3.3 Hierarchical Principles of Model Building
121
We construct a model of this system in generalized coordinates using the Lagrange
equations of the second kind. The second-order Lagrange equations are a system
of ordinary second-order differential equations. They describe the movement of a
mechanical system subordinate to ideal bonds. Lagrange equations of the second
kind can be used to study the motion of any mechanical system with geometric
connections, regardless of how many points or bodies enter the system, how the
bodies move, and what kind of movement is considered.
If the motion of the holonomic system is described by the generalized coordinates
q and the generalized velocities ˙
q, then the equations of motion have the form
d
dt
∂ T (q, ˙
q)
∂ ˙
q
−
∂ T (q, ˙
q)
∂q
= Q ext
where T (q, ˙
q) is the kinetic energy of the system and Q ext is the generalized force.
The difference in the total time derivative of the partial derivative of kinetic energy
with respect to the generalized velocity and the partial derivative of kinetic energy
with respect to the generalized coordinate is equal to the generalized force. We also
note that if the dimension q is length, then Q ext has the dimension of ordinary force;
if the generalized coordinate q is an angle (a dimensionless dimension), then Q ext
has the dimension of the moment of force.
So, in order to write the Lagrange equation, you must sequentially perform the
following steps:
• set generalized coordinates;
• record kinetic and potential energy: T (q, ˙
q) and W (q);
• calculate Lagrangian L(q, ˙
q);
• identify external applied forces Q ext ;
• use L(q, ˙
q) and Q ext to get the expression for the Lagrange equation.
The trolley–pendulum system has two degrees of freedom and is located in the
field of gravity. Let q 1 = y as the generalized coordinates be the offset of the trolley
from the origin, and q 2 = θ is the angle of deviation of the rod from the vertical, as
shown in Fig. 3.24.
We represent the kinetic energy of the system as the sum of the kinetic energy of
the trolley and pendulum:
T = T 1 + T 2 =
1
2
m 1 ˙
y
2
+
1
2
m 2 v
2
2 ,
here v 2 is the absolute speed of the load
− → v 2 = =
v
rel
+ +
v
abs
Given that
v
rel
= l ˙
θ, v
abs
= ˙
y
121
We construct a model of this system in generalized coordinates using the Lagrange
equations of the second kind. The second-order Lagrange equations are a system
of ordinary second-order differential equations. They describe the movement of a
mechanical system subordinate to ideal bonds. Lagrange equations of the second
kind can be used to study the motion of any mechanical system with geometric
connections, regardless of how many points or bodies enter the system, how the
bodies move, and what kind of movement is considered.
If the motion of the holonomic system is described by the generalized coordinates
q and the generalized velocities ˙
q, then the equations of motion have the form
d
dt
∂ T (q, ˙
q)
∂ ˙
q
−
∂ T (q, ˙
q)
∂q
= Q ext
where T (q, ˙
q) is the kinetic energy of the system and Q ext is the generalized force.
The difference in the total time derivative of the partial derivative of kinetic energy
with respect to the generalized velocity and the partial derivative of kinetic energy
with respect to the generalized coordinate is equal to the generalized force. We also
note that if the dimension q is length, then Q ext has the dimension of ordinary force;
if the generalized coordinate q is an angle (a dimensionless dimension), then Q ext
has the dimension of the moment of force.
So, in order to write the Lagrange equation, you must sequentially perform the
following steps:
• set generalized coordinates;
• record kinetic and potential energy: T (q, ˙
q) and W (q);
• calculate Lagrangian L(q, ˙
q);
• identify external applied forces Q ext ;
• use L(q, ˙
q) and Q ext to get the expression for the Lagrange equation.
The trolley–pendulum system has two degrees of freedom and is located in the
field of gravity. Let q 1 = y as the generalized coordinates be the offset of the trolley
from the origin, and q 2 = θ is the angle of deviation of the rod from the vertical, as
shown in Fig. 3.24.
We represent the kinetic energy of the system as the sum of the kinetic energy of
the trolley and pendulum:
T = T 1 + T 2 =
1
2
m 1 ˙
y
2
+
1
2
m 2 v
2
2 ,
here v 2 is the absolute speed of the load
− → v 2 = =
v
rel
+ +
v
abs
Given that
v
rel
= l ˙
θ, v
abs
= ˙
y