Dynamics cheat sheet

close all;
zeta=linspace(0.1, 0.7, 10);
r=linspace(0, 3, 10);
D=@(r,z) (sqrt(1+(2*z*r).^2)./sqrt((1-r.^2).^2+(2*z*r).^2));
figure;
hold on;
for i=1:length(zeta)
plot(r,D(r,zeta(i)));
end
grid on;
legend
3.3 accelerometer
We need transfer function between u and z a where now z a is the amplitude of the ground acceleration.
This device is used to measure base acceleration by relating it linearly to relative displacement of m to base.
Equation of motion. We use relative distance now.
m ( u ′′ + z ′′) + cu ′ + ku = 0
mu ′′ + cu ′ + ku = −mz ′′
Let z ′′ = Re { Z a e iωt} . Notice we here jumped right away to the z ′′ itself and wrote it as Re { Z a e iωt} and we
did not go through the steps as above starting from base motion. This is because we want the transfer function
between relative motion u and acceleration of base.
Now, u = Re { Ue iωt} , u ′ = Re { iωUe iωt} , u ′′ = Re { −ω 2 Ue iωt} , hence the above becomes
Hence |D (r, ζ)| =
m Re { −ω 2 Ue iωt} + c Re { iωUe iωt} + k Re { Ue iωt} = −m Re { Z a e iωt}
−m
U =
−ω 2 m + iωc + k Z a
−1
=
−ω 2 + iω2ζω n + ωn
2 Z a
−1
=
(ωn 2 − ω 2 Z a
) + iω2ζω n
( )
−1
√(ω and arg (D) = −1800 − tan −1 2ωζωn
n−ω 2 2 ) 2 +(2ωζω n) 2 ωn−ω 2 2
When system is very stiff, which means ω n very large compared to ω , then D (r, ζ) ≈ −1
ω 2 n
Z a , hence by
measuring u we estimate Z a the amplitude of the ground acceleration since ω 2 n is known. For accuracy, need
ω n > 5ω at least.
3.4 seismometer
Now we need to measure the base motion (not base acceleration like above). But we still use the relative
displacement. Now the transfer function is between u and z where now z is the base motion amplitude.
Equation of motion. We use relative distance now.
m ( u ′′ + z ′′) + cu ′ + ku = 0
mu ′′ + cu ′ + ku = −mz ′′
Let z = Re { Ze iωt} , z ′ = Re { iωZe iωt} ,z ′′ = Re { −ω 2 Ze iωt} ,and let u = Re { Ue iωt} , u ′ = Re { iωUe iωt} , u ′′ =
Re { −ω 2 Ue iωt} , hence the above becomes
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