114
4 Multi-Degree of Freedom (MDOF) Systems
Problem 4.2 Suppose that the 2-DOF system in Example 4.4 is subjected to an
acceleration a(t) = sin(ν t) applied at its fix end rather than the forcing function
f (t). First, find the steady-state modal and system responses for forcing frequencies
ν in the range [0, 1.5 ω 2 ]. Then, pretend that the modal frequencies are unknown.
Use the steady-state solutions to design an experiment for estimating the modal
frequencies.
Problem 4.3 Plot the time evolution of the displacement vector x(t) of the 2-DOF
system in Example 4.5 for a time interval [0, τ ] and initial conditions of your choice.
Plot also the modal shapes { i } of this system and identify the projections {q i (t)}
of x(t) on { i } at an arbitrary time t.
Problem 4.4 Repeat the calculations of Example 4.3 and assume that the system is
damped with damping matrix given by Eq. 4.14, α = 0.03 and β = 0.05. Compare
the system response with that in Example 4.3.
Problem 4.5 Repeat the calculations of Example 4.5 and assume that the system is
damped with damping matrix given by Eq. 4.14, α = 0.03 and β = 0.05. Compare
the system response with that in Example 4.5.
Problem 4.6 Repeat the calculations of Example 4.4 and assume that the system is
damped with damping matrix given by Eq. 4.14, α = 0.03 and β = 0.05. Compare
the system response with that in Example 4.4.
Problem 4.7 Show that the formulas of Eq. 4.85 are correct by using the orthogonality conditions given by Eq. 4.83.
Problem 4.8 Find the right and left eigenvectors of a in Example 4.8 by direct
calculations and compare with MATLAB solutions. Show that the vectors are
orthogonal in the sense of Eq. 4.83.
Problem 4.9 Show that solution z(t) of Sect. 4.5.4 does not depend on the scaling
of the right/left eigenvectors.
Problem 4.10 Find the response of the 2-DOf system in Example 4.10 by the
matrix exponential method.
References
1. R.W. Brockett, Finite Dimensional Linear Systems (John Wiley & Sons Inc., New York, 1970)
2. A.S. Cakmak, J.F Botha, W.G. Gray, Computational and Applied Mathematics for Engineering
Analysis (Springer, New York, 1987)
3. T.K. Caughey, M.E.J. O’Kelly, Classical normal modes in damped linear dynamic systems. J.
Appl. Mech. 32, 583–588 (1965)
4. M.D. Greenberg, Foundations of Applied Mathematics (Prentice Hall Inc., Englewood Cliffs,
1978)
4 Multi-Degree of Freedom (MDOF) Systems
Problem 4.2 Suppose that the 2-DOF system in Example 4.4 is subjected to an
acceleration a(t) = sin(ν t) applied at its fix end rather than the forcing function
f (t). First, find the steady-state modal and system responses for forcing frequencies
ν in the range [0, 1.5 ω 2 ]. Then, pretend that the modal frequencies are unknown.
Use the steady-state solutions to design an experiment for estimating the modal
frequencies.
Problem 4.3 Plot the time evolution of the displacement vector x(t) of the 2-DOF
system in Example 4.5 for a time interval [0, τ ] and initial conditions of your choice.
Plot also the modal shapes { i } of this system and identify the projections {q i (t)}
of x(t) on { i } at an arbitrary time t.
Problem 4.4 Repeat the calculations of Example 4.3 and assume that the system is
damped with damping matrix given by Eq. 4.14, α = 0.03 and β = 0.05. Compare
the system response with that in Example 4.3.
Problem 4.5 Repeat the calculations of Example 4.5 and assume that the system is
damped with damping matrix given by Eq. 4.14, α = 0.03 and β = 0.05. Compare
the system response with that in Example 4.5.
Problem 4.6 Repeat the calculations of Example 4.4 and assume that the system is
damped with damping matrix given by Eq. 4.14, α = 0.03 and β = 0.05. Compare
the system response with that in Example 4.4.
Problem 4.7 Show that the formulas of Eq. 4.85 are correct by using the orthogonality conditions given by Eq. 4.83.
Problem 4.8 Find the right and left eigenvectors of a in Example 4.8 by direct
calculations and compare with MATLAB solutions. Show that the vectors are
orthogonal in the sense of Eq. 4.83.
Problem 4.9 Show that solution z(t) of Sect. 4.5.4 does not depend on the scaling
of the right/left eigenvectors.
Problem 4.10 Find the response of the 2-DOf system in Example 4.10 by the
matrix exponential method.
References
1. R.W. Brockett, Finite Dimensional Linear Systems (John Wiley & Sons Inc., New York, 1970)
2. A.S. Cakmak, J.F Botha, W.G. Gray, Computational and Applied Mathematics for Engineering
Analysis (Springer, New York, 1987)
3. T.K. Caughey, M.E.J. O’Kelly, Classical normal modes in damped linear dynamic systems. J.
Appl. Mech. 32, 583–588 (1965)
4. M.D. Greenberg, Foundations of Applied Mathematics (Prentice Hall Inc., Englewood Cliffs,
1978)