Define and Convert Units

Load units by usingsymunit.

u = symunit;

Specify a unit by usingu.unit. For example, specify a distance of5meters, a weight of50kilograms, and a speed of10kilometers per hour. In displayed output, units are placed in square brackets[].

d = 5*u.m w = 50*u.kg s = 10*u.km/u.hr

d = 5*[m] w = 50*[kg] s = 10*([km]/[h])

Units are treated like other symbolic expressions and can be used in any standard operation or function. Units are not automatically simplified, which provides flexibility. Common alternate names for units are supported. Plurals are not supported.

Add500meters and2kilometers. The resulting distance is not automatically simplified.

d = 500*u.m + 2*u.km

d = 2*[km] + 500*[m]

Simplifydby usingsimplify. Thesimplifyfunction automatically chooses the unit to simplify to.

d = simplify(d)

d = (5/2)*[km]

Instead of automatically choosing a unit, convertdto a specific unit by usingunitConvert. Convertdto meters.

d = unitConvert(d,u.m)

d = 2500*[m]

There are more unit conversion and unit system options. SeeUnit Conversions and Unit Systems.

Find the speed if the distancedis crossed in50seconds. The result has the correct units.

t = 50*u.s; s = d/t

s = 50*([m]/[s])

Use Temperature Units in Absolute or Difference Forms

By default, temperatures are assumed to represent differences and not absolute measurements. For example,5*u.Celsiusis assumed to represent a temperature difference of 5 degrees Celsius. This assumption allows arithmetical operations on temperature values.

To represent absolute temperatures, use kelvin, so that you do not have to distinguish an absolute temperature from a temperature difference.

Convert23degrees Celsius to kelvin, treating it first as a temperature difference and then as an absolute temperature.

u = symunit; T = 23*u.Celsius; diffK = unitConvert(T,u.K)

diffK = 23*[K]

absK = unitConvert(T,u.K,'Temperature','absolute')

absK = (5923/20)*[K]

Verify Dimensions

In longer expressions, visually checking for units is difficult. You can check the dimensions of expressions automatically by verifying the dimensions of an equation.

First, define the kinematic equation $$v^2=v_0{}^2+2as$$, wherevrepresents velocity,arepresents acceleration, andsrepresents distance. Assumesis in kilometers and all other units are in SI base units. To demonstrate dimension checking, the units ofaare intentionally incorrect.

syms v v0 a s u = symunit; eqn = (v*u.m/u.s)^2 == (v0*u.m/u.s)^2 + 2*a*u.m/u.s*s*u.km

eqn =v^2*([m]^2/[s]^2) == v0^2*([m]^2/[s]^2) + (2*a*s)*(([km]*[m])/[s])

Observe the units that appear ineqnby usingfindUnits. The returned units show that both kilometers and meters are used to represent distance.

findUnits(eqn)

ans = [ [km], [m], [s]]

Check if the units have the same dimensions (such as length or time) by usingcheckUnitswith the'Compatible'input. MATLAB^®assumes symbolic variables are dimensionless.checkUnitsreturns logical0(false), meaning the units are incompatible and not of the same physical dimensions.

checkUnits(eqn,'Compatible')

ans = logical 0

Looking ateqn, the accelerationahas incorrect units. Correct the units and recheck for compatibility again.eqnnow has compatible units.

eqn =(v*u.m/u.s)^2 == (v0*u.m/u.s)^2 + 2*a*u.m/u.s^2*s*u.km; checkUnits(eqn,'Compatible')

ans = logical 1

Now, to check that each dimension is consistently represented by the same unit, usecheckUnitswith the'Consistent'input.checkUnitsreturns logical0(false) because meters and kilometers are both used to represent distance ineqn.

checkUnits(eqn,'Consistent')

ans = logical 0

Converteqnto SI base units to make the units consistent. RuncheckUnitsagain.eqnhas both compatible and consistent units.

eqn =unitConvert(eqn,'SI')

eqn =v^2*([m]^2/[s]^2) == v0^2*([m]^2/[s]^2) + (2000*a*s)*([m]^2/[s]^2)

checkUnits(eqn)

ans = struct with fields: Consistent: 1 Compatible: 1

After you finish working with units and only need the dimensionless equation or expression, separate the units and the equation by usingseparateUnits.

