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Unformatted text preview: ME 563 - Fall 2004
Problem No. 1 – 20 points Name Use the influence coefficient method to determine the flexibility matrix for the threeDOF system shown below.
3k k k 2k x2 x1 2 M E 5 6 3 - F a ll 2 0 0 4
Problem No. 2 – 20 points Name A summer intern working under your direction has found the natural
frequencies and modal vectors for a three-DOF system where the mass matrix
[M] for the system is known to be: "1 0 0%
[ M ] = $0 4 0' kg
#0 0 1'
& ! ! In the process, he has chosen to “stiffness normalize” the modes such that
ˆ (i )T
ˆ (i )
" [K ]" = 1 (i = 1, 2, 3) where [K] is the (unknown) stiffness matrix. The results
of his work are: $#2(
ˆ (1) & & meter
" = % 1)
& & newton
ˆ ( 2) & & meter
" = %1)
& & newton
'* $ 1(
ˆ ( 3) & & meter
" = % 2)
& & newton
'* a) As his supervisor, you are checking over his work. As a result of this ! ˆ ( 3)
check, you have reason to believe that his results for the modal vector "
ˆ ( 3)
are incorrect. Explain why the modal vector " cannot be correct.
Support your explanation with calculations, if necessary.
b) Assuming that the modal vector " is correct, what is the natural
frequency corresponding to this mode?
! 3 ME 563 - Fall 2004
Problem No. 3 – 20 points ! ! Name Consider the vibrational system shown below where the generalized coordinates
(q1, q2, q3 ) = ( x1, x 2, x 3 ) are to be used to describe the motion of the system. All
springs are unstretched when x1 = x 2 = x 3 = 0 . The disk is homogeneous
( IG = mR 2 ) and rolls without slipping on blocks A and B.
a) Write down th! kinetic and potential energy expressions for the system.
b) Determine the elements M12 and K12 of the mass [M] and stiffness [K]
x2 ! A k no slip k m m k R m B m x1 k x3 4 C ME 563 - Fall 2004
Problem No. 4 – 20 points Name A taut string having a mass per length of ρ and tension T is stretched between
two supports separated by a distance of L. On the left end support, the string is
attached to a block of mass M that is free to slide on a smooth vertical guide. A
spring of stiffness is attached between this block and ground. On the right end
support, the string is attached to ground. Ignore the influence of gravity in your
analysis of this system.
a) Derive the boundary condition for the left end of the string. You must
present a correct free body diagram to receive full credit for this derivation.
b) Determine the characteristic equation that governs the natural frequencies
of this system. Use M / "L = 1 and kL / T = 2 . Your characteristic equation
should be in terms of functions of "L , where " = # $ / T . ! ! ! ! M T, ρ k x=0 x=L 5 ME 563 - Fall 2004
Problem No. 5 – 20 points Name A two-DOF system has the equations of motion of: "m 0 %( ˙˙1 , " k /2 k%( x1 , (0,
* * **
') - + $
') - = ) $ 0 2 m'* ˙˙2 * $/2 k 4 k '* x 2 * *0*
&+ x . #
&+ . + . ! where m and k have units of kg and N/m, respectively. This system is given a
set of initial conditions of x1 (0) = x 2 (0) = x1 (0) = 0 and x 2 (0) = v , where v has units
Find the response x1 ( t ) for t > 0.
! ! ! 6 ...
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This document was uploaded on 12/23/2011.
- Fall '09