Homework 2

Mecjanical Vibration

A force of 20 newton stretches a spring 1 meter. A 5 kg mass is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 10 times the instantaneous velocity. (a) Let x denote the downward displacement of the mass from its equilibrium position. [Note that x>0 when the mass is below the equilibrium position. ] Assume the mass is initially released from rest from a point 3 meters above the equilibrium position. Write the differential equation and the initial conditions for the function x(t)(b) Solve the initial value problem that you wrote above.(c) Find the exact time at which the mass passes through the equilibrium position for the first time heading downward. (Do not approximate.)(d) Find the exact time at which the mass reaches the lowest position. The "lowest position" means the largest value of x

For small positive damping, c, in the sinusoidally forced mass-spring system ma" + ca' + ka = Ggcos (wt) there is a value of the forcing frequency w for which the amplitude of the response is greatest - a resonance peak. very light darmping moderate damping very heavy darmping forcing frequency, w Derive a formula for the location of the peak in terms of the parameters in the differential equation (m, c, k; Go). So the location of the peak depends on the damping . Where does the peak move to as the damping constant c is reduced to zero? Does the above frequency have a name? amplitude of response

a mass-spring-damper system has a mass of 100 kg. In 60 s, its free response apmplitude decays such that the amplitude of the 30th cycle is 20% of the amplitude of the 1st cycle. Estimate the damping constant c and the spring constant k.

Question 1: A) Plot the forced-undamped vibration response of a spring-mass system given by Eq. (3.13), page 265 of the Book, for the following sets of data: Set 1: 8st Set 2: 8st Set 3: 8st = 0.17 m, w = 6.1,wn = 7.6 r/s, x₁ = 0.11 m, x = 0.7 m/s = 0.17 m, w = = 0.17 m, w = 7.7,wn = 7.6½, x₁ = 0.11 m, x = 0.7 m/s Set 4: 8st = 0.17 m, w = 7.6 r/s, wn = 7.6 r/s, xo = 0.11 m, x₁ = 0.7 m/s 7.3 r/s, wn = 7.6 r/s, xo = 0.11 m, xo = 0.7 m/s x(t) = xo cos wnt + xo @n sin wnt + st B) Analyze and discuss the results. COS wt 1 cos @nt W Wn (3.13)

Vibrations

Two mass-spring systems are experiencing damped harmonic motion, both at 0.7 cycles per second and both with an initial maximum displacement of 12 cm. The first has a damping constant of 0.6, and the second has a damping constant of 0.2. a. Find functions of the form to model the motion in each case. b. How do they differ? (Hint: graph the two functions)

A spring-mass-damper system is at the equilibrium position. The mass m = 10 kg, the damping coefficient c = 18 N.s/m and stiffness k = 9000 N/m. The initial velocity is 5 m/s. (a) Is the system over, under or critically damped. Why?(b) Find the damped period of the system.(c) Find the position of the mass as a function of time.(d) Calculate the amplitude of acceleration of the mass. Please solve (d)

Find the free-vibration response of a spring-mass system subject to Coulomb damping for the following initial conditions x(0) = 0.5 m, *(0) = 0. Data: m= 10kg, k = 200 N/m, µ = 0.5. Q1 . ...

Many experts got me a lot of answers for this question !!!! , can I get the perfect correct answer ?

umang

Please I want solution of these questions urgently. Big thumb for you. Thanks

Show this complete solution for this force driven without damping equation:

Consider an underdamped mass spring system with mass m = 1 kg spring constant k = 8 N/m, and damping coefficient, stretched past equilibrium by 1 m and released with an initial velocity of 0 m/s. N.S a = 4 m www. Equilibrium Position (a) Write the solution of the initial value problem. x(t) = (b) Identify the pseudo-frequency. Find the circular frequency if damping is removed. m N s m (c) Determine the first time in seconds in which the mass returns to equilibrium position after being set in motion. (Round your answer to two decimal places.)

Mechanical Vibration Lesson A 65 kg industrial sewing machine has an imbalance value of 0.15 kgm in rotation. The machine operates at 125 Hz and is located on the ground with a spring coefficient of 2 x 106 N / m and a dimensionless damping coefficient of 0.12. Find the uniform regime amplitude of the machine.

Task 1 (a) The position of a mass in a spring-mass system is given as x(t) = 2cos4t. Find the period of oscillations and initial conditions. (b) Consider a vertical spring mass-damper-system with mass m = 2 kg, damping coefficient c = 5 N.s/m and stiffness k = 10 N/m. An external force f(t) = sin3t N is applied to the mass. (i) Find the static equilibrium position as measured from the un-stretched length of the spring (ii) Write the equation of motion. (iii) Find the natural and damped frequencies. (iv) In steady state, find the time delay in seconds between the mass position and At). (v) Find the steady state maximum acceleration.

A vehicle weighing 5,000 lb is supported on four identical springs and four identical viscous dampers. The static deflection of the vehicle under its own weight is 10 in. Determine the damping required for each of the dampers so that the critical damping is achieved.

Given an oscillator of mass 2.0kg and spring constant of 180N/m, what is the period without damping?  Use numerical methods to model this oscillator with an additional friction force equal to where c is a positive damping constant.  Using c=5.0, what is the new period of oscillation.  What about for c=10? Assume initial position is 0.2m and initial velocity is zero. Please find the period using the position versus time plot and use the first full cycle of the motion.

Mohammed Mahad Tabook A lathe can be modeled as an electric motor mounted on a steel table. The table plus rotor have a mass of 100 kg. The rotating parts of the lathe have a mass of 10 kg at a distance 0.1 m from the center. The damping ratio of the system is measured to be zeta= 0.06 (viscous damping) and its natural frequency is 7.5 Hz and the driving frequency is 30HZ The natural frequency in rad/s is Choose.. The driving frequency in rad/s is Choose... The frequency ratio is Choose... The amplitude of the steady-state displacement of the motor (in mm) Choose... the phase shift of the steady state displacement (in rad) is Choose... +

A spring with a 8-kg mass and a damping constant 2 can be held stretched 0.5 meters beyond its natural length by a force of 1.5 newtons. Suppose the spring is stretched 1 meters beyond its natural length and then released with zero velocity. In the notation of the text, what is the value c2 – 4mk? m'kg/sec? help (numbers) Find the position of the mass, in meters, after t seconds. Your answer should be a function of the variable t with the general form cieat cos(Bt) + cze"s t sin(&t) help (numbers) a = B = * help (numbers) * help (numbers) 8 = * help (numbers) C1 = help (numbers) C2 = 2 help (numbers) kies help us deliver our services. By using our services, you agree to our use of cookies. OK Learn more CONNECT OFF/ON CAPS Charge Po- 14 F4 F5 1- 1+ F6 F7 F8 F9 F10 F11 F12 7 8 R T Y U 5

Please send me answer of this question immediately and i will give you like sure sir