Tuned mass dampers in low damping regimes: from Lagrangian mechanics to computational optimization

Absorvedores de massa sintonizados em regimes de baixo amortecimento: da mecânica Lagrangiana à otimização computacional

Abstract

Restricting the instruction of coupled oscillators to non-dissipative systems limits the comprehension of real-world phenomena. To bridge this gap, this work proposes a unified pathway for Tuned Mass Dampers on steel structures, combining a Lagrangian derivation, Fixed Point Theory as the analytical approximation, and Differential Evolution as the numerical reference to quantify performance degradation. For combinations of primary damping and mass ratios within ranges typical of real structures, Differential Evolution applied to the frequency response of the non-dimensional two-degree-of-freedom system reveals that classical Fixed Point Theory parameters remain demonstrably adequate in the low-damping regime, while numerical optimization becomes increasingly relevant as primary damping grows. Aimed at advanced undergraduate students in Physics and Engineering, this primarily educational contribution shows quantitatively how to assess, rather than assume, the practical validity of an analytical approximation. The proposed algorithms are self-contained and directly reproducible, facilitating integration into advanced undergraduate physics and engineering curricula.

Keywords:
Physics Education; Coupled Oscillators; Tuned Mass Damper; Computational Simulation; Fixed Point Theory

Resumo

A restrição do ensino de osciladores acoplados a sistemas sem dissipação limita a compreensão de fenômenos reais. Para superar essa lacuna, este trabalho propõe um roteiro unificado que combina derivação Lagrangiana, Teoria do Ponto Fixo como aproximação analítica e Evolução Diferencial como referência numérica para quantificar a degradação de desempenho, aplicado a Absorvedores de Massa Sintonizados em estruturas de aço. Para combinações de razões de amortecimento primário e de massa dentro de faixas típicas de estruturas reais, a Evolução Diferencial aplicada à resposta em frequência do sistema adimensional de dois graus de liberdade revela que os parâmetros clássicos da Teoria do Ponto Fixo permanecem quantitativamente adequados no regime de baixo amortecimento, enquanto a otimização numérica torna-se progressivamente necessária à medida que o amortecimento primário cresce. Voltada a estudantes em fase final de graduação em Física e Engenharia, esta contribuição primariamente educacional mostra quantitativamente como avaliar, e não apenas assumir, a validade prática de uma aproximação analítica. Os algoritmos propostos são autocontidos e diretamente reprodutíveis, facilitando a integração em disciplinas avançadas de graduação em física e engenharia.

Palavras-chave:
Ensino de Física; Osciladores Acoplados; Absorvedor de Massa Sintonizado; Simulação Computacional; Teoria do Ponto Fixo

1. Introduction

The harmonic oscillator is a cornerstone of physics education, occupying a central place in courses from introductory mechanics through advanced classical dynamics. In most curricula, coupled oscillators are introduced through the paradigmatic two-mass-spring system, where the analysis typically assumes ideal, undamped behavior.

While this idealization provides a clear conceptual foundation for understanding normal modes, resonance, and energy transfer, the gap between these textbook models and the real systems encountered in engineering practice, where dissipation is unavoidable and plays a non-negligible role in the dynamics, remains difficult to bridge. For students, this transition offers an opportunity to assess quantitatively where the idealized model ceases to be reliable. This opportunity aligns with Problem-Based Learning (PBL), in which students learn content and thinking strategies by solving problems embedded in real contexts [¹, ²].

Recent literature explores vibration phenomena through various technical lenses: Diniz [³] revisits the harmonic oscillator damped by surface contact friction, providing a comprehensive physical and mathematical treatment of its properties, and separately catalogs twelve distinct analytical methods for the standard SHO [⁴]. Furthermore, Lima [⁵] analyzes the impact of initial conditions and damping in the underdamped regime, and Alves [⁶] introduces an algebraic solution grounded in the Hamiltonian formalism. In the computational domain, Melnik [⁷] describes SHO dynamics through space-time discretization and recurrence matrices. These contributions reflect the enduring role of the fundamental oscillator in physics education, yet none extends the analysis to the regime of coupled, damped oscillators addressed here.

The Tuned Mass Damper (TMD), also known as the Dynamic Vibration Absorber, offers a compelling bridge between the theoretical framework of coupled oscillators and practical structural engineering. A TMD is a secondary mass-spring-damper system deliberately attached to a primary vibrating structure to reduce its oscillation amplitude near resonance [⁸].

The underlying physics is essentially the coupled-oscillator problem studied in classical mechanics: by tuning the secondary system’s natural frequency, one exploits the energy exchange between the two coupled masses to suppress vibrations in the primary structure. Real-world TMDs are found in landmark structures such as the Taipei 101 skyscraper, the Citicorp Center in New York, and numerous slender footbridges worldwide. A Brazilian application is the Rio–Niterói Bridge, where multiple synchronized dynamic attenuators were installed to control wind-induced oscillations [⁹].

The seminal contribution of Ormondroyd and Den Hartog [¹⁰] introduced a damping element into the absorber, and Den Hartog’s Fixed Point Theory (FPT) [¹¹] subsequently provided elegant closed-form design formulas that remain widely used nowadays. These formulas, however, rest on the idealization that the primary structure is undamped (${\zeta }_1=0$), the same simplification routinely encountered in textbook coupled-oscillator problems.

