--- title: "Forced damped two-degree-of-freedom system" subject: "Theory of Mechanism and Machine II" chapter: 7 key: "semester_7/theory_of_mechanism_and_machine_ii:damped-forced-two-dof-equations" source: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/damped-forced-two-dof-equations site: BCT Engineering --- # Forced damped two-degree-of-freedom system Theory of Mechanism and Machine II · Chapter 7: Vibration of Two Degree of Freedom Systems - Asked in: year not given, 6 marks - Video: [Two Degree of Freedom (2DOF) Problem With Damping - Equations of Motion (EOMs)](https://www.youtube.com/watch?v=NV7cd-7Rz-I) — Good Vibrations with Freeball - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/damped-forced-two-dof-equations - Whole chapter: https://www.bctengineer.com/md/semester_7/theory_of_mechanism_and_machine_ii/chapter-7 ## Question For a two-degree-of-freedom spring-mass-damper system with a harmonic force $F_1\sin\omega t$ acting on mass $m_1$, write the equations of motion in matrix form and explain how the steady-state amplitudes are obtained. Explain static and dynamic coupling and principal (normal) coordinates. ## Short answer Matrix equation: [M]{ẍ}+[C]{ẋ}+[K]{x}={F}. Steady‑state amplitudes are obtained from {X}=[Z(ω)]⁻¹{F} with impedance matrix Z = -ω²M + iωC + K. Static coupling involves off‑diagonal stiffness, dynamic coupling off‑diagonal mass; principal coordinates diagonalise M and K. ## Answer #### Equations of motion Take masses $m_1, m_2$, springs $k_1$ (wall to $m_1$), $k_2$ (between masses), dampers $c_1, c_2$ in parallel with them, and force $F_1\sin\omega t$ on $m_1$. $$ \begin{bmatrix} m_1 & 0 \\ 0 & m_2 \end{bmatrix}\begin{Bmatrix} \ddot{x}_1 \\ \ddot{x}_2 \end{Bmatrix} + \begin{bmatrix} c_1+c_2 & -c_2 \\ -c_2 & c_2 \end{bmatrix}\begin{Bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{Bmatrix} + \begin{bmatrix} k_1+k_2 & -k_2 \\ -k_2 & k_2 \end{bmatrix}\begin{Bmatrix} x_1 \\ x_2 \end{Bmatrix} = \begin{Bmatrix} F_1\sin\omega t \\ 0 \end{Bmatrix} $$ or $[M]\{\ddot{x}\} + [C]\{\dot{x}\} + [K]\{x\} = \{F\}$. #### Steady-state amplitudes Write the force as $F_1e^{i\omega t}$ and the response as $\{x\} = \{X\}e^{i\omega t}$. Substituting: $$ \left[-\omega^2[M] + i\omega[C] + [K]\right]\{X\} = \{F\} $$ The bracket is the **mechanical impedance matrix** $[Z(\omega)]$ with entries $Z_{rs} = -\omega^2 m_{rs} + i\omega c_{rs} + k_{rs}$. Then $$ \{X\} = [Z(\omega)]^{-1}\{F\}, \qquad X_1 = \frac{Z_{22}F_1}{Z_{11}Z_{22} - Z_{12}^2}, \quad X_2 = \frac{-Z_{12}F_1}{Z_{11}Z_{22} - Z_{12}^2} $$ The amplitudes are complex: $|X_r|$ is the amplitude, and the argument gives the phase lag behind the force. Damping makes the denominator never vanish, so the resonance peaks are finite. #### Coupling - **Static (elastic) coupling:** the stiffness matrix has off-diagonal terms ($k_{12} \ne 0$); a static force on one coordinate displaces the other. - **Dynamic (inertia) coupling:** the mass matrix has off-diagonal terms ($m_{12} \ne 0$), e.g. a vehicle body with the c.g. not midway between the axles. - The coupling depends on the choice of coordinates, not on the system. #### Principal (normal) coordinates They are special coordinates in which both $[M]$ and $[K]$ become diagonal, so the equations are uncoupled and each is a single-degree-of-freedom equation: $$ \ddot{q}_r + \omega_r^2 q_r = 0, \qquad \{x\} = [\Phi]\{q\} $$ where $[\Phi]$ is the matrix of mode shapes. Each principal coordinate vibrates in one mode only. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The equations of motion of the damped forced 2-DOF system are written as: A. [M]{x} + [C]{x'} + [K]{x''} = {F} B. [M]{x''} + [K]{x'} = {F} C. [M]{x'} + [C]{x''} + [K]{x} = {F} D. [M]{x''} + [C]{x'} + [K]{x} = {F} Answer: D. Inertia, damping and stiffness matrices multiply acceleration, velocity and displacement respectively. (Key point: Matrix form) 2. The steady-state amplitudes are found from {X} = [Z]^-1 {F}, where Z_rs equals: A. -w^2 m_rs + i w c_rs + k_rs B. w^2 m_rs + c_rs + k_rs C. k_rs - c_rs w D. m_rs w + c_rs + k_rs / w Answer: A. Substituting x = X e^(iwt) gives the impedance matrix [-w^2 M + i w C + K]. (Key point: Mechanical impedance) 3. Why are resonance amplitudes finite in the damped system? A. Mass matrix is diagonal B. The force is complex C. The amplitudes are complex numbers so they cannot be infinite D. Damping makes the determinant denominator never vanish Answer: D. The imaginary damping terms keep Z11 Z22 - Z12^2 nonzero for all w, so peaks are finite. (Key point: Finite resonance peaks) 4. Dynamic (inertia) coupling is indicated by: A. off-diagonal terms in the stiffness matrix B. off-diagonal terms in the mass matrix C. a non-zero damping matrix D. diagonal mass matrix only Answer: B. m12 not equal to 0 is dynamic coupling; k12 not equal to 0 is static (elastic) coupling. Coupling depends on the coordinates chosen. (Key point: Static vs dynamic coupling) 5. In principal (normal) coordinates: A. only the damping vanishes B. the system has no mode shapes C. both [M] and [K] are diagonal, so each equation is a single-DOF equation D. the masses are all equal Answer: C. x = [Phi]q diagonalises [M] and [K], uncoupling the equations: q_r'' + w_r^2 q_r = 0. 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