SDOF and MDOF

Presented

by• Ajay Partap

Singh

OUTLINE -1- DOF2- DAMPING3- SDOF4- MDOF

DEGREE OF FREEDOM

Degrees of Freedom

Degrees of Freedom (DOF)Number of Independent Co-ordinates required to define the displaced position of all the masses relative to their all positions are defined as degrees of freedom.

EXAMPLES

RIGID BODY

An unconstrained rigid body in space has Six Degrees of Freedom: three translational and three rotational.

DEGREE OF

FREEDOM

SDOF• SINGLE DEGREE OF FREEDOM

DAMPINGDamper System-

Damping represents loss of energy.

SDOFThe simplest vibratory system can be described by a single mass connected to a spring (and possibly a dashpot). The mass is allowed to travel only along the spring elongation direction. Such systems are called Single Degree-of-Freedom (SDOF) systems and are shown in the following figure,

EQUATION OF MOTION FOR SDOF

f(t)kxxcxm

SDOF vibration can be analyzed by Newton's second law of motion, F = m*a.

UNDAMPED SDOF  If there is no resistance or damping in the system, , the oscillatory motion will continue forever with a constant amplitude. Such a system is termed Undamped

f(t)kxxm

SOLUTION OF UNDAMPED SDOF

• Natural Frequency =

As Natural Frequency depends only on the system mass and the spring stiffness (i.e. any damping will not change the natural frequency of a system

Solution also express as, where,

A- AmplitudeØ -Phase

SAMPLE TIME BEHAVIOR

The displacement plot of an Undamped system would appear as -

 An assumption of zero damping is typically not accurate. In reality, there almost always exists some resistance in vibratory systems. This resistance will damp the vibration and dissipate energy; the oscillatory motion caused by the initial disturbance will eventually be reduced to zero.

DAMPED SDOF SDOF vibration can be analyzed by Newton's second law of motion, F = m*a.

f(t)kxxcxm

SOLUTION OF DAMPED SDOF

• The characteristic equation for this problem is,

• which determines the 2 independent roots for the• equation fall into one of the following 3 cases:-

< 0

=0

> 0

underdamped

critically-damped

overdamped

DAMPED, UNDAMPED ,CRITICALLY DAMPED SYSTEM

underdamped critically-dampedoverdamped

TERMS • To simplify the solutions coming up, we define the critical damping cc, the damping ratio z, and

the damped vibration frequency wd as

Critical damping 

Damping ratio

Damped vibration frequency

Natural Frequency

Note that wd will equal wn when the damping of the system is zero (i.e. Undamped).

The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance

UNDERDAMPED SYSTEM< 0 The system oscillates (at reduced frequency compared to

the Undamped case) with the amplitude gradually decreasing to zero

Note that the displacement amplitude decays exponentially.

CRITICALLY-DAMPED SYSTEM= 0 The system returns to equilibrium as quickly as possible

without oscillating E.g.-The recoil mechanisms in most guns are also critically damped so that they return to their original position, after the recoil due to firing, in the least possible time.

The displacement decays to a negligible level after one natural period, Tn

The displacement solution-

OVERDAMPED SYSTEM> 0 The system returns (exponentially decays) to equilibrium

without oscillating.

The motion of an over damped system is non-periodic, regardless of the initial conditions. The larger the damping, the longer the time to decay from an initial disturbance.

The displacement solution-

MDOFA Multi –Degree of freedom system, as the name suggests, is one that requires two or more independent coordinates to describe its motion.

MDOF

HOW TO SOLVE MDOFConsider the 3 degree-of-freedom system,

To fully characterize the system we must know the positions of the three masses (x1, x2, and x3).

MDOF

In matrix form the equations become,

Then Solve by Matrix Methods

HOPE YOY ENJOY IT