Question

The free damped motion of a mass on a spring at time \(t\) is governed by the equation $$m \ddot{y}+c \dot{y}+k y=0,$$ where the coefficients are constants. The dot, as usual, denotes differentiation with respect to time. The roots of the characteristic equation are $$\lambda_{1,2}=\frac{-c \pm \sqrt{c^{2}-4 m k}}{2 m}$$ Describe the behavior of the solution in the three different cases of \(c^{2}-4 m k\) positive, negative or zero.

Short answer

Answer: The possible behaviors of the mass-spring system depend on the value of the discriminant of the characteristic equation. When the discriminant is positive, the motion is overdamped and the mass returns to equilibrium without overshooting and does not oscillate. When the discriminant is zero, the motion is critically damped, resulting in the fastest possible return to equilibrium without overshooting and no oscillation. When the discriminant is negative, the motion is underdamped and the mass returns to equilibrium with oscillations that decay exponentially.

Step-by-step solution

Identify the characteristic equation

The given motion equation is \(m\ddot{y} + c\dot{y} + ky = 0\). We can write the characteristic equation associated with this equation as: $$m\lambda^2 + c\lambda + k = 0.$$

Analyze the discriminant of the characteristic equation

The discriminant of the given quadratic characteristic equation is: $$D = c^2 - 4mk.$$ We will describe the behavior of the solution based on the three cases of the discriminant, \(D\).

Case 1: \(D > 0\) (discriminant is positive)

For the discriminant being positive, the roots of the characteristic equation are real and distinct: $$\lambda_{1,2}=\frac{-c \pm \sqrt{c^{2}-4 m k}}{2 m}$$ In this case, the general solution for the motion equation is given by: $$y(t) = A e^{\lambda_1 t} + B e^{\lambda_2 t},$$ where \(A\) and \(B\) are constants determined by the initial conditions. Since both \(\lambda_1\) and \(\lambda_2\) are negative, the motion is described as an overdamped system. The mass returns to equilibrium without overshooting and does not oscillate.

Case 2: \(D = 0\) (discriminant is zero)

For the discriminant being zero, the roots of the characteristic equation are real and equal: $$\lambda_{1,2}= -\frac{c}{2m}$$ In this case, the general solution for the motion equation is given by: $$y(t) = (A + Bt) e^{\lambda_1 t},$$ where \(A\) and \(B\) are constants determined by the initial conditions. The motion is described as a critically damped system. This is the fastest possible return to equilibrium without overshooting. There is no oscillation.

Case 3: \(D < 0\) (discriminant is negative)

For the discriminant being negative, the roots of the characteristic equation are complex conjugates: $$\lambda_{1,2} = -\frac{c}{2m} \pm \frac{\sqrt{-c^2+4mk}}{2m}i$$ In this case, the general solution for the motion equation is given by: $$y(t) = e^{-\frac{c}{2m}t}[A \cos(\omega_d t) + B \sin(\omega_d t)],$$ where \(A\) and \(B\) are constants determined by the initial conditions, and the damped angular frequency \(\omega_d\) is given by: $$\omega_d = \frac{\sqrt{-c^2+4mk}}{2m}$$ The motion is described as an underdamped system. The mass returns to equilibrium with oscillations that decay exponentially. In conclusion, the behavior of the mass-spring system depends on the value of the discriminant (\(D\)) of the characteristic equation. When the discriminant is positive, the motion is overdamped; when it is zero, the motion is critically damped; and when it is negative, the motion is underdamped.

Key concepts

Characteristic Equation

In the realm of damped harmonic oscillators, the characteristic equation is a vital tool in understanding the system's behavior. It's a quadratic equation derived from the motion equation
\[m \ddot{y} + c \dot{y} + k y = 0,\]
where the constants \(m\), \(c\), and \(k\) are mass, damping coefficient, and spring constant, respectively. The characteristic equation for this system is:
\[m \lambda^2 + c \lambda + k = 0.\]
Solving this equation for \(\lambda\) gives us the roots, \(\lambda_{1,2}\), which are crucial in determining how the system behaves over time. These roots tell us if the system's response is overdamped, critically damped, or underdamped, which are all distinct types of damping responses.

Discriminant Analysis

Discriminant analysis for a damped harmonic oscillator involves studying the expression under the square root in the quadratic formula. This expression is known as the discriminant. For our characteristic equation, the discriminant is given by:
\[D = c^2 - 4mk.\]
By analyzing \(D\), we can predict the nature of the roots:

• If \(D > 0\), the roots are real and distinct.
• If \(D = 0\), the roots are real and equal.
• If \(D < 0\), the roots are complex conjugates.

Each of these cases corresponds to a different damping behavior in the system. Thus, discriminant analysis is a cornerstone concept that frames our understanding of system responses in damped oscillators.

Overdamped System

An overdamped system occurs when the discriminant \(D\) is positive (\(D > 0\)). This implies the roots of the characteristic equation are real and distinct. Denoting these roots by \(\lambda_1\) and \(\lambda_2\), the general solution of the motion is:
\[y(t) = A e^{\lambda_1 t} + B e^{\lambda_2 t},\]
where \(A\) and \(B\) are determined by the initial conditions. In this solution, both \(\lambda_1\) and \(\lambda_2\) are negative, resulting in a response that smoothly decays to equilibrium over time without oscillations.
The overdamped response means that the system returns to rest position more sluggishly compared to less damped systems, but it remains stable without bouncing back past the equilibrium state.

Critically Damped System

A critically damped system arises when the discriminant \(D\) equals zero (\(D = 0\)). This condition signifies that the roots of the characteristic equation are real and equal, expressed as:
\[\lambda_{1,2} = -\frac{c}{2m}.\]
The general solution for this scenario is:
\[y(t) = (A + Bt) e^{\lambda_1 t},\]
where \(A\) and \(B\) are constants based on initial conditions. Critically damped systems achieve the fastest return to equilibrium without overshooting.
It is important in applications where speed is crucial, yet overshooting would be problematic, such as in automotive suspensions or door dampers designed to close without slamming.

Underdamped System

In an underdamped system, the discriminant \(D\) is negative (\(D < 0\)), leading to complex conjugate roots for the characteristic equation. Represent these roots as:
\[\lambda_{1,2} = -\frac{c}{2m} \pm i \frac{\sqrt{-D}}{2m}.\]
The general solution for the system is:
\[y(t) = e^{-\frac{c}{2m} t} \left[ A \cos(\omega_d t) + B \sin(\omega_d t) \right],\]
where \(\omega_d = \frac{\sqrt{-D}}{2m}\) represents the damped angular frequency. This indicates that the system exhibits oscillations as it returns to equilibrium, with the amplitude of oscillations decreasing exponentially over time due to damping.
Underdamped responses are common in systems where a certain degree of oscillation is acceptable or desirable, such as in musical instruments or some engineering structures.