ME 459 Dynamics of Machinery
Lagranges Equations for Multi-Degree-of-Freedom, Planar Systems
The configuration of systems with N degrees of freedom (DOF) can be defined in
terms of N generalized coordinates, say qk (k = 1,K, N ) . The differential equatio
ME 459 Dynamics of Machinery
Linearization of Differential Equations of Motion
The equations of motion (EOM) derived using Newtons laws or Lagranges
equations may be linear or nonlinear. If they are nonlinear, it may be possible to
linearize the equations
ME 4590 Dynamics of Machinery
Lagranges Equations for Multi-Degree-of-Freedom Systems with
Dependent Generalized Coordinates
Often it is more convenient to use a dependent set of generalized coordinates
qk (k = 1, n) to describe the configuration of a mec
ME 459 Dynamics of Machinery
Lagrange's Equations Examples
Example #1
The system at the right consists of two
bodies, a slender bar B and a disk D, moving
together in a vertical plane. As B rotates about
O, D rolls without slipping on the fixed circular
o
ME 4590 Dynamics of Machinery
Configuration Constraints for Mechanical Systems
Suppose the configuration of a mechanical system is
defined by "n" generalized coordinates, say
qk (k = 1, n) .
These coordinates may all be
independent or they may form a depe
ME 459 Dynamics of Machinery
Bearing Loads on a Simple Crank Shaft
The figure to the right shows a
simple crank shaft consisting of seven
segments, each considered to be a
slender bar. Each segment of length
has mass m .
There are six
segments of length
a
ME 459 Dynamics of Machinery
Angular Momentum of a Simple Crank Shaft
The figure to the right shows a
simple crank shaft consisting of seven
segments, each considered to be a
slender bar. Each segment of length
has mass m .
There are six
segments of lengt
ME 459 Dynamics of Machinery
Angular Momentum Fixed Axis Rotation
The three dimensional body shown in the figure to
the right is affixed to a shaft that rotates about the Zaxis. The X and Y axes are fixed in and rotate with
the body. For simplicity, assum
ME 459 Dynamics of Machinery
Degrees of Freedom of Mechanical Systems
System Configuration and Generalized Coordinates
The configuration of a mechanical system is defined as the position of each of the
bodies within the system. In general, both translatio
ME 459 Dynamics of Machinery
Equations of Motion for Fixed Axis Rotation
The figure at the right shows a rigid body rotating
about a fixed vertical axis. As the body rotates,
bearing loads may be generated in the X and Y
directions resulting from 1) the
ME 459 Dynamics of Machinery
Flywheel Dynamics
Background
To understand the fundamental need for
flywheels, consider the slider-crank
mechanism that moves in a horizontal plane
as shown to the right. The mechanism may
be driven by a torque at O or a force
ME 4590 Dynamics of Machinery
Generalized Forces
General Definition
Given a mechanical system whose configuration is defined by a set of generalized
coordinates qk (k = 1,., n) , we can define a generalized force associated with each of the
"n" generalize
ME 4590 Dynamics of Machinery
Partial Velocities and Partial Angular Velocities
Partial Velocities
If the velocity of some point P within a mechanical system can be written in terms of a set
of generalized coordinates qi (i = 1, n) and their time derivati
ME 4590 Dynamics of Machinery
Partial Velocities and the Slider Crank Mechanism
System Configuration
The figure shows a simple slider
crank mechanism with no offset. Given
the physical dimensions of the links
( R, L) , the configuration of the system
at a
ME 459 Dynamics of Machinery
Moments and Products of Inertia and the Inertia Matrix
Moments of Inertia
A rigid body B is shown in the diagram below. The unit vectors ( e1 , e2 , e3 ) are fixed in
% % %
the body and are directed along a convenient set of a
ME 4590 Dynamics of Machinery
Principle of Virtual Work
Principle of Virtual Work
If a mechanical system whose configuration is defined by a set of independent
generalized coordinates qk (k = 1,., n) is in static equilibrium, then the
generalized force as