When the primary structure possesses inherent damping (${\zeta }_1>0$), the fixed points lose their invariant character [¹²], exact closed-form solutions become analytically intractable, and numerical methods emerge as the natural and necessary tool [¹², ¹³]. Several numerical strategies, including Particle Swarm Optimization (PSO) [¹⁴] and Genetic Algorithms (GA) [¹⁵], have been applied to TMD optimization; here, Differential Evolution (DE) [¹⁶] is adopted for its simplicity and native availability in SciPy [¹⁷], making it directly reproducible.

This intersection between an elegant analytical theory and its natural limits makes the TMD an ideal vehicle for teaching several interconnected physics and computational concepts in advanced undergraduate courses.

Steel structures like footbridges, floors, and slender buildings [¹⁸] represent a particularly instructive case study because their inherent damping ratios are characteristically low (0.3% to 1.5%), placing them in a regime where classical FPT parameters are expected to remain approximately valid, but where deviations become observable and quantifiable through simulation.

While two-degree-of-freedom (2-DOF) TMD optimization is well-documented, there is an excellent opportunity to further integrate its analytical foundations and computational methods in undergraduate education. Recent literature provides robust practical foundations, such as the valuable quantitative TMD assessment via Scilab developed by Souza [¹⁹] and the coupled oscillators approach via Python and RK4 by Rodrigues et al. [²⁰]. Alongside standard textbooks [²¹, ²²] and applied works like Soriano [¹⁸], these resources establish a strong theoretical and practical baseline.

Building upon this solid baseline, this work proposes a unified treatment: a Lagrangian derivation of the 2-DOF equations of motion, a non-dimensional formulation that makes scale invariance explicit, closed-form FPT as a transparent analytical reference, and Differential Evolution as a numerical counterpart, applied across the mass ratios and primary damping levels representative of steel structures. This combination makes it possible to determine, quantitatively and reproducibly, to what extent the classical FPT parameters remain suitable for structures whose primary damping is low but nonzero, and where numerical optimization becomes necessary instead, turning the validity of an analytical idealization from something to be assumed into something to be assessed.

2. Physical Model and Equations of Motion

The system consists of a primary structure (mass $m_1$, stiffness $k_1$, viscous damping $c_1$) coupled to a TMD (mass $m_2$, stiffness $k_2$, viscous damping $c_2$), as depicted in Figure 1. An external harmonic force $f\left(t\right)=f_0{e}^{i\omega t}$ acts on the primary mass. This two-degree-of-freedom (2-DOF) system is the natural generalization of the standard textbook coupled-oscillator problem, extended to include damping and forced excitation.

Figure 1
Schematic of the 2-DOF system: a primary mass ($m_1$) with stiffness ($k_1$) and damping ($c_1$), coupled to a TMD mass ($m_2$) with stiffness ($k_2$) and damping ($c_2$). An external force $f\left(t\right)$ drives the primary mass. Here $x_1$ and $x_2$ denote the absolute displacements of $m_1$ and $m_2$.

2.1. Lagrangian formulation

The Lagrangian formalism is adopted as an energy-based alternative to Newtonian free-body diagrams, simplifying the treatment of interaction forces. The Rayleigh function incorporates damping systematically, while the variational formulation connects directly to the principle of stationary action studied in advanced classical mechanics.

The system’s configuration is described by the generalized coordinates $\mathbf{q}={[q_1,q_2]}^T\equiv {[x_1,x_2]}^T$, representing the absolute displacements of $m_1$ and $m_2$, with $l=2$ degrees of freedom. The system is subjected to an external harmonic force on the primary mass, so the generalized force vector is $\mathbf{Q}={[f_0{e}^{i\omega t},0]}^T$. The kinetic energy $T$, potential energy $V$, and Rayleigh dissipation function $𝒟$ are:

(1) $$T=\frac{1}{2}{m}_1{\dot{q}}_1^2+\frac{1}{2}{m}_2{\dot{q}}_2^2,$$

(2) $$V=\frac{1}{2}{k}_1{q}_1^2+\frac{1}{2}{k}_2{\left(q_2-q_1\right)}^2,$$

(3) $$𝒟=\frac{1}{2}{c}_1{\dot{q}}_1^2+\frac{1}{2}{c}_2{\left({\dot{q}}_2-{\dot{q}}_1\right)}^2.$$

The Lagrangian $ℒ=T-V$ is therefore

(4) $$ℒ=\frac{1}{2}{m}_1{\dot{q}}_1^2+\frac{1}{2}{m}_2{\dot{q}}_2^2-\frac{1}{2}{k}_1{q}_1^2-\frac{1}{2}{k}_2{\left(q_2-q_1\right)}^2.$$

The Euler-Lagrange equations arise from requiring the action $S={\int }_{t_1}^{t_2}ℒ\mathit{d}t$ to be stationary with respect to arbitrary variations of the generalized coordinates that vanish at the endpoints, the principle of stationary action ($\delta S=0$). It is precisely this variational structure that Emmy Noether [²³] exploited to establish that every continuous symmetry of the action generates a conserved quantity, a result explored in the context of physics education by Acevedo and Pimentel [²⁴].

