Most important in this respect is Lagrange’s theorem, from which follows that
for purely dissipative systems the equilibrium x = 0 of (1.129) (with f = 0) is stable
if K is positive definite. Since K involves only the static part of the system, a full
dynamic analysis is then not needed to assess the stability of x = 0.
When the system is not only purely but also completely dissipative (i.e. C S is
positive definite) a stronger statement can be made (Ziegler 1968, Theorem 8): The
equilibrium x = 0 is stable if K is positive definite, and unstable if K is not positive
definite. Again, dynamic analysis is not needed.
Another important theorem says that the same holds for non-gyroscopic conservative systems, i.e. x = 0 is (un)stable if K is (not) positive definite (Ziegler
1968, Theorem 1). Also, it can be shown that gyroscopic forces cannot destabilize
x = 0 for a conservative system, implying that stability can be checked with
gyroscopic forces ignored (Ziegler 1968, Theorem 5).
For most other system classes the stability of x = 0 cannot be assessed by
considering just the static part (i.e. K). Then a full dynamic analysis of stability is
required, involving all terms in (1.129), as described in Chaps. 3-5 of this book.
1.10 Problems
Problem 1.1 Solve € u + 2fx _
u + x
2 u = 0 for u(t) when u(0) = u 0 , _
u(0) = 0, and
0 < f<1, and sketch the solution.
Problem 1.2 Solve _
u + fx _
u + x
2 u = pcos(Xt) for u(t), and sketch the stationary
amplitude of u(t) as a function of X when, respectively, f = 0 and 0 < f ( 1.
Problem 1.3 Solve € u + fx _
u + x
2 u = pd(t − t 0 ) and sketch the solution for
f ( 1.
Problem 1.4 For the model system shown in Fig. 1.4 (Sect. 1.3.1), Calculate the
undamped natural frequencies and associated mode shapes when, respectively,
k 1 = k 2 = k and k 1 = k 2 /2 = k.
Problem 1.5 For the model system shown in Fig. 1.4 (Sect. 1.3.1), Calculate the
forced response when F(t) = F 0 sin(Xt), a = l/2, k 1 = k 2 = k and c 1 = c 2 = c.
Problem 1.6 The mass m in Fig. P1.6 moves in a plane (x, y), restricted by
springs with linear stiffness k.
a) Use Lagrange’s equations for setting up the equations of motion.
b) Expand nonlinear terms to order three to yield polynomial nonlinearities.
c) Linearize the equations of motion for small oscillations near (x, y) = (0, 0).
d) Compute the undamped natural frequencies and mode shapes.
Problem 1.7 The beam in Fig. P1.7 is simply supported, has length l, flexural
stiffness EI and mass per unit length qA. It is subjected to a time-varying load P(t) at
x = x 0 , where a linear spring of relatively small stiffness k restricts the motion.
1.9 Classification of Forces and Systems
45
for purely dissipative systems the equilibrium x = 0 of (1.129) (with f = 0) is stable
if K is positive definite. Since K involves only the static part of the system, a full
dynamic analysis is then not needed to assess the stability of x = 0.
When the system is not only purely but also completely dissipative (i.e. C S is
positive definite) a stronger statement can be made (Ziegler 1968, Theorem 8): The
equilibrium x = 0 is stable if K is positive definite, and unstable if K is not positive
definite. Again, dynamic analysis is not needed.
Another important theorem says that the same holds for non-gyroscopic conservative systems, i.e. x = 0 is (un)stable if K is (not) positive definite (Ziegler
1968, Theorem 1). Also, it can be shown that gyroscopic forces cannot destabilize
x = 0 for a conservative system, implying that stability can be checked with
gyroscopic forces ignored (Ziegler 1968, Theorem 5).
For most other system classes the stability of x = 0 cannot be assessed by
considering just the static part (i.e. K). Then a full dynamic analysis of stability is
required, involving all terms in (1.129), as described in Chaps. 3-5 of this book.
1.10 Problems
Problem 1.1 Solve € u + 2fx _
u + x
2 u = 0 for u(t) when u(0) = u 0 , _
u(0) = 0, and
0 < f<1, and sketch the solution.
Problem 1.2 Solve _
u + fx _
u + x
2 u = pcos(Xt) for u(t), and sketch the stationary
amplitude of u(t) as a function of X when, respectively, f = 0 and 0 < f ( 1.
Problem 1.3 Solve € u + fx _
u + x
2 u = pd(t − t 0 ) and sketch the solution for
f ( 1.
Problem 1.4 For the model system shown in Fig. 1.4 (Sect. 1.3.1), Calculate the
undamped natural frequencies and associated mode shapes when, respectively,
k 1 = k 2 = k and k 1 = k 2 /2 = k.
Problem 1.5 For the model system shown in Fig. 1.4 (Sect. 1.3.1), Calculate the
forced response when F(t) = F 0 sin(Xt), a = l/2, k 1 = k 2 = k and c 1 = c 2 = c.
Problem 1.6 The mass m in Fig. P1.6 moves in a plane (x, y), restricted by
springs with linear stiffness k.
a) Use Lagrange’s equations for setting up the equations of motion.
b) Expand nonlinear terms to order three to yield polynomial nonlinearities.
c) Linearize the equations of motion for small oscillations near (x, y) = (0, 0).
d) Compute the undamped natural frequencies and mode shapes.
Problem 1.7 The beam in Fig. P1.7 is simply supported, has length l, flexural
stiffness EI and mass per unit length qA. It is subjected to a time-varying load P(t) at
x = x 0 , where a linear spring of relatively small stiffness k restricts the motion.
1.9 Classification of Forces and Systems
45