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TOPIC 2: D’ Alembert Principle, Force Analysis of Linkages By Energy Method

LECTURER:
DR AFIFAH BT MOHD ALI

GROUP MEMBERS:

AISYAH MADIHAH BT MUSTAFA  (1526748)

AMIRA SURFEENA BT MANSUR (1320400)

NOOR AFIFAH INSYIRAH BT MOHD GHANI (1320944)

NUR SHAFIHA BT JUHARI (1326100)

SITI NABIHAH BT MOHAMAD ROZY (1322378)

SITI NUR AININA BT MAHAZIS (1329022)

ZAIDATUN AMINAH BT ZAINUDDIN (1328400)

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Two basic approaches:

1. Application of Newton’s laws of motion to free-body diagrams.

2. Use of work.

FIRST APPROACHES

-Deal with vector quantities. E.g. force, displacement.

-Need to resolve in 2D or 3D.

-Need to introduce all internal forces on free-body diagrams.

LINK
First approaches

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Free-Body Force Diagrams - Physics Video by Brightstorm

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Physics video on time-saving-body force diagrams. A time-saving-body force diagram is used to show all the forces acting upon an object to predict the net force and ultimately the path of the object. Each force is drawn as vector arrow.

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-Deal with scalar quantities. E.g. work.

Advantages of Work (Energy Method)
-Avoid of computing the acceleration.

-Lead to velocity changes as function of forces which do work.

-Work energy equation involves only forces which do work.

-Give rise to changes in magnitude of velocities.

LINK
Second approaches

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Here is a SNEAK PEEK of Work Energy Method Video, to watch the entire video log on to www.ezed.in

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Static Force Analysis :
Graphical and analytical methods of force analysis on stationary mechanisms

- Equilibrium method

- Virtual (Energy) method

Dynamic Force Analysis :
Force analysis of mechanisms in motion

- D’ALembert’s principle OR

- Newton’s 2nd Law of motion

D' Alembert's principle is summarized as follows:

1. The vector sum of all external forces and inertia forces acting upon a system of rigid bodies is zero.

2. The vector sum of all external moments and inertia torques acting upon a system of rigid bodies is also separately zero.

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According to Newton’s second law of motion :

For Linear Motion :

1. Static case : ∑F =  0
2. Dynamic case : ∑F = m.a

where : F = resultant force acting on the body
m = mass of the body
a = Linear mass of the acceleration of the body

For Rotational Motion :

1. Static case : ∑M = 0
2. Dynamic case ∑M = Iα

For D'alembert's Principle :

1. Static case : ∑F - ma = 0
2. Dynamic case : ∑M - Iα = 0

where :  ∑M = sum of any external moment
α = angular acceleration

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The work done by all the forces acting on a system, during a small virtual displacement is ZERO.

A virtual displacement is a small displacement of the system which is compatible with the geometric constraints.

A geometric constraint is a limitation placed on an object, which can have two dimensions or more, because there are zero degrees of freedom

Method of Virtual Work is suitable for analysis of multi-link structures (pin-jointed members) which change configuration.

it is most suited for solving equilibrium problems involving a syste of several connected rigid bodies.

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Determine the force P

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Using the concept of virtual displacements, and virtual work, we can derive the equations of motion of lumped parameter systems.

Number of degree of freedom =1
Coordinate to describe the motion is x
No external force (only initial displacement condition)

Formulation using Newton's Law:
F = ma
F-ma=0 (consider m.a to be inertial force)
D’Alembert’s law
F=0
Appears when:
- external force tries to accelerate a body with mass
- direction of inertial force always opposite to the define position direction of body mass position

m (kg) : mass
k (N/m) : spring constant
c (Ns/m) : damping coefficient
F (N) : external force
x (m) : displacement of body

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Fi : inertial force (mass)
Fe : elastic force (spring)
Fd : damping (viscous friction)

Fe = kx
Fi = ma = m.dv/dt = m. d^2x/dt^2
Fd = cv = c.dx/dv
F-Fe-Fd-Fi=0
Thus, the equation of motion is:
F - kx - m. d^2x/dt^2 - c.dx/dv

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7. D’Alembert’s Principle & Energy Method

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7. D’Alembert’s Principle & Energy Method

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Solving the example for D'Alembert's Principle & Energy Method

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Nuevo ambiente de aprendizaje Convergencia 2010 Campus Ciudad de México