Mechanical Vibration

Mechanical VibrationLecture # 02

TopicDegrees of FreedomMEHRAN UNIVERSITY OF ENGINEERING & TECHNOLOGY, JAMSHORO MECHANICAL ENGINEERING, DEPARTMENT13-BATCHDegrees of Freedom (DOF)Degrees of freedom are the number of independent coordinates that describe the position of a mechanical system at any instant of time.

DOF is Selection of parameters for a system that describe the motion of that system. if we know the values of these variables at a particular instant in time then we know the configuration of the system at that time.

The minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time defines the number of degrees of freedom of the system.

In figure:Three points A B CCoordinates are:(xB, yB,) (xC, yC,) AB, BC , And length of both barslAB, lBC,Degrees of Freedom (DOF)If we know the location of pins B and C at any time, then we know the configuration of the entire system at that time, since the lengths of the rigid rods are specified. We know the location of every point of the system.

The location of pins B and C may be characterized in two ways. i) Cartesian coordinates (xB , yB) and (xC , yC), ii) Angular coordinates AB and BC,

Both sets of coordinates describe the configuration of the mechanism completely. Combination of both sets can also be used.

In figure:Three points A B CCoordinates are:(xB, yB,) (xC, yC,) AB, BC , And length of both barslAB, lBC,Degrees of Freedom (DOF)If we choose the angular coordinates (AB, BC )then we only need two coordinates to describe the configuration of the system, If we choose the Cartesian coordinates ((xB, yB,) (xC, yC,))we need four coordinates.if we choose the mixed set of coordinates ((xB, yB,) BC ) or ((xC, yC,) AB ) we need three coordinates.The minimum number of coordinates is referred to as the degrees of freedom of the system. The two angular coordinates may not be expressed in terms of one another. They are said to be independent in this regard. DOF, the number of degrees of freedom of a system refers to the number of independent coordinates needed to describe its configuration at any time.

In figure:Three points A B CCoordinates are:(xB, yB,) (xC, yC,) AB, BC , And length of both barslAB, lBC,Newtons second law of motionNewtons second law of motion and the expression of kinetic energy are presented for three types of motion: 1) Pure translational motion, 2) Pure rotational motion, and 3) Planar motion (combined translational and rotational).

1) Pure translational motionNewtons second law of motion is Forces= Mass x Acceleration.Newton's second Law for a static system EquilibriumF=0 zero

mmThe fundamental kinematical quantities used to describe the motion of a particle are i) displacement, ii) velocity, and iii) acceleration vectors.Displacement(d)=

Velocity (v)=

Acceleration (a) =

Newtons second law of motion1) Pure translational motion (cont)

Potential energy = Kinetic energy =

As we know F=ma and F=0

There are 2 forces on shown spring mass system.i) Mass x Acceleration and ii) Spring force ma + Sf=0Spring force = Spring stiffness x distance = kx

The equation can be written as:

mmk

Newtons second law of motion

EXERCISE # 1Express figure b and c.Single Degree of Freedom Systems:Single degree of freedom systems are the simplest systems as they require only one independent coordinate to describe their configuration. The simplest example of a single degree of freedom system is the mass-spring system.The coordinate x indicates the position of the mass measured relative to its position when the mass less elastic spring is un-stretched. If x is known as a function of time t, that is x = x(t) is known, then the motion of the entire system is known as a function of time.

mmSingle Degree of Freedom Systems:The simple pendulum is also a one degree of freedom system since the motion of the entire system is known if the angular coordinate is known as a function of time. The position of the bob may be described by the two Cartesian coordinates, x(t) and y(t), these coordinates are not independent. That is, the Cartesian coordinates (x, y) of the bob are related by the constraint equation, x2 + y2 = L2. Thus, if x is known then y is known and vice versa. Further, both x(t) and y(t) are known, if (t) is known. In either case, only one coordinate is needed to characterize the configuration of the system. The system therefore has one degree of freedom

Two Degree of Freedom SystemsIn two mass-spring, the configuration of the entire system is known if the position of mass m1 is known and the position of mass m2 is known. The positions are known if the coordinates u1 and u2 are known. u1 and u2 represent the displacements of the respective masses from their equilibrium configurations.

The motion of the double pendulum is known if the angular displacements, 1 and 2, measured from the vertical equilibrium configurations of the masses, are known functions of time.Multi-Degree of Freedom SystemsThe system with more than 1 DOF is known as multi-degrees of freedom.

Two degrees of freedom systems are a special case of multi-degree of freedom systems. In figures consider general N degree of freedom systems, where N can take on any integer value as large as we like.

Examples of such systems are the system comprised of N masses and N + 1 springs shown mass spring system, and the compound pendulum consisting of N rods and N bobs.

EXERCISE

EXERCISE

Continuous SystemsContinuous systems that have a finite (or even infinite) number of masses separated by a finite distance. Continuous systems are systems whose mass is distributed continuously, typically over a finite domain. An example of a continuous system is the elastic beam. For the case of a linear beam (one for which the strain-displacement relation contains only first order terms of the displacement gradient), the transverse motion of the beam is known if the transverse deflection, w(x, t), of each particle located at the coordinates 0 xL along the axis of the beam is known.