Math Problem Statement
A forcing function Fdrive(t)=-40sin(t) N is applied to a mass of 1 kg in a mass on a spring problem in which the spring constant is 5 N/m and there is damping proportional to velocity with coefficient 4 N*s/m. If the initial mass position is -5 m and the initial velocity is 2 m/s, find the unique solution and identify the steady state and transient terms.
Solution
To solve this problem, let's model it with a second-order linear differential equation with forcing, damping, and spring constants involved. The equation of motion is:
where:
- (mass),
- (damping coefficient),
- (spring constant),
- (forcing function).
Given initial conditions:
- ,
- .
Step 1: Set Up the Differential Equation
The equation of motion becomes:
Step 2: Solve for the Homogeneous Solution (Transient Term)
The homogeneous equation is:
Assume a solution of the form . Plugging into the homogeneous equation gives:
Solving this quadratic equation:
Thus, the homogeneous solution is:
This represents the transient solution, which will decay over time due to the factor.
Step 3: Solve for the Particular Solution (Steady-State Term)
Since the driving force is , we look for a particular solution of the form:
Differentiating:
Substitute , , and into the original equation:
Grouping terms involving and :
This gives the system:
Solving these equations, we find:
Thus, the particular solution is:
This represents the steady-state solution, which oscillates with the same frequency as the driving force.
Step 4: Write the General Solution
The general solution is the sum of the homogeneous and particular solutions:
Step 5: Apply Initial Conditions
Using and :
-
Initial Position:
-
Initial Velocity:
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Forced Oscillations
Damping
Harmonic Motion
Formulas
m * d²x/dt² + c * dx/dt + k * x = F_drive(t)
Characteristic equation: λ² + 4λ + 5 = 0
Particular solution form for sinusoidal forcing: x_p(t) = C * sin(t) + D * cos(t)
Theorems
Second-Order Linear Differential Equation Solution
Method of Undetermined Coefficients
Suitable Grade Level
Undergraduate - Physics/Engineering
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