The Euler-Lagrange equation with dissipation [²⁵] for each generalized coordinate $q_j$ reads

(5) $$\frac{d}{dt}\left(\frac{\partial ℒ}{\partial {\dot{q}}_j}\right)-\frac{\partial ℒ}{\partial q_j}+\frac{\partial 𝒟}{\partial {\dot{q}}_j}=Q_j,j=1,2.$$

Here $Q_j$ denotes only the non-dissipative, externally applied generalized force, since dissipative effects are already accounted for by $𝒟$. Computing each term explicitly:

For $j=1$:

(6) $$\frac{d}{dt}\left(\frac{\partial ℒ}{\partial {\dot{q}}_1}\right)=m_1{\ddot{q}}_1,$$

(7) $$-\frac{\partial ℒ}{\partial q_1}=k_1{q}_1-k_2\left(q_2-q_1\right),$$

(8) $$\frac{\partial 𝒟}{\partial {\dot{q}}_1}=c_1{\dot{q}}_1-c_2\left({\dot{q}}_2-{\dot{q}}_1\right).$$

For $j=2$:

(9) $$\frac{d}{dt}\left(\frac{\partial ℒ}{\partial {\dot{q}}_2}\right)=m_2{\ddot{q}}_2,$$

(10) $$-\frac{\partial ℒ}{\partial q_2}=k_2\left(q_2-q_1\right),$$

(11) $$\frac{\partial 𝒟}{\partial {\dot{q}}_2}=c_2\left({\dot{q}}_2-{\dot{q}}_1\right).$$

These equations are assembled into the compact matrix form:

(12) $$\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{Kx}=\mathbf{f}\left(t\right).$$

The mass matrix, equation (13), is a diagonal matrix, indicating that the system is inertially decoupled in the selected coordinate system. It accounts for the inertia of the primary structure and the auxiliary absorber:

(13) $$\mathbf{M}=\left[\begin{array}{cc}{m}_1& 0\\ 0& m_2\end{array}\right].$$

The damping matrix, equation (14), characterizes the dissipative forces within the system. The off-diagonal terms represent the viscous coupling between the two masses:

(14) $$\mathbf{C}=\left[\begin{array}{cc}{c}_1+c_2& -c_2\\ -c_2& c_2\end{array}\right].$$

The stiffness matrix, equation (15), defines the restorative elastic forces. Similar to the damping matrix, the negative off-diagonal terms indicate that the force acting on one mass is dependent on the relative displacement of the other:

(15) $$\mathbf{K}=\left[\begin{array}{cc}{k}_1+k_2& -k_2\\ -k_2& k_2\end{array}\right].$$

The vector on the right-hand side of the governing equation represents the external harmonic excitation applied to the system:

(16) $$\mathbf{f}\left(t\right)=\left[\begin{array}{c}{f}_0{e}^{i\omega t}\\ 0\end{array}\right].$$

2.2. Non-dimensional parameters

Dividing the equations of motion by $m_1{\omega }_1^{\text{2}}$, where ${\omega }_1=\sqrt{k_1/m_1}$ and ${\omega }_2=\sqrt{k_2/m_2}$ are the undamped natural frequencies of the primary structure and the TMD, respectively, and adopting dimensionless time $\tau ={\omega }_{1}t$, collapses the system into a form governed entirely by a small set of ratios. These are: the mass ratio $\mu =m_2/m_1$, representing the ratio of the absorber mass to the primary mass; the tuning ratio $\nu ={\omega }_2/{\omega }_1$, which quantifies how closely the TMD natural frequency matches that of the primary system; the primary damping ratio ${\zeta }_1=c_1/\left(2m_1{\omega }_1\right)$, characterizing energy dissipation in the primary structure; the TMD damping ratio ${\zeta }_2=c_2/\left(2m_2{\omega }_2\right)$; and the normalized excitation frequency $\Omega =\omega /{\omega }_1$.

In these variables, the non-dimensional matrices are

(17) $${\mathbf{M}}_{nd}=\left[\begin{array}{cc}1& 0\\ 0& \mu \end{array}\right],$$

(18) $${\mathbf{C}}_{nd}=\left[\begin{array}{cc}2{\zeta }_1+2\mu {\zeta }_2\nu & -2\mu {\zeta }_2\nu \\ -2\mu {\zeta }_2\nu & 2\mu {\zeta }_2\nu \end{array}\right]$$

and

(19) $${\mathbf{K}}_{nd}=\left[\begin{array}{cc}1+\mu {\nu }^2& -\mu {\nu }^2\\ -\mu {\nu }^2& \mu {\nu }^2\end{array}\right].$$

This normalization carries a concrete physical message: two physically different systems with the same dimensionless parameters behave identically. This is a concrete and striking application of dimensional analysis in structural dynamics. TMD performance is entirely determined by the four numbers $(\mu ,\nu ,{\zeta }_1,{\zeta }_2)$ and the frequency ratio variable $\Omega$. The TMD design problem therefore reduces to selecting the design vector $\Xi =[\nu ,{\zeta }_2]$ for fixed $\mu$ and ${\zeta }_1$ so as to minimize the primary structure’s peak response amplitude.

3. Frequency Response and Fixed Point Theory

3.1. Frequency response function

The quantity of physical interest is the frequency response function (FRF)

$H_{11}\left(\Omega \right)$, defined as the steady-state displacement amplitude of the primary mass normalised by the static deflection $X_{st}=f_0/k_1$:

(20) $$H_{11}\left(\Omega \right)=\frac{X_1}{X_{st}}.$$

To compute $H_{11}$, the steady-state solution of equation (12) under harmonic excitation is sought. Assuming $\mathbf{x}\left(t\right)=\mathbf{X}{e}^{i\omega t}$ and substituting into equation (12) yields the algebraic system

(21) $$\mathbf{Z}\left(\omega \right)\mathbf{X}={\mathbf{F}}_0,$$

where the dynamic-stiffness matrix

(22) $$\mathbf{Z}\left(\omega \right)=-{\omega }^2\mathbf{M}+i\omega \mathbf{C}+\mathbf{K}$$

generalises static stiffness to account for inertial and dissipative effects at each excitation frequency.

