• 沒有找到結果。

# Equilibrium of a Rigid Body

N/A
N/A
Protected

Share "Equilibrium of a Rigid Body"

Copied!
29
0
0

(1)

## in SI Units, 12e

(2)

### Chapter Objectives

• Develop the equations of equilibrium for a rigid body

• Concept of the free-body diagram for a rigid body

• Solve rigid-body equilibrium problems using the equations of equilibrium

2011/5/10 2

(3)

### Chapter Outline

• Conditions for Rigid Equilibrium

• Free-Body Diagrams

• Equations of Equilibrium

• Two and Three-Force Members

Free Body Diagrams

Equations of Equilibrium

• Constraints and Statical Determinacy

(4)

### Two-Force Members

• When forces are applied at only two points on a member, the member is called a two-force member

• Only force magnitude must be determined

2011/5/10 42

(5)

### Three-Force Members

• When subjected to three forces, the forces are concurrent or parallel

(6)

### Example 5.13

The lever ABC is pin-supported at A and connected to a short link BD. If the weight of the members are negligible, determine the force of the pin on the lever at A.

2011/5/10 44

(7)

### Solution

Free Body Diagrams

• BD is a two-force member

• Lever ABC is a three-force member Equations of Equilibrium

Solving,

kN F

kN FA

32 . 1

07 . 1

0 45

sin 3

. 60 sin

; 0

0 400

45 cos 3

. 60 cos

; 0

3 . 4 60

. 0

7 . tan 1 0

F F

F

N F

F F

A y

A x

(8)

### Support Reactions

As in the two-dimensional case:

• A force is developed by a support

• A couple moment is developed when rotation of the attached member is prevented

• The force’s orientation is defined by the coordinate angles , and

2011/5/10 46

(9)

(10)

2011/5/10 48

(11)

### Example 5.14

Several examples of objects along with their associated free- body diagrams are shown. In all cases, the x, y and z axes are established and the unknown reaction components are indicated in the positive sense. The weight of the objects is neglected.

(12)

2011/5/10 50

(13)

### 5.6 Equations of Equilibrium

Vector Equations of Equilibrium

• For two conditions for equilibrium of a rigid body in vector form,

∑F = 0 ∑MO = 0

Scalar Equations of Equilibrium

• If all external forces and couple moments are expressed in Cartesian vector form

∑F = ∑Fxi + ∑Fyj + ∑Fzk = 0

∑MO = ∑Mxi + ∑Myj + ∑Mzk = 0

(14)

### Redundant Constraints

• More support than needed for equilibrium

2011/5/10 52

(15)

### Improper Constraints

• Instability caused by the improper constraining by the supports

• When all reactive forces are concurrent at this point, the body is improperly constrained

(16)

### Procedure for Analysis

Free Body Diagram

• Draw an outlined shape of the body

• Show all the forces and couple moments acting on the body

• Show all the unknown components having a positive sense

• Indicate the dimensions of the body necessary for computing the moments of forces

2011/5/10 54

(17)

### Procedure for Analysis

Equations of Equilibrium

• Apply the six scalar equations of equilibrium or vector equations

• Any set of non-orthogonal axes may be chosen for this purpose

Equations of Equilibrium

• Choose the direction of an axis for moment summation such that it insects the lines of action of as many unknown forces as possible

(18)

### Example 5.15

The homogenous plate has a mass of 100kg and is subjected to a force and couple moment along its edges. If it is supported in the horizontal plane by means of a roller at A, a ball and socket joint at N, and a cord at C, determine the components of

reactions at the supports.

2011/5/10 56

(19)

### Solution

Free Body Diagrams

• Five unknown reactions acting on the plate

• Each reaction assumed to act in a positive coordinate direction

Equations of Equilibrium

0 981

300

; 0

0

; 0

0

; 0

N N

T B

A F

B F

B F

C z

z z

y y

x x

(20)

### Solution

Equations of Equilibrium

• Components of force at B can be eliminated if x’, y’ and z’

axes are used

0 )

3 ( .

