Plot the theta as a function of time. For system with damping for given cases: i) underdamping ii) critical damping iii) overdamping
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Plot the theta as a function of time. For system with damping for given cases:
i) underdamping
ii) critical damping
iii) overdamping
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- Let e be the counterclockwise angular displacement of the disk in the following system. a) Derive the governing differential equation of the system by using the Newton 2d law. b) Find the natural frequency of the system c) For what value of c is the damping ratio of the system equal to 1.15? fy-10 cm f30 cm m2 m1 Jputey10 kg-m m;-10 kg my-25 kg k-1x10' N/mThe equation of motion of a simplified model mechanical system is given below. of a m, o X₁ ki+k₂ k₂ O +1 NA 1/2 FU 10 ma -k₂ k₂ | k₂ = 10,000 N m₁ = 40 kg / m₂ = 130 kg, k₁ = 20,000~ m 0 a) Natural Freequency b) Find the mass normalized mode shapes of the system. c) Drave the mass normalized mode shapes.A mass spring system has a mass of 3/4 kg, a damping constant of 3/2 kg/sec and a spring constant of 15/2 kg/sec. There is no external force. The system is started in motion at y = 4 meters with an initial velocity of 3 m/s in the downward direction. a) Find the differential equation and the initial conditions that describe the motion of this system. b) Solve the resulting initial value problem. c) Is the spring system overdamped, underdamped or critically damped?
- The car's corner mass is m₁ = 220 kg and the unsprung mass is m₂ = 16 kg . The suspension spring stiffness is k = 13000 N/m and the tire stiffness is k, = 150000 N/m. a) Find the ride rate, RR, in N/m. b) Find the bounce natural frequency, f, in Hz for this corner of the car. c) Find the value of the suspension damping coefficient, c, that corresponds to a suspension damping ratio of 5 = 0.5. d) Find the damped natural frequency, fa, in Hz if the damping ratio is 5 = 0.5. m₂ k₂ m₁ III Jizz Z2 z, (t) = A, sin otQ1 The vibration of a car is modelled as a damped single degree-of-freedom system with the following parameters: m = 20 kg, K = 100 N/m, xo = 0.003 m, Vo = 5 m/s, = 0.01. Calculate the following a) Damped natural frequency b) Maximum vibration amplitude of the system c) Vibration phase of the systemA spring-mass-damper system consists of a mass m=120 slug, a spring constant k = 40 lb/in, and a damping coefficient c = 5lb*sec/in. Find the following: a.) Natural frequency b.) The damping ratio c.) Damped frequency. d.) Assume the initial conditions to be Uo = 0.2in, Vo = 0.5 in/sec, find the maximum deflection and plot the response, u(t).
- K Kt m₂ mi For this fig. > with damping Find the following: ~Equations of motion C ~Mass matrix ~Stiffness matrix ~Damping matrix ~Natural freqyancy ~All damping properties.. 72 ZIA) Assuming that no inputs are present to a first-order and a second-order systems, mention two fundamental differences between the two systems. B) Determine the damping ratio and natural frequency of the single-loop RLC electrical circuit system shown below. d²1 dI I + R + = dt² dt с L E(t). C) State two distinctions between odd and even functions, and name and sketch a function that's neither odd nor even. D) What is the frequency in Hertz and the amplitude of the 5th term (i.e. the 4th harmonics) of the following signal? 4 2π 10π == - sin TT 4 бп y(t) (² t) + 37 sin (17) 4 + sin 5π (10+) +. 10 101) Guess the form and solve for the particular solution of a driven oscillator described by the equation: mx" + cx' + kx = Fosin(wt) 2) A mass of 4 kg on a spring with k = 4 N/m and a damping constant c = 1Ns/m is driven by a force following the function Focos(at); where FO = 2 N. Find the driving frequency that causes practical resonance and find the amplitude. 3) Suppose there is no damping in a mass and spring system with m = 5, k = 20, and Fo= 5. Suppose is chosen to be precisely the resonance frequency. a) Find a b) Find the amplitude of the oscillations at time = 100, given that the system is at rest at / = 0.
- Given the vibrating system below: K1 K3 K4 Find the following 1. Keq 2. Ceq 3. Natural angular velocity K2 m C1 C3 C5 C2 C4 4. Damped angular velocity 5. Type of Damping 6. Equation of motion x(t). Assume Initial conditions for displacement and velocity Graph 2 cycles of the vibrating system. You can use third party app for this. 7. M = 10 kg K1=100 N/m K2= 80 N/m K3=75 N/m K4= 120 N/m C1 = 20Ns/m C2= 40 Ns/m C3= 35Ns/m C4= 15 Ns/m C5= 10 Ns/m1. Verify Eqs. 1 through 5. Figure 1: mass spring damper In class, we have studied mechanical systems of this type. Here, the main results of our in-class analysis are reviewed. The dynamic behavior of this system is deter- mined from the linear second-order ordinary differential equation: where (1) where r(t) is the displacement of the mass, m is the mass, b is the damping coefficient, and k is the spring stiffness. Equations like Eq. 1 are often written in the "standard form" ď²x dt2 r(t) = = tan-1 d²r dt2 m. M +25wn +wn²x = 0 (2) The variable wn is the natural frequency of the system and is the damping ratio. If the system is underdamped, i.e. < < 1, and it has initial conditions (0) = zot-o = 0, then the solution to Eq. 2 is given by: IO √1 x(1) T₁ = +b+kr = 0 dt 2π dr. dt ل لها -(wat sin (wat +) and is the damped natural frequency. In Figure 2, the normalized plot of the response of this system reveals some useful information. Note that the amount of time Ta between peaks is…03.: A vibrating system consists of a mass of 5 kg supported by an elastic element and a damping element from above and set vibrating. It is observed that the maximum displacement from the equilibrium position reduces to 5% of its initial value after 2 cycles. It takes 0.5 seconds to do them. Calculate the following :- a) The damping ratio č. b) The natural frequency o. c) The damped frequency w4. d) The spring stiffness k. e) The actual damping coefficient c. (20 Marks)