Dividing equation (21) by $k_1$ and using the non-dimensional matrices and parameters of Section 2 gives the non-dimensional counterpart

(23) $${\mathbf{Z}}_{nd}\left(\Omega \right)\mathbf{X}=\frac{{\mathbf{F}}_0}{k_1},$$

with

(24) $${\mathbf{Z}}_{nd}\left(\Omega \right)=-{\Omega }^2{\mathbf{M}}_{nd}+i\Omega {\mathbf{C}}_{nd}+{\mathbf{K}}_{nd}.$$

Solving equation (23) gives $\mathbf{X}={\mathbf{Z}}_{nd}^{-1}{\mathbf{F}}_0/k_1$. Defining the FRF matrix $\mathbf{H}$ via $\mathbf{X}=\mathbf{H}\left({\mathbf{F}}_0/k_1\right)$ and comparing with the solution identifies

(25) $$\mathbf{H}\left(\Omega \right)={\mathbf{Z}}_{nd}^{-1}\left(\Omega \right).$$

Since only the primary mass is excited (${\mathbf{F}}_0={[f_0,0]}^T$), the quantity of interest is the $(1,1)$ entry of $\mathbf{H}$, which by Cramer’s rule yields

(26) $$H_{11}=\frac{{\left[{\mathbf{Z}}_{nd}\right]}_{22}}{det\left({\mathbf{Z}}_{nd}\right)}.$$

Expanding the non-dimensional matrices of Section 2, the elements of ${\mathbf{Z}}_{nd}$ are

(27) $$Z_{11}=\left(1+\mu {\nu }^2-{\Omega }^2\right)+2i\Omega \left({\zeta }_1+\mu {\zeta }_2\nu \right),$$

(28) $$Z_{12}=Z_{21}=-\mu {\nu }^2-2i\mu {\zeta }_2\nu \Omega ,$$

(29) $$Z_{22}=\mu \left({\nu }^2-{\Omega }^2\right)+2i\mu {\zeta }_2\nu \Omega ,$$

so that $H_{11}=Z_{22}/\left(Z_{11}{Z}_{22}-Z_{12}^{\text{2}}\right)$ is an explicit rational function of $\Omega$ for any $(\mu ,\nu ,{\zeta }_1,{\zeta }_2)$.

A structurally important feature is that ${\zeta }_1$ enters the FRF exclusively through the imaginary part of $Z_{11}$: setting ${\zeta }_1=0$ causes the ${\zeta }_2$-dependent terms to cancel from the FRF ratio at specific frequencies, giving rise to the invariant fixed points exploited in the following subsection.

For the sake of brevity, $H(\Omega ;\mu ,{\zeta }_1;\Xi )$ is denoted simply $H_{11}$, as the idea is to diminish the vibration of the main mass. Figure 2 shows how the shape of $\left|H_{11}\right|$ varies with $\nu$ and ${\zeta }_2$, providing an intuitive grasp of what tuning means physically: a poorly tuned absorber may amplify rather than suppress oscillations, while an optimally tuned one splits the single resonance peak of the bare primary structure into two bounded peaks. This two-peak structure motivates the closed-form design strategy developed in the following subsection.

Figure 2
Normalized amplitude–frequency curves for $\mu =0.05$, ${\zeta }_1=0$, and fixed $\nu =0.900$, with ${\zeta }_2=0.080$ (blue), $0.100$ (orange), and $0.120$ (green); the corresponding peak amplitudes are ${\max }_{\Omega }\left|H_{11}\right|=12.34$, $10.58$, and $9.58$. Black dots mark the two fixed points. An optimal set of $(\nu ,{\zeta }_2)$ yields two peaks of equal height.

3.2. Fixed point theory

For an idealized undamped primary system (${\zeta }_1=0$), a remarkable property holds: all FRF curves for varying ${\zeta }_2$ pass through two common invariant points, independent of ${\zeta }_2$[¹¹]. Den Hartog exploited this property in a two-step argument:

1. Equal fixed points. Set the FRF magnitude at the two frequency-independent points equal to each other; this uniquely determines the optimal tuning ratio ${\nu }_{opt}^a$.

2. Equal peaks. Choose ${\zeta }_2$ such that the two resonant peaks of $\left|H\right|$ coincide with these fixed points in amplitude; this determines ${\zeta }_{2,opt}^a$.

The result is an elegant closed-form set (superscript $a$: analytical; superscript $n$: numerical, introduced in Section 4):

(30) $${}_{}{}^a\Xi =\left[\begin{array}{c}{\nu }_{opt}^a\\ {\zeta }_{2,opt}^a\end{array}\right]=\left[\begin{array}{c}\frac{1}{1+\mu }\\ \sqrt{\frac{3\mu }{8\left(1+\mu \right)}}\end{array}\right].$$

3.3. The $H_{\infty }$ criterion

The design objective is to minimize the peak FRF amplitude:

(31) $${𝒥}_{\infty }\left(\Xi \right)=\underset{\Omega }{\max }\left|H(\Omega ;\mu ,{\zeta }_1;\Xi )\right|,$$

known as the $H_{\infty }$ norm of the transfer function. This criterion is appropriate for vibration attenuation because it directly bounds the worst-case dynamic amplification over all excitation frequencies, the quantity most relevant to structural safety and human comfort.

