Mechanical Control Systems

Overview

Problem 1
Consider the function 𝑦 = ln(√2𝑥).

a) Linearize it about 𝑥∗ = 2 and 𝑥∗ = 4. b) Use Matlab to plot the function and its linear approximations at 𝑥∗ = 2 and 𝑥∗ = 4 . hho these three curves in the same graph in the range 𝑥 ∈ [0,6] and 𝑦 ∈ [0,1.5]. hubmit the figure and your codes. Problem 2.

A kinematic bicycle model is sho n from the top in the figure. (𝑥R, 𝑦R) denote the coordinates of the rear heel’s center, 𝑙 is the distance of the heels, 𝜓 is the ya angle, 𝛾 is the steering angle, 𝐯R and 𝐯F are the velocities of the rear and front heel’s center (points R and F), 𝑣R and 𝑣F are the associated speeds, and 𝜔 is the angular velocity of the frame. a) Express 𝜔 using 𝑣R, 𝑙, and 𝛾. (Hint: relate 𝐯R and 𝐯F using (𝑡, 𝑛) coordinates.) b) Derive the equation of motion (EoM), that is, express 𝑥̇R, 𝑦̇R, and 𝜓̇ using 𝑣R, 𝜓, 𝑙, and 𝛾.

Problem 3. The experiment sho n in the figure is used to measure the mass moment of inertia of a tennis racket. A racket of mass 𝑚 is hung and s ings about point 𝑂. The center of mass 𝐺 of the racket is at distance 𝑑 from 𝑂. The mass moment of inertia of the racket about 𝑂 is 𝐽𝑂. The angle of the racket is 𝜑. a) Determine ho many degrees of freedom (DoF) the racket has. Dra a free body diagram (FBD) and express the moment equation about 𝑂 . Derive the equation of motion (EoM) for 𝜑 (using 𝑚, 𝑔, 𝑑, 𝐽𝑂). b) Linearize the EoM about the equilibrium 𝜑∗ = 0. holve the linearized EoM ith 𝜑(0) = 𝜑0 and 𝜑̇ (0) = 0 analytically. c) The duration of 10 periods of small oscillations (s ings) is measured as 10𝑇 = 20.5 s. Considering 𝑚 = 0.3 kg, 𝑔 = 9.81 m/s2, and 𝑑 = 1 m, determine the mass moment of inertia 𝐽𝐺 of the racket about 𝐺. d) himulate the motion in Matlab by solving the nonlinear EoM numerically for 𝜑0 = 𝜋/6 . llot the numerical solution 𝜑(𝑡) over 𝑡 ∈ [0,10] s. llot the analytical solution 𝜑(𝑡) of the linearized EoM in the same graph. hubmit the figure and your codes. Problem 4. Consider t o coupled mass-spring- damper systems, each ith mass 𝑚, stiffness 𝑘, and damping coefficient 𝑏 . The positions of the masses are 𝑥1 and 𝑥2. The springs are unstretched for 𝑥1 = 0 and 𝑥2 = 0. A force 𝐹(𝑡) acts on the right mass. a) Determine the DoF and derive the EoM for 𝑥1, 𝑥2 using FBD and fundamental la of dynamics (FLD).

Problem 5. The mechanical model of a seg ay is sho n in the figure, hich consists of t o rigid bodies: a rolling heel and a frame. The frame is of mass 𝑚 and mass moment of inertia 𝐽𝐺 about its center of mass 𝐺 that is distance 𝐿 from the heel center 𝐶. The heel is of radius 𝑅, mass 𝑀, and mass moment of inertia 𝐽𝐶 about 𝐶 . The motor of the seg ay exerts a clock ise torque 𝑀d on the heel and a counterclock ise torque 𝑀d on the frame. The heel position is 𝑥𝐶 and the frame pitch angle is 𝜑. a) Determine the DoF. Express the acceleration 𝑎𝐶 of point 𝐶 using the angular acceleration 𝛼w of the heel. Express the acceleration 𝑎𝐺𝑥, 𝑎𝐺𝑦 of point G using the angular velocity 𝜔f and angular acceleration 𝛼f of the frame by relating 𝐚𝐺 to 𝐚𝐶. b) Dra FBDs for the heel and frame (dra the frame as a stick like in the right figure) and rite the FLD equations. Derive the EoM for 𝑥𝐶 and 𝜑. (The EoM should only contain 𝑥𝐶, 𝜑 and their derivatives, and the parameters 𝑚, 𝐽𝐺, 𝑀, 𝐽𝐶, 𝑅, 𝐿, 𝑔, and 𝑀d.