For the translational mechanical system shown in Figure (3). 1. Write the mathematical model in a format of matrices. ✓1. 2. Find the transfer function G(s) = a₁ (s)/T (s) where a is the acceleration.
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- Get the equation of motion by drawing the free body diagram of the given systems. a) Get the system's transfer function and find the unit digit answer. Show all decals in detail. m = 1 kg b= 20 Ns/m k = 125 N/m F ww k b) get the transfer function of the system. X(s)/Pg(s) =? Show all decals in detail. resistance R k Pg massless piston area (A) capacitancell b) Obtain the mathematical model of the system shown in Figure Q2b using Newton's second law of motion, F=ma. k₁ w 3- 777777 C1 7771 k₂ D 7777 Figure: Q2b Page 2 of 7 A C2 11121 An object of mass 125 kg is released from rest from a boat into the water and allowed to sink. While gravity is pulling the object down, a buoyancy force of times the weight of the object is pushing the object up (weight = mg). If we assume that water 40 resistance exerts a force on the object that is proportional to the velocity of the object, with proportionality constant 10 N-sec/m, find the equation of motion of the object. After how many seconds will the velocity of the object be 90 m/sec? Assume that the acceleration due to gravity is 9.81 m/ sec2. Find the equation of motion of the object. X(t) = %3D
- 6. The electro-mechanical system shown below consists of an electric motor with input voltage V which drives inertia I in the mechanical system (see torque T). Find the governing differential equations of motion for this electro-mechanical system in terms of the input voltage to the motor and output displacement y. Electrical System puthiy C V V₁ R bac (0) T bac T Motor - Motor Input Voltage - Motor Back EMF = Kbac ( - Motor Angular Velocity - Motor Output Torque = K₂ i Kbacs K₁ - Motor Constants Mechanical System M T Frictionless SupportFor the mechanical translation system below, find the force-voltage analogy and force-current analogy. Use the following values. K1 = 2 fv, = 1/2 M1 = 1+a %3D K2 = 2 fv2 = 4+b M2 = 5 K3 = 3+c fv3 = 3 a = 0 where a = 3rd digit of your student number %3D b = 5th digit of your student number b =7 C = 7th digit of your student number C = 5 For reference, the 1st digit of your student number is the leftmost number in your student number. Indicate your student number when solving problems.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.
- 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?2) As shown in the figure, a uniform rod of length I and mass m is supported by a pin at one end and is suspended horizontally from a string of length h at a point s from the end. m 1/2 $ G: Ah h 真横からみた図 1) When the string rotates by 0, how much does the position of points on the rod move upward? (Just the answer is enough.) 2) When the string is displaced upward by Ah, how much does the bar's center of gravity displace? (Just write your answer). 3) Find the change in potential energy U at this time. Next, assuming that is small, express U as a power of 0. 4) Find the moment of inertia of mass m around the fulcrum. 5) Find the kinetic energy when the rod rotates by around the fulcrum. Using the relationship h0 = so, express the kinetic energy as the power of the derivative of 0. 6) Find the natural angular frequency of this system using the energy method.Consider the manipulator provided in the figure. Given the Position vector P w.r.t. to the base frame, derive the Analytical Jacobian for the position. Write your answer as J = [dpx/dq1 dpx/dq2 dpx/dq3; dpy/dq1 dpy/dq2 dpy/dq3; dpz/dq1 dpz/dq2 dpz/dq3] %3D Z3, Z4 Z1, Y, b,s, +b;C;82 -b,c +b3S152 -b;C2 X4 YA #3 b2 b3 O2, 02 P = #2 Z, Y3 #0
- You walk along the beach towards a dock while your friend rows a boat towards the same dock on a flat lake. Your friend's boat approaches the dock on a straight course, but also rotates about its center-of-mass since your friend is not pulling evenly on the oars. If you knew your own velocity (vwalker), the magnitude of boat's angular velocity (thetaboat), and radius vector from yourself to your friend in the boat (rfriend) at any given time, you could use the following equation to calculate the your friend's velocity: vfriend = vwalker + (thetaboat)k x rfriend where k is a unit vector in the vertical direction. True FalseA four-bar mechanism is used to transmit power to slider E. Link AC rotates counterclockwise at 100 rpm. A and B are fixed points. The following linkages are measured in cm: AC=35, AB=70, CD=45, BD=45, DE=40. Find the instantaneous velocity of slider E in cm/s 1.Draw the mechanism in appropriate scale on your paper2. Find the instantaneous linear velocity of C3. Find a point on your paper to draw the velocity polygon and scale the magnitude of the computed linearvelocity4. Use the relative velocity equation for finding the linearvelocity of D; ? = ? + ?? ? ?/?5. Remember that although the magnitudes are unknown, their directions can be identified6. Obtain the magnitudes of the unknown velocities from the velocity polygon7. For Link DE, repeat the process by using ? = ? + ??the line as much as needed8. Remember that E is a SLIDER therefore it can only goin the direction where its movement is not constrained? . Add the vector ? to the head of ? and extend?/? ?/??Find the differential equation of the mechanical system in Figure 1(a) To obtain the differential equation of motion of the mass and spring system given in Fig. 1. (a) one may utilize the Newton's law for mass and spring relations defined as shown in Fig. 1. (b) and (c) use f = cv for viscous friction, where v is the velocity of the motion and c is a constant. Z/////// k M F. F, F F F, F, k EF=ma F = k(x, - x,) = kx (b) (c) Figure 1: Mass-spring system (a), Force relations of mass (b) and spring (c)