3.4. FPT approximation

Although FPT yields practical closed-form expressions, its solution is an approximation [¹², ²⁶, ²⁷] even under

the assumption ${\zeta }_1=0$: Den Hartog’s step-2 procedure averages two different optimal ${\zeta }_2$ values, one per fixed point, so the resulting peaks are only approximately equal and approximately coincident with the fixed-point amplitudes. The true $H_{\infty }$ optimum achieves exactly equal peaks at a slightly lower common amplitude, as illustrated in Figure 3 for $\mu =0.05$.

Figure 3
FRF comparison for ${\zeta }_1=0$ and $\mu =0.05$, illustrating FPT as an approximation. Analytical (blue): ${}_{}{}^a\Xi =[0.952381,0.133631]$, with ${𝒥}_{\infty }\left({}_{}{}^a\Xi \right)=6.4084$. Numerical (orange): ${}_{}{}^n\Xi =[0.952374,0.133911]$, with ${𝒥}_{\infty }\left({}_{}{}^n\Xi \right)=6.4079$.

Figure 4 makes this residual gap visible at quantitatively meaningful scale: at the left peak the FPT curve sits slightly higher than the numerically optimized curve; the same asymmetry, with a different magnitude, appears at the right peak, direct evidence of the averaging step. Analytical elegance and strict numerical optimality are related but distinct properties.

Figure 4
Zoomed-in view of Figure 3 near the (a) left and (b) right resonance peaks for ${\zeta }_1=0$ and $\mu =0.05$. At the left peak the analytical design ${}_{}{}^a\Xi$ reaches $\left|H_{11}\right|=6.4075$ and the numerical optimum ${}_{}{}^n\Xi$ reaches $6.4079$; at the right peak, $6.4084$ and $6.4079$, respectively: the numerical solution equalizes the two peaks exactly.

This gap, small at ${\zeta }_1=0$, grows as primary damping increases. The fixed points exist because ${\zeta }_1=0$ decouples ${\zeta }_2$ from their locations; for any ${\zeta }_1>0$ these intersections drift, visually breaking the invariance. This growing deviation motivates the computational treatment developed in Section 4.

4. Computational Simulation

4.1. Computing the FRF

For a given parameter set $\Xi =[\nu ,{\zeta }_2]$, the FRF magnitude is evaluated over a discrete frequency grid ${\left\{{\Omega }_k\right\}}_{k=1}^N$ by solving the complex linear system at each point. Algorithm 1 outlines the procedure.

Algorithm 1
Computation of the primary-mass FRF magnitude

Require:$\mu$, ${\zeta }_1$, $\nu$, ${\zeta }_2$, frequency grid ${\left\{{\Omega }_k\right\}}_{k=1}^N$

Ensure:$\left|H\left({\Omega }_k\right)\right|$ for each $k$ Build non-dimensional matrices ${\mathbf{M}}_{nd}$, ${\mathbf{C}}_{nd}$, ${\mathbf{K}}_{nd}$ for each ${\Omega }_k$do ${\mathbf{Z}}_{nd,k}\leftarrow -{\Omega }_k^2{\mathbf{M}}_{nd}+i{\Omega }_k{\mathbf{C}}_{nd}+{\mathbf{K}}_{nd}$ $\mathbf{H}\left({\Omega }_k\right)\leftarrow {\mathbf{Z}}_{nd,k}^{-1}$ $\left|H_{11}\left({\Omega }_k\right)\right|\leftarrow \left|{\mathbf{H}}_{(1,1)}\left({\Omega }_k\right)\right|$$▷$ First component = primary mass response end for

In practice, the frequency grid is set to $N=10000$ uniformly spaced points over $\Omega \in [0.01,2.0]$, which covers both resonant peaks of the optimally tuned TMD for all parameter combinations considered here. The complex linear system at each grid point is solved by direct inversion of the $2\times 2$ dynamic-stiffness matrix of equation (23); because the matrix is small, no iterative solver is needed and the per-point cost is negligible.

A useful analytical verification of Algorithm 1 is the static limit. At $\Omega =0$, the dynamic-stiffness matrix reduces to $\mathbf{Z}\left(0\right)={\mathbf{K}}_{nd}$, and equation (23) simplifies to $X_1\left(0\right)=K_{nd,22}/det\left({\mathbf{K}}_{nd}\right)$. Since $det\left({\mathbf{K}}_{nd}\right)=\mu {\nu }^2$ and $K_{nd,22}=\mu {\nu }^2$ for any $\mu >0$ and $\nu >0$, one obtains $X_1\left(0\right)=1$ identically, confirming that $\left|H\right|=1$ at zero frequency, the purely static deflection equal to $X_{st}$. This result holds regardless of $(\nu ,{\zeta }_2)$ and provides a parameter-free unit test for any implementation of Algorithm 1: if a numerical code does not return $\left|H\left(0\right)\right|=1$, the matrix assembly or normalization contains an error that would propagate into the optimization of Algorithm 2.

