Consider a second order system given by 7 d²x(t)/dt² + 3 dx(t)/dt + 24 x(t) = t. What is the damping ratio? Do not include units. Write your answer in scientific notation using 3 significant figures. (e.g. 1.23e0)
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- 2- A free vibrations test is run to determine the stiffness and damping properties of an elastic element. A 20 kg block is attached to the element. The block is displaced 1 cm and released. The resulting oscillations are monitored with the results shown in Figure (1). Determine k and c for this element. 0.01 0.008 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008 -0.01 0 0.06 0.12 0.18 Figure (1) 0.24 0.30 0.3The mass of a single degree of freedom damped vibrating system is 75 kg and vertically attached to the spring of a stiffness 10.5 kN/m as shown in Figure 2. The damping coefficient of the system is 600 Ns/m and the system is excited by a force: F=700sin25t in Newtons and t-is the time in seconds 2.1 Calculate the natural frequency of the system in rad/s 2.2. Calculate the periodic time of the system if at free vibrating mode and that of exciting force.2.3. Using the damping ratio criteria, state the damping type of the system. (5)1. Suppose you are riding your bicycle on a bumpy road having a surface profile that varies harmonically with +/- 6 cm undulations. The distance between consecutive peaks of these undulations is 2 m. When you sit on the seat of your bike for your casual ride, the springs deflect 5 cm. When you are seated, the damper under the seat provides an equivalent linear viscous damping of 10% of the critical damping. A simple representation of your ride on the "never-ending" rough road is shown below. (a) If you are riding your bicycle at a horizontal speed of 2.5 m/sec, how much bumping up and down will you experience? (b) Next day, you are carrying a backpack which increases your on-seat weight by 20%. Assuming that you are still able to ride with same speed, will this "loaded" ride be more or less comfortable than your previous, "no backpack" ride? M k/23 k/2 6 cm 2 m
- Question 3 Find the natural frequency (in rad/sec) of the system in In the figure below: Use: m1= 5 kg, k1 = 53 N/m, k2 65 N/m and k3 18 N/m Write your answer to FOUR significant figures. Don't write the units X1 X2 k1 k2 k3 W ww m1Consider the following for a single-degree-of-freedom system with m = 1. X, k = 2.5, and c = 1.8. Here, the value of X indicating of the number of your group. For example, if your group number is 15 gives X = 15, therefore the values of m = 1.15. If your group number is 9 gives X = 9, therefore the values of m = 1.9 Find; (i) Natural frequency, Wn (ii) Linear frequency, fn (iii) Critical damping constant, cc (iv) Damping ratio, 3 (v) Damped frequency, waA body of mass 100 kKg is suspended by a spring of stifness of 30KN/m and a dashpot of damping constant 1000 N.s/m. Vibration is excited by harmonic force ft)=80 cos (67tt) 1- Write the equation of motion of the system using Lagrange's equation. 2- Find the natural frequency and the driving frequency. 3- Write the complete solution of the vibrating system.
- x(t) 0000000000 m kea F Vibration model Fo Fo F(t) T 2T t (sec) The vibration model is subjected to the force shown in the figure. Find the response of the system for time t>2T(Vibration Subject) Predict the fundamental natural frequency using Rayleigh's Method in Figure below: Given, k1 = 10 N/m k2 = 20 N/m k3 = 15 N/m J1 = 1 kg m2 J2 = 2 kg m2For given mass (m=10kg) and two springs systems k1=30 N/m and k2= 20N/m. Find the following 1. Circular Frequency and natural frequency 2. If initial displacement is 50 mm and initial velocity is 0, what is max displacement and max velocity. 3
- t. Design the following application case The mechanical vibration system is shown in Figure (a). When a step input with 3N amplitude is subjected to this system, the movement rule of the mass m is shown in Figure (b). Evaluate the values of mass m, damping constant c and spring constant k. m (a) F-3N XX 1.0 x(1)(cm) 0 2 (b) M,-0.095 ⒸA 2 DOF mass-spring system shown below with n = 1, k = 8, and m initial conditions 2₁ (0) = 1, 2₂(0) = 0, ₁ (0) = 0, and ₂(0) = 1. (a) Find the natural frequencies and normalized modal matrix [X] for the system. T x₁ (1) (b) Using modal analysis and the Cartesian solution form, find the free vibration re- sponse of the system x₁(t) and x2(t). Ţ x2₂ (1) www. 00000 = m m₁ = 00000 k₁ = k 00000 k₂ = nk m₂ = m k3= k = 2 DOF mass-spring system. 2 with theFigure 1 below depicts the popular Spring-Mass-Damper system in which m is the mass, c represented by the dashpot symbol in the figure is called the damping factor, and k is the spring constant. The Spring-Mass-Damper system is with m = 20 kg, c = 20 Ns/m, k = 4000 Px k în с Li m Figure 1: The Spring-Mass-Damper system N/m. Moreover, denote by x(t) the displacement of the spring (from its equilibrium position). The system is acted on by a periodic harmonic force F(t) = Fo sin(wt) where Fo and w are the amplitude and frequency of the harmonic force, respectively. Given that Fo= 100 N and w = 20 rad/s. The Spring-Mass-Damper in Figure 1 is modelled by the following ODE: d²x dx m. +c- + kx = F(t). dt² dt F(t) dx (0) dt (1) Assume that x(0) = 0.01 m and = 0.0. Your duty as an engineer is to analyse this Spring- Mass-Damper system by fulfilling the following requirements: a) Establish the spring displacement trajectory x(t) by solving analytically the ODE equation (1) modelling the…