理论力学第一章 英文ppt

A F
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 body is called to be in the state of equilibrium if it does not move or if it moves with an uniform velocity.
If r is the radius vector at point A then
mO ( F ) r F , mO ( F ) r F sin(r ,F ) F d
The moment of the force about the point o is equal to the vector product of the radius vector of the force at the point and the force. The angle between 12 the two vectors is
Conclusion:
1) the moment of a force about an axis which is parallel to the force is zero. In other words, if the force F and the axis are coplanar the moment is zero.
b q
g
O
magnitude:
FF
Fxy
point of application: the point at which the force acts onto the object.
direction: it can be determined by the three angles , b, g
(3) The method of indirect projection When the angles , b and g are not easy to determine we can project F onto the plane xy firstly and then project the result Fxy onto the axes x X F sing cos Fxy cos F cosq cos and y. Y F sing sin Fxy sin F cosq sin
It mainly studies simplifications and equilibrium conditions
of force systems and applications of the corresponding results.
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: ①motion (external effect) ②deformation (internal effect). 3) The three elements: magnitude, direction and point of application.
2. The moment of a force about an axis
m Definition： z ( F ) mO ( Fxy ) Fxy d 2 area(OA' B' )
It is an algebraic quantity. Direction is + – [Proof] m z ( F ) m z ( Fz ) m z ( Fxy ) mO ( Fxy )
Z F cosg F sinq
(4) Decomposition of a force along the coordinate axes:
If the three rectangular components of a force along the coordinate axes are Fx , Fy and Fz then: F Fx Fy Fz due to Fx Xi ,Fy Yj,Fz Zk We get
Draw an arbitrary axis Z through the point O then
m z ( F ) m z ( Fxy ) 2OA'B'
From the geometrical relationship:
OABcosg OA'B' Hence: 2OABcosg 2OA'B'
Example
Let P=2000N, point C is in the plane Oxy. Determine the moment of force P about the three axes.
Solution:
Pz Psin45 Pxy Pcos45 Px Pcos45sin60 Py Pcos45 cos60
3. The relationship between the moment of the force about a point and the moment of the force about an axis through the point Proof:
mO ( F ) 2 area( AOB )
or by the angle of inclination q and the angle of depression .
(2) The method of direct projection
From the fig we see
X F cos , Y F cos b , Z F cos g
m x ( F )i m y ( F ) j m z ( F )k
The moment of the force about point O is given by
mO ( F ) (mx ( F )) 2 (my ( F )) 2 (mz ( F )) 2
my (F ) mx ( F ) mz ( F ) cos ,cosb ,cosg mO ( F ) mO ( F ) mO ( F )
Hence : mO ( F ) cosg mz ( F )
[mO ( F )] z mz ( F )
Theorem: the projection of the vector of the moment of a force about a point on an axis through this point is equal to the moment of the force about the axis. This is the required relationship between the moment of a force about a point and the moment of the force about an axis through this point. Due to mO ( F ) r F [mO ( F )] x i [mO ( F )] y j [mO ( F )] z k
couples
4. The projection and decomposition of a force on the axes
(1)The representation of force in space: a force is given by its magnitude, its direction and its point (line) of application
Hale Waihona Puke Baidu
In the plane it is an algebraic quantity. In space, it is a vector.
[Example] the spherical hinges on the reflective mirror of a car
mO ( F ) F d 2 area(AOB )
F Xi Yj Zk
Fz Fy Fx
F X 2 Y 2 Z 2
X Y Z cos ,cosb ,cosg F F F
5 The moment of a force about a point or an axis
1. The moment of a force about a point
Chapter 1: Fundamental principles of statics and force analysis
§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
F
A
4) Unit of force: newton (N) (International System of Units)
kilonewton (kN)
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.
coplanar force system
Force system
Force system in space
Coplanar force system: 1 coplanar system of concurrent forces 2 coplanar system of parallel forces 3 general case of a force system in a plane Force system in space 1 Concurrent force system in space 2 parallel force system in space 3 A general force system in space
§1.1 Fundamental concepts in statics
Statics is the branch of mechanics which studies the laws of equilibria of bodies under the action of force systems.
2) The moment of a force about an axis which is parallel to the force is zero. In other words, if the force F and the axis are coplanar the moment is zero.
Because F Xi Yj Zk and
i mO ( F ) r F x X j y Y k z Z
O
r xi yj zk
( yZ zY )i ( zX xZ ) j ( xY yX )k [mO ( F )] x i [mO ( F )] y j [mO ( F )] z k