Advanced Engineering Mathematics

Topic: Mechanical free oscilltion undamping system

Branch: INFORMATION TECHNOLOGYPresented by: Acharya Mansi (150450116001) Prachi Jain (150450116014) Jyoti Mishra(150450116018)

GUIDED BY:- RAJESH JADAV

Mechanical free oscillation

What is mechanical free oscillation ? A mass m attached at the free end of spring. When the body is pulled down by a certain distance and the released, it comes in certain type of motion. we assume that the body moves vertically. We shall determine the differential equation which governs the whole problem, the solution of which describes the motion of the system providing an expression for the displacement as a function of time. (for convenience consider downward direction downward force as positive and the upward ones as negative)

Once the mechanical system is set into motion the following forces acts on the body, we will neglect air resistance

1)the weight of the body of mass m given by ……..(1) Where is the constant acceleration due to gravity. 2)the spring force excreted by the spring when it is stretched(by hookes law)This force is called stress is proportional to the strain where s=stretch of spring k=constant of proportionality called spring modulus

When the body address the position of the body is called static equilibrium. If the spring is stretch by a constant amount of length so such that resulting force cancel with the weight (1) then

let the displacement of the body at any instant be From hooke’s law it follows that the spring force corresponding to a displacement is

When is the spring force when the body is in static equilibrium and additional spring force is caused due to displacement From (1) and (3), the resultant of the force , acting vertically in opposite direction , is given by

FROM ..(2) IT BECOMES = ……(4)SINCE THERE ARE NO EXERTNAL FORCES, THE SYSTEM IS SAID TO BE “FREE”.

=

UNDAMPED SYSTEM

When the damping of the system is negligibly small, then it can be ignored an hence (4) gives all forces acting on the system . Thus by newton second law of motion, we have

Which is the governing differential equation of undamped system. The characteristic equation is:-

, where=

The general solution of (5) is,

= Where AND = Thus , (6) represent the motion of the spring and also called as

harmonic oscillation . The period of the motion is . Thus the body executes cycle per second.

Here , is also called the frequency of the oscillation.`

THANK U……..