Question 2

Question 1

explain why this is a valid and suitable approximation to the displacement of the beam

Answer 1

for pinned, displacement at support is 0
for clamped, displacement and slope is 0 at support
1 - substitute x=l and x=0 (pinned and clamped conditions) to equation and see if it is equal 0
2 - differentiate equation to show slope condition and substitute x=0

Question 2

show that the expression for mass is:

Answer 2

1 - use kinetic energy formula in data sheet
2 - differentiate original formula then square
3 - complete the integration
4 - given that 𝑇=1/2[𝑞̇1 𝑞̇2]𝑴[𝑞̇1/ 𝑞̇2] you can show that the mass matrix is the one given in the question

Question 3

show that the expression for stiffness is:

Answer 3

1 - use strain energy formula in data sheet
2 - differentiate original formula twice then square
3 - complete the integration
4 - given that U=1/2[𝑞̇1 𝑞̇2]K[𝑞̇1/ 𝑞̇2] you can show that the stiffness matrix is the one given in the question

Question 4

estimate the first 2 natural frequencies

Answer 4

1 - use det[−𝜔n^2𝑴+𝑲]=𝟎 and substitute M and K matrices
2 - set 𝛽 as the bhL/EI stuff
3 - solve for a quadratic in 𝛽 for 𝛽1 and 𝛽2
4 - find w1 and w2 using this

Question 5

effective mass matrix when additional mass

Answer 5

1 - additional mass changes kinetic energy 𝑇mr =1/2𝑚r(𝜕𝑤(𝐿/2,𝑡)/𝜕𝑡)^2
2 - solve for Tmr in terms of q’1 and q’2
3 - given that 𝑇=1/2[𝑞̇1 𝑞̇2]𝑴[𝑞̇1/ 𝑞̇2] you can find the mass matrix

Question 6

changing natural frequency find additional mass

Answer 6

1 - use 𝑑𝑒𝑡[−𝜔n^2(𝑴+𝑴r)+𝑲]=𝟎
2 - solve for Mr

Question 7

effective stiffness when additional spring

Answer 7

1 - additional spring changes strain energy 𝑈kr =1/2 𝑘r (𝑤(𝐿/2,𝑡)^2
2 - solve for Ukr in terms of q’1 and q’2
3 - given that U=1/2[𝑞̇1 𝑞̇2] K [𝑞̇1/ 𝑞̇2] you can find the mass matrix

Question 8

changing natural frequency find stiffness of the new spring

Answer 8

1 - use 𝑑𝑒𝑡[−𝜔n^2𝑴+(𝑲+Kr)]=𝟎
2 - solve for Kr

Question 9

change parameter b or h to