Lab Report 1_EELE3314

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2 Find frequenciel and mode shapes Ic of the two 9/ian systems, assumiing m=z ton/m (b) d=らm hz 3m E =210000 o kestlat 1 = 80Đ Cnt ca)

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QUESTION 1 Consider a 2 DOF system shown below. X1 k₁ m1 F₁ k₂ X2 k3 m₂ F2 The modeshape can be written as (1))] What is x for the second modeshape? Use scientific notation with 3 significant digits and omit units. (eg. -0.123) Let m1 = 2, m2 = 2, k1 = 9, k2 = 8, and k3 = 8.

2. The equation of motion for a damped multidegree of freedom system is given by Where [m] = [m]{x} + [c]{x} + [k]{x} = {f} [100 = 0 0 0 10 0 10. 1000-4 {f} [c] 8 –4 0 8 – 4 -4 4 0 8 4 = 100 2 0 = Focos(wt) -2 -2 The value of Fo 50N and w 50 rad/sec. Assuming the initial conditions to be zero and using the modal coordinate approach, find out the steady state solution of the system in the modal coordinate for first mode. Plot the steady state solution using the computational tools. 0 −2], [k] = = 2

H-W Find Time response for 2S+3 Yes)= s° (s+3)

H.w 2: The open loop response, that is, the speed of the motor to a voltage input of 20V, assuming a system without damping is dw 20 = (0.02) + (0.06)w. dt If the initial speed is zero (w(0) = 0) ,and using the Runge-Kutta 4th order method, what is the speed at t = 0.8s? Assume a step size of h = 0.4s.

Q1: (Section A) Considering single degree undamped vibration system and Newton's equation as follow: më +kx=0; find the solution of the displacement equation [(t)=Cietwnt+C2e-lwn] for the case with: Wn = 2 rad/s, x (0) = 1 mm, and x(0) = v5 mm/s. (Section B) Given the matrix equation of motion of a two degree-of-freedom system 3m 2k - k = 0 X2 + as: -k 4k m Determine (a) the natural frequencies, (b) the modes shapes.

A liquid-level process control system is shown in Figure 5a and its closed loop representation is shown in Figure 5b. The system parameters are: A = 2 m², Rf = 15 s/m², H₁= 1 V/m, K₁ = 1 m³/sV, K₁ = 1. What are the values of Ti and when the undamped natural frequency WT = 0.01 rad/s? Answer should be supported with appropriate calculation steps. hul!) Pl hon Controller Control Valve K₁ Pressure transducer H₁ Figure 5a. Tank Area A Outlet valve Resistance R/\10 PI Controller Tank and valve Control valve H₁(s) Ha(s) E(s) U(s) V₁(8) K₁ K₁ R₁ 1+ARS Hm(s) Pressure tranducer H₁ Figure 5b. Ꮵ.

Please show all work for FBD and equatiosn

Question is image. Solution of c) ? Inverse Laplace

A X₁ RA P₁ X₂ P₂ 2 X3 X L 10 C ХА B RB 22 Z P₁ = 30kN; P₂ = 50kN; w= 10kN/m; L = 6m x₁ = 1m; x₂ = 2m; x3 = 3m; x = 5m Find RA and Rg, and draw SFD and BMD

KxM = Jö + bø - co where Ø denotes the angle of attack (output). M is the correction moment (input), K, is the moment gain, J is the moment of inertia along the rotation axis, b is the coefficient of Viscous friction, C is the coefficient of gravitational loading due to offset. J=7400kgm^2c=4670 Nm/rad, b=3550 Nms/rad, Kx =5000 Initial conditions are zero. Design a LEAD-LAG Controller such that the maximum overshoot is 12% and the settling time is 4 seconds and the steady state error is less than 0.3 when we implement unit step as the input. Clearly explain the necessary steps. Use a MATLAB Script for the controller design. Using MATLAB/Simulink, simulate the controller. Use a 2-input scope to plot the output and the step input. Show the Simulink model and its parameters. Make a sketch of the step input and system response Indicate the steady state error.

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- Consider the positions servomechanism is shown in figure below. Assume the following numerical values for the system constant are. Kp = gain of potentiometer detector = 7.64 v/ rad, Kb = back e. m. f. constant -5.5*10-2 V/rad/sec K₁ = motor torque constant = 6*10-5 Nm/A C₁ = 4*10-2 Nm/ (rad /sec). Determine the natural frequency and damping ratio of the system. Amp. % Ra= armature winding resistant =20, Ka=Gain of amplifier =10 Jm= 1*10-5 Kg.m². J₁=4.4*10-³ Kg.m² n = gear ratio=1/10, Neglect Cm and la constant 0050 N₁ DI

A= 8kN/m B=12kN.m C=6kn

For the level process shown in figure, 4 m?/min AT = 3 m? 8 m 4 m³/min R = 1 min/m2 A 3-unit amplitude step-type input change is applied that changes to 6 units after 6 minutes and remains constant. 1) What is the level height that the system will reach after stabilization? 2) What is the level height 15 minutes after the input variation was applied? 3) In how long will the system reach 65% of the final value?

