Question

Exercises 1. (introduction) Sketch or plot the displacement of the mass in a mass-spring system for at least two periods for the case when Wn-2rad/s, 괴,-1mm, and eto =-v/5mm/s. 2. (introduction) The approximation sin θ ะ θ is reasonable for θ < 10°. If a pendulum of length 0.5m, has an initial position of 0()0, what is the maximum value of the initial angular velocity that can be given to the pendulum without violating this smll angle approximation? 3. (harmonic motion) A foot pedal mechanism for a machine is crudely modeled as a pendulum connected to a spring as illustrated in the figure below. The purpose of the spring is to provide a return force for the pedal action. Compute the spring stiffness eeded to keep the pendulum at 1 from the horizontal (when no force is applied by the foot) and then compute the corresponding natural frequency. Assume that the the angular deflections are small, such that the spring deflection can be approximated by the are length ·the pedal may be treated as a point mass . the pendulum rod has negligible mass . the pedal is horizontal when the spring is at its free length. The values in the figure are m = 0.5kg, g-9.8m/s2, 4-0.2m, and 12-0.3m. 4. (harmonic motion) An automobile is modeled as a 100 kg mass supported by a spring of stiffnes k = 400, 000N/m. When it oscillates it does so with a maximum deflection of 0m When loaded with passengers, the mass increases to as much as 1300kg. Calculate the change in frequency, velocity amplitude, and acceleration amplitude if the maximum deflection remains 10cm. 5. (harmonie motion) A machine oscillates in simple harmonic motion and appears to be well modeled by an undamped 1 DOF oscillation. Its acceleration is measured to have an amplitude of 10,000mm/s2 at 8Hz. What is the machine's 6. (viscous damping) A spring-mass-damper system has mass of 100kg, stiffness of 3000N/m, and damping coeflicient of 300kg/s. Calculate the undamped natural frequency, the damping ratio, and the damped natural frequency Does the solution oscillate? 7. (viscous damping) A rough sketch of a valve-and-rocker-arm system for an internal combustion engine is give in the figure below. Model the system as a pendulum attached to a spring and a mass and assume the oil provides viscous daniping in the range of ( = 0.01 Deterinine the equations of motion and calculate an expressen for the natural frequency and the damped natural frequency. Here J is the rotational inertia of the rocker arm about its pivot point, k is the stiffness of the valve spring, and m is the mass of the valve and stem. Ignore the mass of the maximum displacement? pring Rocker arm Follower Valve spring Oil Cam Valve

Answer 1

In this solution some basic concepts and formulas of Vibration Mechanics are used. For more information, refer to any standard textbook or drop a comment below. Please give a Thumbs Up, if solution is helpful.

Solution :

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