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1. Methods for Modeling Mechanical Systems
Two main methods are used in modeling of mechanical systems. In the
first method, positions, velocities and accelerations are found by doing
kinematic analysis. Then, laws of mechanics are applied by drawing free
body diagrams.
1. METHOD
• Kinematic Analysis (position, velocity and acceleration)
• Free body diagrams, Laws of mechanics
In the second method, positions and velocities are found by doing
kinematic analysis. Then, for the system, Lagrange equation is applied by
writing the expressions of total kinetic energy, potential energy, and
virtual work.
2. METHOD
• Kinematic Analysis (position and velocity)
• Kinetic energy, potential energy, virtual work. Lagrange equation
In this course, the second method will be applied.
BASIC MECHANICAL SYSTEM ELEMENTS
In mechanical systems, there are translational and rotational elements.
We can model mechanical systems using translational or rotational
elements or both.
Let’s deal with the basic mechanical system elements.
Translational Elements: Mass
The first one of the translational elements is the mass of m. F is the
force, which is applied to the mass. x is the displacement of the mass. F
is also an element equation which equals to mx . E1 is the kinetic energy,
which is stored in the mass of m and equals to 1 mx 2 .
2
x(t)
1
W  Fx
E1  mx 2
F(t)
F  mx
m
2
W is the virtual work, which is done by external forces acting on the
element of m.
Translational Elements: Spring
The second one of the translational elements is the spring with a spring
constant k. F is the force, which is applied to the spring. x is the
displacement of the spring. Element equation F equals kx. E2 is the
1
potential energy, which is stored in the spring element and equals to kx2 .
2
x(t)
1 2
k
E

kx
F  kx
2
2
F(t)
Translational Elements: Damping
The third one of the translational elements is the damper with a damping
constant c. F is the damping force acting on the element. x is the
displacement of the element. x
 is the velocity which equals the
derivative of x. Damping force equals to cx .
c
x(t)
F(t)
F  cx
W  cx x
In this course, viscous damping is assumed for a damper element.
friction
velocity
c
x (t )
velocity
friction
c
x (t )
Viscous friction is in opposite direction to the velocity; the magnitude
of the friction force is proportional to the magnitude of the velocity.
Virtual work also occurs in the damping element. The cause of the
negative sign in the formula of virtual work is that the direction of
frictional forces is always opposite to the direction of motion.
Rotational Elements
Similar equations can be written for rotational elements too. Here, T is the
moment.  is the angular displacement. dot is the angular velocity. 2dot
is the angular acceleration.
IG is the mass moment of inertia which is resistance to a rotational
motion. Kr is the torsional spring constant, Cr is the torsional damper
constant.
1
E1  I G  2
2
W  T
T  I G 
T  K rθ
1
E2  K r 2
2
T  Cr 
W  Cr  
You should learn the moment of inertia of a circular thin disc having
mass m, radius R which equals to mR2/2.
Also, you should learn the moment of inertia of a homogenous bar
having mass m and length L, which equals to mL2/12.
IG 
1
mL2
12
1
I G  mR 2
2
CAD : Solid Modeling
Mass for Tranlational ve Rotational Motion :
The kinetic energy of a rigid body, which performs translational and
rotational motion in plane, can be written as follows.
1
1
E1  mx G2  I G  2
2
2
x G is the velocity of the body’s center of mass.