4.2. Numerical $H_{\infty }$ optimization

When ${\zeta }_1>0$, the FPT parameters are no longer optimal [¹²]. The true optimum is found by solving:

(32) $${}_{}{}^n\Xi =\arg \underset{\Xi \in 𝒪}{\min }{𝒥}_{\infty }\left(\Xi \right),𝒪=\{\Xi :\nu >0,{\zeta }_2\ge 0\}.$$

Because ${𝒥}_{\infty }\left(\Xi \right)$ is non-smooth (it is a maximum over a continuous set) and may exhibit multiple local minima, gradient-free global optimization is appropriate. Gradient-based methods are unreliable for non-smooth objectives, and single-start simplex methods (e.g., Nelder-Mead) may stall at local optima. DE [¹⁶], a population-based stochastic algorithm that evolves a set of candidate solutions through stochastic difference operations, is employed to provide robust global exploration without derivative information. Algorithm 2 summarizes the approach, implemented in Python via SciPy’s differential_evolution function [¹⁷].

Algorithm 2
$H_{\infty }$ optimization via DE

Require:$\mu$, ${\zeta }_1$, frequency grid ${\left\{{\Omega }_k\right\}}_{k=1}^N$ (same grid as Algorithm 1)

Ensure: Numerically optimal parameters ${}_{}{}^n\Xi =[{\nu }^n,{\zeta }_2^n]$ Set search bounds: $\nu \in [0.5,1.5]$, ${\zeta }_2\in [0,0.5]$ Define objective ${𝒥}_{\infty }\left(\Xi \right)={\max }_k\left|H({\Omega }_k;\mu ,{\zeta }_1;\Xi )\right|$ $▷$ calls Algorithm 1 Set DE hyperparameters: strategy best1bin, population multiplier $=20$, max. iterations $=2000$, tolerance $={10}^{-8}$, random seed $=42$ ${}_{}{}^n\Xi \leftarrow \mathtt{\text{differential\_evolution}}(𝒥,[{\nu }_{\min },{\nu }_{\max }]\times [{\zeta }_{2,\min },{\zeta }_{2,\max }])$

The hyperparameters were chosen with two criteria in mind: numerical precision and reproducibility. The tolerance ${10}^{-8}$ ensures that the optimizer’s residual error is negligible relative to the physical phenomenon under study. A SciPy population multiplier of 20, corresponding to 40 individuals, offers a wide margin for the two-dimensional search space $(\nu ,{\zeta }_2)$, and 2000 iterations provide a sufficient budget even in flat regions of $𝒥$; in practice, convergence is observed well before the iteration limit. The fixed seed $42$ yields the same numerical values at the reported precision in repeated executions. The best1bin mutation strategy is the standard choice for smooth, low-dimensional objectives and requires no additional tuning.

Repeating the optimization for the most sensitive case of Table 1 ($\mu =0.050$, ${\zeta }_1=0.015$) with ten different random seeds produced a maximum variation of $5.9\times {10}^{-8}$ in ${\nu }^n$, $9.8\times {10}^{-7}$ in ${\zeta }_2^n$, and $2.0\times {10}^{-8}$ in ${𝒥}_{\infty }\left({}_{}{}^n\Xi \right)$, confirming that DE converges to the same global optimum regardless of the seed. Likewise, recomputing $\Delta {𝒥}_{\%}$ for all eight cases of Table 1 with frequency grids of 5 000, 10 000, and 20 000 points changes the reported values by at most $7\times {10}^{-4}$ percentage points relative to the $N=\mathrm{10\; 000}$ grid adopted throughout this work, confirming that the chosen discretization is adequate.

Table 1
Percentage performance degradation $\Delta {𝒥}_{\%}$ when using FPT parameters instead of numerically optimal parameters, for selected primary damping ratios ${\zeta }_1$ and mass ratios $\mu$.

$\mu \backslash {\zeta }_1$	0.003	0.005	0.010	0.015
0.010	0.55	0.88	1.62	2.25
0.050	0.54	0.88	1.70	2.46

5. Results and Discussion

5.1. Comparative FRF analysis

Table 1 quantifies the percentage difference between the peak FRF amplitude obtained with classical FPT parameters (${}_{}{}^a\Xi$) and numerically optimal parameters (${}_{}{}^n\Xi$) for representative primary damping levels:

(33) $$\Delta {𝒥}_{\%}=100\times \frac{{𝒥}_{\infty }\left({}_{}{}^a\Xi \right)-{𝒥}_{\infty }\left({}_{}{}^n\Xi \right)}{{𝒥}_{\infty }\left({}_{}{}^n\Xi \right)}.$$

The numerically optimal response ${𝒥}_{\infty }\left({}_{}{}^n\Xi \right)$, rather than the analytical one, is taken as the reference in the denominator because it is, by construction, the true minimum achievable peak response for the given $(\mu ,{\zeta }_1)$; $\Delta {𝒥}_{\%}$ therefore measures how far the FPT design falls from the best attainable performance.

The four values of ${\zeta }_1$ are not arbitrary: they correspond to commonly accepted equivalent viscous damping ratios for distinct classes of steel structures [¹⁸] – footbridges (${\zeta }_1\approx 0.003$), bridges with spans exceeding 20 m (${\zeta }_1\approx 0.005$), buildings taller than 100 m (${\zeta }_1\approx 0.010$), and buildings up to approximately 100 m in height (${\zeta }_1\approx 0.015$). Each column of Table 1 therefore maps to a distinct structural class, allowing the student or practicing engineer to read off the degradation relevant to their specific application and decide whether FPT remains adequate or numerical optimization becomes necessary. The two mass ratios carry analogous meaning; $\mu =0.010$ and $\mu =0.050$ mark the lower and upper ends of the interval covered by this study, spanning the values commonly adopted in TMD installations on steel structures, so the rows of Table 1 bracket the practical design range while Figures 5 and 6 cover the intermediate mass ratios continuously.

