Dynamic Modelling of Mechanical Systems

Transcription

1 Dynamic Modelling of Mechanical Systems Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering g IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD

2 Hints of the Last Assignment g The Governing EOM may be written as: 0 ) ( ) ( ) ( B K K M t f B K M a Now, you may consider the following states for the system: 0 ) ( B K K M. X C h d d O i f fi d O d b i h. X Covert the two second order ODEs into four first order ODEs and obtain the state space representation.

3 This Lecture Contains Modelling of a Mechanical System Basic Elements of a Mechanical System Eamples to Solve Joint Initiative of IITs and IISc - Funded by MHRD

4 Mechanical Systems Mechanical systems are generally modeled as a lumped parameter 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 the system; springs the potential energy and dashpots the energy that gets dissipated from a mechanical system in the form of heat/sound etc.

5 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 ais. The moment of inertia will be defined as: J m r dm Where r denotes the distance of an elemental mass dm from the centroidal ais. For a rotation about an ais which is at a distance d from the centroidal ais, following parallelais theorem the moment of inertia could be epressed as: J new J m d

6 For translatory mechanical systems, stiffness is represented by spring element k, while for rotational systems this is represented by torsional spring element k t. For eample: diameter diameter

7 A few more translational spring constants 7

8 Torsional Spring Constants 8

9 Damping Element 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 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 Velocity is low, ensuring a laminar fluid flow. 9

10 Friction Damping Element Another type of common damping force is the so-called dry friction force between two solid interfaces. This is known as Coulomb damping. In this model, the magnitude of damping force is assumed to be a constant, which is independent of the relative velocity (or slip velocity) at the interface. The direction of the damping force is opposite to that of the relative velocity. In a physical model, a Coulomb damper is represented by the symbol shown below. The nature of change of the friction force with respect to displacement of the system is shown net. The area under this curve represents the amount of energy dissipated from the system. 0

11 Concept of Degrees of Freedom 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 eample, a particle in three dimensional space may have 3 DOF, hence two such particles may have total 6DOF. However, if they are connected together by a rigid link, this will come down to 6-=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.

12 Eamples and Assignments Consider the first two cases: there are two links of identical lengths but subjected to different boundary constraints. Find out the DOF in each case. (A) Now, consider the following assignments and find out the governing EOM of the mechanical systems.

13 (B) (C)

14 Special References for this Lecture System Dynamics for Engineering Students Nicolae Lobontiu, Academic Press Fundamentals of Mechanical Vibrations S Graham Kelly, McGraw-Hill 4