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Problem 3) A uniform rod of length L is pinned at both ends. Show that the...
Problem 3) A uniform rod of length Lis pinned at both ends. Show that the frequencies...
Problem 3) A uniform rod of length Lis pinned at both ends. Show that the frequencies of longitudinal vibrations are conic/L, where c= is the velocity of longitudinal waves in the rod, and n = 0,1,2,3,4....... Note: You must show all the steps. b) Plot the first three natural modes.
A uniform rod of length L is pinned at both ends. Show that the frequencies of...
A uniform rod of length L is pinned at both ends. Show that the frequencies of longitudinal vibrations are n = nc/L, where c=√?? is the velocity of longitudinal waves in the rod, and n = 0,1,2,3,4……. Note: You must show all the steps. b) Plot the first three natural modes.
(10 pts) The mode shape of a uniform rod of length L fixed at both ends in axial vibration is 1. ...
(10 pts) The mode shape of a uniform rod of length L fixed at both ends in axial vibration is 1. e,(x)-a,sin/m) Find the value of the coefficient a, such that the mode is orthonormal. Assume the mass per unit length of the rod is m
(10 pts) The mode shape of a uniform rod of length L fixed at both ends in axial vibration is 1. e,(x)-a,sin/m) Find the value of the coefficient a, such that the mode is...
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...
A 24 lb uniform slender rod AB of length L= 36 in is fixed to a...
A 24 lb uniform slender rod AB of length L= 36 in is fixed to a
12 lb uniform disk of radius, R= 12 in , and both are pinned at B
as shown. A belt is attached to the rim of the disk and then to a
spring with spring constant, k= 40 lb/in which holds the rod/disk
at rest in the equilibrium position shown.
natural period is 0.446 s
If end A is displaced a distance of 0.9...
Problem 4 [8 pts] A long pipe, length L, is closed at both ends, and filled...
Problem 4 [8 pts] A long pipe, length L, is closed at both ends, and filled with a gas with speed of sound v. The pipe is excited in some fashion in order to produce standing waves. (a) Sketch the standing wave pattern for the four lowest frequencies supported by this pipe. Label the nodes and antinodes. (b) Make a table of the wavelengths and frequencies of the sound waves that are formed by these four excitations, in terms of...
A uniform rod with a mass m and length L has one end attached to a...
A uniform rod with a mass m and length L has one end attached to a pivot. The rod swings around on a frictionless horizontal table with angular speed wo. A ball with mass m (so same mass as the rod) is placed on the table a distance d from the pivot. The ball is made of clay so when the rod strikes it, the ball sticks to the rod (i.e., inelastic collision). If the final (post-collision) angular velocity of...
Solve the heat equation ut = 10uct for a rod of length 1 with both ends...
Solve the heat equation ut = 10uct for a rod of length 1 with both ends insulated for all time (zero Neumann boundary conditions), if the initial temperature is given by (2) = x+sin ax. First, formulate the mathematical problem and complete the three steps as described. Mathematical Formulation Step 1: Derive an expression for all nontrivial product (separated) solutions including an eigenvalue problem satisfying the boundary conditions Step 2: Solve the eigenvalue problem Step 3: Use the superposition principle...
A thin metal rod of length L is rotating counterclockwise with an angular velocity omega on...
A thin metal rod of length L is rotating counterclockwise with an angular velocity omega on the plane of the paper about an axis through one of its ends, as shown in the Figure. A uniform magnetic field B_0 points into the plane in the region where the rod is rotating. Calculate the magnitude and direction of the magnetic force F_B that acts on the rod's charge carriers, resulting in the accumulation of opposite charges at the two ends of...
Problem 3: The system in Figure 3 consists of a double pendulum where both masses are...
Problem 3: The system in Figure 3 consists of a double pendulum where both masses are m and both lengths are L 2 Figure 3: System for Problem 3 (a) Derive the differential equations of motion for the system. The angles a(t) and θ2(t) can be arbitrarily large. (b) Linearize the equations by assuming that a (t) and 02(t) are small. Write the linearized differential equations in matrix form (c) Obtain the natural frequencies and modes of vibration. (d) Plot...
Problem 3) A uniform rod of length Lis pinned at both ends. Show that the frequencies of longitudinal vibrations are conic/L, where c= is the velocity of longitudinal waves in the rod, and n = 0,1,2,3,4....... Note: You must show all the steps. b) Plot the first three natural modes.
(10 pts) The mode shape of a uniform rod of length L fixed at both ends in axial vibration is 1. e,(x)-a,sin/m) Find the value of the coefficient a, such that the mode is orthonormal. Assume the mass per unit length of the rod is m
(10 pts) The mode shape of a uniform rod of length L fixed at both ends in axial vibration is 1. e,(x)-a,sin/m) Find the value of the coefficient a, such that the mode is...
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...
A 24 lb uniform slender rod AB of length L= 36 in is fixed to a
12 lb uniform disk of radius, R= 12 in , and both are pinned at B
as shown. A belt is attached to the rim of the disk and then to a
spring with spring constant, k= 40 lb/in which holds the rod/disk
at rest in the equilibrium position shown.
natural period is 0.446 s
If end A is displaced a distance of 0.9...
Problem 4 [8 pts] A long pipe, length L, is closed at both ends, and filled with a gas with speed of sound v. The pipe is excited in some fashion in order to produce standing waves. (a) Sketch the standing wave pattern for the four lowest frequencies supported by this pipe. Label the nodes and antinodes. (b) Make a table of the wavelengths and frequencies of the sound waves that are formed by these four excitations, in terms of...
A uniform rod with a mass m and length L has one end attached to a pivot. The rod swings around on a frictionless horizontal table with angular speed wo. A ball with mass m (so same mass as the rod) is placed on the table a distance d from the pivot. The ball is made of clay so when the rod strikes it, the ball sticks to the rod (i.e., inelastic collision). If the final (post-collision) angular velocity of...
Solve the heat equation ut = 10uct for a rod of length 1 with both ends insulated for all time (zero Neumann boundary conditions), if the initial temperature is given by (2) = x+sin ax. First, formulate the mathematical problem and complete the three steps as described. Mathematical Formulation Step 1: Derive an expression for all nontrivial product (separated) solutions including an eigenvalue problem satisfying the boundary conditions Step 2: Solve the eigenvalue problem Step 3: Use the superposition principle...
A thin metal rod of length L is rotating counterclockwise with an angular velocity omega on the plane of the paper about an axis through one of its ends, as shown in the Figure. A uniform magnetic field B_0 points into the plane in the region where the rod is rotating. Calculate the magnitude and direction of the magnetic force F_B that acts on the rod's charge carriers, resulting in the accumulation of opposite charges at the two ends of...
Problem 3: The system in Figure 3 consists of a double pendulum where both masses are m and both lengths are L 2 Figure 3: System for Problem 3 (a) Derive the differential equations of motion for the system. The angles a(t) and θ2(t) can be arbitrarily large. (b) Linearize the equations by assuming that a (t) and 02(t) are small. Write the linearized differential equations in matrix form (c) Obtain the natural frequencies and modes of vibration. (d) Plot...
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