Figure 5
Maximum amplitude magnification factor ${𝒥}_{\infty }\left({}_{}{}^n\Xi \right)$ of the primary system equipped with a numerically optimized TMD, as a function of mass ratio $\mu$ and primary damping ratio ${\zeta }_1$, shown as a three-dimensional surface. The surface represents the minimized peak response achieved.

Figure 6
Two-dimensional contour map of the same minimized peak response ${𝒥}_{\infty }\left({}_{}{}^n\Xi \right)$ surface shown in Figure 5, as a function of mass ratio $\mu$ and primary damping ratio ${\zeta }_1$.

The performance degradation when relying on classical FPT is minimal for very low primary damping (around 0.5% for ${\zeta }_1=0.003$), as shown in Table 1, but it grows steadily as ${\zeta }_1$ increases, reaching up to 2.46% for ${\zeta }_1=0.015$. While these deviations might seem small, they quantitatively confirm that FPT is strictly an approximation for damped primary structures. As the inherent dissipation of the host structure grows, the equal-peaks assumption loses its strict optimality, making numerical minimization increasingly relevant for precise $H_{\infty }$ tuning. The near-identical values across both rows further indicate that, for the two mass-ratio values considered, performance degradation is more sensitive to variations in ${\zeta }_1$ than in $\mu$, which justifies the compactness of Table 1. The companion repository provides the same comparison and complete design parameters for intermediate mass ratios.

Figure 5 presents the global performance landscape of the numerically optimized TMD as a smooth surface over the design space $(\mu ,{\zeta }_1)$, with Figure 6 giving the corresponding contour view. Each point on the surface represents the true minimum achievable peak response obtained by DE for that combination of mass ratio and primary damping. The dominant feature is the strong dependence on $\mu$: as the mass ratio increases, the achievable peak response decreases substantially, because a larger secondary mass extracts more vibrational energy from the primary structure per cycle and sustains a broader suppression bandwidth.

This trade-off is the central design constraint of any TMD application, with a heavier absorber performing better but imposing a larger mass penalty on the host structure. Along the ${\zeta }_1$ direction, for fixed $\mu$, the surface reveals how the optimal achievable performance evolves as the primary structure’s inherent dissipation changes, illustrating the interplay between the TMD’s active suppression mechanism and the structure’s own energy dissipation. For the low-damping regime examined here (${\zeta }_1\le 0.015$), the surface varies smoothly and without abrupt features, consistent with the expectation that FPT parameters remain near-optimal throughout this domain.

Instructors may use this surface as a global design chart: given a mass ratio and a structural damping estimate, it immediately provides the best attainable peak response.

5.2. Educational discussion

This PBL unit targets courses in advanced classical mechanics, structural dynamics, or computational physics, assuming only the Euler–Lagrange equation with dissipation, complex algebra, and basic Python. Deriving the coupled equations builds physical modeling as flexible knowledge [¹], assessing FPT exercises metacognitive reasoning about model validity [²], and implementing DE demands self-directed learning [²⁸]. The computational results suggest the following directions for course design.

(1) Quantifying the validity of an approximation. For ${\zeta }_1\le 0.005$, FPT parameters deliver performance within 1% of the numerical optimum, an excellent approximation. This concretely illustrates that idealized models can remain practically valid well beyond their strict domain of validity, and that quantifying the error is as important as recognizing the model’s assumptions.

A productive exercise is to reproduce the $\mu =0.05$ row of Table 1 from scratch using Algorithms 1 and 2.

(2) When numerical methods become necessary. As ${\zeta }_1$ grows, the analytical approximation deteriorates. At ${\zeta }_1=0.015$ the gap reaches $\approx 2.46\%$, small in absolute terms but potentially significant in sensitive applications. This motivates numerical optimization as a natural, incremental extension of the analytical approach, rather than an alternative to it.

An exercise is to plot, on the same axes, the FRFs obtained with FPT parameters and with numerically optimal parameters for ${\zeta }_1\in \{0.003,0.010,0.015\}$ and $\mu =0.05$, using Algorithms 1 and 2. Following the approach of Figure 3, the exercise invites visual identification of the value of ${\zeta }_1$ below which the two curves become indistinguishable, then comparison with the quantitative boundary implied by Table 1.

(3) Mode splitting and resonance visualization. The FRF plots of Figure 2 make visible how the single resonance peak of the undamped primary structure splits into two peaks when the TMD is added. Figure 5 further shows how the minimized peak response varies with $\mu$ and ${\zeta }_1$, revealing the role of mass ratio in reducing the achievable peak amplitude. These are computational manifestations of the mode-splitting phenomenon well known from coupled-oscillator theory.

A complementary exercise is to verify the peak locations analytically by solving the secular equation for the undamped coupled system. For the footbridge case ($\mu =0.05$, $\nu =0.952381$, ${\zeta }_1={\zeta }_2=0$), setting $det\left[{\mathbf{K}}_{nd}-{\Omega }^2{\mathbf{M}}_{nd}\right]=0$ yields a quadratic in ${\Omega }^2$ with roots ${\Omega }_{-}^2\approx 0.762$ and ${\Omega }_{+}^2\approx 1.191$, corresponding to coupled eigenfrequencies ${\Omega }_{-}\approx 0.873$ and ${\Omega }_{+}\approx 1.091$.

