Q2) a vibratory system with viscous damping: Mass = 2.5 kg; spring constant k = 3 N/mm and the amplitude X decreases to 0.25 of the initial value after five consecutive cycles. Determine the damping coefficient of the damper in the system. k spring ww + b damper m mass x(t) Time

A force of F = 50 N is applied to the rope that causes the angle 0₁ = 60 degrees to keep the system at equilibrium. The N spring constant is k = 100 m B a 0₁ с a b с Variable Value 2 m 2 m 2 m F b Values for the figure are given in the following table. Note the figure may not be to scale. cc i❀O BY NC SA 2013 Michael Swanbom

5B Material point of mass m moves under the influence of force F-kr=-krî With in other words, the mass m is at the tip of an isotropic harmonic oscillator with equilibrium position at the origin of the axes. c) To qualitatively study the movement of the mass m for all its permissible values of total energy of E and L 0. d) To qualitatively study the movement of the mass m for all its permissible values of total energy of E and L = 0.

Q1. Consider a 3-meter long pendulum placed on the moon and given an initial angular displacement of 0.4 rad and zero initial velocity. Note that gravity on the earth's moon is g,m g is the acceleration due to gravity on earth. g/6, where m mg (A) Write down the equation of motion for the pendulum (B) Calculate the amplitude of the angular displacement of the swinging pendulum (C) Calculate the maximum angular velocity of the swinging pendulum (D) Calculate the maximum angular acceleration of the swinging pendulum

A mass-spring damper system has the following parameters: • mass 250g damping constant 1/36000 Nh/m • spring stiffness 500 N/m Calculate the damped frequency. O 1.55.2 rad/s O 2.25.8 rad/s O 3. Not possible to calculate. O 4.79.4 rad/s O 5.44.7 rad/s

QUESTION 3 Vibration absorber (also called tuned mass damper) is used to reduce* the vibration at the natural frequency of the original system. Consider a M-K system shown below. Original System x(t) Modified System x₁(1) X2(t) K K MA M 0000 M F(1) F(t) absorber K This system has a natural frequency at rad. The absorber mass and the spring are attached to the original system as shown in the figure, and the values are selected so that (1) the amplitude of the original mass, M. at the original natural frequency is zero, and (2) the new natural frequencies are moved away from the original natural frequency. The selection of absorber mass and stiffness, m and k, is easy. They have the same ratio as the ratio of the original system. That is, M/m = Klk. Find the transfer functions X1(s)/F(s) and X2(s)/F(s) before you can answer questions. What is the fundamental frequency (first natural frequency) of the modified system in rad/s? Use scientific notation with 3 signifincant figures. Omit…

1- Two mass-spring-damper systems A & B. System A has = 0.1 and @n = 20 rad/sec, and system B has < = 0.3 and wn = 5 rad/sec. Which system will vibrate longer in time? Why?

2. Consider a car suspension, modeled as a mass/spring/damper system (mass m, stiffness k, damping b). Suppose the height of the chassis is lo at rest, the height of the terrain below the driver varies as h(t), and the height of the chassis is denoted lo + y(t). (i.e., spring deflection away from rest is y(t) – h(t)). 2 (a) Give the transfer function G(s) = H(s) · = (b) Suppose the ground follows an oscillatory profile h(t) A cos(wx (t)) with magnitude A (in meters) and frequency w (measured in radians per meter). Suppose the car is traveling at a constant forward speed v. Using a frequency response analysis strategy, give the amplitude of oscillations experienced by the driver at steady state as a function of m, k, b, A, w, and v. Hint: You can't simply consider |G(iw)| to get the amplification in this case. (c) Suppose the ground varies by A = 5cm, w = 2 rad/m, and you are driving at v = 15 m/s. Using your answer to part (b), what amplitude of oscillation is felt by the driver when m…

A vibratory system consists of a spring of stiffness 5 N/mm and a mass of 10 kg. Mass of spring is 3 kg. The natural frequency of the system in rad/sec is

P3). A heavy block of m = 40.0 kg is connected to a spring. The natural frequency of the blockspring system is 0.30 Hz. The system is forced to vibrate sinusoidally. The amplitude of the driving force and its frequency is 5N and vibrating system is Q= 1. 0.190 Hz , respectively. The Quality Factor of this а) Determine the spring constant k of the spring. b) Determine the block's amplitude of motion for the above condition (i.e., at f= 0.190 Hz) c) The driving force suddenly becomes zero. Determine the time it takes for the vibration amplitude to decrease to 1/2 its initial value.