![4. (30%) Consider a 2-dof system with 8 -2 1 0 [m] = 0 4 [k] = -2 2 (a) Find the natural frequencies and mode shapes of the s](http://img.homeworklib.com/questions/a979fb50-ef5a-11eb-9580-6f1a98d64e69.png?x-oss-process=image/resize,w_560)
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Homework 7: Undamped, 2-DOF System 1. A system with two masses of which the origins are...
Homework 7: Undamped, 2-DOF System 1. A system with two masses of which the origins are at the SEPs is shown in Figure 1. The mass of m2 is acted by the external force of f(t). Assume that the cable between the two springs, k2 and k3 is not stretchable. Solve the following problems (a) Draw free-body diagrams for the two masses and derive their EOMs (b) Represent the EOMs in a matrix fornm (c) Find the undamped, natural frequencies...
Problem 2) For a 2 DOF system the equations of motion are given as: m9 [m...
Problem 2) For a 2 DOF system the equations of motion are given as: m9 [m OX -M29 2 m29 0 mal -29 Where m=m2 Em g -gravity and L =length a) Determine the frequencies and mode shapes. b) Verify that the natural modes are orthogonal. c) Determine the response (0) (0)) Note: xi(t) = XO, xa(t) = 0 and 1) = xo.* () = 0 d) If the system is excited by a harmonic force F. (t) =F, sinot,...
Homework 8: Modal and Direct Solution Approaches Figure 1 shows a system with two masses. The two...
Homework 8: Modal and Direct Solution Approaches Figure 1 shows a system with two masses. The two coordinates of which the origins are set up at the unstretched spring positions are also shown in Fig. 1. The system is excited by the force f(t) 1. (a) Draw the FBDs for the system and show that the EOMs can be written as (b) Find the undamped, natural frequencies and the corresponding mode shapes of the system for the given system parameters...
Problem 2) For a 2 DOF system the equations of motion are given as: [mi 0...
Problem 2) For a 2 DOF system the equations of motion are given as: [mi 0 0 m2 (X2 mig L -m29 L -m29 L m29 L Where m1 =m2 =m g=gravity and L =length a) Determine the frequencies and mode shapes. b) Verify that the natural modes are orthogonal. c) Determine the response fX:(0) Note: x1(t) = xo , x2(t) = 0 and xi(t) = xo , iz(t) = 0 d) If the system is excited by a harmonic...
2. For the following 3-DOF spring-mass system: (a) Derive the equations of motion. (b) Assuming ki-k2-k3-k...
2. For the following 3-DOF spring-mass system: (a) Derive the equations of motion. (b) Assuming ki-k2-k3-k and mi-m2-m3-m, determine the natural frequencies and mode shapes. rt
2. Assuming for a 2-DOF system the following eq uations of motion, andg so kip, g 386.4 in/s, k1 100 kip/in, Pi(t) 10 kip. P2(t) a. The two natural frequencies of the system. (25%) b. The two eig...
2. Assuming for a 2-DOF system the following eq uations of motion, andg so kip, g 386.4 in/s, k1 100 kip/in, Pi(t) 10 kip. P2(t) a. The two natural frequencies of the system. (25%) b. The two eigenvectors normalized with respect to mass and the 10 kip, determine the following: corresponding checks. (25%) c. Assuming a modal damping ratio ξ equal to 0.02, express numerically (as b, and N10) the uncoupled two equations of motion as shown below assuming classical...
For the following 2DOF linear mass-spring-damper system r2 (t) M-2kg K -18N/m C- 1.2N s/m i(t) - ...
For the following 2DOF linear mass-spring-damper system r2 (t) M-2kg K -18N/m C- 1.2N s/m i(t) - 5 sin 2t (N) f2(t)-t (N) l. Formulate an IVP for vibration analysis in terms of xi (t) and x2(t) in a matrix form. Assume that the 2. Solve an eigenvalue problem to find the natural frequencies and modeshape vectors of the system 3. What is the modal matrix of the system? Verify the orthogonal properties of the modal matrix, Ф, with system...
with steps please 04. For the system shown in Figure 4 where m-10 kg, k-100 kN/m, the governing equations has been d...
with steps please
04. For the system shown in Figure 4 where m-10 kg, k-100 kN/m, the governing equations has been derived as (1) Find the natural frequencies of the system; (2) Determine the associated mode shapes; and (3) Obtain the vibration response if the initial conditions are given as x,(0)-0,x,(0)-0.001 m, 2kE TIITTTUITTU Figure 4
04. For the system shown in Figure 4 where m-10 kg, k-100 kN/m, the governing equations has been derived as (1) Find the natural...
For the system shown in Figure 5, a. How many degrees of freedom is this system and why? (5) b. If x3-0 (the upper end...
For the system shown in Figure 5, a. How many degrees of freedom is this system and why? (5) b. If x3-0 (the upper end is fixed and K1 and K2=K (5) Write the equations of motion. Set the necessary matrix to find the natural frequencies and mode shapes Determine and explain how to get the natural frequencies 1. (5) (5) 2. 3. Figure 5 ww ww-
For the system shown in Figure 5, a. How many degrees of freedom...
