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5th principle,principle of solidification
If a deformable body subjected to the action of a force system is in
equilibrium,the state of equilibrium will not be disturbed if the
body solidifies(becomes rigid)
The 5th principle tells us
that the equilibrium of a
balanced deformable body
can be described by the
theory for a rigid body,
Soft string
draw
draw
Solidified rigid rod
draw
draw
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§ 1-3 constraints and their reaction forces
1,Concepts
Free body,A body not connected with other bodies and capable
of displacement in any direction is called a free body,
Constrained body,
A body whose displacement in space is restricted by
other bodies,either connected to or in contact with it,
is called a constrained body,
Constraint,Anything that restricts the displacement of a given
body in space.(here,constraint is a noun,not a verb),
Reaction force of a constraint,
The force by which a constraint acts on a body,thereby
restricting its displacements,is called the force of
reaction of the constraint,
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§ 1-3 约束与约束反力

（这里，约束是名词，而不是动词的约束。）
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① their magnitude is usually unknown;
② their direction is always opposite to the direction in which
the given constraint prevents the displacement of the body;
③ the point of application is the contact point between the
body and the constraint,
The characteristics of reaction force of constraints,
G G
N1
N2
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① 大小常常是未知的；
②方向总是与约束限制的物体的位移方向相反；
③作用点在物体与约束相接触的那一点。

G G
N1
N2
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Strings can only be stretched,therefore the reaction force applies
at the point of contact and the direction of it is away from the
body,
2,Some common types of constraints and their reaction
force direction,
1) Constraints given by a string,
a chain or a strap
P P
T S1 S'1
S'2 S2
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1.由柔软的绳索、链条或皮带构成的约束
P P
T S1 S'1
S'2 S2
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the reaction force is applied at the point of contact and directed
normal to both surfaces,
2) Smooth contact plane (smooth,not taking friction into account )
P
N N
P
NA
NB
Common normal
Common
tangent
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2.光滑接触面的约束 (光滑指摩擦不计 )
P
N N
P
NA
NB
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3) Smooth cylindrical pin
① cylindrical pin
Pin
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3.光滑圆柱铰链约束
① 圆柱铰链
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A XA
YA
A
A
Direction is uncertain
Magnified
picture
Pin
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A
A
XA
YA
A
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② fixed hinged support
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② 固定铰支座
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fixed hinged support
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runner and pin
(double surface
constraints)
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（双面约束）
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double force
equivalent rod
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Kinetic hinged support (step bearing)
The actual
direction of N can
be downward,
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N的实际方向也

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Kinetic hinged support (step bearing)
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1,Force analysis
To solve a problem of statics,First select the body to be studied,
Secondly,in accordance with the given conditions and constraints,
analyze the forces which act on the bodies using the basic concepts
and principles,This process is called force analysis,
The forces acting on a body can be classified as,
1,active forces,such as the gravitational force,the force of wind
power,the pressure of gases,etc,
2,passive forces,such as reaction force of constraints,
§ 1-4 Force analysis and force diagrams
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§ 1-4 物体的受力分析和受力图
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The main steps to draw the force diagrams,
① select the object under study;
② divide the object into several parts and
study the parts one by one;
③ draw the active forces;
④ draw the reaction force of the constraints,
2,Force diagrams
[Example 1]
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③画上主动力；④画出约束反力。

[例 1]
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[Example 2] Please,draw the force diagrams for
all components in the picture,
Q A
O
B
C
D
E
Rod
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[例 2] 画出下列各构件的受力图
Q A
O
B
C
D
E
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Q A
O
B
C
D
E
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Q A
O
B
C
D
E
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Q A
O
B
C
D
E
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Q A
O
B
C
D
E
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[Example 3] Please,draw the force diagrams for
all components in the picture,
Notice,three balanced forces
must intersect at a point,But
if they are parallel,they
intersect at infinity,This is a
special case,
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[例 3] 画出下列各构件的受力图

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[Example 4] Sharp point contacts
Get rid of constraints Get rid of constraints
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[例 4] 尖点问题

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[Example 5] Please,draw the force diagrams for
all components in the picture,
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[例 5] 画出下列各构件的受力图
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3,Some problems which should be taken into account when
drawing a force diagram,
Mutual mechanical action forces exist,only if
bodies contact,except for gravitation and
electromagnetic force,So the bodies which are
in contact with the object under study should be
made clear,There must have force at the contact
point,the direction of the force depends on the
type of constraints,
Mind you,the force is the mutual mechanical
action between bodies,Thereby,every force the
body subjected,relatives to a body who applies
the force,We should make it clear,
1) Don’t leave
out any force
2) Don’t draw
redundant
force
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2、不要多画力

1、不要漏画力
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The direction of constraint reaction force must be strictly
according to the type of constraint,never be simply supposed
by direct-view or the direction of the active force,When
analyzing the action forces and the reaction forces between
two bodies,please give attention to the direction of the forces,
If the direction of the action force has been determined,the
direction of the reaction force is opposite to the action force,
Don’t draw the direction of forces wrong,
3) Don’t draw the direction of force wrong
4) There are no constraints on the force diagrams
The force diagrams can be draw only on a part of the object,
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3、不要画错力的方向
4、受力图上不能再带约束。

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Whether a force is an external force or an internal one depend
on the special objects,When a body system is divided into
several parts,some internal forces of old system may turn into
the external forces of the new object under study,
If the direction of the reaction force of a constraint is defined,it
should be kept to the same in the analysis of the whole body as
in the analysis of the components,
5) no internal forces but only external forces can be drawn on
a force diagram
6) The force diagram for the whole object should be in
accordance with the force diagrams for any part,they
should assort with each other,but not conflict,
7) Make a exact judge for the double forces components,
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5、受力图上只画外力，不画内力。
6,同一系统各研究对象的受力图必须整体与局部一致，相

7,正确判断二力构件。
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