• Define the equations of motion for the system shown on the right •Then, solve for the free vibration solution using an eigen solution when: m2 • k₁ = 1000 N/m • k₂ = = 1000 N/m • = 10 kg K2 ⚫ m2 = 20 kg k₁ m₁
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- 2 – Response of the two-degree-of-freedom system shown in to an initial displacements of uj 0.915 and u2 = 2.0 is such that q2 = 0. Find the mode shapes and a, where ki = a k2. Normalize each mode so that the modal mass has unit value. Write the equation for the force in the springs. k, 8m kz 2m U2 uiset up the integrals but do not solve them. Assume that the system is damped and you know ζ (only do the damped solution and not the undamped solution)3) Here is a simplified model of vehicle suspension system. The vehicle is being driven on the road below. The profile of the road and the velocity of the car in the horizontal direction are given. A road with such a profile will create an oscillation in the body of the car (the oscillation is denoted by y(t)). 1 y 1) Please obtain the magnitude of y(t) as a function of k and c. 2) Assume the mass of the car as 1 kg, k=10 N/m and c=2 Ns/m. What should the velocity of the car be to make ly(t)| less Vx = 3 m/s than 0.001 m? X : horizontal distance in meters sin(4x) the profile of the road in meters
- For the next two questions, consider the spring-mass system shown below. The system consists of two bodies of mass on three springs. Damping is assumed to be practically zero. X2 k2 14 m, W 5. The linear system from the previous question can be transformed into an eigenvalue problem to solve for the solutions x1 and x2 by assuming a trial solution æ = aeat. if k1 = k2 = k3 and m1 = m2 1kg, which of the following are the eigenvalues? %3D {2k, k} {-2k, -2k} (-3k,-k} {3k, k}SHOW YOUR STEP BY STEP SOLUTION. 1. Assuming that the displacement (x) resulting from the application of a force F at a point A is small, find the equivalent spring constant of the system that relates the applied force F to the displacement x. k- 2kSAMPLE PROBLEM Determine the equivalent spring stiffness of the system using the displacement of the block as the generalized coordinate. E = 200 x 10 N/m 1 = 1.6 x 10 m 6 x 10 N/m 3 m 2 x 10 N/m m 3 x 10' N/m S x 10' N/m mm.
- Consider the double mass/double spring system shown below. - click to expand. Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of F friction We can write the resulting system of second-order DEs as a first-order system, w'(t) = Aw(t), with w = (x₁, x₁, x₂, x₂) T For values of k = 4, m = 1 and c = 21,2 = 0.5±3.2i, V₁ 23,4= - 0.5 ± 1.13i, V3 1, the resulting eigenvalues and eigenvectors of A are -0.039 0.248i 0.813 0.024 +0.153i -0.502 -0.134-0.302i' 0.409 -0.216 - 0.489i 0.661 and (a) Find a set of initial displacements x₁(0), x₂(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero. (x₁(0), x₂(0)) Enter your answer using angle braces, (and ). cx (friction proportional to velocity). (b) At what frequency will the masses be oscillating in this mode? Frequency = rad/sA machine is supported on four steel springs. By neglecting damping, the natural frequency of vertical vibration the machine-spring system runs at f=200 cycles per minute. The machine generates a vertical force p(t)=posinot. When the machine runs at f₁ =30 cycle per minute the amplitude of vertical vibration is measured as u₁=0.2cm, at f₂=170 cycle per minute is u₂=1.042cm, and at f3-400 cycle per minute is u3=0.0248cm. Calculate the amplitude of vertical motion of the machine if the steel springs are replaced by four rubber isolators which provide the same stiffness, but introduce damping equivalent to -0.25 for the system. Evaluate the response of the system for frequency ratios r, respectively.Q1) A dynamic vibration absorber is shown in Figure. This system is representative of many situations involving the vibration of machines containing unbalanced components. The parameters M2 and k12 may be chosen so that the main mass M1 does not vibrate in the steady state when F(t) = a sin wo(t). Obtain the differential equations describing the system. Force F(t) M1 M2
- Q4/ Using the Dunkerley's method prove that natural frequency of transverse vibrations of a system with two loads attached to the same shaft is given by (neglect the weight of the shaft) 0.4985 f. =Consider the two-degrees-of-freedom mass-spring system below. Assume m₁ = 30 kg, m2 = 35 kg. If the second natural frequency of the system is 100 rad/s, the value of stiffness k is www. k₁=k k₂ = k k3= k 174.41 kN/m 107.38 kN/m 140.93 kN/m 39.24 kN/m 73.62 kN/m Next 000 00 Previous 4 x₁ (t) x₂ (t) m₁ m₂. mm2) If the under-damped spring mass system shown at right is CON subject to the force F(t)= Fe (is the radial frequency of excitation), find the steady-state response of the system by the method of Laplace transformation. k2 m FO k/2