2. The equations of motion of this system are; 12Y+7Y+24Y-62-122-0 62"+62+122-6Y-12Y=f(t) 7W+7W+14W-7Y-14Y=ft) Where Y,Z and W are deformations of the masses and springs. Put these equations into state variable form and express the model as matrix vector equation if output of the system is Y. Energy Storage Element m₂ m₂ M3 k₁ k₂ State Variable X₁ = Y' X₂ = Z' X3 = W' X₁ = Y X5 = Z X = W mi Suspension d Wheo Road Datum les

2. The equations of motion of this system are; 12Y+7Y+24Y-62-122-0 62"+62+122-6Y-12Y=f(t) 7W+7W+14W-7Y-14Y=ft) Where Y,Z and W are deformations of the masses and springs. Put these equations into state variable form and express the model as matrix vector equation if output of the system is Y. Energy Storage Element m₂ m₂ M3 k₁ k₂ State Variable X₁ = Y' X₂ = Z' X3 = W' X₁ = Y X5 = Z X = W mi Suspension d Wheo Road Datum les

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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3. Jacobian Matrix: a) Derive the equation for Joint Force of the given planar 1R robot. As it's a planar robot, forces at the end-effector are Ex and Ex. b) Find the force (FX) when the link length is 0.5 m, Joint force is 1 Nm, Joint variable, 0 = 60 and Fx = 0 N. T P₂ y r F -F₂ ee 0,0 F Р. X 7 ४
diagram, F, is the input force exerted on the cart by the voltage applied to the motor, m, is the mass of the cart. The encoder is used to keep track of the position of the cart on the track. Cart Encoder Mater March 7, 2023 Track X Figure 5: Free body diagram of the cart (ignoring friction) Using Figure 5 and basic Newtonian dynamics you can derive the equations governing the system. 3.3 Motor Dynamics The input to your system is actually a voltage to the cart's motor. Thus, you need to derive the dynamics of the system that converts the input voltage to the foren exertal on the cart. These are the dynamics of the motor. Figure 6 shows a diagram of the electrical components of the motor. Figure 6: Clasic armature circuit of a standard DC motor For this derivation of the motor dynamics we assume the following: • We disregard the motor inductance: IR, so we can use the approximation 0. • Perfect efficiency of the motor and gearbox: -- 1. The torque generated by the motor is proportional…
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 Support
2. Try to write the motion equations of the system under the following conditions, and judge whether it is acceleration, deceleration or uniform speed(The arrows in the figure show the actual direction of torque). TX-IL IN CIL IW TW TL TO TL (4) IL TV TL TH= TL (1) Judge whether the intersection in the figure below is a stable working point? "Tx. Iz K
1. The equations of motion of this system are ÿ + 3y + 4y - 32 - 4Z = 0 Ż +52 +6Z-5ý - 6y = f(t) * = A + Bū y = Cx+Dū Put these equations into state variable form and express the model as a matrix vector equation if output of the system is y. Energy storage element m1 m2 k₁ k₂ State variable *1=ý x₂ = Ż x3 = y x4 =Z k₁ my D k₂ C₂H m₂
= = Problem #2 - Modeling, Mass-Spring System Two masses are connected by springs in series as shown in Figure 1 below where k₁ 20 N/m, k₂ = 10 N/m, m₁ 10 kg, and m₂ = 5 kg. Solve the system of ODEs in matrix form using the determinant to obtain the characteristic equation (it will make sense to substitute > for w² when you get to your characteristic equation). Determine the eigenvalues (frequencies) and eigenvectors (modes) of oscillation. Assume that no damping is present and that the masses slide without friction. On your diagram, sketch the modes of oscillation (think of how the masses could oscillate with respect to each other). What determines which mode or combination of modes that the masses oscillate at? i.) ii.) iii.) iv.) v.) vi.) vii.) Sketch the system and include a free body diagram. Use a conservation equation to develop your differential equation model (you should end up with two ODEs to describe the motion of both masses). Write your system of ODEs in matrix form. Set…
Given the vibrating system below: K1 K3 K4 Find the following 1. Keq 2. Ceq 3. Natural angular velocity K2 m C1 C3 C5 C2 C4 4. Damped angular velocity 5. Type of Damping 6. Equation of motion x(t). Assume Initial conditions for displacement and velocity Graph 2 cycles of the vibrating system. You can use third party app for this. 7. M = 10 kg K1=100 N/m K2= 80 N/m K3=75 N/m K4= 120 N/m C1 = 20Ns/m C2= 40 Ns/m C3= 35Ns/m C4= 15 Ns/m C5= 10 Ns/m
Given the vibrating system below: K4 Y(t) =Ysin30t where for = 30 and Y=20mm Find the following K1 K2 m C3 H C2 C1 C5 C4 1. Frequency Ratio 2. Displacement Transmissibility Ratio 3. Absolute displacement of the mass 4. Type of Damping 5. Equation of motion x(t). Assume Initial conditions for displacement and velocity 6. Graph 2 cycles of the vibrating system. You can use third party app for this. M = 10 kg K1=100 N/m K2= 80 N/m K3=75 N/m K4= 120 N/m C1 = 20Ns/m C2=40 Ns/m C3= 35Ns/m C4= 15 Ns/m C5= 10 Ns/m
Lagrange's equation can be used to find the dynamic model of the robot. Select one: True O False
Consider the following spring system. m, C3 Cy m₂ C₂ Write the stiffness matrix K Write the matrix M ¹K C₁ = 6, m₁ = 6, C₂ = 12, m₂ = 3 Find the eigenvalues and eigenvectors of M ¹K: Smaller eigenvalue = with eigenvector • Larger eigenvalue = with eigenvector C3 = 18, C₁=6 If this spring system oscillates without any external forces present, then the position of each mass satisfies the following general formula: X 8 u(t) = (a₁ cos( t) + b₁ sin(t)) + (a2 cos (t) + b2 sin(t)) If the system begins oscillation with initial position u(0) = [] | and initial velocity (0) = [] then the position of the masses at time t is given by u₁(t): u₂(t):
ll 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 1112