ES386 Assignment 2024-25 - Modal analysis
Student ID: […]
You may find it helpful to insert a figure when answering some of the questions below, see Briefing.
(a) For the continuous beam that the lumped-mass model is aiming to capture, a fundamental
frequency of 42.9 rad/s was determined. How does this compare to your results for the natural
frequencies and mode shapes of the lumped-mass model?
The lumped-mass model’s first natural frequency (35.33 𝑟𝑎𝑑/𝑠, Fig. 1a) underestimates the
continuous beam’s fundamental frequency (42.9 𝑟𝑎𝑑/𝑠) by 17.6%. This discrepancy arises because
the 9-DOF lumped model approximates an infinite-DOF continuous system, introducing spatial
discretization errors. While lower modes (Figs. 1a-c) reasonably approximate the beam’s behaviour,
higher modes (Figs. 1d-i) exhibit unrealistic jaggedness due to limited resolution. The Euler-Bernoulli
beam theory predicts exact frequencies for continuous systems, whereas the lumped model
provides an engineering approximation. Increasing the number of masses would improve accuracy,
but the 9-DOF model suffices for practical low-frequency analysis, balancing computational
efficiency with acceptable error.
[99 words]
[100 words max]
, Figure 1 – first nine eigenmodes of the 9-DOF lumped-mass beam, showing mode shapes and nodal displacements. Natural
frequencies range from 𝜔! = 35.33 𝑟𝑎𝑑/𝑠 to 𝜔" = 4025.23 𝑟𝑎𝑑/𝑠.
(b) Why do we use proportional damping? By comparing with measurements made in the
(experimental) laboratory workshop, explain why ⍺ = 0.1 and β=0.0001 (as used in the plot) are
reasonable choices for C = ⍺M+ βK. Finally, state which damping model (e.g., viscous, air, Coulomb)
was assumed in this assignment.
Proportional damping (𝐶 = 𝛼𝑀 + 𝛽𝐾) is employed because it maintains modal orthogonality,
enabling decoupling into single-DOF systems via the modal matrix. The values 𝛼 = 0.1 and 𝛽 =
" %$#
0.0001 yield a damping ratio 𝜁! = #$ + #
= 0.0032 for 𝜔! = 35.33 𝑟𝑎𝑑/𝑠, matching
#
experimental data for steel beams. Unlike structural damping (which predicts infinite damping
at 𝜔 → 0), Rayleigh damping ensures frequency-proportional 𝜁. The time constant (𝜏 = 8.9 𝑠)
prevents over-damping while accommodating higher modes. Viscous damping is confirmed by
velocity-proportional decay in all figures. This model outperforms other alternatives such as
Coulomb damping by avoiding nonlinearity and modal coupling, making it ideal for industrial
superposition analyses.
[93 words]
[100 words max]
Student ID: […]
You may find it helpful to insert a figure when answering some of the questions below, see Briefing.
(a) For the continuous beam that the lumped-mass model is aiming to capture, a fundamental
frequency of 42.9 rad/s was determined. How does this compare to your results for the natural
frequencies and mode shapes of the lumped-mass model?
The lumped-mass model’s first natural frequency (35.33 𝑟𝑎𝑑/𝑠, Fig. 1a) underestimates the
continuous beam’s fundamental frequency (42.9 𝑟𝑎𝑑/𝑠) by 17.6%. This discrepancy arises because
the 9-DOF lumped model approximates an infinite-DOF continuous system, introducing spatial
discretization errors. While lower modes (Figs. 1a-c) reasonably approximate the beam’s behaviour,
higher modes (Figs. 1d-i) exhibit unrealistic jaggedness due to limited resolution. The Euler-Bernoulli
beam theory predicts exact frequencies for continuous systems, whereas the lumped model
provides an engineering approximation. Increasing the number of masses would improve accuracy,
but the 9-DOF model suffices for practical low-frequency analysis, balancing computational
efficiency with acceptable error.
[99 words]
[100 words max]
, Figure 1 – first nine eigenmodes of the 9-DOF lumped-mass beam, showing mode shapes and nodal displacements. Natural
frequencies range from 𝜔! = 35.33 𝑟𝑎𝑑/𝑠 to 𝜔" = 4025.23 𝑟𝑎𝑑/𝑠.
(b) Why do we use proportional damping? By comparing with measurements made in the
(experimental) laboratory workshop, explain why ⍺ = 0.1 and β=0.0001 (as used in the plot) are
reasonable choices for C = ⍺M+ βK. Finally, state which damping model (e.g., viscous, air, Coulomb)
was assumed in this assignment.
Proportional damping (𝐶 = 𝛼𝑀 + 𝛽𝐾) is employed because it maintains modal orthogonality,
enabling decoupling into single-DOF systems via the modal matrix. The values 𝛼 = 0.1 and 𝛽 =
" %$#
0.0001 yield a damping ratio 𝜁! = #$ + #
= 0.0032 for 𝜔! = 35.33 𝑟𝑎𝑑/𝑠, matching
#
experimental data for steel beams. Unlike structural damping (which predicts infinite damping
at 𝜔 → 0), Rayleigh damping ensures frequency-proportional 𝜁. The time constant (𝜏 = 8.9 𝑠)
prevents over-damping while accommodating higher modes. Viscous damping is confirmed by
velocity-proportional decay in all figures. This model outperforms other alternatives such as
Coulomb damping by avoiding nonlinearity and modal coupling, making it ideal for industrial
superposition analyses.
[93 words]
[100 words max]
