Assigment 1

Assignment 1 (To be submitted; on A4 size sheets) Q1: A mass of 0.5 kg hangs on a spring. The stiffness of the spring is 10 0N/m, and the mechanical resistance is 1.4 kg/s. The force (N) driving the system is f=2cos 5t. (a) What will be steady state values of the displacement amplitude, speed amplitude, and average power dissipation? (b) What is the phase angle between speed and force? (c) What is resonance frequency and what would be the displacement amplitude, speed amplitude, and average power dissipation at this frequency and for the same force magnitude as in (a)? (d) What is the Q of the the system, and over what range of frequencies will the power loss be at least 50% of its resonance value? Q2. Figure shows a two DOF freedom undamped model of a vibrating structure. The physical  properties are: m1=m2=9.8 Kg; k1=k2= 8820 N/m and kc= 3430 N/m.

a) Find the two natural frequencies and the corresponding mode shapes. Sketch the mode shapes. Find the motion of the two masses as a function of time in each of the following cases: if b) both the masses are displaced to the right by 5mm and then released. c) both the masses are displaced  by 5mm; m1 to the right and m2 to the left and then released d) the mass m1 is displaced by 5mm to the right and the mass m2 is held fixed fixed and then the two masses released simultaneously. simultaneously. e) Write your observations on the results in b), c) and d). Q3: A stretched string of length L, fixed at both the ends, is plucked at a position L/3 by producing an initial displacement h and then releasing the string. Take L=1m, h=0.01m, density=1000 kg/m3, diameter of string wire=1mm; tension T=10N. Determine the resulting amplitudes for the fundamental and the first three harmonic overtones. Sketch the wave shapes of these individual waves and the shape of the string resulting from the linear combination of these waves at t=0. Repeat for t=L/c, where c is the transverse wave speed on the string. Q4: Evaluate the mechanical impedance seen by the applied force driving an infinite string at a distance L from a fixed end. Interpret the individual terms in the mechanical impedance.

Problems for practice (Not to be submitted) Q5: The inertial switch that activates an airbag in an automobile can be modeled as a spring of stiffness s stiffness s with one end attached to a case fixed to the vehicle and the other end attached to a mass m free to move within the case. When the case is decelerated, the mass compresses the spring to activate a switch that releases the air into airbag. The case is decelerated at a constant rate a. (a) Find the equation for the motion of mass. (b) By direct substitution, show that the motion of mass 



2

 

is  =  2 + (

)(co )(coss 0  + 1), where 02 =

 

. (c) Find the minimum deceleration required

to compress the spring a distance X , and express the answer in terms of force necessary to statically compress the spring the same distance X. Q6: Evaluate the mechanical impedance seen by the applied force driving a simple harmonic oscillator with an infinite string extending transversely from the mass. Q7: A string of density 0.01 kg/m is stretched with a tension of 5 N from a rigid support at one end to a device producing transverse periodic vibrations at the other end. The length of the string is 0.44 m, and it is observed that, when the driving frequency has a given value, the nodes are spaced 0.1 m apart and the maximum amplitude is 0.02 m. What are the (a) frequency and (b) amplitude of the driving force? Q8: A string of length L, tension T, and linear density ρL is fixed at one end and free at the other. Find the resonance frequencies if the string is driven in the middle.

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