Homework Answers
Rotate it to the left by 90 degrees, we get a classic case of Transverse Vibrations in a string
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Problem 2: Hanging cable A cable of uniform mass per unit length p(x)-ρ constant, hangs freely fr...
A uniform horizontal beam OA, of length a and weight w per unit length, is clamped horizontally at O and freely supporte...
A uniform horizontal beam OA, of length a and
weight w per unit length, is clamped horizontally
at O and freely supported at A. The transverse
displacement y of the beam is governed by the
differential equation
where x is the distance along the beam measured
from O, R is the reaction at A, and E and I are
physical constants. At O the boundary conditions
are y(0) = 0 and . Solve the
differential equation. What is the boundary...
Problem 1: Axial vibrations of a rod The rod of length L is fixed at ends x = 0 and x = L. The de...
Problem 1: Axial vibrations of a rod The rod of length L is fixed at ends x = 0 and x = L. The density of the rod is ρ(x), stiffness k(x) being subjected to a force f(x, t). Let's derive the equations for axial vibrations of a rod using almped model. We express the rod niy mol 41 in as a chain of masses m,mm, connected to each other through springs as shown in the figure. Let's say each...
I need help with this problem, please and thank you! A cable of length ?and density...
I need help with this problem, please and thank you!
A cable of length ?and density ρ[mass/length] is stretched under
a tension τ. One end of the cable is connected to a mass ?, which
can move in a friction-less slot, and the other end is fastened to
two springs of stiffness ?/2, as shown in the figure. Write down
its governing equation,boundary conditions and then calculate its
natural frequency.
cele ni
1. A uniform horizontal beam OA, of length a and weight w per unit length, is...
1. A uniform horizontal beam OA, of length a and weight w per unit length, is clamped horizontally at O and freely supported at A. The transverse displacement y of the beam is governed by the differential equation d2y El dx2 w(a x)- R(a - x) where x is the distance along the beam measured from O, R is the reaction at A, and E and I are physical constants. At O the boundary conditions are dy (0) = 0....
Problem 4 (20%) Figure 5 shows a uniform elastic bar fixed at one end and attached to a mass M at the other end. The cr...
Problem 4 (20%) Figure 5 shows a uniform elastic bar fixed at one end and attached to a mass M at the other end. The cross sectional area for the bar is A, mass density per unit length p, modulus of elasticity E and second moment of area I. For the longitudinal vibration: S Set the necessary coordinate system, governing equation of motion and boundary conditions a. b. Derive the general solution. Explain how you can obtain the natural frequencies...
2. What is the function y(x) that describes the shape of a cable of length L and mass per unit le...
2. What is the function y(x) that describes the shape of a cable of length L and mass per unit length stretched between the points (0, 0) and (D, 0)? This is the famous catenary which is the shape of the cable that supports a suspension bridge. (a) What is the functional that we want to minimize? (b) What is the constraint that y(x) must obey? (c) What new functional must be minimized to solve this problem? (d) Find the...
Problem 2. The ball with mass m is attached to two elastic cords each of length...
Problem 2. The ball with mass m is attached to two elastic cords each of length L. The ball is constrained to move on a horizontal, frictionless plain. The cords are stretched to a tension T When t 0, your intrepid instructor gives the ball a very small horizontal displacement x (a) Derive the equation of motion and find expressions for the natural circular frequency, the frequency, and the period of vibration. (b) For m - 2 kg, L 3...
A uniform flexible chain of length L, with mass per unit length A, passes over a small (radius is negligible), frictionless pulley, as shown in figure 1. One side of the chain with length x is tied to a block with mass m. The chain and block are released
A uniform flexible chain of length L, with mass per unit length
A, passes over a small (radius is negligible), frictionless pulley,
as shown in figure 1. One side of the chain with length x is tied to
a block with mass m. The chain and block are released from a
rest position at t = 0 s. (Hint: the chain can be treated as a rope
with non-negligible mass)a. Find x such that the chain and the block are...
Elementary Differential Equation Unit Step Function Problem Project 2 A Spring-Mass Event Problenm A mass of...