[eqn,units] = separateUnits(eqn)

eqn =v^2 == v0^2 + 2000*a*s units = 1*([m]^2/[s]^2)

You can return the original equation with units by multiplyingeqnwithunitsand expanding the result.

expand(eqn*units)

ans = v^2*([m]^2/[s]^2) == v0^2*([m]^2/[s]^2) + (2000*a*s)*([m]^2/[s]^2)

To calculate numeric values from your expression, substitute for symbolic variables usingsubs, and convert to numeric values usingdoubleorvpa.

Solveeqnforv. Then find the value ofvwherev0 = 5,a = 2.5, ands = 10. Convert the result to double.

v = solve(eqn,v); v = v(2); % choose the positive solution vSol = subs(v,[v0 a s],[5 2.5 10]); vSol = double(vSol)

vSol = 223.6627

Use Units in Differential Equations

Use units in differential equations just as in standard equations. This section shows how to use units in differential equations by deriving the velocity relationsv=v₀+atand $$v^2=v_0{}^2+2as$$starting from the definition of acceleration $$a=\frac{dv}{dt}$$.

代表symbolica加速度的定义lly using SI units. Given that the velocityVhas units,Vmust be differentiated with respect to the correct units asT = t*u.sand not justt.

syms V(t) a u = symunit; T = t*u.s; % time in seconds A = a*u.m/u.s^2; % acceleration in meters per second eqn1 = A == diff(V,T)

eqn1(t) = a*([m]/[s]^2) == diff(V(t), t)*(1/[s])

Because the velocityVis unknown and does not have units,eqn1has incompatible and inconsistent units.

checkUnits(eqn1)

ans = struct with fields: Consistent: 0 Compatible: 0

Solveeqn1forVwith the condition that the initial velocity isv₀. The result is the equationv(t)=v₀+at.

syms v0 cond = V(0) == v0*u.m/u.s; eqn2 = V == dsolve(eqn1,cond)

eqn2(t) = V(t) == v0*([m]/[s]) + a*t*([m]/[s])

Check that the result has the correct dimensions by substitutingrhs(eqn2)intoeqn1and usingcheckUnits.

checkUnits(subs(eqn1,V,rhs(eqn2)))

ans = struct with fields: Consistent: 1 Compatible: 1

Now, derive $$v^2=v_0{}^2+2as$$. Because velocity is the rate of change of distance, substituteVwith the derivative of distanceS. Again, given thatShas units,Smust be differentiated with respect to the correct units asT = t*u.sand not justt.

syms S(t) eqn2 = subs(eqn2,V,diff(S,T))

eqn2(t) = diff(S(t), t)*(1/[s]) == v0*([m]/[s]) + a*t*([m]/[s])

Solveeqn2with the condition that the initial distance covered is0. Get the expected form ofSby usingexpand.

cond2 = S(0) == 0; eqn3 = S == dsolve(eqn2,cond2); eqn3 = expand(eqn3)

eqn3 (t) = S (t) = = t * v0 * [m] + ((* t ^ 2) / 2) * [m]

You can use this equation with the units in symbolic workflows. Alternatively, you can remove the units by returning the right side usingrhs, separating units by usingseparateUnits, and using the resulting unitless expression.

[S units] = separateUnits(rhs(eqn3))

S(t) = (a*t^2)/2 + v0*t units(t) = [m]

When you need to calculate numeric values from your expression, substitute for symbolic variables usingsubs, and convert to numeric values usingdoubleorvpa.

Find the distance traveled in8seconds wherev0 = 20anda = 1.3. Convert the result to double.

S = subs(S,[v0 a],[20 1.3]); dist = S(8); dist = double(dist)

dist = 201.6000