Comparing these values with the peaks visible in Figure 2 confirms that the two-peak structure is a direct consequence of mode splitting in the coupled eigenvalue problem rather than an artifact of the optimization criterion.

These directions align with findings in the education literature beyond the PBL framework itself. Medina and Velazco address conceptual and mathematical difficulties of the conventional damped-oscillation model [²⁹]. Review-level evidence indicates that problem-based formats favor deep learning [³⁰] and that collaborative problem solving improves students’ critical thinking [³¹], while the integration of computational thinking into STEM instruction constitutes an emerging research line of its own [³²].

5.3. Instructional extension

The following cross-disciplinary connection is offered as optional enrichment for instructors seeking to extend the course beyond structural dynamics.

Cross-disciplinary connection: avoided crossing in quantum mechanics. The coupling-induced splitting of the TMD’s two eigenfrequencies is the mechanical instance of the quantum avoided crossing [³³]: as the tuning ratio $\nu$ is swept through unity, the two eigenfrequencies of the coupled system approach but never cross, repelling one another just as interacting quantum states form an energy gap proportional to their coupling strength. Both the undamped TMD and the two-level quantum system are instances of the same $2\times 2$ eigenvalue problem, and the two hybridized eigenfrequencies of the TMD are the mechanical analogues of the repelled energy levels $E_{\pm }$.

The formal parallel becomes explicit when both problems are reduced to their $2\times 2$ secular equations. For the undamped TMD (${\zeta }_1={\zeta }_2=0$, $\mathbf{f}=\mathbf{0}$), requiring $det\left[\mathbf{K}-{\omega }^2\mathbf{M}\right]=0$ gives

(34) $$\left(k_1+k_2-m_1{\omega }^2\right)\left(k_2-m_2{\omega }^2\right)-k_2^2=0,$$

while the two-level quantum system with unperturbed energies $E_1$, $E_2$ and coupling element $V_{12}$ yields

(35) $$\left(E_1-E\right)\left(E_2-E\right)-{\left|V_{12}\right|}^2=0.$$

Both equations share the same algebraic structure: a product of diagonal factors minus the square of an off-diagonal coupling. Note, however, that the isomorphism is algebraic rather than term-by-term physical: in the TMD, $k_2$ appears in both diagonal factors (shifting the effective stiffness of each mass), whereas $V_{12}$ is strictly off-diagonal and leaves $E_1$, $E_2$ unchanged. The two roots of equation (34) are the hybridized eigenfrequencies of the coupled TMD; the two roots of equation (35) are the repelled energy levels of the coupled quantum states, separated by a gap $\Delta E\propto \left|V_{12}\right|$ at the degeneracy point $E_1=E_2$. The splitting between the two resonant peaks is the mechanical counterpart of this energy gap.

Instructors may use this analogy to demonstrate that the physics of mode splitting transcends any single discipline, deepening engagement and providing a concrete example of how the same linear-algebraic structure governs coupled systems in both classical mechanics and quantum theory.

6. Conclusion

This work presented a structured pathway for teaching coupled-oscillator physics through TMDs applied to steel structures. Starting from first principles via the Euler-Lagrange formulation, the article developed the frequency-domain analysis of the driven 2-DOF system and introduced a non-dimensional parameter framework, $(\mu ,\nu ,{\zeta }_1,{\zeta }_2)$, that makes the physics portable across scales. The article then introduced Den Hartog’s FPT as a transparent and historically important closed-form reference, and used computational simulation to quantify its limits when the primary structure possesses small but finite damping.

The quantitative findings confirm that, for steel structures with ${\zeta }_1\le 0.005$ and for both mass ratios examined ($\mu =0.01$ and $\mu =0.05$), chosen as representative of typical TMD applications, the classical FPT parameters deviate less than 1% from the numerical optimum. Intermediate values can be generated directly from Algorithms 1 and 2. As Table 1 shows, at ${\zeta }_1=0.005$ the deviation reaches $\approx 0.9\%$ and grows to 2.46% at ${\zeta }_1=0.015$, making numerical optimization increasingly relevant as primary damping grows, a practical approximation whose validity can now be assessed rather than merely assumed.

Beyond the numerical results, the principal contribution of this work is a single reproducible pathway combining four elements into a unified treatment: a complete Lagrangian derivation of the 2-DOF equations of motion, a non-dimensional formulation that makes scale invariance explicit, closed-form FPT results as a transparent analytical reference, and quantified performance degradation $\Delta {𝒥}_{\%}$ as an explicit function of primary damping. The proposed algorithms are self-contained, facilitating their direct integration into Physics and Engineering courses and interactive notebooks. Through this computational approach, this pathway enables actively quantifying the limitations of theoretical approximations, a cognitive step that a purely analytical treatment cannot provide.

For instructors wishing to extend the treatment beyond structural dynamics, Section 5.3 shows that the mode splitting observed here is the mechanical instance of the avoided-crossing phenomenon in quantum mechanics, an illustration that the physics of coupled oscillators transcends any single discipline.

Data Availability

The code supporting the results of this study is openly available at Zenodo: https://doi.org/10.5281/zenodo.21312499.

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*

Correspondence email address: [e-mail]

Editor-in-Chief:

Marcello Ferreira https://orcid.org/0000-0003-4945-3169

Publication Dates

• Publication in this collection
  09 Oct 2026
• Date of issue
  2026

History

• Received
  09 May 2026
• Reviewed
  15 July 2026
• Accepted
  23 Aug 2026