Physics · Oscillations and waves · Grade 10

Spring pendulum

Why does a car without shock absorbers keep bouncing for so long after a bump?

The situation

On Tashkent's streets drivers slow down in front of a “Speed bump” sign. Over the same obstacle some cars glide smoothly, while others rock several times. The spring between the wheel and the body softens the shock, and the shock absorber damps the oscillation. Which spring and shock absorber carry a passenger most comfortably — and why does a car shake especially hard at one particular speed on a bumpy dirt road?

Topic

Spring pendulum. Car shock absorber: driving over a pothole

Key ideas

Hooke's law
Damped oscillations
Resonance

Formulas

m \ddot{x} = - k (x - y) - c (\dot{x} - \dot{y})

Quarter-car model: body mass $m$ , spring $k$ , shock absorber $c$ , road profile $y (t)$ .

T = 2 π \sqrt{\frac{m}{k}}, T = 2 π \sqrt{\frac{I}{m g d}}

Periods of a spring pendulum and a physical pendulum (undamped, small amplitude). For a simple pendulum $I = m l^{2}$ , $d = l$ : $T = 2 π \sqrt{l / g}$ .

ζ = \frac{c}{2 \sqrt{k m}}, β = \frac{c}{2 m}

Damping: $ζ < 1$ — underdamped (oscillatory), $ζ = 1$ — critically damped, $ζ > 1$ — overdamped (aperiodic); $β$ is the damping coefficient, the logarithmic decrement is $δ = \ln (A_{n} / A_{n + 1}) = β T$ , and the quality factor is $Q \approx π / δ$ .

v_{r e s} = f_{0} λ, f_{0} = \frac{1}{2 π} \sqrt{\frac{k}{m}}

The resonance speed on a wavy road ( $λ$ is the wavelength).

\frac{X}{Y} = \frac{\sqrt{1 + {(2 ζ r)}^{2}}}{\sqrt{{(1 - r^{2})}^{2} + {(2 ζ r)}^{2}}}, r = \frac{f}{f_{0}}

The ratio of the body amplitude to the road amplitude in the steady state.

Theory

Spring pendulum

A mass hanging on a spring oscillates about its equilibrium position; the restoring force is $F = - k x$ by Hooke's law. We write Newton's law $m \ddot{x} = - k x$ as $\ddot{x} + ω_{0}^{2} x = 0$ , $ω_{0}^{2} = k / m$ — this is the differential equation of harmonic oscillations, with the solution $x = A \cos (ω_{0} t + φ_{0})$ . The period $T = 2 π \sqrt{m / k}$ depends neither on the amplitude nor on $g$ : gravity only shifts the equilibrium position down by $m g / k$ . In Laboratory mode compare the measured period with the formula. The addition of oscillations of the same frequency — in the “Coupled oscillations” experiment.

The simple and the physical pendulum

The laboratory has two more pendulums. A simple pendulum is a small ball on a long thread: $T = 2 π \sqrt{l / g}$ . A physical pendulum is a rigid body oscillating about an axis: $T = 2 π \sqrt{I / (m g d)}$ , where $I$ is the moment of inertia about the axis and $d$ is the distance from the axis to the centre of mass. It can be replaced by a simple pendulum of length $l_{k} = I / (m d)$ — the reduced length. The reversible pendulum (lab work): a rod carries two knife-edges $O_{1}$ , $O_{2}$ and a movable mass. We position the mass so that the periods are equal when the pendulum is hung from either knife-edge in turn — then $l_{k} = l_{0}$ (the distance between the knife-edges, measured precisely) and $g = 4 π^{2} l_{0} / T^{2}$ . There is no need to know $I$ or the centre of mass — that is the advantage of the method. At large angles the pendulum is nonlinear: the period increases with amplitude (≈ 0.05 % at 5°) — in the “Nonlinear oscillations” experiment.

The shock absorber and damping

Real oscillations always die away: friction and the resistance of the medium turn energy into heat. A shock absorber is a cylinder filled with oil that gives a force proportional to speed: $F = - c \dot{x}$ . ζ < 1 — the body rocks several times; ζ = 1 (critical damping) — it returns fastest without oscillating; ζ > 1 — it “creeps” back slowly and the shock is felt more sharply. In passenger cars ζ ≈ 0.2–0.4 is usual — a compromise between comfort and the grip of the wheel on the road.

Logarithmic decrement and quality factor

For small damping $x = A_{0} e^{- β t} \cos ω t$ : the amplitude decreases exponentially (the dashed line on the graph), $β = c / (2 m)$ is the damping coefficient, and the frequency drops slightly: $ω = \sqrt{ω_{0}^{2} - β^{2}}$ . The logarithm of the ratio of two successive maxima is constant — the logarithmic decrement $δ = \ln (A_{n} / A_{n + 1}) = β T$ . The quality factor $Q \approx π / δ$ — after about $Q / π$ oscillations the amplitude falls by a factor of $e$ . In the laboratory, raise $β$ (as if the mass were dipped in oil) and compare the measured $δ$ with $β T$ . In Car mode $ζ$ is likewise found from successive maxima after a bump.

Resonance

On a wavy road the wheel moves up and down at the frequency $f = v / λ$ . When $f$ approaches the natural frequency of the body $f_{0}$ , the body amplitude exceeds that of the road several times over — resonance. The shock absorber lowers the resonance peak. That is why on such a road either increasing or decreasing the speed reduces the shaking.

Model simplifications

A quarter car with one degree of freedom: a quarter of the body ( $m$ ); the wheel and tyre are rigid — they follow the road profile exactly and do not leave the road. The front and rear axles are calculated independently; the rear wheel reaches the obstacle after a time $L / v$ — this is the pitching of the body. The acceleration felt by the passenger is the body acceleration $\ddot{x}$ . The mass of the spring is neglected; the equations are solved by the RK4 method.

Did you know?

In 1678 Robert Hooke announced the law of elasticity in the Latin words “Ut tensio, sic vis” — “as the extension, so the force”.
In passenger cars the natural oscillation frequency of the body is usually chosen to be about 1–1.5 Hz; in sports cars the springs are stiffer and the frequency higher.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Swing — Grade 10
Waves on a rope and a spring — Grade 10
Sound: speed, pitch, ultrasound — Grade 10
The dutar string and sound — Grade 11
Waves and the Doppler effect — Grade 11
Bridge resonance — Grade 11
Coupled oscillations — Grade 11
Nonlinear oscillations — Year 2
Fourier: the spectrum of sound — Year 3