θ θ θ

Midterm Examination ISE 2142-212 Dynamics and Vibrations 2^nd semester, 2007 International School of Engineering Chulalongkorn University NAV and PPT

Name... ID...CR No……….

1

Problem 1 (10 pts, 45 min)

The bar is in static equilibrium in the upright position at which the springs exert no force. Force ^{F t}

( )

is applied at the top so the bar rotates with small angle θ clockwise.

(a) Determine the deflection of the left and right spring (b) Linearize the answers in (a) using the Taylor series

( ) ( ) ( ) ( )

^...

e

e e

x x

f x f x d f x x x

dx ₌

= + − +

(c) Use the results in (b) to find the equation of motion for small angle motion of the bar. Arrange the equation in standard form; Mθ+Cθ+Kθ =G.

θ

Midterm Examination ISE 2142-212 Dynamics and Vibrations 2^nd semester, 2007 International School of Engineering Chulalongkorn University NAV and PPT

Name... ID...CR No……….

3

Problem 2 (10 pts, 45 min)

A gun barrel weighing 5300 N has a recoil spring of stiffness 290 kN/m. If the barrel recoils 1.22 m (to a complete stop) on firing, then get hooked to a damper as shown in the figure. Determine

a) the initial recoil velocity of the barrel,

b) the critical damping coefficient of a dashpot that is engaged at the end of the recoil stroke,

c) the time (measured after firing) required for the barrel to return to a position 50 mm from its initial position.

W = 5300 N k = 290 kN/m

x x = 1.22 m

Neglect friction force W = 5300 N k = 290 kN/m

x x = 1.22 m

Neglect friction force

Midterm Examination ISE 2142-212 Dynamics and Vibrations 2^nd semester, 2007 International School of Engineering Chulalongkorn University NAV and PPT

Name... ID...CR No……….

5

Problem 3 (10 pts, 45 min)

Consider the typical mass-spring-damper system subjected to the harmonic input F t

( )

=F0cosωt. The equation of motion is well-known as mx+cx+kx=F₀cosωt.

(a) Derive the frequency response function. It will be used in part (b).

(b) If the steady-state response of the system to a harmonic force 10sinωt N is observed to lag the excitation by 90° when the frequency is 50 Hz. At 48 Hz, the response is observed to be

3cos 4sin

x= ωt− ωt mm. Determine the stiffness, mass, and damping ratio of this system.

Midterm Examination ISE 2142-212 Dynamics and Vibrations 2^nd semester, 2007 International School of Engineering Chulalongkorn University NAV and PPT

Name... ID...CR No……….

7

Problem 4 (10 pts, 45 min)

An accelerometer is used to measure the oscillation of an airplane wing caused by the plane’s engine operating at 6000 rpm. The magnitude of actual acceleration (measured from another device) of the wing is 1g (g = 9.81 m/s²). However, the accelerometer reads 10 m/s². If the accelerometer has a 0.01- kg moving mass and a damped natural frequency of 100 Hz,

a) Write down EOM in terms of w(t) and y(t) coordinates, where w(t)= x(t)−y(t) b) Determine the ratio

Y

nW

2 2

ω

ω in terms of ζ and r

c) Find the damping constant c and the stiffness k.

Hints: The actual acceleration is y(t) and the measured acceleration from the accelerometer is w(t).