the method degenerates progressively for higher natural frequencies. The lower
frequency approximations may be very accurate.
1.5 Energy Methods for Setting up Equations
of Motion
Among numerous energy methods for setting up equations of motion, we present
here only the two to be employed in subsequent chapters: Lagrange’s equations and
Hamilton’s principle. For more complete treatments of these and other methods see,
e.g., Arczewski et al. (1993), Chen (1966), El Naschie (1990a), Greenwood (2003),
Lanczos (1962), or Tabarrok and Rimrott (1994).
Energy methods are occasionally preferred over force–balance methods (e.g.,
Newton’s 2nd law). This is because scalar measures of energy are typically easier to
calculate than forces, being vectorial quantities. For a particular system there will
typically be common agreement upon the energies involved, whereas the decision
of which forces are relevant, and how they relate to the system state, may take some
discussion. In particular this holds true when rotating coordinate systems and/or
nonlinearities are involved. Using energy or force methods, the efforts spent in
setting up equations of motion may be the same. However, with energy methods the
main efforts are spent on performing trivial mathematical operations, the correctness
of which may be easily checked.
1.5.1 Lagrange’s Equations
Lagrange’s equations are applicable for systems having a finite number of degrees
of freedom, those we call multi-DOF, discrete or lumped systems.
With N degrees of freedom, choose a set of generalized coordinates q i (t), i = 1,
N, uniquely defining the state of the system. Express the potential and kinetic
energy V and T of the system in terms of generalized coordinates q i and generalized
velocities _
q i . Consider p vectors of external forces F k which are not derivable from
potentials, and denote by r k the natural coordinates (Cartesian, e.g.) of the associated system masses. Lagrange’s equations then take the form:
d
dt
@L
@ _
q i
À
@L
@q i
¼ Q i ; i ¼ 1; N;
ð1:72Þ
where
L T À V; Q i
X p
k¼1
F k Á
@r k
@q i
:
ð1:73Þ
20
1 Vibration Basics
frequency approximations may be very accurate.
1.5 Energy Methods for Setting up Equations
of Motion
Among numerous energy methods for setting up equations of motion, we present
here only the two to be employed in subsequent chapters: Lagrange’s equations and
Hamilton’s principle. For more complete treatments of these and other methods see,
e.g., Arczewski et al. (1993), Chen (1966), El Naschie (1990a), Greenwood (2003),
Lanczos (1962), or Tabarrok and Rimrott (1994).
Energy methods are occasionally preferred over force–balance methods (e.g.,
Newton’s 2nd law). This is because scalar measures of energy are typically easier to
calculate than forces, being vectorial quantities. For a particular system there will
typically be common agreement upon the energies involved, whereas the decision
of which forces are relevant, and how they relate to the system state, may take some
discussion. In particular this holds true when rotating coordinate systems and/or
nonlinearities are involved. Using energy or force methods, the efforts spent in
setting up equations of motion may be the same. However, with energy methods the
main efforts are spent on performing trivial mathematical operations, the correctness
of which may be easily checked.
1.5.1 Lagrange’s Equations
Lagrange’s equations are applicable for systems having a finite number of degrees
of freedom, those we call multi-DOF, discrete or lumped systems.
With N degrees of freedom, choose a set of generalized coordinates q i (t), i = 1,
N, uniquely defining the state of the system. Express the potential and kinetic
energy V and T of the system in terms of generalized coordinates q i and generalized
velocities _
q i . Consider p vectors of external forces F k which are not derivable from
potentials, and denote by r k the natural coordinates (Cartesian, e.g.) of the associated system masses. Lagrange’s equations then take the form:
d
dt
@L
@ _
q i
À
@L
@q i
¼ Q i ; i ¼ 1; N;
ð1:72Þ
where
L T À V; Q i
X p
k¼1
F k Á
@r k
@q i
:
ð1:73Þ
20
1 Vibration Basics