Chapter 2
Single Degree of Freedom (SDOF)
Systems
We review briefly the second law of Newton and its impulse-momentum and workenergy versions and use them to write equations of motion for single degree of
freedom systems. These equations are solved by analysis in the time and frequency
domains. Analytical and numerical methods are used for solution. Multi-degree of
freedom and continuous systems are considered in Chaps. 4 and 5.
2.1 Newton’s Second Law
Consider a material point with mass m which is subjected to a, generally, timedependent force f . The point moves with velocity v along the real line according to
the ordinary differential equation
d
dt
m v
= f or, equivalently, d
m v
= f dt, (Newton’s law (NL)).
(2.1)
It is common to denote the point position, velocity, and acceleration by x or x(t),
˙
x, ˙
x(t), v, or v(t), and ¨
x, ¨
x(t), a, or a(t). Similarly, the time argument of the mass
and force may or may not be written so that they are denoted by m or m(t) and f or
f (t). If the mass is time-invariant, the Newton law becomes
m a(t) = f (t) or, with the above convention, m a = f.
(2.2)
The impulse-momentum (IM) and the work-energy (WE) relationships result
from Newton’s law (NL) by elementary manipulations. They are useful in applications since the solutions of some problem by these relationships are more efficient
than that by Newton’s law.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6_2
5
Single Degree of Freedom (SDOF)
Systems
We review briefly the second law of Newton and its impulse-momentum and workenergy versions and use them to write equations of motion for single degree of
freedom systems. These equations are solved by analysis in the time and frequency
domains. Analytical and numerical methods are used for solution. Multi-degree of
freedom and continuous systems are considered in Chaps. 4 and 5.
2.1 Newton’s Second Law
Consider a material point with mass m which is subjected to a, generally, timedependent force f . The point moves with velocity v along the real line according to
the ordinary differential equation
d
dt
m v
= f or, equivalently, d
m v
= f dt, (Newton’s law (NL)).
(2.1)
It is common to denote the point position, velocity, and acceleration by x or x(t),
˙
x, ˙
x(t), v, or v(t), and ¨
x, ¨
x(t), a, or a(t). Similarly, the time argument of the mass
and force may or may not be written so that they are denoted by m or m(t) and f or
f (t). If the mass is time-invariant, the Newton law becomes
m a(t) = f (t) or, with the above convention, m a = f.
(2.2)
The impulse-momentum (IM) and the work-energy (WE) relationships result
from Newton’s law (NL) by elementary manipulations. They are useful in applications since the solutions of some problem by these relationships are more efficient
than that by Newton’s law.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6_2
5