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Dynamics - Mechanical

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A Brief about
Dynamics is a branch of mechanics which deals with the motion of bodies under the action of forces.

History of Dynamics
The beginning of rational understanding of dynamics credited to Galileo (1564-1642), who made careful observations concerning bodies in free fall, motion on an inclined plane and motion of the pendulum.
Newton (1642-1727), guided by Galileoʼs work, was able to make an accurate formulation of laws of motion. Important contributions to dynamics were made by Euler, DʼAlembert, Lagrange, Laplace, Poinsot, Einstein and other.
Application of Dynamics
Dynamics principles are basics to analysis and design of moving structure, to robotic devices, to automatic control devices, to rockets, missiles, spacecraft, to ground and air transportation vehicles, to machinery of all types such as turbines, pumps, machine tools.
Basic Concept
Space is the geometric region occupied by bodies.
Time is a measure of the succession of events and is considered an absolute quantity in Newtonian mechanics.
Mass is the quantitative measure of inertia or resistance to change in motion of a body.
Force is the vector action of one body on another.
Particle is a body of negligible dimensions.
Rigid Body is a body whose changes in shape are negligible compared with overall dimensions of the body.

Newtonʼs Laws
Law 1. A particle remains at rest or continues to move with uniform velocity if there is no unbalanced force acting on it.
Law 2. The acceleration of a particle is proportional to resultant force acting on it and is in the direction of this force.
Law3. Every action has equal and opposite reaction.
Equation of Motion
When a particle of mass m is subjected to the action concurrent forces F1, F2, F3……whose vector sum becomes
 = ma
Work and Energy
Figure shows a force F is acting on particle A which move along path shown. The position vector r measured from origin O and dr is the differential displacement associated with an infinitesimal movement from A to A ʼ.
The work done by force F during displacement dr is defined as
dU= F. dr
The magnitude of this dot product is dU= F ds cosα, where α is the angle between F and dr and where ds is the magnitude of dr.
Examples of Work
·      Work associated with a constant External force.

 Consider the constant force P acting to the body as it moves from position 1 to position 2. With the force P and differential displacement dr written as vectors, the work done on body by the force is
= ʃ [(P cosα) i + (P cos α ) j].dx i
·      Work associated with a spring force.
We consider here the common linear spring of stiffness k, where the force required to stretch or compress the spring is proportional to the deformation
x.
The force exerted by spring on the body is F= -k xi,
U=ʃ F.dr = ʃ(-k xi). dxi

Principle of Work and Energy
The kinetic energy T of a particle is defined as
The principle of work and energy states that the total work done by all forces acting on a particle as it moves from one point to another point equals to corresponding change in kinetic of particle.

= ∆T
Power
The capacity of a machine is measured by the time rate at which it can do work or deliver energy.
Accordingly, the power P developed by a force F which does an amount of work U is P = dU/dt= F.dr/dt. Because dr/dt is the velocity v of the point of application of the force, we have
P=F.v

Linear Momentum
Consider the general curvilinear motion in space of a particle of mass m, particle is located by its position vector r measured from a fixed origin O. The velocity of particle is v and tangent to its path. The resultant force  of all  force on m is in direction of its acceleration a, so we have

= ma = (mv)
Where the product of mass and velocity is defined as the linear momentum G=mv.
So above equation states that the resultant of all the forces acting on a particle equals its time rate of change of linear momentum.

References
·      Engineering Mechanics Dynamics (7th Edition)- J.L. Meriam, L. G. Karige
·      Engineering Mechanics Dynamics (12th Edition)- R.C. Hibbeler
·      Advance Engineering Dynamics – H. R. Harison, T. Nettleton
Read more
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