Numerical: A door along with door-closing system shown is shown in the figure below. It has a moment of inertia of 25 k g ⋅ m 2 25 \ kg \sdot m^2 25 k g ⋅ m 2 about the hinge axis. If the stiffness of torsional spring is 20 N m / r a d 20 \ N m/rad 20 N m / r a d , find the most suitable value of the damping coefficient.
Solution:
Given data:
Moment of inertia, I = 25 k g ⋅ m 2 \ I = 25 \ kg \sdot m^2 I = 25 k g ⋅ m 2
Stiffness of torsional spring = 20 N ⋅ m / r a d = 20 \ N \sdot m/rad = 20 N ⋅ m / r a d
C c = ( ? ) C_c = (?) C c = ( ?)
According to D'Alembert's principle;
Σ [ i n e r t i a t o r q u e + e x t e r n a l t o r q u e ] = 0 \Sigma \bigg[ \ inertia \ torque + \ external \ torque \bigg] = 0 Σ [ in er t ia t or q u e + e x t er na l t or q u e ] = 0
∴ I ⋅ θ ¨ + ( c c x ˙ ) ⋅ a + K t θ = 0 ∴ I ⋅ θ ¨ + c c a θ ˙ a + K t θ = 0 ( x = a θ a n d x ˙ = a θ ˙ ) ∴ I ⋅ θ ¨ + c c a 2 θ ˙ + K t θ = 0 \begin{aligned}
\therefore \ I \sdot \ddot \theta + (c_c \dot x) \sdot a + K_t \theta &= 0\\
\therefore \ I \sdot \ddot \theta + c_ca \dot \theta a + K_t \theta &= 0 \quad (x = a \theta \ and \ \dot x = a \dot \theta)\\
\therefore \ I \sdot \ddot \theta + c_ca^2 \dot \theta + K_t \theta &= 0
\end{aligned} ∴ I ⋅ θ ¨ + ( c c x ˙ ) ⋅ a + K t θ ∴ I ⋅ θ ¨ + c c a θ ˙ a + K t θ ∴ I ⋅ θ ¨ + c c a 2 θ ˙ + K t θ = 0 = 0 ( x = a θ an d x ˙ = a θ ˙ ) = 0
The above equation can be written as:
∴ I ⋅ θ ¨ + c t θ ¨ + K t θ = 0 where c t = c c a 2 \therefore \ I \sdot \ddot \theta \ + c_t \ddot \theta \ + K_t \theta = 0 \quad \quad \text {where} \ \ c_t = c_ca^2 ∴ I ⋅ θ ¨ + c t θ ¨ + K t θ = 0 where c t = c c a 2
Now,
c t = 2 ⋅ I ⋅ ω n ∴ c t = 2 ⋅ I K t I ∴ c c ⋅ a 2 = 2 K t I ∵ ( c t = c c a 2 ) ∴ c c = 2 a 2 K t I = 2 ( 0.05 ) 2 20 × 25 ∴ c c = 17888.54 N ⋅ s / m \begin{aligned}
c_t &= 2 \sdot I \sdot \omega_n\\
\therefore \ c_t &= 2 \sdot \ I \sqrt{K_t \over I}\\
\therefore \ c_c \sdot a^2 &= 2 \sqrt{K_tI} \quad \because (c_t = c_ca^2)\\
\therefore \ c_c &= {2 \over a^2} \sqrt{K_tI}\\
&= {2 \over (0.05)^2} \sqrt{20 \times 25}\\
\therefore \ c_c &= 17888.54 \ N \sdot s/m
\end{aligned} c t ∴ c t ∴ c c ⋅ a 2 ∴ c c ∴ c c = 2 ⋅ I ⋅ ω n = 2 ⋅ I I K t = 2 K t I ∵ ( c t = c c a 2 ) = a 2 2 K t I = ( 0.05 ) 2 2 20 × 25 = 17888.54 N ⋅ s / m
Comments:
All comments that you add will await moderation. We'll publish all comments that are topic related, and adhere to our Code of Conduct.
Want to tell us something privately? Contact Us
Post comment