Structural Dynamicsassignment

Bruce Lyu, 28 September 2017

Problem Set #1

Problem 1-1

Question: The weight of the building of Fig.E2-1 is and the building is set

into free vibration by releasing it from a displacement of . If the maximum

displacement on the return swing is at time , determine

(a) the lateral spring stiffness (b) the dampling ration (c) the damping coefficient

Solution:

The equation of motion for a single degree of freedom system without driven force is

given by

Divide each side by , we have

Denote

The solution is in the form of

The equation of motion becomes

1

Solve the quadratic equation we have

Because in civil engineering, is always much smaller than , we can take

As a result, the complex solution is

And the real solution is given by taking the linear combination of real and imaginary

part

and are arbitrary constant.

Part(a)

We know and

Then

Part(b)

After a period T

We know

Then we have

Take the exponential function on both side of the first equation and using Taylor Series

to approximate the value, and solve for , we have

2

Part(c)

Problem 1-2

Question: Assume that the mass and stiffness of the structure of Fig.2-1a are as follows:

. If the system is set into free vibration with the

initial condition and ,determine the displacement and

velocity at , assuming

(a) (b)

Solution:

Part(a)

The equation of motion is

Take Laplace Transform in both sides we have

Solve it for , we have

3

Take the Inverse Laplace transform we can get the form of solution

And

Part(b)

The equation of motion becomes

Apply Laplace Transformation on both sides

Solve it for

Take the Inverse Laplace Transform, we can get the solution

And

）

Problem 1-3

Question: Assuming that the mass and stiffness of the system of Fig2-1a are

and , and that it is undamped. If the initial

displacement is , and the displacement at is also ,

4

determine:

(a) the displacement at (b) the amplitude of free vibration

Solution:

We know . From Problem1-2 we have

Plug

We can solve for

So

And amplitutde is calculated by

Appendix

Some rules about Laplace Transform

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