P:- Solve the System of Linear algebroie EXuation G.E-H. X- y +7-2 W = -1 - 2K+2y-7+2w = 3 3X -3y +27-4e k! = -4 -4X +4y=3Z+6w = 5 %3D

Problem 3: The RPH robot of Figure 3 is shown in its zero position. Determine the end- effector zero position configuration M, and the screw axes S; in {s}. n. F -inta. Sorowo- Zs {s} Xs L₁ ŷs Lo 0₁5 L2 02 {b} zb 103 pitch h = 0.1 m/rad Ấb ŷb L3 Figure 3: An RPH open chain shown at its zero position. All arrows along/about the joint axes are drawn in the positive direction (i.e., in the direction of increasing joint value). The pitch of the screw joint is 0.1 m/rad, i.e., it advances linearly by 0.1 m for every radian rotated. The link lengths are Lo = 4, L₁= 3, L2= 2, and L3= 1 (figure not drawn to scale).

The following EOMS describe the behavior of an electric motor attached to a torsional spring and damper: [Li + Rz + vb = Vin(t) IÖ = k0b0 + T Importantly, the equations are coupled by the fact that 7 = k₁z and v₂ = k₂0. Here, L, R, kv, I, k, b, and kt are constants. z is the current in the motor, and is the angular displacement of the motor (clearly then, is the motor's angular velocity). Finally, the input to the system is vin (t). Your tasks: A Substituting in the coupling terms (7 = k₁² and v = k„0) into the EOMs, write the state space form of the system x = Ax + Bu, assuming your states are 2₁=2, 22 = 0, and x3 = 0. Clearly identify your matrices. B Assume that the desired outputs of the system (to be measured) are the current z, the angle and the input Vin all as separate elements in the vector y. Write the output equation y = Cx + Du. Hint: D will NOT be zero in this case. Clearly identify your matrices Assume represen angular velocity and 9(s) to

The following EOMS describe the behavior of an electric motor attached to a torsional spring and damper: [Li + Rz + vb = Vin(t) IÖ= k0b0 + T Importantly, the equations are coupled by the fact that 7 = k₁z and v₂ = k₂0. Here, L, R, kv, I, k, b, and kt are constants. z is the current in the motor, and is the angular displacement of the motor (clearly then, is the motor's angular velocity). Finally, the input to the system is vin (t). Your tasks: A Substituting in the coupling terms (7 = ktz and v k0) into the EOMs, write the state space form of the system x = Ax + Bu, assuming your states are x₁ = z, x₂ = 0, and x3 = 0. Clearly identify your matrices. B Assume that the desired outputs of the system (to be measured) are the current z, the angle and the input Vin all as separate elements in the vector y. Write the output equation y = Cx + Du. Hint: D will NOT be zero in this case. Clearly identify your matrices C Assume there is no stiffness to the spring (k = 0). Using w(t) to represent…

Given the torsional system shown below actuated by a step input torc T(t): TO) 00) lll fro> K D 1- Write first the equation of motion, and then: 2- for K=2 N.m/rad, find J and D to yield 15% overshoot and a 2% settling time of 5 seconds, and then calculate the resultant natural frequency and damping ratio. 3- for K=2 N.m/rad, find J and D to yield a peak time of 2 seconds and a 2% settling time of 10 seconds, and then calculate the resultant natural frequency and damping ratio. 4- for D=1 N.m.s/rad, find K and J to yield 20% overshoot and a peak time of 2 seconds, and then calculate the resultant natural frequency and damping ratio. 5- for J=0.5 kg.m², find K and D to yield 5% overshoot and a peak time of 1 second, and then calculate the resultant natural frequency and damping ratio.

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You are tasked to enhance the performance of an analog plant using a digital controller PID. The controller is in connected in series with the plant in a closed-loop configuration. The pulse transfer K(z+a) function is given as GzohGp(s) =- Use sampling time Ts=0.6 second, K=8, a=0.78, b=-0.602 (0+z)(q+z) and c=-0.705. Do the following: i. Obtain the design point in z-coordinates to achieve the requirement of having a system settling time of 3 second, and damping ratio of zeta (z=0.705).

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