This paper presents a modified harmonic balance solution method incorporated with Vieta’s substitution technique for nonlinear multimode damped beam vibration. The aim of the modification in the solution procedures is to develop the analytic formulations, which are used to calculate the vibration amplitudes of a nonlinear multimode damped beam without the need of nonlinear equation solver for the nonlinear algebraic equations generated in the harmonic balance processes. The result obtained from the proposed method shows reasonable agreement with that from a previous numerical integration method. In general, the results can show the convergence and prove the accuracy of the proposed method.
Over the past decades, many solution methods were developed for various engineering modelling problems (e.g., [
The governing equation of motion of nonlinear vibration of the EulerBernoulli theory is as follows [
Then the governing equation is discretized using the modal reduction approach,
Consider substituting (
If the beam is simply supported, then
The solution form of the modal amplitudes is given by [
By inputting (
For simplicity, let
Using the substitution of
Note that
Consider the harmonic balances of
Again, the 2nd level governing equation can be set up by inputting (
Note that
Consider the harmonic balances of
Finally, the overall amplitude and overall modal amplitude are defined as
In this section, the material properties of the simply supported beams in the numerical cases are considered as follows: Young’s modulus
(a) Convergence study for various excitation magnitudes,
Normalized overall amplitude 

7.5  15  25 

Zero level solution  99.92  101.00  101.93  102.69 
1st level solution  100.00  100.03  100.13  100.32 
2nd level solution  100.00  100.00  100.00  100.00 
Normalized overall amplitude 





Zero level solution  99.79  100.44  96.43  79.43 
1st level solution  100.03  100.03  100.00  100.00 
2nd level solution  100.00  100.00  100.00  100.00 
(a) Modal contributions for various excitation frequencies,
Normalized modal contribution 

7.5  15  25 

1st mode  99.94  99.88  99.67  99.54 
2nd mode  0.05  0.10  0.29  0.40 
3rd mode  0.01  0.01  0.04  0.06 
Normalized modal contribution 





1st mode  99.59  99.78  96.03  78.20 
2nd mode  0.36  0.19  3.56  20.25 
3rd mode  0.05  0.03  0.42  1.55 
(a) Comparison between the results from the proposed and numerical integration methods,
Frequencyamplitude curve for various excitation levels.
(a) The 1st level residues remained at the 1st and 2nd modal equations,
In this study, the analytic solution steps for nonlinear multimode beam vibration using a modified harmonic balance approach and Vieta’s substitution have been developed. Using the proposed method, the nonlinear multimode beam vibration results can be generated without the need of nonlinear equation solver. The standard simply supported beam case has been considered in the simulation. The solution convergences and modal contributions have been checked. The theoretical result obtained from the proposed method shows reasonable agreement with that from the previous numerical integration method.
The author declares that there is no conflict of interests regarding the publication of this paper.
The work described in this paper was fully supported by a grant from City University of Hong Kong (Project no. 7004362/ACE).