2.9 Problems
Problem 2.1 The clamped-hinged elastic column in Fig. P2.1 has bending
stiffness EI and length l, and is centrally loaded by a compressive force P.
a) Set up an EVP for the determination of buckling loads and associated buckling
modes.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.2 A mass M hangs in a chain having length l and mass per unit length
m. The chain is vertically suspended in a gravity field of strength g.
a) Set up an EVP for the determination of natural frequencies and mode shapes for
small transverse oscillation of the chain.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.3 The beam in Fig. P2.3 has bending stiffness EI, and is clamped in
one end and supported at the other end by springs having linear stiffness c.
a) Set up an EVP for the determination of a critical buckling load P.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.4 The clamped-free torsional rod in Fig. P2.4 has axially varying
moment of inertia I(x), density q(x) and torsional stiffness GK(x).
a) Set up an EVP for the determination of natural frequencies associated with small
torsional vibrations of the rod.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.5 The clamped-free beam in Fig. P2.5 has mass per unit length
qA(x) and bending stiffness EI(x), and is oriented vertically in a gravity field g.
a) Set up an EVP for the determination of natural frequencies associated with small
amplitude bending vibrations of the beam.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.6 The elastic rope in Fig. P2.6 has constant density q and varying
cross-sectional area A(x) = A 0 (1 + x
2 ), and is pre-tensioned by a force P between
two rigid walls at x = ±1.
a) Set up an EVP for the determination of natural frequencies associated with small
transverse vibrations of the rope.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.7 The stiff cylindrical tube in Fig. P2.7 has inner radius R, and rotates
about its axis with angular speed X. A flexible column of circular cross-section has
one end rigidly clamped at the inner rim of the rotating tube and the other end free.
The column has length l, bending stiffness EI and mass per unit length qA. The
undeformed axis of the column coincides with a radius of the tube.
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2 Eigenvalue Problems of Vibrations and Stability
Problem 2.1 The clamped-hinged elastic column in Fig. P2.1 has bending
stiffness EI and length l, and is centrally loaded by a compressive force P.
a) Set up an EVP for the determination of buckling loads and associated buckling
modes.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.2 A mass M hangs in a chain having length l and mass per unit length
m. The chain is vertically suspended in a gravity field of strength g.
a) Set up an EVP for the determination of natural frequencies and mode shapes for
small transverse oscillation of the chain.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.3 The beam in Fig. P2.3 has bending stiffness EI, and is clamped in
one end and supported at the other end by springs having linear stiffness c.
a) Set up an EVP for the determination of a critical buckling load P.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.4 The clamped-free torsional rod in Fig. P2.4 has axially varying
moment of inertia I(x), density q(x) and torsional stiffness GK(x).
a) Set up an EVP for the determination of natural frequencies associated with small
torsional vibrations of the rod.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.5 The clamped-free beam in Fig. P2.5 has mass per unit length
qA(x) and bending stiffness EI(x), and is oriented vertically in a gravity field g.
a) Set up an EVP for the determination of natural frequencies associated with small
amplitude bending vibrations of the beam.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.6 The elastic rope in Fig. P2.6 has constant density q and varying
cross-sectional area A(x) = A 0 (1 + x
2 ), and is pre-tensioned by a force P between
two rigid walls at x = ±1.
a) Set up an EVP for the determination of natural frequencies associated with small
transverse vibrations of the rope.
b) Examine whether the EVP is self-adjoint and completely definite.
Problem 2.7 The stiff cylindrical tube in Fig. P2.7 has inner radius R, and rotates
about its axis with angular speed X. A flexible column of circular cross-section has
one end rigidly clamped at the inner rim of the rotating tube and the other end free.
The column has length l, bending stiffness EI and mass per unit length qA. The
undeformed axis of the column coincides with a radius of the tube.
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2 Eigenvalue Problems of Vibrations and Stability