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Experiments to identify precision grip dynamics between the index finger and thumb have been performed using a ball-drop experiment. A subject holds a device with a small receptacle into which an object is dropped, and the response is measured (Fagergren, 2000). Assuming a step input, it has been found that the response of the motor subsystem together with the sensory system is of the form
Convert this transfer function to a state-space representation.
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Control Systems Engineering
- Harmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x 0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kr = 0. (1) k Lui Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d)…arrow_forward1. For the linkage system show, Link 1 moves along the vertical direction via a linear motor. Link 2 and 3,,dependent on the motion of Link 1, will move accordingly. Perform position analysis using Chace Mehod: Draw the loop closure vectors for the problem. (a) (b) Determine the unknowns. (c) Determine which kinematic case it is (d) Determine equations for the unknowns as function of known variables. (e) Assume link 1 is located 3 inches above the x-y coordinates, and link 2 has a length of 5 inches. Determine the numerical values of the two unknowns for this specific configuration. Link 1 X Link 2 Link 3arrow_forward1. A cylinder has a cross-sectional area of 'A' and a length of 'L'. It is made of a material that has a modulus of 'E'. Show that stiffness of the cylinder can be expressed as K=(EA)/L for the cylinder when the cylinder is undergoing tension. 2. A) Plot the time response of deformation for a 100 kDa globular protein under the effect of a step force input of amplitude 25 pN. B) What is the time constant for the particle to reach a steady state deformation. 3. Estimate the size of a protein that will result in the overdamping criterion to be unity. In your estimation, the shape of the protein can be assumed to be spherical. 4. Plot the overdamping criterion for bending of a rod-shaped protein assuming that length of the protein ranges from 1 nm to 1000 nm. What is your conclusion on the likelihood of a protein to vibrate in bending with increasing length of the protein?arrow_forward
- A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): A. Use Laplace transform of the differential equation to determine the transfer function of the system.arrow_forwardExample # 1 For the mechanical system shown, a force of 2 lb( step input) is applied to the system, the mass oscillates, as shown in fig. determi from this response curve. The displacement x is measured from the equilibrium position m, b, k of the system 0.0095 ft 0.1 ft 4 5 (b)arrow_forwardQuestion 1: Determine the mathematical model of the linear mechanical system shown below, given f(t) is input force and x₁ (t), x₂ (t) and x3 (t) are output displacement for each mass. -x1(1) -X₂ (1) +x3(1) 1 N/m 1 N-s/m 1 N/m 1 N-s/m I oooo PAN 1 kg 1 kg0000 1 kg f(t)- HHHHHarrow_forward
- A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): 1. What is the order of this system?arrow_forwardQ2 The position of an elevator h(s) is controlled by means of lifting cables. A feedback control system is used to control the force applied by the cables to the elevator. The transfer function of the plant is, 1 Gp(s) = Controller Elevator hr(s) h(s) E(s) G.(s) GP(s) A unit step input is provided. If only proportional control is used, show that the position of the elevator oscillates about the reference value of 1. Find the period (a) of this oscillation. (b) Show that the addition of derivative action to the system can ensure a non- oscillatory response. Find a relation between the derivative and proportional gains that ensures the response is non-oscillatory. (c) When the proportional and derivative gains are set to Kp = 9 and Kd = 6, respectively, find the damping ratio and natural frequency of the system. Derive the time-domain response of the elevator for a unit step input and confirm that it is not oscillatory.arrow_forwardIn Figure.4, a car wheel balancer device is used to estimate the location/direction and mass value needed to be attached to the wheel in order to statically balance it and prevent vibration during driving. Your role is to calibrate the system measurement by constructing theoretical calculation of a sample wheel and compare it with the results of the computer. The unbalanced wheel can be modelled as in Figure.5 and the four distributed point-mass as in Table.2.arrow_forward
- Q2 The angular position c of a mass is controlled by a servo system through a referen signal °r. The moment of inertia of moving parts referred to the load shaft, J, is 150 kebo and damping torque coefficient referred to the load shafi(B, is 4.5 x 103 Nw-m fraden The torque developed by the motor at the load s 7.2 x 10* Nw-m per radian of error (a)Obtain the response of the system to a step input of 1 rad and determine the peak time, peak overshoot and frequency of transient oscillations. Also find the steady state error for à constant angular velocity of 1 revolution / minute. (b)If a steady torque of 1000 Nwm is applied at the load shaft, determine the steady Kan state error. 5(J5+ B) 03 For the system shown, apply a proportional controller to move the time constant to a sixth of its open loop value. R(s) + E(s) U(s) Kp 'Y(s) s +0.333 10arrow_forwardA 5 kg object is hung from a spring whose other end is attached to a rigid support. The transient response of the object when immersed in a liquid is shown in Figure QA2a. The liquid is at room temperature. Determine: i. ii. the damping ratio, the stiffness of the spring and the damping coefficient; The damped frequency and natural frequency (in Hz). Displacement (m) 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.5 mi Time (s) 1.5 2arrow_forwardMECHANICAL VIBRATIONS An unknown mass m is attached to the end of a linear spring with unknown stiffness coefficient k. The system has natural frequency of 30 rad/s. When a 0.5 kg mass is added to the unknown mass m, the natural frequency is lowered to 20 rad/s. Determine the mass m and the stiffness coefficient k.arrow_forward
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