Damped oscillator position calculator
With amplitude 1, decay rate 0.2, angular frequency 2 and phase 0, the position at time 1 is −0.340712212877 inside an envelope of 0.818730753078. Enter the five values to evaluate the decaying cosine at any moment. Free in your browser.
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Run — free
Worked example
- Amplitude
- 1
- Decay rate
- 0.2
- Omega
- 2
- Phase
- 0
- Time
- 1
- Envelope
- 0.8187307531
- Argument
- 2
- Position
- -0.3407122129
Underdamped harmonic oscillators appear wherever a restoring force competes with light friction: mass-spring dampers, RLC circuits rewritten as second-order ODEs, seismic instruments, and every differential-equations course that treats free vibration. The closed-form position is an exponentially decaying sinusoid, not a pure sine, so hand evaluation mixes an exponential envelope with a cosine of the damped frequency and phase. This damped oscillator position calculator accepts amplitude A, decay rate β, damped angular frequency ω, phase φ in radians, and time t, then returns the numerical position together with the envelope and the cosine argument.
The underdamped closed form and what each symbol means
In that regime the natural frequency ω₀ equals the square root of k over m, the decay rate β equals ζ ω₀ (or c over 2m in viscous notation), and the damped angular frequency is ω equals ω₀ times the square root of one minus ζ squared. The free-response position that satisfies the initial conditions can always be written as x of t equals A times e to the minus β t times cosine of ω t plus φ, where A and φ absorb the initial displacement and velocity.
Amplitude A may be positive or negative and simply scales the whole motion. Decay rate β must be non-negative so the envelope shrinks or stays constant rather than exploding. Angular frequency ω must be strictly positive so the motion is truly oscillatory; critical and overdamped cases use different real-exponential formulas and are out of scope.
Phase φ is measured in radians, matching standard calculus and physics textbooks, and time t may be any finite real number so you can evaluate past or future samples on a simulation grid.
What you can do with it
Classroom free-vibration checks
Let students submit A, β, ω, φ, and t for underdamped x(t) and verify the position, envelope, and cosine argument against textbook methods.
LMS and exam auto-grading
Mark underdamped oscillator items with a deterministic oracle that rejects negative decay rates and non-positive ω as invalid input.
Design notebooks and clearance sampling
Sample closed-form damped free response on a time grid when sizing dampers or checking mechanical envelopes without a full ODE integrator.
FAQ
What if omega is zero or negative?
Underdamped oscillation requires a positive damped angular frequency ω so the cosine term oscillates in real time.
Is phase in degrees or radians?
Radians only, matching standard calculus and physics notation for cos(ωt+φ). Convert degrees to radians before calling if your data is in degrees.
How do I get the decay rate and frequency from mass, damping and stiffness?
The natural frequency is the square root of k over m, the decay rate is c divided by twice m, and the damped frequency is the natural frequency times the square root of one minus the damping ratio squared. Enter those reduced values here.
What is the envelope in the result?
It is the shrinking bound the cosine swings inside, set by the amplitude and the decay rate. With amplitude 1 and decay rate 0.2 it is 0.818730753078 at time 1, and the position equals the envelope times the cosine.
Can I evaluate the position at a negative time?
Yes. Time may be any finite real number, so you can sample past or future points on a simulation grid. The amplitude may also be negative or zero; it simply scales the whole motion.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/calculus/damped-oscillator-position \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"amplitude":1,"decay_rate":0.2,"omega":2,"phase":0,"time":1}'const res = await fetch("https://api.kit.forhosting.com/calculus/damped-oscillator-position", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"amplitude": 1,
"decay_rate": 0.2,
"omega": 2,
"phase": 0,
"time": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/calculus/damped-oscillator-position",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"amplitude": 1,
"decay_rate": 0.2,
"omega": 2,
"phase": 0,
"time": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/calculus/damped-oscillator-position", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"amplitude":1,"decay_rate":0.2,"omega":2,"phase":0,"time":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"amplitude":1,"decay_rate":0.2,"omega":2,"phase":0,"time":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/calculus/damped-oscillator-position", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"amplitude": 1,
"decay_rate": 0.2,
"omega": 2,
"phase": 0,
"time": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "calculus.damped_oscillator_position",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Invalid decay rates or non-positive ω fail with invalid_input instead of inventing a growing or non-oscillatory answer.
The underdamped closed form and what each symbol means
A linear mass-spring-damper with equation mẍ + cẋ + kx = 0 is underdamped when the damping ratio ζ is strictly less than one. This endpoint does not re-solve the characteristic equation from m, c, and k; it evaluates that already-reduced underdamped solution at a chosen time.
Input contract, response fields, and validation errors
On success the body echoes all five inputs and adds envelope equal to A times e to the minus β t, argument equal to ω t plus φ in radians, position equal to envelope times cosine of the argument, and a formula string restating x of t equals A e to the minus β t times cosine of ω t plus φ. POST a JSON body with five required finite numbers: amplitude, decay_rate, omega, phase, and time. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, and non-finite values fail with invalid_input naming the field. If omega is zero or negative the motion is not an underdamped oscillation with a real damped frequency, so the call fails instead of returning a pure exponential or a complex cosine. Example defaults amplitude one, decay_rate zero point two, omega two, phase zero, time one yield a decaying cosine sample you can verify with any scientific calculator. Failed validation is not billed; successful parse calls use the published $0.002 base price with no model tokens and no outbound network.
Where a damped oscillator position API earns its keep
Classrooms and tutoring platforms use a damped oscillator position calculator so students can check free-vibration homework without misplacing the decay factor or converting degrees for the phase by accident. Instrumentation scripts that convert measured ζ and ω₀ into β and ω can still call this capability once those parameters are known, keeping the numerical core tiny and testable. Exam generators and LMS plugins need a deterministic oracle that rejects non-positive ω and negative β rather than silently returning a nonsense success payload. Mechanical design notebooks sample x of t along a time grid when sizing dashpots and checking clearance envelopes before CAD; a tiny parse endpoint avoids pulling a full ODE solver for a single closed-form evaluation.
Technical questions
How do I compute underdamped oscillator position x(t)?
Send finite numbers amplitude A, decay_rate β (≥0), omega ω (>0), phase φ in radians, and time t. The API returns x = A·e^{-βt}·cos(ωt+φ) plus the envelope and cosine argument.
Can decay_rate be zero?
Yes. β = 0 is the undamped limit of the same formula (pure cosine). Negative β is rejected because a growing envelope is not the passive free response.
What does the damped oscillator position calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.