Question 13.P.5: Assume that the vibration shape of the beam in Problem 13.3 ...

Dynamics of Structures

Assume that the vibration shape of the beam in Problem 13.3 is given by $\psi(x)=1+d \sin (\pi x / L)$ where $d$ is a constant. Select an appropriate value for $d$ so as to satisfy the four boundary conditions at the two ends of the beam. Using the resulting value of $\psi(x)$ obtain the fundamental frequency and the corresponding mode shape. Compare your results with those obtained in Problem 13.3.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

Related Answered Questions

Question: 13.P.1

Using Rayleigh Ritz method and the following Ritz shapes obtain estimates of 2 frequencies and the associated mode shapes of the four-story frame shown in Figure 11.14. ...

Verified Answer:

\begin{aligned}\omega_{1} &=9.56 \math...

Question: 13.16

The two-degree-of-freedom system of Figure E13.16a is subjected to an applied force given by p = [1 1/2]g(t) where g(t) is the rectangular pulse function shown in Figure E13.16b. Obtain the response of the system by using a modal analysis through frequency domain. ...

Verified Answer:

The stiffness and mass matrices of the system are ...

Question: 13.15

Find the response of the system of Example 13.9 using the central difference method. In the integration of the equation corresponding to degree-of-freedom 1, use a time step h while for the equation corresponding to degree-of-freedom 2 use a time step h/2. Select h = 0.1 s. ...

Verified Answer:

As in Example 13.14, the iteration equation corres...

Question: 13.12

Find the response of the system of Example 13.9 using the Wilson-θ method and a time step of (a) 0.1 s, and (b) 10.0 s. Select θ = 1.4. ...

Verified Answer:

(a) Again, the initial conditions are as in Exampl...

Question: 13.11

Find the response of the system of Example 13.9 using the linear acceleration method and a time step h of (a) 0.1 s, and (b) 0.5 s. ...

Verified Answer:

(a) The initial conditions are the same as in Exam...

Question: 13.P.9

Assuming that Guyan’s reduction gives acceptable accuracy, obtain the response of the frame of Problem 13.8 to a lateral load of 200 kips applied suddenly to the fourth floor level. Neglect damping. ...

Verified Answer:

\begin{aligned}&u_{1}=0.1667-0.1679 \co...

Question: 13.P.7

Repeat Problem 13.6 considering only the first mode. Compute the value of error norm by using Equation 13.45. Improve the estimate of story forces obtained by you by adding static corrections for the responses in the second and third modes. ...

Verified Answer:

In first mode

\mathbf{f}_{S}=\left[\begin{a...

Question: 13.P.3

Using Rayleigh Ritz method obtain estimates of 2 frequencies of a uniform flexible beam supported at each end by a spring of stiffness k as shown in Figure P13.3. Use the following shape function ψ1(x)=1 ψ2(x)=sin⁡πx/L Plot the mode shapes corresponding to the calculated frequencies. Note that ...

Verified Answer:

\begin{aligned}&\omega_{1}=1.025 \sqrt{...

Question: 13.10

Find the response of the system of Example 13.9 using the average acceleration method and a time step h of (a) 0.1 s, and (b) 10 s. ...

Verified Answer:

(a) The initial displacement and velocity vectors ...

Question: 13.17

Obtain the response of the system of Example 13.16 by a frequency-domain analysis but without using a normal coordinate transformation. ...

Verified Answer:

As in Example 13.16, we will determine the respons...