Linear Vibrations Lab

Worked examples › Example 7

Example 7Keeping a base-driven mass below a limit

base excitationfrequency responsedesign

Problem

Block A (mass $m$) is connected by a spring $k$ to a base B that moves as $y(t)=y_0\sin\omega t$, and by a second spring $k$ to a fixed wall.

Use $m=4$ kg and $k=800$ N/m.

Find: the range of frequencies $\omega$ for which the amplitude of A does not exceed $1.5\,y_0$.

InteractiveThe system in motion

Displacements in the drawing are scaled up so the motion is visible; the plot shows the true values.

Presets:
x(t)±1.5 y₀ limit (green); steady state (orange)

Steady-state amplitude vs forcing frequency; the dot is your current ω.

Solution

Every solution follows the same five steps: coordinates, free-body diagrams, kinetics, kinematics, solve. Try the problem first, then open one step at a time to check your work.

Step 1 · Choose a coordinate system

Let $x$ be the block's position, positive to the right, with both springs unstretched when $x=0$ and $y=0$. The base motion $y(t)=y_0\sin\omega t$ is prescribed.

Step 2 · Draw the free-body diagrams
Block Amk(x − y)kxmgN+xleft spring: stretched by x − yright spring: compressed by x
Free-body diagram of the block. Both spring forces point in −x when x and x − y are positive.
Step 3 · Kinetics (Newton–Euler equations)

Newton's second law for the block (horizontal):

$$m\ddot x=-k(x-y)-kx$$
Step 4 · Kinematics (constraints)

The base motion is given, so the only kinematic relation needed is the stretch of each spring:

$$\text{left spring: } x-y,\qquad \text{right spring: } x,\qquad y=y_0\sin\omega t$$
Step 5 · Solve

Rearranged:

$$m\ddot x+2kx=ky_0\sin\omega t,\qquad \omega_n=\sqrt{\frac{2k}m}=\sqrt{\frac{1600}{4}}=20\ \text{rad/s}$$

Steady-state amplitude, with $r=\omega/\omega_n$:

$$X=\frac{ky_0}{2k-m\omega^2}=\frac{y_0/2}{1-r^2}$$

Require $|X|\le1.5\,y_0\iff|1-r^2|\ge\tfrac13$:

Below resonance: $r\le\sqrt{2/3}=0.816\Rightarrow\omega\le16.3$ rad/s. Above resonance: $r\ge\sqrt{4/3}=1.155\Rightarrow\omega\ge23.1$ rad/s.

Answer

$$\omega\le16.3\ \text{rad/s}\quad\text{or}\quad\omega\ge23.1\ \text{rad/s}$$

Avoid the band $16.3<\omega<23.1$ rad/s around resonance ($\omega_n=20$ rad/s).

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