Question

engineering structural resilience
a tuned mass damper (tmd) is a device mounted in structures to reduce the maximum extent of a vibration. in some ways, a tmd is like a shock absorber in a car—it absorbs mechanical kinetic energy. as wind, earthquakes, and other events perturb the building, the resulting vibrations can be intense, causing the occupants to become ill and even threatening the structural integrity of the building. the john hancock tower needed to be retrofitted with a tmd for this reason.
engineers analyze the number of real solutions of the equation $mx^{2}+bx + k = 0$ to determine whether a tmd is underdamped (0 real solutions), critically damped (1 real solution), or overdamped (2 real solutions).
the number of solutions is determined by the three parameters, $k$, a spring constant ($kg/s^{2}$), $b$, a constant that describes the damping force ($kg/s$), and $m$, a mass ($kg$).
a. graph the equation $y = mx^{2}+bx + k$ with a variety of values for the parameters, and categorize different combinations as underdamped, critically damped, or overdamped.
b. describe what it means to be underdamped, critically damped, and overdamped.
c. write an algebraic condition that ensures a given combination of parameters will define a critically damped system.
d. research the three parameters and summarize your findings by describing the relationship between them. use your research as evidence when explaining your reasoning.

Explanation:

Step1: Recall the discriminant formula

For a quadratic equation \(ax^{2}+bx + c=0\) (in our case \(a = m\), \(b=b\), \(c = k\)), the discriminant \(\Delta=b^{2}-4ac=b^{2}-4mk\).

Step2: Analyze under - damped condition

When \(\Delta=b^{2}-4mk<0\), the quadratic equation \(mx^{2}+bx + k = 0\) has 0 real solutions. This corresponds to an under - damped system. Physically, in an under - damped TMD, the system oscillates with a gradually decreasing amplitude over time.

Step3: Analyze critically - damped condition

When \(\Delta=b^{2}-4mk = 0\), the quadratic equation \(mx^{2}+bx + k=0\) has 1 real solution. This is a critically damped system. In a critically damped TMD, the system returns to equilibrium as quickly as possible without oscillating.

Step4: Analyze over - damped condition

When \(\Delta=b^{2}-4mk>0\), the quadratic equation \(mx^{2}+bx + k = 0\) has 2 real solutions. In an over - damped TMD, the system returns to equilibrium without oscillating, but more slowly than in the critically damped case.

A.

To graph \(y=mx^{2}+bx + k\), we can use graphing software or a graphing calculator. For different values of \(m\), \(b\), and \(k\):

• If \(b^{2}-4mk<0\), the parabola \(y = mx^{2}+bx + k\) does not cross the \(x\) - axis (under - damped).
• If \(b^{2}-4mk = 0\), the parabola touches the \(x\) - axis at exactly one point (critically damped).
• If \(b^{2}-4mk>0\), the parabola crosses the \(x\) - axis at two points (over - damped).

B.

• Underdamped: The system oscillates with a decaying amplitude. The damping force is not strong enough to prevent oscillations, but it gradually reduces their magnitude over time.
• Critically damped: The system returns to its equilibrium position as quickly as possible without oscillating. It is on the boundary between oscillatory and non - oscillatory behavior.
• Overdamped: The system returns to equilibrium without oscillating, but more slowly than the critically damped case. The damping force is so strong that it prevents any oscillations from occurring.

C.

The algebraic condition for a critically damped system is \(b^{2}-4mk = 0\). This is because for the quadratic equation \(mx^{2}+bx + k = 0\), when the discriminant \(\Delta=b^{2}-4mk\) is zero, there is exactly one real root.

D.

The spring constant \(k\) determines the stiffness of the spring in the TMD. A higher \(k\) value means a stiffer spring. The damping coefficient \(b\) represents the strength of the damping force. A larger \(b\) implies a stronger damping effect. The mass \(m\) is the mass of the TMD. The relationship \(b^{2}-4mk\) determines the nature of the system's response. If \(b\) is relatively small compared to \(m\) and \(k\), the system is likely to be under - damped. If \(b\) is large enough such that \(b^{2}-4mk=0\), it is critically damped, and if \(b\) is even larger, it is over - damped.

Answer:

A. Graphs of \(y = mx^{2}+bx + k\) with different \(m\), \(b\), \(k\) show non - crossing (under - damped), single - crossing (critically damped), and double - crossing (over - damped) of \(x\) - axis.
B. Underdamped: Oscillates with decaying amplitude; Critically damped: Returns to equilibrium quickly without oscillation; Overdamped: Returns to equilibrium without oscillation, slower than critically damped.
C. \(b^{2}-4mk = 0\)
D. \(k\) is spring stiffness, \(b\) is damping strength, \(m\) is mass. \(b^{2}-4mk\) determines system response type.