Question

(cantilever beam)

In most practical applications the dimensions do not scale down by the same amount. Using the formula’s given in class/lectures derive the overall scaling factor for the mass and deflection if the width, thickness, and length is scaled down by 10x, 100x, and 5x respectively.

Answer 1

1.scaling factor for mass.

we know that, mass=volumedensity

density will be constant as it is intensive property

let original width ,thickness,length is given by W,T,L, respectively

original volome=WTL=WTL

original mass =WTL .........(=densitywhich is constant)

new width,thicknwss,length is given by   ,, respectively

new volume =

new mass =  

mass is scaled down by factor 5000x .................ans.

2.scaling factor for deflection.

as we know that deflection at free end of cantelever beam is given by

and here two cases arise

(i) when line of action of force is parellel to y-axis then

(I is second moment of area)

(ii) when line of action of force is parellel to z-axis then

for case (i)

=

here deflection is scaled up by 80000x

for case (ii)

here deflection is scaled by 800x