The maximum permissible recoil distance of a gun is specified as 0.5 m. If the initial recoil velocity is to be between 8 m/s and 10 m/s, find the mass of the gun and the spring stiffness of the recoil mechanism. Assume that a critically damped dashpot is used in the recoil mechanism and the mass of the gun has to be at least 500 kg

A mass-spring system of mass m = 0.2 kg and elastic constant k = 5 N/m is immersed in a viscous fluid which exerts a damping force on the system. Furthermore, the system is subjected to a harmonic forcing of intensity F_0 = 10 N and frequency w rad/s. The value of the damping constant is b = 1.5 Ns/m. If the system starts at rest and in its equilibrium position, find: a) the frequency w for which the system is in resonance; b) Make a graph of position v/s time where the amplitude and period of the system can be seen explicitly if the frequency w is equal to twice the resonance frequency; c) Make a graph of amplitude v/s frequency where the maximum frequency and the resonance frequency of the system can be seen explicitly.

You are to investigate the vibration properties of the shock absorber system of a 2000 kg car. The shock absorbers drop 10 cm when the car is lowered from the lift and stands on its wheels. In addition, the amplitude decreases by 50% with each oscillation. Assume that the weight of the car is distributed evenly over all wheels and use the above information to determine the values ​​for the spring constant k and the damping coefficient b. What error does the calculation deliver if the natural frequency of the shock absorbers ω is approximated by that of the undamped system ω_0?

A 5.0 kg weight is hanging from an ideal spring with a k of 250N/m. The block is pulled down .075m from equilibrium.   Will the new period for the motion of the block be greater than, less than, or equal to the period for the motion of the block if it were pulled down .15m?  Why?   The first block is brought to rest at equilibrium again.  An identical block is launched up into the first block. The new block is moving with an initial speed of v=5.0m/s when it collides with and sticks to the original block. Calculate the maximum compression of the spring after the collision of the two blocks.

Consider the double mass/double spring system shown below. - click to expand. Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of F friction We can write the resulting system of second-order DEs as a first-order system, w'(t) = Aw(t), with w = (x₁, x₁, x₂, x₂) T For values of k = 4, m = 1 and c = 21,2 = 0.5±3.2i, V₁ 23,4= - 0.5 ± 1.13i, V3 1, the resulting eigenvalues and eigenvectors of A are -0.039 0.248i 0.813 0.024 +0.153i -0.502 -0.134-0.302i' 0.409 -0.216 - 0.489i 0.661 and (a) Find a set of initial displacements x₁(0), x₂(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero. (x₁(0), x₂(0)) Enter your answer using angle braces, (and ). cx (friction proportional to velocity). (b) At what frequency will the masses be oscillating in this mode? Frequency = rad/s

An undamped free vibration system consisting of mass m = 5 kg and stiffnessa spring of 400 N/m, is given an initial displacement of 0.02 m and is kicked with an initial velocity of 0.1 m/s.Determine the equation of motion for the system.

A four-cylinder automobile engine must be supported on three shock mounts,as shown in the picture. The engine block assembly weighs500 lbs. If the unbalanced force produced by the engine is given by200 sin 100 t lb, design three shock mounts (each stiffness k andviscous damping constant c) such that the vibration amplitudemaximum 0.1 in

A mass weighing 4 pounds is attached to a spring whose constant is 2 Ib/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equilibrium position with a downward velocity of 14 ft/s. Determine the time (in s) at which the mass passes through the equilibrium position. (Use g = 32 ft/s? for the acceleration due to gravity.) Find the time (in s) after the mass passes through the equilibrium position at which the mass attains its extreme displacement from the equilibrium position. What is the position (in ft) of the mass at this instant? ft

Consider a mass-and -spring system containing two masses m1=1 and m2=1 whose displacement functions x(t) and y(t) satisfy the differential equations : x'' = -10x + 2y y'' = 3x -15y Assume that the two masses  start in motion  with the initial  conditions x(0) =32, x'(0)=6, and y(0)=36, y'(0)=3 and are acted on by the same force F1(t)=F2(t)=-1320cos(7t). Describe the resulting motion as a superposition of oscillations at the three different frequencies.  The three frequencies consist of lower, higher, and the forcing frequency.  For each one please compare the magnitude of m2 to m1.

A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at t = 0, an external force equal to f(t) = 2sin(4t) is applied to the system. Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity. Hint: Use g = 32 ft/s2 for the acceleration due to gravity.