200 )

5 . 1 ( 981 )

5 . 1 ( 300

; 0

0 ) 2 ( )

2 ( 300 )

1 ( 981

; 0

0 .

200 )

3 ( )

3 ( )

5 . 1 ( 981 )

5 . 1 ( 300

; 0

0 ) 2 ( )

1 ( 981 )

2 (

; 0

' '

m T

m N m

N m

N M

m A

m N

m N

M

m N m

A m

B m

N m

N M

m B

m N

m T

M

C y

z x

z z

y

Z C

x

2011/5/10 58

(21)

### Solution

Solving,

Az = 790N Bz = -217N TC = 707N

• The plate is partially constrained as the supports cannot prevent it from turning about the z axis if a force is applied in the x-y plane

(22)

2011/5/10 60

(23)
(24)

2011/5/10 62

(25)
(26)

2011/5/10 64

(27)
(28)

2011/5/10 66

(29)

Arise from various neuroendocrine cells NE cells are present in endocrine glands. throughout the body produce hormones , diffused in all

 There is a well-defined unilocular ovoid shaped circumcoronal radiolucency with corticalted margin over the impacted tooth 38 of the posterior mandibular body, extending from

Axial (A), sagittal (B) and coronal (C) CBCT sections demonstrating a well deﬁned radiolucency, with the attenuation of air, at the inferior margin of the C2 vertebral body,

z If all options are European a box spread is worth the present value of the difference between the strike prices. z If they are American this is not necessarily so (see

Listen, point and repeat... Little EFL Phonics

2.4 a Number of schools by supervisory body/individual, level of instruction and type of institutions 2.5 a Number of students as at end of the academic year by gender, level

• Density is varying in the medium and the volume scattering properties at a point is the product of scattering properties at a point is the product of the density at that point

♦ The action functional does not discriminate collision solutions from classical solutions...

The best way to picture a vector field is to draw the arrow representing the vector F(x, y) starting at the point (x, y).. Of course, it’s impossible to do this for all points (x,

11[] If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle (J as the

To proceed, we construct a t-motive M S for this purpose, so that it has the GP property and its “periods”Ψ S (θ) from rigid analytic trivialization generate also the field K S ,

The purpose of this talk is to analyze new hybrid proximal point algorithms and solve the constrained minimization problem involving a convex functional in a uni- formly convex

• Many-body physics is essential for sub-keV detectors of neutrinos and dark matter. • High-quality many-body calculations can substantially reduce

Since the generalized Fischer-Burmeister function ψ p is quasi-linear, the quadratic penalty for equilibrium constraints will make the convexity of the global smoothing function

• When a number can not be represented exactly with the fixed finite number of digits in a computer, a near-by floating-point number is chosen for approximate

 understand and use the English terms for describing the animal types, external features, body parts, feeding habits, movement and habitats of a panda, a crocodile and a crab

The remaining positions contain //the rest of the original array elements //the rest of the original array elements.

A system is said to be in stable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in a direction opposite to the direction of the displacement..

To convert a string containing floating-point digits to its floating-point value, use the static parseDouble method of the Double class..

Furthermore, given a set of nets in a bus on routing grids with obstacles, and the length constraints for all the nets in the bus, based on this proposed longest path generation,

26 PRADIER Jean-Marie：La Scène et la fabrique des corps – Etnologie du spectacle vivant en Occident ( V ème siècle av. – XVIII ème siècle ), Bordeaux FRANCE, édition de Presses

Based on the “ Wife-tester, Wife-killer ” by the National Guo-Kwung Troup Honan Drama Group, this article explores the notable evidences that the modern drama in Taiwan

The goal of this research is to design a method to solve a joint optimization problem to maximize the spectral efficiency of GFDM systems under performance constraints, in-