How to convert radian to degree from answer (or equation) generated from syms

%%%%%%%%%%%%%%% Equation of Motion: Undamped SDOF System %%%%%%%%%%%%%%%%%%
%%%%%% Example Problem - 2.2 (Dynamics of Structures - Ashok K.Jain %%%%%%%
close all;
clear all;
clc;
%% INPUTs:
E = 2*10^11;     % Young's Modulus (in N/m/m)
I = 3*10^-3;     % Moment of Inertia (in m^4)
x0 = 0.02;       % Initial Displacement (in m)
v0 = 0.01;       % Initial Velocity (in m/s)
L = 3;           % Length of Column in Frame (m)
W = 4*10^5;      % Lumped Weight Supported by Column (N)
%% OUTPUTs:
k = 2*3*E*I/(L^3) % Total Stiffness (in N/m)

k = 1.3333e+08

m = W/9.81      % Lumped Mass (Kg)

m = 4.0775e+04

wn = sqrt(k/m)  % Natural Circular Frequency or Angular Frequency (rad/s)

wn = 57.1839

f = wn/(2*pi());% Natural Cyclic Frequency (Hz)
T = 1/f         % Fundamental Time-Period (sec)

T = 0.1099

A = sqrt((x0^2) + (v0/wn)^2)    % Amplitude (m)

A = 0.0200

vm = A*wn       % Maximum Velocity (m/s)

vm = 1.1437

am = vm*wn      % Maximum Acceleration (m/s/s)

am = 65.4025

Phi = atand(x0*wn/v0) % Phase Angle (in degree)

Phi = 89.4990

syms X(t)
F = diff(X,t,2) + (wn^2)*X == 0;
x = dsolve(vpa(F))   % C1 & C2 are constant and can be determined by BCs

x = 

dX = diff(X,t);
conds =[X(0)==x0,dX(0)==v0];
x = dsolve(vpa(F),conds) 

x = 

%% Plots:
fplot(x,[0 2],'k','LineWidth',1.25);
xlabel('displacment (in m)');
ylabel('time (t)');
title('Displacement Response Curve');

[Walter Roberson]

More exactly

Q = @(v) sym(v);
Pi = sym(pi);

%% INPUTs:
E = Q(2)*10^11;     % Young's Modulus (in N/m/m)
I = Q(3)*10^-3;     % Moment of Inertia (in m^4)
x0 = Q(0.02);       % Initial Displacement (in m)
v0 = Q(0.01);       % Initial Velocity (in m/s)
L = Q(3);           % Length of Column in Frame (m)
W = Q(4)*10^5;      % Lumped Weight Supported by Column (N)
%% OUTPUTs:
k = Q(2)*Q(3)*E*I/(L^3) % Total Stiffness (in N/m)

k = 

m = W/Q(9.81)      % Lumped Mass (Kg)

m = 

wn = sqrt(k/m)  % Natural Circular Frequency or Angular Frequency (rad/s)

wn = 

f = wn/(2*Pi);% Natural Cyclic Frequency (Hz)
T = 1/f         % Fundamental Time-Period (sec)

T = 

A = sqrt((x0^2) + (v0/wn)^2)    % Amplitude (m)

A = 

vm = A*wn       % Maximum Velocity (m/s)

vm = 

am = vm*wn      % Maximum Acceleration (m/s/s)

am = 

Phi = atand(x0*wn/v0) % Phase Angle (in degree)

Phi = 

syms X(t)
F = diff(X,t,2) + (wn^2)*X == 0;
x = dsolve(F)   % C1 & C2 are constant and can be determined by BCs

x = 

dX = diff(X,t);
conds =[X(0)==x0,dX(0)==v0];
x = dsolve(F,conds) 

x = 

%% Plots:
fplot(x,[0 2],'k','LineWidth',1.25);
xlabel('displacment (in m)');
ylabel('time (t)');
title('Displacement Response Curve');

Answer 1

[Walter Roberson]

Q = @(v) sym(v);
Pi = sym(pi);

%% INPUTs:
E = Q(2)*10^11;     % Young's Modulus (in N/m/m)
I = Q(3)*10^-3;     % Moment of Inertia (in m^4)
x0 = Q(0.02);       % Initial Displacement (in m)
v0 = Q(0.01);       % Initial Velocity (in m/s)
L = Q(3);           % Length of Column in Frame (m)
W = Q(4)*10^5;      % Lumped Weight Supported by Column (N)
%% OUTPUTs:
k = Q(2)*Q(3)*E*I/(L^3); % Total Stiffness (in N/m)
m = W/Q(9.81);      % Lumped Mass (Kg)
wn = sqrt(k/m);  % Natural Circular Frequency or Angular Frequency (rad/s)
f = wn/(2*Pi);% Natural Cyclic Frequency (Hz)
T = 1/f;         % Fundamental Time-Period (sec)
A = sqrt((x0^2) + (v0/wn)^2);    % Amplitude (m)
vm = A*wn;       % Maximum Velocity (m/s)
am = vm*wn;      % Maximum Acceleration (m/s/s)
Phi = atand(x0*wn/v0); % Phase Angle (in degree)
syms X(t)
F = diff(X,t,2) + (wn^2)*X == 0;
x = dsolve(F);   % C1 & C2 are constant and can be determined by BCs
dX = diff(X,t);
conds =[X(0)==x0,dX(0)==v0];
x = dsolve(F,conds) 

x = 

%% Plots:
fplot(x,[0 2],'k','LineWidth',1.25);
xlabel('displacment (in m)');
ylabel('time (t)');
title('Displacement Response Curve');

syms CosD(t) SinD(t)
xD = mapSymType(mapSymType(x, 'sin', @(V) SinD(vpa(children(V,1)*180/Pi, 15))), 'cos', @(V) CosD(vpa(children(V,1)*180/Pi, 15)))

xD = 

[Parvesh]

I've two problems associated with the solution you provided:

P1 - How to plot with the given solution in form of sinD with respect to 't' ?

P2 - How can one find a specific value of xD(t) at a particular time instant 't' ?

[Walter Roberson]

In order to graph or evaluate xD, you would need to convert it back to x, the sin() and cos() version.

SinD and CosD are here as placeholders, suggestive of the actual operations. If you were to convert them,

XX = mapSymType(mapSymType(xd, 'SinD', @(V) sind(children(V,1))), 'CosD', @(V) cosd(children(V,1)))

then the sind() and cosd() would immediately get evaluated to sin() and cos() times the appropriate conversion factor.

You can either have a placeholder like SinD and CosD that are suggestive of what is to be done but do not actually do anything, or else you can convert back to radians... in which case you might as well stay with radians in the first place.