Q2Control System
Question
Q.2. Obtain the differential equations describing the complete dynamics of the mechanical system shown in figure (a two-mass translational spring-mass-damper system: mass M1 with applied force F(t) and displacement X1 connected to a fixed wall through spring K1 and damper B1, and to mass M2 (displacement X2) through the same K1-B1 pair; M2 is connected to a fixed ceiling through spring K2 and damper B2).
Answer
Applying Newton's second law to each mass in the two-mass mechanical network gives two coupled second-order differential equations: M1ẍ1 + B1(ẋ1-ẋ2) + K1(x1-x2) = F(t), and M2ẍ2 + B2ẋ2 + K2x2 + B1(ẋ2-ẋ1) + K1(x2-x1) = 0.
The mechanical system consists of mass M1 (displacement x1, driven by external force F(t)) connected to a fixed wall through spring K1 and damper B1, with M1 also connected through the same K1 and B1 to mass M2 (displacement x2); M2 in turn is connected to a fixed upper support through a separate spring K2 and damper B2.
Free body diagram for M1: The forces acting on M1 are the applied force F(t), the spring force from K1 (proportional to the relative displacement between M1 and M2, since K1 connects them: force = K1(x1 - x2), opposing extension), and the damping force from B1 (proportional to relative velocity: B1(ẋ1 - ẋ2)). Applying Newton's second law (ΣF = Mẍ):
Free body diagram for M2: M2 experiences the reaction spring/damper forces from K1, B1 (pulling it toward M1, i.e., proportional to (x2-x1) and (ẋ2-ẋ1)) plus its own spring K2 and damper B2 connecting it to the fixed upper support (proportional to its absolute displacement x2 and velocity ẋ2, since the far end is fixed). No external force acts directly on M2. Applying Newton's second law:
These two coupled linear second-order ODEs completely describe the dynamics of the two-mass, two-spring, two-damper mechanical translational system. They form a direct electrical-mechanical analogy (force-voltage or force-current analogy) with a two-loop RLC electrical network, where mass M corresponds to inductance L (or capacitance, depending on analogy), damping B corresponds to resistance R, and stiffness K corresponds to the reciprocal of capacitance 1/C (or inductance), which is the standard basis for representing mechanical systems using equivalent electrical circuit analysis and transfer function techniques in control system design.
Transfer function form. Taking the Laplace transform of both equations (assuming zero initial conditions) converts them into an algebraic pair in terms of X1(s), X2(s) and F(s):
These two simultaneous algebraic equations can be solved for X1(s)/F(s) or X2(s)/F(s) using Cramer's rule, giving the transfer function relating the applied force to either mass's displacement as a ratio of polynomials in s — this is the standard procedure by which the physical mechanical system is converted into the s-domain block-diagram or signal-flow-graph representation used throughout the rest of control system analysis and design.