1. Consider the two degree of freedom system shown. (a) Find the natural frequencies for the system (b) Determine the modal fraction for each mode. (c) Draw the mode shapes for each mode and iden...
1. Consider the two degree of freedom system shown. (a) Find the natural frequencies for the system (b) Determine the modal fraction for each mode. (c) Draw the mode shapes for each mode and identify any nodes for each mode. (d) Demonstrate mode shape orthogonality. (e) If F- and the motion is initiated by giving the mass whose displacement is a velocity of 0.2 m/s when in equilibrium, determine 0) and ,0 (f) Determine the steady-state solution for both *)...
Homework 7: Undamped, 2-DOF System 1. A system with two masses of which the origins are at the SEPs is shown in Figure 1. The mass of m2 is acted by the external force of f(t). Assume that the cable between the two springs, k2 and k3 is not stretchable. Solve the following problems (a) Draw free-body diagrams for the two masses and derive their EOMs (b) Represent the EOMs in a matrix fornm (c) Find the undamped, natural frequencies...
Problem 2) For a 2 DOF system the equations of motion are given as: m9 [m OX -M29 2 m29 0 mal -29 Where m=m2 Em g -gravity and L =length a) Determine the frequencies and mode shapes. b) Verify that the natural modes are orthogonal. c) Determine the response (0) (0)) Note: xi(t) = XO, xa(t) = 0 and 1) = xo.* () = 0 d) If the system is excited by a harmonic force F. (t) =F, sinot,...
Homework 8: Modal and Direct Solution Approaches Figure 1 shows a system with two masses. The two coordinates of which the origins are set up at the unstretched spring positions are also shown in Fig. 1. The system is excited by the force f(t) 1. (a) Draw the FBDs for the system and show that the EOMs can be written as (b) Find the undamped, natural frequencies and the corresponding mode shapes of the system for the given system parameters...
Problem 2) For a 2 DOF system the equations of motion are given as: [mi 0 0 m2 (X2 mig L -m29 L -m29 L m29 L Where m1 =m2 =m g=gravity and L =length a) Determine the frequencies and mode shapes. b) Verify that the natural modes are orthogonal. c) Determine the response fX:(0) Note: x1(t) = xo , x2(t) = 0 and xi(t) = xo , iz(t) = 0 d) If the system is excited by a harmonic...
2. For the following 3-DOF spring-mass system: (a) Derive the equations of motion. (b) Assuming ki-k2-k3-k and mi-m2-m3-m, determine the natural frequencies and mode shapes. rt
2. Assuming for a 2-DOF system the following eq uations of motion, andg so kip, g 386.4 in/s, k1 100 kip/in, Pi(t) 10 kip. P2(t) a. The two natural frequencies of the system. (25%) b. The two eigenvectors normalized with respect to mass and the 10 kip, determine the following: corresponding checks. (25%) c. Assuming a modal damping ratio ξ equal to 0.02, express numerically (as b, and N10) the uncoupled two equations of motion as shown below assuming classical...
For the following 2DOF linear mass-spring-damper system r2 (t) M-2kg K -18N/m C- 1.2N s/m i(t) - 5 sin 2t (N) f2(t)-t (N) l. Formulate an IVP for vibration analysis in terms of xi (t) and x2(t) in a matrix form. Assume that the 2. Solve an eigenvalue problem to find the natural frequencies and modeshape vectors of the system 3. What is the modal matrix of the system? Verify the orthogonal properties of the modal matrix, Ф, with system...
with steps please
04. For the system shown in Figure 4 where m-10 kg, k-100 kN/m, the governing equations has been derived as (1) Find the natural frequencies of the system; (2) Determine the associated mode shapes; and (3) Obtain the vibration response if the initial conditions are given as x,(0)-0,x,(0)-0.001 m, 2kE TIITTTUITTU Figure 4
04. For the system shown in Figure 4 where m-10 kg, k-100 kN/m, the governing equations has been derived as (1) Find the natural...
For the system shown in Figure 5, a. How many degrees of freedom is this system and why? (5) b. If x3-0 (the upper end is fixed and K1 and K2=K (5) Write the equations of motion. Set the necessary matrix to find the natural frequencies and mode shapes Determine and explain how to get the natural frequencies 1. (5) (5) 2. 3. Figure 5 ww ww-
For the system shown in Figure 5, a. How many degrees of freedom...
1. Consider the two degree of freedom system shown. (a) Find the natural frequencies for the system (b) Determine the modal fraction for each mode. (c) Draw the mode shapes for each mode and identify any nodes for each mode. (d) Demonstrate mode shape orthogonality. (e) If F- and the motion is initiated by giving the mass whose displacement is a velocity of 0.2 m/s when in equilibrium, determine 0) and ,0 (f) Determine the steady-state solution for both *)...
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