--- title: "Two-mass system X2/F with F-V and F-I analogy (D3)" subject: "Control Systems" chapter: 2 key: "semester_3/control_systems:tf-fv-fi-m1-m2-2" source: https://www.bctengineer.com/q/semester_3/control_systems/tf-fv-fi-m1-m2-2 site: BCT Engineering --- # Two-mass system X2/F with F-V and F-I analogy (D3) Control Systems · Chapter 2: Component Modeling - Asked in: 2071 Shrawan, 8 marks - Video: [2.Force Current and Force Voltage Analogy solved problem with system equation](https://www.youtube.com/watch?v=jgCfYudsIFY) — ELECTRICAL HUB - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/tf-fv-fi-m1-m2-2 - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-2 ## Question Find the transfer function X2(s)/F(s) for the mechanical system of figure below. Also draw the F-V and F-I analogy circuit of the system. [Figure: wall – spring K1 – mass M1 (force F(t) applied, displacement x1(t), friction D1 with ground); M1 connected to mass M2 (displacement x2(t), friction D2 with ground) through spring K2 in parallel with damper D3; M2 – spring K3 – wall] ## Short answer X2(s)/F(s) = (D3 s + K2) / Δ, with Δ = [M1s^2+(D1+D3)s+K1+K2][M2s^2+(D2+D3)s+K2+K3] - (D3s+K2)^2. The F-V analogy maps force to voltage, mass to inductance, damper to resistance, spring to 1/capacitance; the F-I analogy maps force to current, mass to capacitance, damper to 1/resistance, spring to 1/inductance. ## Answer Displacements $x_1$ ($M_1$, force $F$, ground friction $D_1$) and $x_2$ ($M_2$, ground friction $D_2$). $K_1$ ties $M_1$ to the left wall, $K_3$ ties $M_2$ to the right wall, and $K_2 \parallel D_3$ couples them. #### Equations of motion (Laplace) $$ \begin{aligned} \left[M_1s^2 + (D_1 + D_3)s + K_1 + K_2\right]X_1 - (D_3s + K_2)X_2 &= F \\ -(D_3s + K_2)X_1 + \left[M_2s^2 + (D_2 + D_3)s + K_2 + K_3\right]X_2 &= 0 \end{aligned} $$ (Sum of impedances at each mass on the diagonal; coupling impedance off the diagonal.) #### Transfer function $$ \frac{X_2(s)}{F(s)} = \frac{D_3s + K_2}{\Delta} $$ $$ \Delta = \left[M_1s^2 + (D_1+D_3)s + K_1 + K_2\right]\left[M_2s^2 + (D_2+D_3)s + K_2 + K_3\right] - (D_3s + K_2)^2 $$ Expanded: $$ \begin{aligned} \Delta =\; & M_1M_2s^4 + \left[D_1M_2 + D_2M_1 + D_3(M_1 + M_2)\right]s^3 \\ & + \left[D_1D_2 + D_1D_3 + D_2D_3 + M_1(K_2 + K_3) + M_2(K_1 + K_2)\right]s^2 \\ & + \left[D_1(K_2 + K_3) + D_2(K_1 + K_2) + D_3(K_1 + K_3)\right]s \\ & + K_1K_2 + K_1K_3 + K_2K_3 \end{aligned} $$ #### F-V analogy (mesh) $F\to e$, $M\to L$, $D\to R$, $K\to 1/C$, velocity $\to$ current. ```text L1=M1 R=D1 C=1/K1 L2=M2 R=D2 C=1/K3 +-LLLL-/\/\--||---+-------+-LLLL-/\/\--||---+ | | | (~) e R3 = D3 | | | | | i1 C2 = 1/K2 i2 | +-----------------+-------------------------+ ``` #### F-I analogy (node) $F\to i$, $M\to C$, $D\to 1/R$, $K\to 1/L$, velocity $\to$ voltage. ```text v1 L2∥R3 v2 +----+----+----+---+--LLLL--+---+----+----+ | | | | +-/\/\/--+ | | | (i) C1 R1 L1 C2 R2 L3 | | | | | | | +----+----+----+----------------+----+----+ gnd ``` ($C_1 = M_1$, $R_1 = 1/D_1$, $L_1 = 1/K_1$; $L_2 = 1/K_2$ in parallel with $R_3 = 1/D_3$; $C_2 = M_2$, $R_2 = 1/D_2$, $L_3 = 1/K_3$.) The mesh (F-V) and node (F-I) equations have exactly the same form as the two mechanical equations above. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the Laplace equation for M1, what is the coefficient of X1? A. M1 s^2 + D1 s + K1 B. M1 s^2 + (D1 + D3) s + K1 + K2 C. M1 s^2 + (D1 + D3) s + K1 D. M1 s^2 + D3 s + K2 Answer: B. The diagonal term is the sum of all impedances attached to M1: its inertia, D1, K1 and the coupling pair K2 and D3. (Key point: Mass equation diagonal term) 2. What appears as the off-diagonal (coupling) term between the two equations? A. M2 s^2 + K3 B. D3 s + K2 C. D1 + D2 D. K1 + K3 Answer: B. K2 in parallel with D3 joins the masses, so their combined impedance D3 s + K2 is the coupling term, with a negative sign. (Key point: Coupling impedance) 3. Why is the numerator of X2(s)/F(s) equal to D3 s + K2? A. Force passes from M1 to M2 only through the K2 || D3 coupling B. F acts directly on M2 C. It is the sum of all spring constants D. It equals the determinant of the system Answer: A. By Cramer's rule, X2 = (coupling term x F) / determinant, since F acts on M1 and reaches M2 only through the coupling. (Key point: Numerator of X2/F) 4. In the F-V analogy used here, what do mass, spring and damper become? A. Capacitance, inductance, conductance B. Inductance, 1/C (elastance), resistance C. Resistance, inductance, capacitance D. Inductance, resistance, 1/L Answer: B. F to e, M to L, D to R, K to 1/C, and velocity to current. (Key point: F-V analogy rules) ## Close variants set in other exams - [Three-mass system X2/F with F-V and F-I analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-fv-fi-three-mass) — Control Systems, chapter 2, asked 3× - [Two-mass system X2/F with F-V analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-fv-m1-m2-k1k2-b1b2b3) — Control Systems, chapter 2, asked 3× ## Related in Control Systems - [Two-mass system X1/F, X2/F with F-V analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afv-tf-m1-m2-k1b1-k2b2) — Control Systems, chapter 2, asked 3× - [Two-mass spring system X1/F and F-V analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-fv-m1-m2-k1-k2) — Control Systems, chapter 2 - [Two-mass system X2/F by free body diagram, F-V analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afbd-fv-m1-m2-k1-b1) — Control Systems, chapter 2 - [Two-mass system with numeric values: F-V analogy and X1/F](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afv-tf-m1m2-k1k2k3) — Control Systems, chapter 2, asked 3× - [Two-mass system transfer function and F-V analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-two-mass-k1k2b1b2) — Control Systems, chapter 2 - [Two-mass system X2/F with F-V analogy (B1, K3 link)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-fv-m1-m2-b1k3-k1) — Control Systems, chapter 2 ## Related in other subjects - [AASHTO layer thicknesses D1, D2, D3](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aaashto-thickness-d1-d2-d3-esal) — Transportation Engineering II, chapter 2