4. Calculate and solve the characteristic equation for the following system:62-1[ 2 ] *0 + [37] ×(t) | ] x (t) = 04-1 1

Answered: 4. Calculate and solve the…

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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Consider the system shown in the figure below. Assume that when z is zero, that the spring is at its unstretched length. That is, the restoring force in the spring only develops when z, (the figure is a little misleading here). Note that z, is defined as positive to the left and zg is defined as positive downwards. Syntax notes: when entering your responses, use x[1] for z1and x2] for z2. m[1] for m1. m[2] for mg, and g for gravity. Use T for the tension in the rope. You can use the loupimagnifying glass to check that what you entered is what you intended. Kinematics The change in z is related to the change in zgsince they are part of the same string. What is the kinematic relationship between the two of them? 21- -2x(2] Kinetic Energy Write the total system kinetic energy. (ie write it in terms of both z1and zgand their velocities). Use diffio(1]t).) for ż and diffx(2]t).1) for ż KE = (12)m[1]*(difix[1]{t),f)^2+(1/2}*m[2]*(diff(x{2]{t),f)^2 Now use your kinematic relationship to write…
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Q-3 mechanical forces (Shear type). Find the propagation speed (vs) (for both longitudinal and transverse vibrations) of the elastic waves propagating along [111] direction to dissipate the gained elastic energy through the sample. The necessary equations can be obtained from the equation given below for the displacement vector A continuum bcc structure is deformed elastically by x, y, and z directional R = R,& + Ryỹ + R22 p(ô²R/@t) = C11(8²R$ /ôx²)+ C44 (a²R¾/@y²+ ô²Rx /ô z²)+(C12+C4)(ô²Ry/ Ô x ô y+ ô²Rz/@xð z) Here, C11, C44, C12 and p are the moduli of elasticity (Stiffness constants) and the mass density of the bcc structure, respectively.
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4.78. Demonstrate that the system modes are orthogonal with respect to the [M] and [K] matrices for the following system: 10 0 0 (, 8-8 500 -50 X1 2 0 *2 + -50 400 -20 0 4 0 - 20 100 X3
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Where the general equation of vibration is (NKI- *[M][X]. Find eigenvalues and eigenvectors. Q.3. 1.5 1.5 50 --35 IK] =-35 -35 50 (M] =|0 70 1.5 0 -35
Q-4: Let T:R3 → R? be a linear transformation such that T T and T a) [8 marks] Find a formula for T (X),X E R³. b) [4 marks] Find a matrix A such that T(X) = AX. c) [8 marks] Describe the null space and the range of T.

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4. Calculate and solve the characteristic equation for the following system: 62 -1 [ 2 ] *0 + [37] × (t) | ] x (t) = 0 4 -1 1