Elementary Differential Equation Unit Step Function Problem
Project 2 A Spring-Mass Event Problenm A mass of magnitude m is confined to one-dimensional motion between two springs on a frictionless horizontal surface, as shown in Figure 4.P.3. The mass, which is unattached to either spring, undergoes simple inertial motion whenever the distance from the origin of the center point of the mass, x, satisfies lxl < L. When x 2 L, the mass is in contact with the spring on the...
I need help with problem #3, please and thank you! Problem #2 (25 points) - The...
I need help with problem #3, please and thank you!
Problem #2 (25 points) - The True Hanging String Shape After solving for ye(2) for the scenario in Problem #1, show that the mag- nitude of the tension in the string is given by the expression T(X) = To cosh (Como) where To = Tmin is the minimum tension magnitude in the string which occurs at the bottom point of the string, and then show that the maximum tension magnitude...
A uniform horizontal beam OA, of length a and
weight w per unit length, is clamped horizontally
at O and freely supported at A. The transverse
displacement y of the beam is governed by the
differential equation
where x is the distance along the beam measured
from O, R is the reaction at A, and E and I are
physical constants. At O the boundary conditions
are y(0) = 0 and . Solve the
differential equation. What is the boundary...
Problem 1: Axial vibrations of a rod The rod of length L is fixed at ends x = 0 and x = L. The density of the rod is ρ(x), stiffness k(x) being subjected to a force f(x, t). Let's derive the equations for axial vibrations of a rod using almped model. We express the rod niy mol 41 in as a chain of masses m,mm, connected to each other through springs as shown in the figure. Let's say each...
I need help with this problem, please and thank you!
A cable of length ?and density ρ[mass/length] is stretched under
a tension τ. One end of the cable is connected to a mass ?, which
can move in a friction-less slot, and the other end is fastened to
two springs of stiffness ?/2, as shown in the figure. Write down
its governing equation,boundary conditions and then calculate its
natural frequency.
cele ni
1. A uniform horizontal beam OA, of length a and weight w per unit length, is clamped horizontally at O and freely supported at A. The transverse displacement y of the beam is governed by the differential equation d2y El dx2 w(a x)- R(a - x) where x is the distance along the beam measured from O, R is the reaction at A, and E and I are physical constants. At O the boundary conditions are dy (0) = 0....
Problem 4 (20%) Figure 5 shows a uniform elastic bar fixed at one end and attached to a mass M at the other end. The cross sectional area for the bar is A, mass density per unit length p, modulus of elasticity E and second moment of area I. For the longitudinal vibration: S Set the necessary coordinate system, governing equation of motion and boundary conditions a. b. Derive the general solution. Explain how you can obtain the natural frequencies...
2. What is the function y(x) that describes the shape of a cable of length L and mass per unit length stretched between the points (0, 0) and (D, 0)? This is the famous catenary which is the shape of the cable that supports a suspension bridge. (a) What is the functional that we want to minimize? (b) What is the constraint that y(x) must obey? (c) What new functional must be minimized to solve this problem? (d) Find the...
Problem 2. The ball with mass m is attached to two elastic cords each of length L. The ball is constrained to move on a horizontal, frictionless plain. The cords are stretched to a tension T When t 0, your intrepid instructor gives the ball a very small horizontal displacement x (a) Derive the equation of motion and find expressions for the natural circular frequency, the frequency, and the period of vibration. (b) For m - 2 kg, L 3...
Elementary Differential Equation Unit Step Function Problem
Project 2 A Spring-Mass Event Problenm A mass of magnitude m is confined to one-dimensional motion between two springs on a frictionless horizontal surface, as shown in Figure 4.P.3. The mass, which is unattached to either spring, undergoes simple inertial motion whenever the distance from the origin of the center point of the mass, x, satisfies lxl < L. When x 2 L, the mass is in contact with the spring on the...
I need help with problem #3, please and thank you!
Problem #2 (25 points) - The True Hanging String Shape After solving for ye(2) for the scenario in Problem #1, show that the mag- nitude of the tension in the string is given by the expression T(X) = To cosh (Como) where To = Tmin is the minimum tension magnitude in the string which occurs at the bottom point of the string, and then show that the maximum tension magnitude...
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