Respuesta :
Answer:
The mass = m = 2 kg
The spring constant = k = 4 N/m
The damping = c = 6 kg/s
Explanation:
u(t) = 4
y(t) = 1 - 2e⁻ᵗ + e⁻²ᵗ
my" + cy' + ky = u
y = 1 - 2e⁻ᵗ + e⁻²ᵗ
y' = 2e⁻ᵗ - 2e⁻²ᵗ
y" = -2e⁻ᵗ + 4e⁻²ᵗ
u = 4
Substituting all of these into the differential equation
my" + cy' + ky = u
m(-2e⁻ᵗ + 4e⁻²ᵗ) + c(2e⁻ᵗ - 2e⁻²ᵗ) + k(1 - 2e⁻ᵗ + e⁻²ᵗ) = 4
-2me⁻ᵗ + 4me⁻²ᵗ + 2ce⁻ᵗ - 2ce⁻²ᵗ + k - 2ke⁻ᵗ + ke⁻²ᵗ = 4
collecting like terms
(4m - 2c + k)e⁻²ᵗ + (-2m + 2c - 2k)e⁻ᵗ + k = 4
Comparing the RHS and the LHS
(4m - 2c + k)e⁻²ᵗ + (-2m + 2c - 2k)e⁻ᵗ + k = 0e⁻²ᵗ + 0e⁻ᵗ + 4
It is evident that;
4m - 2c + k = 0
-2m + 2c - 2k = 0
k = 4
Substituting the value of k into the first two expressions,
4m - 2c + k = 0
-2m + 2c - 2k = 0
4m - 2c + 4 = 0
-2m + 2c - 8 = 0
we obtain the simultaneous equation
4m - 2c = -4
-2m + 2c = 8
Solving the simultaneous equation,
m = 2 kg and c = 6 kg/s
Hence,
The mass = m = 2 kg
The spring constant = k = 4 N/m
The damping = c = 6 kg/s
Hope this Helps!!!
Explanation:
Given a mass spring damper system
u = my'' + cy' + ky
where,
y is the position of the mass
m is the mass of the attached object
k is the spring constant
c is the damping.
All initial condition are zero.
Given that,
Input: u(t) = 4
Position: y(t) = 1 − 2e−t + e−2t
m, c and k?
Let find y' i.e. the dy/dt
Also, y'' i.e d²y/dt²
y(t) = 1 − 2e−t + e−2t
y'(t) = 0 + 2e-t — 2e-2t
y'(t) = 2e-t — 2e-2t
Also,
y"(t) = —2e-t + 4e-2t
Since
u(t) = my'' + cy' + ky
u(t) = m(-2e-t + 4e-2t) +c(2e-t - 2e-2t) + k(1 − 2e−t + e−2t)
Collect likes terms
u(t) = (4m - 2c + k)e-2t + (-2m + 2c -2k)e-t + k
Given that u(t) = 4
Therefore,
4 = (4m - 2c + k)e-2t + (-2m + c -2k)e-t + k
Comparing coefficient
The constant
k = 4 N/m
The e-2t is 0
4m - 2c + k =0
4m - 2c + 4 = 0
4m - 2c = -4
Divide both side by 2
2m - c = -2. Equation 1
Also the e-t is also 0
-2m+c-2k = 0
-2m + 2c -8 = 0
-2m + 2c = 8. Equation 2
Add equation 1 and 2 together
2m-c-2m+2c = -2+8
c = 6 kg/s
From equation 2
-2m+2c = 8
-2m = 8-2c
-2m = 8 - 2×6
-2m = 8 -12
-2m = -4
m = -4/-2
m = 2kg
Then, mass of the object attached is m = 2kg
The damping constant is c = 6 kg/s
The spring constant is k = 4N/m
