NA spring-mass system has a spring constant of 3 A mass of 2 kgmis attached to the spring, and the motion takes place in a viscousfluid that offers a resistance numerically equal to the magnitude ofthe instantaneous velocity. If the system is driven by an externalforce of 15 cos(3t) — 10 sin(3t) N, determine the steady-state responsein the form R cos(wt - 8).R =W=S||.=

Given Spring constant (k)= 3 Nm mass (m)= 2 kg External force (F) =15cos3t-10sin3t

Answered: N A spring-mass system has a spring…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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An object stretches a spring 6 inches in equilibrium. a) Find its displacement for t > 0 if it is initially displaced 4 inches below equilibrium and given an upward velocity of 8 ft/s. y(t) = = b) State the natural frequency of the oscillation. Wo = rad/s c) State the period of the oscillation. T S = d) State the amplitude of the oscillation. R ft = e) State the phase angle of the oscillation. The phase angle should be between - T and π radians. = Notes: 1. Use g 32 ft/s². = rad 2. Assume there is no damping in the system and displacement and velocity are positive upward.
A mass of 1 kg is attached to a spring and a damper, set in motion, and observed to move as x(t) = 3 e-2t sin(3t). Determine the spring constant k (in kg/s²) and the damping constant c (in kg/s), where c= = 2 mwo S.
8 A mass weighing 16 pounds stretches a spring feet. The mass is initially released from rest from a point 2 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to the instantaneous velocity. Find the equation of motion x(t) if the mass is driven by an external force equal to f(t) = 20 cos(3t). (Use g = 32 ft/s² for the acceleration due to gravity.) 2 (√4) x(1) - e -(5) [ - 14 cos (1) ( (VT),) - 134 sin(e) ((7),)] + 20 cos (3) +- 20 sin (38) = е 3 2 3√4t 2 3 3 ft
A force of 15 Newtons stretches a spring 20 centimeters. After a 3 kilogram mass is attached to the spring, the mass is pulled down to a position √3 meters below equilibrium and released with an upward velocity of 5 meters per second. Find a. The equation of motion. b. The amplitude, period, and phase shift of the simple harmonic motion.
2. A mass that weighs 64 lb stretches a spring 0.8 ft. The mass is attached to a dashpot mechanism that has a damping coefficient of 8 lb s/ft and is acted on by an external force of 60 cos(4t) lb. The mass is set in motion from its equilibrium position with no initial velocity. Let y(t) denote the displacement (in ft) of the mass from its equilibrium position after t seconds. (a) Find an initial-value problem which describes this process. (You may use the fact that g = 32 ft/s' and 1 ft=12 in.) (b) Is the system underdamped, overdamped, or critically damped? Justify your an- swer.
*) Find y as a function of x if y" 11y" + 18y' = 32e*, y(0) = 29, y'(0) = 12, y"(0) = 22. y(x) =
a) A mass, m = 200N is attached to the free end of a vertical spring as shown in Figure Q3(a), where the damping coefficient, c 10N.s/m and the spring constant, k = 20N/m. When displaced from the equilibrium position, the mass is subjected to free vibration response which can be described by the pendulum motion: d2x m at? dx + kx = 0 dt where x is the horizontal displacement from the equilibrium position and t is time. The displacement (x) as a function of time t is given below, where the initial velocity y(t=0) is zero. + x(t) x (m) 0.005 0.005 0.005 -0.0575 -0.08875 t (s) 10 15 20 Figure Q3(a) i) Assume y = dt' dx reduce the system to first order differential equation. ii) Determine the first order differential equation over the time period of Os t s 20s seconds using the Euler's method. (CO2,PO2)(C3)
A mass of 6 kg is attached to the end of a spring with spring constant k = 18 N/m. It is set into motion with initial position xo = 2 m and initial velocity vo = -3 m/s. Find the position function x(t) of the mass, as well as the amplitude C and circular frequency wo of its motion.
A-kg mass is attached to a spring with stiffness 16 N/m. The damping constant for the system 8 is 2 N-sec/m. If the mass is moved m to the left of equilibrium and given an initial leftward 15 56 15 m/sec, determine the equation of motion of the mass and give its damping factor, quasiperiod, and quasifrequency. velocity of
Using linear approximation (or differentials) I estimate cos 30.5° Select one: O a. 0.861662 O b. 0.861629 О с. V3 1 O d. 1 e. 0.616025 O f. 720

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