Governing equation of motion and modal transformation
A two DOF system consists of two equal masses, m, two springs having stiffnesses k and 2k, and two viscous dampers each having damping coefficient c, see the figure.
Here; k = m = c =1 [N/m, kg, Ns/m].
a) Write the governing equation of motion in matrix form.
b) Calculate the undamped eigenfrequencies and their corresponding mass normalized mode shapes.
c) Derive the governing equations (in matrix form) in the modal domain, ie. use modal transformation. Let p(t)=10 N. Set up the equations including initial conditions so that they could be solved.
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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...
6. Consider two coupled oscillators of mass mi and m2 that are attached by springs and are unstre...
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Please answer the questions for Part 1 and Part 2 showing all steps, using the provided...
Please answer the questions for Part 1 and Part 2 showing all
steps, using the provided data values.
Many thanks.
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characteristic of the train system based on the mode shape
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Elementary Differential Equation Unit Step Function Problem Project 2 A Spring-Mass Event Problenm A mass of...
Elementary Differential Equation Unit Step Function Problem
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The equations of motion for a certain mechanical system with two degrees of freedom, can be...
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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...
6. Consider two coupled oscillators of mass mi and m2 that are attached by springs and are unstretched when x,-12-0. The damping force viscous dampers. The springs is proportional to the speed of the displacement and acts in the direction opposite the motion, Fd--cv T1 k1 k2 m1 m2 C1 C2 C3 a) Find the equations of motion in and i2 b) Since the equations are coupled, write them in matrix form: ME+ CE+ K 0
6. Consider two coupled...
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...
Figure 1 shows a system with two masses of which the origins are set up for the springs of \(k_{1}, k_{2}\), and \(k_{3}\) to be unstretched. The system is excited by the base motion of \(x_{b}(t)\).(a) By drawing the FBDs of the two masses and applying the Newton's \(2^{n d}\) Law of Motion, find a matrix equation of motion.(b) Find the undamped, natural frequencies and the corresponding mode shapes of the system for the given system parameters of \(k_{1}=k_{2}=k_{3}=100 \mathrm{kN}...
Please answer the questions for Part 1 and Part 2 showing all
steps, using the provided data values.
Many thanks.
M2 2 C2 2' 2 2 C2 2'2 Spring steel Mi k1 C1 2'2 1 C1 Base y(t) Base movement Figure 2 shows a shear building with base motion. This building is modelled as a 2 DOF dynamic system where the variables of ml-3.95 kg, m2- 0.65 kg, kl-1200 N/m, k2- 68 N/m, cl- 0.40 Ns/m, c2- 0.70Ns/m The base...
3.2 Pre-Lab Assignment When deriving the governing equations for an electromechanical system, it is often beneficial to examine the electrical and mechanical components independently. Looking at only the electrical components of the QUBE-Servo DC motor (as shown in Figure 3.2): R v00 C e, (00 Figure 3.2: Electrical curcuit of the QUBE-Servo DC motor Q1. Write the differential equation in the form of Kirchoff's voltage law) in the Laplace domain for the electrical circuit (do not use parameter values given...
a)by using newtons 2nd law,derive the equation of
motion for the vibration system in matrix form
b)diferrentiate between the 1st,2nd and 3rd vibration modes
characteristic of the train system based on the mode shape
diagram
A three coaches train system shown in Figure 3(a) can be simplified as a three degree of freedoms semi definite mass-spring system as illustrated in Figure 3(b). The masses of the three coaches are /m = 15000 kg, m-10000 kg and m-15000 kg. The three...
Elementary Differential Equation Unit Step Function Problem
Project 2 A Spring-Mass Event Problenm A mass of magnitude m is confined to one-dimensional motion between two springs on a frictionless horizontal surface, as shown in Figure 4.P.3. The mass, which is unattached to either spring, undergoes simple inertial motion whenever the distance from the origin of the center point of the mass, x, satisfies lxl < L. When x 2 L, the mass is in contact with the spring on the...
The equations of motion for a certain mechanical system with two degrees of freedom, can be written as a pair of coupled, second-order, differential equations: (M + m)x - 1/2 mL theta^2 sin(theta) + 1/2 mL theta cos(theta) + k(x - L_0) = 0 1/3 mL^2 theta + 1/2 mLx cos(theta) + 1/2 mgLsin(theta) = 0 We can rewrite them in matrix form, A*qdd - b, to be solved simultaneously: [M + m 1/2 mL cos theta 1/2 mL cos...
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