2
1 Introduction
The book has three parts. The first part (Chap. 2) is on single degree of freedom
(SDOF) systems. The third part (Chaps. 4 and 5) is on the multi-degree of freedom
(MDOF) and continuous systems. The insertion of Chap. 3 on the eigenvalue
problem between developments on SDOF systems of Chap. 2 and MDOF and
continuous systems of Chaps. 4 and 5 is intentional for the following reason. This
nonstandard approach has been successful with the Cornell students, since, if one
masters the theory of SDOF systems and of the eigenvalue problem, he/she will be
capable to reconstruct developments in the third part of the book (Chaps. 4 and 5)
by viewing the displacement functions of MDOF/continuous systems as elements
of the vector spaces spanned by the eigenvectors/eigenfunctions of these systems
whose components are defined by equations of the types satisfied by SDOF systems.
The number of degrees of freedom is equal to the number of parameters
(functions of time) needed to specify the position of the masses of a system at all
times. Consider a massless blade fixed at one end and free at the other end with
a concentrated point mass, which vibrates in the plane of the paper. If the blade
deformation is small, the position of its concentrated mass (the only mass in the
system) can be described with good approximation by a single function of time,
e.g., the distance of the concentrated mass to the undeformed blade, see Fig. 1.1a.
The system has a single degree of freedom. If the blade deformation is large, two
functions are required to describe the position of the concentrated mass, e.g., the
distance of the concentrated mass to the undeformed blade and its distance to the
fixed end of the blade, see Fig. 1.1b. The system has two degrees of freedom. We
refer to systems that require n ≥ 2 functions of time to describe the position of their
masses at all times as multi-degree of freedom systems with n degrees of freedom.
A blade with n > 1 point masses undergoing small deformation is another example
of MDOF system with n degrees of freedom, see the system in Fig. 1.1c which has
3 masses so that n = 3. If the assumption that the blade is massless is removed,
the blade has masses at all spatial locations so that an infinite number of functions
of time is needed to describe the position of the system masses at all times, see
Fig. 1.1 Number n of
degrees of freedom:
(a) n = 1, (b) n = 2,
(c) n = 3, and (d) n = ∞
(a)
(b)
(c)
(d)
1 Introduction
The book has three parts. The first part (Chap. 2) is on single degree of freedom
(SDOF) systems. The third part (Chaps. 4 and 5) is on the multi-degree of freedom
(MDOF) and continuous systems. The insertion of Chap. 3 on the eigenvalue
problem between developments on SDOF systems of Chap. 2 and MDOF and
continuous systems of Chaps. 4 and 5 is intentional for the following reason. This
nonstandard approach has been successful with the Cornell students, since, if one
masters the theory of SDOF systems and of the eigenvalue problem, he/she will be
capable to reconstruct developments in the third part of the book (Chaps. 4 and 5)
by viewing the displacement functions of MDOF/continuous systems as elements
of the vector spaces spanned by the eigenvectors/eigenfunctions of these systems
whose components are defined by equations of the types satisfied by SDOF systems.
The number of degrees of freedom is equal to the number of parameters
(functions of time) needed to specify the position of the masses of a system at all
times. Consider a massless blade fixed at one end and free at the other end with
a concentrated point mass, which vibrates in the plane of the paper. If the blade
deformation is small, the position of its concentrated mass (the only mass in the
system) can be described with good approximation by a single function of time,
e.g., the distance of the concentrated mass to the undeformed blade, see Fig. 1.1a.
The system has a single degree of freedom. If the blade deformation is large, two
functions are required to describe the position of the concentrated mass, e.g., the
distance of the concentrated mass to the undeformed blade and its distance to the
fixed end of the blade, see Fig. 1.1b. The system has two degrees of freedom. We
refer to systems that require n ≥ 2 functions of time to describe the position of their
masses at all times as multi-degree of freedom systems with n degrees of freedom.
A blade with n > 1 point masses undergoing small deformation is another example
of MDOF system with n degrees of freedom, see the system in Fig. 1.1c which has
3 masses so that n = 3. If the assumption that the blade is massless is removed,
the blade has masses at all spatial locations so that an infinite number of functions
of time is needed to describe the position of the system masses at all times, see
Fig. 1.1 Number n of
degrees of freedom:
(a) n = 1, (b) n = 2,
(c) n = 3, and (d) n = ∞
(a)
(b)
(c)
(d)