dynamic modelling of mechanical systems

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 5

Dynamic Modelling of Mechanical Systems

Dr. Bishakh Bhattacharya

Professor, Department of Mechanical Engineeringg g

IIT Kanpur

Joint Initiative of IITs and IISc - Funded by MHRD


Hints of the Last Assignmentg

The Governing EOM may be written as:

0)(

)()(...

.

11211

..

11

BKKM

tfxBxxKxM a

Now, you may consider the following states for the system:

0)( 222212122 xBxKxxKxM

.

1

1

x

x

X

C h d d O i f fi d O d b i h

.

2

2

x

xX

2

Covert the two second order ODEs into four first order ODEs and obtain the state space representation.

Joint Initiative of IITs and IISc ‐ Funded by MHRD


This Lecture Contains

Modelling of a Mechanical SystemModelling of a Mechanical System

Basic Elements of a Mechanical Systemy

Examples to Solve

Joint Initiative of IITs and IISc - Funded by MHRD


Mechanical Systems

Mechanical systems are generally modeled as a lumped parameter system such that a distributed system like a beamparameter system, such that a distributed system like a beam could be considered to be a system consisting of an array of rigid inertia elements linked by a combination of mass‐less spring and dashpot elements.

The inertia elements represent the kinetic energy stored in p gythe system; springs the potential energy and dashpots the energy that gets dissipated from a mechanical system in the form of heat/sound etcform of heat/sound etc .



For translatory mechanical systems, inertia is represented by mass ‘ m’, while for rotational systems this is represented by moment of inertia ‘J’.

Consider a rotor of mass ‘m’, rotating about it’s centroidal axis. The moment of inertia will be defined as:

dmrJm 2

Where ‘r’ denotes the distance of an elemental mass dm from the centroidal axis. For a rotation about an axis which is at a distance ‘d’ from the centroidal axis, following parallel‐axis theorem the moment of inertia could be expressed as:

2dJJ 2dmJJ new



For translatory mechanical systems, stiffness is represented y y , pby spring element ‘k’, while for rotational systems this is represented by torsional spring element ‘kt ’. For example:

diameterdiameterdiameter



A few more translational spring constants

7Joint Initiative of IITs and IISc ‐ Funded by MHRD


Torsional Spring Constantsp g



Damping Elementp g

There are two common damping elements used to model energy dissipation from a mechanical system. These are Viscoelastic Damping and Friction Damping.Viscous damping model is most common; here, the damping force is taken to be proportional to the velocity across the damper, acting in the direction opposite to that of the velocitythat of the velocity.Linear damping force is represented by a viscous dashpot, which shows a piston moving relative to a cylinder containing a fluid. The ideal linear relationship between the force and the relative velocity holds good so long as the relative y g gVelocity is low, ensuring a laminar fluid flow.



Friction Damping Elementp gAnother type of common damping force is the so-called dry friction force betweentwo solid interfaces. This is known as Coulomb damping. In this model, themagnitude of damping force is assumed to be a constant, which is independent ofthe relative velocity (or slip velocity) at the interface. The direction of the dampingthe relative velocity (or slip velocity) at the interface. The direction of the dampingforce is opposite to that of the relative velocity. In a physical model, a Coulombdamper is represented by the symbol shown below. The nature of change of the frictionforce with respect to displacement of the system is shown next. The area under thiscurve represents the amount of energy dissipated from the systemcurve represents the amount of energy dissipated from the system.



Concept of Degrees of Freedomp g

An important element in describing the dynamics of a system consisting of multiple lumped parameters is the Degrees of Freedom (DOF) for the system. This is defined as the number of kinematically independent variables required to describe completely the motion of the system.

It may be noted that the number of degrees of freedom of a particle/lumped mass gets reduced if it is subjected to constraints. For example, a particle in three dimensional space may have 3 DOF, hence two such particles may have total p y , p y6DOF. However, if they are connected together by a rigid link, this will come down to 6-1=5 DOF. Thus, the actual number of DOF of a system equals to the difference between the numbers of unconstrained DOF and the constraining conditions.



Examples and Assignmentsp g

Consider the first two cases: there are two links of identical lengths but subjected ff Oto different boundary constraints. Find out the DOF in each case.

(A)

N id th f ll i i t d fi d t th i EOM f thNow, consider the following assignments and find out the governing EOM of the mechanical systems.



(B)

(C)



Special References for this Lecturep

• System Dynamics for Engineering Students – Nicolae Lobontiu Academic Press• System Dynamics for Engineering Students – Nicolae Lobontiu, Academic Press

• Fundamentals of Mechanical Vibrations – S Graham Kelly, McGraw-Hill

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dynamic modelling of mechanical systems

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