G is given by

Dynamics · FE Reference Handbook section

Dynamics

60 formulas

Reference page

~60 min

All Dynamics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

Thus the undamped circular natural frequency for a system with a solid round supporting rod may be written as
Similar to the linear vibration problem, the undamped natural period may be written as
Figure Mass & Centroid Mass Moment of Inertia (Radius of Gyration)2
Figure Mass & Centroid Mass Moment of Inertia (Radius of Gyration)2

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

G is given by

Handbook notes for this section

Core formulas for this FE topic

kt=GJ/Lkt = GJ/Lkt=GJ/L

n=GJ/IL~n = GJ/IL n=GJ/IL

xn = 2r/~n = =Needs technical review

y M = ρLANeeds technical review

xc=L/2Ix=Ixc=0rx2=rx2c=0xc = L/2 I x = I xc = 0 rx_{2} = rx_{2}c = 0xc=L/2Ix=Ixc=0rx2​=rx2​c=0

yc=0yc = 0yc=0

zcIyc=Izc=ML212z c I yc = I zc = ML_{2} 12zcIyc=Izc=ML2​12

zc=0ry2c=rz2c=L212zc = 0 ry_{2}c = rz_{2}c = L_{2} 12zc=0ry2​c=rz2​c=L2​12

A=cross−sectionalareaofIy=Iz=ML23A = cross-sectional area of I y = I z = ML_{2} 3A=cross−sectionalareaofIy=Iz=ML2​3

Lxry2=rz2=L23L x ry_{2} = rz_{2} = L_{2} 3Lxry2​=rz2​=L2​3

ρ=mass/vol.\rho = mass/vol.ρ=mass/vol.

y M = ρs ANeeds technical review

xc=R=meanradiusIxc=Iyc=MR22rx2c=ry2c=R22xc = R = mean radius I xc = I yc = MR 2 2 rx_{2}c = ry_{2}c = R 2 2xc=R=meanradiusIxc=Iyc=MR22rx2​c=ry2​c=R22

yc=R=meanradiusIzc=MR2r22c=R2yc = R = mean radius I zc = MR 2 r22c = R 2yc=R=meanradiusIzc=MR2r22c=R2

Rzc=0R zc = 0Rzc=0

cA=cross−sectionalareaofIx=Iy=3MR22rx2=ry2=3R22c A = cross-sectional area of I x = I y = 3MR 2 2 rx_{2} = ry_{2} = 3R 2 2cA=cross−sectionalareaofIx=Iy=3MR22rx2​=ry2​=3R22

ringIz=3MR2rz2=3R2ring I z = 3MR 2 rz_{2} = 3R 2ringIz=3MR2rz2​=3R2

ρ=mass/area\rho = mass/areaρ=mass/area

R M = πR2ρhNeeds technical review

xc=0xc = 0xc=0

Ixc=Izc=M3R2+h212)(I xc = I zc = M 3R 2 + h 2 12 ) (Ixc=Izc=M3R2+h212)(

rx2c=rz2c=3R2+h212)rx_{2}c = rz_{2}c = 3R 2 + h 2 12)rx2​c=rz2​c=3R2+h212)

cyc=h/2Iyc=Iy=MR22ry2c=ry2=R22c yc = h/2 I yc = I y = MR 2 2 ry_{2}c = ry_{2} = R 2 2cyc=h/2Iyc=Iy=MR22ry2​c=ry2​=R22

hzc=0h zc = 0hzc=0

ρ=mass/vol.\rho = mass/vol.ρ=mass/vol.

Ix=Iz=M3R2+4h212)(I x = I z = M 3R 2 + 4h 2 12 ) (Ix=Iz=M3R2+4h212)(

rx2=rz2=3R2+4h212)rx_{2} = rz_{2} = 3R 2 + 4h 2 12 )rx2​=rz2​=3R2+4h212)

M=πR12−R22ρhIxc=Izc(M = \pi R12 - R22 \rho h I xc = I z c (M=πR12−R22ρhIxc=Izc(

rx2c=rz2c=3R12+3R22+h212)rx_{2}c = rz_{2}c = 3R12 + 3R22 + h 2 12 )rx2​c=rz2​c=3R12+3R22+h212)

R1xc=0(R_{1} xc = 0 (R1​xc=0(

=M3R12+3R22+h212)r=r=(R+R)2= M 3R12 + 3R22 + h 2 12 ) r = r = (R + R ) 2=M3R12+3R22+h212)r=r=(R+R)2

yc=h2(yc = h 2 (yc=h2(

Iyc=Iy=MR12+R222)ycy12I y c = I y = M R12 + R22 2 ) yc y 1 2Iyc=Iy=MR12+R222)ycy12

hzc=0Ix=Izrx2=rz2h zc = 0 Ix = Iz rx_{2} = rz_{2}hzc=0Ix=Izrx2​=rz2​

ρ=massvol.(\rho = mass vol. (ρ=massvol.(

=M3R12+3R22+4h212)(= M 3R12 + 3R22 + 4h 2 12 ) (=M3R12+3R22+4h212)(

=3R12+3R22+4h212)= 3R12 + 3R22 + 4h 2 12 )=3R12+3R22+4h212)

M = πR ρNeeds technical review

3Ixc=Ix=2MR25rx2c=rx2=2R253 I xc = I x = 2 MR 2 5 rx_{2}c = rx_{2} = 2 R 2 53Ixc=Ix=2MR25rx2​c=rx2​=2R25

Rcxc=0R c xc = 0Rcxc=0

Iyc=Iy=2MR25ry2c=ry2=2R25I yc = I y = 2 MR 2 5 ry_{2}c = ry_{2} = 2 R 2 5Iyc=Iy=2MR25ry2​c=ry2​=2R25

yc=0yc = 0yc=0

zxzc=0Izc=Iz=2MR25rz2c=r2z=2R25z x zc = 0 I zc = I z = 2 MR 2 5 rz_{2}c =r 2z = 2 R 2 5zxzc=0Izc=Iz=2MR25rz2​c=r2z=2R25

ρ=mass/vol.\rho = mass/vol.ρ=mass/vol.

Ix′x′=Iy′y′=80Ix'x' = Iy'y' = 80Ix′x′=Iy′y′=80

M=3rRhtM = 3 r R htM=3rRht

rxx2=ryyr xx_{2} = r yyrxx2​=ryy

=34R2+h2i= 3 _4R 2 + h 2 i=34​R2+h2i

V=13πR2hxc=yc=0380V = 13 \pi R 2 h xc = yc = 0 3 80V=13πR2hxc=yc=0380

Izz=10MR2I zz = 10 MR 2Izz=10MR2

c y' h r zz2 =Needs technical review

hzc=4110h zc = 4 1 10hzc=4110

RIyy=Ixx=20M3R2+2h2iR Iyy= Ixx = 20 M _3 R 2+ 2h 2iRIyy=Ixx=20M3​R2+2h2i

4t=mass/vol.4 t = mass/vol.4t=mass/vol.

M=rR2tsM = rR_{2}tsM=rR2​ts

Ixx=Iyy=4MRIxx = Iyy = 4 MRIxx=Iyy=4MR

rxx2=ryyr xx_{2} = r yyrxx2​=ryy

=1R2= 1 R_{2}=1R2​

Rxc=yc=zc=01R xc = yc = zc = 0 1Rxc=yc=zc=01

Izz=2MRI zz = 2 MRIzz=2MR