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Chapter 2: Problem 33

A screen door of mass \(m\), height \(L\) and width \(\ell\) is attached to a door frame as indicated. A torsional spring of stiffness \(k_{T}\) is attached as a closer at the top of the door as indicated, and a damper is to be installed near the bottom of the door to keep the door from slamming. Determine the limiting value of the damping coefficient so that the door closes gently, if the damper is to be attached a distance a from the hinge.

Short Answer

Expert verified
Answer: The limiting value of the damping coefficient for the door to close gently is given by the expression: \(c = \frac{(k_{T} + mg\frac{L}{2}) \theta}{a}\), where \(k_{T}\) is the torsional spring constant, \(m\) is the mass of the door, \(g\) is the gravitational acceleration, \(L\) is the length of the door, \(\theta\) is the angular displacement, and \(a\) is the distance of the linear damper from the hinge.

Step by step solution

01

Define the forces acting on the door

There are three major forces acting on the door: 1. Torsional spring force, which is proportional to the angular displacement of the door and is given by \(k_{T} \theta\). 2. Damping force, which is proportional to the angular velocity of the door and the chosen damping coefficient. This force is given by \(c \dot{\theta}\). 3. Gravitational force, which acts on the center of mass of the door. The gravitational torque due to this force will be given by \(mg\frac{L}{2} \sin{\theta}\).
02

Set up the equation of motion for the door

Using Newton's second law for rotational motion, we can write the equation of motion as: $$I \ddot{\theta} = - k_{T} \theta - c \dot{\theta} - mg\frac{L}{2} \sin{\theta}$$ Here, \(I\) is the moment of inertia of the door about the hinge, which is given by \(I = \frac{1}{3}mL^2\).
03

Determine the equilibrium condition for gentle closing

For the door to close gently, it should not slam shutand there should be no rapid angular acceleration just before reaching the closed position (\(\theta = 0\)). So, the forces should balance in such a way that: $$- k_{T} \theta - c \dot{\theta} - mg\frac{L}{2} \sin{\theta} = 0$$ When the door is near the closed position, \(\sin{\theta} \approx \theta\) and \(\dot{\theta}\) is small. Hence, we can approximate the equation as: $$- k_{T} \theta - c \dot{\theta} - mg\frac{L}{2} \theta = 0$$
04

Express the damping force in terms of the linear damper

The damping force acting on the door is provided by the linear damper attached a distance \(a\) from the hinge. The damping torque due to this force is given by: $$F_d = c \dot{\theta}a$$
05

Solve for the limiting damping coefficient

Using the expressions from Steps 3 and 4 and setting \(F_d = c \dot{\theta}a\), we can solve for the limiting damping coefficient as: $$- k_{T} \theta - F_d - mg\frac{L}{2} \theta = 0$$ $$F_d = (k_{T} + mg\frac{L}{2}) \theta$$ Now substituting the expression for \(F_d\) based on the linear damper, we have: $$c \dot{\theta}a = (k_{T} + mg\frac{L}{2}) \theta$$ As we are looking for the limiting value of \(c\), we can replace \(\dot{\theta}\) with 0, since the door is closing gently and moving slowly towards the closed position. This gives us: $$c = \frac{(k_{T} + mg\frac{L}{2}) \theta}{a}$$ This expression gives us the limiting value of the damping coefficient \(c\) for the door to close gently.

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