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matlab代写-ME 163

时间：2021-03-05

Engineering Vibrations (W21)

ME 163

Homework #6

Due date: 03/04/2021

Problem 1 (10 points) Consider the two degrees of freedom mechanical system shown

in figure 1 such that k1 = k2 = k and m2 = 10m1 = 10m.

1. Derive the normal modes.

2. Find the complete solution of the system. Why is it difficult to calculate the con-

stants in the solution using the initial conditions? Explain.

3. Discuss the relationship between the frequencies and the normal modes ampli-

tudes. Provide a physical interpretation.

Figure 1: Two degrees of freedom System

1

Problem 2 (10 points) Consider the weakly coupled mechanical system shown in figure

2. Let k be the stiffness of the spring and m1 = m2 = m. Given that the initial conditions

are:

θ1(0) = 0

θ˙1(0) = A

θ2(0) = 0

θ˙2(0) = 0

1. Compute the complete solution of the system linearized around θ1 = θ2 = 0.

2. Given the numerical values in the following table, plot θ1(t) and θ2(t) on the same

figure for 0 < t < 100s. Give a physical interpretation of what is happening.

Parameter Numerical Value Unit

l 1 [m]

d 0.5 [m]

m 3 [Kg]

g 9.8 [m/s2]

A 5 [rad/s]

k 10 [N/m]

Figure 2: Weakly Coupled System

2

Problem 3 (10 Points)

1. Simulate the Nonlinear model of problem 2 with the same initial conditions. Plot

θ1(t) and θ2(t) on the same figure and compare with the results of problem 2.

2. Simulate the Nonlinear model of problem 2 with the following initial conditions.

θ1(0) = −pi/6

θ˙1(0) = pi

θ2(0) = pi/6

θ˙2(0) = pi

Plot θ1(t) and θ2(t) on the same figure and compare with part 1.

3

Problem 4 (Extra credit-10 points) Consider the two degrees of freedom mechanical

system shown in figure 3. A rod of mass m2, uniformly distributed along its length

2l, is hinged at its center of mass. One of its two ends is attached to spring 1 whose

stiffness constant is k1. Also, spring 1 is attached to a body with mass m1 which in turn

is attached, via spring 2 whose stiffness constant is k2, to a wall. The unstretched length

of the two springs is l/2. Clearly, this is a two degrees of freedom system having x (the

stretched length of spring 2) and θ (the angle between the rod and the vertical line) as

the variables.

1. Show that the equations of motion are given by

m1x¨ = f1(x, θ), f1(x, θ) = −(k1 + k2)x− k1l(1−

√

3− 2(cosθ − sinθ))

m2l2

3

θ¨ = f2(x, θ), f2(x, θ) = −k1l(l − x+ l√

3− 2(cosθ − sinθ) )(sinθ + cosθ)

Hint: Use the law of cosines and law of sines.

2. Linearize the two functions f1(x, θ) and f2(x, θ) around (x¯ = 0, θ¯ = 0).

3. Put the linearized equations of motion in matrix form and give the mass matrix

M and the stiffness matrix K. Now, let k1 = k2 = k and m1 = m2 = m. Find the

solution for the normal modes using the matrix method.

Note: This device is in the horizontal plane, so the force of gravity is not a factor in this

problem.

Figure 3: Two degrees of freedom System

4

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ME 163

Homework #6

Due date: 03/04/2021

Problem 1 (10 points) Consider the two degrees of freedom mechanical system shown

in figure 1 such that k1 = k2 = k and m2 = 10m1 = 10m.

1. Derive the normal modes.

2. Find the complete solution of the system. Why is it difficult to calculate the con-

stants in the solution using the initial conditions? Explain.

3. Discuss the relationship between the frequencies and the normal modes ampli-

tudes. Provide a physical interpretation.

Figure 1: Two degrees of freedom System

1

Problem 2 (10 points) Consider the weakly coupled mechanical system shown in figure

2. Let k be the stiffness of the spring and m1 = m2 = m. Given that the initial conditions

are:

θ1(0) = 0

θ˙1(0) = A

θ2(0) = 0

θ˙2(0) = 0

1. Compute the complete solution of the system linearized around θ1 = θ2 = 0.

2. Given the numerical values in the following table, plot θ1(t) and θ2(t) on the same

figure for 0 < t < 100s. Give a physical interpretation of what is happening.

Parameter Numerical Value Unit

l 1 [m]

d 0.5 [m]

m 3 [Kg]

g 9.8 [m/s2]

A 5 [rad/s]

k 10 [N/m]

Figure 2: Weakly Coupled System

2

Problem 3 (10 Points)

1. Simulate the Nonlinear model of problem 2 with the same initial conditions. Plot

θ1(t) and θ2(t) on the same figure and compare with the results of problem 2.

2. Simulate the Nonlinear model of problem 2 with the following initial conditions.

θ1(0) = −pi/6

θ˙1(0) = pi

θ2(0) = pi/6

θ˙2(0) = pi

Plot θ1(t) and θ2(t) on the same figure and compare with part 1.

3

Problem 4 (Extra credit-10 points) Consider the two degrees of freedom mechanical

system shown in figure 3. A rod of mass m2, uniformly distributed along its length

2l, is hinged at its center of mass. One of its two ends is attached to spring 1 whose

stiffness constant is k1. Also, spring 1 is attached to a body with mass m1 which in turn

is attached, via spring 2 whose stiffness constant is k2, to a wall. The unstretched length

of the two springs is l/2. Clearly, this is a two degrees of freedom system having x (the

stretched length of spring 2) and θ (the angle between the rod and the vertical line) as

the variables.

1. Show that the equations of motion are given by

m1x¨ = f1(x, θ), f1(x, θ) = −(k1 + k2)x− k1l(1−

√

3− 2(cosθ − sinθ))

m2l2

3

θ¨ = f2(x, θ), f2(x, θ) = −k1l(l − x+ l√

3− 2(cosθ − sinθ) )(sinθ + cosθ)

Hint: Use the law of cosines and law of sines.

2. Linearize the two functions f1(x, θ) and f2(x, θ) around (x¯ = 0, θ¯ = 0).

3. Put the linearized equations of motion in matrix form and give the mass matrix

M and the stiffness matrix K. Now, let k1 = k2 = k and m1 = m2 = m. Find the

solution for the normal modes using the matrix method.

Note: This device is in the horizontal plane, so the force of gravity is not a factor in this

problem.

Figure 3: Two degrees of freedom System

4

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